Computational Materials Science. Surfaces, Interfaces, Crystallization

Computational Materials Science

Computational MaterialsScienceSurfaces, Interfaces, Crystallization

A. M. Ovrutsky and A. S. ProkhodaDepartment of Physics of Metals, Faculty of Physics,Electronics and Computer Systems,Oles Gonchar Dnipropetrovs’k National University,Dnipropetrovs’k, Ukraine

M. S. RasshchupkynaDepartment of Low-Dimensional and Metastable Materials,Max Planck Institute for Intelligent Systems,Stuttgart, Germany

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Acknowledgments

Authors would like to thank Professor O.Yo. Sokolovsky, Department of

Theoretical Physics, Dnipropetrovs’k National University; Assistant Professor

I.G. Rasin, Department of Chemical Engineering of Technion, Israel; Assistant

Professor O.I. Kushnerev, Department of Physics of Metals, Dnipropetrovs’k

National University for their fruitful discussion on a number of principal questions.

Authors want to gratefully acknowledge Professor V.F. Bashev, the head of the

Department of Physics of Metals for his support of the researches in the field of

Computational Materials Science which resulted in this book, and for the whole-

some discussions.

Authors are thankful to the reviewers of the book Computer Simulation of Phase

Transitions and Surface Phenomena: textbook published in Ukrainian language in

2011, which was a precursor to this book: Professor V.V. Maslov� (deceased),

the head of the Department of Crystallization, G.V. Kurdyumov Institute for Metal

Physics, National Academy of Sciences, Ukraine (Kiev) and Professor V.V. Girson,

the head of the Department of Physics of Metals, Zaporozhye National University,

Ukraine for their preview of the manuscript and useful advices.

Preface

Simulation is one of the main means for development of our ideas of outward

things and theoretical description of various phenomena and processes. History of

knowledge clearly shows that new, more complicated models come to replace the

old, simple ones to provide a better description of the real processes. Simple mod-

els such as ideal gas model are easily analyzable. Complication of models leads

to the increase of difficulties in their analysis and expects application of advanced

mathematical methods.

Mathematical physics and computational mathematics have evolved due to the

need for development of analysis and computer techniques. The latter was “translated”

into the language suitable for computers and became a useful instrument for the

scientists in different fields of knowledge.

Analysis of the sufficiently realistic models is an extremely hard task, and it is

not always possible to reduce results to a form suitable for application of the

computational mathematics technique. For example, analytical solutions of the

boundary problems of heat and mass transfer could be derived only for bodies of a

very simple shape under some certain simplified boundary conditions. At the same

time, numerical solution of the initial equations by the finite-difference method

(one of the simulation techniques) allows to obtain a full picture of changes in

temperature and concentration fields, to take into account movement of the phase

boundaries and changes in their shapes. At the same time, simulation program is an

analogue of both an analytical solution and its finite expressions. Using calculations

provided on computer, it is enough to change the input parameters of the system

under consideration in order to obtain corresponding results with complete visuali-

zation of the ongoing processes.

An algorithm and a program provided that they are correct and that results of

their application are proved at least for simplified models are none the worse for

analytical solutions and could be much simpler for usage in practice. For example,

now nobody tries to obtain an analytical solution to the many-body problem of

celestial bodies, instead appropriate programs for calculations are used.

Hence it is clear why the simulation methods find their place in curricula of

famous universities. A good many books are dedicated to the simulation methods

at a different level of complexity. Those written by mathematicians are mostly

focused on the methods themselves. In textbooks written by theoretical physicists,

most attention is given to the phenomenological problems. But those who want to

apply simulation methods should bear in mind that in order to be able to do it they

need to master the subject itself and to understand the relevant phenomena at the

level of latest advances in science and technology. Therefore, it is better not to

separate courses in simulation from the main course.

In this book, we yield to the theoretical basis necessary for understanding atomic

surface phenomena and processes of phase transitions, especially crystallization.

Theoretical basis for computer simulation by different methods and simulation

techniques for modeling of physical systems are also presented, as well as addi-

tional information concerning their accuracy. A number of results are discussed

concerning modern studies of crystallization: processes of thin film formation,

kinetics of crystal growth, stability of crystal shapes including crystallization front,

and nanocrystal formation during solidification from the supercooled melts.

In the last chapter of this book, several computer experiments from the list

proposed to the students of the Dnipropetrovs’k National University are described.

Explicit instructions to contents of these works and detailed explanations of the main

procedures of programs (Delphi, C11, Visual C# environments, and the Pascal

codes of several programs are also included) should help everyone understand the

essence of simulations. Open access to executable files (the website of Elsevier

http://booksite.elsevier.com/9780124201439/) makes it possible for everyone to

achieve a better understanding of the main phenomena described in this book.

A description of programs is sufficient for their reconstruction in any programming

environments.

Owing to the specific structure of the book, lists of references to its first chapters

are considerably reduced. Some educational stuff is given without source references

if it was previously presented in some textbooks and it is hard to figure out where it

was published for the first time. The following sources were the most often used for

the preparation of the book:

D.W. Heermann, Computer Simulation Methods in Theoretical Physics, second ed.,

Springer, 1990.

Experiment on a Display, Moscow, Science, 1989, 99 p. (in Russian).

M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1989,

385 p.

D. Frenkel, B. Smit, Understanding Molecular Simulation. From Algorithms to Applications,

Academic Press, New York, London, Tokyo, 2002, 628 p.

D.K. Belashchenko, Computer Simulation of Liquid and Amorphous Matters MISSIS,

Moscow, 2005, 407 p. (in Russian).

H. Gould, J. Tobochnik, An Introduction to Computer Simulation Methods: Applications

to Physical Systems Parts 1 and 2, Addison-Wesley, Reading, MA, 1988.

H. Gould, J. Tobochnik, W. Cristian, An Introduction to Computer Simulation Methods:

Applications to Physical Systems, third ed., Addison-Wesley, Reading, MA, 2007, 813 p.

V.I. Rashchikov, A.S. Roshal, Numerical Methods in Solution of Physical Problems, Lan’,

St. Petersburg, 2005 (in Russian).

xvi Preface

http://booksite.elsevier.com/9780124201439/

Other editions wherein mathematical fundamentals of simulation methods are

described in step-by-step fashion.

The book Physics of Surface by A. Zangwill (Cambridge University Press,

1988) remains the most consistent on the subject of surface physics; some materials

from this book were used in Chapter 4. More recent researches of the surface

structure are represented in the book Introduction to the Physics of Surface by

K. Our, V.G. Livshitz, A.A. Saranin, A.V. Zotov, G. Katayama (in Russian, Nauka,

Moscow, 2006, 490 p.).

Our book does not cover all aspects of simulations in Materials Science.

Simulations of mass crystallization that give information on microstructure

formation in materials during crystallization, especially in the high and very high

supercooling ranges, are not presented here. Another large direction in modeling,

which is of a special importance for production and exploitation of engineering

materials, is application of computational methods in continuum mechanics.

There are some very useful books dealing with the questions of continuum

mechanics. Continuum-based simulation approaches in the continuum scale and

atomic scale are described in the book by Dierk Raabe (Computational Materials

Science. The Simulation of Materials Microstructures and Properties, Wiley-

VCH, Weinheim, New York, Toronto, 1998, 326 p.) and the book edited by

Dierk Raabe, Franz Roters, Frederic Barlat, Long-Qing Chen (Continuum Scale

Simulation of Engineering Materials: Fundamentals—Microstructures—Process

Applications, Wiley-VCH Verlag GmbH & Co. KGaA, 2004, 845 p.). The book

of S. Schmauder and L. Mishnaevsky Jr. (Micromechanics and Nanosimulation of

Metals and Composites, Springer-Verlag, Berlin Heidelberg, 2009, 421 p.) con-

tains descriptions of different experimental and computational analysis methods

of micromechanics of damage and strength of materials.

This book will be useful for everyone who has interest in applying modern

simulation techniques for development and analysis of more realistic models of

physical processes in Materials Science.

xviiPreface

1 Computer Modeling of PhysicalPhenomena and Processes

1.1 Application of Computers in Physics

1.1.1 Role of Models in Theoretical Study

Models of phenomena or processes underlie any physical theory. Such models are

simple enough as a rule; their complication hampers the theory by elaborating.

If results obtained in a simple model framework are in satisfactory agreement with

experimental data, there is no need to complicate it. However, if the essential dis-

agreement in results takes place, it is necessary to choose another model, which

would correspond better to a nature of phenomena under consideration.

The ideal gas model is the simplest. Gas is considered as a set of noninteracting

mass points, which can move in any direction. It is sufficient to use the ideal gas

model in order to find the relationship between the gas pressure on the vessel walls

and such characteristic of the molecules movement as their mean-square velocity.

We will note that determination of relationships between parameters of the system

state and characteristics of the molecule movement is the main subject of the

kinetic molecular theory.

In order to determine the pressure of an ideal gas, it is supposed that some

velocity distribution of gas molecules exists such that mean-square speed of mole-

cules for the given conditions is a constant value. Hence, the question puzzles, if

molecules move freely and do not collide (a mass point has no sizes), how could

any certain velocity distribution of molecules be set? Consequently, a better-

adjusted model of gas should consider the size of molecules. One of the widely

used models of gas considers molecules as solid spheres. This model is used for the

description of transport phenomena in gases, such as diffusion, thermal conductiv-

ity, and interior friction.

If concentration of gas molecules is high, interaction of molecules mostly

defines physical properties. Real gas models consider attraction of molecules.

If distance between molecules is small, repulsive forces also should be taken under

consideration.

Balance between attractive and repulsive forces determines the average distance

between atoms in liquid or in solid body. Resultant forces appear when molecules

shift from equilibrium positions; they are in direct proportion with deviation dis-

tances. Therefore, the simplest and the most widely used model of a solid body

Computational Materials Science. DOI: http://dx.doi.org/10.1016/B978-0-12-420143-9.00001-6

© 2014 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/B978-0-12-420143-9.00001-6

(crystal) is the crystal lattice with atoms disposed in its knots and interacting with

each other by elastic forces.

Simple models allow one to perform analytical study easily. Analysis of more

realistic models of matters is carried out with application of the special mathemati-

cal methods developed by physicists-theorists. As a rule, it is necessary to evaluate

complicated integrals to find solutions of algebraic or transcendental equations,

their systems, etc., and finally, to compute the matter properties. For example, sta-

tistical theories of system ordering (the system of magnetic moments or electric

dipole moments, or atoms of different types) are based on searching analytical

expressions for the free energy of systems in the framework of the considered

model. Calculating the statistical sum is necessary for the Gibbs free-energy deter-

mination; the Helmholtz free energy is determined through an internal energy (U)

and entropy (S), F5U2 TS. The free energy minimum corresponds to the equilib-

rium state of systems. Minimization of main parameters in the analytical expression

for the free energy (order parameter, probability of certain configurations of atoms

or dipoles) results in the transcendental equations. Their solutions are usually per-

formed using standard computing procedures. In this case, the computer is still

used as a powerful calculator.

1.1.2 Methods of Computer Modeling of Physical Processes

Mathematical models play a great role in the scientific study. With their help, a

physical phenomenon is transformed by the means of equations into a discrete alge-

braic form, which can be used for a numerical analysis. Discrete algebraic equa-

tions describe a calculated model. Translation of the latter into machine codes is a

computer program. The computer and the program allow exploring evolution of a

modeled physical system in computing experiments [1].

Mathematical modeling is a kind of theoretical problem on the numerical solu-

tion of the Cauchy boundary value problem. At the instant t5 0, the initial state of

a system is set in some bounded spatial area (simulated volume) on whose surface

some given boundary conditions are retained. Modeling consists of observing evo-

lution of the system state. The basic part of evaluation is a cycle with a certain

timestep (Δt), during which the state of physical system progresses over this time.

Even the simplest modeling calculation generates a huge amount of data and

demands an experimental approach for obtaining desired outcomes (from which the

name “computing experiment” originates). However, even if the amount of infor-

mation which can be treated by computers is large, their capability is not limitless.

Three methods giving the best performance for modeling physical processes

have received the widest application. These methods are: the method of nets for

solution of the transport equation (i.e., partial differential equations), the Monte

Carlo (MC) method (including its modifications for kinetic modeling), and the

method of molecular dynamics (MD method) for modeling of classical statistical

and quantum statistical systems. In all cases, it is a question of approximating a

continuous environment by a discrete model with local interaction. The choice of

method, the search for a model of the substance structure which is adequate to

2 Computational Materials Science

reality, working out algorithms and programs for model performance, carrying out

numerical experiments, and analyzing their outcomes comprise the essence of sim-

ulation of physical phenomena.

When modeling a large system, the model is loaded into the computer’s memory

in a convenient way for calculations, and a parallel execution of noninteracting spa-

tial domains (or interacting is neglected during the timestep) is provided. These

evaluations are essentially simple but being distributed over a large volume, they

demand many resources. In order to accelerate the execution, the simultaneous

work of several computers (a “cluster” of computers) is organized. The main com-

puter called the host machine rules the cluster.

Modeling by the MC method does not require complicated mathematics because

it comes almost from the first principles—probabilities of states or transitions of

particles from one state into another are defined by the Boltzmann factor of ener-

gies (taken with the negative sign) in units of kT. Modeling by the MC method sup-

poses consideration of the substance models, which are more complicated, than

models that are analyzable in the framework of the modern theoretical physics.

Statistical modeling by the MC method allows studying equilibrium states of sys-

tems. Kinetic modeling by the MC method allows analysis of the course of physical

processes.

The MD method develops most intensively now. It is already applied to systems

consisting of many thousands of atoms (systems of many millions of atoms are

already executed in some research). The method consists of numerical solution of

Newton’s equations for all atoms with a timestep smaller than 10214 s. For this

timestep, increments of coordinate values and velocities of all particles are calcu-

lated, taking into account their values on the previous timestep. Though the level

of adequacy of calculated outcomes to the real physical picture of a yielded process

or phenomenon cannot be guaranteed, because dependences of interaction energy

of atoms on distance between them are not defined with adequate accuracy; the

method is extremely valuable and perspective, due to the exclusively “first” princi-

ples used in it.

1.1.3 Influence of Computers on Methods of Physical Researches

Purposes and means of science were changed due to the computer facilities devel-

opment. Long-time theoretical physics aspired to analytical solutions of the pro-

blems. It seemed to be the single possible method of full description of

phenomena. Unfortunately, the most important and actual problems cannot be

solved analytically. Computer modeling has proved to be very effective in the case

of such problems; its development is connected with efficiency. This progress has

now come so far that analytical solutions are not required in many cases. “The

problem of three bodies”—movement of three bodies in the total gravitational field—

is not solved analytically yet. However, it does not prevent astronomers from

calculating trajectories not only for three but also for any number of bodies by the

means of computer modeling. Essentially, an algorithm allowing any accuracy to

calculate trajectories using computers is no worse than “explicit analytical”

3Computer Modeling of Physical Phenomena and Processes

solution. Numerical solutions allow answering any questions, to which it would

possibly answer by means of formulas, when they will be obtained.

Elaboration at the end of twentieth century of software packages that could exe-

cute algebraic transformations was unexpected and shocking for some physicists-

theorists. It meant that intellectual operations became accessible to computers.

Reports became known, in which from the beginning to the end, all the formulas

and theorems were obtained (were deduced and proved) by machine. The essence

of the conflict was that those machines had inevitably invaded the field of science,

which was considered traditionally as belonging to the most qualified scientists,

namely, to theorists.

It is also possible to give examples of the lowering of the status of experts own-

ing perfectly some theoretical methods. The qualified experts in the field of the

heat and mass transport, which did not master in time new numerical methods,

have discovered with surprise that their huge wealth of theoretical knowledge is

substantially depreciated. And still it does not allow solving transport equations

for complex boundary conditions varying with time; and young researchers, not

theorists at all, can do it by means of rather simple programs. The qualified

physicists-theorists in the field of statistical physics have discovered that their

young colleagues applying the MC method do not simply check its reliability, but

obtain already much more powerful outcomes, better mapping structure of sub-

stances, and different processes (first of all, the phase transitions), which occur

with them. On the other hand, this is a normal phenomenon, that when new people

come, they are able to work in a new way. However, they must still learn the pro-

cess. In addition, the main point is that it is necessary to be a good expert in a cer-

tain field of knowledge. The computer can facilitate analyzing of processes or

phenomena; it allows one to work with the models as an experimentalist, obtaining

outcomes for different initial and boundary conditions (unlike real experiments

with the material system, these conditions are known precisely in the case of com-

puter experiment). However, the principle of theoretical work does not vary: it is

model development and execution, in this case, with application of the computer.

In cases of application of direct methods, such as the MC or MD methods, based

on the most common principles, the core basis of modeling becomes the competent

elaboration of algorithms and validation of solutions. For the planning of comput-

ing (machine) experiments and fulfilling analysis of outcomes, the core is the

knowledge of theoretical researches in this branch of science and of outcomes of

the newest experimental research.

The important direction in physics is the modeling of large systems or any sys-

tem in extreme situations. These are situations when the system differs qualitatively

from the total of independent small subsystems, that is, the cases when the radius

of correlations is large in it. These are also critical phenomena of different kinds,

such as turbulence and wave function collapse. Methods developed for the analysis

of such systems find interesting applied applications, sometimes in unexpected

branches of knowledge.

Use of modeling in scientific knowledge is caused, as is known, by that circum-

stance that the immediate object of research is either difficult to access or generally


inaccessible for direct research on any physical properties [2]. There is a difference

between physical and mathematical modeling. Physical model operation is based

on study of the phenomena on models of one physical nature with the original. For

example, wind tunnels are used to test small models of airplanes in the air flow.

Crystallization of the transparent organic matter (salol) in small vessel works as a

model of crystallization of steel ingot. The mathematical model is more generalized

than the physical model: it is not required more of physical similarity between the

original and the model is not required any more, as the parameters of mathematical

model, which have the mathematical description, are only studied, and they are

connected by the mathematical relations concerning both the model and the

original.

Mathematical models, as they come from the mathematical similarity of the

original to the model and are used for studying of the quantitative characteristics

and the quantitative correlation of different parameters, may be considered as math-

ematical computers. On the contrary, the computer (after corresponding program-

ming) is the generalized model of those processes, equations of which can be

solved by this machine. The up-to-date computers are used as simulators of objects

and processes of the diversified character. Computers with large number of ele-

ments (1012�1014), according to Neumann’s theorem, are the universal automatic

machines, capable of performing the operation of any automatic machine [1].

It is important to see the COMPUTER as a source of creative pleasure, which

can be ensured by an increase of the intellectual (game) part in scientific work. It

became possible as a result of the radical increase of labor productivity at so-called

routine stages of the working process. Losing time connected with routine opera-

tions strongly narrows down creative possibilities of researchers. Besides, many

operations inherently are inaccessible to people because of the huge volume of nec-

essary transformations (or logic steps). It is impossible to fulfill them without the

participation of machines. For such tasks, the COMPUTER should possess artificial

intellect. Its creation includes the software engineering, allowing solving the tasks

of intellectual nature by means of computers, for example, proving theorems with

application of operations of the formal logic, pattern recognition, and use of natural

language for tutoring of robots.

1.1.4 The Basic Aspects of Computer Application in Physics

Generally speaking about the use of computers in physics, it is necessary to discuss

four aspects:

1. Numerical analysis (computational mathematics)

2. Symbolic transformations

3. Mathematical modeling

4. Controlling the physical equipment in real time.

In the numerical analysis, evaluations are preceded by the simplifying physical

reasons. Solution of many physical problems can be reduced to the solution of a

system of linear equations. The analytical solution can be fulfilled for sets of two,


three, and four equations. If the number of variables becomes very large, it is nec-

essary to apply numerical methods and computers. In this case, the computer serves

as the numerical analysis tool. It is often necessary to calculate a many-

dimensional integral to perform operations with big matrixes or to solve a complex

differential equation. This stipulates wide application of computers in physics.

This increasing importance gains computer application in theoretical physics for

analytical (symbolical) transformations. Analytical transformations are already

included in many up-to-date mathematical packages, e.g., Mathcad, Matlab, Maple,

and Mathematics. For example, let us suppose that we want to find out the solution

of the quadratic equation ax21 bx1 c5 0. The program of analytical transforma-

tions can produce the solution in the formula form x1,25 (2b6 (b22 4ac)1/2)/2a,

or in the usual numerical form for definite values a, b, and c. Thus the computer

can already deduce equations. It is especially important when equations contain

many terms or when their deduction needs many operations. A person most likely

will make a mistake but the computer will yield the right answer.

By means of typical programs for analytical transformations, it is possible to ful-

fill such mathematical operations as differentiation, integration, solution of equa-

tions, and series expansion.

Mathematical modeling is characterized by the feature that only the most com-

mon physical laws (principles) with minimum analysis are included in algorithms.

As an example, let us determine energy distribution in the system with a great num-

ber of particles. To answer the question, what is the probability that the value of

energy of the particle is in the range from E to E1ΔE? One of the ways to find the

answer to such a question is in carrying out the experiment, for example, by defini-

tion of velocities of gas molecules. Such experiments were carried out. But they are

not easy, and they answer the question of distribution of gas molecules on energies

only. The problem can be solved precisely analytically. And it is solved by statistical

physics. It is the problem about energy distribution of particles in systems making

up the microcanonical ensemble of systems (consisting of a constant number of par-

ticles with the constant total energy). However, problems like this cannot always be

solved analytically. It is much easier to act differently: to introduce the game rules

into the computer program, to simulate a large number of energy exchanges between

particles, and to calculate probabilities for distribution determination.

Computers can also be used to answer a “what if” question. For example, how

the distribution of particles on energies would be modified, if the maximum possi-

ble value of the exchange energy were varied?; what would be, if exchange of

energy occurred in discrete portions? The specified type of modeling finds applica-

tion for ordering problems in many-particle systems, consisting of dipole or mag-

netic moments (dielectrics and magnets).

In all varieties of the use of computers in physics, the main purpose is usually

“understanding not the numbers.” Computers have very much influenced physical

researchers and the choice of physical systems for study especially. The numerical

analysis and modeling are connected with some simplifying approximations, that

is, with the choice of the model that allows solving the problem numerically. Thus,

a creative work of the researcher is in the foreground.


Computers are also the important tool in experimental physics—controlling the

physical equipment. Often they are linked to all phases of laboratory experiment:

instrumentation design, controlling of this instrumentation during experiment,

obtaining and evaluating data sets. The use of computer facilities not only has

allowed experimentalists to sleep better at night but also has made possible experi-

ments which otherwise would be impossible. Some of problems mentioned above,

for example, instrumentation design or the data evaluation, are close to problems

that scientists come across in theoretical work. However, the tasks connected with

controlling and interactive analysis of data differ qualitatively; they demand pro-

gramming in real time and joining of different types of devices to computing

equipment.

1.1.5 Computational Experiments and Their Role in Modern Physics

Why are simulations important for physics? This question was considered in Refs.

[1,3,4]. One of the reasons is that the majority of analytical tools, such as differen-

tial calculus, suit mostly examination of the linear problems. For example, it is

easy to analyze oscillations of one particle solving the equation of its movement

(Newton’s second law) in the supposition of the linear restoring force. However,

the majority of natural processes are nonlinear, so small changes in one variable

can lead to large changes in value of other variables. Nonlinear problems can be

solved analytically only in special cases, and the computer gives a possibility of

examination of the nonlinear phenomena. Another direction of numerical modeling

is the analysis of behavior of systems with many degrees of freedom (consisting of

large number of particles or many variables).

Development of computer techniques results in the new sight on physical sys-

tems. The statement of the question: “How to formulate the task for the computer?”

has led to the modification of the formulation of some physical laws. Therefore, it

is quite practical and it is natural to express laws in the form of rules for the com-

puter, instead of language of the differential equations [1]. Now this new vision of

the physical processes leads some physicists to review the computer as a certain

physical system and to elaborate the novel architecture of computers, which can

simulate natural physical systems more efficiently. Often numerical modeling is

termed computational experiment as it has a lot of common with laboratory experi-

ments. Some analogies are shown in Table 1.1, taken from Refs. [1,3].

The advantage of the computational experiment is that the conditions, at which

certain process runs, are set precisely in it. As a rule, it is very difficult to define

them in real experiments.

The basic point of numerical modeling is creating the model of the idealized

physical system. Then it is necessary to elaborate algorithms and procedures for

the model realization on the computer. The computer program simulates a physical

system and features the computing experiment. Such computational experiment is a

bridge between laboratory experiments and theoretical calculations. For example,

we can obtain in essence exact outcomes for the idealized model, which does not

have a laboratory analog. Comparison of results of modeling with corresponding


theoretical calculations stimulates development of computing methods. On the

other hand, it is possible to check and improve the model using realistic parameters

for more direct comparison of simulation results with the results of the laboratory

experiments.

The method of finite differences, the Monte Carlo method or method of mole-

cular dynamics for systems with the great number of particles depends not only on

the wish of the researcher. At the choice of the modeling method, the main condi-

tions are: the size (the number of particles in the system) and the environment of

the physical system, accuracy of calculation and possibility of interpreting the

results obtained with certain reliability; and duration of the computing experiment.

For every single case, it is necessary to introduce certain corrective deductions, to

choose some correlative coefficients, to consider the nature of particle interaction,

to estimate errors. In any case, it is necessary to make many tests of the computing

experiments with previously known outcomes. Only under the condition of getting

positive outcomes, can you assert that the mathematical model created by you and

realized in the computer program is effective and possible for use and prediction of

physical properties of such systems.

Numerical modeling, as well as laboratory experiments, does not substitute the

theory and is the tool used for comprehension of the complex phenomena.

However, the purpose of all our examinations of the fundamental physical phenom-

ena consists of searching of such explanations, which can “be noted on the enve-

lope underside” or which is possible “to present on fingers” [1], that is, consists of

searching for the simplest explanations.

1.2 Determination of Statistical Characteristics of Systemsby the MC Method

1.2.1 Determination of Average Values of Physical Quantities

Statistical physics usually deals with systems consisting of a large number of parti-

cles. Statistical descriptions of such systems, basically, were developed long ago in

the works of Maxwell, Boltzmann, and Gibbs. The description is stochastic. The

probability of realization of a certain configuration of all degrees of freedom in the

state of the thermodynamic equilibrium of the system is defined by the mechanics

Table 1.1 Analogies Between Computational and Laboratory Experiments

Laboratory Experiment Computational Experiment

The sample The model

The physical device The program for the computer

Calibration Program testing

Measuring Calculation

Data analysis Data analysis


laws (either classical or quantum), which control the system evolution in time. One

can consider as degrees of freedom a set of all independent physical variables of

the mechanical system (three-dimensional coordinates of particles ri5 (xi, yi, zi),

where i5 1, ..., n, for gas consisting of n point particles), which determines its con-

figuration at the present moment of time. Values of all degrees of freedom define

the system microstate (its certain configuration); we will designate it by letter s.

According to the basic postulate of statistical physics, all microstates of macro-

scopic system with the constant energy and the number of particles (their quantity,

Wo, is huge) have equal probabilities. Therefore, the probability of realization of a

definite microstate s is very small, ps5 1/Wo. The probability of the macrostate m

that is defined by macroscopic parameters (P, T, V, etc.) for such systems is

pm5Wm/Wo, where statistical weight Wm is the number of microstates correspond-

ing to the macrostate m.

Statistical physics tells us how to calculate probability distributions on states of

different systems (e.g., those which contact with the thermostat—a very large sys-

tem). These distributions can be used for the evaluation of physical values.

However, except for some simple cases, such as ideal gas of noninteracting parti-

cles or a system of linearly coupled harmonic oscillators (e.g., a spring oscillating

system), evaluation of physical values is connected with huge mathematical

difficulties.

Assume that for gas of n interacting particles with coordinates ri, i5 1, ..., n, the

probability distribution is known in every single case as the normalized function

f(r1, ..., rn), dependent on all these variables. We will designate through A(r1, ..., rn)

the certain physical quantity, which is the certain function of variables ri. Then the

average value hAi in the thermodynamic equilibrium state of the system is the fol-

lowing multiple integral:

Ah i5ðN2N

dr1

ðN2N

dr2?ðN2N

drnAðr1; . . .; rnÞf ðr1; . . .; rnÞ: ð1:1Þ

The problem of the evaluation of such integrals in the case of large n or for

n!N is extremely difficult.

Often there are situations when variable degrees of freedom take over the dis-

crete set of values. Then instead of multiple integrals, it is necessary to calculate

the multiple sums. It is also practically impossible in the case of systems with

many particles. As an example, we will describe one of the most simple from the

point of view of formulation and most well-studied models of statistical physics—

Ising’s ferromagnetic model.

It is known that there are areas of preferentially identical orientation of elemen-

tary “magnets” in the real ferromagnetic. Such areas are termed domains.

Orientations of magnetic moments of different domains differ; therefore, the aver-

age moment of magnet of macroscopic system, in absence of the exterior magnetic

field, is equal to zero.

The Ising ferromagnetic model is an extremely simplified model of the real fer-

romagnetic model. Onsager has found the exact solution of the problem of


definition of equilibrium states for two-dimensional variant of the Ising model. Let

us imagine a two-dimensional square lattice, in each knot of which an elementary

magnetic moment, a spin, is placed, and it can have only one of two possible direc-

tions (Figure 1.1). Figure 1.1 shows Ising’s spins by arrows that are oriented

upward or downward. In this case, each spin represents a separate degree of free-

dom, and the variable corresponding to it (qi) takes over only two values: qi5 1

corresponds to the spin directed upward and qi521—to the spin directed down-

ward. Assume that only the nearest neighbor spins qi and qj interact with each

other, and interaction energy Eij may be written as follows:

Eij 52 Jqiqj: ð1:2Þ

If ith and jth, the nearest neighbor spins, have the same orientation, Eij52J, and

in case of the opposite orientation Eij5 J. According to the laws of statistical

mechanics, in the thermodynamic equilibrium state, the configuration of all spins

of the system, characterized by the set of variables qi, has the probability defined

by the Gibbs distribution function. According to Gibbs’ theory, the canonical distri-

bution function for the system that is in contact with the thermostat can be found

from the equation:

f ðq1; . . .; qnÞ5 1

Zexp 2

1

kT

Xbondsði; jÞ

Eij

" #; ð1:3Þ

where T is the Calvin (absolute) temperature; k is the Boltzmann’s constant; Z is

the normalizing factor which is usually termed as statistical sum, and is determined

from the condition:

Xq1 5 6 1

Xq2 5 6 1

?X

qn 5 6 1

Wðq1; . . .; qnÞ5 1; ð1:4Þ

i.e.,

Z5X

q1 5 6 1

Xq2 5 6 1

?X

qn 5 6 1

exp 21

kT

Xbondsði; jÞ

Eij

" #: ð1:5Þ

Figure 1.1 Two-dimensional Ising ferromagnetic model: (A) arbitrary orientations of spins

and (B) minimum-energy spin configurations.


Let us now suppose that we want to calculate, for example, average energy hEiat the temperature T. The energy of the certain configuration of spins is

Esðq1; . . .; qnÞ5X

bondsði;jÞEij 52 J

Xbondsði;jÞ

qiqj: ð1:6Þ

Average energy hEi (and thus the average from any other physical quantity) can

be found by the equation:

hEi5X

q1 5 6 1

?X

qn 5 6 1

Eðq1; . . .; qnÞf ðq1; . . .; qnÞ

51

Z

Xq1 5 6 1

?X

qn 5 6 1

2JX

bondsði;jÞqiqj

!3 exp

1

kT

Xbondsði;jÞ

qiqj

24

35: ð1:7Þ

As each spin variable qi can take on one of values, the configuration space of

system of n spins (it is routinely termed the phase space) consists of 2n possible

configurations. The sum (1.7) can be calculated precisely at any finite n. However,

for the study of properties of macroscopic systems, it is necessary to take a large

number of spins. For example, in the case of the Ising model, the number of spins

should be, at least, order 103 that computed results of average energy and other

physical quantities were close to their values in a system with an infinite number

of spins.

It might seem that computer should be immediately put in charge of the evalua-

tion of the sum (1.5). However, summation of 21000 terms of sum (1.7) is a problem

inaccessible for any computer. This does not mean, however, that the analysis of

statistical models by means of computers is essentially impossible.

From consideration of amounts of configurations corresponding different ener-

gies of the system, it becomes clear that the contribution of different summands to

the sum (1.7) is not equal at all. For example, there is only one configuration of

spins in which any two neighbors are oriented in the same way as it is shown in

Figure 1.1. Despite the small value of energy of the system, the probability of

realization of the state with such energy is very small. There are many high-

energy microstates when many pairs of the next spins are oppositely directed.

However, a value of the exponent from summarized energy (from 2E/(kT)) is

very small for such configurations, and they are not realized at moderate tempera-

tures. Thus, there are also plenty of states with smaller energy, and the sum of

their exponents may be significant in spite of the small values of the exponents. In

that case, one can say that the system in states with such energies has high

entropy, sufficient to compensate a little value of all these terms. Entropy of these

states, with the certain energy E, S(E)5 ln W(E), that is, the logarithm of number

of such states W(E).

To each macrostate of the system with energy E responds W(E) microstates.

Therefore, the probability of configuration, that is, the probability to hit in a certain


microstate with a certain set of coordinates (degrees of freedom) and the total

energy is

f ðEÞBWðEÞexp½2E=ðkTÞ�5 exp½SðEÞ2E=ðkTÞ�: ð1:8Þ

This entropic-energy reasoning certainly should be considered in every effective

computing algorithm. Such principle of choice of configurations according to their

importance is used in the majority of the modern algorithms for MC simulations—

in Metropolis’s algorithm, in the thermostat algorithm, and in Creutz’s algorithm

for the isolated system.

1.2.2 Application of the MC Method to Physical Problems

In MC methods, the solution of problems determined in principle is replaced by the

approximate consideration based on introduction of stochastic elements, sequences

of which allow finding the approximate solutions. The essence of the MC method

in application to statistical physics problems is described in a great many mono-

graphs and reviews [5�7].

As all states of system are achievable (state s is characterized by a set of values

of microscopic parameters, for instance, of coordinates and impulses of all parti-

cles), there are transition probabilities from one state to another, p(s!s0). It is

assumed that the ensemble of all possible states s forms the Markov chain with

constant probabilities of transitions from the state s to the state s0 that satisfy the

normalizing condition for all s, including s5 s0:

Xs0

pðs ! s0Þ5 1: ð1:9Þ

According to the theory of the homogeneous Markov chains [8,9], transition

from the state s into the state s0 in n steps can be realized through different transient

states. We will designate as p(n)(s!s0) the summarized probability of realization of

such transition by all possible paths from n steps, and p(s!s0) is the summarized

probability of transition in all possible ways in n steps. If all s states form one ergo-

dic class, which means that transition from any state s into state s0 is possible in

some finite number of steps, then there are the limiting probabilities that s0 transi-tion is possible for some finite number of steps and there are the limiting

probabilities

pðs0Þ5 limn!N

pðnÞðs ! s0Þ; ð1:10Þ

for all s and s0 is independent of s. Thus, p(s0)$ 0,P

s0p ðs0Þ5 1. Equation (1.10)

means that system transition to the stationary distribution of states does not depend

on initial state choice. It is proved in the Markov chain theory that values p(s0) arerelated by the system of linear equations with p(s!s0)


pðs0Þ5Xs

pðsÞpðs ! s0Þ for all s0: ð1:11Þ

For the MC method, on the contrary, p(s0) is known, and value p(s!s0) is

required. Equations (1.9) and (1.11) can be considered as a set of equations con-

cerning unknown quantities p(s!s0). In this system, the number of unknown quan-

tities exceeds the number of equations, therefore a choice of p(s!s0) in a different

way is possible. In the space of states s there are great many Markov chains which

realize transitions s!s0. A definite choice of p(s!s0) is made for simplicity rea-

sons and also on the assumption of what sequence of transient states (what of possi-

ble Markov chains) reaches the stationary state faster. Relation

pðsÞpðs ! s0Þ5 pðs0Þpðs0 ! sÞ for all s and s0; ð1:12Þ

which is the expression of the principle of microscopic convertibility in system or,

in other words, of detailed equilibrium, transforms Eq. (1.11), taking into account

Eq. (1.9), into identity. Therefore, in practice, while constructing the Markov

chains, it is possible to start from any state. As Eqs. (1.11) and (1.12) are homoge-

neous concerning p(s), then for definition of p(s!s0) it is sufficient to know the

distribution p(s) to within a constant factor.

The explicit form of p(s!s0) depends on the s state. Using the known value of

p(s!s0), one can realize on the computer the corresponding sequence of states as fol-

lows. Any arbitrary s state is taken as the initial state, then the s0 state is randomly cho-

sen from the states, the transition probability to which is p(s!s0) 6¼ 0, and this

probability is calculated. Knowing the value p(s!s0), a trial concerning whether the

s!s0 transition will come true, is fulfilled according to the principle of choice of con-

figurations on importance degree. Such choice is ensured by the application of the

majority of the modern algorithms realized by the MC method: Metropolis, thermo-

stat, Creutz algorithm for an isolated system. In the sequences realized during model-

ing, some states can be repeated (mainly due to the unrealized transitions). However,

the considerable majority of states s from all their possible quantity are not realized at

all. Despite this, sequences appear to be representative enough. Calculations with their

use result in exactness precisions of evaluation of average values of the order of 1% or

even better. Just the small amount of terms in sequence of transient states in compari-

son with the total of possible s states stipulates the advantage of algorithm of essential

sampling in the MC method in comparison with other numerical methods.

Figure 1.2 graphically presents the principle of choice of states (configurations).

The constitution diagram of some systems with the microstates grouped on energies

is shown. It defines the summarized probabilities for the system to be in states with

certain energy (the diagram of function Pm(E)). Each rectangle corresponds to a

certain configuration. The height of rectangles is higher the less the energy of the

configuration (it is proportional to exp[2E/(kT)], that is, the Boltzmann probability

of the configuration with energy E for the system that contacts with the thermo-

stat); the number of rectangles in the yielded column corresponds to the number of

configuration with this energy.


The polygonal curve symbolizes the random walk in the configuration space at

consecutive modifications of one of the coordinates, which characterizes the system

microstate, in agreement with the above-mentioned algorithms. This broken curve

goes more often through microstates, which are the most favorable from entropic-

energy points of view (not so high energy and great many microstates with such

energy). Only one microstate corresponds to the minimum energy and accordingly,

the entropy is zero. Thus the majority of configurations of macroscopic systems are

not realized at all. Therefore, average values of physical quantities can be calcu-

lated using those configurations only, which have been realized as a result of the

random walk in the course of modeling. The longer this walk, the more configura-

tions it has trapped (or as it is routinely described, the better the statistics col-

lected); the more exact will be the evaluation.

1.2.3 The Metropolis Algorithm and the Thermostat Algorithm

The Metropolis method has been formulated in work devoted to the equation of

state calculation for fluid consisting of hard disks [10]. The essence of the method

consists of the generation of a set of consecutive configurations of system. On each

step of this process, the following configuration is considered, which differs from

the previous by the modification of one of the degrees of freedom. A new confi-

guration can be accepted into the set or not, according to the procedure of the sta-

tistical trial. If by results of the statistical trial the new configuration is not

accepted, the previous configuration is once again joined to the ensemble on this

step. Thus, depending on its importance, the weight of the certain configuration in

the ensemble can grow.

Having the mode of walk in the system phase space defined, it is necessary to

set transition probabilities between two configurations at each step of the walk.

Thus, it is necessary to begin consideration that the system would go into the equi-

librium state in the limiting case of total number of walks. It means that the aver-

age of any physical quantity A over the time may be calculated by the equation:

Ah iN 51

N

XNi51

AðtiÞ; ð1:13Þ

Figure 1.2 Schematic

representation of distribution

of system configurations on

energies.


where ti is the ith moment (step) of time, at which measuring of the quantity A is

made. The mean value of A should coincide with the statistical average from this

value for the canonical assembly in the case of large but limited number N of

measurements.

Procedure of the statistical trial in the Metropolis method consists of following:

the new configuration is accepted in ensemble with the probability equal to ratio of

the Gibbs weights (Boltzmann’s exponents) of the new and old configurations. For

example, for the Ising model, the old and new configurations differ by the turning

of one spin. Probability of acceptance of the new configuration

pð1 ! 2Þ5 ½expð�E2=ðkTÞÞ�½expð�E1=ðkTÞÞ�

5 expE1 � E2

kT

� �ð1:14Þ

in the case E2.E1, and p(1!2)5 1, if E2,E1 (E2 and E1 are the energies of the

new and old configurations accordingly). As the probability of the following con-

figuration is defined by only previous configuration and does not depend on the

previous history, consecutive configurations form the Markov chain. We will term

the number of the realized steps of such process as Markov time.

An essential and very important component of the Metropolis algorithm and

thermostat algorithm is the generation of pseudorandom numbers. These numbers

are termed pseudorandom, instead of random, as any generator of such numbers

programmed on the computer has a finite period, after which these numbers start to

repeat. It happens because the number of digits to the right of the decimal point in

representation of real numbers on the computer is finite. The quality of the random

numbers generator essentially influences the accuracy of evaluations by the MC

technique.

The main property of the Metropolis algorithm is that it “guides” the system

into area of the most probable states in the phase space. The majority of configura-

tions is skipped at the Markov process construction, and the average values of

physical quantities are calculated taking into account the configurations that

respond to the most probable states.

Any starting configuration (any values of variables) can be used to construct the

Markov chains. The outcomes should not depend on this choice: the system in

equilibrium “forgets” about the history of establishing the equilibrium. However, at

real calculations, the Markov time of reaching of typical equilibrium configura-

tions, for which it is possible to realize measuring, can depend on the successful

choice of the starting configuration. It can happen in the case of so-called hystere-

sis phenomena, when some different phases are available, which are realized

depending on the initial state. Suppose, at “cold” start (spins in the initial configu-

ration are oriented to one side), we will come to the ordered state, and at “hot”

(spins in the initial configuration are oriented randomly), to disorder.

Application of the Metropolis algorithm or the thermostat algorithm to the

ensemble of the systems consisting of two spins, results in changes of numbers of

systems in possible configurations Nmm, Nmk, Nkm, and Nkk toward their equilibrium


values corresponding to the canonical distribution: Nmk/Nmm5Nmk/Nkk5Nkm/

Nmm5Nkm/Nkk5 exp(2 2J). According to the Metropolis algorithm, probabilities

of transitions

pðmm ! mkÞ5 pðmm ! kmÞ5 pðkk ! mkÞ5 pðkk ! kmÞ5 expð22JÞð1:15Þ

pðmk ! mmÞ5 pðmk ! kkÞ5 pðkm ! mmÞ5 pðkm ! kkÞ5 1: ð1:16Þ

The change in number of systems Nmk for one Markov step due to expense of

transitions mm!mk over mk!mm will be such

ΔNmk5 nðmm ! mkÞ2 nðmk ! mmÞ5Nmmðexpð2 2JÞ2Nmk=NmmÞ:ð1:17Þ

If, for example, the ratio Nmk/Nmm exceeds the equilibrium value exp(22J), the

number of systems Nmk exceeds the equilibrium value of the ratio Nmk/Nmm, transi-

tions mk!mm will occur more frequently, so that the number of systems Nmk will

decrease, until the ratio Nmk/Nmm will correspond to the Gibbs weights

(ΔNmk5 0). If the ratio Nmk/Nmm is less than the equilibrium value, transitions

mm!mk will occur more frequently that leads the ensemble to the proper ratio

between numbers of systems.

Practically, we usually have one Markov chain at numerical modeling.

Segments (the next sites) of this long chain can substitute the considered ensemble

of systems. The average from any quantity on the ensemble of such segments is

also the average on Markov time. This statement concerns any system with discrete

or continuous degrees of freedom.

In the thermostat algorithm [11] on each step of the Markov time, one of the

degrees of freedom is brought to the thermal equilibrium with the exterior “thermo-

stat” having the temperature T; other degrees of freedom are fixed for this step.

As a result, of the multiple recurring of such procedure for all degrees of freedom

and for a long Markov time, establishing full thermodynamic equilibrium of the

system with the thermostat occurs. For example, in the Ising model, orientation of

the chosen spin on each step is chosen with probability determined only by temper-

ature of the thermostat and by the configuration of neighbor spins. The normalized

probabilities of states with energies E1i and E2

i in the ith configuration of the

neighbor spins are equal correspondingly:

p1i 5expð2βE1

i Þexpð2βE1

i Þ1 expð2βE2i Þ

ð1:18Þ

p2i 5expð2βE2

i Þexpð2βE1

i Þ1 expð2βE2i Þ

: ð1:19Þ

Denominators are introduced into these equations in order to make the sum of

probabilities p1i and p2i equal to 1.


The thermostat algorithm, as well as the Metropolis algorithm, moves the system

quickly into the area of the most probable configurations in the phase space. The

system state will come nearer to equilibrium after no ,100 trials for each of its

degrees of freedom (particle coordinates). After that, values of physical quantities

of the system fluctuate only, and it is possible to use the chain of these values for

the evaluation of averaged characteristics of the system.

The lattice systems, which contain up to 107 discrete or continuous degrees of

freedom, are simulated using the Metropolis algorithm or the thermostat algorithm.

Computer clusters or multiprocessing complexes are used in many scientific centers

of the world. At present, it is efficient to run simulations using the Graph proces-

sors (GPU of graphics cards), which consist of sufficient big numbers of enclosed

processors. NVIDEA Corporation offers a special programming environment

(Cuda) and demo-programs for the Visual Studio package. Studying these programs

is sufficient for understanding their work and then elaborating one’s own programs.

1.2.4 Boundary Conditions

Good quality atomic simulation should include all characteristic features of the

considered physical system. The ultimate goal of simulations is obtaining estimates

of behavior for macroscopic systems that are the systems containing N�(1023�1025) particles (per 1 cm3). For such systems, the part of the surface parti-

cles is small; therefore, surfaces do not influence essentially the bulk structure. For

the system of 1000 molecules with the free spherical surface, the ratio of the num-

ber of surface molecules to their total number has an an order of magnitude of

N2/3/N5N21/35 0.1 (or (4πR2/Ω2/3)/(4πR3/(3Ω))5 3Ω1/3/R, where Ω is the vol-

ume of molecules). While computer technique allowed the simulation of small

volume systems only (order of 1022 104 particles), there was the problem of choice

of boundary conditions and extrapolation of the results to the macroscopic systems.

Using rigid walls is undesirable in most cases, if an atom is reflected from the

rigid wall, its position and interaction potential energy vary strongly without any

change in its kinetic energy. Therefore, the presence of rigid walls would mean that

the system total energy is not conserved.

One of the methods to minimize surface effects and more precisely to simulate

properties of macroscopic system more precisely consists of the use of periodic

boundary conditions. Until recently, researchers used only periodic boundary con-

ditions in many works because computer facilities did not allow to model systems

with a large enough number of atoms. Actually, studying many processes with

phase transitions, periodic boundary conditions in all directions do not give a good

approach [12]. It is desirable for the system to have a free surface. In the case of

application of free boundary conditions, a part of atoms forms a free surface con-

tacting with a vacuum; some atoms can evaporate, others move in the bulk of con-

densed phase.

The short descriptions of various boundary conditions, applying of which

depends on nature of phenomenon studied, will be discussed below.


1. “Periodic” boundary conditions.

In the majority of research by the MC and MD methods, the periodic boundary condi-

tions were used to simulate areas with a constant number of particles as elements of the

infinite volume. In this case, the first and last degrees of freedom on each of the direc-

tions of the model interact as if they were neighbors. Thus, on each of the directions, the

model is folded in the ring in each of directions (it forms a torus of corresponding

dimensionality).

Figure 1.3A shows that in the case of the periodical boundary conditions, the atom A0

interacts not only with atom A from the basic cell but also with the imaginary atom A0,shifted relative the A-atom on distance that is equal to the cell size (the identity period).

If the A-atom is in the left lower quarter, its reflection A0 is determined by the shift on the

identity period in two directions (Figure 1.3B).

In the case when particles can migrate, additional conditions are usually used. If the

particle A goes out outside the limits of the cubic basic cell through the certain edge in

the result of movement, the particle A0, which is termed an “image” will enter through

the opposite edge. Thus the injected particle will occupy the location, the coordinates of

which are calculated by the following relation:

x0A 5xA 2 L; xA .L

xA 1 L; xA , 0

� �ð1:20Þ

where xA is the x-coordinate of the particle after moving, x0A is the “image” coordinate, L

is the linear size of the cell. Other coordinates also are transformed in the same way. As a

result, instead of the particle, its image with the same number enters the cell. Thus the

total number of particles in the system does not vary. Figure 1.4 shows the image enter-

ing. Thus each particle after moving and leaving the basic cubic cell appears in the cell

A

(A) (B)A′

A′Y Y

y1

y1

X

A0

A

0 0 1/21/2x1

x1

A0

x′1 x′1

Figure 1.3 Interaction

scheme for an atom placed

near the border.

A′

A

Figure 1.4 Shifts of the particle and its images at application

of periodic boundary conditions, the cutoff sphere is shown.


from other side as the “image.” It compensates the influence of boundaries at the interac-

tion energy calculations.

An advantage of the periodic boundary conditions is that all degrees of freedom

become equivalent. The simulation volume is built so that undesirable surfaces are elimi-

nated and the quasi-infinite volume is formed, which allows consideration of the basic

cell as the element of the infinite system. However, periodic boundary conditions can

effect (and really do) the considerable additional order in condensed systems. Periodic

boundary conditions are expedient, if it is possibly revealing the identity period in a cer-

tain direction of the model that allows imitating the infinite propagating of substance

(a crystal) in a certain direction. In the case of multiphase systems, one can apply them in

directions where there are no interfaces. In addition, it is necessary to bear in mind that

occurrence of the crystalline nucleus near to the periodic boundary initiates immediately

the propagation (growth) from the opposite side. Now when personal computers allow

simulating systems of a 100,000 atoms, there is the possibility of not applying periodic

conditions at all, and considering systems with free surfaces, or applying periodic condi-

tions only for one or two directions.

2. “Rigid” boundary conditions.

Coordinates of boundary atoms are fixed. In this case, it is supposed that the suffi-

ciently great number of layers with mobile atoms compensates the influence of such con-

ditions on simulation results. This type of boundary condition is attractive in its

simplicity, but for it, a large number of atoms is needed, and it does not allow solving the

problems that require essential modification of thermodynamic parameters. This type of

boundary condition can be applied in the MC and MD methods in combination with other

types of boundary conditions.

3. The “elastic” boundaries.

The elastic force appears after the atom crosses the boundary; it returns the atom into

the simulation volume (main cell). The system total energy is conserved at application of

these conditions. Such boundary conditions are convenient at modeling of real gases.

4. The “free” boundary conditions.

The part of atoms of the compact model forms free surfaces “far” from the cell walls

and edges, in Figure 1.5A; these surfaces contact with a vacuum, and they can move.

Such a type of boundary condition ensures establishing of equilibrium interatomic dis-

tances for any temperature and pressure. The structure of areas near to the surface differs

from the bulk structure. Such boundary conditions are the most convenient for the study

of the surface phenomena and the modeling of the phase transformations. In many cases,

it is better to set the periodic boundary conditions in one or two directions; two direc-

tions, for example, at studying of thin film formation. The presence of at least one free

surface allows the system to reach the equilibrium state. For determination of diffusivities

or for study of crystal growth, periodic boundary conditions in one or two directions also

may be set.

Figure 1.5 Models with the free surface: (A)

iron after crystallization (FCC crystal lattice)

and (B) the scheme illustrating mirror

boundary conditions.


5. “Mirror” boundary conditions.

Boundary planes map the calculated cell as in the mirror. Particles cannot cross the

boundary planes, as they are repelled by their “images” when approaching them. An

interesting variant is to consider the calculated cell as follows: the model is a 1/8 part of

a full sphere (1/4 in the plane, Figure 1.5B), and it maps in three mirror planes, forming

the complete full sphere. Thus, the exterior surface of such full sphere is free. These con-

ditions “increase” the number of particles in the model with the free surfaces in eight

times. The shortage of such models is the actual excluding of free crossings of the mirror

planes by particles, and there are difficulties in the calculation of forces, which act on

particles near to the vertices of segments of the full sphere.

1.2.5 The Classical Atomic Interaction Potential Functions

The problem of choice of the interaction potential is essential at modeling of

atomic structure of substances. The choice should depend on the type of bonds in

the certain substance, physical concepts about nature of particle interaction in it.

Potentials are calculated in different ways. Theoretical potentials are calculated

according to the laws of quantum mechanics; the density of charge distributions

(the electronic density) are usually considered (see Chapter 6) [13]. Empirical

potentials are calculated based on the scattering experiment of the matter structure

(structure factor, radial pair distribution function (RPDF)) [14]. Those and other

potentials are corrected so that results of modeling would map the physical proper-

ties of a certain matter, such as elastic constants, temperature dependence of the

expansion coefficient, melting point, and heat capacity. To satisfy all these require-

ments is not always possible. Moreover, the statement that the potential corre-

sponds to this matter is not always justified. Research of common laws of these or

those physical processes has special value because inaccuracies of the potentials do

not change the character of the physical processes.

Two-, three-, and many-particle potentials are used depending on the nature of

the simulated substance. In cases of the many-particle potentials, the potential

energy of the system is a function not only of distances to atoms, which contain the

certain atom, but also of angles between the atom and its neighbor atoms. In a case

of pair potentials, only distances between atoms are considered. Results of simula-

tions with such potentials correspond to reality if the interaction energy of two

particles does not depend on the position of other particles. Such a situation takes

place for simple fluids and for different states of metals. Three-particle potentials are

applied for the study of systems with covalent bonds. Many-particle potentials

are useful for descriptions of more complex systems, for example, solid bodies

with structure imperfections. In many simple cases, interaction is pairwise. Thus the

interaction energy of system is

Uðr1; r2; . . .; rNÞ5Xi, j

ϕðrijÞ; ð1:21Þ

where ri is the position vector of the i-particle, rij is the distance between i- and j-

particles.


Function ϕ(r) for electroneutral atoms can be constructed by detailed calcula-

tion, which is based on the fundamental laws of quantum mechanics. Such calcula-

tion is very complicated and often it is sufficient to choose a simple

phenomenological formula of the function ϕ(r).The strong repulsion at small r and the weak attraction at large distances are the

most important features of ϕ(r) for simple fluids. Repulsion at small r is predeter-

mined by the exclusion rule. In other words, if electron clouds of two atoms over-

lap, one of the electrons should increase the kinetic energy to be in the different

quantum state. The feeble attraction at the large r is mainly caused by the cross-

polarization of atoms; the resultant force of the attraction is termed the Van der

Waals’ forces.

Most widespread of the continuous potentials which is applied to systems consist-

ing of electrically neutral atoms or molecules is the Lennard-Jones two-parameter

potential:

ϕLJðrÞ5 εr0

r

� �122 2

r0

r

� �6� �or ϕLJðrÞ5 4ε

d

r

� �122

d

r

� �6" #; ð1:22Þ

where the value ε5ϕLJ (r5 r05 21/6d) is the depth of the potential well, d is the

coordinate of zero of the potential, i.e., ϕLJ(d)5 0.

Figure 1.6 shows the graph of the Lennard-Jones potential. The dependence r26

in Eq. (1.22) is obtained theoretically and dependence r212 is chosen only for rea-

sons of convenience reasons. Notice that the yielded potential is short range, that

is, practically ϕLJ(r)5 0 for r. 2.5d.

More common is the Mi potential:

ϕMiðrÞ5ε

m2 nn

r0

r

� �m2m

r0

r

� �nh ið1:23Þ

2.0

1.0

00.5 1.5 2.5

r/d

V/ε

–1.0

Figure 1.6 The plot of the Lennard-Jones potential.


and the Morse’s three-parameter potential:

ϕMðrÞ5 ε exp 22αr

r02 1

� �� 2 2exp 2α

r

r02 1

� �� : ð1:24Þ

All these potentials have a strong repulsive branch and asymptotically approach

zero at large distances. For reduction of evaluation time, the potentials are trun-

cated at a certain distance rc, so interaction with the remote particles is not consid-

ered [13].

Such important structural characteristics as structure factors and pair correlation

functions can be obtained from X-rays, or electron or neutron scattering experi-

ments. However, it is necessary to construct corresponding atomic models, which

figure the spatial arrangement of the system particles for more complete representa-

tions about the substance structure. The potentials corresponding to certain matters

should be applied for the building-up of a model. Empirical potentials should be

chosen if their application gives results concerning the structure, which coincide

with experimental data for the structure factor and the RPDF [14].

The empirical interatomic potential is one of the most simple and available

research techniques to study dynamic and structural properties of substances. Such

potential contains a certain number of adjustable parameters. Results of computa-

tion experiments will be as exact as far as it may be achieved after the parameters

are corrected in order to make the results match the experimental data for certain

conditions.

Potentials applied in the case of covalent crystal semiconductors having about

four bonds per atom take into account angles of each atom with all pairs of its near-

est neighbors. At least, three-body potentials are used in this case. The expression

for the interaction energy of the system has the following form:

U5U0 1C1

Xi

Xj

uðri; rjÞ1C2

Xi

Xj

Xk

υðri; rj; rkÞ: ð1:25Þ

Summation is fulfilled over the nearest neighbors of the central atom. Here U0 is

the energy of the ideal crystal, C1, C2 are the coefficients which are corrected for

better correspondence of the model to the real system; uðri; rjÞ, υðri; rj; rkÞ are the

functions for the description of energy modification of in consequence of changes

of bond lengths or valence angles, accordingly. Keating [15] has offered the sim-

plest form of these functions:

uðr1; r2Þ5 uðjr1 2 r2j2 2 d20Þ; ð1:26Þ

where r1 and r2 are the position vectors of the neighbor ions, d0 is the equilibrium

bond length,

υðri; rj; rkÞ5 ðcos θijk2cos θ0ijkÞ2; ð1:27Þ


where θijk is the angle between bonds i�j and i�k, θ0ijk is the corresponding equi-

librium angle.

The Keating potential was successfully applied for study of energy of elastic

deformation in diamond-like crystals.

The empirical interatomic Stillinger�Weber [16] potential is also widely used

for study of structural and dynamic properties of silicon. The interaction energy of

the system appears as follows:

U51

2

Xi; j

ϕðrijÞ1X

i, j, k

gðrijÞgðrikÞ cos θijk11

3

� �2; ð1:28Þ

where g(rij) is the decreasing function with the cutoff radius between the first and

second coordination shells.

The Tersoff potential [17] is applied to study vibration spectrums and relaxation

dynamics of imperfections in crystal lattices of semiconductors:

U5Xi, j;k

Uij;k; ð1:29Þ

where Uij;k 5 fcðrijÞðaijfRðrijÞ1 bijfAðrijÞÞ; fcðrijÞ is the truncating function;

fRðrijÞ5Ae2λ1r and fAðrijÞ52Be2λ2r are the components of repulsion and attrac-

tion (indexes R and A originate from words “repulsive” and “attractive”); bij5b(rij, rjk, θijk).

The bond i�j may be weakened because of incorrect directions of other bonds

i�k, which the i-atom has also. The angular components appear necessary for the

building-up of a realistic model. Certainly, because of enough great many of

parameters, their choice, and, hence, building-up of a realistic potential, is a hard

task.

The function mentioned above in the Tersoff potential for silicon appears as fol-

lows [17]:

bij 5 11βnξnij� �21=2n

; ξij 5P

k 6¼i;j fcðrijÞgðθijkÞexp λ33ðjrij2rikjÞ3

� ;

gðθijkÞ5 11c2

d22

c2

d2 1 ðh2cos θijkÞ2:

Key parameters of the potential are: A5 3264.7 еV, B5 95.373 еV,λ15 3.2394 A21, λ25λ35 1.3258 A21, β5 0.33675, c5 4.8381, d5 2.0417,

n5 22.956, h5 0.

The Stillinger�Weber, Keating, and Tersoff potentials are quite successfully

applied for silicon. The common disadvantage of all potentials is an inaccuracy

of their application under those conditions than for which they have been

adjusted.


1.2.6 Typical Errors in the MC Method

As well as each approximate method, the MC technique needs estimation of accuracy

and precision of the results obtained, that is, estimation of the relative error of calcu-

lated average value. Use of the MC technique is connected with systematic errors

(accuracy), which are proper to the model, and the peculiar errors of another sort—

statistical errors (precision). The first errors are connected with model idealization, for

example, with finiteness of degrees of freedom, with round-off errors at evaluations,

with inaccuracies of the potentials, with imperfection of random numbers generators,

etc. The second type of error arises for the reason that the average on the finite interval

of the Markov time, i.e., on the finite number of steps, is the casual value of a ran-

domly fluctuating value, and it converges to the exact average only in the limiting

case of the infinite time of measuring. For example, the value hEiN is a random value;

the width of its distribution decreases as N increases. Quantitatively, the dispersion

width δEN is calculated as the standard error in the error theory:

δEN 51

NðN21ÞXNi51

½Ei2hEii�( )1=2

5σffiffiffiffiN

p ; ð1:30Þ

where Еi is the instantaneous value of the measurand; hEii is the instantaneous

value of the average energy after i measurements; σ is the dispersion, that is, the

standard deviation of the energy from its average value.

Equation (1.30) shows that the dispersion of values of the average energy is

equal to the dispersion of its random values divided by on the radical from the

measuring number. The main property of the MC technique consists of: the error of

evaluations wanes proportionally to the radical from the real computational in the

certain statistical experiment. Thus, for magnification of the exactitude of the eval-

uation of any physical quantity on one decimal sign, a 100 times faster computer is

needed. Numerical modeling has objectively several key difficulties. One of them

is the choice of the optimal interaction potential and the scaling of quantities con-

nected with its functional form. At the modeling of atomic ensembles, it is neces-

sary to operate with very small magnitudes (mB10226 kg, rB10210 m,

ЕB10220 J). Positions of particles and energy of the system are usually scaled in

the MC method. It is possible to apply equations in reduced units, as other physical

quantities are connected to those two. Positions of particles are normalized by the

minimum distance between particles in real crystal lattices. Energy of systems is

normalized by the characteristic parameter, which enters into the interaction poten-

tial between particles.

1.3 The MD Method and Its Application

Alder and Wainright [18] offered the MD simulation method in 1957. It is consid-

ered in detail in many reviews [3,14,19,20]. The MD method allows us to explore


the systems at temperatures Т. 0. An ensemble of particles in some volume is a

model of the substance (phase). Forces acting on the particles from other particles

are determined on each timestep (the step of iterations �10215 s) and the resultant

forces are found as their sums, then accelerations of the particles and incremental

values of velocities are calculated, and then all the particles are simultaneously dis-

placed to new positions. Forces acting on the particles can be set analytically in the

form of equations, which include distances between atoms or also angles between

triples of atoms. In other variants, forces are given in tabular form corresponding to

certain consecutive distances with a constant step (let us say, δrD0.01 A). In this

case, an interpolation of the table data to the actual value of interparticle distance

for a certain pair of particles is necessary (data of long tables may be used without

of interpolation). Interparticle forces can be calculated according to the laws of

quantum mechanics or found from known diffraction data on structure of the sys-

tem under consideration [14,20].

For determination of the resultant forces (projections of the forces to coordi-

nates), it is necessary to fulfill calculations for all pairs (or triples) of atoms that

take the main part of evaluation time in the MD method. In the case of short-range

forces, it is possible to introduce a cutoff radius for interactions, rс, and to take into

account those pairs, distances between which do not exceed rс.

Real systems consist of a great many particles that interact with each other.

Though intermolecular forces generate complex trajectories of movement of each

molecule, properties of substance do not depend directly on them. They are con-

nected with averaged characteristics of the molecular movement. No supercom-

puter of the future will be able to solve microscopic equations of motion for 1025

particles interacting with each other. However, many phenomena occur in systems

from several thousand atoms just as it takes place in macroscopic systems. The MD

method is now applied to systems which routinely include from several thousand to

several tens of millions of particles, and it has already helped to advance under-

standing of observable properties of gases, fluids, and solids.

What basic properties do the systems of many particles have; what correlations

take place in them? What parameters should be used for description of such sys-

tems? Such questions are considered in statistical physics and the researcher who

uses the MD method should know its basic principles.

1.3.1 Algorithms for Numerical Solution of the Equation of Motion

The main meaning of Newton’s second law is that it expresses the equation of

motion

ma5mðd2r=dt2Þ5FðrðtÞÞ; ð1:31Þ

where r(t) is the particle position vector. The second law becomes valuable only at

known function F(r(t)), written down in Eq. (1.31). Then its integration is possible.

If initial velocity and particle coordinates are set, their values may be defined in


other instants of time. The simplest method of numerical solution of this equation

(Euler’s method) consists of the following: having chosen a small step on time Δt,

to define increments of position vectors and vectors of velocity under the

equations:

Δ~r 5 ~VUΔt;

Δ~V 5~aΔt51

m~Fð~rðtÞÞΔt: ð1:32Þ

Repeating such calculations for each particle, we obtain consecutive values of

coordinates and components of the velocity vectors at the instant of time

tn115 tn1Δt, for example, for coordinate x,

xn11 5 xn 1 υnUΔt: ð1:33Þ

For accuracy of numerical solution of the equation of motion by Euler’s method,

other similar methods depend on the timestep value Δt. This step should be small

enough that errors of the numerical solution are small and results satisfy laws of

conservation of both energy and impulse. In other ways to increase accuracy of cal-

culations, the equations become somewhat more complicated, for example, in

expression for coordinates the term with acceleration is introduced, an averaged

timestep acceleration is used for the velocity calculation.

A simple modification of expression (1.33) consists in defining xn11 through

υn11—end-point velocity in the finishing point of the timestep, instead of in initial

point. We will write down Euler’s method modified in such way (for simplifica-

tion, we will consider the one-dimensional motion):

υn11 5 υn 1 anΔt; ð1:34aÞ

xn11 5 xn 1 υn11Δt: ð1:34bÞ

As the algorithm (1.34) has been considered in detail by Cromer, it is termed

the Euler�Cromer method. Cromer has termed expressions (1.33) and (1.34) as

start-point and end-point approximations.

The purpose of all finite-difference methods consists in calculation of values vn11

and xn11 (a point in the phase space) at the instant of time tn115 tn1Δt. The time-

step value τ5Δt must be chosen so that integration yields the right solution. One

way to verify correctness of the method consists of monitoring the magnitude of the

total energy value and providing that it does not essentially differ from the initial

value. The overestimated magnitude of the timestep Δt results in numerical solu-

tions, which are more and more different from the true. The Euler and

Euler�Cromer algorithms do not ensure a conservation of energy during a compara-

tively large time of modeling. The timestep reduction increases the accuracy of cal-

culations, but after a large number of timesteps, the errors of rounding accumulate.


The distinctive feature of the MD algorithms consists of the method for calculat-

ing new coordinates and velocities at the end of a step. It is a standard problem of

numerical integration of equations of motion. One can apply the algorithms of a

different order of accuracy for its solution, e.g., the Runge�Kutta formulas [4], the

predictor�corrector method, etc. The Verlet algorithm is simple and exact enough.

It allows calculating coordinates of particles at the end of the timestep through their

coordinates at the beginning of this step and the previous step. It is important to

understand that the successful use of a numerical method is defined not only by

that how well it approaches derivatives in each timestep but also how well it

approximates integrals of motion, for example, total energy.

The essence of many algorithms can be understood after taking Taylor of

υn115 υ(tn1Δt) and xn115 x(tn1Δt):

υn11 5 υn 1 anΔt1OððΔtÞ2Þ; ð1:35Þ

xn11 5 xn 1 υnΔt11

2anðΔtÞ2 1OððΔtÞ3Þ: ð1:36Þ

The well-known Euler method is equivalent to keeping terms to the order of

O(Δt) in Eq. (1.36). Therefore, the magnitude of the local error (the error on the

step) is O(Δt)2. However, errors accumulate from step to step. The number of

steps, into which the time interval is divided, is proportional to 1/Δt. Therefore, the

global error of the Euler method obtained by summing the errors on the interval of

time is O(Δt). Hence, the global error is larger than local one. It is considered to

be that the method has n-order of approximating, if its local error is equaled

O((Δt)n11); therefore Euler’s method is a first-order method [3].

The Euler�Cromer algorithm (1.34) (end-point approach) has the same disad-

vantages as the Euler’s method [3]. The obvious way to improve the Euler

method is to use the timestep averaged velocity for evaluation of the new value

of the coordinate. The corresponding method of the average point can be written

as follows:

υn11 5 υn 1 anΔt; ð1:37aÞ

xn11 5 xn 11

2ðυn11 1 υnÞΔt: ð1:37bÞ

Upon substituting expression (1.37a) for v in (1.37b), we obtain


2anðΔtÞ2: ð1:38Þ

Hence the method of the average point is a method of the second order of accu-

racy for coordinates and the first order of accuracy for velocities.


The half-step method relates to methods of higher order of accuracy with

restricted error:

υn1

1/25 υ

n21/21 anΔt; ð1:39aÞ

xn11 5 xn 1 υn11/2

Δt: ð1:39bÞ

Here a change of coordinates is determined through average velocity on the certain

step. The half-step method is not self-starting, therefore, for starting the program

and for the program to run, it is necessary to set values of υ1/2. For the first step, it

is supposed that υ1/25 υ01 (1/2)a0Δt.

Verlet [21] has developed one of the best known algorithms of high order of

accuracy with respect to coordinates; it is termed the Verlet algorithm. By an anal-

ogy with Eq. (1.36), we will take Taylor of xn�1

xn21 5 xn � υnΔt11

2anðΔtÞ2: ð1:40Þ

Combining forward and backward integration equations forward and back

(expressions (1.36) and (1.40) accordingly), we obtain

xn11 5 2xn � xn21 1 anððΔtÞ2Þ1OððΔtÞ3Þ: ð1:41Þ

In addition, velocities are determined as follows:

υn 5xn11 � xn21

2Δt: ð1:42Þ

The local error due to the Verlet algorithm (1.41, 1.42) is of the third order in

coordinate and of the second order in velocity. However, the velocity does not par-

ticipate in integration of equations of motion. The Verlet algorithm (1.42) is termed

the implicit symmetric difference scheme in the literature on numerical analysis.

A shortcoming of the last algorithm is that it is not a self-starting, and it is nec-

essary to use another algorithm for deriving the several first points of the phase

space. Another disadvantage is that the new velocity is calculated by Eq. (1.42)

through odds of values (coordinates), which are close in magnitude. Such an opera-

tion stipulates losses of significant numerals and can result in the considerable

growth of the round-off error.

The following scheme is the version of the Verlet algorithm, its mathematical

equivalent:


2anðΔtÞ2; ð1:43aÞ

υn11 5 υn 11

2ðan11 1 anÞΔt: ð1:43bÞ


The scheme (1.43) is termed the velocity form of the Verlet algorithm. It is self-

starting and does not result in accumulation of the round-off error.

The Verlet method has properties of the predictor�corrector method known

from the educational literature. Namely, at first new values of coordinates are

predicted:

~xn11 5 xn21 1 2υnΔt: ð1:44aÞ

These coordinates are used for determination of accelerations ~an11, and then,

using ~an11, the corrected values vn11 and xn11 are obtained as

Corrector: υn11 5 υn 11

2ð ~an11 1 anÞΔt;

xn11 5 xn 11

2ðυn 1 υn11ÞΔt:

ð1:44bÞ

The predictor�corrector methods of higher orders of accuracy are often used.

This means that velocities are repeatedly corrected after determination of xn11 and

of the new values an11, and then the improved values of the coordinate xn11 are

determined again.

Other algorithms are also used in practice. For example, in Beeman’s algorithm

[3], not only coordinates but also velocities and accelerations are saved at the cer-

tain current timestep and the previous timestep. Coordinates and velocities in the

following instant of time are calculated by the equations:

riðt1 τÞ5 riðtÞ1 υiðtÞτ1 1

64aiðtÞ � aiðt � τÞ½ �τ2; ð1:45aÞ

υiðt1 τÞ5 υiðtÞ1 1

62aiðt1 τÞ1 5aiðtÞ � aiðt � τÞ½ �τ: ð1:45bÞ

Unlike the Verlet algorithm in the velocity form, the Beeman algorithm needs

one additional array of accelerations of all atoms. However, errors of evaluations at

its application are much smaller only in the case of applying of the Lennard-Jones

potential.

Algorithms of a higher order are also known: e.g., the Ralston�Wilf algorithms

[3,19] (data of four previous instants are used), Raman and Stillinger [3,19] (five

derivatives of coordinates are determined on each timestep), and Berne and Harp

[3] (a variant of the Runge�Kutta method). However, it makes sense to use such

algorithms of higher order only in special cases, for example, for returning into the

past when the split-hair accuracy of calculations is necessary.

There is no need to prefer any certain algorithm. It is better to consider all

aspects of the problem, such as a minimum: a number of particles required, a num-

ber of arrays, optimization of computation speed, necessary accuracy of the energy


conservation, method of temperature stabilization, etc., and only then to select an

algorithm. Concerning the outcomes of modeling, the errors due to connection with

the choice of the interaction, as a rule, are larger than those due the choice of the

algorithm if it is not too bad, and the timestep is small enough.

1.3.2 Near-Neighbor Calculations

As mentioned in Section 1.3.2, forces which act on a particle from other particles

are usually taken into account within the cutoff radius sphere, rc, for the particle

interaction potential when calculating its acceleration. Therefore, it is useful to

periodically create an array of the near-neighbors of each atom (the Verlet neighbor

list), which are in the orb with radius rv slightly exceeding the truncation radius rс.

If such an array is updated, for example, every k05 10 timesteps that for this period

any other particle should not penetrate into the sphere with the radius rc. The more

k0, the higher should be the difference rv2 rc. The higher is the difference rv2 rc,

the longer are the series of iterations, which can be made with the same list of the

nearest neighbors. There is considerable growth in computing performance while

using an array of Verlet lists, as a search of all pairs of atoms is not necessary, but

only the pairs with the atoms are considered, which are brought into a certain list.

For set rv and rc, the series length k0 is defined experimentally. Usually rv is chosen

so that this length exceeds rс by 10�20%. The more is k0, the less is the average

time of calculations, which is necessary on average for one iteration, but the longer

is the list of the nearest neighbors.

An increase of the computation speed is connected with an increase in the vol-

ume of occupied operative memory. If, for example, the model contains 10,000

particles and in the Verlet’s array are 100 neighbors per each atom, the arrays of

neighbors will contain 1,000,000 integers on the average. Thus, at the great number

of atoms in the model, a computer with a large operative memory is necessary for

application of the Verlet’s arrays.

Another way to decrease the number of trials of atom pairs concerning on the

value of distance between them consists of partition of the computational volume

of the model on cubic cells, the size of which is close to the cutoff radius rс of the

potential (Figure 1.7). Thus, each atom is defined not only by its number (denoted

by i) but also by three integer coordinates of the cell and the number cj in the list

of atoms for this cell (cx, cy, cz, cj). The pairs of atoms should be taken from the

cell, in which there is the chosen atom with the number i, and from the 26 nearest

small cells (from the 8 nearest cells in the two-dimensional case). If the distance

between atoms is less than the radius rc (see Figure 1.7), their interaction needs to

be considered for calculation of acceleration. It is clear that the enclosed cycles on

i, cx, cy, cz, and cj accordingly are needed for calculations of accelerations of all

atoms. However, the summarized size of these lists is considerably smaller, we will

say 103 103 103 605 60,000, than total of pairs of atoms (we will say,

50003 5000/2). Lists on cells, similarly to Verlet’s lists, should be updated through

a certain number of timesteps. For this purpose, an atom at first should be excepted

from the list of its previous cell, the coordinates of the new cell should be defined


through the new coordinates of the atom, and the number assigned to it is the next

after the last number in the new cell, in which the atom is placed. It is also possible

to include own numbers of atoms into the cell list.

1.3.3 Typical Elements of the Program for MD Modeling

To study qualitative properties of systems of many particles, it is enough to con-

sider that dynamics of atoms movement being classical, and atoms or molecules

being chemically inert blobs. If the force of interaction of any two atoms depends

only on the distance between them, the system full potential energy U is deter-

mined by the sum of energies over pairs of atoms. The pair interaction in the form

(1.22) corresponds to “simple” fluids, which consist of neutral atoms or molecules,

such as liquid argon.

The parameters σ and ε of the Lennard-Jones potential for the liquid argon are

ε/k5 119.8 K and σ5 3.405 A. It was accepted to express the energy, lengths, and

mass in units of ε, σ, and m, where m is the mass of a particle. In this case, measur-

ing of velocity in units of (ε/m)1/2 and time in units of τ5 (mσ2/ε)1/2 is convenient.The atom mass of argon is mAr5 6.693 10226 kg, and hence τ5 2.173 10212 s.

Now the main unit for tabulated potentials is electron volt. The use of natural units

for time and velocity (s and m/s) does not increase the evaluation time.

For solution of equations of motion, it is convenient to use the Verlet algorithm

in the velocity form (1.43) or Beeman’s algorithm (1.45). These equations should

be written for three coordinates x, y, and z accordingly for three components of

velocities and accelerations.

For reducing the text of the program, it is convenient to define operations with

vectors and to write down the equations of motion in the vectorial form.

Considering the two-dimensional case and using the elastic boundaries is enough

for qualitative analysis with an educational goal. Program interface is usually cre-

ated using tools of the installed software. Certainly, there should be procedureFormCreate and the procedures which provide functioning of the basic operating

elements of the Form.

Setting of the initial coordinates and velocities, as well as acceleration of parti-

cles in the case of Beeman’s algorithm, is mandatory. Let us call the corresponding

procedure—Init. For calculation of the forces acting on the particle and their ener-

gies, corresponding procedures or functions should be introduced in the program.

The basic procedure is the procedure in which the basic algorithm for calculation

rc

Figure 1.7 The scheme of the choice of particles

from the nearest cells m, for determination of the

resultant force of particle interaction.


of new velocities and coordinates is written down. It repeatedly calls the procedure,

which adds forces acting on each atom, and the last calls the procedure of calcula-

tion of forces. The experience shows that the deep structuration of the program,

which is convenient for understanding of all connections during its work, often

results in a decrease of the computing performance. In scientific programs, for

which the computing performance is determinative, researchers avoid multiple calls

of procedures and functions, including them as the following operators in a certain

large basic procedure.

Windows for mapping the structure of the simulated system in the course of cal-

culations, the current values of the basic physical characteristics, and the planned

diagrams should be present in the main Form or on its additional pages. It is clear

that additional procedures are necessary for reflection of outcomes of modeling.

The typical elements of the program for simulations by the MD method (elabo-

rated using “C11 Builder”) are described in the Application 10.

Nowadays, researchers often use the LAMMPS software (large-scale atomic-

molecular massively parallel simulator) [22] for MD and MC simulations. It is in

open access in the Internet. It allows using modern potentials of interanomic inter-

action, which take into account the electron density distribution. It allows construc-

tion of many variants of the acting program with different boundary conditions and

different regimes of the thermal treatment of the model in consideration. And it

allows parallel calculations by means of Graph processors (graphics cards). With

this, many researchers with only personal computers can fulfill serious scientific

studies. Application 13 gives information on how to prepare directives for simula-

tions with LAMMPS.

References

[1] R.Z. Sagdeev (Ed.), Experiment on the Display, Nauka, Moscow, 1989 (in Russian).

[2] B.A. Glinskij, B.S. Grjaznov, B.S. Dynin, E.P. Nikitin, Model Operation as the

Method of Scientific Examination, Moscow State University, Moscow, 1965.

[3] H. Gould, J. Tobochnik, Parts 1 and 2 An Introduction to Computer Simulation

Methods. Applications to Physical Systems, Addison-Wesley, Reading,

Massachussetts, USA, 1988.

[4] H. Gould, J. Tobochnik, W. Cristian, An Introduction to Computer Simulation

Methods: Applications to Physical Systems, third ed., Pearson & Addison-Wesley,

Reading, Massachussetts, USA, 2007, 813 p.

[5] D.W. Heerman, Computer Simulations Methods in Theoretical Physics, Springer-

Verlag, Berlin, 1986.

[6] V.M. Zamalin, G.E. Norman, V.S. Filinov, The Monte Carlo Technique in Statistical

Thermodynamics, Nauka, Moscow, 1977 (in Russian).

[7] V.A. Kazakov, Experiment on the Display, Nauka, Moscow, 1989, pp. 45�96.

[8] W. Feller, An Introduction to Probability Theory and its Applications, Wiley, New

York, 1971, 683 pp.

[9] J.A. Shrejder (Ed.), Method of Statistical Trials, Fizmatgiz, Moscow, 1962 (in

Russian)


http://refhub.elsevier.com/B978-0-12-420143-9.00001-6/sbref1



















[10] N. Metropolis, A.E. Rosenbluth, et al., J. Chem. Phys. 21 (1953) 1087.

[11] M. Creutz, Phys. Rev. Lett. 43 (1979) 553.

[12] A.M. Ovrutsky, A.S. Prokhoda, Sov. Phys. Crystallogr. 54 (3) (2009) 537.

[13] D.K. Belashchenko, Computer Modeling Liquid and Amorphous Substances, MISIS,

Moscow, 2005 (in Russian).

[14] A.P. Shpak, A.B. Melnik, Micro-inhomogeneous Structure of Unordered Metal

Systems, Akademperiodika, Kiev, 2005 (in Russian).

[15] P.N. Keating, Phys. Rev. 145 (1966) 637.

[16] F. Stillinger, T.A. Weber, Phys. Rev. B 31 (1985) 5262�5271.

[17] J. Tersoff, Phys. Rev. B 37 (1988) 6991�7000.

[18] B.J. Alder, T. Wainright, J. Chem. Phys. 27 (1957) 1208.

[19] D. Frenkel, B. Smit, Understanding Molecular Simulation from Algorithms to

Applications, Academic Press, the Netherlands, NY, Boston, London, Sydney, Tokyo,

2002.

[20] V.A. Poluhin, N.A. Vatolin, Modeling of Amorphous Metals, Nauka, Moscow, 1985

(in Russian).

[21] L. Verlet, Phys. Rev. 159 (1) (1967) 98.

[22] S. Plimpton, J. Comput. Phys. 117 (1995) 1.






















2 Basic Concepts of Theory of PhaseTransformations

2.1 The Method of Thermodynamic Functions

The aggregate of a large number of atoms or molecules with certain physicochemi-

cal properties and structure is termed a phase. Thermodynamics defines the phase

as a homogeneous part of system, which is separated from other parts of this sys-

tem by certain boundaries (boundary surfaces) and can be mechanically removed

out of it.

Phase states of the substance differ essentially in structure. Crystalline phases are

characterized by ordered placement of molecules in space, which is described by the

concept of crystal lattice. A smaller order concerning placement of molecules takes

place in amorphous phases; so-called short-range order exists in them. It can be

described by the probabilities of the nearest environment of every atom. The concept

“amorphous phase” concerns both liquid and solid bodies with corresponding struc-

ture. Therefore, solid glasses and resins are amorphous phases. During heating, they

gradually transform into the liquid state (they soften at first, and then start to flow).

The transition from the crystalline into liquid state, on the contrary, is very abrupt; it

occurs with a jump change of a matter volume and absorption of a heat or emission.

The same indications have such phase transitions as evaporation and condensation,

polymorphous transitions with a change of crystal structure of solids or transitions

in the state of liquid crystals. Transitions into the state of superfluidity of liquid

helium (4He and 3He) and superconductivity of some alloys occur without jump of

the volume and absorption of the heat or emission, but matter properties change

with a jump at the certain temperature. Ordering in positions of atoms in alloys, in

location of magnetic or dipole moments (ferromagnetic, antiferromagnetic, and

ferroelectric materials) can occur as with the jump of volume and the heat effect as

well as without them (only jumps of the matter properties).

Thermodynamics establishes general laws of phase equilibrium and offers how

to evaluate the driving force of phase transformations. Many thermodynamic rela-

tions can be deduced considering the Carnot’s cycle. However, the method of ther-

modynamic functions offered by Gibbs is more convenient. Each function that is

very convenient for use in certain conditions can be expressed through other

functions.



http://dx.doi.org/10.1016/B978-0-12-420143-9.00002-8

2.1.1 Internal Energy

According to the first law of thermodynamics,

δQ5 dU1 δA;

where U is the internal energy, Q is the heat. In the case of reversible processes,

the elemental heat δQ is connected with a change of entropy, δQ5 T dS. The united

form of the first and second laws of thermodynamics follows from these two

equations:

T dS5 dU1 δA: ð2:1Þ

The elemental macroscopic work δA5P dV for the system with P, T, V para-

meters only; therefore,

dU5 T dS� P dV : ð2:2Þ

It is convenient to consider the internal energy as a function of parameters S, V.

These parameters are termed characteristic. For isoentropic process (S5Const), a

change of internal energy is equal to work of a system with the opposite sign. The

change of internal energy is equal to the value of heat if the process occurs at con-

stant volume (V5Const). Writing down the total differential of internal energy as

a function of parameters S, V:

dU5 ð@U=@SÞVdS1 ð@U=@VÞSdV : ð2:3Þ

Comparing this expression with Eq. (2.2), we will obtain

T 5 ð@U=@SÞV ; P52ð@U=@VÞS: ð2:4Þ

If function U(V,S) is known, the expressions in Eq. (2.4) are the equations of

state because they express interdependence between parameters of state. The deri-

vatives of parameters on other parameters determine the properties of the matter.

They can be expressed through second derivatives from the internal energy:

CV 5 T@S

@T

� �V

5T

ðð@2UÞ=ð@S2ÞÞV; χS 52

1

V

@V

@P

� �S

51

Vðð@2UÞ=ð@V2ÞÞS:

As the internal energy is the state function that is a single-valued function of

state parameters, according to the Cauchy relation, its mixed derivatives do not

depend on order of differentiation:

@2U

@S @V5

@2U

@V @S:


Taking into account the expressions in Eq. (2.4) for the first derivatives, we

obtain the relation:

ð@T=@VÞS 5 � ð@P=@SÞV : ð2:5Þ

This is one of four Maxwell’s equations, which determines connections between

properties of substance.

2.1.2 The Helmholtz Free Energy

The Helmholtz free energy is determined through internal energy:

F5U � TS: ð2:6Þ

Its total differential

dF5 dU � dðTSÞ5 T dS� P dV � T dS� S dT ;

dF52S dT � P dV : ð2:7Þ

Function F(T,V) is convenient for consideration of processes at a constant vol-

ume. In the case of isothermal processes (T5Const),

dF52P dV 52dA; ð2:8Þ

ΔF52

ðV2

V1

P dV 52ΔA: ð2:9Þ

Hence, the free energy is the part of internal energy, which can be transformed

into a work during isothermal processes. The free energy change at the isothermal

process is equal to work of the exterior forces: ΔF5ΔAexter52ΔA. Product TS

is often termed linked energy. From expression U5F1 TS, equivalent to

Eq. (2.6), it follows that the system internal energy consists of the sum of free and

linked energy. The last cannot be transformed into a work in case of isothermal

changes.

The first derivatives of the free energy yield the equation of state:

S52ð@F=@TÞV ; P52ð@F=@VÞT : ð2:10Þ

The second equation for pressure is routinely used, when it is possible to calcu-

late free energy as outcome of the elaborated molecular kinetic theory. The equa-

tion of state, if it is already found, allows calculating properties of matter. If they

coincide with experimental data, the deduction about sufficiency of the developed

37Basic Concepts of Theory of Phase Transformations

molecular kinetic theory will be valid. One can express properties of the matter

through second derivatives from the free energy

CV 52T@2F

@T2

� �V

; χT 5 V@2F

@V2

� �T

� �21

:

Equating the second mixed derivatives of free energy with a different order of

differentiation,

@2F

@T @V5

@2F

@V @T;

we will obtain

ð@S=@VÞT 5 ð@P=@TÞV : ð2:11Þ

This is the Maxwell second thermodynamic relation.

Enthalpy

Enthalpy of the matter is defined through internal energy:

H5U1PV : ð2:12Þ

It is often designated also by the letter I. Its total differential

dH5 T dS1V dP: ð2:13Þ

The function H(S,P) is convenient for consideration of isobaric processes.

If P5Const, dH5 T dS5 dQ and ΔH5ΔQ. Thus, the heat obtained by a body in

case of isobaric process is equal to the increment of its enthalpy. Values of

enthalpy often are used for different physicochemical calculations. Therefore, the

values of enthalpy calculated for different temperatures are given in tables, which

are in corresponding manuals. For instance, values of integral

H2H0 5

ðT0

CμPðTÞdT

are given in the tables for crystals (counting per one mole of substance at atmo-

spheric pressure). During fusion or boil, a change of enthalpy is equal to the heat

of corresponding transformation.

The first derivatives of enthalpy on characteristic parameters yield equations of

state:

T 5 ð@H=@SÞP; V 5 ð@H=@PÞS: ð2:14Þ


Let us remember that function H(S,P) is the caloric equation of state, as well as

the functions U(S,V) and F(T,V), if they are known.

Equating the second mixed derivative of enthalpy with different order of

differentiation:

@2H

@S @P5

@2H

@P @S;

we will obtain the Maxwell third equation

ð@T=@PÞS 5 ð@V=@SÞP: ð2:15Þ

2.1.3 The Gibbs Free Energy

The Gibbs free energy, G, is determined through enthalpy or the Helmholtz free

energy:

G5H � TS; G5F1PV : ð2:16Þ

It is convenient for considering isobaric processes because the corresponding

total differential is connected with differentials from T and P:

dG5 dH � dðTSÞ5 T dS1V dP� T dS� S dT 5 dF1 dðPVÞ;dG5 2S dT 1V dP:

ð2:17Þ

That is, the parameters T and P are convenient for considering them as main

parameters of the Gibbs free energy, G(T,P). Then,

dG5 ð@G=@TÞPdT 1 ð@G=@PÞTdP: ð2:18Þ

Comparing Eq. (2.17) with Eq. (2.18), we will obtain equations of state:

S52ð@G=@TÞP; V 5 ð@G=@PÞT : ð2:19Þ

Further, we express properties of the substance:

CP 5 T@S

@T

� �P

52T@2G

@T2

� �P

;

χT 52ð1=VÞð@2G=@P2ÞT :

In addition, we write the Maxwell fourth equation

ð@S=@PÞT 52ð@V=@TÞP: ð2:20Þ


The Gibbs�Helmholtz relations connect thermodynamic functions with each

other:

U5F1 TS5F2 T @F@T

!V

52T2 @

@TFT

!V

;

H5U1PV 5G2 TS5G2 T @G@T

!P

52T2 @

@TGT

!P

:

After inverse integration, we will obtain

FV5Const 52 T

ðU

T2dT; GP5Const 52 T

ðH

T2dT : ð2:21Þ

2.2 Thermodynamic Functions of One-Component Systems

As at constant pressure dH5 T dS1V dP5 dQP,

HðTÞ5ðT0

CPðTÞdT1Hð0Þ: ð2:22Þ

Accordingly for entropy:

dS5δQT

5CP dT

T; SðTÞ5

ðT0

CPðTÞdTT

1 Sð0Þ: ð2:23Þ

In thermodynamic tables, the values of functions CPðTÞ; SðTÞ;HðTÞ2Hð0Þ andof functions connected with them are given in reference state (a pressure P05 1

physical atmosphere): C0PðTÞ; S0ðTÞ;H0ðTÞ2H0ð0Þ,

H0ðTÞ5ðT0

trans

0

Csol1P dT 1ΔHtrans 1

ðT0f

Ttrans

Csol2P dT 1ΔH0

f 1

ðT0ev

T0f

CliqP dT 1H0ð0Þ

(the specific heats are different for two polymorphic phases of solid state below

and above Ttrans). Else, one can find the reduced Gibbs free energy

Φ0ðTÞ52G0ðTÞ2H0ð0Þ

T5 S0ðTÞ2H0ðTÞ2H0ð0Þ

T:

Obviously,

F0ðT ;VÞ5G0ðT ;PÞ2PV0ðTÞ5ðT0

C0PðTÞ dT 1H0ð0Þ2 TS0ðTÞ2PV0ðTÞ:


PV0ðT ;VÞ{G0ðT ;PÞ in the case of condensed states. Therefore, functions

G0(T,P) and F0(T,V) differ slightly for condensed bodies.

For vapor of ideal monatomic gas:

dH5CP dT 55

2R dT ;

HðTÞ5 5

2RT 1Hð0Þ;

SðT ;VÞ5CμV ln T 1R ln V1 S00:

The entropy temperature dependence for gas in its reference state (P05 1 atm)

is determined with taking into account of the known theoretical value S0gasð0Þ of gasentropy at temperature 0 K,

S0gasðTÞ5 2:5R ln T 1 1:5R ln μ2 1:165R: ð2:24Þ

Accordingly, we shall have at pressure P05 1 atm for G0 (T):

G0ðTÞ5 3:665RT 2 2:5RT ln T 2 1:5RT ln μ1H0ð0Þ; ð2:25Þ

where μ is the molar mass.

At constant temperature, dGðT ;PÞ5V dP5RT d ln P. Therefore,

ΔGðT ;PÞ5RTÐ PP0d ln P5RT lnðP=P0Þ;

GðT ;PÞ5G0ðTÞ1RT ln P:ð2:26Þ

For real gases, dGðT ;PÞ5RT d ln aðPÞ, where a(P) is the volatility;

limP!0

aðPÞ5P for ideal gas, a(P05 1 atm)DP0. Therefore,

ΔGðT ;PÞ5ðPP0

RT d ln aðPÞ5RT lnðaðPÞ=P0Þ: ð2:27Þ

2.3 Conditions of Equilibrium in the ThermodynamicSystem

In the case of irreversible processes, which lead the system to equilibrium, entropy

increases: dS$ dQ/T or T dS$ dU1P dV.

From here,

dU# T dS� P dV : ð2:28Þ

Heat would get out from the system for the irreversible process at stationary

value of S to compensate the entropy growth; the internal energy of such system


will decrease. Thus, the minimum of internal energy at stationary values of the S

and V parameters is a condition of the system equilibrium.

If we add the product S dT to inequality (2.28), and subtract, we will obtain the

expression:

dU � T dS1P dV 1 S dT � S dT# 0 or dF1 S dT 1P dV # 0: ð2:29Þ

That is, at stationary values T and V parameters, dF# 0. The equality sign con-

cerns reversible processes; the inequality sign concerns irreversible processes. The

free energy will decrease, until the system does not come to equilibrium state.

Thus, the Helmholtz free energy for the system with stationary values of the T and

V parameters has the minimum value in the equilibrium state. It is important,

because this statement concerns isolated systems.

In the same way, we will deduce else two conditions of the equilibrium state of

systems. If we add the product V dP to Eq. (2.29) and subtract, we will have

dF � S dT 1P dV 1V dP� V dP# 0 or dG1 S dT 1V dP# 0: ð2:30Þ

That means, dG# 0 in case of irreversible processes with the constant T and P

parameters. We come to the conclusion that the Gibbs free energy has the mini-

mum value in the system equilibrium state in case of the time-invariant T and P

parameters. Such a system is not isolated. For example, if water is supercooled

below 0�C, crystals of ice with the smaller Gibbs energy per unit mass are precipi-

tating, and the Gibbs free energy of the system is decreasing. However, taking

away heat from the system is necessary, so that such process would occur in it.

To inequality (2.28), we add product V dP and subtract it:

dU1P dV � T dS1V dP� V dP# 0:

From here,

dH � T dS� V dP# 0: ð2:31Þ

Hence, enthalpy decreases at stationary values of the S and P parameters, if the

process is nonreversible. Enthalpy will have the minimum value in the equilibrium

state after the irreversible processes will be finished.

2.4 Equilibrium Conditions for Multiphase Systems

Homogeneous systems are physically homogeneous systems in which all thermody-

namic functions are identical in all points in the absence of field of forces; they

vary continuously in the presence of such fields. These are systems of gas

mixtures, liquid, or solid solutions. There are chemical reactions, dissociation of


gases, diffusion, processes of ordering, and so forth in such systems. Passing of

processes in one direction will be stopped at approaching equilibrium.

Breaks of continuity of thermodynamic functions take place on the boundary

surfaces. Homogeneous systems can transfer into heterogeneous and, on the con-

trary, the heterogeneous systems can become homogeneous. All number of identi-

cal physically homogeneous parts of thermodynamic systems are termed “phase.”

All drops of fluid are one phase. Phase equilibrium is the state of thermodynamic

system, in which the different phases of the substance having common boundary

surfaces do not vary quantitatively.

It is necessary to consider modification of thermodynamic functions for many-

component systems at adding of particles of a certain type:

dU5 T dS2P dV 1Xi

μi dNi; dF52 S dT 2P dV 1Xi

μi dNi;

dG52 S dT1V dP1Xi

μi dNi; dH5 T dS1V dP1Xi

μi dNi:

Here, μi is the chemical potential of the i-component:

μi 5@U

@Ni

� �S;V ;Nj

5@F

@Ni

� �T ;V ;Nj

5@H

@Ni

� �S;P;Nj

5@G

@Ni

� �T ;P;Njðj6¼iÞ

:

It follows from the condition dG5 0 for equilibrium in case of stationary values

of the parameters T and P that the interphase equilibrium condition is

Xi

μi dNi 5 0: ð2:32Þ

Let us consider a chemical reaction, for example 2H2 1O2 5 2H2O, which gen-

eralized equation isP

iυiAi 5 0, that takes place in the gas phase. In the yielded

example, υ15 2, υ25 1, υ3522. From the condition (2.32) and connection condi-

tions for values dNi (according to the reaction equation, dNi 5 υi dN), we will find

the equilibrium condition, which determines equilibrium concentrations of all com-

ponents:P

iυiμi 5 0.

2.5 Different Types of Phase Transformations

As mentioned above, evaporation, the transition from the liquid to its vapor under

condition of isothermal expansion at stationary pressure, happens with a change

of volume and entropy (ΔS5ΔH/T). The heat of the phase transition ΔH is con-

nected with the interior potential energy increase; the potential energy of vapor is

much higher than its value for liquid. An execution of work is necessary for


removal of molecules, which are attracted to each other, to the considerable dis-

tances at such transition. Changes of volume and entropy are indications also for

crystal melting and polymorphous transition. Such transitions are termed phase

changes of the first order. The transitions, which are connected with atomic

ordering or with ordering of magnetic or electrical dipole moments, have often

such indications too, that is, they are phase changes of the first order. However,

it happens frequently that the processes of ordering occur not so sharply.

Figure 2.1 shows two possible temperature dependences of magnetization of sub-

stances (the magnet moment per volume unit), which are ferromagnetic at low

temperatures. The curve 1 corresponds to phase changes of the first order—such

that the magnetization and volume of substances vary abruptly in the transition

point. The curve 2 corresponds to so-called phase change of the second order,

at which magnetization wanes gradually in the wide interval of temperatures, but

in the point θ (the magnetic transformation temperature or Curie’s point), where

a wane ends, its velocity is the greatest. Above the point θ, some small magneti-

zation still takes place, but it is already connected with paramagnetic state of

substance.

The characteristic indication of the phase transition of the second order is the

λ-kind dependence of the substance heat capacity on the temperature (Figure 2.2).

The heat capacity varies abruptly in the point θ. The heat capacity is the infinitely

large (CP5 (δQ/dT)P, dT5 0) in the transition point in a case of phase changes of

the first order. However, it is necessary to note that the infinite magnification of

heat capacity cannot always be determined experimentally.

Transitions to the state of the superfluidity or superconductivity fall into phase

changes of the second order. However, they are very complex and until now have

not been studied definitively. Especially, many obscure questions are connected

with the nature of high-temperature superconductors.

Ehrenfest has offered classification of phase transitions. According to this classi-

fication, the transition, at which the first derivatives of the Gibbs thermodynamic

potential on temperature T and pressure P vary abruptly, relates to phase changes

I

T

1

2

θ

Figure 2.1 Magnetization of substances dependences on the temperature: curve 1 for the

phase transition of the first order; curve 2 for the phase transition of the second order.


of the first order. Correspondingly, stepwise changes of entropy S and volume V of

the substance take place:

S52 ð@G=@TÞP; V 5 ð@G=@PÞT : ð2:33Þ

The thermodynamic potential in the transition point, to be exact, in equilibrium

point between two phases, is identical for both phases. However, dependences G(P,T)

for two phases are different. Figure 2.3 shows qualitatively such kind of dependences

of thermodynamic potentials of two phases, G1(T) and G2(T), on the temperature

under condition of stationary pressure.

Phase transitions of the second order are such transitions, at which the free

energy, entropy, and volume of both phases are identical still in equilibrium point,

that is, the dependences of these magnitudes on the pressure and temperature are

continuous. Hence, the first derivatives of the free energy on temperature and pres-

sure for both phases are identical in equilibrium point:

2S1 5@G1

@T

� �P

5@G2

@T

� �P

52 S2; V1 5@G1

@P

� �T

5@G2

@P

� �T

5V2: ð2:34Þ

Figure 2.2 The heat capacity dependence on the temperature for the phase transition of the

second order.

G

T

G2(T )

G1(T )

Tm

Figure 2.3 The temperature dependences of the Gibbs free energy of two phases (1 and 2)

in the case of the phase transition of the first order.


At the same time, second derivatives from the free energy on the same variables

are not identical; their sharp changes take place during transition from one phase to

another. It means the sharp change of all thermodynamic coefficients, which char-

acterize properties of substance, for example: heat capacity under condition of sta-

tionary pressure

CP 5 Tð@S=@TÞP 5 �Tð@2G=@T2ÞP; ð2:35Þ

coefficient of volume thermal expansion

αP 5 ð1=VÞð@V=@TÞP 5 ð1=VÞð@2G=@T @PÞ; ð2:36Þ

compressibility factor

χT 5 �ð1=VÞð@V=@PÞT 5 �ð1=VÞð@2G=@P2ÞT : ð2:37Þ

Unlike phase changes of the first order, the dependences of free energy on the

temperature and pressure can be considered as some uniform functions, which have

singularity in the transition point, in the case of the second-order transition.

However, the transition point is not a point of discontinuity G(T); there is an inflec-

tion point and a second derivative is equal to zero in it. Figure 2.4 shows something

similar for dependence G(T), at constant pressure, P5Const. The heat of transition

is absent, as entropy does not vary abruptly at the transition point.

2.5.1 Equilibrium Conditions for the First-Order Phase Transitions

In the case of phase changes of the first kind, the free energies of two phases are

equal to each other in equilibrium point (Figure 2.3). The Gibbs free energy is con-

nected with enthalpy by relation G5H2 TS. It follows from the condition G15G2:

H1 � TS1 5H2 � TS2; ΔS5 S2 � S1 5 ðH2 � H1Þ=T 5ΔH=T : ð2:38Þ

G

TTm

1

2

Figure 2.4 The single temperature dependence of the Gibbs free energy in the case of the

phase transition of the second order.


The system free energy is equal to the sum of free energies of both phases tak-

ing into account their amount:

G5 ðm1=μ1ÞG1 1 ðm2=μ2ÞG2: ð2:39Þ

Below the transition temperature, the free energy of phase 1 is less than the free

energy of phase 2. Therefore, the system free energy will decrease, if the phase 2 is

transforming into the phase 1. The phase 2 should be absent in the equilibrium

state. However, such transition will take place if the system is open for heat

exchange with environment. Otherwise, at the expense of heat of phase transition,

the temperature will increase to equilibrium point and the process will stop. The

similar situation takes place when the temperature is above the equilibrium.

Existence only of the phase 2 responds to the system equilibrium state. However,

heat is absorbing during transition from the phase 1 to the phase 2, and the

certain quantity of heat should be inserted into the system in order that such transi-

tion has come true.

The chemical potential is determined as the potential of Gibbs referred to one

molecule (or to one mole) in the case of one-component system; and the equilib-

rium condition is the equality of chemical potentials

μα 5μβ ; ð2:40Þ

where letters α and β designate two phases. This follows from the equilibrium con-

dition dG5μα dNα1 μβ dNβ5 0 that comes out right at stationary values T, P.

The equilibrium conditions in the case of many-component systems are the

equations of equality of chemical potentials of each component in all phases. We

will demonstrate that on example of the two-component two-phase system. In equi-

librium state, the Gibbs potential is minimum; therefore, dG5 dGα1 dGβ5 0 (as

G5Gα1Gβ). In case of stationary values of T and P,

dG5μαA dNα

A 1μαB dNα

B 1μβA dN

βA 1μβ

B dNβB 5 0: ð2:41Þ

Numbers of particles NαA, N

βA and Nα

B, NβB are connected with each other:

NαA 1N

βA 5NA; Nα

B 1NβB 5NB;

where NA and NB are the numbers of molecules of A- and B-components accord-

ingly. Therefore,

dNβA 52dNα

A; dNβB 52 dNα

B :

Let us rewrite Eq. (2.41), considering these relations:

ðμαA 2μβ

AÞdNαA 1 ðμα

B 2μβBÞdNα

B 5 0:


As incremental values of number of molecules dNαA and dNα

B do not depend on

each other, last equality is valid always, if

μαA 5μβ

A; μαB 5μβ

B: ð2:42Þ

Let us consider in more detail the equilibrium condition for the one-component

system. The change of pressure on dP and temperature on dT will not break equi-

librium (G15G2), if the corresponding increment values of free energies of two

phases are identical (1�A, 2�B):

dG1 5 dG2: ð2:43Þ

Writing down the increment values of the potentials dG1 and dG2 as total differ-

entials, we will obtain the expression:

ð@G1=@TÞPdT 1 ð@G1=@PÞTdP5 ð@G2=@TÞPdT 1 ð@G2=@PÞTdP: ð2:44Þ

Taking into account Eq. (2.43) in the following view:

�S1 dT 1V1 dP5 � S2 dT 1V2 dP;

we shall obtain

dP

dT5

ΔS

ΔV5

ΔH

TðV2 2V1Þ: ð2:45Þ

The last equation is termed the Clausius�Clapeyron equation. With its help, it

is possible to find the plurality of equilibrium points that is the equilibrium curve.

Equation (2.45) actually follows from the condition of equality of chemical poten-

tials for the one-component system case.

Let us write the Clausius�Clapeyron equation for one mole of the substance:

dP

dT5

λμ

TðVμ22Vμ1

Þ ; ð2:46Þ

and apply it to the transition from the condensed body (liquid or crystal) to vapor

(λμ is the evaporation heat per one mole). In this case, a volume of the second

phase (gas) is much larger than the volume of condensed phase. We will neglect

the volume Vμ1, and we will express the volume Vμ2

of gas through P, T, according

to ideal gas law. Thus,

dP

dT5

λμ

RT2=P: ð2:47Þ


After integration, we come to

P5Const expð�λμ=RTÞ: ð2:48Þ

This equation features in an explicit form the curve of equilibrium of condensed

phase with gas or, in other words, the dependence of the saturation pressure on the

temperature (Figure 2.5). The equilibrium temperature T1 corresponds to the pres-

sure P1. If the pressure increases at a constant temperature, the vapor will become

supersaturated. The state with supersaturated vapor will not exist long in the closed

vessel. After condensation of part of the vapor, the pressure will decrease to the

equilibrium value.

The supersaturated vapor can be obtained also by way of temperature reducing.

The saturated pressure P2 answers the smaller temperature T2. If in the vessel,

where there is no fluid, the vapor pressure will decrease at reducing of temperature

according to the gas law (see the dashed straight line in Figure 2.5), its change will

be much smaller, and it will reach the value P02. As the pressure P0

2 is more than

saturated vapor pressure P2, the vapor will be supersaturated. Because of vapor

condensation, the system will approach the equilibrium, and the vapor pressure will

wane to value P2. Hence, the fogs in the mornings reflect different dependences of

the pressure of gas and saturation pressure on the temperature.

2.5.2 The Ehrenfest Equations

The Clausius�Clapeyron equation loses sense for phase transformations of the sec-

ond kind because both the numerator and the denominator in Eq. (2.45) are equal

to zero. Instead of this equation, Ehrenfest has offered several equations, which

connect the changes of the temperature and pressure values with the changes of the

thermodynamic coefficients. We will consider the conditions of the continuity of

the first derivatives of the Gibbs potential in the transition point for the one-

component system:

ð@G1=@TÞP 5 ð@G2=@TÞP.dð@G1=@TÞP 5 dð@G2=@TÞP; ð2:49Þ

P1

P

P2

P2

T2 T1 T

′

Figure 2.5 The pressure dependences on the temperature: the solid line shows the

dependence for saturated vapor pressure; the dotted line shows the vapor pressure in the case

of absence of liquid phase in the vessel.


ð@G1=@PÞT 5 ð@G2=@PÞT.dð@G1=@PÞT 5 dð@G2=@PÞT : ð2:50Þ

Let us write total differentials:

ð@2G1=@T2ÞPdT 1 ð@2G1=@T @PÞdP5 ð@2G2=@T2ÞPdT 1 ð@2G2=@T @PÞdP;ð@2G1=@P @TÞdT 1 ð@2G1=@P2ÞTdP5 ð@2G2=@P @TÞdT 1 ð@2G2=@P2ÞTdP:

Taking into account relations (2.35)�(2.37), we will obtain

2CP1

TdT 1VαP1

dP52CP2

TdT 1VαP2

dP; ð2:51Þ

VαP1dT 2VχT1

dP5VαP2dT 2VχT2

dP: ð2:52Þ

Designating jumps of thermodynamic coefficients:

ΔCP 5CP2 2CP1; ΔαP 5αP2 2αP1; ΔχT 5χT2 2χT1;

we will definitively obtain

dP=dT 5ΔCP=ðTV ΔαPÞ; ð2:53Þ

dP=dT 5ΔαP=ΔχT : ð2:54Þ

As we see, the changes of pressure and temperature in the case of maintenance

of equilibrium are really connected with the changes of the thermodynamic coeffi-

cients. Having equated right members of Eqs. (2.53) and (2.54), we find relations

between these changes:

ΔCP 5Δα2PTV=ΔχT : ð2:55Þ

2.5.3 The Gibbs Phase Rule

The equilibrium condition for the many-component system at stationary values of

temperature and pressure is

dG5Xi;j

μji dN

ji 5 0; ð2:56Þ

where μji is the chemical potential of ith component in the jth phase; dN

ji are the

differentials of particle numbers of the components. The same as for the binary sys-

tem (see Eq. (2.40)), the equilibrium condition (2.56) leads to the set of equations.

In total, κ �φ of chemical potentials (κ is the number of components, φ is the


number of phases) enter into these equations, but not all of them are independent.

At first, chemical potentials of each component should be equal in all phases:

μ1i 5μ2

i 5μ3i 5?5μφ

i : ð2:57ÞThere are κ(φ2 1) of such equations. Second, chemical potentials depend on

concentrations of components, which are connected with each other in each phase

by the condition that their total is a magnitude of the stationary value; it means that

φ equations are in addition (generally, the chemical potential is the function of tem-

perature, pressure, and concentrations). Therefore, we have φ1 κ(φ2 1) equations

for a system with κ components and φ phases, and the number of independent para-

meters (concentrations) is

κ � φ2φ2κ � ðφ2 1Þ5 κ2φ:

Besides, temperature and pressure are also the independent parameters. Hence,

total of independent parameters or degrees of freedom of the system is

C5κ� φ1 2:

This amount cannot be negative (C. 0). The Gibbs rule of phases follows from

φ#κ1 2: ð2:58Þ

It spots the maximum quantity of the phases existing simultaneously at equilib-

rium conditions.

Along the curve of equilibrium of condensed one-component substances with

the vapor, the system has two phases, and thus the degree of freedom is one.

Any change of pressure completely determines the temperature change necessary

for maintenance of equilibrium (or on the contrary). Three phases can be in equilib-

rium in the one-component system, for example: the crystal, the liquid, and the

gas. However, in this state, the system has no degrees of freedom. One of phases

disappears in the case of change of the temperature or pressure. It is so-called the

triple point.

Phase Diagrams

The plurality of curves of the phase equilibrium is termed a constitution diagram

(or state diagram). The state diagrams are considered in both the general course of

physics, and in special courses of thermodynamics and kinetics of phase changes.

We will consider only two diagrams, concerning equilibriums of crystalline and liq-

uid phases.

Figure 2.6 shows the phase diagram of binary system of the cigar type.

Formation of solutions at any concentration of components both in liquid and in

solid phase takes place in this case. Such is the phase diagram of the system

copper�nickel, for example. The melting points of pure components are pointed on


the temperature axes of this diagram. On the abscissa axis, the concentration of

B-component is put aside. The curve 1 is termed the liquidus line, and the curve 2

the solidus line. The liquid with concentration Cl1 and the crystal with concentra-

tion Ccr1 are in equilibrium at the temperature T1. During the gradual dropping of

temperature, the amount of fluid in the system is decreasing, and the amount of

crystalline phase is increasing, their equilibrium concentrations vary according to

the curves of equilibrium—1 and 2.

The phase diagram shown in Figure 2.7 is termed the diagram of eutectic type.

Such is the phase diagram of the system lead�tin, for example. Here solid solu-

tions can also be, but only as the additive of the second component to the crystal

phase on the basis of the main component with the structure of pure component.

If the concentration of B-component in the system is less than Cevt, the solid solu-

tion α with concentration Cα1and liquid with concentration Cl1 can be in equilib-

rium at temperature T1. If the concentration exceeds Cevt, the solid solution β with

concentration Cβ1and liquid with concentration C0

l1will be in equilibrium. Point E

is termed the eutectic point. In it, there is fluid and two solid solutions α and βare in equilibrium. If we consider also the gas mixture of components A and B,

which is in the system, it becomes clear that the point E is the quadruple point.

T

T1

TfA

TfB

A 0.5

1

2

Liquid + crystal

BCl1Ccr1 CB

Figure 2.6 The cigar type constitution diagram of binary system. There are liquid or solid

solutions at any concentrations.

T

T1

TfATfB

Tevt

Liquid

A

E

Cβ1

Cα1Cl1

Cevt CBB

α

α + β

β

Cl1′

Figure 2.7 The eutectic type constitution diagram of binary system. There are solid

solutions only at concentrations that are close to the pure elements.


Below temperature Tevt, only two solid solutions and gas can be in equilibrium.

Crystals of α- and β-phases grow simultaneously from the melt during crystalliza-

tion of the alloy of eutectic composition. More often, they form the regular struc-

tures, the so-called eutectic colonies.

2.6 Influence of the Interfacial Tension on Crystallizationof Liquids

Interfacial tension plays a rather considerable role in the phase transformations. For

example, it provides an existence of metastable states of substance, such as super-

saturated vapor and supercooled liquid.

Suppose that liquid is in the closed vessel with the capillary tube, the walls of

which cannot be moistened by liquid; so the vapor over the liquid is saturated

(equilibrium). However, the vapor pressure depends on height. It is higher over the

convex surface of liquid in the capillary tube (the level is below, rather than in the

vessel) than the saturated vapor pressures over the flat liquid surface Po:

Pr 5Po 1 ρvgh5Po 1ρvρl

2σR

; ð2:59Þ

where R is the curvature radius of liquid surface, ρl is the liquid density. It is a very

important equation. If a liquid drop will be entered into the vessel, where there is the

saturated vapor, it will evaporate, because saturation pressure for the drop is more

than the saturation pressure for the flat surface (vapor is not saturated for the drop).

In order that the liquid drops would exist in the vessel, it is necessary to reach the

higher vapor pressure. Nuclei of condensation, that is, small droplets of liquid, which

are usually formed on the dust particles, may be in equilibrium with vapor only at its

certain supersaturation. Moreover, very large supersaturation of vapor is necessary

for forming of liquid nuclei, if there is no dust particles in vapor.

Equation (2.59) can be rewritten as

r� 52σ

P2Po

ρvρl

:

The last equation allows calculating the drop size r�, for which the dynamic

equilibrium between the number of condensing molecules and the number of mole-

cules evaporated for the same time is fulfilled. However, this equilibrium is labile.

If the drop size becomes a little smaller because of fluctuations of amounts of

deposited molecules and those, which evaporate, the vapor over the drop becomes

nonsaturated for it: P5Pr� ,Pr0, and the drop will be evaporated further. On the

contrary, if the drop size increases, it becomes a center of condensation

(P5Pr� .Pr0), and its volume will increase further.

The surface tension influences not only on equilibrium of liquid with its vapor.

Forming of nuclei of a new phase is connected with interface formation for any


phase transition of the first kind. The classical picture of nucleation assumes the

statistical particle density fluctuations in the metastable melt, which make possible

the forming of ordered clusters. Whether this initial cluster is stable (and grows) or

unstable (and disappears) is governed by competition between the surface energy

of the cluster dependent on the (macroscopic) fluid-crystal surface tension, σ, andthe gain of energy upon crystallization. The radius of the critical cluster, r�, can be

calculated by minimizing the free energy. If a crystal nucleus in the form of full-

sphere in radius r appears in liquid, the Gibbs free energy of the system will vary;

its change depends on the radius:

ΔG5ΔNcrμcr 1ΔNlμl 1 4πr2σ

or

ΔG5 � ð4πr3=ð3ΩÞÞðμl � μcrÞ1 4πr2σ; ð2:60Þ

where μcr and μl are the chemical potentials of the solid and liquid phase;

ΔNcr52ΔNl; 4πr3/(3Ω) is the number of molecules in the nucleus; Ω is the vol-

ume of one molecule; σ is the specific free surface energy; 4πr2 is the area of the

nucleus surface. The first term in Eq. (2.60) is negative, as for supercooled liquid

μl.μcr.

The second term in Eq. (2.60) always is positive; it is larger than the first term

in the case of small sizes of nuclei. The change of the Gibbs free energy (ΔG) of

the system in result of nucleus formation depends on its size. Figure 2.8 features

such dependence. If the nucleus size is very small, the magnitude ΔG is positive,

and the more the size r is the more ΔG becomes. At the certain size, which we

will name the critical size and designate r�, the value of ΔG passes the maximum;

and then the decreasing of ΔG begins. If the nucleus size exceeds the value r�,new molecules are joining to it that results in a decrease of free energy of the sys-

tem. Such a nucleus is termed the crystallization center. The critical size r� can be

found from the condition of extremum of the function ΔG(r). We shall fulfill dif-

ferentiation on r and shall equate the first derivative to zero; then,

Figure 2.8 The changes of the Gibbs free energy (ΔG) of the system in result of nuclei of

different size formation; r� is the critical size.


r� 52Ωσ

μl 2μcr

52VμσTfL ΔT

; ð2:61Þ

whereΔT5 Tf2 T is the supercooling, Vμ5ΩNA is the molar volume of the crystal-

line phase, NA is the Avogadro number. The chemical potential of one-component

systems is the Gibbs free energy in counting per one molecule; chemical potentials of

two phases are equal at equilibrium conditions (melting point). For the supercooled

system,

Gμl � Gμ

cr 5Hl � TSl � ðHcr � TScrÞ5ΔHμ � T ΔSμ;

where H is the enthalpy; S is the entropy; ΔHμ5 L is the heat of phase transition

(the crystallization heat) per one mole of the substance. From the condition of

equality of the free energies Gμl 5Gμ

cr at the melting point Tf, the relation follows:

ΔHμ 5 Tf ΔSμ:

Therefore,

μl 2μcr 5 ðGμl 2Gμ

crÞ=NA 5L ΔT

NATf5

ΩL ΔT

VμTf:

Equation (2.61) is termed Thomson’s formula. It allows determining the size of

crystal, which is in equilibrium (unstable balance) with supercooled liquid. Having

substituted expression for r� in Eq. (2.60), we will find the work of formation of a

nucleus of critical size:

ΔGðr�Þ5 �ð8=3Þπσðr�Þ2 1 4πr�σ5 ð4=3Þπσðr�Þ2: ð2:62Þ

In the case of small supercoolings, the equilibrium size of crystals is large and

work of formation of such large nuclei is rather considerable. Therefore, the proba-

bility of their formation during a finite time is too small. Experiments in the study

of so-called homogeneous crystallization (crystallization of small drops, which may

be free from contaminations and the solid particles weighed in liquid) have shown

that very large supercoolings congruent with the value of the melting temperature

are necessary for nucleation, ΔT5 Tf2 T� 0.2Tf.

The probability of nucleation is connected also with the value of liquid volume.

During fast cooling of small drops of different substances, the crystalline centers

are not arising at all in many of them, and the liquid turns into a solid amorphous

body. The amorphous materials obtained by the method of fast crystallization are

widely applied now in different fields of science and manufacturing. Scientists of

Dnepropetrovsk University, I.V. Salli, M.I. Varich, and I.S. Miroshnichenko, were

pioneers who applied the method of the superrapid liquid cooling for obtaining of

metal alloys in metastable (nonequilibrium) states.


It is not possible to reach the considerable supercoolings during cooling of

liquids with great volume, because nuclei are formed not only in liquid volume, but

also on walls of the vessel or solid particles, which are in liquid. In this case, so the

saying goes, heterogeneous crystallization takes place. The heterogeneous crystalli-

zation takes place nearly always under usual conditions because the work of nucle-

ation on the walls is much less than the work of nucleation far from them. It is

possible to influence the amount of solid impurities and the value of supercooling

by the filtering of liquid. Overheating of a liquid above melting point before the

start of cooling leads also to an increase of the supercooling, at which crystalliza-

tion begins.

The crystallization of substances does not begin without the supercooling of liq-

uid; therefore, curves of melting and solidification (dependences of temperature on

the time during heating and cooling) differ qualitatively (Figure 2.9). The fusion

curve (shown in the drawing by a solid line) has a horizontal segment correspond-

ing to the value T5 Tf; it is connected with crystallization heat absorption (the heat

entered into the system, goes on magnification of internal potential energy). The

dotted cooling curve testifies that crystallization begins when the temperature of

the substance is less than the melting point. Owing to the crystallization heat, the

substance is warming up nearly to melting point. When crystallization ends, the

temperature starts waning.

A crystallization process is influenced essentially by the amount of nuclei. If a

lot of nuclei are formed, the substance has a polycrystalline structure in a solid

state. During crystallization from one center, that practical difficulty is realizable,

the substance becomes single crystal. Single crystals are used for manufacturing

the majority of electronic devices. This stimulates a development of physics of

crystal growth and industry of fabrication of single crystals.

Works of the academician Danilov and his students (Professors Miroshnichenko,

Salli, Ovsienko, Alfintsev, Maslov, Lesnik, Kamenetskaya) at Dnepropetrovsk

University and in the Institute of Physics of Metals of Ukraine Academy of

Sciences (Kiev) were very important for the development of physics of crystalliza-

tion processes.

T

Tm

t

Figure 2.9 The curves of heating (solid) and cooling (dotted) of a system, in which melting

or crystallization take place.


2.7 Phenomena Connected with Formation of Solutions

2.7.1 Heat Effects at the Solution Formation

Solutions are the homogeneous mixtures of two or more substances. If the amount of

one of these substances is much more than of others, it is termed solvent and others

are dissolved in solution. Solutions composed of two components are termed the

binary solution. Depending on the phase state, the solution can be liquid or solid.

The substance equilibrium state at constant temperature and volume is deter-

mined by the minimum of the Helmholtz free energy F5U2 TS. The free energy

will be decreasing, if the internal energy is decreasing or entropy is increasing.

During mixing of different substances, entropy is always increasing, because the

thermodynamic probability of the system is increasing (according to the Boltzmann

equation, S5 k lnWT). Therefore, the solution can be formed also in that case,

when the internal energy increases. Depending on the sign of the internal energy

change in the course of the solutions formation, the heat can get out (the exother-

mic process) or be absorbed (endothermic process).

The change of internal energy in the case of formation of binary solution is con-

nected with a number of pairs of different atoms NAB (A and B are designations of

components) and excess energy of their interaction determined by the equation

εAB 5VAB � ðVAA 1VBBÞ=2; ð2:63Þ

where VAA, VBB, and VAB are the interaction energies of corresponding pairs of

atoms (all of them are subzero because the free atoms have zero potential energy).

If excess energy εAB is equal to zero, the solution is termed the ideal solution. In

the course of its formation, the heat effect is absent. If 2jVABj. jVAAj1 jVBBj, thatmixing heat W52NAB, εAB is more than zero (exothermal reaction of the solution

formation, ΔQ, 0). If 2jVABj, jVAAj1 jVBBj, formation of the solution leads to

increase of internal energy (W52NAB, εAB, 0—heat is absorbed, ΔQ. 0, and

reaction of the solution formation is endothermic). Such solutions are formed only

at sufficiently high temperatures.

Nonideal solutions, as a rule, are slightly ordered. If εAB, 0, it is energetically

favorable that atoms of one component have atoms of other component as the near-

est neighbors; thus, the number NAB increases. On the contrary, if εAB. 0, it is

more favorable that each atom has the nearest environment of the same type at

itself. Thus, entropy decreases. The equilibrium state corresponds to a certain

ordering level in respect of placement of atoms.

2.7.2 The Raoult’s and Henry’s Laws

Osmotic Pressure

Let us consider some vessel C (Figure 2.10) containing pure liquid component A

(solvent). The tube T with solution is closed from below by a semipermeable


membrane D and is lowered into the pure liquid. The membrane D allows mole-

cules of solvent (A-component) to pass through it and does not allow to pass

through molecules of B-component of the solution poured into the tube T. Both the

solvent and solute put pressure upon walls of the tube T. The total of these pres-

sures is also equal to the sum of atmospheric and hydrostatic pressures: Pa1 ρgh(ρ is the solution density). Pressure of solvent over the membrane D

PA 5Pa 1 ρgH2Π; ð2:64Þ

where H is the height of fluid in the tube and Π is the, so-called, osmotic pressure

of the solute. If the pressure PA of A-component is less than Pa, to which the pres-

sure of pure solvent below the membrane is equal, solvent molecules will transfer

through the membrane from the vessel into the tube T. This process will prolong

until pressure PA reaches the value Pa. From this condition, it follows that the cer-

tain height of the solution in the tube T corresponds to dynamic equilibrium on the

semipermeable membrane D:

H5Π=ðρgÞ: ð2:65Þ

In the case of sufficiently small concentration of B-component in the solution,

one can consider it as ideal gas (i.e., to neglect by interaction of B-molecules).

Therefore, the pressure of the solute can be calculated under the gas law (the van’t

Hoff equation)

Π5PB 5 ðmB=μBÞRT=V 5 ðρB=μBÞRT ; ð2:66Þ

where mB and μB are the mass and molar mass of the B-component accordingly;

ρB5mB/V is its concentration. In cases when the concentration of B-component is

several percents (e.g., the salt or sugar solution in water), its pressure reaches to a

level of tens of atmospheres. To make things clear concerning the sense of the

T

C

Pa

PaD

H

Figure 2.10 The scheme of the vessel for observation of the osmosis process.


phenomenon of osmosis, we point out that solvent can be in the solution under a

considerable negative pressure. The certain density of the solvent corresponds to

the atmospheric pressure. The solvent density in the solution lies under this value.

Hence, the solvent is in the expanded state.

The First Raoult’s Law and the Henry’s Law

It is clear that dynamic equilibrium between solvent and its vapor will be upset after

addition of solute. The amount of molecules, which fly out from solvent, becomes

smaller, and a smaller value of equilibrium vapor pressure will correspond to this

amount. For its determination, let us imagine that the vessel C with tube T

(Figure 2.10) is placed in a large closed vessel. Vapor pressure in such system

(we consider its equilibrium state) depends on a height. The pressure at the height H,

P5Po � ρvgH; ð2:67Þ

where Po is the vapor pressure over pure solvent and ρv is the density of its vapor.

We will express the height H through osmotic pressure from Eq. (2.65) considering

also the relation (2.66). Then,

Po 2P

Po

5ρvgPo

Πρg

5ρvρBRTρvRTρ

μA

μB

5ρBμB

μA

ρ5

υBυA

;

where Po5 ρvRT/μA; υA and υB are the numbers of moles of the components

A and B accordingly.

For small concentrations of the B-component, υA� υA1 υB; therefore, the last

equation can be rewritten as follows:

Po 2P

Po

5υB

υA 1 υB: ð2:68Þ

This is the first Raoult’s law: the relative pressure decrease of the vapor of sol-

vent over the solution is directly proportional to the relative molarity of the solute.

Henry has offered the similar relation for concentration of the solute and its

pressure over the solution:

υBυA 1 υB

BPB: ð2:69Þ

Henry’s law in reading from right to left,

PB 5PoB

υBυA 1 υB

; ð2:70Þ

is termed the Raoult’s law for solute.


Concerning ideal solutions, Raoult’s and Henry’s laws are valid for all

concentrations of the components and in case of real solutions are valid only

for small concentrations of one of two components. Figure 2.11 shows depen-

dences of partial pressure of components on their concentration in the solution.

Straight lines 1 and 2 feature the ideal solution case. The straight line 3, which

joins points PoA and Po

B (PoA and Po

B are the vapor pressures over pure compo-

nents), shows the total pressure over the ideal solution. The curve 4 maps the

dependence on concentration of the solvent vapor pressure in the case of the

real solution.

The Second Raoult’s law

The second Raoult’s law correlates the boiling points with the concentration of

solute in the solution.

Boiling begins at such temperature Tb, at which the pressure of saturated vapor

reaches the value of exterior (atmospheric) pressure. After adding the second com-

ponent into the liquid, the vapor pressure of solvent decreases. Therefore, boiling

becomes possible only at a higher temperature, which will ensure an increase of

the solvent vapor pressure to the value of exterior pressure.

It follows from the Clausius�Clapeyron equation (2.46) and the first Raoult law

that

ΔTb 5RT2

λμ

υBυA 1 υB

5EmB=μB

mA=μA

; ð2:71Þ

where E5RT2/λμ is constant for a certain substance, λμ is the evaporation heat per

one mole, and υB5mB/μB, υA5mA/μA.

P

Pid

PAPA

PB

42

3

1

10.5 νB/(νA + νB)

oPA

oPB′

Figure 2.11 Dependences of partial pressures of components in the gas phase on their

concentration in the solution. The line 1 shows the pressure of A-component upon the ideal

solution, P0A is the pressure of A-component upon the real solution; the line 2 shows the

pressure of B-component upon the ideal solution; the line 3 shows the total pressure upon

the ideal solution.


The Third Raoult’s Law

The third Raoult’s law establishes the relation between the drop of melting point

and concentration of solute in the liquid solution; it is written as follows:

ΔTf 5KυB

υA 1 υB; ð2:72Þ

where K5RT2/L is the cryoscopic constant; L is the melting heat per one mole.

Equation (2.72) follows from the first and second Raoult’s laws.

2.7.3 Partial Thermodynamic Functions

Partial functions are convenient for the theoretical description of solutions [1]. We

will consider first the dependence of the solution volume on the solute concentra-

tion (the curve V(C) in Figure 2.12). For the ideal solution, the volume depends lin-

early on the concentration between values of volumes for pure components V0A and

V0B. The partial volumes are defined by segments (VA and VB), which are cut off

on axis of ordinates by the line tangential to the curve V(C) in the point corre-

sponding to a certain concentration (the relative molarity xB5CB5 υA/(υA1 υB),υA and υB are the numbers of moles).

Considering the geometrical meaning of the derivative dV/dx, we will write

down the following obvious equations:

V 5 xAVA 1 xBVB ; dV 5 dxAVA 1 dxBVB ; ð2:73Þ

VA 5V2dV

dxxB; VB 5V 1

dV

dxxA: ð2:74Þ

Designate through ΔV the amount of deviation of the solution volume from the

magnitude of volume of the ideal solution: ΔV5V2Vid. Obviously,

ΔV 5 x1 ΔV1 1 x2 ΔV2 ;

Ideal solution

BA 0.5 CB, mol.%

0

dVdx

V(C)VA

VB

VA

0VB

Figure 2.12 Volume dependences of real and ideal solutions on the solute concentration.


where

ΔV1 5V1 2V01 ; ΔV2 5V2 2V0

2 ; x1 � xA; x2 � xB:

Other partial functions are similarly introduced:

dS5 S1 dx1 1 S2 dx2; dH5H1 dx1 1H2 dx2; dG5μ1 dx1 1μ2 dx2:

In the last equation, chemical potentials are written instead of Gi because

μi5 (@G/@xi)T,P,xj(j 6¼ i) (μi5 (@G/@Ni)T,P,Nj(j 6¼ i) counting per one molecule). As the

equations G5μ1x1 1μ2x2 and dG5μ1 dx11μ2 dx21 x1 dμ11 x2 dμ2 are also

fulfilled, the Gibbs�Duhem relation follows from this:

x1 dμ1 1 x2 dμ2 5 0: ð2:75Þ

2.7.4 Ideal Solutions; the van’t Hoff Equation; the DistributionCoefficient

In the case of ideal solution, ΔV5V2Vid5 0, ΔH5 0, ΔS5 ðΔH=TÞ5 0, but

Sid 6¼ 0.

As the first Raoult’s law is fulfilled for all concentrations,

P01 2P1

P01

5 x2;P02 2P2

P02

5 x1;

the chemical potential of the i-component depends simply on its concentration.

We have expression for μi(Pi): μi 5RT ln Pi 1ϕiðTÞ and in the reference state for

the pure components μ0i 5RT ln P0

i 1ϕiðTÞ. The Gibbs molar potential change due

to the solution formation at stationary temperature from the corresponding amounts

x1 and x2 of pure components is

ΔGsol 5Gsol 2G0 5 x1 Δμ1 1 x2 Δμ2 5RT x1 lnP1

P01

� �1 x2 ln

P2

P02

� �� :

Therefore,

ΔGsol 5RTðx1 ln x1 1 x2 ln x2Þ: ð2:76Þ

As dG52S dT 1V dP1Xi

μi dNi,

ΔSsol 5@ΔGp

@T52Rðx1 ln x1 1 x2 ln x2Þ: ð2:77Þ


Enthalpy during ideal solution formation does not vary:

ΔHp 5ΔGp 1 T ΔSp 5 0:

The Van’t Hoff Equation

Let us consider the equilibrium of the liquid regular solution with the pure

A-component in the crystalline state. The condition of phases equilibrium is

μsAðTÞ5μl

AðTÞ. It follows from Eq. (2.76),

μlAðTÞ5 ð@G=@xlAÞT ;P;xB 5RT ln xlA:

Therefore,

RT ln xlA 5μsAðTÞ2μl

A;xB50ðTÞ52ðΔHA 2 T ΔSAÞ52HA 12T

Tf

� �:

This equation obtained by van’t Hoff defines slope of the liquidus line (equilib-

rium concentrations in the liquid phase). Usually, it is written as follows:

ln xlA 5ΔHA

R

1

Tf2

1

T

� �at T , Tf : ð2:78Þ

Equation (2.78) coincides with the third Raoult’s law at small concentrations of

the second B-component.

Distribution Coefficient

Distribution coefficient, κ5 xs2=xl2, is connected with slope of the solidus and liqui-

dus lines. We will find expression for the distribution number in the case of both

ideal solutions, liquid and solid. We take the component A (index 1�A) in the

amount xl1 in the solid state at temperature T and transform it into liquid state,

μl1 2μs

1 5ΔHA 12T

TfA

� �:

We form the liquid solution from liquid A and B components in the amount xl1and xl2

ΔGl 5 xl1 ΔHA 12T

TfA

� �1RTðxl1 ln xl1 1 xl2 ln x

l2Þ:


Then we take liquid component B in the amount xs2 at temperature T and trans-

form it in solid state,

μs1 2μl

2 52ΔHB 12T

TfB

� �:

We form the solid solution from solid A and B components in the amount xs1and xs2

ΔGs 52 xs2 ΔHB 12T

TfB

� �1RTðxs1 ln xs1 1 xs2 ln xs2Þ:

As the equilibrium condition of phases (crystalline and liquid) is the equality of

the chemical potentials of components in each phase, we will equate the derivative

of expressions of the Gibbs potentials for two phases on one of components (deri-

vatives on other component will yield the same equality):

ΔHA 12T

TfA

0@

1A1RTðln xl1 2 ln xl2Þ5

ΔHB 12T

TfB

0@

1A1RTðln xs1 2 ln xs2Þ:

At high concentration of A-component, it is possible to neglect the terms

RT ln xs1 and RT ln xl1, and also ΔHA(12 T/TfA). Therefore,

lnð xs2=xl2Þ5ΔHB 12T

TfB

� �: ð2:79Þ

2.7.5 Real Solutions and Regular Solutions

If the excess interaction energy of unlike atoms εAB5VAB2 (VAA1VBB)/2 6¼ 0,

the solution will not be ideal and, accordingly, ΔQ 6¼ 0, ΔVsol 6¼ 0 (the change

of volume), ΔHsol 6¼ 0. If internal energy during solution formation decreases

(εAB, 0), heat precipitates out (ΔQ, 0)—exothermic reaction. In the case of mag-

nification of the internal energy, solution formation is possible, if the free energy

nevertheless decreases at the expense of entropy. The higher the temperature is, the

more the role of entropy is. Therefore, such solutions are formed at comparatively

high temperatures, and heat is captured—the heat-absorbing reaction.

Real solutions have the cluster structure. At the subzero values εAB, atoms of

one type have in the environment of more atoms of other type. At plus εAB, clustersare formed of atoms of one and other types (clusters from atoms of type A and

clusters from atoms of type B).


In case of the real solution, the Raoult’s law is not correct; therefore, the partial

Gibbs potentials (the chemical potentials) are expressed through values of activity

or activity coefficients:

ðPi=Pi0Þ5 ai 5 fixi;

where ai is the thermodynamic activity, or volatility of the ith component; fi is the

activity coefficient; xi is the relative molarity. Hence, the change of the Gibbs

potential at formation of one mole of the solution at stationary values of T and P is

ΔGsol 5 x1RT ln a1 1 x2 RT ln a2 5 x1 RT lnðf1x1Þ1 x2 RT lnðf2x2Þ: ð2:80Þ

Accordingly,

μ1 5RT ln a1 1μ10ðTÞ; μ2 5RT ln a2 1μ20

ðTÞ;

ΔSsol 52@ΔGsol

@T52 x1R lnðf1x1Þ2 x2R lnðf2x2Þ2RT x1

@ lnðf1Þ@T

1 x2@ lnðf2Þ@T

24

35

and

ΔHsol 5ΔGsol 1 T ΔSsol 52RT2 x1@ lnðf1Þ@T

1 x2@ lnðf2Þ@T

� �:

Connection of Henry’s and Raoult’s Laws

According to Henry’s law, x25 const �P2 if x1 ! 0. It means that the activity coef-

ficient of the second component f25 const, f2 6¼ f2ðx2;TÞ. We will write down a

change of the Gibbs potential during solution formation through values of activity

coefficients and concentration:

ΔGsol 5 x1RT lnðf1x1Þ1 x2RT lnðf2x2Þ:

According to the Gibbs�Duhem relations (2.75),

dμ1 52ðx2 dμ2Þ=ð12 x2Þ;

and

dμ2 5RT

x2dx2 ðμ2 5RT lnðx2f2Þ1μ0ðTÞÞ:

Therefore,

dμ1 52RTdx2

12 x25RT

dx1

x1; μ1 5RT ln x1 1 const:


As we see, the activity coefficient is equal to 1 in the last expression for the

chemical potential of the first component. It means that, if Henry’s law is right for

the second component (its concentration is small), then Raoult’s law holds true for

the first component:

P1=P10 5 x1:

Regular Solutions

Solutions are termed regular if there are heat effects (ΔHsol 6¼ 0) at their forma-

tion, but they differ feebly from ideal solutions in structure—clusters are practi-

cally absent. Therefore, the entropy change at formation of such solution may be

calculated under the equation for the ideal solution (2.77). Actually, the solutions

containing the second component in amount less than 1 mol% may be considered

as ideal and the solutions with concentrations from 1 to 10 mol% as regular.

There are little clusters, because the atoms of the second component, the amount

of which is small, have preferentially atoms of the first component as the nearest

neighbors.

It is easy to make sure that correspondence of Eq. (2.77) to validity means inde-

pendence of enthalpy of the solution formation on the temperature

(ΔHp 6¼ΔHp(T)). As S5 Sid, the excess entropy ΔS5 0 (compared with the ideal

solution). Hence,

ΔG5ΔH2 T ΔS5ΔH; ΔGA 5ΔHA; ΔGB 5ΔHB; ΔHid 5 0; ΔHP 5ΔH;

ΔS5 ð@ΔH=@TÞP 5 0; ΔHA 6¼ ΔHAðTÞ; ΔHB 6¼ ΔHBðTÞ:

According to Eq. (2.80), the excess free energy

ΔG5ΔGsol 2ΔGid 5 x1RT ln f1 1 x2RT ln f2:

Therefore, ΔHA 5ΔGA 5RT ln f1 and ΔHB 5RT ln f2 are the values of the

partial excess enthalpy, and the activity coefficients may be determined:

f1 5 expðΔH1=ðRTÞÞ; f2 5 expðΔH2=ðRTÞÞ: ð2:81Þ

2.7.6 The Basic Positions of the Quasi-Chemical Theory of Solutions

In the quasi-chemical theory, the solution potential energy is expressed through

energies of pair interaction of atoms V11;V22;V12 (all of them are subzero) consid-

ering numbers of corresponding pairs:

E5N11V11 1N22V22 1N12V12: ð2:82Þ


Total number of different pairs is caused by the average number of the nearest

neighbors (the coordination number z):

zN

25

zðN1 1N2Þ2

5N11 1N12 1N22:

The numbers of bonds of atoms of one and other kinds are determined too by

value of the coordination number:

zN1

25N11 1

1

2N12; z

N2

25N22 1

1

2N12: ð2:83Þ

From here, we find connections between numbers of atom pairs:

N11 5zN1 2N12

2; N22 5

zN2 2N12

2:

Designate through E0 binding energy of all atoms in the primary state (before

mixing):

2E0 51

2zN1V11 1

1

2zN2V22:

Rewrite energy of the solution as follows:

E51

2zN1V11 1

1

2zN2V22 1N12ε12;

where ε125V122 (V111V22)/2. Therefore, energy of formation of the solution

ΔH5E � E0 5N12ε12: ð2:84Þ

The basic method of determination of the solution structure in the equilibrium

state is minimization of expressions for the Gibbs or Helmholtz free energy. By

this way, it is possible to find the equilibrium numbers of atoms pairs of a certain

kind or probabilities of such pairs. If there is the possibility to express all quantities

of the pairs through one parameter of ordering [2], minimization of expressions for

the free energy should be made on this parameter, and its equilibrium value can be

determined.

If the solution is completely disordered (the ideal or regular solution), the number

of pairs of kind N12 depends only on the concentration:

N12 5 x1ðzN=2Þ � ω12 1 x2ðzN=2Þ � ω21 5 ðx1x2 1 x2x1ÞðzN=2Þ5 x1x2zN:


(ω12 and ω21 are the probabilities to have the atom of other kind as neighbor).

Therefore,

ΔH5 x1x2zNε12 5 x1x2A: ð2:85Þ

For the solution with certain concentration, the value of ΔH is connected with

partial excesses:

ΔH5 x1 ΔH1 1 x2 ΔH2;ΔH1 5H1 2H10;ΔH2 5H2 2H20:

From geometrical reasons (see Figure 2.12), they can be expressed through the

tangents of the curve H(x) slopes:

ΔH1 5ΔH � ð1� x1ÞðdΔH=dx2Þ5 ð1� x21ÞA5 x22A;

ΔH2 5ΔH � ð1� x2ÞðdΔH=dx1Þ5 ð1� x22ÞA5 x21A:

Considering Eq. (2.81), we write the expressions for the activity coefficients of

components in case of regular solutions:

f1 5 expðx22A=ðRTÞÞ; f2 5 expðx21A=ðRTÞÞ: ð2:86Þ

2.7.7 Calculation of Interatomic Binding Energies

It is possible to find the expression for the distribution number κ5 xs2=xl2 in a case

of regular solutions by analogy to the relation (2.79) considering the modifications

of the Gibbs potentials at formation of liquid and solid solutions:

ΔGμl 5ΔH

μf1xl1 12

T

Tf1

0@

1A1 xl1x

2l2Al 1 xl1RT ln xl1 1 xl2x

2l1Al 1 xl2RT ln xl2;

ΔGμs 5ΔH

μf2xs2 12

T

Tf2

0@

1A1 xs1x

2s2As 1 xs1RT ln xs1 1 xs2x

2s1As 1 xs2RT ln xs2:

Here, ΔHμf1 and ΔH

μf2 are the molar melting heats of the first and second compo-

nents accordingly. Constants Al and As that enter into these equations are the mixing

heats in liquid and solid states accordingly. They are connected with the excess

energies of pair interaction εs12 and εl12:Al 5NAzlεl12 and As 5NAzsεs12; NA is the

Avogadro number; zl and zs are the numbers of the nearest neighbors in liquid and

solid states accordingly; εl125 (V122V11/22V22/2)l, εs125 (V122V11/22V22/2)S;

V11, V22, V12 are the potential energies of pair interaction (binding energies).


According to the phase equilibrium conditions, μs15μl1, μs25μl2. We will

equate derivatives on corresponding concentrations from the written expressions

for the free energy of formation of solutions. Then we will come to the equation

lnxs2xl1

xl2xs1

0@

1A52

ΔHf1

k

1

T2

1

Tf1

0@

1A1

ΔHf2

k

1

T2

1

Tf2

0@

1A2

ð12 2xl2Þzlεl12kT

1ð12 2xs2Þzsεs12

kT;

where ΔHf1, ΔHf2 are the melting heats of first and second components per one

atom.

The approximate variant of this equation given in Ref. [3] was obtained in

assumption that concentration of one of components is very small. As excess ener-

gies are connected with the values of energies of pair interactions Vij, it is possible

using the state transition diagram data (the values of equilibrium concentrations of

liquid and solid phases) for determination of values of energies Vij. It is clear that

for determination of two excess energies, it is necessary to take the values of con-

centrations for two temperatures and to solve the set of two transcendental equa-

tions. It is better to assign such work to a computer. Authors of this book

developed and used the program with the interface built in the Delphi environment

for such calculations (see the program Pair Energies on the web-page http://

booksite.elsevier.com/9780124201439). The values of energies Vij were necessary

for modeling by the Monte Carlo technique. Calculations were made using several

concentrations, which did not exceed 10 at. % of the second component. It was

assumed that the interaction energy of atoms of different type through the boundary

surface was equal to the energy of their interaction in the melt, Vsl125Vll12.

References

[1] E.I. Kharkov, V.I. Lysov, V.E. Fedorov, Thermodynamics of Metals, Vysha shkola,

Kiev, 1982 (in Russian).

[2] A.A. Smirnov, The Molecular-Kinetic Theory of Metals, Nauka, Moscow, 1966

(in Russian).

[3] A.G. Lesnik, Models of Atom Interaction in the Statistical Theory of Alloys, Fizmatgiz,

Moscow, 1962 (in Russian).


http://booksite.elsevier.com/9780124201439

http://booksite.elsevier.com/9780124201439







3 Diffusion Problems of CrystalGrowth: Methods of NumericalSolutions

3.1 Differential Equations for the Heat and MassTransport Processes

A lot of physical phenomena, which are used in technologies of production of mate-

rials for engineering and electronic devices, are connected with processes of heat and

mass transfer, including convection. These processes are described by the second-

type differential equations in partial derivatives. Their solutions depend on initial and

boundary conditions. Analytical solutions can be obtained only for simple boundary

conditions: stationary values of temperature and concentration at stationary boundary

surfaces, which separate a body in consideration from its environment.

3.1.1 Diffusion

Assume that some substances contain impurities. Moving randomly from one place

to another, molecules transfer themselves and their mass. If the concentration of

impurity molecules is nonuniform in the matter, the mass flow will take place.

According to the Fick’s diffusion law, the mass, which is transferred in a direction

of axis x through some plane (built perpendicularly to the axis x) with square δS, isproportional to a gradient of the impurity concentration,

dM5 �DðdC=dxÞdS dt: ð3:1Þ

The minus sign in this equation means that the substance is transferring in the

direction of the decrease of its concentration. The mass dМ is measured in corre-

sponding concentration units: kg/m3, mol/m3, or number of molecules in the unit

volume. Diffusivity D is measured in m2/s. The Fick’s law is a result of generaliza-

tion of numerous experimental outcomes.

Let us consider a small volume between two planes having equal areas δS and

coordinates x and x1 dx (Figure 3.1). The mass dM1 that enters through the first

plane into the volume during time dt,

dM1 52DðxÞ � ðdC=dxÞxdS dt ð3:2ÞComputational Materials Science. DOI: http://dx.doi.org/10.1016/B978-0-12-420143-9.00003-X


http://dx.doi.org/10.1016/B978-0-12-420143-9.00003-X

(it is negative in the case of going out). The mass, which goes through the second

plane,

dM2 52Dðx1 dxÞðdC=dxÞx1dxdS dt: ð3:3Þ

If dM1 6¼ dM2, the concentration in the small volume will vary:

dC5dM1 2 dM2

dx δS5

Dðx1 dxÞðdC=dxÞx1dx 2DðxÞðdC=dxÞxdx

dt: ð3:4Þ

The relation follows from the last equation,

@C

@t5

@

@xD@C

@x

� �: ð3:5Þ

It is named the time-dependent diffusion equation or Fick’s second law. This

second-type differential equation in partial derivatives has tremendous significance:

it is one of the basic equations of mathematical physics.

If the diffusion coefficient D does not depend on coordinates (through C(x)),

one can consider the equation

ð@C=@tÞ5Dð@2C=@x2Þ ð3:6Þ

instead of Eq. (3.5). The following equation gives generalization of Eq. (3.6) for

the case of a concentration dependent on three coordinates x, y, and z:

ð@C=@tÞ5D½ð@2C=@x2Þ1 ð@2C=@y2Þ1 ð@2C=@z2Þ�5D r2C; ð3:7Þ

where r25 @2=@x2 1 @2=@y2 1 @2=@z2 is the Laplace operator, which is shortly

written as Δ�r2.

The time-dependent concentration field C(x,y,z,t), in all points of some volume,

can be found by solving Eq. (3.7). Consideration of Eq. (3.5) or Eq. (3.6) is enough

for determination of the concentration distribution along one coordinate. For example,

C

C(x)

C(x + dx)

xdx

δS

Figure 3.1 The concentration distribution along the

x-axis, the concentration in the parallelepiped dS�dxdepends on the time if the concentration gradient

changes along the x-axis.


in the case of the junction of two rods, in one of which there are two components

A and B, and the concentration of the B-component is equal to Co, and in the other

rod is only the A-component, concentration Co/2 is immediately established on

their boundary. The concentration of the B-component in the second rod varies dur-

ing time according to the equation

Cðx; tÞ5 ðCo=2Þ½12 erfðωÞ�; ð3:8Þ

where ω5 x=ð2 ffiffiffiffiffiDt

p Þ; erfðωÞ5 ð2= ffiffiffiπ

p Þ Ð ω0e2α2

dα is so-called the error integral.

Equation (3.8) is a solution of the differential equation (3.6) at given initial and


Diffusion Coefficients

Several diffusion coefficients are used depending on the experimental conditions.

The simplest situation takes place at the consideration of diffusion of the radioac-

tive isotope impurity, while all other properties of the matter are homogeneous in

all other relations. In this case, the single factor yielded a flux of radioactive atoms;

this is their concentration gradient. The diffusion coefficient measured in such con-

ditions is designated D�. The term self-diffusion coefficient is used in the case

when there is only one substance to which radioactive atoms are added in a small

quantity.

When forces are acting in some direction, the probability of molecules moving

in the force direction, say 1x, is on average higher than in the opposite one. Thus,

the forces give the additional average velocity ,V. f of atoms, in directions 1 x,

and predetermine the contribution C,V. f into a flux of atoms. The main diffu-

sion equation will get the form:

j5 �D�ðdC=@xÞ1ChVif ; ð3:9Þ

if the acting force is taken in consideration (it can be, for example, the force that

acts on ions in an electric field).

The force F and, accordingly, the drift velocity hVif are proportional to the con-

centration gradient (dC/dx) in some important cases. Therefore, it is possible to

unite two terms in the right part of Eq. (3.9) and to write the expression:

ji 5 �DIi ðdCi=@xÞ; ð3:10Þ

in which

DIi 5D� � CihVif=ðdCi=dxÞ:

DIi is named the partial diffusion coefficient; it is the characteristic coefficient of

the i-component diffusivity.

73Diffusion Problems of Crystal Growth: Methods of Numerical Solutions

The force proportional to the concentration gradient emerges in the next cases:

(i) appearance of the electric potential in the ionic crystals due to different mobility

of cations, (ii) appearance of the electric potential during diffusion of divalent

impurity in the univalent atom lattices, and (iii) diffusion in nonideal solid

solutions.

When atoms of two sorts get mixed up, a mixing rate depends on diffusivities of

both components. The “interdiffusion” coefficient can be defined as parameter,

which characterizes the mixing rate. In the case of diffusion in the isolated system, it

defines the homogenization rate (vanishing of the initial gradient of concentration).

An uncompensated flux of atoms arises through any plane in the diffusion zone

of the two-component system, provided that the two components have different dif-

fusivities DIi . It causes a crystal bloating on the one side of the diffusion plane and

shrinkage on the other. Therefore, each plane in the diffusion zone is traveling with

the certain velocity relative to the plane fixed on the edge of the sample.

Interdiffusion coefficient ~D is expressed through a flux ji of ith component relative

to the fixed plane:

j0i 52 ~Dð@Ci=@xÞ:

The following equations take place for two-component crystals with the

coordinate-independent atom density:

CA 1CB 5Const; ð@CA=@xÞ52ð@CB=@xÞ; j0A 5 � j0B:

And consequently, there is the same interdiffusion coefficient in equations for

fluxes j0A and j0B. It can be expressed through the partial diffusivities by Darken’s

formula:

~D5DIAxB 1DI

BxA; ð3:11Þ

where xA5CA/(CA1CB); xB5CB/(CA1CB).

When solving problems of crystallization, it is necessary to deal with the inter-

diffusion coefficient as a rule, which is usually designated simply as D.

Interior Friction

The Newtonian friction law defines interior friction macroscopically in the follow-

ing way:

Fz 5 ηjduz=dxjδS; ð3:12Þ

where Fz is the frictional force between two adjacent layers of gas or liquids, which

move in the direction of the z-axis with different velocities uz(x) along the x-axis

(Figure 3.2). This force is directly proportional to an area of plane δS, to the coeffi-

cient of viscosity η, and to the velocity gradient along the x-axis, which is


perpendicular to the flow. The friction force that acts on the layer from adjacent

layers changes the layer impulse. Thus, the impulses of adjacent layers also vary.

This process can be considered as impulse transport in the x-direction. After multi-

plying both parts of Eq. (3.12) by time interval dt, we will obtain:

Fz dt5 ηjduz=dxjdS dt

or

dK52ηðduz=dxÞdS dt; ð3:13Þ

where dК is the transferred impulse; the minus sign is present because the direction

of transport and a direction of a gradient of a velocity are opposite to each other.

The equation obtained is completely equivalent to the diffusion equation (3.1)

from the point of view of mathematics.

3.1.2 Thermal Conductivity and Heat Emission

The equation, which allows calculating a quantity of heat, which is transferring in

the matter through the area δS during time dt in the conditions of practical absence

of convection and thermal radiation, is named the Fourier law. Analytically, it can

be written as follows:

dQ52κðdT=dxÞdS dt; ð3:14Þ

where dQ is the quantity of heat; κ is the thermal conductivity coefficient; dT/dx is

the temperature gradient in the x-direction. The minus sign means that the heat

transfers from more heated to less heated parts of a system. Fourier’s equation is

the same as the diffusion equation and the impulse transport equation from the

point of view of mathematics.

The second Fourier equation follows from considering the balance of heat,

which enters into some small volume and leaves it, using the same scheme shown

in Figure 3.1. We will divide the sum of those heats, which enter and leave by the

heat capacity of the considered volume, c � ρ δS dx (c is the specific heat per unit

mass, ρ is the density, and δS dx is the volume). This gives a change of temperature

during time dt. A corresponding relation

z

x1 x2 x3 x

uz1uz2 uz3

Figure 3.2 The velocity of liquid layers in the

z-direction distribution along the x-axis.


@T

@t5

@

@xαT @T

@x

� �ð3:15Þ

is the time-dependent thermal conductivity equation. The value αT5κ/(cρ) is

named thermal diffusivity. It has the same dimensionality as the mass diffusivity

(m2/c) and is the substance characteristic.

Equation (3.15) coincides with Eq. (3.5) from the point of view of mathematics.

Therefore, the equations for the thermal conductivity, such as Eqs. (3.6) and (3.7),

also take place. If conditions for diffusion or thermal conductivity are such that the

concentration or temperature does not depend on the time but only on coordinates,

the corresponding concentration or temperature distribution will be stationary. Such

temperature or concentration fields can be defined by solution using Laplace’s

equation r2C5 0 or r2T5 0 at certain boundary conditions.

Therefore, all transport phenomena obey to the same law from the point of view

of mathematics:

I52βðdG=dxÞ; ð3:16Þ

where I is the flux of mass, heat, or impulse: I5 dM/(δS dt) in the case of diffusion,

I5 dQ/(δS dt) in the case of thermal conductivity, and I5 dK/(δS dt) in the case of

impulse transport. In Eq. (3.16), β is a certain coefficient for corresponding pro-

cess; dG/dx is the gradient of a certain physical value: temperature, concentration,

velocity of liquid or gas layers; its value predetermines the value of the correspond-

ing flux.

Heat Emission

Heat emission from a solid body into liquid or gas at their convective agitation is

connected with thermal conductivity through the thin contact layer of the static

matter adjoining at the body surface. As the temperature gradient is proportional to

temperature difference of the surface temperature and the bulk temperature of fluid

or gas, the equation of heat emission has the form of Newton’s heat emission law:

dT=dt5 �αðT � TsÞs; ð3:17Þ

where T is the body temperature, Ts is the temperature of liquid or gas, α is the

heat emission coefficient, and s is the area of the surface. The heat emission coeffi-

cient depends on the heat transfer coefficient, area of the body surface, which is in

contact with the environment, and roughness of the body surface. Try to integrate

Eq. (3.17) and to obtain the temperature dependence on the time.

Equation (3.17) is an example of the differential equation of the first order as it

contains only the first derivative of unknown function T(t). Many processes that

happen in the nature are described by the differential equations, and it is important

to know how to solve these equations. We will consider the extended equation of

the first order


dy=dx5 gðxÞ: ð3:18Þ

Generally, an analytical solution of Eq. (3.18) expressed through well-known

function does not exist. Besides, even in that case when analytical solution never-

theless exists, it is necessary to present this solution in graphics to understand

results. The approximate numerical solutions obtained by means of computers

allow considering much more complex equations in which Eq. (3.18) can express

one of the boundary conditions. The graphic interface of the up-to-date mathe-

matical packets allows featuring all stages of processes of the heat and mass

transport.

3.1.3 Differential Equations of Convective Heat Transfer

Convection is always accompanied by heat transport as there are always mixing

and collisions of particles with different energies at flow of liquid or gas.

Equations for energy transport should take into account the convective heat

exchange [1,2]

~q5~qheat conduct: 1~qconv: 52κr!T 1 ρ w! h; ð3:19Þ

where h is the specific enthalpy (counting on a mass unit), w!

is the vector of local

fluid flow velocity of a fluid.

According to Eq. (3.19), projections of a heat flux on the coordinate axes Ox,

Oy, Oz are the following:

qx552κð@T=@xÞ1ρwxh; qy552κð@T=@yÞ1ρwyh; qz552κð@T=@zÞ1ρwzh:

Taking into account the heat balance, we will obtain instead of Eq. (3.15):

cρρ@T

@t5κ r2T2 ρ wx

@h

@x1wy

@h

@y1wz

@h

@z

� �2 ρh divðw!Þ1 ρqρ: ð3:20Þ

According to a continuity equation for incompressible liquids (ρ5Const),

divðw!Þ5 ð@wx=@xÞ1 ð@wy=@yÞ1 ð@wz=@zÞ5 0: ð3:21Þ

Equation of Motion

Let us consider an elementary volume with edges dx, dy, and dz in a flow of vis-

cous fluid. We will assume that the velocity varies only in the direction x. It is pos-

sible to separate forces acting upon the fluid element on those that operate in the

fluid bulk and those that act from the surface. One of the forces in the fluid bulk


that acts upon the volume dV is the force of gravity, df15 ρ � g � dV. The effective

force of pressure in the direction x can be calculated in the following way:

df2 5P dy dz� ðP1 ðdP=dxÞdxÞdy dz5 � ðdP=dxÞdV :

Find by analogy the resultant forces of friction acting between two planes of

volume dV which are at the distance dy from each other,

df3 5 ffrjy1dy � ffrjy 5 ηðdwx=dyÞy1dydx dz� ηðdwx=dyÞydx dz5 ηð@2wx=@y2ÞdV ;

where η is the viscosity coefficient. It is assumed here that the flow velocity wx

increases with coordinate y and the force ffr coincides in direction with the flow

direction. Summing forces and considering Newton’s second law,

df 5 ρ dVðdwx=dtÞ;

we obtain the Navier�Stokes equation of motion

ρðdwx=dtÞ5 ρgx � ðdP=dxÞ1 ηð@2wx=@y2Þ:

Generally, the field of velocities in three-dimensional incompressible liquid with

constant physical properties is presented by three Navier�Stokes equations of

motion. They are expressed as one equation in the vector form

ρðd~w=dtÞ5 ρ g!

2 r!P1 η r2 w

!: ð3:22Þ

Approach of a constant density (ρ5Const) in liquid with the nonuniform tem-

perature and pressure is not correct. We will consider the linear density dependence

on the temperature, ρ5 ρо(12β(T2 To)). It is possible to present the product ρogas a hydrostatic pressure gradient r!P0 in static fluid with the density ρo. After divi-sion of the left and right sides of Eq. (3.22) by ρ, we gain the following equation

of motion:

ðd~w=dtÞ52 g!βðT 2 ToÞ2 ð1=ρÞ r

!ðP2P0Þ1 υr2 w

!: ð3:23Þ

where υ5 η/ρ is the kinematic viscosity.

As not only wx, wy, wz but also the temperature T and the pressure P0 are

unknown in Eq. (3.23), a set of equations equivalent to Eq. (3.23) (for three axes)

should be solved simultaneously with the continuity equation (3.21).

Shapes and sizes of boundary surfaces essentially influence flows in fluid

and convective heat exchange. It is known that there are two basic types of

fluid flows: laminar and turbulent. If the laminar or, in other words, the “strati-

fied” flow takes place, it is possible to find out tubes of a current, which do not

intersect. At a turbulent (sinuous) flow, chaotic velocity veerings and changes


of their values occur, and fluid gets mixed up. If the temperature field and

the heat flux were calculated taking into account the flow of fluid, the

heat emission coefficient for the convective heat exchange can be found from

Eq. (3.17).

The Dimensionless Variables, Numbers of Similitude, andthe Similitude Equation

Many serious difficulties must be overcome to solve analytically the complete set

of equations of the convective heat transfer. Therefore, experimental research of

natural or model systems is important. By means of them, it is possible to find

numerical values of required variables. To answer a question, whether it is possible

that results, obtained by means of any certain device or by the means of computer

modeling, may be related to other analogous processes, a similarity theory helps.

The similarity theory has been developed in the works of Kirpichev [3], Guhman

[4], and Mikheev and Mikheeva [5]. The equations of motion and thermal conduc-

tions should be written down for dimensionless physical values [2�4]: X5 x/lo,

Y5 y/lo, Z5 z/lo, Wx5wx/wo, Wy5wy/wo, Wz5wz/wo, θ5 (Ts2 T)/(Ts2 To).

Along with the dimensionless coordinates, velocities and temperatures in the

following dimensionless groups [2�4] appear in the equations as coefficients:

Re5wolo/υ is the Reynolds number;

Nu5αlo/κ is the Nusselt number (convective heat exchange dimensionless factor);

Pe5wolo/αT is the Pekle coefficient (a relationship between the heat transferred by a

convection and the heat transferred by a thermal conduction), Pe5Re Pr, where Pr5 υ/αT is the Prandtl number;

Gr5 gβ(Ts2 To)lo3/υ2 is the Grashof number (characterizes an elevating force which ori-

ginates in a fluid as a result of inhomogeneity of density).

It is possible to consider dimensionless magnitudes θ, Wx, Wy, Wz, X, Y, Z, Nu,

Pe, Re, Gr as new variables. From them, the dimensionless coordinates are inde-

pendent, and the following magnitudes are dependent: θ, Wx, Wy, Wz, Nu. For their

determination for a certain problem, it is necessary to know the values of Pe, Re,

and Gr.

3.1.4 Euler’s Algorithm for the Numerical Solution ofDifferential Equations

The typical method of the numerical solution of the differential equations includes

transformation of the differential equation into a usual difference equation. We will

analyze Eq. (3.18). Let us assume that function y takes a value y0 at x5 x0. As

Eq. (3.18) gives a value of the function y in a point x0, it is possible to find an

approximate value of function y in a neighboring point x15 x01Δx if the argu-

ment increases a little (Δx{x). The first approximation is valid if the function

g(x), or the velocity of the function y modification, has a stationary value on the

segment from x0 to x1. In this case, the approximate value of function y in a point

x15 x01Δx is spotted by the expression


y1 5 yðx0Þ1Δy5 y0 1 gðx0ÞΔx: ð3:24Þ

We can iterate this procedure once again and find value y in a point x2

y2 5 yðx1 1ΔxÞ5 y1 1 gðx1ÞΔx: ð3:25Þ

This rule makes it possible to extend and compute an approximate value of the

function in any point xn5 x01 nΔx using the iterative equation

yn 5 yn21 1 gðxn21ÞΔx ðn5 0; 1; 2; . . .Þ: ð3:26Þ

The described method is termed a method of tangents or Euler’s method. The

method will yield a good approach to the “true” value of function y, if a change Δx

of argument value is small enough. The Euler method is entirely valid if the velocity

of the function y(x) changes on the segment from xn21 to xn has a constant value.

In a case when the function g(x) varies on some segment, there is a deviation from

the precise solution. This deviation will reduce, if the smaller value ofΔx is chosen.

3.2 Boundary Value Problems

Equations of heat and mass transport are applied for describing different physical�chemical and manufacturing processes. These are the second-order differential

equations in partial derivatives. However, their analytical solution is possible only

for the simplified boundary conditions—stationary surfaces of simple shape, for

example, the sphere or cylinder.

3.2.1 Boundary Conditions

Certain initial and boundary conditions should be taken into account to obtain the

unequivocal solution of differential equations of the heat and mass transport.

Starting conditions are spotted by setting distribution of temperature or concentra-

tion at an initial instant of time, that is at t5 0,

Tðx; y; z; 0Þ5 T0ðx; y; zÞ; ð3:27Þ

where T0(x,y,z) is the known function.

The boundary condition can be set in different ways.

The boundary condition of the first kind (the Dirichlet boundary condition) con-

sists of setting of the temperature distribution on the solid body surfaces at any

point of time, that is,

Ts 5 Tboundaryðx; y; z; tÞ: ð3:28Þ


The boundary condition of the second kind (the Neumann boundary condition)

consists of setting fluxes through each element of a boundary surface as functions

of its coordinates and of time, that is,

qs 5 qboundaryðx; y; z; tÞ: ð3:29Þ

The boundary condition of the third kind consists of setting the temperature of

an environment, Tenv, and the heat interchange law between the surface and the

environment. The heat emission spotted by Newton’s law of cooling must be equal

to the heat, which is brought to the surface from the bulk of body:

qs 5 �κð@T=@~nÞs 5αðTs � TenvÞ: ð3:30Þ

It is necessary to take into account the heat flux from surface in total form,

when there is essential heat interchange by radiation energy:

q5 �κð@T=@~nÞs 5αðTs � TcÞ1Cn½Ts=100� Tenv=100�; ð3:31Þ

where α is the coefficient of convective heat interchange (heat emission), and Cn is

the reduced radiation coefficient.

The boundary condition of the fourth kind answers to the heat interchange between

two phases when temperatures of adjoining surfaces are equal. It is set by relationships:

Ts;1 5 Ts;2;

�κð@T1=@~nÞs 5 � κð@T2=@~nÞsð3:32Þ

(n!

is the normal vector to the interface).

The differential equation of a nonstationary diffusion coincides in the form with

the thermal conductivity equation. For definition of concentration fields in different

instants of time, solution of Eq. (3.7) simultaneously with the initial (at t5 0) and

boundary conditions is necessary. The last can be the condition of the first, second,

or fourth kind (see conditions for the thermal problem (3.28), (3.29), and (3.32)).

Fluxes of mass are absent on the exterior boundary surfaces of isolated system;

therefore, the main thing is the correct consideration of boundary conditions on

interior phase to phase boundary surfaces.

3.2.2 The Boundary Value Problem in the Dimensionless Variables

Set of equations should be considered:

ρcρð@T=@tÞ5 divðκ grad TÞ1 ρqρ; ð3:33Þ

at t5 0 T 5 T0ðx; y; zÞ; ð3:34Þ


�κð@T=@~nÞs 5αkðTs � Tenv:Þ1Cn½Ts=100� Tenv:=100�; ð3:35Þ

cρ 5 cpðTÞ; κ5κðTÞ; ρ5 ρðTÞ; ð3:36Þ

in the case of the nonlinear boundary condition (3.31) of the third kind. The condi-

tion (3.35) can be reduced to conditions of the first or second kind in cases

Ts5 Tenv. or at known qs(x,y,z). The solution of such a boundary problem has not

been found until now in a general view. Therefore, it gains a special importance

for using a principle of similitude, which allows obtaining a number of the impor-

tant aftereffects only based on a formulation of the problem.

The linear transformations are applied for transferring to the dimensionless vari-

ables: l5 lol1, T5 ToT1, κ5κoκ1, ρ5 ρoρ1, cρ,15 cρ,ocρ,1, qρ,15 qρ,oqρ,1, where

magnitudes with an index “o” are the dimensional units, and those with an index

“1” are dimensionless. The boundary problem takes a form

ρ1cρ;1ð@T1=@FÞ5 divðκ1 gradðT1ÞÞ1 ρ1qρ;1;

at F5 0 T1 5 To;1ðx1; y1; z1Þ;

�κ1ð@T1=@~nÞs 5BikðTs;1 � Tc;1Þ1Bo½Ts;1=100� Tenv:;1=100�;

κ1 5 κ1ðT1Þ; ρ1 5 ρ1ðT1Þ; cρ;1 5 cρ;1ðT1Þ:

Po5 ðρoqoρl2o=κoToÞ is Pomerancev’s number;

Bi5αкlo/κ1 is Biot’s number;

Bo5CnTo3lo/10

8 is Boltzmann’s number;

Fo5 ðκot=ρocoρl

2oÞ is Fourier number used instead of time.

The gained dimensionless set of Eqs. (3.33)�(3.36) allows formulating necessary

and sufficient conditions of similitude of temperature fields in the course of the heat

transport. Temperature fields will be similar, if they are presented by identical equa-

tions and identical boundary conditions. The dimensionless systems of equations are

applicable for similar geometric fields; they ensure that the dimensionless numbers

of similitude gained in result of the linear transformations are equal.

3.3 Analytical Solutions of Heat and Mass TransportProblems for Crystal Growth

3.3.1 Stefan’s Problems

The whole class of problems about the transport connected with differential thermal

conductivity or diffusion equations in the case of mobile boundary surfaces [6,7] is

united under the title “Stefan’s problems.” Stefan studied a growth rate of ice in


polar seas. A crystallization heat of ice is taken away through off a stratum of a

solid phase, and it is absorbing by cold air, which has a temperature below the

freezing temperature of water. Heat taken from the crystallization front becomes

smaller, when the ice stratum becomes thicker. Thanks to that, the growth of ice is

slowed down; it makes this process nonstationary.

The principal complexity of this problem is that the phase boundary position is

not known; it depends on time and is the subject for definition during the solution.

Afterward, all problems about the heat and mass transport at crystal growth, for

example, problems about heat flows in the overheated melts during crystal growth

(by techniques of Bridgman, Czochralski, and Chalmers), have been termed

Stefan’s problems; also such problems as diffusion at crystal growth (from a solute,

from a vapor in the presence of noble gas, from melts in the presence of impurities

or from a solute in the melt).

The shape of the solidifying front and its temperature are considered as they are

known, and the velocity is the subject of determination in traditional statement of

Stefan’s problems. The modified problem of Stefan, which is differently termed the

reverse Stefan problem, is devoted to calculation of temperature of a solidified

front in the case when the shape of a front and a velocity of its moving are set. The

most difficult alternative of the problem will contain, apparently, the full problem

of crystal growth with definition of the shape, temperatures, and velocities of

boundaries between phases for the given temperature or concentration field in ini-

tial conditions and known dependence of the growth velocity on the local surface

supersaturation (supercooling).

3.3.2 Stefan’s Problem in the Initial Statement

Stefan published a solution of the unidirectional solidification problem (with a flat

front of growth) by the first, though statements are known that at first it was

obtained by Neumann. Stefan assumed that water is at one temperature TN. Tf(Tf is the fusion temperature), and the temperature of a flat ice�water surface is

equal to Tf, that is, T5 0. For any matter, thermal conductivity equations for an iso-

tropic crystal and melt in laboratory coordinates with the axis x directed along a

normal line to a cooling surface, thus, may be written as:

ð1=αTl Þð@Tl=@tÞ5 ð@2Tl=@x2Þ; ð3:37Þ

ð1=αTcrÞð@Tcr=@tÞ5 ð@2Tcr=@x2Þ: ð3:38Þ

Here αT is the thermal diffusivity (m2/s); αT1 5κl/(ρ � cl); αT

cr 5κcr/(ρ � ccr); κland κcr are the coefficients of thermal conductivity of liquid and solid (crystalline)

phases accordingly; ρ is the density of the phases, cl and ccr are their specific heats.

The temperature of the crystal�melt interface with coordinate x5X(t) is equal to

the melting temperature Tl(X)5 Tcr(X)5 Tf.


The condition of a heat balance on the phase boundary is

κcrð@Tcr=@xÞx5X � κlð@Tl=@xÞx5X 5 Lρð@X=@tÞ; ð3:39Þ

where L is the latent heat of fusion. This condition also causes nonlinearity of the

set of differential equations.

Other initial and boundary conditions of the problem are

Tl ! TN at x ! N; Tcr ! Tair at x ! 0; X5 0 at t5 0;

TlðxÞ5 TN for all x at t5 0:

If to introduce the dimensionless parameter ul 5 x=ffiffiffiffiffiffiffiαTl t

p, the equation of heat

conductivity will take the form:

ð@2Tl=@u2l Þ5 � ðul=2Þð@Tl=@ulÞ: ð3:40Þ

Considering that η5 @Tl/@ul and making integration, we find

η5C expð�ul2=4Þ; TlðulÞ � TlðNÞ5 �

ðNul=2

ηðzÞdz52A1 erfcðul=2Þ;

where A1 is the constant value, and

1� erfðxÞ5 erfcðxÞ5 ð2= ffiffiffiπ

p ÞðNx

expð2 ξ2Þdξ:

Thus, the following equations describe the distributions of temperatures:

in the melt:

Tlðx; tÞ5 TN � A1 erfc x= 2

ffiffiffiffiffiffiffiαTl t

q� �� : ð3:41Þ

In a like manner, integration for the crystal from x5 0 up to X gives

Tcrðx; tÞ5 Tair � B1 erf x= 2

ffiffiffiffiffiffiffiffiαTcrt

q� �� : ð3:42Þ

Considering the boundary condition at x5X

Tair 1B1 erf X= 2

ffiffiffiffiffiffiffiffiαTcrt

q� �� 5 TN 2A1 erfc X= 2

ffiffiffiffiffiffiffiαTl t

q� �� 5 Tf 5Const;

ð3:43Þ


we will find that the distance X is proportional toffiffit

p, as X=

ffiffit

p5Const,

X5 2λffiffiffiffiffiffiffiffiαTcrt

q; ð3:44Þ

where λ is the constant dependent on thermal properties of phases and the values

of differences Tair2 Tf and TN2 Tf. It is possible to connect constants B1 and A1

with the value of λ using Eq. (3.43):

B1 5 ðTair � TfÞ=erfðλÞ; A1 5 ðTN � TfÞ= erfðλÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiαTcr=α

Tl

q� �

and then gain expression for definition λ considering Eq. (3.39).

Equations (3.41) and (3.42) with these constants define the temperature distribu-

tions in the crystal and melt. Parker [6] has fulfilled the numerical analysis of this

solution taking into account thermal parameters of water. He obtained that ice stra-

tum with thickness of 1 cm is forming during t5 1.483 103 s at Tair525�C, andthe stratum of 10 cm thickness is forming during 42 h.

For crystallization with constant velocity, instead of its reducing with time (the

revertive modified problem of Stefan), it is necessary that temperature on the sur-

face x5 0 will not be kept a stationary value, but will be downgraded gradually.

The Case of the Supercooled Melt

Let the temperature in the bulk of melt become below the melting temperature

(TN, Τf), and crystal thin layer with the temperature Tf appears at the instant

t5 0. As the heat removal through a crystal is absent, all the heat of crystallization

is transferred into liquid, which gets warm from it. Condition Tcr5 Tf is thus satis-

fied at 0# x#X(t).

After solving the set of Eqs. (3.37)�(3.39) with the changed boundary condi-

tions, we will obtain

X5 2λffiffiffiffiffiffiffiαTl t

q; ð3:45Þ

Tlðx; tÞ5 TN 1λL

ffiffiffiπ

p

clexpð2λ2Þ erfc x=2ffiffiffiffiffiffiffiαTl t

q� �� ; ð3:46Þ

where the constant λ is the solution of equation

λ expðλ2ÞerfcðλÞ5 ðTf � TNÞcl=ðLffiffiffiπ

p Þ: ð3:47Þ

Here the dependence X(t) is obtained the same in the form, as in the case of not

supercooled liquid with the heat that is removed through a crystal phase.


3.3.3 Boundary Conditions for the Diffusion Problem of Crystal Growth

At crystal growth, concentration fields are routinely nonstationary, and they should

be found from solution of the time-dependent diffusion equation (Fick’s second

law (3.5))

ð@C=@tÞ5 divðD rCÞDD r2C; ð3:48Þ

where the second equality takes place, if the diffusion coefficient does not depend

on concentration.

Fluxes are usually absent on the exterior boundary surfaces of the volumes

under consideration excepting of special cases with feeding of solutes over such

boundary. Therefore, the setting of an initial concentration field C(x,y,z,t5 0) is

sufficient as a rule. The basic nonlinearity is connected with the boundary condi-

tions on the interfaces (surfaces of a crystal with an environment). These surfaces

can have the complex shapes, and besides, they move. Therefore, boundary condi-

tions on these surfaces must be considered only locally.

The condition of the mass balance on the boundary surface is

VðCcr � CsÞ5Dð@C=@ n!Þ; ð3:49Þ

where Ccr is the concentration in a crystal, Cs in the noncrystalline environment

near the crystal surface, and @C/@~n is the concentration gradient in a normal direc-

tion to the surface. This condition expresses equality of mass fluxes, which is trans-

ferring by diffusion from liquid and which is entering into a crystal at the phase

boundary movement.

For the solution of the crystal growth problem, it is necessary to know a law of

the growth rate dependence on the solution supersaturation, Cs2Ce (Ce is the equi-

librium concentration), at the crystallization front. If this dependence is nonlinear

(as a rule, it is so), generally it is impossible to gain analytical solutions of the

problem.

A more exact condition of mass balance, in comparison with Eq. (3.49), should

consider also a diffusion flux to the interface from the bulk of crystal. However,

analysis of crystal growth problems is fulfilled, as a rule, considering the condition

(3.49) only. The Laplace equation r2C5 0 is more often considered instead of a

diffusion equation (3.7) in the case of growth of spherical crystals from feeble solu-

tions. However, for other shapes of crystals, gained functions do not result in con-

sent with boundary conditions.

The most full analytical theory of diffusion growth is described in the book of

Lubov [7]. As it follows from Ref. [7], the exact solution of a nonstationary prob-

lem of diffusion in case of spherical symmetry (apparently, and for cylindrical) can

be found by a number of successive approximations at consideration of the

integral�differential equations, which takes into account a total mass balance.

Solutions obtained by Ivantsov [8] (see Eq. (3.55)) for the case of stationary con-

centration on the phase boundary have a known peculiarity. For them, the product


of a crystal radius on the gradient of concentration near the crystal surface is a con-

stant value, ρ(dC/dn)ρ5Const, and accordingly, the crystallization center radius

ρ is proportional to the radical square root from time, ρB(t)1/2.

3.3.4 Growth of a Cylinder and a Sphere from Solution atConstant Surface Concentration

Let us consider a case of growth of a spherical crystal from a solution. The analyti-

cal solution of such a problem is gained by Ivantsov [8] for a case of constant

surface concentration (it does not happen actually at the free crystal growth).

Figure 3.3 shows the concentration distribution in fluid for the case when the crys-

tal growth is determined by both the surface process of atom attaching and diffu-

sion in liquid solution. For describing crystal growth in that case, it is necessary to

consider the time-dependent diffusion equation

@C

@t5

@

@xD@C

@x

� �or ð@C=@tÞ5Dð@2C=@x2Þ ð3:50Þ

(dependence of the diffusion coefficient on the concentration is neglected in the

second equation) at boundary conditions, which take into account initial distribu-

tion of concentration and the condition of the mass balance at the interface:

VðCcr � CsÞ5Dð@C=@ n!Þ: ð3:51Þ

The solution of diffusion problem with rather complex boundary conditions is

connected with heavy mathematical difficulties. Therefore, analytical solutions of

Eq. (3.48) are gained only for simple crystal shapes: spheres, cylinders, paraboloids

of revolution [8], and simple bodies with small distortions of their shape [9], for

cases of the linear or square-law dependence of the growth velocity on the super-

saturation at the crystallization front.

C

Ccr

Cs

C∞

C∞

Ce

CeCs Cr A

(A) (B)

ΔT

Tf

T

LS

Figure 3.3 The concentration distribution at crystal growth (A) and relationship of

supersaturation with supercooling of a binary melt (B); Cs is the surface concentration

(in liquid near the crystal).


Laplace’s equation, r2C5 0, is often considered instead of the diffusion equa-

tion (3.50) for problems of growth of polyhedrons or the distorted simple shapes.

The Laplace equation gives correct distributions of concentration or temperature

near the surface of a growing sphere in cases of slow growth with a constant con-

centration or temperature at the interface. Such cases take place at small supersa-

turations, when the relationships are fulfilled: Sc5 (CN2Cs)/(Ccr2Cs){1 for

growth from solution or ST5 cv(Tf2 TN)/Lv{1 (cv is the specific heat capacity

per unit volume and Lv is the latent heat of fusion per unit volume) for growth

from melt.

Let us solve the Laplace equation for the case of growth of the spherical crystal

from supersaturated solution under condition of constant concentration at its surface

Cs5Ce (Ce is the equilibrium concentration at considered stationary temperature):

@c

@t5D

@2c

@r21

2

r

@c

@r

� �.

@2c

@r22

r

@c

@r5 0: ð3:52Þ

The condition of mass balance at the interface Eq. (3.51) has the following form:

VðCcr 2 ~NsÞ5@C

@r

� �ρ;

where ρ is the crystal radius.

Generally, the growth rate is determined by the surface supersaturation

σs 5Δμs=kT 5 ðCs 2CeÞ=Ce(Δμ5μl2μs), that is, by the value of concentration

near the crystal surface: (V5 f(σs)). But Cs!Ce at increase of the crystal size,

and C!CN on the large distance from the crystal. Let us introduce in the

Laplace equation a new variable: we designate ð@C=@rÞ5U and write

ð@U=@rÞ1 ð2=rÞU5 0. Then we separate the variables and integrate the equation:

ðdU

U52

ð2

rdr.ln U52 2 ln r1 ln A;.U5A=r2;

ðdc

dr5

ðA

r2; C52

A

r1B:

We find the constant value B from the boundary condition on infinity: at

r5~.B5C~, and constant A from the condition that the interface concentration

is equilibrium: Cs5Ce (growth is limited by diffusion)

r5 ρ5 rcryst:.C5Ce.Ce 52A

ρ1CN;

A5 ðCN 2CeÞρ; C5CN 2CN 2Ce

rρ: ð3:53Þ


The growth rate in purely diffusion regime can be found from the condition of

the mass balance

V 5D1

ρðCN 2CeÞðCcr 2CeÞ

; ð3:54Þ

C~ is the initial solution concentration.

Ivantsov [8] obtained the next solution of the diffusion equation (3.48) for

growth of sphere at the constant surface concentration:

C2CN 5 ðCs 2CNÞ ðρ=rÞexpð2 ðλ2r2Þ=ðρ2ÞÞ2λffiffiffiffiπ

perfcðλr=ρÞ

expð2λ2Þ2λffiffiffiπ

perfcðλÞ ; ð3:55Þ

where the boundary condition C5CN at r5N is taken into account. Based on

this solution, we find the dependence of the crystal size on the time from

Eq. (3.13): r5 2 λffiffiffiffiffiDt

p, where λ is defined by the equation

2λ2 12 2ffiffiffiπ

pexpðλ2ÞerfcðλÞ

5 Sc: ð3:56Þ

Equation (3.56) is reduced to the equality λ5ffiffiffiffiffiffiffiffiffiSc=2

pat Sc5 (CN2Cs)/

(Ccr2Cs){1. Let us neglect the term of the orderλ in comparison with the term of

order 1 (λ{1) in Eq. (3.55) and also consider that λr/ρ{1. For this case, we have

from Eq. (3.55) that C2CN5 (Cs2CN) ρ/r. That is in this case, concentration dis-

tribution near to a crystal coincides with the solution (3.53) of the Laplace equation.

3.3.5 On the Heat and Mass Transport During Growth of Single Crystals

This problem was considered in the book by Konakov [1]. Most often for growing of

single crystals, the Czochralski method is applied. In the course of pulling of single

crystal, the crucible and the growing crystal are twirled with certain velocities, and the

single crystal realizes its vertical headway. After melting of the material entrained in

the crucible and transition to steady conditions, the heat gained from a calefactor is

spent for compensation of a different sort of losses (refrigeration of current taps, radia-

tion heat interchange of a crystal, and cooling walls of the chamber, and so forth).

Heat interchange problem during the growing of single crystals by the

Czochralski method is usually considered in cylindrical axes r, ϕ, and z. We will

connect a z-axis with the oblong crystal. Let us designate the radial component of

the melt movement velocity through wr, azimuthal through wϕ, and axial compo-

nent through wz. We will consider that the melt movement has a rotational symme-

try, that is, derivatives from components of the melt movement velocity on the

coordinate ϕ are equal to null. Assume that the melt density ρ is constant. During

the certain time after turning on of calefactor, nonstationary fields of the velocities

wr, wϕ, and wz, pressures P, melt temperatures T, and partial density of impurities

are approaching their stationary values.


The state of melt is defined by differential equations:

a. of continuity

1

ρ@ρ@t

1@wr

@r1

wr

r1

@wz

@z5 0; ð3:57Þ

b. of movement

@wr

@t1wr

@wr

@r2

w2ϕ

r1wz

@wr

@z52

1

ρ@P

@r1 gr 1 η

@2wr

@r21

@

@r

wr

r

� �1

@2wr

@z2

� �; ð3:58aÞ

@wϕ

@t1wr

@wϕ

@r1

wrwϕ

r1wz

@wϕ

@z5 η

@2wϕ

@r21

@

@r

wϕ

r

� �1

@2wϕ

@z2

� �; ð3:58bÞ

@wz

@t1wr

@wz

@r1wz

@wz

@z52

1

ρ@P

@z1 η

@2wz

@r21

1

r

@wz

@r1

@2wz

@z2

� �1 gz; ð3:58cÞ

c. of temperature field

@T

@t1wr

@T

@r1wz

@T

@z5αT @2T

@r21

1

r

@T

@r1

@2T

@z2

� �; ð3:59Þ

d. of diffusion

@C

@t1wr

@C

@r1wz

@C

@z5D

@2C

@r21

1

r

@C

@r1

@2C

@z2

� �: ð3:60Þ

The following designations are applied in this set of equations:

ρ is the density;

η is the melt kinematic viscosity coefficient;

P is the pressure in the melt;

αT is the melt thermal diffusivity;

D is the diffusivity of impurities in the melt, and C is their concentration.

The set of six equations includes six required functions: wr, wϕ,wz, P, T, C.

Initial fields in an instant of time t5 0 and boundary conditions should be put for

an unambiguity of the solution.

Besides, it is necessary to determine a temperature field in the crystal, as

mechanical stresses and formation of imperfections depend on it. The single crystal

temperature field is also nonstationary. It is presented by the equation

@Tcr@t

1wz

@Tcr@z

5αT @2Tcr@r2

11

r

@Tcr@r

1@2Tcr@z2

� �: ð3:61Þ


The concentration field in the single crystal is also nonstationary and is spotted

by the equation

@Ccr

@t1wz

@Ccr

@z5Dcr

@2Ccr

@r21

1

r

@Ccr

@r1

@2Ccr

@z2

� �: ð3:62Þ

The following equation of a heat balance is valid for the curved crystallization front

Ð rcr0

κ2πrdT

dzdr1

ðhfr0

κ2πrdT

drdz 1 ðwcr 1wÞπr2ρ5

Ð rcr0

κcr2πrdTcr

dzdr 1

ðhfr0

κcr2πrdTcr

drdz

; ð3:63Þ

where hfr is the distance, which the interface occupies along the z-axis; κ and κcrare the coefficients of thermal conductivity in the melt and crystal accordingly; Lρ

is the heat of crystallization; and Vcr is the velocity of crystal pulling.

The phase boundary curvature is determined locally by the Laplace equation

1

R1

11

R2

5ΔP

σ; ð3:64Þ

where R1 and R2 are the main radii of curvature of the interface; σ is the capillary

constant; and ΔP is the local pressure difference (in the melt and crystal).

Manifold reasons may generate the liquid movement. The nonuniform field of

temperatures creates a field of density in the melt. Interaction of gravitational

forces with a field of the density in the inhomogeneous liquid results in the appear-

ance of currents termed a free convection. The free convection arises also in the

presence of the centrifugal or Coriolis forces, which take place at liquid rotary

movement. A liquid flow can be generated also by the electromagnetic forces or by

mechanical factors of a different type.

The melt movement in the crucible is described by the set of equations: the conti-

nuity equation, the conservation equations for an impulse and energy, and the equa-

tion, which takes into account the boundary conditions. In most cases, a gaining of

solution of these equations is not possible. Therefore, it is necessary to make different

sorts of assumptions to develop the model of the explored process, which differs in

more simple mathematical description. By means of models, it is possible to obtain a

number of mathematical dependences, which represent a practical interest [1].

However, at enough complex boundary conditions, the analytical solution of the equa-

tions becomes impossible. Therefore, it is necessary to obtain numerical solutions of

the equations, which describe the melt movement, diffusion, and thermal conductivity.

At the complex shape of the crystallization front, it is necessary to choose shallow

nets for the approximation of solution by the net functions. Thus, calculations can be

very long-term, even using modern computers.


3.4 Numerical Solutions for the Heat and MassTransport Problems

3.4.1 The Finite Difference Schemes for Solution of theHeat and Mass Transport Problems

Application of the COMPUTER for solution of problems of the heat and mass

transport including the solution of boundary problems of crystal growth has proved

to be very effective. The literature on this question is extensive. A great number of

books are devoted to questions of choice of finite difference schemes for solution

of the thermal conduction problem, for example, Refs. [10�12]. Nevertheless, it

will be useful to reflect some main concepts here.

We will build for the Cauchy problem

@u

@t2

@u

@x5ϕðx; tÞ; 2N, x,N; 0, t, T ;

uðx; 0Þ5ΨðxÞð3:65Þ

one of possible finite difference schemes that approximate it. As the type of a grid

we use an assemblage of intersection points of straight lines:

x5mh; t5 nτ; m5 0; 6 1; . . .; n5 0; 1; . . .; ½T=τ�;

[T/τ] is the whole part of ratio T/τ. We will consider that a time step (time inter-

val) τ is connected with a distant step h by the relation, τ5 rh, so the grid Dh

depends only on one parameter h. The solution of the problem (3.65) is a grid func-

tion, instead of continued function u(x,t), given as a table [u]h5 [u(mh,nτ)] with

values in points of the grid Dh.

Let us begin building-up the finite difference scheme, approximated by the prob-

lem (3.65). We will designate as unm the values of the grid function uh in the grid

point (xm,tn)5 (mh,nτ). We will replace derivatives (du/dt) and (du/dx) with differ-

ence relationships:

ð@u=@tÞx;t �uðx; t1 τÞ2 uðx; tÞ

τ; ð@u=@xÞx;t �

uðx; h1 tÞ2 uðx; tÞh

;

in addition, values u0m are determined by boundary conditions u(x,0)5ψ(x) at a

start time t5 0.

The problem of searching of the grid function u(h)5 unm 5 u(mh,nτ) is describedin a general view as follows:

LhuðhÞ 5 f ðhÞ; ð3:66Þ


where Lh is the finite difference operator (3.65) for the grid function; f(h) is pair of

the grid functions, one of which is set on the two-dimensional net

un11m 2 unm

τ2

unm11 2 unmh

5ϕðmh; nτÞ; ð3:67Þ

and another on the one-dimensional: u0m 5ψ(mh).The difference equation (3.67) can be solved relative to un11

m :

un11m 5 ð1� rÞunm 1 r unm11 1 τϕðmh; nτÞ: ð3:68Þ

Hence, knowing value unm in net points m5 0, 6 1. . . at t5 nτ, one can compute

the value un11m in net points at t5 (n1 1)τ. As the values u0m at t5 0 are set by

equality u0m 5ΨðmhÞ, we can compute step by step values of the solution u(h) in the

net points on straight lines t5 τ, t5 2τ,. . ., t5 nτ, that is everywhere on the Dh.

Approximation

The difference equation (3.66) approximates Eq. (3.65), if in equality,

Lh½u�h 5 f ðhÞ 1 δf ðhÞ; ð3:69Þ

a discrepancy δf(h), which originates at putting of the grid function [u]h in the dif-

ferential boundary value problem (3.65) instead of the real function, decreases to

the zero at h!0. Suppose that the solution of the problem (3.65) has the bounded

second derivatives and we can apply the Taylor’s formulas

uðxm 1 h; tnÞ2 uðxm; tnÞh

5@uðxm; tnÞ

@x1

h

2

@2uðxm 1 ξ; tnÞ@x2

;

uðxm; tn 1 τÞ2 uðxm; tnÞτ

5@uðxm; tnÞ

@t1

τ2

@2uðxm; tn 1 ηÞ@τ2

;

where ξ and η are the some numbers that depend on m, n, and h, which satisfy to

the inequalities 0, ξ, h and 0, η, τ. Then it is not difficult to find[16] that the

discrepancy is

δf ðhÞ 5τ2

@2uðxm; tn 1 ηÞ@t2

2h

2

@2uðxm 1 ξ; tnÞ@x2

:

Therefore, for everywhere on the Dh

jδf ðhÞj# τ2sup

@2u ðx; tÞ@t2

1 sup

1

2

@2uðx; tÞ@x2

� �h: ð3:70Þ


Thus, our difference scheme has the first order of approximating relatively h on

solution u(x,t) if required function has the bounded second derivatives.

Convergence and numerical stability. The difference boundary value problem

(3.65) by definition is convergent, if there are numbers δ. 0 and h0. 0 such, that at

any h, h0 and to any δf(h) from F, that fits to an inequality jjδf(h)jjF# δ, the differ-

ence boundary value problem (3.65) has one and only one solution z(h), and the

condition

jjzðhÞ � uðhÞjjU #Cjjδf ðhÞjjF ð3:71Þ

is satisfied, where C is the some stationary value independent on h. Property of stabil-

ity can be accepted as the uniform relative h sensitivity of solution of the difference

boundary value problem (3.66) concerning perturbation f(h) in a right member (3.69).

In case of the linear Lh operator, the formulated definition is equivalent to the

following:

The difference boundary value problem (3.66) is numerically stable, if exists h0. 0 such,

that at h, h0 and any f(h) it is unequivocally solvable, so that

jjuðhÞjjU #Cjj f ðhÞjjF ; ð3:72Þ

where C is the some constant value independent on h and f(h).

The difference scheme (3.67) is numerically stable at r, 1 [12]. In the case of

r. 1, the difference scheme (3.67) approximates as before the problem (3.65), but

in this case there is no convergence of solution u(h) to solution u(x,t) of the differen-

tial problem (3.65). In the case of equations with partial derivatives, unfitness of the

finite difference scheme taken by guess-work is a rule, and a choice of the stable

(and, therefore, convergent) difference scheme—permanent anxiety of researcher.

Explicit and Implicit Difference Schemata

Consider two difference schemata, which approximate the Cauchy problem for a

heat conduction equation:

@u

@t2

@2u

@x25ϕðx; tÞ; 2N, x,N; 0, t, T ;

uðx; 0Þ5ΨðxÞ:ð3:73Þ

Most simplest of them is the explicit scheme:

Lð1Þh uðhÞ � un11

m 2 unmτ

2unm11 2 2unm 1 unm21

h25ϕðmh; nτÞ;

at �N, x,N; 0, t, T ;u0m 5ψðmhÞ at �N, x,N; t5 0;

ð3:74Þ


it is gained at replacement of derivatives by the difference relationships, which

include values of the grid function on the previous time step.

If we take values of the grid function in the following instant for determination

of the second derivatives, we will come to the other scheme:

Lð2Þh uðhÞ � un11

m 2 unmτ

2un11m11 2 2un11

m 1 un11m21

h25ϕðmh; nτÞ;

at �N, x,N; 0, t, T ;u0m 5ψðmhÞ �N, x,N; t5 0:

ð3:75Þ

The last scheme is named implicit.

These two schemes differ essentially. Solution for u(h) scaled on the first of

them does not add difficulties and is made by the explicit formula:

un11m 5 ð1� 2rÞunm 1 rðunm 1 unm11Þ1 τϕðmh; nτÞ; ð3:76Þ

sequentially on each new time step.

The second scheme is deprived such convenience. As Eq. (3.75) includes many

values of the grid function on a following time step, it is necessary to make up and

solve a set of algebraic equations. However, the scheme (3.75) is applied widely as

it is stable at r. 1. If τ5 rh2 at r5Const, both schemes have the second order of

approximation relatively h and discrepancy jjδf(h)jjF5 5O(h2) [12].

In case of not very fine meshes (with comparative small number of nodes), the

second scheme yields an economy of a computing time in result of using of large

time step τ. From the physical point of view, the mesh should be small in the area

of large variable temperature gradients or concentration and near surfaces with

large curvature. In a case of the mobile phase boundaries, the time step should be

essentially less, as a rule, than it is determined by stability conditions for the sta-

tionary boundary surfaces.

Different meshes and different modes of approximation are applied to solve the

boundary value problems of the heat and mass transport using variable steps on

coordinates. The convergence testing in these cases is very complicated, as a rule.

Quality of such schemes is verified by comparing numerical solutions with analyti-

cal solutions for cases of simplified boundary conditions. It is well, if the method is

allowed to describe the known physical results and to predict the new. If there is a

possibility, the complex experiments for measurements of thermal fields are ful-

filled and data of physical modeling are compared with the numerical solutions.

Coincidence of outcomes provides a basis for applying the technique developed for

study of similar problems.

3.4.2 Boundary Conditions at Interfaces during Crystal Growth

The boundary conditions of type (3.39) or (3.49) of the heat or mass balance at the

interfaces (generally for two-dimensional or three-dimensional problems)


complicate essentially using numerical methods. It is not possible, as a rule, to gain

the stable computing in finite differences, because of the incorrect representation of

starting conditions, which are not known for the interfaces. The cause of this lies in

that, the time step does not enter in boundary conditions, which interrelate the

growth velocity with fluxes. Therefore, the homogeneous differential schemes

[13�16] with heat capacity dependent on temperature in special manner had wide-

spread occurrence in 60�70 years. In them, the crystallization heat was taken into

account by introduction in the area near the crystallization front of some efficient

heat capacity. Mainly implicit schemes were used [16]. Such approach relates to

so-called methods of a phase field, when the problem of the heat and mass transfer

is solved as for one phase, and a position of the phase boundary is determined by

the value of some parameter, for example, the melting temperature.

Two groups of methods of the numerical solution are used, as a rule, for a solu-

tion of Stefan’s problems with crystallization (melting) of different materials [17].

The first group of methods considers artificially an entering of the latent heat of

crystallization (fusion) into the equation of the heat transport—the method of the

phase field [16], for instance. To eliminate nonlinearity connected with the bound-

ary conditions, it uses “smearing” of the latent heat into values of the special heat

capacity. These methods are well accommodated for determination of the general-

ized solution of the Stefan problem, in particular, for calculating a two-phase zone

(the area occupied both by solid and liquid phases) if it exists. The second group of

methods gets out of a classical statement of the Stefan problem and requires exact

determination of the crystallization front position [5,6]. Both groups of methods

have self-sufficient value and should develop independently as there are cases

when the problem solution in the generalized statement does not coincide with the

classical solution [17].

Thus, each of the specified approaches can yield the essentially different numer-

ical solutions of the problem. For example, the first group of methods is inconve-

nient for numerical solving of the problem of crystal growth from the supercooled

melts. On the contrary, at the second approach, it is practically impossible to obtain

solutions with the presence of two-phase area. Avdonin and Ivanova[15] have con-

sidered different variants of approximation for nonlinear members of equations of

the one-dimensional problems in the generalized setting and compared the results

obtained by simulations with the analytical solutions of such problems.

3.4.3 The First Numerical Solutions of Diffusion Problemsof Crystal Growth

Tanzilly and Heckel [18] first obtained numerical solutions of the problem of

α-phase nucleus center growth in the β-phase matrix for cases of a plane, cylindri-

cal, or spherical phase boundary. The authors used the finite difference method for

solution of the time-dependent diffusion equation. They found that choosing a

small enough distant step Δx is necessary for obtaining of solutions of high accu-

racy. The time step Δt should be rather small—its value should fit the condition


Δt#Δx2=ð2DÞ ð3:77Þ

(D is the diffusion coefficient). Small time and distance steps require a considerable

computing time in a case when the bath volume is not small. Moreover, solutions for

crystal growth from large and small volumes differ appreciably by results as in the

latter case, a mean supersaturation of solute noticeably decreases during growth.

For solution of a problem with cylindrical symmetry on a rectangular net

(boundary in the form of a circle), Samarsky and Moiseenko [13] had written

Eq. (3.37) in the form:

cðUÞð@U=@xÞ5 divðκðUÞgradðUÞÞ1 f ; ð3:78Þ

f is the power of thermal radiants; U� T; c(U)5 cρρ.The interface position R(t) was found from the condition U(r,t)5U�. Generally,

the interface position may be found from equation: Φ(U(r,t))5Φ(r,t)5 0; grad(Φ)is perpendicular to the phase boundaries. Therefore, the condition (3.39) may be

rewritten as follows:

½κgradU1 2κgradU2�gradΦ1λð@Φ=@tÞ5 0 ð3:79Þ

(indexes 1,2 designate different phases).

The specific volume energy W undergoes a saltus λ (enthalpy of crystallization)

at U5U�

W 5

ðU0

cðUÞdU1ληðU2U�Þ;

cðUÞ5 c1ðUÞ at U,U�;

cðUÞ5 c2ðUÞ at U.U�;

U � U� 5UðξÞ; ηðξÞ5 1 at ξ. 0; ηðξÞ5 0 at ξ, 0:

Considering Eq. (3.78) in the form: dW(ξ,t)/dt5 div(κ � grad(U))1 f and taking

into account the condition dU(ξ,t)/dξ5 δ(T) (delta function), one can write:

½cðUÞ1λδðU � U�Þ�ðNU=NtÞ5 divðκ gradðUÞÞ1 f : ð3:80Þ

Equation (3.80) includes Eq. (3.78) and the condition of the heat balance on the

interface (3.79).

A flattening of the coefficient [c(U)1λδ(U2U�)] consists of replacement of

the function η(ξ) having the sharp change at the interval [U� 2Δ,U�1Δ] by the

continuous function η(U2U�,Δ) such that η0(ξ,Δ)5 δ(ξ,Δ). Thus, the effective

heat capacity is introduced for flattening,


cefðUÞ5 cðUÞ1λδðU � U�;ΔÞ;

cefðUÞ5 cðUÞ out of an interval ½U� �Δ;U� 1Δ�; andðU�1Δ

U�2ΔcefðUÞdU5λ1

ðU�

U�2Δc1ðUÞdU1

ðU�1Δ

U�c2ðUÞdU:

Now the homogeneous difference scheme can approximate Eq. (3.80), and it is

applicable for the open calculations. The problem described here was solved in

Ref. [11] by the method, which consists in step by step solution for different space

variables (x,y) of the heat conduction equations by means of the unconditionally

stable implicit difference scheme [2,13].

3.4.4 A Technique for the Numerical Analysis of Growth or Dissolutionof Spherical or Cylindrical Crystals

Tanzilly and Heckel [18] had taken into account movement of the interface, but the

surface concentration was assumed a stationary value that is equal to the equilib-

rium value. The diffusion problem for cases of growth or dissolution of a spherical

or cylindrical particle was solved in Ref. [19] taking into account changes of the

surface concentration. The time-dependent diffusion equation for spherical or cylin-

drical symmetry written in finite differences has the form

ΔCin 5DΔt

@2 ~Ni21

n

@r21

K

r

@ ~Ni21

n

@r

" #1ΔC0; ð3:81Þ

where ΔCin is the concentration change in the point n during the current time step Δt

(of number i), K5 1 or 2 for cases of cylindrical or spherical symmetry accordingly.

The distanceΔr between net nodes was chosen not uniform, but such, which was incre-

mented in an arithmetical progression (Figure 3.4) with distance from the interface

n + 1n + 1

n – 1

nR

ρ1

Figure 3.4 The scheme of a net for simulation of growth of

two-dimensional crystal.


Δrn 5Δrn21 1 δ; δ5 2ðR� ρÞ=½NðN � 1Þ�; ð3:82Þ

where ρ is the crystal radius; R is the bath radius; N is the number of intervals into

which the distance R2 ρ is divided. Such net allows us to approximate a concen-

tration distribution in the area, where it fast varies, by large number of discrete

values. Far from the crystal, where the derivatives of concentration are small, larger

distance intervals were set according to (3.82). Coordinates of the net nodes were

being settled up anew on each time step Δt as the particle radius was changing.

After writing finite differences in derivatives of concentration,

@Ci21n

@r5

@Ci21n11 2

@Ci21n21

@r

rn11 2 rn21

;

@2Ci21n

@r25

ðCi21n11 2Ci21

n =rn11 2 rnÞ2 ðCi21n 2Ci21

n21=rn 2 rn21Þ0:5ðrn11 2 rn21Þ

;

(i is the number of time step—the explicit finite difference scheme) and corrections

to the concentrations dependent on changes of coordinates of all net nodes in result

of movement of the interphase boundary:

ΔC0 5@Ci21

n

@rðrin 2 ri21

n Þ;

it is possible to calculate sequentially all new concentrations in each time step, i.e.,

by the sweep method.

Cin 5Ci21

n 1DΔt@2Ci21

n

@r21

K

r

@Ci21n

@r

� �1ΔC0: ð3:83Þ

The value CN was set at first to almost all points except the nodes, which are

the nearest to the surface of a very small crystal given in the initial conditions.

In the case of the constant concentration on the interface (it is equal to the equi-

librium concentration when a large enough crystal grows in a diffusion mode),

growth rate is defined by the condition of mass balance (3.51)

VðCcr 2Cn51Þ5D@C

@r

ρ; ð3:84Þ

(ρ is the crystal radius), in which the derivative @C/@r is usually replaced with the

ratio ðCi1 2Ci

eÞ=ðri1 2 ρiÞ. Accordingly,

ρi 5 ρi21 1Vi Δt:

In the common case, the surface velocity is a certain function of the surface

supersaturation—V5V(Cn512Ce).


To satisfy a boundary condition, which eliminates a mass flux through exterior

boundary of a solute, the additional orb N1 1 is introduced, in which concentration

is not being calculated, and the value found for the N2 1 orb is assigned to it.

While the crystal is much less than the size of a spherical bath, numerical solution

corresponds to the case of growth from the infinite bath and can be checked by

comparing it with appropriate analytical solutions. Later, it is a case of growth

from the finite volume, when the concentration is changing considerably on the

edge of the bath.

Application of nonuniform intervals allows us to reduce by 10 times the number

of points (layers) used for calculations and to decrease essentially a computing

time. Ovrutsky and Chuprina [20] offered the procedure for analysis of crystal

growth in a wide interval of sizes. The size of bath was increased periodically,

when the concentrations in the N2 1 node became smaller than the value CN

(with the certain small difference ε). When this happened, the special procedure

fulfilled a change of the net with increase of distances between nodes.

Concentrations corresponding to the old distribution were assigned to points with

the coordinate r that is smaller than the radius of previous bath, and the value

C5CN was assigned for all other nodes.

Figure 3.5 shows, for example, the calculated concentration distributions for

cases of growth of spherical and cylindrical crystals from the solution with 80 at %

of the basic component (Ce5 0.785) in comparison with analytical solutions of the

Laplace equation—stationary distribution for the case of not growing crystal. The

calculated distributions precisely coincide with Ivantsov’s analytical solutions for

cases of stationary values of the surface concentration [8]. The solution of the

Laplace equation for the infinite in radius cylindrical bath does not exist. The ana-

lytical solution for sphere (the curve 3) testifies that the concentration line varying

smoothly at a very large radii that does not answer to reality in the case of mobile

interface.

C1

12

23

0.795

(A) (B)

0.790

0.7850 0.2 0.4 r, mm r, mm0 0.2 0.4

Figure 3.5 Concentration distributions during growing of spherical (A) and cylindrical

crystals (B). Curve 1—initial, curve 2—after the certain time, curve 3—according to the

Laplace equation; CN5 0.8, Cρ5 0.785, D5 33 1029 m2/s.


As calculations testify, the smallest intervals on distance near the crystal surface

must be approximately 10 times smaller of its size to ensure enough high accuracy

of solutions. Only very small values of intervals between nodes of nets are admissi-

ble. The time step Δt should not exceed the value corresponding to Eq. (3.77) to

provide stability of calculations. Implicit schemes of calculations mentioned above

remain stable at comparatively great values of Δt. In this case, errors increase all

the same quickly with Δt, if phase boundaries are mobile. However, calculations

are much more complex at such procedures.

The presented scheme of the numerical solution is applicable also for the case

of crystal growth from the melt. In this case, it is necessary to substitute concentra-

tions by temperatures, a diffusivity, by the thermal diffusivity. The condition of

balance of heat should be satisfied at the phase boundary. The equation of the heat

balance should be taken in the form of Eq. (3.79) at considering temperature fields

both in liquid and in solid phases. The simultaneous solution of a thermal and dif-

fusion problem in a case of growth from alloy melts will be more exact. It is neces-

sary to do (and it is rationally, taking into account an increase of computing time)

for cases of growth from alloy melts with a small concentration of the second com-

ponent (no more 2% of impurity). At large concentrations, the thermal problem can

be neglected for metal alloys.

The thermal problem can be neglected in the case of growth from the binary

melt with a large enough concentration of the second component, as the crystal

growth is limited in this case not by taking off a heat from the crystallization front

but diffusion.

Two programs described in Sections 9.3 and 9.4 use the technique described

earlier.

3.4.5 Study of the Transport Phenomena in the Framework ofthe Lattice Boltzmann Method

To consider the heat and mass transfer in melts, it is necessary taking into account

the convective transfer. For example, for determination of changes of the concen-

tration field, it is necessary to fulfill calculations according to the equation:

@C=@t1rC u!

5r � ðD rCÞ; ð3:85Þ

where u!

is the flow rate; D is the diffusivity routinely dependent on concentration

and consequently on coordinates; r� div is the divergence. However, determina-

tion of the velocity field u!(x,y,z,t) by the numerical solution of Eqs. (3.57) and

(3.58) for boundary surfaces, which vary in their shape, is a very difficult problem.

The lattice Boltzmann method, based on Boltzmann’s kinetic equation, is applied

now for such problems in the case of microscopic sizes of systems under consider-

ation. The basis of method is in following: if nets are small enough and very small

volumes are associated with their nodes, it is possible to neglect differences in

travel velocities of matter within small volumes. Then these volumes can be


considered as particles and coordinate and velocity distribution functions can be

introduced. Therefore, Boltzmann’s kinetic equation in the form proposed in Ref.

[21] may be applied to such particles:

fið r! 1Δtv!i; t1ΔtÞ5 fið r!; tÞ1 ð1=τÞðf eqi ð r!; tÞ2 fið r!; tÞÞ; ð3:86Þ

where the index i characterizes the discrete directions of velocity, it can vary from

null to z; Δt is the time step; τ is the relaxation time; feqi is the equilibrium distri-

bution function, which characterizes probability for “particles” with radius vector r!

to have velocity ~vi.The concentration and flow are connected with fi as follows:

n5Xz

i50

fi; n u!

5Xz

i50

fi~υi:

The velocity vector v!i should have one of six triples of components: (0,0,0),

(61,61,61), (61,0,0), (0,61,0), (0,0,61) in the lattice parameter. Relaxation

time is connected with kinematic viscosity by the relationship [21]:

υ5 cðτ � 0:5Þδr2=Δt; ð3:87Þ

where δr is the lattice parameter and c is the coefficient dependent on topology of

the lattice.

Equation (3.49) is reduced to the Navier�Stokes equation for nonideal gas, if

we apply the function feqi of a certain type [22]. Coefficients before terms in such

functions should be selected to satisfy the continuity equation (3.57), the normaliz-

ing condition for the distribution function, and the boundary conditions. In particu-

lar, the free surface energy and wetting according to a model of Caen [23] were

taken into account in Ref. [21]. Caen has suggested to consider the additional sur-

face energy connected with a density on the surface: ψc(ns)5φ02φ1ns1 . . .. Thecontact angle θ between a liquid drop and solid substrate can be connected with

magnitude φ1, and then it is possible to determine a density in surface layers nl and

ng through surface energies σlg, σsg, and σsl for boundary surfaces a liquid�gas, a

substrate�gas, a substrate�liquid. A result of Caen’s model is a gaining of the

pressure tensor that defines the drop spreading. In the case, when the contact angle

is not in accordance with the Young equilibrium condition,

σlgcos θ5σsg � σsl;

the drop will spread out.

Dupuis et al. [21] have studied the spreading of the microscopic drop (of meso-

scale size, such as drops in jet printers have) on the substrate by the lattice

Boltzmann method. They considered a lattice 803 803 40, in which the spherical

drop with the radius 16 was set so that it touched the flat substrate with coordinate


z5 0. The phase density was nl5 4.128 and ng5 2.913, and an equilibrium wetting

angle was equal 60� at the dimensionless temperature T5 0.4. Figure 3.6 shows

consecutive images of the drop depending on number of calculation cycles.

References

[1] P.K. Konakov, The Heat and Mass Transfer at Growing of Single Crystals,

Metallurgiya, Moscow, 1971 (in Russian).

[2] P.K. Konakov, Similarity Theory and Its Application in Heating Engineering,

GosEnergoizdat, Moscow, 1969 (in Russian).

[3] M.V. Kirpichov, Similarity Theory, Izd. AN USSR, Moscow, 1953 (in Russian).

[4] A.A. Gukhman, Introduction in Similarity Theory, Vyshaya Shkola, Moscow, 1963 (in

Russian).

[5] M.A. Mikheev, I.M. Mikheeva, Foundations of Heat Transfer, Energiya, Moscow,

1973 (in Russian).

[6] R.L. Parker, Crystal Growth Mechanisms: Energetics, Kinetics and Transport, Solid

State Physics, vol. 25, Academic Press, New York, London, 1970.

[7] B.Ya. Lubov, Kinetic Theory of Phase Transitions, Metallurgiya, Moscow, 1969, 263

p.

[8] G.P. Ivantsov, Crystal Growth, vol. 1, Izd. AN USSR, Moscow, 1957, 98.

[9] W.W. Mullins, R.F. Sekerka, J. Appl. Phys. 34 (1963) 323.

[10] K.W. Morton, D.F. Mayers, Numerical Solution of Partial Differential Equations: An

Introduction, Cambridge University Press, New York, 2005, 293 p.

[11] O. Rubenkonig, The Finite Difference Method (FDM) � An Introduction, Albert

Ludwigs University of Freiburg, Freiburg, 2006.

[12] A. Kaw, E.E. Kalu, Numerical Methods with Applications, Lulu, 2008, 708 p.

[13] A.A. Samarsky, B.D. Moiseenko, J. Comput. Math. Math. Phys. 5 (5) (1985) 816.

Figure 3.6 Kinetics of a drop spreading [21].


http://refhub.elsevier.com/B978-0-12-420143-9.00003-X/sbref1




















[14] B.M. Budak, V.N Solov’eva, A.N Uspensky, J. Comput. Math. Math. Phys. 5 (5)

(1985) 828.

[15] N.A. Avdonin, G.F. Ivanova, Mathematical Modeling. Production of Single Crystals

and Semiconductor Elements, Nauka, Moscow, 1986, pp. 31�39.

[16] A.N. Tikhonov, A.A. Samarsky, Equations of Mathematical Physics, Nauka, Moscow,

1966724 p.

[17] N.A. Avdonin, Mathematical Description of Crystallization Processes, Zinatne, Riga,

1980.

[18] R.R. Tanzilly, R.W. Heckel, Trans. Metall. Soc. AIME 242 (11) (1968) 2313.

[19] A.M. Ovrutsky, Sov. Phys. Crystallogr. 24 (3) (1979) 571.

[20] A.M. Ovrutsky, L.M Chuprina, Sov. Phys. Crystallogr. 22 (1977) 686.

[21] A. Dupuis, A.J. Briant, C.M. Pooley, J.M. Yeomans, arXiv:cond-mat 0303082 v1,

March 5, 2003.

[22] M.R. Swift, E. Orlandini, W.R. Osborn, J.M. Yeomans, Phys. Rev. E 54 (1996) 5051.

[23] P.B. Papatzacos, J.M. Yeomans, Phil. Trans. R. Soc. A 360 (2002) 485.
















4 Structure of the Boundary Surfaces

4.1 Surface Phenomena

Surface science concerns a big variety of different problems [1�5]. Each substance

has surfaces, which separate it from other substances or other phases of the same

substance. We will consider surfaces between crystal and gas, or between crystal

and a fluid. The concept of an ideal surface concerns actually only the internal

properties of solid or liquid substances. If considering a surface in contact with the

environment, it is necessary to consider a multitude of processes and phenomena:

adsorption, catalysis, evaporation, condensation, diffusion, structural transforma-

tions, and so on. The state of the surface influences many properties: electrical,

magnetic, and optical properties. The reflection factor and the light transmission

depend on the state of surfaces; mechanical properties: the hardness, plasticity, and

wear resistance of metals vary at deposition of thin layers of another substance.

Adsorption is one of the major phenomena. Atoms or molecules of the environ-

ment medium stay longer near the surface because of interaction with it and, conse-

quently, their concentration at the surface becomes increased compared with the

concentration in the bulk of contacting phases. Many important techniques in

industry are associated with adsorption, e.g., air drying and purification, purifica-

tion of oils, recovery of organic solvents, gas mixtures separation, distilling of oil

at rather low temperatures. Thin film deposition on the surfaces of earth metals is

applied in the manufacture of photo cathodes and thermionic cathodes; this results

in a decrease of electronic work functions.

The catalysis phenomenon, which consists in acceleration of chemical reactions

of substances in result of their contact with a certain surface (often after adsorption

of another substance at the surface), is very important. Oil cracking, operation of

ferments, assimilation in green plants are processes which have the character of

catalytic reactions on the surface.

The modern manufacturing techniques of electronic components require knowl-

edge of the phenomena and processes occurring in the surface ultrathin layers of

materials. These engineering processes include a surface cleaning, adsorption, pro-

cesses of creation of thin films (condensation, crystallization)—they are a set of

processes, which must be fulfilled before creation of electronic elements with the

necessary parameters at the surfaces. During operation of electronics, the para-

meters of electronic components can vary because of diffusion and other physico-

chemical processes. Therefore, manufacturing techniques of electronic units should

ensure a sufficient long-term period of their stable work.



http://dx.doi.org/10.1016/B978-0-12-420143-9.00004-1

4.2 The Major Discoveries Contributing to the Developmentof Surface Science

Some important laws concerning surfaces were discovered in the nineteenth cen-

tury. In 1833, Michael Faraday discovered that the oxidizing reaction of hydrogen

runs at a lower temperature in the presence of platinum. Berzelius in 1836 formu-

lated the catalytic action theory based on the analysis of a series of experiments by

Faraday.

In 1874, Charles Ferdinand Brown noted deviation from Ohm’s law at measure-

ments of electrical resistance of metal sulfides. The resistance of the passage of an

electric current through a Cu and FeS sandwich depended on the direction of the

current. Some years later, it became clear that the rectifying effect was actually due

to a thin transition layer between copper and iron sulfide.

In 1877, John Willard Gibbs published in Transactions of the Connecticut

Academy of Sciences the second part his fundamental work: “On the Equilibrium

of Heterogeneous Substances,” which provided the basis for thermodynamics and

statistical physics. Gibbs has completely featured thermodynamics of the surface

processes in the part of this work. Gibbs’ work still remains the most fundamental

work in this branch of science.

Surface science split into the separated branch of science after Irving

Langmuir’s investigations. In 1906 in Germany, he ended his study of dissociation

of gases by a heated platinum wire. After 3 years in America (General Electric

Company), he developed instruments for obtaining a vacuum and its measurement.

Then he investigated adsorption of gases on solid surfaces at the new higher level.

He introduced the concept of the surface adsorption grid, accommodation coeffi-

cient, preadsorption states. Then he measured a work function of electrons and for-

mulated the surface ionization laws. In 1932, Langmuir received the Nobel prize

for outstanding work in the branch of surface chemistry.

In 1928, Albert Einstein received a Nobel Prize for photoeffect explanation.

In 1937, Clinton Davisson received the Nobel Prize for research of electron dif-

fraction. Along with his colleague Lester Halbert Germer, he confirmed the wave

nature of electrons. Davisson and Germer understood that they obtained diffraction

from surfaces when applying electrons of small energies. But only 30 years later,

the LEED (low-energy electron diffraction) method became one of the main tools

for studying the surface structure. The most precise (high-precision) method for

studying chemical composition of a surface at the present time is the Ozhe spec-

troscopy method, which is also based on application of electrons having sufficiently

low.

In the first 30 years of the twentieth century, Tamm, Maue, and Goodwin

proved the existence and studied properties of the electronic states localized at

crystal surfaces. Bardeen developed the electronic theory of the free metallic sur-

face. Almost simultaneously, Mott et al. offered the theory of rectifying junction.

And many American and European scientists have been attracted to military

research.


Three important works were published in 1949 that stimulated the subsequent

development of important theories in the surface science.

1. The theory of crystal growth of Burton and Cabrera with development of conceptions

about the surface structure of crystals.

2. Smitt’s paper “Grains, Phases and Interfaces.”

3. Bardeen’s and Brattain’s publication about elaboration of the transistor with the point-

type contact.

The last publication caused the widest interest in the questions of surface phys-

ics. And this has triggered the investment in surface science.

The most important events in the physics of surfaces have taken place in the late

1960s. Methods of obtaining an ultrahigh vacuum have been developed.

Instruments for application LEED and Ozhe spectroscopy have been worked out.

Development of digital computers has allowed carrying out calculations for more

real models of surface phenomena and processes.

4.3 On the Experimental Research Techniques of Surfaces

A major component of the surface science research consists of determining the struc-

ture and the chemical composition of ultrathin surface layers, including the atomic

layers obtained during adsorption. It is clear that for designing electronic compo-

nents it is important to define energy levels of the electrons localized at the surface,

to measure work functions of electrons. Measurements of optical or magnetic prop-

erties associated with surfaces are often also necessary. It is possible to enumerate at

least 30 techniques, which are applied most often for studying surfaces [1,5].

LEED is applied for determination of the surface structure. It allows defining

the type of the surface lattices. Application of X-ray diffraction for definition of

the volume crystal structure was very successful. In order to perform diffraction

analysis of crystal structure, it is necessary that the wavelength of radiation is less

than typical interatomic distances, let us say 1 A. Electrons used for investigation

of the surface structure should have a kinetic energy E5 (h/π)2/(2m)B150 eV.

Such electrons of relatively small energy penetrate into a substance only on some

interatomic distances that allow studying diffraction from a surface. Electrons of

the energy in the range of 20�500 eV form the Fraunhofer diffraction pattern,

which is the Fourier mode of layout of atoms on the surface, after elastic scattering

in the opposite direction from a crystal surface.

Figure 4.1 shows the typical scheme of the LEED device. Electrons fall on the

sample from the left; their certain part is dissipated in the opposite direction toward

the half-spherical grid G1. The odds of potential for retarding is applied between

G1 and the second grid G2. Therefore, only elastically dissipated electrons (nearly

1% from the complete yield) can reach grid G2. The major positive potential pro-

vides an acceleration of electrons, which excite a luminophore at blows on the fluo-

rescent screen S. The camera or video camera records the diffraction reflexes

(Figure 4.2).

107Structure of the Boundary Surfaces

Patterns LEED obtained are the plotting of the surface reciprocal lattice at obser-

vation of a crystal from a big distance along a normal to the surface. To explain

this, we will remind that the distance between adjacent points in the reciprocal lat-

tice is in inverse proportion with the distance between points of the direct lattice in

an appropriate direction. For a two-dimensional lattice space, the period in the

z-direction is infinite. Therefore, “points” of the reciprocal lattice along the normal

to a surface are allocated densely; therefore, we speak about rods in reciprocal

space. However, invariance of translations in two measurements ensures diffraction

in directions, for which Laue’s conditions for the two-dimensional case:

ðki 2 kfÞas 5 2πm and ðki 2 kfÞbs 5 2πn; ð4:1Þ

Vacuum SG2C1

Electronicgun

Window

Externaldetector

V

Figure 4.1 The device for observation of LEED [1].

Rows of revertivelattice

–4a –2a 0 2a 4a

Evald’ssphera

(A) (B) (04)(03)(02) (00) (02) (01)

K1

k1

Figure 4.2 The LEED reflexes from the surface of the single copper crystal (A), the Ewald

graph for electrons, which fall normally on the surface (B) [1]. (A) An energy of electrons is

36 eV, (B) nine rays are shown, for which the diffraction conditions are satisfied.


are fulfilled, where ki and kf are the wave vectors of an incident and dissipated

electron; accordingly, m and n are integers.

It is better to illustrate Laue’s conditions with the known Ewald’s scheme

(Figure 4.2A). The rods of the reciprocal lattice go from each point of the surface

reciprocal lattice gs5 hAs1 kBs. The amplitude of a wave vector of an electron

defines the sphere radius. The diffraction condition is fitted for each wave vector

kf, which is routed in a direction from the sphere center to a point of its intersection

with the rod of the reciprocal lattice. As well as in a three-dimensional case, bun-

dles are determined by indexes of that vector of the reciprocal lattice, which stipu-

lates diffraction. Diffraction mottles in Figure 4.2A are designated according to the

vector gs values.

Only several planes of a crystal lattice are determined by the method LEED, and

rays which quit are visible at any energy of electrons, if the appropriate rod of a

reciprocal lattice is within Ewald’s sphere. Existence of the diffraction pattern

reveals the presence of the ordered surface structure and gives the direct informa-

tion on its symmetry. Therefore, the equipment for LEED study is present in almost

every laboratory for research of surfaces.

It is not easy to define the relative position of the surface atoms using LEED.

Intensity of each Laue’s reflex at volume scattering of the X-ray is defined by mul-

tiplying the atomic scattering factor and the geometrical arrangement factor.

Researchers vary positions of atoms within one cell, until geometrical factors will

not take over such values, at which intensity of each ray has the correct value.

Such simplicity is stipulated by the fact that the X-ray very feebly interacts with

substance; each quantum disperses in the opposite direction after single collision

with a lattice ion, and intensity of reflexes depends neither on the energy of the

incident ray nor on falling lateral angle. Neither of these two conditions is met at

LEED. Each electron undergoes the multiple scattering within the first several

layers of a crystal, because unlike an X-ray scattering experiment, the section of

elastic scattering for electrons is very large (B1 A2). The second (or the third, etc.)

scattering, which has high probability, deflects an electron from the direction of its

primary diffraction.

As the LEED method does not reveal arrangement of atoms in cells of the sur-

face crystalline structure, it is supplemented compulsorily with methods performing

immediate mapping of atomic structure, such as electron microscopy and field

microscopy.

The ionic projector was the first instrument that allowed observing separate

atoms of some elements. A sample for structural studies is fabricated in the form of

a very thin needle. It is placed in a vessel with rarefied noble gas, and an electric

field of high voltage is applied to it. The highest voltage is near the edge of the

needle (Figure 4.3) and especially near separate atoms on its surface. In these

points, an ionization of molecules of gas takes place, which after acceleration in

the electric field, yields the plotting of the needle edge on the screen with huge

magnification. On the image (Figure 4.3B), there are dark circles (planes on the

edge of the needle, which are smooth on the atomic scale), an agglomerate of light


spots between them, that map atomic roughness and vagabonding light specks,

which map movement on the surface of separate atoms.

The most efficient instrument of field microscopy is the scanning tunnel micro-

scope, in which the tunnel current between a sharp needle and the surface (distance

,50 A) is measured. This current is sensitive to the separate atoms on the surface.

The numerical values of this current are obtained during the surface scanning

(Figure 4.4A). Computer processing of these data allows obtaining the image

(Figure 4.4B) of the surface structure on the atomic scale.

Computer processing of outcomes of measurement is often performed using data

of the electron microscopy. If the organic sample can be destroyed by the electron

beam, many images are usually obtained at small intensity of an electron beam. In

this case, single snapshots (Figure 4.5A) do not map atomic structure as the move-

ment of electrons is rather random. The structure of the crystal surface may be dis-

played after computer processing of several hundreds of such images (Figure 4.5B).

Therefore, novel experiments are performed with computer support.

It is impossible to measure the chemical composition of surface layers by chem-

ical methods, which require substances in significant quantities. The spectral analy-

sis method requires surface melting at considerable depth. The mass spectrometry

method is very sensitive, but it results in a variety of additional outcomes (appear-

ance of different chemical compounds in gas during damage of the surface by a

spark); it is difficult to distinguish between them. The most effective method for

chemical composition studying is the Ozhe spectroscopy method. It yields one-

valued outcome concerning presence at the surface atomic layers of one or another

element.

The Ozhe spectroscopy method is based on the effect, which consists of particu-

lar interatomic transition of electrons from one energy level to another at the dis-

posal of the nearest to the nucleus electron with the highest binding energy

(K-level—1s). Afterward, one of electrons of L-level (2s—the main quantum num-

ber is equal to 2) migrates to the K-level, another one that was at the close energy

The He ionmoving to the screen

(A) (B)

(110)

(112)(111)

(100)

+

The He atom±

Metallic edgecharged posit

Figure 4.3 The principle of work of the field ionic projector (A) [1] and the image of the

W needle with radius of 12 nm (B) [6], planes (100) and (111) look different.


level (2p) leaves the atom. The kinetic energy of the last electron is strictly fixed

and it is only one value for each chemical element. The energy spectrometry of

these electrons defines exactly a chemical composition of surfaces. Low-energy

electrons are used (,1000 eV) for realization of the Ozhe effect; they do not

deeply permeate into substance. Therefore, data of research reflect a presence of

basic elements in the surface layers.

4.4 Features of the Surface Phase Transitions

Studying phase transitions plays a central role in modern condensed matter physics.

The characteristic feature of the second-order phase transitions is the temperature

dependence of λ shape of the heat capacity of substances (Figure 2.2). The heat

capacity dependence on the temperature has a jump in the Curie point Tc (magnetic

(110)

(001)

(A) (B)

(C)

A S

S

B

A

B

(110)–

Figure 4.4 A current entry at scanning of the Au surface by the tunnel microscope (A) [7]

and a computer reconstruction of the Si atomic surface structure (B) [8].

Figure 4.5 The electron microscopic image of an organic crystal at low intensity of the

electron beam (A) and outcome of the sum of 200 snapshots (B).


transformation temperature). The heat capacity becomes infinite (CP5 (δQ/dT)P,dT!0) in the transition point in the case of phase transitions of the first type.

However, it is necessary to mark that the infinite magnification of heat capacity

cannot always be defined experimentally.

Phase transitions are caused by the fact that all systems, which are in the

thermodynamically equilibrium state, tend to transfer in the state with the minimum

free energy. Thus, one phase displaces another phase. It is convenient to character-

ize competitive phases using so-called order parameter. By definition, the order

parameter varies from zero in one phase, which corresponds to the lower tempera-

tures or to higher symmetry, and it is equal to zero in another phase (high tempera-

ture). If fluid is transformed into vapor, the difference between the density in the

liquid and the vapor phase can be considered as the order parameter. The homo-

geneous magnetization may play a role in the order parameter at transformation

from ferromagnetic to a nonferromagnetic state. The role of order parameter can be

taken by the amplitude of the certain phonon mode for structural phase transition.

At phase transition of the first type, change of an order parameter by a jump takes

place. In this case, two independent curves of the free energy dependence on the tem-

perature are simply crossed (Figure 2.3). The system sharply transfers from one certain

equilibrium phase state into another certain equilibrium state. Transformations of the

first type are characterized by such known phenomena as coexisting of two phases,

nucleation, and growth of a new phase in the primary phase. In the case of the continu-

ous phase transition, on the contrary, two competing phases change so with tempera-

ture that they do not differ at the temperature Tc. In this case, the order parameter

grows continuously, when the temperature lowers, though fluctuations of its value

concerning average value are possible. At the continuous transformation, the order

parameter change is routinely featured by the following dependence (T2 Tc)β for tem-

peratures T near to Tc. Moreover, now it is known that numerical value of a critical

index β (and of several other indexes linked to this problem) depends only on a small

number of physical properties of system, e.g., symmetries of the system, dimensional-

ity of the order parameter: a scalar, a vector, etc., dimensionalities of the space. Such

property is called universality. Surface phase transitions are an interesting phenomena

because their efficient dimensionality is closer to two than to three.

Depending on the obtaining technique, handling type, and temperature, the sur-

face structure can vary. Different transformations of the surface structure are

possible:

1. Reconstruction—changes of equilibrium structure of the surface as polymorphism;

2. Disordering transformation—appearance of an atomic roughness of the surface, which

causes transition from faceted to round forms of crystal growth;

3. The surface melting, which consists in high mobility of the surface atoms, and such struc-

tural change of several surface layers, which makes them similar to liquid.

According to the Landau theory, structural transitions can occur as phase transi-

tions of the first or second type depending on symmetry of the crystal lattice of a

new phase. If the symmetry group of one of two phases is a subgroup of symmetry

of another phase, it will be a phase transition of the second type.


4.5 Reconstruction

There can be different reconstructive phase transitions on the surface of solid bod-

ies during thermal treatments. Unfortunately, a number of cases is rather restricted,

for which it is possible to construct reliably the surface structural phase diagram,

as it may be done for a similar volume case. It is connected with difficulties in

defining crystallography of the surface, determining the real structure. Besides,

many surface phases are actually metastable; this means that the surface is not in

the state of the true equilibrium. Energy of the system is not minimal, as there are

broken bonds at the interface; this causes structural transformations. If the part of

broken bonds is recovered during the heat treatment of crystals, the driving force

can be not sufficient for transition of the surface atoms to another favorable config-

uration with lower energy. The long-term annealing of the sample at the appropri-

ate temperature is often necessary for achievement of thermodynamic equilibrium.

Therefore, description of the phase diagram of a surface often reminds consider-

ation of history of its previous treatment.

Additional information is needed for description of the arranged surface struc-

tures, both about a unit cell, and about placement of basis atoms. Figure 4.6 shows

the elementary cells of the main types of two-dimensional crystal lattices (except

for scalene lattice): square (A), hexagonal (B), rectangular (C), and rectangular

centered (D). The last lattice, however, is not complex because it can be formed by

translations of the simple scalene unit cell with the base a0, b0. Except for transmit-

ting symmetry, crystal lattices are also characterized by other elements of symme-

try. Each simple space lattice has the symmetry center. Each vertex of a basis

parallelepiped, its center, the middles of its edges, and centers of faces are the sym-

metry centers. Simple lattices are also characterized by the presence of the rotary

and mirror-rotary axes, which can be of the second, third, fourth, and sixth orders

(the 180�, 120�, 90�, and 60� rotational turns). It results in existence of the 10-point

group symmetries of the surface lattices, and taking into account translations—13

space groups.

Ideal or relaxed surfaces can be easily identified, marking planes of rupture, e.g.,

Ni (110) or MgO (100). In this case, the surface lattices own the same period and

orientation as the adjoining volume lattices. So the structure is designated as the

13 1 structure. Primitive vectors of translation of the surface lattice differ from

vectors of translation for an ideal surface in the case of the typical reconstructed

surface. These vectors are connected by relations as5Na and bs5Mb. In this case,

denotations of type R(hkl) N3M; for example, Au (110) 23 1 or Si (111) 73 7 are

used. If the surface lattices are turned on some angle ϕ in relation to volume lat-

tices, this angle is added in the denotation, thus:

RðhklÞN3M2ϕ:

The surface (100) of iridium (transition metal with the not filled up 5d subshell)

undergoes the transformation from structure 13 1 to structure 13 5 at the


temperature above 800 K. It is a bright example of phase transition of the first type

from the metastable to the stable state. The metastable structure 13 1 is typical for

a metal surface: it is an ideal breakaway of the lattice, in which at least the high

layer undergoes relaxation. The ground state structure 13 5 may be featured as a

dense packing of atoms, which “sit” on the ideal face-centered substrate with orien-

tation (100) (Figure 4.7). The energy barrier, which hinders the transition between

the specified two configurations of the surface, arises when atoms in rows are dis-

placed relative atoms located in the following atomic plane.

The described scheme is based on the measurements fulfilled with application

of high-quality instrumentation for LEED that includes a video camera, which

allows register reconstruction in real time. Time and temperature dependences of

diffraction reflexes intensity increase, which is stipulated by superlattices 13 5,

have been determined. These measurements have shown that the activation energy

of transition is close to 0.9 eV/atom. The reconstruction is stipulated by the fact

that the surface formed is densely packed and has smaller energy than the initial

open surface (Figure 4.7). For iridium, the described effect turns out to be consider-

able (in the absolute value) because this metal is characterized by the greatest inter-

facial tension among all elements. Reconstruction is also accompanied by loss in

the energy, linked to the lattice mismatch between the square structure of the sub-

strate and hexagonal structure of the upper layer. The competition between the

specified two effects plays the key role in the epitaxy phenomenon (the oriented

growth of crystalline films).

Unlike the example with the surface (100) of Ir, the detailed information on

atomic geometry for phase transitions at reconstruction is routinely absent.

Square

(A) (B)

(D)(C)

a

a

ab

b

aa

bb

ϕϕ = 120°b

Rectangular, ϕ = 90° Rectangular, centered

Hexagonal

′

′

Figure 4.6 The basic types of unit cells of the surface lattices.


More than 15 years of investigations by the best laboratories all around the world

were required for decoding of the reconstructed structure of the silicon surface.

Researchers used all the above-mentioned methods. Basically, thanks to scanning

tunnel microscopy (STM) (Figure 4.8), the reconstructed surface was established to

have the structure 73 7 (Figure 4.9).

As mentioned above, investigation of the diffraction reflexes in LEED allows

determining only symmetry of the high- and low-temperature phases. However,

Landau and Lifshitz [12] concluded that this information is sufficient to define

what type of the phase transition takes place: continuous (the second type) or

transition of the first type. Their definition for the surfaces is the following.

If ρ(r) is the surface density of atoms, which corresponds to the crystal structure

of the high symmetrical phase, this function is invariant with respect to the sym-

metry operations of the corresponding space group of the surface structure, which

we will designate Go. After phase transition, the reconstructed crystal surface is

described by the new density function ρ0(r)5 ρ(r)1 δρ(r), which is invariant with

respect to the symmetry operations of certain new space group G. According to

the Landau and Lifshitz rule, the transition can be continuous, only if G is a sub-

group of Go and function δρ(r) will be transformed according to unique represen-

tation of a group Go. If this condition is not fulfilled, we have the phase

transition of the first type.

Hexagonal surface lattice Si (111) corresponds to this category. The direct anal-

ysis based on the concept about types of symmetry shows that reconstruction of

Figure 4.7 Allocation of atoms on the Ir surface (100) at phase transition from structure

13 1 to structure 13 5 [9]: (A) ideal structure 13 1 and (B) the reconstructed quasi-

hexagonal structure 13 5.


the specified surface from structure 13 1 to 73 7 cannot happen as the continuous

phase transition. Therefore, this transition should display typical attributes of phase

transitions of the first type. Electron microscope images (Figure 4.10) of the vicinal

surface Si (111) (with steps), obtained in geometry “on reflexion” clearly show

that nucleation and growth of the reconstructed phase (with the structure 73 7)

take place at cooling of the sample below the transition temperature. Complete

transformation is realized in the temperature band 20�30 K (lower than Ttrans).

Figure 4.8 The STM image of the vicinal face of silicium (close to (111) plane), structure

73 7, it is visible on terraces between steps of atomic height [10].

The upper layerThe second layer

The third layer

Figure 4.9 The structure (111) 73 7 of the silicium surface [11] (the view on the upper

three layers of the unit cell is shown).


Such “delayed” behavior is also observed for the volume phase transitions of the

first type in the solid state, when mechanical strains play an essential role [14].

Consider the reconstruction phase transition 13 1!O23O22R45� for the

(100) surface of W. In this case, we deal with a very feeble transformation com-

pared with the previous examples. In the low-temperature phase, atoms of tungsten

at the surface are slightly shifted with respect to the ideal positions and form a zig-

zag chain (Figure 4.11). The analysis of symmetry of this surface, according to

Figure 4.10 The electron microscope images of the Si surface (111) [13]: areas with the

structure 73 7 appear on vertexes of polyatomic steps (more dark sites) and propagate along

terraces (in figures, A and B, arrows specify the growth direction).

Figure 4.11 Structure of the (100) surface of W [1]. (x) The high-temperature phase 13 1

(light circles); (�) the low-temperature phase O23O22R45� (dark circles).


Landau and Lifshitz, shows that this phase transition can be continuous; such a

result is also obtained experimentally.

It is possible as well that the surface atoms of tungsten are also shifted at tem-

peratures above Ttrans also, but in arbitrary directions. And correlation of displace-

ments with the observed structure turns out to be energetically favorable at

sufficient low temperatures (taking into account interactions at the surface).

Reconstruction phase transition from the structure 23 1 to 13 1, which occurs on

the (110) surface of Au, is a vivid example confirming the concept of universality.

Experiments (Figure 4.12) show that at temperatures near 650 K, the high-temperature

structure 13 1 is inversely transformed to a structure with “the disappeared row.”

However, without knowing all details of structure, only based on symmetry reasoning,

one can assert that the continuous phase transition (if it occurs) should detect the criti-

cal properties such as the two-dimensional Ising model has. A contamination of the

surface (or even of the volume of gold) should not influence the character of the tran-

sition. Temperature dependence of the order parameter (intensity of the diffraction

reflexes in LEED from a phase superlattice (23 1)) should be characterized by a criti-

cal index β that is equal to the known exact value (β5 1/8) obtained by Onsager.

Experiment yields β5 0.136 0.02 (Figure 4.12).

4.6 Transition from an Atomically Smooth to an AtomicallyRough Surface Structure

At first it was proved by Frenkel that the surface of crystals should be smooth in

atomic scale at low temperatures and only on the steps, if they exist, can there be

1

0.8 Au (110)

I0.6

0.4

0.2

0350 450 550 650 750

T,K

Figure 4.12 Temperature dependence of the order parameter for the transition from 23 1 to

13 1 structure for the (110) surface of Au [15]. There are intensity of LEED reflexes

corresponding to the structure 23 1 (circles) and the Onsager exact solution for the two-

dimensional Ising model (full curve).


single breaks. With increase in temperature, the number of breaks on steps grows,

and steps cease to be rectangular. Adsorbed atoms and holes appear on them. The

subsequent increase in temperature results in the atomic roughness of the surface

and in the appearance of round areas in the forms of crystals. This transition from

faceted to round forms of the crystal growth is stipulated by the disordering of the

surface atomic structure.

More detailed analysis of the atom structure of crystal surfaces will be given in

Chapter 7. Note that the temperature at which the disordering occurs depends on the

heat ΔH of the phase transition. Well-faceted crystals grow from the gas phase

because entropy of the phase transition ΔS5ΔH/T is high in this case. The melting

heat is much lower than the evaporation heat. For the majority of metals, the change of

entropy at melting is ,3 J/(mol K), and their crystals have round forms during growth

from their own melts. Crystals of semimetals (bismuth, gallium), semiconductors, and

semiconductor compounds have faceted forms while growing from the melts.

In the case of growth of metal crystals from the two component melts at

reduced temperatures (the liquidus temperature depends on the concentration of

components), their forms may vary dramatically in some narrow intervals of tem-

perature. Increasing the concentration of the second component in alloys Au�Bi,

Au�Pb, Ag�Bi, Ag�Pb, Cu�Bi, Cu�Pb, Al�Sn, Sn�Ga and accordingly

lowering equilibrium temperature, it is possible to find crystals of gold, silver,

copper, aluminum, and tin with distinct facets (the roundish dendrite and a fac-

eted skeletal crystal of tin are shown in Figure 4.13A and B). The matter is that

the growth mechanism depends on a surface roughness on an atomic scale. If the

surface is rough, crystals grow by the so-called normal mechanism, at which

anisotropy of growth rate is very small; all sites of the surface are moving along

the local normal, forming the round forms. The point is that the growth mecha-

nism depends on the surface roughness on an atomic scale. If the surface is rough,

crystals grow by the so-called normal mechanism, at which anisotropy of growth

rate is very small; all sites of the surface move along the local normal, forming

the round forms. After lowering temperature and ordering of the surface structure,

the surface becomes atomically smooth, and growth forms become faceted. If the

driving force of crystallization is not very large, growth occurs due to the tangen-

tial movement of steps on the crystal surface under these conditions.

The equilibrium forms of crystals are the forms of the crystal nuclei, which are

in equilibrium with fluid. A very big crystal can be in equilibrium with a melt at

the tabular value of the melting point or liquidus temperature. A small crystal can

be in the dynamic equilibrium with the melt at lower temperatures (neither to grow

nor to dissolve), that is the crystalline nucleus of the critical size. Crystal forms can

be faceted, semifaceted, and round. Even in the case of the faceted form, edges of

crystals are a little bit rounded in the microscale. If a very thin anisotropic crystal

is formed, its form depends on the thickness. Thin naphthalene and diphenyl crys-

tals have plane basis facets and round (rough on an atomic scale) lateral surfaces.

Therefore, their shapes in microscopic snapshots are rounded. Facets appear on

their lateral surfaces with increase of the crystal thickness. Figure 4.14 shows

shapes of diphenyl crystals in equilibrium (A�C) and during growth (D and E).


On the surfaces of crystals there are the permanent steps linked to the screw dis-

locations. Steps can be formed also spontaneously at the origin of clusters or nuclei

of the new layer on the surface. If many steps are formed, the surface will be

rough. The temperature of surface disordering may be estimated, for example, by

calculation of the free edge energy of the steps. If the latter approaches to zero, the

surface roughness will be considerable.

4.7 Surface Melting

The reconstructive structural transitions result in two classes of the surface phase tran-

sitions at sufficient high temperatures: disordering and melting. Disordering transition

Figure 4.14 Equilibrium forms and forms of growth of diphenyl crystals [17]: (A and B)

equilibrium (in small drops); (C) in the wedge bath, that is, the shapes of crystals of

different thickness; (D and E) growth of crystals toward each other: thin (gray) and thick

(white).

Figure 4.13 Forms of crystal growth of tin from the melts with gallium [16]: (A) at the

temperature 25�C and (B) background is the solidified eutectic at 8�C.


is characterized by the temperature, at which the free energy of monatomic steps

becomes very small. Spontaneous origin of such steps stipulates instability of crystal

face in relation to long-wavelength fluctuations of local position of the surface. On

the contrary, the melting can be linked to the disordering process due to the short-

wave fluctuations of atom displacements.

Melting is probably one of the best-known examples of the first-order phase

transition. At the temperature corresponding to the fusion point Tf of a bulk crystal,

some thermodynamic characteristics vary as a jump function. Such behavior takes

place, when symmetry abruptly varies: instead of a space group, which charac-

terizes a crystalline state, we deal now with gyration invariance; such behavior is

intrinsic to fluids. Simple estimation of Tf follows from the Lindemann test: the

crystal is melting, when the value hu2i—the mean-square atomic deviation (from

the equilibrium positions) stipulated by thermal oscillations, reaches a considerable

portion (B25%) of the lattice parameter. It is instructive to fulfill the simple evalu-

ation of the temperature at which such deviation takes place.

The complete displacement of a separate atom in the crystal can be written as

superposition of contributions from each independent phonon mode. The complete

mean-square displacement of atoms hu2i in harmonic approach is (from expressions

for total energy)

hu2i5ΣjUqj2 5Σðnq1 1/2Þh=ðNmωqÞ;

where Uq is the amplitude of the normalized oscillation with the phonon mode ωq

that corresponds to the wave vector q; m is the atomic mass.

At high temperatures, the Bose�Einstein factor of the particular phonon mode

ωq is nq5 kT/(h� ωq). Therefore,

hu2i5 kT

Nm

Xq

1

ω2q

:

It is assumed in the Debye model that ωq5 cq up to the energy kθD5 h� cqD, atwhich the spectrum is truncated, and Ωq3D 5 6π2 (Ω5V/N is the atomic volume).

Therefore, the mean-square displacement of atoms,

hu2i5 kTωmð2πÞ3

ðd3q

c2q25

3h2T

mkθ2D;

increases linearly with the temperature.

The value of the mean-square displacement is measured directly in the dif-

fraction experiments as the thermal oscillations reduce the intensity of the dif-

fraction reflexes; it is determined by the so-called Debye�Waller factor,

expð2 jΔkj2hu2i=4Þ, where Δk is the value of an impulse change of dissipated

quantum. Therefore, comparison of X-ray diffraction and LEED data allows

defining the amplitude ratio of atom thermal oscillations at the surface and in


the bulk. The results of the LEED investigation are routinely represented as the

Debye temperature dependence on the electron energy in the initial electron

beam (Figure 4.15). As a rule, experiments show that thermal displacements of

the surface atoms perpendicularly to the surface are 50�100% higher than in

crystal volume at the same temperature. It takes place because the returning

forces acting on the surface atoms deviated from equilibrium positions are twice

as low as the forces acting on the atoms in the bulk.

The simplified application of the Lindemann condition allows concluding that

the surface lattice “melts” at essential lower temperature than a bulk phase. If this

is the case, the appropriate process can occur as follows. The perpendicular return-

ing force, which acts on atoms of the second layer from the side of the “partially

fused” surface layer, has the intermediate value between values for the arranged

surface layer and vacuum. Accordingly, the second layer melts at a little bit higher

temperature than the surface layer, but still below the bulk melting point. A similar

reasoning is applicable to the third layer and so on. Each layer melts abruptly as

soon as the Lindemann condition is locally realized for it. The melting front propa-

gates into the depth of the crystal, and melting point with each layer increases until

the process will be completed at Tf.

It is possible to visualize this process, applying molecular dynamics simulation.

Figure 4.16 shows such result [19] for the model of ice having free surfaces at the

temperature T below of the melting point. At the lower temperature ,260 K, all

atoms in the planes which were parallel to the free surfaces, both (0001) and (1010)

faces of ice were doing simple harmonic motions with respect to the equilibrium posi-

tions. At a higher temperature (Figure 4.16), thus still below the bulk fusion point, the

surface layer of the (1010) face becomes almost disordered. Raising the temperature,

it is possible to observe movement of the melting front deep into the crystal.

Is melting of a two-dimensional crystal similar to the melting of a three-

dimensional crystal? The answer to this question should be negative, as

50 75

Pb(110)

100 E, eV

75

θD The volume value

50

2525

Figure 4.15 The Debye temperatures (of surface layers) versus the energy of electrons

(corresponding to the depths of penetration) in the initial electron beam [18].


a two-dimensional solid body differs essentially from a three-dimensional one. Let

us suppose that we want to use the Lindemann rule. The evaluations mentioned

above must be slightly modified as integration on the wave vectors is restricted

now by two dimensions, so it is necessary to substitute differential d3q by d2q.

However, this tiny change results in the logarithmic divergence of the integral at

the lower limit. As it is considered, this indicates that long-wavelength phonons

destroy ordered arrangement of particles in two-dimensional “solid body.”

Of course, in any sample of the finite sizes, the mean-square displacements of

atoms ,u2. are not actually infinite. Thus a two-dimensional solid body can be

characterized rather by the long-range orientation order, instead of long-range

translational order [20]. It means that orientation of crystalline axes is conserved

on the large distances, whereas strict periodicity of lattice sites along axes misses.

Earlier, the opinion existed that mechanical firmness of two-dimensional crystals is

absent. Recently, it was discovered that two-dimensional carbon films (graphene,

with thickness on an atomic size) are sufficiently stable due to the nature of cova-

lent bindings. Two-dimensional films of carbon (flakes of graphite) are present in

the lines made by a simple pencil.

Melting in “two dimensions” may take place due to the thermal generation of

topological defects in lattices. Figure 4.17 shows elementary dislocation in

Figure 4.16 Simulated trajectories of water molecules in the ice crystal [19], T5 265 K.

Figure 4.17 Defects in two-dimensional triangular lattices: (A) an isolated dislocation and

(B) a bounded pair of dislocations [21].


triangular lattices. The Burgers vector represents the directional segment needed to

draw a closed line around the dislocation. Calculations of energy of such disloca-

tion are based on the elastic theory [22]:

Ud 5μðλ1μÞa20ðλ1 2μÞ4π ln

A

A0

;

where μ and λ are the Lame constants for a certain substance, a0 is the lattice

parameter and A0Ba20, A is the area of the surface. The long-range tension field of

the dislocation causes the logarithmic dependence of the energy on the surface area

(A) of solid body. Dislocations increase the internal energy of the two-dimensional

crystal, but simultaneously increase the entropy, as there are many sites of their

possible appearance.

Therefore the energy barrier for their forming (the free energy change) can be

small, and the crystal can be “melted” as a result of spontaneous generation of dis-

locations at the temperature defined by condition Ud�TdSd5 0, which is below the

fusion point. Pairwise-bounded dislocations, which have opposite directions of the

Burgers vectors, can appear at temperatures lower Tf. Melting occurs by the ther-

mal bond breaking between pairs of dislocations.

References

[1] A. Zangwill, Physics of Surface, Cambridge University Press, 1988, 536 pp.

[2] S.R. Morrison, The Chemical Physics of Surfaces, Plenum Press, London, New York,

1977.

[3] V.F. Kiselyov, S.N. Kozlov, A. Zoteev, Bases of Physics of the Surface of Solids,

LAN, Petersburg, 1999, 284 pp. (in Russian).

[4] A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, sixth ed., Wiley, New

York, 1997, 800 pp.

[5] M. Prulton, Introduction to Surface Physics, Oxford University Press, New York, 1994.

[6] M. Tsukada, T. Hashina, J. Phys. Soc. Jpn. 51 (1982) 2562.

[7] G. Binnig, H Rohrer, Ch. Gerber, E. Weibel, Surf. Sci. 131 (1983) 1379.

[8] R.S. Becker, J.A. Golovchenko, E.G. McRue, D.S. Swartzentruber, Phys. Rev. Lett. 55

(1985) 2028.

[9] V. Heine, G. Schmidt, L. Hammer, Phys. Rev. B 32 (1985) 6214.

[10] R.I. Hamers, R.M. Tramp, J.E. Demuth, Phys. Rev. Lett. 56 (1986) 1972.

[11] M.W. Robinson, W.K Waskieviwicz, P.H. Fuoss, J.B. Stark, P.A. Bennett, Phys. Rev.

B 33 (1986) 7013.

[12] L.D. Landau, E.M. Lifshitz, Statistical Physics, Nauka, Moscow, 1964, 567 pp. (in

Russian).

[13] N. Osacabe, J. Tanishiro, K. Yagi, J. Honjo, Surf. Sci. 109 (1981) 353.

[14] R. Caehn (Ed.), Physical Metallurgy, vol. 2, North-Holland Physics, Amsterdam, 1968,

480 pp.

[15] J.C. Campuzano, M.S. Foster, J. Jennings, R.F. Willis, W. Unertl, Phys. Rev. Lett. 54

(1985) 2684.

























[16] A.M. Ovrutsky, Capillary and Adhesion Properties of Melts, Naukova Dumka, Kiev,

1987, 66 (in Russian).

[17] A.M Ovrutsky, Physics of Metals 10, N4, Naukova Dumka, Kiev, 1988, 109.

[18] H.H. Furel, G.A. Somorjai, Adv. Chem. Phys. 20 (1971) 215.

[19] H. Nada, Y. Furukava, Trans. Mat. Soc. Jpn. 16A (1994) 453.

[20] N.D. Mermin, Phys. Rev. B 19 (1968) 5194.

[21] D.R. Nelson, B.I. Halperin, Phys. Rev. 19 (1979) 2457.

[22] R. Caehn, P. Haasen (Eds.), Physical Metallurgy, vol. 3, North-Holland Physics,

Amsterdam, 1968.











5 Adsorption. The Gibbs AdsorptionEquation

5.1 Adsorption on Solid Surfaces

5.1.1 Physical and Chemical Adsorption. Different Types of AdsorptionIsotherms

Adsorption on solid bodies has a large practical value, because all adsorbents and

catalytic agents applied in the industry and chemical covers are solids. At the

beginning of the twentieth century, Langmuir fulfilled the first experimental

research of adsorption. This concerned in the core the questions of a sorption of

gases on the surfaces of solid bodies. The adsorption isotherms and isobars were

determined by measurement changes of gas volume after its passing near the sur-

face or by measurements of the body weight during experiment. Langmuir estab-

lished that the volume of adsorbed gas is proportional to the pressure at its small

values, and there is saturation at increasing of the pressure: gas ceases to adsorb

further. Langmuir offered the adsorption isotherm equation (5.1) deduced from the

condition of kinetic equilibrium, that is, the number of molecules, which evaporate

from the surface, is equal to that number of molecules, which join to it.

Estimations of a maximum quantity of molecules, which has joined the surface, tes-

tified that it is less than it is necessary for formation of the dense molecular layer.

Therefore, Langmuir supposed that the adsorption occurs on active sites and only

one molecular layer can be formed on the surface.

Later research fulfilled by many scientists [1] showed that adsorption depends

essentially on the chemical nature of gas and adsorbent and on the state of the

body surface. In the 1930s, many researchers established an existence of two types

of adsorption: the low-temperature adsorption and high-temperature adsorption

(chemisorption). For the first type, it is typical of the quick installation of equilib-

rium, low heats of activation (the work of desorption is less than half of the evapo-

ration heat for liquid state of the substance). These types of adsorption differ

distinctly in adsorption isobars, that is, in graphs of dependences of the surface part

coated with molecules on the temperature at constant pressure of gas (Figure 5.1).

Low-temperature adsorption is a physical adsorption. The adsorption is reducing

when increasing the temperature, because it becomes easier for molecules to over-

come an energy barrier connected with their interacting by a surface. Figure 5.1

shows the adsorption isobar. At the temperature 150�C, the majority of gas



http://dx.doi.org/10.1016/B978-0-12-420143-9.00005-3

molecules have left the surface already. However, the matter adsorbs again after

magnification of temperature. It is connected with the chemical reaction between

molecules of hydrogen and the surface. The reaction does not occur at low tem-

peratures because overcoming the energy barrier is necessary for its course. As a

result of reaction, the strong interacting of molecules with the surface is estab-

lished, and desorption becomes possible only at rather high temperatures.

The theoretical Langmuir isotherm does not present a variety of processes of the

adsorption found in experiments. Dependences of volume of adsorbed gas on pres-

sure can be different. Brunauer [2] considered that there are five basic types of

adsorption isotherm, shown in Figure 5.2. Type 1 is the Langmuir type. Type 2 is

the very common case of physical adsorption; it corresponds to multilayer forma-

tion [1]. In dependence of state of the surface, the types 3�5 of the adsorption

curves may also be realized.

It is natural that several theories have been offered, in which certain types of

isothermals were explained by the effect of certain factors. Some theories analyzed

a capillary condensation, that is, a condensation in pores of the adsorbent. This fac-

tor influences the form of isotherms at comparatively large pressures. The

Pollyanna theory of adsorption considers that the field of forces of adsorbent is

decreasing sluggishly from a surface and consequently an adsorption layer is multi-

molecular in thickness, and its density diminishes along a normal line from the sur-

face. The Parkinson�Yore theory took into account a change of state of adsorbed

molecules from the surface gas to the surface liquid. In the 1960s, de Bur explained

some types of adsorption isotherms from concepts about the possibility of squeez-

ing of a monomolecular layer, in which there is a transition from the surface gas to

the surface liquid. In 1929, de Bur and Cvikker offered the polarization theory

accepting that polarization of molecules promotes deposition of molecules of a fol-

lowing layer. De Bur came to the conclusion, based on consideration of experimen-

tal data, that Langmuir’s theory is applicable only for some cases of reversible

chemisorption with rather small heat of activation (�40,000 J/mol). However, the

Langmuir theory is important for understanding of other theories, and their results

in a case of low pressures should respond to Langmuir’s equation.

–200 0 200

V

400t, °C

Figure 5.1 The experimental isobar of

adsorption on the activated Fe-catalyst

for synthesis of ammonia.


5.1.2 Langmuir’s Equation

According to Langmuir’s theory, each active center of a surface has only the

molecular orb of action as a latent valency on the surface that defines the adsorp-

tion. From this point of view, adsorption is, in fact, the chemical process. In the

course of adsorption, the adsorbed molecules remain linked on the active centers

some time (the lifetime in adsorbed state), and then they come off again. Thus, bal-

anced state at adsorption is defined by equality of a velocity of condensation and

evaporation of molecules.

Let us designate the part θ of the adsorbent surface occupied by adsorbed atoms

or molecules, and the part 12 θ of the surface, which remains free. The θ value is

equal to the ratio of quantity of adsorbed molecules Γ to greatest possible quantity

Γmax at the complete filling-up of the surface, θ5Γ/Γmax. At a constant tempera-

ture, the number of molecule collisions with the surface is proportional to the pres-

sure P or volume concentration of molecules in gas or solute. The amount of

molecules, which falls from gas on the unit area of the surface during unit time, is

proportional to the concentration of molecules n and the mean-arithmetic velocity:

I5 nv=4:

As n5P=ðkTÞ; I5P=ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πmkT

p; where m is the mass of molecules, k is the

Boltzmann constant, and T is the temperature.

According to Langmuir’s opinion, the molecules form only one layer on the sur-

face when being adsorbed; therefore, they cannot be adsorbed on already occupied

parts of the surface. The adsorption velocity is proportional to the concentration

(pressure) and free part of the surface 12 θ.

I1 5K1Pð12 θÞ;

where K1 is the coefficient of proportionality, K1 5 1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πmkT

p. On the contrary,

desorption (or evaporation) molecules occur only from the occupied part of the sur-

face, and consequently, the velocity of this process I2 is proportional to θ, I2 5K2θ.The K2 value depends on the concentration of the surface atoms ns, frequencies

of their oscillations ν, Boltzmann factor k, and from the value of adsorption energy

V1 2 3 4 5

P0 P0 P0 P0 P0 P

Figure 5.2 Types of adsorption isotherms [2]. P0 is the saturation pressure over liquid of the

substance, which is adsorbed.

129Adsorption. The Gibbs Adsorption Equation

Ua: K25 ν exp(2Ua/kT)/a2, where 1/a2 is the number of surface atoms per unit

area (here it is accepted that all atomic places are the active centers of adsorption).

Equating the velocities of both processes I1 and I2, we will obtain for the equilib-

rium state:

K1Pð12 θÞ5K2θ;

or

θ5Γ

Γmax

5K1P

ðK1P1K2Þ5

bP

ð11 bPÞ ; ð5:1Þ

where b5K1/K2. It is Langmuir’s equation that is obtained for the surface with a uni-

form distribution of the homogeneous active centers in absence of interactions

between adsorbed molecules. It can be rewritten through the concentration of atoms:

Γ=Γmax 5bn

ð11 bnÞ ; ð5:2Þ

where Γ is the adsorption value, n is the concentration of molecules in gas

(P5 nkT), and b is the relative velocity of desorption and adsorption. This equation

is named the Langmuir isotherm (equation of the isothermal adsorption). At small

concentrations n (P,,P0, P0 is the saturation pressure for the matter in the liquid

state), the value of adsorption is proportional to concentration or pressure

(Figure 5.3), and at high concentrations (P.P0), the adsorption value comes nearer

to limiting value Γmax.

5.1.3 Model for the Computer Analysis of the Adsorption

We have a substrate of certain area; molecules can be joined to it. The place of

molecule falling or breakoff is chosen in a random way—a site of a two-

dimensional net with random coordinates. The algorithm for simulations by the

θ = ns/n

P

Figure 5.3 The Langmuir isotherm

curve.


Monte Carlo method consists of trials on breakoff or deposition of molecules

depending on whether the selected place is occupied or free. We will determine

pressure P01, at which the dynamic equilibrium between numbers of those mole-

cules, which are deposited, and those which are evaporated, takes place for each

active center on the surface. For this purpose, we will equate the numbers of mole-

cules, which fall on one empty seat with the area Sa5 a2 (the area occupied by one

adsorbed molecule) in a unit of time, and those which evaporated from the occu-

pied place:

I5P01Saffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πmkT

p 5 ν e2UakT ; P01 5

ν expð2Ua=kTÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πmkT

p

Sa:

If the value of pressure P is more than the pressure P01 (P.P01), we will spot

time τdep, during which one molecule will be on the average joined to the surface

(I5 1) from the condition:

I5PSaτdepffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πmkT

p 5 1; τdep 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πmkT

p

PSa:

Determine the probability of breakoff of a molecule in time τdep:

ωev 5 ν e2UakTτdep;

where Ua is the adsorption energy and ν is the vibration frequency of a particle on

the substrate. If the chosen place is free, a molecule will be attached. If the place is

occupied, the test on breakoff will be carried out. The random number [0:1] is gen-

erated for this purpose; if it is less than ωev value, a molecule will be evaporated

(P.P01).

If (P,P01), less than half of places will be filled by adsorbed molecules when

coming nearer to equilibrium. We will determine time τev, during which the mole-

cule will be evaporated from the substrate with probability equal to 1:

ωev 5 ντev expð�Ua=ðkTÞÞ 5 1; τev 5 expðUa=ðkTÞÞ=ν;

and also the probability of deposition of a molecule for this time τev:

ωdep 5Pτev Saffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πmkT

p :

If the place has been occupied, it is released. The random number [0:1] is gener-

ated for case of the empty place; if ωdep is more than this number the molecule will

join to the surface (P,P01).

The breakoff or deposition of atoms is shown on the screen with the help of pro-

cedure “Show” (see Section 9.7). Hence, it is possible to observe a course of


adsorption process on the screen. Equilibrium between the numbers of atoms,

which have joined and those which have come off, is established after the suffi-

ciently large number of trials.

We find magnitude θ after a certain number of the Monte Carlo steps. Then we

change pressure, observe the process, and spot new value θ. Iterating the experi-

ment for different pressures, we spot the adsorption isotherm. In the program given

in Section 9.7, the change of pressure and buildup of the graph of the adsorption

isotherm is provided in a cycle on values of the pressure.

5.1.4 The BET Isotherm of the Multimolecular Adsorption

If there is an attraction of molecules to the surface coated with an adsorbate, the

multimolecular adsorption is possible. First, Brunauer, Emmett, and Teller (BET)

[3] have offered the improved analysis and obtained the isotherm equation of the

multimolecular adsorption. They have supposed that the interaction energy of mole-

cules with a substrate is constant within the first layer, and the interaction energy of

molecules from the next layers with molecules from the previous layers, which are

closer to the substrate, has another value, but the same for all next layers.

Let us designate: Q1 is the part of the surface coated with a layer of molecules

with the thickness equaled to their size; Q2 is the part of the surface on which there

are two layers of molecules, and so on. Apparently, the number of adsorbed mole-

cules of molecules

Na 5NQ1 1 2NQ2 1 3NQ3 1?1 iNQi 1?;

where N is the number of molecules in the filled monolayer (per unit area). The

total number of adsorbed molecules can be defined as Na 5NPN

i51 iQi and a part

of the free surface Q0 is equal to 12P

Qi.

A conservation condition for Q0 takes place in equilibrium:

nV

4Q0 5 νQ0 5 fNQ1;

where ν is the intensity of deposition of molecules and f is the frequency of break-

off of molecules from the first layer, if there are no molecules over them.

The condition of conservation of the first monolayer filling is

νQ0 1 f1NQ2 5 fNQ1 1 νQ1:

Taking into account the condition of conservation of the free part Q0 of the sur-

face, νQ05 fNQ1, we will obtain:

νQ1 5 f1NQ2:


Writing down in the same way the equilibrium conditions for all layers with

consideration of equilibrium for the previous layer, we will come to analogous

relationships:

νQi21 5 fi21NQi:

Let us enter a designation τi5 1/fi; it is the lifetime in an adsorbed state in the

i-layer (for the first layer τ5 1/f). We have a set of equations:

NQ1 5 νQ0τNQ2 5 νQ1τ1^NQi 5 νQi21τi21:

8>><>>:

BET supposed that the lifetime in an adsorbed state does not depend on the

layer number (starting with second layer), τ1 5 τ2 5?5 τi 5?.

Then,

Q2 5 xQ1

Q3 5 xQ2 5 x2Q1; x5ντ1N

^Qi 5 xi21Q1

8>>>><>>>>:

Besides, Q1 5 ðντ=NÞQ0 5 ðxτ=τ1ÞQ0.

For the total number of adsorbed molecules,

Na 5NXNi51

iQi 5XNi51

ixi21Q1 5Nττ1

Q0

XNi51

ixi;

Q0 5 12XNi51

Qi 5 12ττ1

Q0

XNi51

xi:

8>>>><>>>>:

From here,

Q0 5 11ττ1

XNi51

xi

!21

:

We will designate

k5ττ1

;


then

Na 5

NkPNi51

ixi

11 kPNi51

xi5

Nkxðd=dxÞPNi51

xi

11 kPNi51

xi:

Having written down sums of the geometrical progressions, we will obtain the

equation

Na 5Nkx

ð12 xÞð12 x1 kxÞ ;

ν5nV

45

Pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πμRT

p 5βP.x5βτ1N

P; kx5βτN

P:

Rewrite the last equality in the form:

Na

N5Q5

kx

ð12 xÞð12 x1 kxÞ : ð5:3Þ

If x{1; Q5 ðkx=ð11 kxÞÞ, that is, we have Langmuir’s equation; if

x� 0.5�1.0, kxD(12 x1 kx) and Q � ð1=ð12 xÞÞ.Thus, the equation of BET allows obtaining the different types of isotherms

depending on the parameter k, which value is connected with the physicochemical

properties of the surface and gas. However, it is impossible to explain all variety of

isotherms within the limits of the simplified model.

The magnitude ðN=βτ1Þ5 q has the dimension of pressure, x5 ðP=qÞ. If a point

P5 q is reached at increase of P, then x5 1. Thus, NA ! N; that means the vapor

is saturated (a fluid on the substrate) and q is the meaningful saturation pressure P0

for the adsorbed gas at the certain temperature. For pressures smaller than q,

x5 ðP=qÞ, 1, it is possible to calculate the number of adsorbed molecules under

the following equation:

Na 5NkP

ðq2PÞð11 ðk2 1ÞðP=qÞÞ ; ð5:4Þ

where N is the number of the adsorption “centers.” Having designated a volume of

the precipitated molecules as V 5NaΩ, and a monolayer volume as Vm 5NΩ, wewill come to expression for the deposited volume

V 5kPVm

ðP0 2PÞð11 ðk2 1ÞðP=P0ÞÞ: ð5:5Þ


The precipitated volume can be found experimentally from measurements of

product PV of gas before and after its contact with the surface. We will rewrite

Eq. (5.5) as follows:

P

VðP0 2PÞ 51

kVm

1k2 1

kVm

UP

P0

:

It is possible to find the magnitude Vm from this equation, and, hence, the area

of the matter, which is used as an adsorbent. The graph of the left part of the last

equation in dependence on the magnitude P/P0 must be built for its determination.

Such a technique of definition of the area has been used until now.

5.2 The Gibbs Adsorption Equation

5.2.1 The Physical Phase Boundary

The object of interfacial phenomena study is the anisotropic and inhomogeneous in

one direction zone of the continuous modification of local properties; it is disposed

between contiguous phases. Gibbs termed this zone a discontinuity surface. The

term “surface layer” suits for the case when a density of one of phases is much less

than the density of another, for example, the boundary surfaces between liquid and

gas or solid and gas. It is preferentially applying the term “interfacial layer” at con-

sideration of the boundaries between solid and liquid or solid and solid phases of

one matter.

The local density varies in the interfacial layer. In the homogeneous phases αand β the density is invariable (Figure 5.4). Let z-axis be normal to the surface.

The dependence of a local property on the coordinate in the interfacial layer is

termed a property profile. Boundaries of the superficial layer (zα and zβ) should be

out of zone of an essential change of the yielded property (for more details, see

Ref. [4]). Generally speaking, an acting of one phase is passed around to all other

phases; therefore, it is necessary to speak about effective thickness of the interfacial

layer, outside of which deviation of local properties from their volume values

become insignificant. However, a question on delimitation and, consequently, the

effective thickness of the surface layer remains debatable [5].

The following asymptotic equation, which presents a density modification in the

interfacial layer, was obtained by methods of statistical physics in Ref. [6] for liq-

uid with the van der Waals interacting between molecules:

ρ5 ρ0 1πρ20χ0ða0ρ0 � aρ0Þ=ð6 �z3Þ; ð5:6Þ

where ρ0 and χ0 are the density and isothermal compressibility of single liquid

accordingly, a and a0 are the van der Waals constants of interactions between mole-

cules of liquid with each other and with molecules of other phase accordingly, ρ0 is


the density of other phase, and z is the distance from the boundary surface.

Calculations under this equation have shown that the density odds of the surface

layer and the volume phase at distances of order of several intermolecular distances

in liquid become significant. Analogous expressions are gained for a tangential

component of the pressure tensor; it is found that the pressure varies more slowly

than the density. From here it follows that the effective thickness of the interfacial

layer should be determined for a certain property and, as a whole, it will differ for

different properties.

The dynamic nature of equilibrium in the interfacial layer is not taken into con-

sideration at its thermodynamical study. However, molecules in the interfacial

layer, especially if one of the phases is fluid, are in constant movement. If phases

are in a thermodynamic equilibrium, there is a balanced passing of molecules from

both phases into the surface layer and back. Calculations show that each square

centimeter (cm2) of a water plane every second accepts and returns in a vapor

phase of 1.23 1022 molecules at middle temperatures. Analogously, an intensive

interchanging of molecules between the interfacial layer and the liquid volume

phase occurs because of heat motion. The surface molecules between two solid

phases are less mobile in this sense.

In a many-component system, the profiles of local concentration of different

components can differ essentially. As an example, we will consider how local con-

centrations of components vary in the ternary two-phase system. It is obvious that

the magnitude of local concentration depends on the coordinate z along the normal

to the surface (Figure 5.5). In the thermodynamic equilibrium state, the average

ρρα

ρ β

δz

z α0 zz β

Figure 5.4 The density profile in the

interfacial layer δz.

n

n3α

n3β

n1β

n2β

n2(z) n1(z)

n3(z)

n1α

n2α

z

Figure 5.5 Possible profiles of local

concentration in a ternary system.


number of molecules of any kind in the unit of volume of homogeneous phases

remains a stationary value. Any property changes its value at moving from one

phase to another in the interfacial layer from the value of this property in one

phase, we will call α, to the value in other phase, β. It is possible to differ condi-

tionally the profiles of local concentration shown in Figure 5.5 among three groups

of functions:

1. the monotonic function without an extremum with one point of inflexion (n1(z));

2. the function with a maximum and with two points of inflexion (n2(z));

3. the function with a minimum and with two points of inflexion (n3(z)).

In the second case, the concentration of the component is higher than a mean of

volume concentrations, and it is termed the surface active component. If the com-

ponent content in interfacial layer is less than the mean value (the third case), it

will be termed the inactive component.

5.2.2 The Elementary Strain Energy: Interfacial Tension

In the case of the absence of acting external fields, an internal energy of two-phase

many-component open system

dU5 T dS1 dAd 1Xi

μi dNi;

where Ad is the elementary energy of deformation, which is carried out over the

system by external forces.

Find the expression for an energy of deformation of flat surface layer in thick-

ness Z and the area ω. We enter the Cartesian coordinates with the z-axis, normal

to the surface and plane xy lying in the surface layer. In a general view, the energy

of deformation appears thus:

dAd 52

ðððV

δV 0Xj;k

Pj;kεj;k; ð5:7Þ

where Pj,k and εj,k are the components of tensors of pressure and strain accordingly.

It follows from symmetry conditions in absence of external fields and at large

extension of an isotropic (liquid) layer:

Px;y 5Py;x 5Pz;x 5Px;z 5Pz;y 5Py;z 5 0; Px;x 5Py;y 5Pτ ; Pz;z 5Pn;

where Pn is the normal pressure that has a stationary value for all coexistent phases

in equilibrium and Pτ is the tangential component of pressure that depends on the

coordinate z normal to the surface.

If electrical and magnetic fields are present, it will be necessary to introduce

additional terms into expression for work and, accordingly, electrical and magnetic


interfacial tension. Taking into account the relationships of symmetry expression,

Eq. (5.7) can be rewritten as follows:

dAd 52ÐÐÐ

VδV 0½Pτðεx;x 1 εy;yÞ1Pnεz;z�5

2ÐÐÐ

VδV 0½Pnðεx;x 1 εy;y 1 εz;zÞ2 ðPn 2PτÞðεx;x 1 εy;yÞ�:

It is easy to understand that εx,x1 εy,y1 εz,z is the relative change in volume, and

εx,x1 εy,y is the relative change in surface area. In the right part of the previous

equation, the normal pressure may be excluded, and the integral by volume (the first

term in the integral) will give its increase dV. It is possible to exclude the area dωin the second integral (the second term) as Pn2Pτ depends only on z. Therefore,

dAd 5 �P dV 1 γ dω;

where the designation is entered

γ5ðZ0

ðPn 2PτðzÞÞðεx;x 1 εy;yÞdz:

As in the homogeneous phases Pn2Pτ5 0, integration limits can be changed in

the last expression. Then we will obtain for the two-phase system:

γ5ðN2N

ðPn 2PτðzÞÞðεx;x 1 εy;yÞdz: ð5:8Þ

Magnitude γ is the interfacial tension. Apparently, from the last equation, the

interfacial tension is caused by infraction of the Pascal law in the interfacial layer

(Pn 6¼Pτ). Thus, interfacial tension of flat surface is determined by expression

(5.8), from which it is clearly seen that this is the integrated magnitude. Equation

(5.8) can be applied to the cases when other forces (magnetic, electrical, and so on)

act on the system. Corresponding components will appear in the tensor of pressure;

thus, it will yield magnetic or electrical interfacial tension.

Considering Eq. (5.8), we will write a constitutive equation of thermodynamics

for the two-phase open system in the following form:

dU5 T dS2Pn dV 1 γ dω1Xi

μi dNi: ð5:9Þ

5.2.3 The Gibbs Method in Thermodynamics of Surface Phenomena

The thermodynamic theory of surface phenomena created by Gibbs more than 100

years ago is based on such basic concepts as an interfacial layer (a real separating sur-

face as it was termed by Gibbs), the system of comparison, which is idealized so that

it contains only a geometric separating surface, and excess extensive magnitudes.


Consider a multiphase, many-component system with the flat interface (the

interfacial layer) with the area ω and thickness Z (Figure 5.6A) which is in a ther-

modynamic equilibrium state. The interface is characterized by a gradient of prop-

erties, along a normal line to it, and cannot be considered as a self-contained phase,

as it cannot exist without coexisting phases. Application of the thermodynamic

laws to this surface becomes possible if it enters conception of the idealized ther-

modynamic system of comparison with the geometric separating surface.

In a letter to the American Academy, written concerning the awarding to him of

Rumford’s grant, Gibbs wrote: “One of primal problems theoretical research in any

field of knowledge is statement of such point of view from which an object of

researches is manifested with the greatest simplicity.” Such a point of view in the

theory of capillarity of Gibbs is the concept about the separating surface [4].

The separating surface is a geometrical surface, which has no thickness

(Figure 5.6B), mentally placed between coexisting phases parallel with the disconti-

nuity surface (Figure 5.6A), in such a manner that it reproduces the shape of the

last. This notion is important in the capillarity theory. The concept of a thermody-

namic system of comparison, that is, an idealized one, is directly connected with

the notion about the separating surface. In the Gibbs theory of capillarity, it deter-

mines the thermodynamic system with the volume and shape, which are the same as

in the real system; however, properties of phases (density, concentration of compo-

nents, entropy, and energy in unit volume) are constant up to the separating surface.

Thus, between phases, there is the geometrical separating surface instead of non-

uniform interfacial layer in the comparison system, and properties vary abruptly in

mathematical sense at passing from one phase to another. For determinacy, it is

accepted that the volume of the system of comparison remains equal to the volume

of the real system, even in the case of strain of the last. Thus, the comparison sys-

tem is chosen in such a manner that the relationship Vα1Vβ5V is valid at any

modifications in real system. Here, Vα and Vβ are the volumes of phases in the sys-

tem of the comparison, the values of which depend on the position of the separating

surface; V is the volume of the real two-phase system.

β β

ω

ω

αα

0

z

z(A) (B) Figure 5.6 The schemes of the interphase boundary

in the real system (A) and in the system of

comparison (B). The phases α and β are separated

by transition layer in the real system; ω is the area of

the interphase boundary.


Thus, the matching system in the Gibbs theory of capillarity is the idealized

thermodynamic system, the properties of which are compared with properties of a

real system. The volume and shape of the matching system (of the system of com-

parison) are the same, as the real system has. However, the density of extensive

properties (a concentration of components, mass density, entropy, energy) in it is

constant up to the chosen separating surface. The two phases contact, but the inter-

facial layer is absent (Figure 5.6B). It is obvious that properties of phases cannot

remain to be stationary values up to the separating ideal surface mentally spent by

researcher in the area of heterogeneity in any real system with the interfacial layer.

Therefore, values of extensive properties of real system will always differ from

properties in the matching system. The Gibbs theory characterizes this difference

quantitatively by excess values, which represent differences between corresponding

extensive values in these systems.

Consider the two-phase many-component system with the interfacial layer

between phases. Each phase is characterized by its own number of components.

We will consider further the phases α and β with interfacial layer between them.

Write at first the fundamental equation of thermodynamics for two-phase system as

a whole for area, which contains the transition layer (from 0 to Z) and the layers of

contacting phases (Figure 5.6A):

dU5 T dS2Pn dV 1 γ dω1Xi;φ

μφi dN

φi ; ð5:10Þ

where the index φ takes over the value α or β (under conditions of thermodynamic

equilibrium μαi 2μβ

i ). The differential equation (5.10) has that useful property that

is equally applicable both to all system as a whole, and to its any part. We will

write similar expressions for phases α and β in the matching system:

dUα 5 T dSα 2Pn dVα 1

Xi;φ

μαi dNα

i ; ð5:11Þ

dUβ 5 T dSβ 2Pn dVβ 1

Xi;φ

μβi dN

βi : ð5:12Þ

Find the quantitative difference between the real system and the idealized sys-

tem of matching. For this purpose, we will deduct from Eq. (5.10) the expressions

(5.11) and (5.12). Taking into account a condition d(V2Vα2Vβ)5 0, which fol-

lows from equality V5Vα1Vβ, we have:

dðU2Uα 2UβÞ5 T dðS2 Sα 2 SβÞ1 γ dω1Xi

μi dðNi 2Nαi 2N

βi Þ:

Introducing denotations:

Uω 5U � Uα � Uβ; Sω 5 S� Sα � Sβ; Nω 5N � Nα � Nβ ;


we will write down:

dUω 5 T dSω 1 γ dω1Xi

μi dNωi ; ð5:13Þ

where Uω, Sω, and Nωi are the excess values, which characterize distinction between

the real system and the matching system. Their values depend on position of the sepa-

rating surface. It follows from the received equations that if U is characteristic func-

tion of variables S, V, Ni, the excess internal energy Uω is characteristic function of

new variables Sω, ω, Nωi . Therefore, no less than in the case of bulk properties, inten-

sive properties can be expressed through derivatives of parts on extensive parameters:

T 5 ð@Uω=@SωÞω;Njðj 6¼iÞ;

γ5 ð@Uω=@ωÞSω ;Njðj6¼iÞ;

μi 5 ð@Uω=@Nωi ÞSω ;Njðj 6¼iÞ;

and the corresponding Maxwell relations are valid.

As the internal energy U and the excess internal energy Uω are homogeneous

functions of the first degree concerning extensive parameters Sω, ω, Nωi , it follows

from Euler’s theorem:

Uω 5 TSω 1 γω1Xi

μiNωi : ð5:14Þ

Formally writing a total differential dUω and comparing the received expression

with Eq. (5.13), we come to the Gibbs�Duhem equation

Sω dT 1ω dγ1Xi

Nωi dμi 5 0: ð5:15Þ

Write down the expressions for others excess potentials:

Fω 5Uω 2 TSω 5 γω1Xi

μiNωi ;

Hω 5Uω 1PVω 5 TSω 1 γω1Xi

μiNωi 5Uω ðVω 5 0Þ;

Gω 5Hω 2 TSω 5 γω1Xi

μiNωi 5Fω:

As we see, expressions for the excess free energies of Helmholtz and Gibbs are

identical, also for the excess enthalpy and internal energy. No less than thermody-

namic potentials of volume phases, excess thermodynamic potentials are connected


with each other by the fundamental equations and, therefore, they can be consid-

ered as if they determine the certain phase.

Then write the equations for the excess interior energy and excess free surface

energy in counting per unit area:

uω 5 TSω 1 γ1Xi

μiΓωi ; ð5:16Þ

σ5 f ω 5 γ1Xi

μiΓωi 5 uω 2 Tsω; ð5:17Þ

besides,

dσ5 duω 2 dðTsωÞ52sω dT 1 γ dω=ω1Xi

μi dΓi:

Here Γi 5Nωi =ω5 ðNi 2Nα

i 2Nβi Þ=ω is the adsorption value, that is, integral

excess of ith component in comparison with matching system counting per unit

area. The free surface energy σ is equal to interfacial tension γ in the case of one-

component systems; the equality will take place also in the case of a many-

component system ifP

iμiΓi 5 0—the equimolar surface. For such surface, the

Gibbs�Gelmgoltz relation looks like for pure substance as follows:

uω 5 σ1 Tsω 5σ2 Τð@σ=@TÞω; ð5:18Þ

as sω5 (@σ/@T)ω51.

It follows from the above-stated that interfacial tension γ (Eq. (5.17)) is neither

the internal energy nor the potential energy per unit of area of the interfacial layer.

Adsorption Equation

Intensive properties of volume phases are connected between themselves by the

known Gibbs�Duhem equation

Sα dT 2Vα dP1Xi

Nαi dμα

i 5 0:

The similar equation (5.15) is obtained for the excess internal energy of the

interfacial layer. We will divide this equation by the area ω and rewrite as follows:

dγ52sω dT 2Xi

Γi dμi: ð5:19Þ

This Gibbs�Duhem equation for a excess internal energy is named the Gibbs

adsorption equation (5.17). It is obvious that

Γi 52ð@γ=@μiÞT : ð5:20Þ


To find the odds of any property X in the real system and in the system of com-

paring (Xmatch), it is necessary to consider volumes of the α and β phases:

X � Xmatch 5X � xαVα � xβVβ ;

where xα and xβ are its specific values. As V5Vα1Vβ, we can rewrite the last

equation as follows:

X � Xmatch 5X � xβV � ðxα � xβÞVα:

Thus, excess values depend on the volume of the α phase, that is, on disposition

of the interlayer in the system of comparing. It is necessary to analyze this depen-

dence to understand better the Gibbs theory.

5.2.4 Different Ways of Choice of the Separating Surface

The Dependence of the Excess Thermodynamic Values on the Position of theSeparating Surface

Unlike thermodynamic functions of volume phases, the excess thermodynamic func-

tions depend on the position of the dividing surface in the matching system.

However, despite the dependence of the excess functions on position of the separating

surface, the common form of the thermodynamic equations obtained above, except

for several special cases, is invariant relative displacements of the dividing surface.

Any excess extensive thermodynamic value can be presented in the following

form:

Xω 5 ðX � Xα � XβÞ=ω:

Besides, a density of this property xα5Xα/Vα, xβ5Xβ/Vβ in the system of

matching is remained invariable up to the dividing surface. The following expres-

sion is valid for any extensive property of the real two-phase system:

X5ωðzβzαxðzÞdz;

where zα, zβ are the coordinates of phase boundaries and x (z) is the function which

yields dependence of a local density of this property on the normal coordinate to

the surface z. In particular, it can be written for the total number of particles in the

binary system:

N5ωðzβzαnðzÞdz:


We will write for adsorption of ith component

Γi 5Nωi =ω5

ð0zα½niðzÞ2 nαi �dz1

ðzβ0

½niðzÞ2 nβi �dz;

and for any reduced excess thermodynamic value

Xωðz5 0Þ5ð0zα½xðzÞ2 xα�dz1

ðzβ0

½xðzÞ2 xβ�dz: ð5:21Þ

Expression (5.21) is written for the case when the geometric separating surface

is in the position with the coordinate z5 0. We will move mentally this surface on

distance Δz toward a phase β (Figure 5.6). Then the expression (5.21) for excess

property concerning new position of the separating surface will be rewritten as:

Xωðz5ΔzÞ5ðΔz

zα½xðzÞ2 xα�dz1

ðzβΔz

½xðzÞ2 xβ�dz:

The expression obtained can be transformed easily:

Xωz5Δz 5

ð0zα½xðzÞ2 xα�dz1

ðzβ0

½xðzÞ2 xβ�dz1ðΔz

0

½ðxðzÞ2 xαÞ2 ðxðzÞ2 xβÞ�dz

or

Xωz5Δz 5Xω

z50 2 ðxα 2 xβÞΔz: ð5:22Þ

Thus, the dependence of excess thermodynamic properties on the normal coordi-

nate z in the case of the flat separating surface is linear. It allows disposing the sep-

arating surface concerning any excess property, depending on the considered

problem.

Choice of the Separating Surface

Practically valuable and the most important outcome of the Gibbs thermodynamic

theory is gained above the adsorption equation (5.20). As shown above, sω and Γi

are the linear functions of the normal coordinate z that characterizes a position of

the geometric separating surface between coexisting phases. Figure 5.7 shows the

graphs of adsorption of components in binary system. It is clear from this drawing

that the magnitude of the adsorption depends on where we will choose the separat-

ing surface. Hence, it is senseless to speak about magnitude of adsorption while the

position of the separating surface is not fixed unequivocally. From the mathemati-

cal point of view, the position of the separating surface can be any, that is,


Eq. (5.22) is valid at any position of the separating surface. However, as Gibbs

noted, from the physical point of view, it is more convenient to have dividing sur-

face in the area of inhomogeneity, that is, in the discontinuity area, or not so far off

from it. The reasonableness of such choice specifies also that interfacial tension is

independent of the separating surface disposition.

Gibbs has suggested computing adsorption of components concerning the sepa-

rating surface chosen so that adsorption of one of components was equated to null,

Γj5 0. Adsorption of components relative to this surface is termed the Gibbs rela-

tive adsorption, and it is designated Γi(j). The index j underlines that adsorption of

the jth component on this surface is equal to null.

Later, analyzing the adsorption phenomena in a liquid solution, Guggenheim

and Adam offered a different way for choosing the separating surface. Apparently,

from Figure 5.7, it is always possible to find such position of the separating surface

near to the discontinuity surface so that the conditionPk

i51 Γi 5 0 will be satisfied.

Adsorption relatively of such surface is termed the Guggenheim�Adam N-variant

of adsorption in the literature; it is designated Γi(n) (see Figure 5.7). Guggenheim

and Adam also offered other ways of defining the separating surface [2]. In particu-

lar, it is offered to compute adsorption concerning the separating surfaces, which

corresponds to the conditionPk

i51 Γiμi 5 0 orPk

i51 ΓiVi 5 0, where μi is the molar

mass and Vi is the partial molar volume of ith component. The variantPki51 Γiμi 5 0 would be convenient for many problems; however, as a rule, abso-

lute values of μi are unknown. Variants of definition of a point z5 0 such that

uω5 0 or sω5 0 were offered too. However, the most convenient choice of the sep-

arating surface position for consideration of the adsorption in the equilibrium multi-

phase system is the variant offered by Gibbs, as it is the very best for calculation of

the magnitudes Γi(j).

5.2.5 Adsorption Equilibrium in Multi-Component Systems

If the phases α and β separated by flat boundary layer are in a thermodynamic equi-

librium state, the following conditions will be satisfied: Pα5Pβ, Tα5 Tβ, μαi 5μβ

i .

Γ1(2)

Γ

Γ1(n)

Γ2(n)

Γ2(1) Γ2

Γ1

Γ1 = 0

Γ2 = 0

Γ1 = –Γ2

z0

Figure 5.7 The different positions of the

separating surface used for calculation of

adsorption.


The following relationships are also valid at small shifts along the phase equilib-

rium curves:

dPα 5 dPβ; dTα 5 dTβ; dμαi 5 dμβ

i : ð5:23Þ

According to the relationships by Gibbs�Duhem, we will write for each phase:

dPα 5 sα dTα 1Xki51

nαi dμαi ;

dPβ 5 sβ dTβ 1Xki51

nβi dμβ

i :

Based on these relationships, we write the Gibbs�Duhem relationship for the

equilibrium two-phase system

ðsα 2 sβÞdT 1Xki51

ðnαi 2 nβi Þdμi 5 0: ð5:24Þ

Thus, it is possible to consider together Eqs. (5.19) and (5.24) for an equilibrium

two-phase system,

2dμj 5 dTðsα 2 sβÞ=ðnαj 2 nβj Þ1

Xki51

ðnαi 2 nβi Þdμi=ðnαj 2 n

βj Þ: ð5:25Þ

After substitution of Eq. (5.25) in Eq. (5.19), we will gain:

2dγ5 sω 2Γj

sα 2 sβ

nαj 2 nβj

" #dT 1

Xki51

Γi 2Γj

nαi 2 nβi

nαj 2 nβj

" #dμi; ð5:26Þ

where dγ is a total differential in the gained expression. Hence, coefficients at dT

and dμi in this equation are a partial derivative on temperature and a chemical

potential:

sω 2Γj

sα 2 sβ

nαj 2 nβj

52 ð@γ=@TÞμi; ð5:27Þ

Γi 2Γj

nαi 2 nβi

nαj 2 nβj

52 ðdγ=dμiÞT ;μjði 6¼jÞ: ð5:28Þ

As shown above, interfacial tension of a flat interfacial layer does not depend on

the position of the separating surface; hence, derivatives (5.27) and (5.28) should

not depend on the position of the separating surface also. It is easy to prove that


numerical values of coefficients at dT and dμi in Eq. (5.26) coincide with the

values of corresponding magnitudes concerning the geometrical separating surface,

which satisfies the condition Γj5 0. Really, Γjð1=ðnαj 2 nβj ÞÞ5 zðΓj50Þ is the distance

from the chosen position of the separating surface to the position of the surface,

which corresponds to the condition Γj5 0. Then,

Γi 2Γj

nαi 2 nβi

nαj 2 nβj

5Γi 2 zðΓj50Þðnαi 2 nβi Þ5ΓiðjÞ: ð5:29Þ

Analogously,

sω 2Γj

sα 2 sβ

nαj 2 nβj

5 sω 2 zðΓj50Þðnαi 2 nβi Þ5 sωðjÞ: ð5:30Þ

Hence, a choice at the description of the adsorption processes of such geometri-

cal separating surface, for which the adsorption of the certain (jth) component

Γj5 0, is equivalent to taking into account of equilibrium conditions for the interfa-

cial layer and coexisting phases. It is essentially that derivatives from γ in

Eqs. (5.27) and (5.28), which can be found from experimental data, allow finding

the relative values sω and Γi (j). Other separating surfaces adsorption can be com-

puted according to Eq. (5.22).

Adsorption Calculation for Binary Systems

Simple object for application of the Gibbs adsorption equation is the two-phase

binary system, for which the relationship takes place:

2dγ5 sω 2Γ2

sα 2 sβ

nα2 2 nβ2

" #dT 1 Γ1 2Γ2

nα1 2 nβ1

nα2 2 nβ2

" #dμ1 ð5:31Þ

or

2dγ5 sωð2Þ dT 1Γ1ð2Þ:

We will obtain accordingly

sωð2Þ 52 ðdγ=dTÞμ1; Γ1ð2Þ 52 ðdγ=dμ1ÞT :

If attempting to find experimentally or theoretically the temperature dependence

of the surface tension (also on the chemical potential), it is possible to determine

from the last relationships the excess entropy density and excess number of parti-

cles. According to entered terminology, the surface entropy and the Gibbs relative

adsorption may be determined.


As a major variable in the gained expressions is the chemical potential. It is not

entirely convenient, because the chemical potential directly in experiments is not

determined. Therefore, we will express the chemical potential (of a mole fraction)

through the thermodynamic activity coefficients. For this purpose, we will take

advantage of the known expression for the chemical potential:

μi 5μi0 1RT ln ai 5μi0 1RT ln fixi;

where ai5 fixi is the thermodynamic activity and fi is the thermodynamic activity

coefficient. It is obvious that

dμi 5RT ln ai 5RTðdai=aiÞ

at T5Const. Then for adsorption, according to Gibbs,

Γ1ð2Þ 52 ðdγ=dμ1ÞT 52a1

RTð@γ=@a1ÞT 52

f1x1

RTð@γ=@ðf1x1ÞÞT :

Taking into account that da15 (x1(@f1/@x1)1 f1)dx1, write down for the adsorption

Γ1ð2Þ 52f1x1

RTðx1ð@f1=@x1Þ1 f1Þ@γ@x1

� �T

: ð5:32Þ

For a feeble solute, close to ideal, f1D1; therefore,

Γ1ð2Þ 52x1

RT

@γ@x1

� �T

: ð5:33Þ

This expression is given in many textbooks with the simplified description of

adsorption.

Consider the surface characteristics of the binary melt system thallium�bismuth

[7]. Figure 5.8 shows the interfacial tension and bismuth adsorption at the melt-gas

450(A) (B)σ, mJ/m2

430

410

390

370

T1 0.2

1

2

0.4 0.6 0.8 at. % Bi T1 0.2 0.4 0.6 0.8 at. % Bi

ΓBi ΓBi(T1)ΓBi(n)

Figure 5.8 Interfacial tension isotherms and the adsorption in the system Tl�Bi: (A) curve

1—623 K, curve 2—773 K; (B) 623 K, ΓBi(Tl)5Γi(j), Γ(n)—n-variant of adsorption.


boundary surface in dependence on Bi concentration. Adsorption was determined,

according to Eq. (5.32), for two positions of the separating surface—at first,

according to the Gibbs theory (proportionally to the derivative ((dγ/dxBi)T)), thenfor equimolar surfaces (n-variant), according to Guggenheim and Adam.

Apparently, from the drawing, the bismuth adsorption is positive at all concentra-

tions. Small additives of bismuth result in magnification of its adsorption. The depen-

dences of adsorption on the concentration, according to Gibbs and Guggenheim with

Adam, differ essentially at the significant concentrations of bismuth.

References

[1] A.W. Adamson, A.P. Cast, Physical Chemistry of Surfaces, sixth ed., Wiley

Interscience, New York, Toronto, 1997.

[2] S. Brunauer, The Adsorption of Gases and Vapors, vol. 1, Princeton University Press,

Princeton, New York, 1945.

[3] S. Brunauer, P.H. Emmett, E. Teller, Adsorption of gases in multimolecular layers.

J. Am. Chem. Soc. 60 (1938) 309.

[4] A.I. Rusanov, The Phase Equilibrium and Surface Phenomena, Chemistry, Leningrad,

1967 (in Russian).

[5] R.H. Dadashev, Thermodynamics of Surface Phenomena, Checheno-Ingush State

University, Grozny, 1988 (in Russian).

[6] F.M. Kuni, A.I. Rusanov, Asymptotic of the molecular distribution functions in the sur-

face layer of liquid. Rep. Acad. Sci. USSR 174 (1967) 406.

[7] V.I. Nizhenko, L.I. Flock, Interfacial Tension of Rare Metals and Alloys: The

Handbook, Metallurgiya, Moscow, 1981, 340 p.














6 Simulation Techniques for AtomicSystems

6.1 Nonclassical Potentials of Atomic Interaction

6.1.1 The Empirical Pseudopotential Method

Necessity of application of potentials of interatomic interaction arises at buildup of

models of substance by the molecular dynamic (MD) and Monte Carlo (MC) meth-

ods. Classical effective potentials reduce the quantum-mechanical interactions of

electrons and nuclei in a solid to an effective interaction between atom cores. This

greatly reduces the computational effort in MD simulations; classical MD calcula-

tions with many millions of atoms are routinely performed. Such system sizes are

possible because molecular dynamics with short-range interactions scales linearly

with the number of atoms. Moreover, it can easily be parallelized using a geometri-

cal domain decomposition scheme [1,2], thereby achieving linear scaling also in

the number of CPUs.

Some potentials of interatomic interaction for modeling crystals of noble gases

(the Lennard�Jones pair potential) and semiconducting substances Si, C, Ge (the

three-particle potentials of the Stillinger�Weber and Tersoff) were considered in

Section 1.2.5. They do not ensure the correct values of many physical properties

simultaneously in the case of modeling of metals. First of all, it concerns the melt-

ing heat and evaporating heat. The low values of the evaporating heat lead at

modeling to incorrect structure of the boundaries of condensed matters with gas or

vacuum. It is known for a solid described by a purely pairwise interaction that the

elastic stiffness constants C12 and C44 are equal. This is known as the Cauchy rela-

tion; but these constants differ roughly two times for FCC metals. In a solid with

purely pairwise interactions, the ratio of vacancy formation energy (Evf) to cohe-

sive energy (Ecoh) is once again 1:0. The metals have values closer to 0:35, again

significantly deviating from pairwise bonding. The ratio between cohesive energy

and melting temperature for FCC metals differs significantly from a strictly pair-

wise interaction model.

Accurate description of metallic bonding requires the consideration of coordina-

tion dependent bonding, but at the same time must be computationally efficient to

implement. Many techniques for definition of potentials of interatomic interaction,

which are based on quantum mechanics laws, are developed for the investigation

of metals. They are described in many recent relevant transactions and review



http://dx.doi.org/10.1016/B978-0-12-420143-9.00006-5

papers, for instance, Refs. [3�5]. And only some common information about them

is given below.

In the early quantum mechanics theories, interaction between electrons was fea-

tured by some average potential that depends on the state of electrons, and the per-

turbation method was used for determination of wave functions and corresponding

electronic states. In the pseudopotential model, the total potential energy is

assumed to be composed of a large density-dependent but structure-independent

term U(Ω), and a structure-dependent term represented by the pair potential, i.e.,

Σi, j φ(rij) [6]. Only valence electrons have to be considered when using pseudopo-

tentials. The core electrons are treated as if they are frozen in an atomic-like

configuration. As a result, the valence electrons are thought to move in a weak

one-electron potential. The pseudopotential method is based on the orthogonalized

plane wave (OPW) method developed by Herring. In this method, the crystal wave-

funtion ψk is constructed to be orthogonal to the core states. This is accomplished

by expanding ψk as a smooth part of symmetrized combinations of Bloch functions

ϕk, augmented with a linear combination of core states. The Phillips�Kleinman

cancellation theorem [7] provides a means for the energy band problem to be sim-

plified into a one-electron-like problem. To obtain a wave equation for ϕk, the

Hamiltonian operator was applied in which the attractive core potential and a

short-range, non-Hermitian repulsion potential are taken into account. The new

effective potential was termed the pseudopotential. To simplify the problem further,

smoothly varying pseudopotenials were used instead of the actual potentials. They

differed mainly by their constant value in the core region.

The following model pseudopotentials were applied for calculations: the

Ashcroft pseudopotential of empty ionic skeleton [8], Hejne�Abarenkov�Animalu

nonlocal pseudopotential [9], two-parameter pseudopotential of Gursky and Krasko

[10], and Leribo�Anzhel pseudopotential, which was applied for calculations of

properties of light metals with the strong exchange interaction [11].

However, effective potentials of atom interaction calculated by different authors

(with different model potentials and interchanging-correlative corrections) differed

too much.

6.1.2 DFT and Ab Initio Calculations

Although the empirical pseudopotential method can provide a better description for

the bulk properties of perfect crystals than the classical pair potential, it cannot be

used to study the lattice defects, where the atomic volume is poorly defined.

Furthermore, even by including an additional density-dependent term, the pair

potential cannot provide an adequate description of the metallic systems. Therefore,

researchers have tried to figure out how to improve the description of the potential

by incorporating many-body effects into interatomic potentials.

The majority of up-to-date methods for calculations of interaction potentials of

atoms from the first principles are based on the density functional theory (DFT). The

description of its essence is in Refs. [3�5,12,13]. The following simplified scheme

for estimation of electronic density in metals [14] gives the first notion of DFT.


At first, a set of trial wave functions of electrons and trial initial electronic density

ρ0(r) introduced into the Hamiltonian is considered. Then the wave functions

are improved by a method of iterative diagonalization of matrix, and the new elec-

tronic density ρ(r) is calculated. The new density ρ(r) is admixed to the initial

density ρ0(r), and the new effective potential is calculated. The mixing proce-

dure depends on what problem is studied: modeling of physical processes in bulk

volume of the sample, or at its surface, or research of separate molecules. The itera-

tions are retried, until changes of the electronic density become inappreciable

(ρn11(r)Dρn(r)).Classical molecular dynamics methods require interatomic potentials to calculate

the forces acting on atoms/ions, whereas the ab initio molecular dynamics (AIMD)

methods compute those forces from electronic structure calculations, which are

performed as the MD trajectory is generated. DFT [15,16] underlies most AIMD

methods. Starting with a collection of atoms/ions at given nuclear positions, DFT

makes it possible to calculate the ground state electronic energy and, via the

Hellmann�Feynman theorem [17], the forces acting on the atoms/ions. Within

DFT, most methods use the Kohn�Sham (KS) orbital representation of DFT (KS-

AIMD methods) [16], which demands a large computational burden, whereby

allowing the study of small sample sizes (one or two hundreds of particles) during

only short real simulation times (few tens of picoseconds). However, these con-

straints may be somewhat overcome [18] by the so-called orbital-free ab initio

molecular dynamics (OF-AIMD) method. Using the Hohenberg�Kohn (HK) repre-

sentation of the DFT [15], it eliminates the electronic orbitals and permits perfor-

mance simulations with “large” samples (up to a few thousands of particles) and

for long times (hundreds of picoseconds).

In recent years, ab initio calculations have made a profound impact on the inves-

tigation of material properties. The main reason for the enormous success of ab

initio methods lies in the fact that they are parameter free and require no other input

than the atomic number. In addition, improvements in computer performance and

algorithms allow applying these methods to a steadily increasing number of physi-

cal and chemical phenomena. The most successful method currently tractable—the

local density functional (LDF) theory proposed by Kohn and Sham [16] allows

simultaneous investigation of the ions of structural, electronic, and dynamic proper-

ties. The first successful ab initio calculation in this context goes back to a seminal

paper written by Car and Parrinello [19]. In their work, Car and Parrinello proposed

a simulated annealing approach, in which electrons and ions are treated on the

same footing via a quasi-Newtonian equation of motion. This approach not only

allows for an efficient simultaneous update of electrons and ions but also possesses

some serious restrictions: The time step for the technique of Car and Parrinello is

limited by the requirement that the electrons are always close to the exact electronic

ground state. Indeed, it can be shown that this is only the case if the typical excita-

tion frequencies of the electronic subsystem are much higher than that of the ionic

system [20] (in this case, electrons and ions decouple adiabatically, and the elec-

trons oscillate around the real electronic ground state). This also implies that the

time step in a simulation is determined by the electronic degrees of freedom, and

153Simulation Techniques for Atomic Systems

usually the time step is an order of magnitude smaller than that necessary to simu-

late the ionic subsystem.

A straightforward alternative to the simultaneous update of electrons and ions is

the exact calculation of the electronic ground state after each ionic move. This is

possible if the algorithms for calculating the electronic ground state are sufficiently

efficient. For a plane-wave basis set, Car and Parrinello introduced an efficient way

to calculate the action of the Hamiltonian onto the electronic wavefunctions. They

proceeded from the fact that the Kohn�Sham energy functional is minimal at the

electronic ground state. Therefore, minimization of the functional with respect to

the variational degrees of freedom leads to a convenient scheme for calculating the

electronic ground state [21]. Stich et al. [22] by Gillan [23], by Arias et al. [24],

and by Kresse and Furthmiiller [21], elaborated some new techniques and algo-

rithms for minimization of the functional later.

6.1.3 Embedded Atom Method and Modified Embedded Atom Method

Accurate ab initio studies of the structural stability, elastic properties, and the nature

of interatomic bonding have been reported already for many pure metal and binary

alloys, for instance, γ-TiAl [25] as well as other stoichiometric alloys of the Ti�Al

system [26�28]. However, the application of ab initio methods to atomistic studies

of diffusion, deformation, and fracture are limited due to the prohibitively large

computational resources required for modeling point defects, dislocations, grain

boundaries, and fracture cracks. Such simulations require large simulation cells and

long-enough computational time. Semiempirical methods employing model poten-

tials constructed by the embedded atom method (EAM) [29�31] or the equivalent

Finnis�Sinclair method [32] the modified embedded atom method (MEAM)

[33,34] the second-moment approximation of tight-binding potential (SMA-TB)

[35�38], are particularly suitable for this purpose. The many-body effects in these

methods are included implicitly through an environmental dependence of the two-

body terms. Raeker and DePristo [39] have compared all the potentials in detail.

Listed semiempirical methods share the principal view that the cohesive energy

of an atom is largely determined by the local electron density at the site where the

atom is located and that the contribution to the electron density at the site is due to

the neighboring atoms. The de-cohesion energy in most of these potentials is repre-

sented by a pair interaction, i.e., a two-body term, which largely reflects the elec-

trostatic repulsion. Although there are a variety of names associated with these

potentials, they all provide very similar expressions for the total energy of a system

consisting of N atoms of the metals:

Etotal 51

2

Xi; j

φijðrijÞ1Xi

FiðρiÞ; ð6:1Þ

ρi 51

2

Xj6¼i

ΨijðrijÞ: ð6:2Þ


where φij(rij) is a two-body term and Fi(ρi) is a many-body term presenting the energy

of an atom i as a function of a generalized coordinate ρi, where ρi is a measure of the

local electron density constructed as a superposition of ψij(rij), i.e., the contributions

from neighboring atoms. These pair-function potentials are all valuable for studying

complex systems that are intractable with more rigorous methods, and for studying

generic properties that do not depend so much on energetic details [40].

The force fi acting on atom i due to interactions with other atoms and is given

by the gradient of the energy Etotal:

fi 52ΔriEtotal;

fi 52Xj6¼i

φ0ijðrijÞ1 ðF0

iðρiÞ1F0jðρjÞÞΨ0

ij

h i rijrij

: ð6:3Þ

The models listed above have similar analytical forms, but they differ vastly in

the procedures to build the potential functions, often resulting in rather different

parameterizations for the same material. In many cases, researchers could guess the

functions and fit the parameters to available and reliable experimental data. It is

considered in EAM that distribution of electronic density is spherically averaged;

the function ψ(rij) is short-range and monotonically decaying, it is localized in the

area of several coordination orbs. It is clear that the sum in Eq. (6.3) is counted

within certain orb of operation—the cutoff radius (see Section 1.3). The cutoff

radius defines a maximum distance of interatomic interactions in simulation, and

interactions of atoms beyond the cutoff radius are simply ignored. Most currently

used potentials for real materials are designed with a cutoff radius; they go to zero

at rC together with several first derivatives in the potentials [41].

Calculation of forces at every step of the MD modeling can be fulfilled accord-

ing to analytical expressions for all necessary functions: φ0(rij), ψ0(rij), F0(ρ). It ismore convenient using of arrays of tabulated data. One can find such data on the

Internet for all potentials published in known scientific journals (usually the func-

tions φ(rij), ψ(rij), and F(ρ) are tabulated; see, for instance, the official site of

LAMMPS). At every timestep, calculations of the array with data of electronic den-

sity for all atoms have to be fulfilled at first.

The EAM was able to reproduce physical properties of many metals and impuri-

ties. The EAM was applied to hydrogen embrittlement in nickel [29], and to nickel

and palladium with hydrogen [30]. Cherne et al. [42] made a careful comparison of

MEAM and EAM calculations for the liquid nickel.

The MEAM is a semiempirical method with uniform formalism for different

crystalline structures (FCC, BCC, HCP, diamond-structured materials, and even

gaseous elements). Its use gives a good agreement with experiments or first-

principles calculations [43�46]. Baskes [43�47] was the first to propose MEAM,

and its following improvement. MEAM is also an improvement to the EAM in the

case of angular dependence of contributions of electronic density from different

atoms. Angular dependence of the electronic density is connected with the devia-

tion of the Fermi surface in metals in the crystalline state from the spherical shape;

that is the reason of occurrence of noncentral forces.


Total energy E of the system of atoms in MEAM also as well as in EAM is

approximated as the sum of energies of atoms. The embedding energy FiðρiÞ (theenergy necessary for removal of atom i from its place ri with local electronic den-

sity ρi) is calculated under the equation:

FiðρiÞ5AiE0ρi ln ðρiÞ; ð6:4Þ

where E0 is the energy of sublimation, parameter Ai depends on type of atom i, and

electronic density ρi at the point of its placing. It is calculated under complicated

equations [47], which consider different functions of screening for several coordi-

nation orbs and the anisotropic components of functions of the electronic density.

For searching the integral local electronic density in semiconductors, authors,

for instance [48,49], take into account angular dependence of the contribution to

the electronic density on different atoms using three partial functions, the same

which were used by Stillinger and Tersoff for the potentials. The following equa-

tion gives the general form of the function that is similar to the function used by

Tersoff:

ρi 5Xj 6¼i

fjðrijÞ" #20

@1A

1=2

5Xj 6¼i

½fjðrijÞ�212Xk. j

Xj. i

fjðrijÞfkðrikÞ !1=2

: ð6:5Þ

The electronic density ρi in the point i is calculated as the linear sum of local

contributions of the electronic density from all next atoms j, moreover, the relative

positions of atoms j and k, which enter into the calculation orb, are also considered.

Atomistic simulations of a wide range of elements and alloys have been per-

formed using the MEAM potentials. Baskes [43] first proposed the MEAM method

to obtain realistic shear behavior for silicon. Baskes et al. [44] provided the

MEAM model of silicon, germanium, and their alloys. The MEAM was also

applied to 26 single elements [45].

6.1.4 Definition of Potentials of Atomic Interaction for Mixed Systems

Multicomponent systems in the amorphous or nanocrystal state, with valuable

physicochemical properties inherent to them are very important for modern materi-

als technology. For example, among high-strength aluminum alloys today, special

attention is given to amorphous alloys, which contain rare earths in the complex

with transition metals. These new alloys have high mechanical characteristics, and

they have the wide-enough temperature range of thermal stability [50,51].

Owing to the importance of study of the mixed systems, potentials of inter-

atomic interaction are developed in many works for atoms of different kinds. For

example, Shpak et al. [52] carried out calculations of the pair interaction potentials

from the first principles for system Al�Y�Ni based on hypothetical crystalline

structures, which corresponded to the ranked superstructures of substitution with


atoms Y and Ni in the FCC crystal lattice of Al. However, in the majority of the

up-to-date research authors use EAM and its new modifications. For example,

potentials were developed by EAM for systems Al�Ni [53] and Al�Ti [16,54] and

by MEAM for the system Al�Mg [47].

Mishin [53] offered the EAM expression for the energy of systems in the case

of binary alloys,

Etot 51

2

Xi;j

φαi ;αjðrijÞ1

Xi

FαiðρiÞ; ð6:6Þ

where φαiαjðrijÞ is the pair interaction potential as function of the distance rij

between atoms i and j of different chemical kind αi and αj (A and B); Fαiis the

energy of embedding of atom of chemical kind αi as function of the local elec-

tronic density ρi in the point i, induced by all atoms of system. The local electronic

density ρi would be calculated according to the relation:

ρi 5Xj 6¼i

ραjðrijÞ; ð6:7Þ

where ραjðrijÞ is the function of the electronic density determined for atom of chem-

ical type αj. This model includes seven functions of interactions of atoms, which

has to be calculated accordingly parted on three groups:

1. ϕA;A; ρAðrÞ;FAðρÞ—for atoms of the kind A and pairs A�A;

2. ϕB;B; ρBðrÞ;FBðρÞ—for atoms of the kind B and pairs B�B;

3. ϕA;B characterizes interaction of atoms of the different kind (of pairs A�B).

Instead of Eq. (6.3), the force acting on an i-atom has to be calculated according

to equation [41]:

fi 52Xj 6¼i

φ0ijðrijÞ1 ðF0

iðρiÞΨ0ij 1F0

jðρjÞΨ0jiÞ

h i rijrij

: ð6:8Þ

In this expression, atoms i and j can be of one and different types. The derivation

F0j depends on the electron density in the placement of the j-atom and the value Ψ0

ji.

Nowadays, many potentials are already elaborated for binary and ternary sys-

tems [41]. Earlier, Baskes et al. [56] applied MEAM to silicon�nickel alloys and

interfaces. Jelinek et al. [47] calculated energies of formation of alloy AlMg with

different crystalline structures based on MEAM, according to the equation:

Hf 5Etot 2NMgεMg 2NAlεAl

NMg 1NAl

: ð6:9Þ

where Etot is the system total energy, NMg and NAl are the numbers of atoms of Mg

and Al in the system accordingly, εMg and εAl are the total energies per atom Mg


and Al accordingly, which are calculated for the case of the ideal lattice of the cor-

responding phase. The results obtained have been compared with those calculated

within the limits of DFT; see Figure 6.1. It is clear from Figure 6.1 that the depen-

dence of the heat of phase formation on the volume of unit cell calculated by the

MEAM method for the case of the B1 structure type coincides better with out-

comes of calculation under TFD theory in comparison with cases of B2 and B3

structure types.

The potentials are checked more often by comparing the results of simulations

with experimental data and with calculations from the first principles. Many prop-

erties are usually under consideration: the energy of crystalline structures as a func-

tion of volume (total or calculated per one atom), formation heat of crystalline

phases in two-component alloys, temperature coefficient of volume expansion,

elastic constants, and compression modulus. In addition, they are the temperature

of melting and the melting heat, the energy of formation of vacancies, energies of

interstitial or substitutional atoms, and phonon spectrums.

6.1.5 The Problem of Choice of the Pair Potential Function

Evaluations testify that such integral characteristics as binding energy of atoms,

evaporation heat, melting point and polymorphic transformation, enthalpy of

fusion, elastic constants, the compression modulus, and so on strongly depend on

the interaction potential choice. Many studies are known now which were devoted

to refinement of potentials so that they give correct values of many physical char-

acteristics, for example, Refs [55,57].

However, it is not possible for anybody to obtain during simulation by the MD

method satisfactory values of great many of physical characteristics. Certainly,

there is the problem of the choice of the potential of the optimal version. As for

simple classical pair potentials, it is known that the presence of asymmetry of the

basic potential well ensures thermal expansion of the material. Melting heat and

15 20 25 30 35

B1 (DFT)

B2 (DFT)

B3 (DFT)

B1 (MEAM)

B2 (MEAM)

B2 (MEAM)

40 45

Atomic volume (Å3)

10

1.2

0.8

0.4

Hf (

eV)

0

Figure 6.1 The heat of

formation per atom for MgAl

alloys in the B1, B2, and B3

crystal structures; B1 (NaCl

—prototype), B2 (CsCl—

prototype), and B3 (ZnS—

prototype).


evaporation heat of the matter model depend on the cutoff radius of the potential as

the number of the neighbor atoms thus varies, interaction with which has to be con-

sidered. This factor influences interatomic distance, which is not equal exactly to

the position of minimum on the curve of potential energy. Melting point and elastic

modules are connected first with the potential curvature near to the basic minimum.

The character of oscillations of the calculated potentials is connected with crystal-

line structure. On the contrary, it is quite possible to simulate different crystalline

structures, changing the placement of oscillations on the axis of distances.

The melting process results in diffusing of the Fermi surface of electronic states,

approximately 15%, in comparison with the substance in crystalline state. Oscillat-

ing character of effective potentials should become less indicative at transition

from solid to fluid. There is the problem of application of the potentials calculated

for the crystalline state of the substance at simulation its liquid or amorphous state.

6.2 Finding the Equilibrium Structures by the MC Methodand Their Analysis

6.2.1 Searching for Equilibrium Structures

An algorithm for relaxation, which consists in minimization of potential energy, is

routinely used for definition of geometrical structure of substance (configuration of

atoms). Interatomic or intermolecular potentials should be known. Any initial con-

figuration is set for a system; and as a result of algorithm operations, the structure

responded to energy minimum can be found. Minimization of potential energy

U (r1, r2, . . ., rN) as functions of many variables (coordinates of atoms) may be ful-

filled at relaxation by the descent algorithm. Thus, the potential of interaction of

atoms should be known. Initial values of position vectors r01 ; r01 ; . . .; r

0N may be cho-

sen by any method. Search of minimum of function U (r1, r2, . . ., rN) is carried out

along each of the axes. Values of functions U (r11 dr1, r2, . . ., rN) and U (r12 dr1,

r2, . . ., rN) are should be compared for this purpose, and that direction of displace-

ment 1dr1 or 2dr1, in which the value of potential energy decreases, must be cho-

sen. Further process of a search of local minimum is retried until a very small

modification of potential energy in one step of iteration is reached. If in both direc-

tions 6dri values U (r1, r2, . . ., rN) grows, it means that the minimum is closer

than 6dr, hence, it is necessary to reduce step dri.

A search can be realized also not in turn along each of axes, and selecting new

location of jth particle after such displacement, which is determined by the equa-

tion: r!j 5 r

!j 1 random � δ � r!1, where δ is the amplitude, r

!1 is the any casual unit

vector.

There is modification of this method—the method of the fastest descent. As it is

known that the gradient direction is the direction of the fastest growth of function, it

is clear that the opposite direction is the direction of the fastest falling. Thus, a

search of minimum at application of method of the fastest descent is carried out in


antigradient direction, that is in direction 2riU(r1, r2, . . ., rN)5Fi. Within the lim-

its of the linear approach, for one of components of the full force Fx which operates

on ith atom from all others within orb of operation of potential, it may be written:

Fxðri 1 driÞ5FxðriÞ1Xj

@FxðriÞ@rji

drji: ð6:10Þ

It is assumed that Fx(ri1 dri)5 0 for an atom in position, which corresponds to

local minimum of energy, and it is necessary to solve the system of N (three) linear

equations for determining dri. Certainly, a shift of atom position should affect all

environmental atoms. However, at iterative repetition of the relaxation procedure,

the calculation process should converge, and displacements of atoms should

become less than some value set beforehand.

Unlike the descent method, at realization of relaxation by the MC method, the

new configuration is selected casually (it is generated).

The algorithm for relaxation by the MC method consists from following

steps [1]:

1. The definition of initial configuration.

2. Generation of new configuration.

3. Evaluation of modification of energy δE.4. Acceptance of new configuration in case of condition realization δE, 0 and performance

of additional steps if this condition is not fulfilled:

a. Calculation of exp[2 δE/(kT)];b. Random number generation RandA[0;l];

c. Acceptance of new configuration in case Rand, exp[2δE/(kT)], otherwise—returning

to step 2 without configuration modification.

Working this algorithm, the system goes to the state with the minimum energy,

which corresponds to constant macroscopic parameters: the number of particles, volume

and temperature (NVT). Step 4 is provided that by new configuration with less than pre-

vious energy is always accepted. Configurations which raise an energy of system are

accepted only with the Boltzmann probability (step 4c). Transition to a new configura-

tion is realized by means of casual displacement of casually selected particles.

6.2.2 Evaluation of Structural Properties

Metropolis fulfilled the first modeling by the MC method of the system of

hard disks. Later, Rotenberg spent model operation for system of rigid orbs, and

Wood—for the Lennard�Jones potential. The NVT-ensemble was considered;

the data obtained in result of simulations were analyzed and function P(V) was

determined [58].

The following characteristics are counted more often for the analysis of structure

of the simulated systems.

Radial pair distribution function (RPDF) g(r) is connected with probabilities of

certain distances between atoms (see below).


Partial radial pair distribution function (PRPDF) gA2B(r) determines the radial

distribution of atoms of type B, which are placed in spherical layers round of atoms

of type A:

gA2BðrÞB 1

MN

XMm51

XNn51

Pnm; ð6:11Þ

where N is the number of atoms of type A; M is the quantity of independent config-

urations on which the yielded function is counted, Pnm is the number of atoms of

type B around nth atom of type A in mth independent configuration.

Average configuration potential energy:

hEi5 1

M

XMm51

Em; Em 51

2

Xi; j5 1

Ei; j; ð6:12Þ

where Ei,j is the energy of ijth pair of atoms in the mth configuration.

Three partial correlative function G(θ):

GðθÞ5 1

Ni;j;k

Xi;j;k

r!i;jr!i;k

jr!i;jj � jr!i;kj

; ð6:13Þ

where Ni,j,k is the number of triples of nearest atoms. However, the angular distri-

butions are more often analyzed. Angles are determined through components of

position vectors in directions to the two nearest neighbors of an atom under consid-

eration: (rijrik)5 jrijjjrikj cosθijk.

6.2.3 The Radial Pair Distribution Function

Placements of atoms in crystals feature by means of space lattices. However, this

method for liquids is unsuitable. The distribution functions are used in case of

liquids. To make representation the distribution function, we will consider place-

ment of atoms in crystal from the point of view of the environment of the certain

chosen atom. We take in consideration the plane (100) of the face centered cubic

lattice. Figure 6.2 shows the flat net, which corresponds to the yielded crystalline

plane; some positions of atoms are shown and the characteristic distances between

the chosen atom and those, which surround it, are specified.

If the chosen atom is disposed in point 0, which is the blanket vertex for eight

unit cells, the atoms proximate to it are disposed at the centers of facets of the

neighbor unit cells (points A1, A2, A3, and A4) on identical distances from the cho-

sen atom. We will draw the circle in radius r1, and we will find four atoms on this

circle. If we draw the orb in radius r1 (so-called, the first coordination orb), n15 12

atoms will be on it. Draw still the circle in radius r2 5ffiffiffi2

pr1. There will be four


atoms on it also, and n25 6 atoms will be placed on the orb in radius r2 (the second

coordination orb). Similarly, we will find that n35 24 for the third coordination

orb (r3 5ffiffiffi3

pr1), n45 12—for the fourth orb (r45 2r1), and so on. These outcomes

are mapped pictorially in Figure 6.2B. The ratio r2=r21 values are put along the

abscissa ordinate axis and the numbers of atoms n corresponding them—along

the ordinate axis. Such graph term as the graph of radial distribution of atoms in

the ideal crystal. In the real crystal, in which there are different imperfections, the

long-range order is broken the more notably if the larger is a distance from the

selected atom. In the absence of imperfections, the placement of atoms in each

instant also differs from ideal because of thermal oscillations of atoms.

In the case of fluids and amorphous solid bodies, it is better to consider number

of atoms not on the surface of the specified orbs, but in spherical interlayers. If to

spend round the chosen atom two orbs with radiuses r and r1Δr, the volume of

the spherical layer will be equal 4πΔr2Δr. We will determine the atomic radial

density function through the number of atoms in such layer:

ρðrÞ5 nr

4πr2Δr: ð6:14Þ

It is the average quantity of particles in the unit volume of the layer. We will

designate the average density of atoms through ρ0. The ratio:

gðrÞ5 ρðrÞρ0

ð6:15Þ

is termed the RPDF. It characterizes probability of occurrence of atoms in some

layer on the certain distance from any chosen atom.

Figure 6.3A shows the function g(r) for liquid sodium, which is obtained based

on results of the X-ray examination. This function grows at first with magnification

r from the value g(r)5 0 on small distances, and then varies so, that it becomes

r1 r2 r3 r4

r 2/r12

r5A2 A3

A4

(A)

A1

n

(B)

2 4 6 8

Figure 6.2 The scheme of the relative positioning of atoms (A) and the graph of radial

placement of atoms (B).


alternately more or less than unity. The greater the distance from atom, which is

considered as initial, the smaller the deviations of the g(r) function values from

unity. The graph of the atomic distribution function 4πr2ρ(r) given in Figure 6.3B,

is convenient for comparing with the graph of radial placement in the ideal crystal

(Figure 6.2B). Product 4πr2ρ(r) Δr determines the number of atoms in the spheri-

cal layer formed by orbs with radiuses r and r1Δr. Therefore, areas under the

graph peaks define numbers of atoms in corresponding coordination orbs. The area

under the first peak (it is shaded in Figure 6.3B) defines the mean number of the

nearest atoms. The solution of the problem, on how to separate the part of intersec-

tion of peaks between them, is complex enough. Therefore, determination of num-

bers of atoms only for the first two coordination orbs with the adequate accuracy is

possible only. The dotted curve in Figure 6.3B shows the function 4πr2ρ0.To build up RPDF according to data of simulation, it is necessary to simply sum

up numbers of atoms in spherical interlayers around each of the atoms chosen by

turns. The ratio (6.15) has to be evaluated as the ratio of averaged values. The

RPDF is the fundamental characteristic of structure, both for single-component,

and for many-component (partial RPDF) systems. In the case of the one-

component system, calculations have to be fulfilled according to the relation:

gðriÞ5ρiρ0

5nðri;ΔrÞR3

Nðr3i11 2 r3i Þ; ð6:16Þ

where N is the number of measurements, which correspond to the orb with radius

R; n(ri, Δr) is the number of measurements, which correspond to the spherical

layer from ri to ri11. If writing down data concerning the environment of every

atom in separate arrays, it is also possible to construct the distribution of atoms on

the number of environmental nearest neighbors.

The following relation allows calculating the partial RPDF:

gijðrÞ5VnijðrÞ

Nj4πr2Δr; ð6:17Þ

where nij(r) is the number of particles of type j, which are in the spherical layer Δr

at the distance r from the particle of type i; Nj/V is the average density.

4πr2 p

(r)

g(r)

4 8 12 r, Å 4 8 12 r, Å

(A) (B)

Figure 6.3 The RPDF (A) and atomic distribution function (B).


The laconic code (the language C11) for calculation of the unnormalized

partial RPDF is shown below (type[i] sets the type of the component, type[i]51 or 2):

for (int i 5 0; i , COUNT; i11)if (type[i]551)

{ atom &a15a[i]; //the first atom

NCOUNT11;for (int j 5 0; j , COUNT; j11)

{ if (j55i) continue;if (type[j]551) continue;

atom& a2 5 a[j]; //the second atom

double dr5(a1.r2a2.r).abs(); // the distance between atoms

int nr 5 floor(dr�10);if (nr . intR) continue; //intR�is the radius of the orb of calculation

g12[nr]11; //accumulation of data on layers in thickness of 0.1 A

} }for (int ir 5 0; ir ,5 intR; ir11)g12[ir] 5 g12[ir]/NCOUNT/(pow(ir11,3)-pow(ir,3));,

where COUNT is the number of all atoms, NCOUNT is the number of atoms of the first

component; pow(ir,3)5ir�ir�ir;ir is the layer number.

The important information can be obtained also from the Fourier transformation

of the RPDF, that is, from the structure factor. Such data allow us to compare the

results obtained by modeling with experimental data of the X-ray investigation and

study in more detail the modification of structural characteristics at atomic level

during heat treatments of simulated systems. In such a way, the outcomes of

modeling of heat treatments of amorphous iron and alloys of the Fe�B system [59]

were compared with the experimental outcomes given in Ref. [60].

A set of the space correlation functions, isotropic and anisotropic, characterizes

the structural characteristics of simulated objects. However, such structure charac-

teristic as RPDF does not yield the full representation about the substance structure.

The subsequent step is in buildup of atomic models, which map the spatial arrange-

ment of the system particles more visually. However, there is the number of the

difficulties in the case of noncrystalline systems connected with the insufficient

precision of output data and with possible ambiguity of the problem solution.

Therefore, Shpak and Melnik [60] improved the reversible MC technique, to allow

the construction of atomic models of fluid or amorphous phases with the structure

characteristics corresponding to diffraction data.

6.2.4 The Topological Analysis of the Simulated Atomic Configurationsby the Voronoi�Delone Method

The up-to-date examination of structure requires the mathematical method for study

of the relative positioning of molecules in the space, not leaning on chemical


bonds, as at study of structure of separate molecules, but on principles and proper-

ties of transmitting symmetry, which are developed in crystallography. Such a

method has been developed for a long time already by mathematicians [61,62].

Voronoi was one of its principal creators. Delone explained the essence of

Voronoi’s works and extended the basic theorems [62]. Bernall for the first time

applied the Voronoi method to the examination of structure of the fluid [63]. The

geometrical building up, on which the method bases, is connected with so-called

Voronoi’s polyhedrons—spatial areas, which are the proximate to the yielded

atom. For its definition in the middle of the segments spent to the nearest atoms,

perpendicular planes to them (Figure 6.4A) are constructed. Actually, it is the

Vigner�Zejtts cell. The shape of Voronoi’s polyhedrons depends on the certain

disposition of neighbors around the yielded atom.

Buildup of Voronoi’s polyhedrons allows transferring from the list (file) of coor-

dinates of atoms to geometrical fashions. Figure 6.5 shows Voronoi’s polyhedrons

of some three-dimensional packing of full spheres. For simple crystalline struc-

tures, there are only a few polyhedrons, which reflect the nearest environment of

atoms of the yielded crystal. Voronoi’s polyhedrons differ for different atoms in

the unregulated phase, but they have the features which correspond to the structure

of the yielded system. It is shown by up-to-date simulations that a lot of clusters of

icosahedron type exist in amorphous phases (20th polyhedron, formed by triangular

pyramids, Figure 6.5E). Voronoi’s polyhedron for them is shown in Figure 6.5D.

Icosahedrons are usually the centers of quasi-crystals, the sizes of which are

restricted owing to lack of transmitting symmetry.

5

6

7

8

2

3

(A) (B)

4

1

Figure 6.4 Illustration of building up of

the Voronoi polyhedron (A); the two-

dimensional Voronoi construction and

Delone simplex (B).

Figure 6.5 Voronoi’s polyhedrons for some systems: simple cubic lattice (A); the face-

centered cubic lattice (B); the diamond lattice (C); the icosahedron configuration (D) (the

icosahedron is shown by the scheme (E), its basic symmetry axes of the 5 order, the atom at

center has 12 nearest neighbors); arbitrary environment of atom (F).


Building up of Voronoi’s polyhedrons is part of the Voronoi�Delone method.

Then the complete mosaic of Voronoi’s polyhedrons can be defined and analyzed.

The important point is that Voronoi’s polyhedrons fill space without superpositions

and slots; the method realizes, as speak mathematics, the space partition.

Figure 6.4B shows the two-dimensional variant of such decomposition. The Delone

simplex is shown by dashed lines. The centers of atoms are the vertexes of Delone’s

simplex. Delone’s simplexes of the system fill space without superpositions and

slots, making Delone’s decomposition. The orb featured around Delone’s simplex is

empty; centers of other atoms are not present in it. It is simultaneously possible to

divide all system of atoms into groups of atoms—Delone’s simplexes. They are tri-

ples of atoms in two-dimensional space, in three-dimensional—quadruples

(Figure 6.6). Centers of atoms determine vertexes of polyhedrons, the most simple

in space of the yielded dimensionality (in mathematics, such a polyhedron is termed

a simplex). They are the delta circuit on the plane, in space—the tetrahedrons.

Their shape is determined by the certain disposition of atoms and can serve as the

characteristic of structure of system.

Except study of structural characteristics, decomposition of Voronoi�Delone

allows us to initiate with the study of hollows between atoms. This aspect of exam-

inations is even more important than the analysis of laws in the disposition of

atoms in itself. Many of the important physical processes, for example, diffusion of

impurities, passing of small atoms through granulose sponges are connected

directly with interatomic hollows. Delone’s simplex defines rather large empty seat

between atoms, where as a rule, narrower hollows carry on (Figures 6.4B and 6.6).

From the geometrical point of view, each simplex Delone defines the elemental

concavity in system of atoms.

Delone’s simplexes are curved tetrahedrons for disordered and thermally per-

turbed systems. Using the quantitative measure of the shape of the simplex,

Alinchenko et al. [64] evolved those atomic configurations, which are closer to the

perfect tetrahedron. Such simplexes are characteristic for the dense crystalline

structure. They exist also in the fluid phase, however, the character of their relative

positioning essentially differs from the crystalline. Medvedev [65], in his book,

considered the problems of structure analysis by the Voronoi�Delone method.

Figure 6.6 Illustration of Delone’s

simplex in three-dimensional space; (A)

simplicial configuration of atoms; (B)

the empty space between atoms defines

the elementary simplicial pore.


6.2.5 Evaluation of Pressure and Definition of the State Equation

The method of pressure evaluation follows from the virial theorem. We will con-

sider its deduction given in the book of Hansen and McDonald [66]. We will write

the magnitude of the total from scalar products:

G5Xi

riFi; ð6:18Þ

where the sum undertakes on all particles of system, and Fi there is the full force

acting on i particle. The G-value averaged on time can be written as follows:

hGi5 limτ!N

1

τ

ðτ0

dtXi

riðtÞ � FiðtÞ5 limτ!N

1

τ

ðτ0

dtXi

riðtÞ � md2riðtÞdt2

: ð6:19Þ

Considering that

rd2r

dt25

1

2

d2r2

dt22

dr

dt

� �2.

mv2

252

rF

21

1

4md2r2

dt2;

(v is the velocity of particles), we will obtain after the average of last equality on

time:

mv2

2

� �5

1

τ

ðτ0

mυ2

2dt52

rF

2

� �1

1

4md2r2

dt2

��τ

0

:

In the case τ!N, d2r2=dt2 6¼ N, therefore,

mυ2

2

� �52

rF

2

� �and hGi52 3NkT ; ð6:20Þ

where the integration by parts is spent and the theorem of the equidistribution is

used. Expression for G can be segmented: the first part is connected with forces of

particle interaction and the second part is predetermined by exterior force acting

from walls. The exterior force is connected with pressure and it may be determined

from:

dFi ext 52Pn dA; ð6:21Þ

where n is the normal line unit vector to the plane dA. Therefore,

Xi

ri dFi ext 52P

ðrn dA52P

ðdivr dV 523PV ;


where we have applied the Gauss theorem and have considered, that div(r)5 3.

As a result, it may be written down:

23PV 1 hGinti52 3NkT :

So,

P5NkT

V1

1

3V

Xi5 1

riFi

* +; ð6:22Þ

where ri is the position vector of ith particle, Fi is the full force that acts on the

particle i from all other particles, and the sum undertakes to all N particles (in the

two-dimensional case the multiplier 1/3 should be replaced by 1/2).

By data of MC simulations, it is convenient to present the force virial through

RPDF, g(r), and the derivative of the potential. We will consider the sum of pro-

ducts of position vectors and forces taking into account Nk atoms concerning all the

ith chosen atom (directions of position vectors and forces coincide):

Xi;k

rikFik 5N

2

Xk

rkFkðrkÞ;Xk

rkFkðrkÞDðN0

rF dN;

where dN5 4πr2g(r)ρ0 dr, ρ05N/V. Therefore,

Xi;k

rikFkðrikÞ5N2

2V

ðN0

F4πr3gðrÞ dr522πN2

V

ðN0

dU

dr

� �r3gðrÞ dr:

Hence, it follows for the equation of the state of the simulated system:

P5NkT

V2

2πN2

3V2

ðN0

dU

dr

� �r3gðrÞ dr: ð6:23Þ

Thermodynamic properties of simulated system can be defined through its total

energy expressed through the pair potential (in case of metal or molecular crystals)

and RPDF,

E53

2NkT 1

2πN2

V

ðN0

UðrÞgðrÞr2 dr: ð6:24Þ

6.3 Kinetic MC Modeling

6.3.1 The Basic Relations for the Transition Probabilities

The Monte Carlo kinetic modeling (MCKM) is the extremely efficient method for

realization of dynamic simulations for processes, which are activated thermally in

the atomic scale. The method is applied for simulation of manifold processes: from

catalysis to growth of thin films.


The procedure MCKM of crystal growth simulations has been developed in

researches Chernov [67], Bennema et al. [68], Gilmer [69], Leamy and Jackson

[70], Ovrutsky et al. [71�73]. Bennema et al. [68] considered the lattice gas model

and proceeded from the principle microscopic revertive (Eq. (6.26)). Thus, it was

actually considered the two-dimensional case, but the model allowed featuring

of three-dimensional growth of the crystal [69]. The (100) crystal face net is

represented as the double array of integers in the model of Kossel’s crystal

(the prime cubic lattice). Each number connected with the net site defines the

atom position along the column ,100., normal to the net plane. Vacancies and

hanging configurations are excluded (admission “solid-on-solid”) and consequently

such integers also represent the surface height over each position in the cross

section (100).

The kinetics of the crystal growth is simulated by adding and removal of atoms

on vertexes of columns ,100.. The knot of such lattice can represent either solid

(s) or fluid (l) atom. The system configuration s is set by distribution of “solid”

atoms in the lattice. System evolution in the course of time is yielded by probabili-

ties of transitions p(s!s0), p(s!s0) is the probability of the system in the instant

t1 τ will be in the new configuration s0 if in the instant t it was in the configura-

tion s. Bennema et al. [68] considered the system evolution with time as the

Markovian process. Existence of such small (characteristic) time slice τ for which

configuration discontinuous variations are possible only, it was supposed, each of

which takes terminating time, and the system configuration, on the average, does

not vary. At known probabilities p(s!s0) for the yielded transition and dependent

on time probability pt(s) that the system is in the configuration s in the instant t, the

probability of that the system in the instant t1 τ will be in the state s0, pt1τ(s0) is

defined by the expression:

pt1τðs0Þ5Xs

ptðs0Þpðs ! s0Þ; ð6:25Þ

that satisfies to the principle of microscopic reversibility:

pðs ! s0ÞpðsÞ5 pðs ! s0Þpðs0Þ: ð6:26Þ

The large canonical assembly was considered in Ref. [68]—each system con-

tacts to the thermostat and can interchange with it particles. The temperature, vol-

ume, and equilibrium concentrations in “solution” are assumed to be stationary

values, whereas the number of “solid” atoms can vary. At equilibrium conditions,

the microstate probability (of the certain configuration):

ptðsÞBexp 21

kTEðsÞ2μs

eNsðsÞ2μleNlðsÞ

� � �; ð6:27Þ

where k is the Boltzmann constant, T is the temperature; Ns and Nl are the numbers

of “solid” and “fluid” atoms accordingly; μse and μl

e (μse 5μl

e) are the corresponding


chemical potentials under equilibrium conditions; E(s) is the total energy of particle

interaction in the system:

EðsÞ5NssVss 1NllVll 1NslVsl 1NsAs 1NlAl; ð6:28Þ

where Nss, Nll, and Nsl are the number of pairs of type “solid�solid,” “fluid�fluid,”

and “solid�fluid” accordingly (only horizontal bonds); Vss, Vll, Vsl, 0 are the cor-

responding potential energies of pair interaction (binding energy), As, Al are the

interior free energies of solid and fluid atoms (these summands to energy are

artificial).

As in the course of crystallization or fusion, the number of particles in a system

does not vary, so we will consider the canonical assembly. We remove restriction

solid-on-solid considering the case of any configurations in system (the three-

dimensional variant). In addition, we throw out the last two summands in the equa-

tion for energy (6.28):

E5NssVss 1NllVll 1NslVsl: ð6:29Þ

The system interchanges only an energy with the thermostat (does not inter-

change particles). Therefore, the canonical distribution, that is the Boltzmann distri-

bution law, is correct in this case. Moreover, statistical weights of the thermostat

may only be considered for determination of the system states and their modifica-

tion. For the phase equilibrium description, it is necessary to consider yet the

number of states, which correspond to the same geometrical configuration.

Degeneration of states is connected, for example, with distinction of vibration

spectrums of atoms in different phases. To take into account degeneration, factors

K1, K2 are inserted to the expression for the ratio of probabilities:

p1

p25

K1pðS0ÞK2pðSÞ

: ð6:30Þ

This equation follows from Eq. (6.26) if to put that p(s!s0) is the probability

(p1) of attachment of one atom to the crystal and p(s!s0) is the probability (p2) of

detachment of one atom. The K1/K2 ratio can be expressed through the ratio of num-

bers of different states of considered atom, that is, through the exponent from the

system entropy change at modification of atom belonging to the certain phase state.

p1

p25 exp

2ΔE

kT

� �� exp ΔS

k

� �; ð6:31Þ

where ΔS52ΔHf/Tf is the entropy of fusion, ΔHf is the enthalpy of fusion, the

sign “2 ” is written down because oscillatory entropy decreases at addition of

atom to the crystal.


Let us consider the model of any lattice system. Numbers N, Ns, and Nl and

numbers of pairs of atoms Nss, Nsl, and Nll are connected by relations:

Ns 1Nl 5N5Const:;

Nss 51

2Nsz2

1

2Nsl;

Nll 51

2Nlz2

1

2Nsl;

9>>>>>>=>>>>>>;

ð6:32Þ

where z is the coordination number (number of the nearest neighbors of each

atom). Taking into account relations (6.32),

E52z

2NsðVll 2VssÞ1Nslεsl 1

zN

2Vll; ð6:33Þ

where εsl 5 ðVsl 2 ðVss 1VllÞ=2ÞDðVll 2VssÞ=2 � ϕ is the excess energy.

Then the difference of energies between states s and s0:

ΔE52ΔNs ΔHf 1ΔNslεsl; ð6:34Þ

where ΔHf 52ðz=2ÞðVll 2VssÞ is the enthalpy of fusion; ΔNs and ΔNsl are the

changes of numbers of “solid” particles and pairs of the type “solid�fluid” between

states s and s0. For the case ΔNs5 1, Eq. (6.31) will take the form:

p1

p25 exp

2ΔHf

kT

� �exp

ΔNslεslkT

� �exp

2ΔHf

kTf

� �: ð6:35Þ

For temperatures which approach to the melting point, this equality can be

rewritten so:

p1

p25 exp

ΔHfðTf 2 TÞkT2

� �exp 2

ΔNslεslkT

� �; ð6:36Þ

where ΔHfðTf 2 TÞ=kT2 5Δμ=kT 5σ; Δμ5μl 2μs is the thermodynamic driv-

ing force of crystallization.

Following relations for probabilities of addition and detaching of one atom satis-

fies Eq. (7.21):

p1 5 ν0τ2 expΔμkT

� �; ð6:37aÞ

p2 5 ν0τ2 expΔNslεslkT

� �; ð6:37bÞ


where ν0 is the vibration frequency of atoms; τ2 is the mean lifetime on the sur-

face of an atom with minimum number of the “solid�solid” (s�s) bonds (with one

bond in case of the simple cubic lattice). According to the original paper [68], it is

better to rewrite Eqs. (6.37a,b) in the form:

p1 5K1ν0τ2 expΔμkT

� �; ð6:38aÞ

p2 5 ν0τ2exp2nsεslkT

� �; ð6:38bÞ

where ns is the number of s�s bonds, that is bonds between “solid” atoms at the

interface, which can change their phase belonging; K15 exp(2ΔHf/kT).

6.3.2 Developing More Realistic Models for Study of the SurfaceProcesses

Crystal Growth Modeling Within the Condition solid-on-solid

Authors of Refs. [68�70] used Eqs. (6.38a,b) for simulation of crystal surfaces.

Leamy and Jackson [70] researched the structure of the crystal-melt surface study-

ing the disposition of solid atoms in consecutive atomic layers (a number of atoms

was such that it was possible to fill them with the half of the model volume).

As they found, the equilibrium structure depends on magnitude of parameter

α (α5 ξ(ΔH/kT) is Jackson’s parameter of roughness; ΔH is the melting heat or

dissolution; ξ5 η/z is the package density degree in planes, parallel to boundary

surface, η and z are the coordination numbers in the plane and in crystal volume

accordingly; for the simple cubic lattice ξ5 2/3). The outcomes obtained are featured

in Chapter 7. They coincide with outcomes of analytical calculations for the model

solid-on-solid at not too small values of the parameter α.The basic shortage of the solid-on-solid models is the impossibility of natural

formation of vacancies and “canopies.” Dalla Torre et al. [74] made an attempt to

overcome this shortage by introduction of the additional torn-off bonds for some

atoms, which are considered as imperfections of the surface structure. Studying

relaxation of two-dimensional islets on the triangular lattice by the kinetic Monte

Carlo (KMC) method, Ovrutsky and Rasshchupkyna [75] considered diffusion of

the surface atoms. Authors supposed that the energy barrier for diffusion depends

only on the beginning number of the nearest neighbors (it is proportional to them,

and it is not connected with number of neighbors after the jump).

Nurminen et al. [76] used the KMC modeling to study incipient states of two-

dimensional islet formation on the substrates with contour, which is applied to

making of semiconducting quantum points. K.E. Khor, S. Das Sarma [77] studied

by means of KMC modeling the formation of thin semiconducting films (with the

islet formation), the lattices of which do not coincide with lattices of the substrates.


Numbers of atoms in layers over the substrate (with other parameter of lattice)

were set, and the equilibrium structure was researched for these quantities of

atoms.

Interaction with Neighbor Atoms from Several Coordination Orbs

The MC variant similar to that used in Ref. [68] was applied to simulate crystalli-

zation of thin films in Refs. [78,79]. The probability of detaching randomly chosen

s-atoms from crystalline cluster during time τ2 is expressed through the exponen-

tial factor from energy U of its bonds with the nearest s-atoms.

p2 5 ν0τ2 exp 2U=kT�

; ð6:39Þ

where ν0 is the vibration frequency of atom, k is the Boltzmann constant, T is the

temperature. U5 2nsϕ in case of taking into account interaction with the nearest

neighbors only; τ25 τ0 exp(2ϕ/kT), τ05 1/ν0, 2ϕ is the modification of two atom

energy bond at fusion, 2ϕ5 2Vsl2 (Vss1Vll), and ns is the number of the nearest

“solid” atoms. In this approach, the binding energy Vsl (“solid�liquid”) is equal to

the binding energy Vll of two “liquid” atoms, VslDVll, ϕ5ΔHf/z, ΔHf is the melt-

ing heat counting per one atom, z is the total number of the nearest neighbors for

the yielded structure.

To take into account the interaction with neighbor atoms from the first, second,

and third coordination orbs, the change of energy at melting was written in such

form: U5 2(n1ϕ11 n2ϕ21 n3ϕ3), n1, n2, and n3 are the corresponding numbers of

neighbor “solid” atoms, a ϕ1, ϕ2, and ϕ3 are the corresponding energies. For

detaching an atom, which is in position of semicrystal (a step break), it is necessary

to perform the work ΔHf5 z1ϕ11 z2ϕ21 z3ϕ3, where z1, z2, and z3 are the corre-

sponding coordination numbers. Dependence of the interaction energy on the dis-

tance was chosen the same, as well as in Ref. [67]: ϕ1=ϕ2=ϕ3 5 ð1=r31Þ=ð1=r32Þ=ð1=r33Þ, where r1, r2, and r3 are the radiuses of coordination orbs.

Overcoming Activation Barriers and Diffusion of Atoms

Outcomes of crystal growth simulations by the traditional method [68�70] in the

framework of the “solid-on-solid” model without activation at joining of atoms are

in agree, as shown in Ref. [68], with the results of statistical analysis of the equilib-

rium structure. The method can be applied to definition of kinetics of growth from

gas or from very dilute solution. However, both analytical theories, and MC simu-

lations [69,71] result in strongly overestimated (in comparison with experimental

data) values of the dimensionless entropy of fusion (ΔS5ΔHf/(kTf)� 4.8) or

Jackson’s parameter of roughness (α� 3.2), which respond to transition from

smooth to rough interfaces in atomic scale. For elimination of this discordance in

Refs. [72,73], it was accepted that the model of the lattice liquid, in which


probability of joining of “liquid” atoms to crystalline clusters depends on number

nl their bonds with other nearest atoms in fluid phase:

p1 5 τminν0 expΔμ2 2nlηϕ

kT

� �exp

2ΔHf

kT

� �; ð6:40Þ

where 2nlηϕ is the magnitude of the additional barrier to joining of atoms to crys-

tal; (η is the factor which gets out so that magnitude 2zηϕ was equal to activation

energies of self-diffusion in fluid phase); τmin is the time step.

Correspondingly, the probability of atom breakoff from crystal is equal:

p2 5 τminνo expð�2nsð11 ηÞϕ=ðkTÞÞ; ð6:41Þ

where ns is the number of s�s (“solid�solid”) atom bonds. The energy barrier is

f5 2(11 η)ϕ per one s�s bond; it consists from odds in interaction energy for

atoms solid and fluid phase (Vll�Vss)� 2ϕ and additional barrier 2ηϕ.Diffusion in the lattice liquid is carried out by the vacancy mechanism, and

vacancies were introduced in small quantity (B3%). It was accepted that the energy

barrier for jumps of atoms into vacancies depends on number of the nearest neigh-

bors in old and new position. The probabilities of interchanging places of an atom

with the close vacancy were computed according to the equation:

pjump 5 τminν0exp2ΔE2 umin

kT

� �; ð6:42Þ

where umin 5 nl2ηϕ is the minimum magnitude of the jump activation energy

(2ηzϕ5QD is the activation energy of self-diffusion); ΔE5P

i;jεα;βi;j Δn

α;βi;j 1P

i;jVα;βi;j Δn

α;βi;j is the additive to the magnitude of activation energy, if the energy

of atom in new state is above than in previous one; i and j are the indexes of com-

ponents; α and β are the indexes of phases; Δnα;βi;j are the changes of numbers of

neighbors of certain type; Vα;βi;j are the corresponding bond energies. The excess

energies εα;βi;j and also energiesVα;βi;j , connected with values of the evaporation heats,

were calculated in Refs. [78�80] through tabular and diagram data [81] according

to equations for regular solutions [82]. Accordingly, calculations of numbers of all

types of bonds in the previous and possible new positions of atoms were provided

in programs. The step on time τmin was calculated from the condition that if there

can be several simultaneous events with the atom chosen casually, the sum of prob-

abilities of their realization cannot exceed unity.

Study of atom structure of the crystal surface by modeling of crystal growth is

carried out in [78] for approbation of the lattice liquid model. The initial crystal

with direct step was in size 803 803 10 of interatomic distances, and an area from

conditionally “liquid” atoms was over it. In two directions (x, y), periodic boundary

conditions were applied.


6.4 Particularities in Application of the MolecularDynamics Method in the Case of Phase Transitions

6.4.1 Application of the Molecular Dynamics Method in DifferentEnsembles

NPT ensemble (Gibbs’ canonical assembly � the number of particles, pressure and

temperature are constant) must be considered for modeling of the liquid�solid

phase transition at stationary values of number of particles, of pressure and temper-

ature of the sample [60,83]. Actually, the physical processes observed in real

experiments routinely happen at constant temperature and pressure. To keep up the

constant pressure and temperature, it is necessary to fine-tune sizes of modeling

boxing and velocities of atoms [84,85].

However, thermodynamic NVT ensemble (canonical assembly of Helmholtz �the number of particles, volume and temperature are constant) was used more

often, in which the volume (average density) of sample is conserved instead

of pressure. Volume of the basic cell is set in initial conditions. The dependences

of pressure and energy on the volume, P(V) and U(V), determined by results of

modeling define equation of state; such results allow us to find the compressibility

of a system, expansion coefficient, and its temperature dependence.

Quite often, the NVE ensemble (microcanonical assembly � the number of par-

ticles, energy and temperature are constant) with constant the total energy of sys-

tems is used for molecular-dynamic modeling. Use of this ensemble appears

sufficient for many “applied” examinations in case the account of thermodynamic

particularities is not so important. This ensemble realizes that conservative classical

mechanical system as for its realization special tools, which are necessary for reali-

zation of other ensembles, are not necessary.

NPE ensemble (is isobaric-isoenergetic ensemble with constant number of parti-

cles, pressure and energy) is sometimes applied and also ensemble with N 6¼Const.

(major canonical assembly), in which it is probable to realize interchanging of par-

ticles with thermostat.

Consider some techniques most often applied to support stationary values of

necessary parameters.

NVTensemble:

1. The thermostabilization technique consists of periodically resetting to atoms of new

velocities at random—according to the Maxwell�Boltzmann distribution [86]. Resetting

can be made for separate atoms, or for all systems as a whole.

2. Berendsen’s algorithm [87] consists of multiplication of velocities of all atoms to certain

coefficient λ on each step on time, or through a fixed little number (5�10) of time steps:

λ5 11τtc

T

TM21

� � �1=2; ð6:43Þ

where τ is the time step; tc is the characteristic time in unities t0 (which needs to be

picked up); TM is the instantaneous temperature; T is the given temperature.


3. The Nose�Hoover [88,89] algorithm consists of introduction in the dynamic equations of

an additional variable, which is connected with the thermal tank:

_ri 5pi

m; ð6:44aÞ

_pi 5 fi 2 ξpi; ð6:44bÞ

ξ5P

iαp2iα=m2 gkBT

Q� η2T

Piαp

2iα=m

gkT2 1

�5 η2T

TM

T2 1

�; ð6:44cÞ

where pi is the impulse of ith particle; _pi is the derivative on time from the impulse; fi is the

force which operates on ith particle; ξ is the friction coefficient which can vary in due course;

Q is the thermal lag, this parameter can be substituted to ηT5 gkT/Q, that is, the parameter of

relaxation for thermal oscillations; g5 3N is number of degrees of freedom TM is the instanta-

neous temperature. In this case, the distribution function for ensemble is proportional:

expð2W=kTÞ;where

W 5H11

23NkTξ2=η2T; H5

Xiα

p2iα=m1U;

U is the potential energy of the system.

Energy of system depends on time:

_H5Xiα

piα _piα=m2X

fiα _riα 52 ξXiα

p2iα=m: ð6:45Þ

If TM. T, the system is too hot, the friction coefficient ξ becomes plus and the

system starts to be cooled. If the system is too cold, there is an inverse—the fric-

tion coefficient becomes subzero and the system starts to heat up.

There are other algorithms for maintaining of stationary temperature, for exam-

ple, Andersen’s [86] and Lanzaven’s [90] algorithms. They consider the systems

that have stationary interaction with virtual particles and interchanges with them by

energy. Practically, these interactions are necessary for replacement of the velocity

of atoms on new velocities, according to the Boltzmann distribution law by ener-

gies [86] or to the normal distribution on velocities [90].

NPT ensemble:

Andersen [86] offered the method of stabilization of pressure in which the addi-

tional variable V is the cell volume. It is equivalent to piston operations on real sys-

tem. The piston has “mass” Q (last has dimensionality kg/m4) and the kinetic

energy EV 5 0:5QV2, and the potential energy UV5PV connected with the volume

of calculated cell, where P is the yielded pressure. Potential and kinetic energies of

atoms are connected with variables r and v, which are standardized according to

the following relations:

r5V1=3s; ð6:46aÞ


v5V1=3_s: ð6:46bÞ

Potential energy of particles is a function of coordinates of particles U5U (V1/3s),

and accordingly, the first component of the Lagrangian (Lv5Ek1 EV2U2UV)

has the following form:

Ek 5 1=2mXi

υ2i 5 1=2mV2=3Xi

_s2i : ð6:47Þ

Equations of motion will be such:

€si 5 fi=ðmV1=3Þ2 ð2=3Þ_si _V=V ; ð6:48Þ

€V 5 ðPM 2PÞ=Q; ð6:49Þ

where €si is the acceleration ith particle, fi is the reduced force, PM is the pressure

instantaneous value. The Hamiltonian of this system, H5Ek1EV1U1UV, is

conserved; it is equaled an enthalpy with the additional factor kT, connected with

the kinetic energy of fluctuations of volume. Trajectories, which correspond to the

isobaric-isoenthalpic NPH ensemble (the number of particles, pressure and enthalpy

are constant), satisfy the equations of motion.

Application of this method in the MD model is described in Ref. [91]. As the

differential equation for volume is included in the dilated equations of motion,

forces and pressure are calculated, with use of not normalized coordinates. Authors

used the equation for not standardized velocities when calculating the trajectory of

atom movement:

_r5V1=3 _s1 ð1=3ÞV22=3 _Vs:

The method is not realized precisely in circuits of calculations as the value _swhich enters in Eq. (6.46) do count on each iterative time step δt. Parameter Q is

selected concerning a certain system; too small value Q result ins to fast oscilla-

tions of volume of the calculated cell that makes it impossible to maintain station-

ary pressure. A large mass leads to slow oscillations of the volume; and infinitive

mass leads to the usual MD modeling. Value of magnitude Qσ4=m5 0:0027 for the

system of atoms with the Lennard�Jones potential, where σ is the parameter of the

Lennard–Jones potential, was offered in refs. [92,93].

The isobaric�isoenthalpic ensemble almost does not apply. More often, the

method of stationary pressure is combined with the certain method for keeping of sta-

tionary temperature—NPT ensemble. With this purpose, Berendsen [87] modified the

Lagrangian so that it has led to occurrence of additional equations of motion (addi-

tional to Eq. (6.49)), which ensure the return of pressure to the stationary value:

dPM

dt5

P2PM

tp; ð6:50aÞ


_r5χ1=3r; ð6:50bÞ

χ5 12βT

δttpðPM 2PÞ; ð6:50cÞ

where P is the necessary pressure, tp is the time constant. The volume of the cell,

which is connected with the factor χ1/3 is continually calculated (every time step),

and coordinates of center of mass are connected with magnitude χ1/3, βT is the iso-

thermal compressibility. A precisely definition βT is unessential, as this factor can

be included in the temporary constant tp. Berendsen [87] has found that use of

values tp, from the interval from 0.013 10212 to 0.13 10212 s, results in to satis-

factory outcomes during water simulation. The method essentially does not influ-

ence dynamic trajectories and is easily implemented in programs. However, it is

difficult to determine a type of ensemble, which is applied.

Hoover [89] offered a set of equations for supporting the NPT ensemble in the

spirit of the procedure of Andersen:

_s5 p=mV1=3; ð6:51aÞ

_p5 f 2 ðχ1 ξÞp; ð6:51bÞ

_ξ5 fkðTM 2 TÞ=Q; ð6:51cÞ

χ5 _V=3V ; ð6:51dÞ

_χ5 ðPM 2PÞV=t2pkT ; ð6:51eÞ

where Ti 5P

i pi�� 2=ðmfkÞ is the instantaneous temperature, PM is the instantaneous

pressure, tp is the relaxation time for the pressure.

6.4.2 Reaching the Equilibrium State and Measuring MacroscopicParameters

During simulation by the MD method, the sweeping process often borrows the

lion’s share from a blanket time of calculation. As a rule, it is more convenient to

choose in initial conditions the “equilibrium” configuration from any former calcu-

lation fulfilled for the temperature and density, close to those parameters which are

required. The molecular dynamics develop the molecular configuration of a finite

size, changing it in time step by step.

There are limits for typical periods of time and scales of length, for which the

system can be explored. 103�105 steps on time (#10215 s) fulfilled at modeling by

the MD algorithm correspond usually to several nanoseconds of real time. It is nec-

essary to check whether in system the equilibrium has been reached before to


calculate average values of magnitudes. Besides, it is necessary to fulfill statistical

analysis of results of simulations to estimate the possible errors realistically. How

long is it necessary to carry out calculation? It depends on the system and physical

properties, which are of interest.

Suppose the variable a, defined so that hai5 ð1=NÞPiai 5Const: in equilibrium

state, (e.g., ai is the velocity ith atom). Enter temporary function of correlations of

values a2P

iaiðt0Þaiðt0 2 tÞ, which connects the values calculated through time

terms t. Designate it as ,a(t0) a(t01 t).. If the system is in equilibrium state, this

function does not depend on the choice of the time origin; so it may be defined as

,a(0) a(t).. It is clear that ,a(0) a(0). 5, a2.. In case of large time t:

limt!N

hað0ÞaðtÞi5 limt!N

hað0ÞihaðtÞi5 0; ð6:52Þ

as variables a(0) and a(t) any more do not correlate. Hence, the correlation function

decays, it is decay happens for the characteristic time τa. It is formally possible to

determine time of correlation from the relation:

τa 5ðN0

hað0ÞaðtÞiha2i dt: ð6:53Þ

If correlations decay in time under the exponential law, time τa can be deter-

mined by the equation:

hað0ÞaðtÞi~ expð2 t=τaÞ: ð6:54Þ

High correlation of value a takes place within the characteristic time τa.The similar statement can be made also about properties, for which it is possible to

find space averages under the simulated volume L3. Space correlation functions

are determined through values of considered quantities in points with coordinates

roi—the initial coordinates of particles in the instant t5 0, and values of this quan-

tities in different points ri, rather close to roi. We will designate such correlations

,a(r0)a(r)..

As a rule, a considerable time of simulation is borrowed for reaching the equi-

librium state and cumbersome arithmetic, which is necessary for evaluation of

forces and energies. If the cutoff radius is small enough, a number of modes are

used to reduce the system relaxation time. For example, if it is necessary to simu-

late three-dimensional system of 8000 particles, one can simulate at first the smal-

ler system of 1000 particles and to give a chance to this small system to come to

the equilibrium at the necessary temperature. After the equilibrium is reached, the

small system can be doubled in each space direction to create the necessary system

of 8000 particles. After that, procedure of the establishing of the equilibrium is iter-

ated. The equilibrium is usually reached quickly in the complete system.

Potashnikov et al. [94] have given evidence that using the Graphics Processing

Unit (GPU) for simulations allows us to explore systems which contain


B1054 106 atoms during periods of time up to tens of microseconds that in some

approaches answers real experiments.

The equilibrium macrostate is characterized by several parameters, such as, for

example, the Kelvin temperature T, middle pressure P, volume V, and total energy E.

Kinetic definition of temperature follows from the theorem of the equidistribu-

tion: each square-law term which enters into expression for energy of classical sys-

tem in the equilibrium state at temperature T, has average value 1/2kT. From here, it

is possible to determine temperature T of system in d-measuring space by the

relation:

d

2NkT 5

X 1

2hmυ2di; ð6:55Þ

in which the sum is undertaken with all N particles of system and d velocity com-

ponents. Brackets h. . .i mean the average on time. Expression (6.55) is an example

of connections of macroscopic magnitude, in this case of temperature, with the

temporary average on trajectories of particles (the instantaneous kinetic energy of

system fluctuates). The relation (6.55) is valid in such form in the event that a

velocity of center of mass of system is equal to zero.

One more thermal value of a system is specific heat at constant volume

CV5 (@E/@T)V, CV is the measure of the heat necessary for the temperature change

in 1 K. As heat capacity depends on system sizes, it is convenient to define specific

heat per the particle, namely cV5CV/N. It is easier to obtain cV by determination

of the average potential energy and the average total energy at the close tempera-

tures T and T1ΔΤ. Temperature dependence cV is predetermined by the tempera-

ture dependence of the potential and kinetic energies, the last is directly

proportional to temperature B(d/2)kT.

The method of definition of pressure based on the virial theorem is featured in

Section 6.3.5. In the MD program for simulation (Section 9.10) the pressure is

defined through impulses of molecules intersected boundaries of the basic cell. We

will consider the surface element ΔA; and it is admissible that the average momen-

tum, which is transferred in unit of time through the surface from left to right, will

be K1 and K2 is the average momentum, which is transferred through the surface

from right to left. Then average force F is equal F5 0.5(K11 jK2j), and middle

pressure is determined by expression:

P5Fn=ΔA; ð6:56Þ

where Fn designates the force component, normal to the surface element. In the

two-dimensional case, the pressure is equal to flux of impulse through the unit seg-

ment, instead of through the unit surface.

6.4.3 Kinetic Properties

Consider the properties of atoms, which are connected with their mobility. We will

present that we watch the trajectory of the certain particle. Let in any way selected


instant t1, its location is defined by the vector r1. We will define displacement of

the particle to some following instant t2. It is known that if the summarized force

operating on particle Fi, is equal to zero, and its displacement grows linearly with

time. However, each particle tests many collisions in the condensed state, and on

the average, its integral displacement is practically equal to zero. Interest is repre-

sented with magnitude of average quadrate of displacement determined by the

equation:

hΔr2it 51

N

XNi51

½riðtÞ2rið0Þ�2; ð6:57Þ

where h?i are the brackets of the statistical average on all particles of system. As

the system is in equilibrium, time datum is arbitrary and the average in Eq. (7.44)

depends only on the time slice Δt. It is known that the functions ,Δr2 (t).dependence on t allows us to calculate diffusivities. Diffusivity D at some tempera-

ture T can be calculated through declination of the dependence ,Δr2 (t). of

mean-square displacement of particles from time t:

,Δr2ðtÞ. 5 6Dt1C; ð6:58Þwhere C is the constant, which characterizes the system dynamics deviation from

the Markov type dynamics (for the Markovian processes C5 0).

Velocity autocorrelation function (VAF) [95] is calculated to characterize

dynamic properties of particles. vi(t) is the random value of the velocity of the par-

ticle in the instant t provided that in the initial instant its velocity equaled vi(0).Average value ,v(t). 5 0. If the integral force, which operates on ith particle, is

equal to zero, the velocity of the last will be the stationary value. However, the par-

ticle velocity will vary at the expense of interaction with other particles, and it is

reasonable that through a certain time, the velocity will not correlate noticeably

with the initial velocity. By definition, VAF is written below function Z(t):

ZðtÞ5 hvið0ÞviðtÞihvið0Þvið0Þi

; ð6:59Þ

where vi(0) and vi(t) are the vectors of velocities in the initial instant and in instants

t, h?i are the brackets of the statistical average with all particles of system.

Dynamics of atoms can be characterized by the spectrum of atom oscillations g

(ω), obtained by the Fourier transformation of VAF:

gðωÞ5 1

π

ðN0

ZðtÞexpð2iωtÞ dt: ð6:60Þ

Let us separate the true part of the Eq. (6.60) for pictorial map,

gðωÞ5 2

π

ðN0

ZðtÞcosðωtÞ dt: ð6:61Þ


The form of function Z(t), namely its decay, contains information about dynamic

relaxation in materials. One can see from Figure 6.7A and B that time of the

dynamic relaxation in materials with nanocrystals of different size is different. In

case of samples with nanocrystals of small size (Figure 6.7A) and amorphous sam-

ples, oscillation character VAF and their spectrums almost do not differ.

Uncorrelated oscillations are characteristic for atoms in these samples. In the case

of samples with nanocrystals of large size, the VAF oscillations decay more slowly,

and in spectrum legible bifurcation of the maxima (Figure 6.7B) is observed.

The VAF is connected with diffusion mobility of atoms of system. The equation

for calculation of diffusivity D is known,

D51

3N

ðN0

XNi51

hvið0ÞviðtÞi* +

dt; ð6:62Þ

where N is the number of atoms in system, vi(0) is the velocity ith atom in the ini-

tial instant, vi(t) is the velocity ith atom in the instant t. Actually, this equation can

be applied to the liquid state, for which diffusivities are large enough.

The important information concerning evolution of the short-range order in the

spatial distribution of atoms of amorphous and liquid materials can be obtained

from calculations of the van Hove spatio-temporal correlation function (dependent

on time) [97]. Rahman [98] gave evidence that the van Hove correlation functions,

calculated during time of simulation by the MD method, can be compared with

experimental data, namely with intensity of scattering of slow neutrons by the sub-

stance. He offered the simplified variant of calculation of the one of the van Hove

functions Gd(r,t),

Gdðr; tÞ5Ω0½nðr; tÞ=4πr2Δr�; ð6:63Þ

20(A) (B)

z(t

)

z(t

)

1.0

0.5

0.0

–0.5

1.0

0.5

0.0

–0.50 2 4

t, 10–13S t, 10–13S6 8 10 0 2 4 6 8 10

12

10

8

6

4

2

0

15

10

f(ν)

/f(0

)

f(ν)

/f(0

)

5

00 2 4 6 8 0 2 4 6 8

ν (1013S–1) ν (1013S–1)

Figure 6.7 Autocorrelation functions of the velocity and their spectrums from nanocrystals

of different size in systems of pure iron [96], T5 300 K; (A) small nanocrystals (D350

atoms), (B) large nanocrystals ($2000 atoms).


where n(r,t) is the number of particles in the instant t, placed in the range of dis-

tances from r to r1Δr, (r is the distance from the position of the reference parti-

cles in the instant t5 0), Ω0 is the volume in counting per one particle.

The function Gd(r,t) coincides with the g(r) function in the instant t5 0—the

equilibrium RPDF (particle distribution). The time dependence of function Gd(r,t)

reflects relaxation of the system structure, the initial spatial correlations become

lost in due course that is connected with displacement of particles.

One more function Gs(r,t) yields the density of probability of finding of the par-

ticle in the point r in the instant t, if it was at the beginning of coordinates at t5 0.

According to Ref. [99],

Gsðr; tÞ5 1

N

XNi51

δ½r2 jriðtÞ2 rið0Þj�* +

; ð6:64Þ

where jri(t)-ri(0)j are the modules of relocating of atoms, and function

δ½r2 jriðtÞ2 rið0Þj� is equal to unity in cases, when jri(t)-ri(0)j belongs to the inter-

val from r to r1Δr, differently this function is equal to zero.

It is clear that dependence of mean-square displacement of particles on time is

defined by the function Gs(r,t):

hr2i5ðr2Gsðr; tÞ dr: ð6:65Þ

Considering that function Gs(r,t) partially falls under the partition Gauss law,

some authors study in what measure this function differs from the normal distribu-

tion in all intervals of the (r,t) values. More information about van Hove’s func-

tions is in reviews [99,100].

References

[1] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford Science

Publications, Clarendon Press, Oxford, 1987.

[2] D.M. Beazley, P.S. Lohmdahl, N. Gronbech-Jensen, R. Giles, P. Tamayou, Parallel

algorithms for short-range molecular dynamics, Annual Reviews of Computational

Physics, vol. 3, World Scientific, Singapore, 1995, pp. 119�175.

[3] G. Kotliar, S.Y. Savrasov, K. Haule, V.S. Oudovenko, O. Parcollet, C.A. Marianetti,

Rev. Mod. Phys. 78 (2006) 865�951.

[4] H. Eschrig, The Fundamentals of Density Functional Theory, Teubner, Stuttgart, 1996.

[5] J. Callaway, N.H. March, Solid State Phys. 38 (1984) 135.

[6] R.A. Johnson, Phys. Rev. B 6 (1972) 2094.

[7] J.C. Phillips, L. Kleinman, Phys. Rev. 116 (1959) 287.

[8] N.W. Ashcroft, Phys. Lett. 23 (1) (1966) 48.

[9] A.O.E. Animalu, Phys. Rev. B 8 (8) (1973) 3542.

[10] Z.A. Gursky, G.L. Krasko, Rep. AN SSSR 197 (4) (1971) 810.


















[11] H.R. Lelibaux, A.W. Engel, J. Chem. Phys. 68 (1) (1978) 1.

[12] R.G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford

University Press, New York, NY, 1989.

[13] N.H. March (Ed.), Electron Correlation in the Solid State, ICP, London, 1999.

[14] G. Kresse, J. Non-Cryst. Solids 192 (1995) 222.

[15] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864.

[16] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133.

[17] R.P. Feynman, Phys. Rev. 56 (1939) 340.

[18] D.J. Gonzalez, L.E Gonzalez, Condens. Matter Phys. 11 (1) (53) (2008) 155�168.

[19] R. Car, M. Parrinello, Phys. Rev. Lett. 55 (1985) 2471.

[20] G. Pastore, E. Smargiassi, F. Buda, Phys. Rev. A 44 (1991) 6334.

[21] G. Kresse, J. Furthmiiller, Comput. Mater. Sci. 6 (1996) 15�50.

[22] I. Stich, R. Car, M. Parrinello, S. Baroni, Phys. Rev. B 39 (1989) 4997.

[23] M.J. Gillan, J. Phys. Condens. Matter 1 (1989) 689.

[24] T.A. Arias, M.E. Payne, J.D. Joannopoulos, Phys. Rev. Lett. 69 (1992) 1077.

[25] R.R. Zope, Y. Mishin, Phys. Rev. B 68 (2003) 024102.

[26] T. Hong, T.J. Watson-Yang, X.Q. Guo, A.J. Freeman, T. Oguchi, J. Xu, Phys. Rev. B

43 (1791) 1940.

[27] T. Hong, T. Watson-Yang, A. Freeman, T. Oguchi, J. Xu, Phys. Rev. B 41 (N12)

(1990) 462.

[28] C.L. Fu, J. Mater. Res. 5 (1990) 971.

[29] M.S. Daw, M.I. Baskes, Phys. Rev. Lett. 50 (1983) 1285.

[30] M.S. Daw, M.I. Baskes, Phys. Rev. B 29 (1984) 6443.

[31] M.S. Daw, S.M. Foiles, M.I. Baskes, Mater. Sci. Rep. 9 (1993) 251.

[32] M.W. Finnis, J.E. Sinclair, Philos. Mag. A 50 (1984) 45.

[33] B.J. Lee, M.I. Baskes, H. Kim, Y.K. Cho, Phys. Rev. B 64 (2001) 184102.

[34] Y.F. Ouyang, B.W. Zhang, S.Z. Liao, Z.P. Jin, Z. Phys. B 101 (1996) 161.

[35] F. Ducastelle, F. Cryrot-Lackmann, J. Phys. Chem. Solids 31 (1971) 1295.

[36] C. Massobrio, V. Pontikis, G. Martin, Phys. Rev. B 41 (1990) 10486.

[37] F. Willaime, C. Massobrio, Phys. Rev. B 43 (1991) 11653.

[38] F. Cleri, V. Rosato, Phys. Rev. B 48 (1993) 22.

[39] T.J. Raeker, A.E. DePristo, Phys. Rev. B 49 (1994) 8663.

[40] D. Raabe, Computational Materials Science, Wiley-VCH, New York, NY, 1998.

[41] J.H. Li, X.D. Dai, S.H. Liang, K.P. Tai, Y. Kong, Phys. Rep. 455 (2008) 1�134.

[42] F.J. Cherne, M.I. Baskes, P.A. Deymier, Phys. Rev. B 65 (2001) 024209.

[43] M.I. Baskes, Phys. Rev. Lett. 59 (1987) 2666.

[44] M.I. Baskes, J.S. Nelson, A.F. Wright, Phys. Rev. B 40 (1989) 6085.

[45] M.I. Baskes, Phys. Rev. B 46 (1992) 2727.

[46] M.I. Baskes, R.A. Johnson, Modell. Simul. Mater. Sci. Eng. 2 (1994) 147.

[47] B. Jelinek, J. Houze, S. Kim, M.F. Horstemeyer, M.I. Baskes, Phys. Rev. B 75 (2007)

054106.

[48] A.M. Dongare, L.V. Zhigilei, A.M. Rajendran, B. LaMattina, Composites Part B 40

(2009) 461.

[49] B.J. Thijsse, Phys. Rev. B 65 (2002) 195207.

[50] A. Inoue, S. Banerjee, Adv. Phys. Metall. 127 (1996) 147.

[51] A. Inoue, Mater. Sci. Eng. A 57 (1994) 179.

[52] A.P. Shpak, V.V. Maslov, A.B. Melnik, A.N. Timoshevsky, Met. Phys. Adv. Technol.

25 (11) (2003) 1461.

[53] Y. Mishin, Phys. Rev. B 65 (2002) 224114.






















































[54] D. Farkas, Modell. Simul. Mater. Sci. Eng. 2 (1994) 975.

[55] S.M. Foiles, MRS Bull. 31 (1996) 24�28 (special ed.).

[56] M.I. Baskes, J.E. Angelo, C.L. Bisson, Modell. Simul. Mater. Sci. Eng. 2 (1994) 505.

[57] J.B. Sturgeon, B.B. Laird, Phys. Rev. B 62 (2000) 14720.

[58] H.N.V. Temporally, J.S Rowlinson, G.S. Rushbrooke, Physics of SimpleLiquids, North

Holland, Amsterdam, 1968, 400 p.

[59] A.M. Ovrutsky, A.S. Prokhoda, Met. Phys. Adv. Technol. 30 (8) (2008) 1119 (in

Russian).

[60] A.P. Shpak, A.B. Melnik, Micro-Inhomogeneous Structure of Unordered Metal

Systems, Academperiodika, Kiev, 2005. 322 p. (in Russian).

[61] G.F. Voronoi, Collected Works, Ukraine Academy of Sciences, Kiev, 1952, 239 p.

[62] B.N. Delone, Petersburg School of Theory of Numbers, USSR Academy of Sciences,

Moscow, Leningrad, 1947, 196 p.

[63] J.D. Bernal, Nature 183 (1959) 141.

[64] M.G. Alinchenko, A.V. Anikeenko, N.N. Medvedev, V.P. Voloshin, M. Mezei,

P. Jedlovszky, J. Phys. Chem. B 108 (2004) 19056.

[65] N.N. Medvedev, The Voronoi�Delone Methods in Study of Noncrystalline Systems,

Siberian Branch of USSR Academy of Sciences, Novosibirsk, 2000. 352 p. (in Russian).

[66] J.P. Hansen, I.R. McDonald., Theory of SimpleLiquids, third ed., Academic Press,

London, 2006, 416 p.

[67] A.A Chernov, Modern Crystallography III: Crystal Growth, Springer, Berlin, 1984.

[68] J.P. van der Eerden, C. van Leeuwen, P. Bennema, et al., J. Appl. Phys. 48 (6) (1977)

2124.

[69] G.H. Gilmer, J. Cryst. Growth 42 (1977) 3.

[70] H.J. Leamy, K.A. Jackson, J. Appl. Phys. 42 (5) (1971) 2121.

[71] A.M. Ovrutsky, I.G Rasin, J. Surf. Invest. 12 (2002) 65.

[72] A.M. Ovrutsky, M.S. Rasshchupkyna, A.A. Rozhko, DNU Reports Ser. Phys. Radio

Electron. 12 (2) (2004) 100 (in Russian).

[73] A.M. Ovrutsky, M.S. Rasshchupkyna, A.A. Rozhko, Phys. Chem. Solid State 5 (3)

(2004) 498.

[74] J. Dalla Torre, M.D. Rouhani, R. Malek, et al., J. Appl. Phys. 84 (10) (1998) 5487.

[75] A.M. Ovrutsky, M.S. Rasshchupkyna, Mater. Sci. Eng. A 495 (2008) 292.

[76] L. Nurminen, A. Kuronen, K. Kaski, Phys. Rev. B 63 (2000) 35407.

[77] K.E. Khor, S. Das Sarma, Phys. Rev. B 62 (2000) 16657.

[78] A.M. Ovrutsky, M.S. Rasshchupkyna, A.A. Rozshko, S.S. Antropov, DNU Reports

Ser. Phys. Radio Electron. 13 (2) (2006) 107.

[79] A.M. Ovrutsky, M.S. Rasshchupkyna, A.A. Rozshko, J. Surf. Invest. (1) (2006)

85�91.

[80] A.M. Ovrutsky, V.N Sheludko, Surf. Ser. Phys. Chem. Mechanics 12 (1989) 36

(in Russian).

[81] M. Hansen, K. Anderko, Constitution of Binary Alloys, vol. 2, McGraw-Hill, New

York, NY, 1958.

[82] A.G. Lesnik, Models of Atom Interaction in the Statistical Theory of Alloys,

Fizmatgiz, Moscow, 1962 (in Russian).

[83] D.K. Belashchenko, Computer Simulation of Liquid and Amorphous Matters, MISSIS,

Moscow, 2005. 407 p. (in Russian).

[84] C.G. Gray, K.E. Gubbins, Theory of Molecular Fluids. 1. Fundamentals, Clarendon

Press, Oxford, 1984, 351 p.

[85] M.P. Allen, D.J. Tildesley, NATO ASI Ser. C 397 (1993) 211.






















































[86] H.C. Andersen, J. Chem. Phys. 72 (1980) 2384.

[87] H.J.C. Berendsen, J. Chem. Phys. 81 (1984) 3684.

[88] S. Nose, Mol. Phys. 52 (1984) 255.

[89] W.G. Hoover, Phys. Rev. A 31 (1985) 1695.

[90] J. Kohanoff, A. Caro, M.W. Finnis, J. Chem. Phys. 6 (2005) 1848.

[91] J.M. Haile, H.W. Graben, J. Chem. Phys. 73 (1980) 2412.

[92] D. Brown, J.H.R. Clarke, Mol. Phys. 51 (1984) 1243.

[93] J.R. Fox, H.C. Andersen, J. Chem. Phys. 88 (1984) 4019.

[94] S.I. Potashnikov, A.S. Boyarchenkov, K.A Nekrasov, Altern. Power Eng. 49 (N5)

(2007) 86.

[95] Rahman, J. Chem. Phys. 45 (7) (1964) 2585.

[96] A.M. Ovrutsky, A.S. Prokhoda, Physical and Chemical Foundations of Micro- and

Nano-Structures Modifications, Karasin Kharkov University, Kharkov, 2008. p. 467

(in Russian).

[97] L. van Hove, J. Phys. Rev. 95 (1954) 249.

[98] A. Rahman, J. Phys. Rev. 136 (2) (1964) A405.

[99] I.Z. Fisher, R.M. Yul’ment’tv, Adv. Phys. Sci. 87 (10) (1965) 374.

[100] A.N. Lagar’kov, V.M. Sergeev, Adv. Phys. Sci. 125 (3) (1978) 409.




















7 The Surface Processes DuringCrystallization

7.1 Surface Energy and Equilibrium Forms of Crystals

7.1.1 Surface Energy in the First Approximation and its Anisotropy

As we know, the interfacial tension coincides with the free surface energy in the

case of one-component systems, σ5Fω/ω5 uω�Tsω, where uω and sω are the sur-

face internal energy and entropy related to the unit of area. In the case of rather

ordered surfaces of crystals, the contribution of entropy term is small. We will esti-

mate the energy change at surface formation according to Ref. [1]. Let us separate

by imagined surfaces two individual phases, crystalline and liquid (Figure 7.1A).

Break-off phases and, accordingly, bonds of atoms ss and ll by the tearing sur-

face then move the upper parts (will change their places) and link them with the

lower parts (Figure 7.1B). Thus, there are two phase boundaries, and new bonds of

type sl are formed: 1ss1 1ll!2sl. Therefore, modification of the internal energy

(per unit of area) will make γDuω5 nsεsl, εsl5Vsl2 (Vss1Vll)/2, where ns is the

surface concentration of atoms, εsl is the excess energy of sl bonds counted per one

bond of one atom. For an estimation of superficial energy of the crystal�melt

boundary surface the approach VslDVll is usually used. Then εslD(Vll�Vss)/

25ΔH/z, where ΔH is the heat (the change of enthalpy) of phase transformation,

z is the coordination number. A melting heat of metals is rather small, 8�13 kJ/

mol. In the case of the crystal�vapor boundary surface, bonds of the surface atoms

are unsaturated, εs5VssDΔH/z, where ΔH is the evaporation heat.

Evaporation heats of metals ΔH5 100�300 kJ/mol. Let us take the value

ΔH5 166 kJ/mol. For the FCC crystal lattice z5 12, ε5ΔH/NA/

125 2.33 10220 J; each surface atom of the facet (111) has three uncompensated

bonds, and its area is a2ffiffiffi3

p=8D43 10220 m2 (a is the parameter of a crystal

lattice). Therefore, taking into account bonds only of the nearest neighbor atoms,

we have γ111 5 8ffiffiffi3

pΔH=ðza2Þ5 2 J=m2. Molecular crystals have evaporation heats

two to three times lower than metals, and intermolecular distances two to three

times larger. Therefore, surface energies are 20�50 times less than the energies of

the metal crystal�vapor boundary.

Surface energy of crystals, γ, is anisotropic. Values γ have minima for the sur-

faces, which coincide with densely packed planes of a crystal lattice (F-facets).

Calculate an energy of the surfaces feebly declined from orientation of the F-facets.



http://dx.doi.org/10.1016/B978-0-12-420143-9.00007-7

Such surfaces are termed vicinal (from “vicinus,” meaning “close” in Latin). The

vicinal surface shown in Figure 7.2 is built of flat terraces having the unity width

and height a. Write down the total surface energy of the surface section with length

l designating the free surface energy of the F-facets as γ010 and γ001,

γl5 l sin θ γ010 1 l cos θ γ001 5

lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ2010 1 γ2001

psin θ

γ010ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ2010 1 γ2001

p 1 cos θγ001ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

γ2010 1 γ2001p

0@

1A

lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ2010 1 γ2001

pðsin θ sin Ψ1 cos θ cos ΨÞ:

Imagine a rectangular triangle with values of sides γ010 and γ001 (a hypotenuse

is equal toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ2010 1 γ2001

p). The ratio of sides to the hypotenuse in the last equation

is designated as sin ψ and cos ψ. As a result, we have

γ5λffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ2010 1 γ2001

qcosðθ2ΨÞ: ð7:1Þ

If γ0105 γ001, γ takes over maximum value for the orientation ψ5 45� and for

angles 135�, 225�, and 315�. Hence, the surface energy increases at magnification

of the angle θ orientations of a surface concerning the F-facet orientation. But mag-

nification of θ will lead in due course to approach of orientation of a surface to ori-

entation of other F-facet and reduction of the surface energy. The surface energy

(001)

l

α(010)θ

Figure 7.2 The scheme of the vicinal surface that has an angle θ relatively the singular

orientation.

S

S

(A)

ω ω

L

L

L

S

(B)

S

L

Figure 7.1 Hypothetical formation of boundary surface.


has minimum for the F-facets. If such minima are acute, the dependence γ(θ) hasespecial singular points, in which derivatives γ0(θ) test tearing up. Therefore, the

surfaces with the F-facet orientation are termed singular. The dependence γ on the

surface orientation is mapped usually by means of polar diagrams. Such diagrams

are three dimensional, as surface orientations are determined by two angles.

Figure 7.3 shows the two-dimensional section of the polar diagram of the surface

energy constructed according to Eq. (7.1), and Figure 7.4 shows the equilibrium

shape of a crystal, which corresponds to this polar diagram.

7.1.2 Equilibrium Forms of Crystals

Actually, polar diagrams are much more complex [1]; they have more minima even

in their sections. Knowing the polar diagram gives the possibility to construct the

equilibrium shape of crystals. On the other hand, it is possible to find the polar dia-

gram from experimental data on the equilibrium shape. Definition of the equilib-

rium shapes is complicated in experiments because the crystal needs to be standing

long in equilibrium with the environment. But the crystal of limited size, which is

Figure 7.3 The two-dimensional polar diagram of the surface free energy.

Figure 7.4 The equilibrium shape of a crystal that corresponds to the polar diagram shown

in Figure 7.3.

189The Surface Processes During Crystallization

in equilibrium with environment, is a critical nucleus (see Figure 2.7), and this

equilibrium is labile. Crystal habitus is defined by the set of its facets. The equilib-

rium shape can contain facets and rounded parts between them. There is often a

disordering of the surface structure at increase of temperature; the crystal surface

becomes rough in atomic scale. Therefore, the polar diagram also varies, the sharp

points disappear on its minima, and thus the singularity disappears. Accordingly,

the rounded areas in the equilibrium form appearance. The area of facets decreases

at the subsequent rise in temperature, and then they disappear completely. The

equilibrium shape becomes round, and crystals grow in the form of rounded den-

drites in the supersaturated medium.

Herring investigated the local equilibrium of sites of the crystal surface with the

environment in cases of any orientation, which is determined by the angles θ and ϕcharacterized declinations from orientation of the F-facets [1]. He found that the

local values of temperatures and concentrations near the surface, at which the

dynamic equilibrium takes place (between a number of particles, which come off,

and those which join), are determined by the equation

μenv 5μcr 1ΩR1

γ1@2γ@θ2

� �1

ΩR2

γ1@2γ@ϕ2

� �; ð7:2Þ

where μenv and μcr are the chemical potentials of the environment (the local value)

and crystal; R1 and R2 are the surface radiuses of curvature in two perpendicular

cross sections, which comprise a normal line vector to the surface.

According to Herring’s theory, if the polar diagram of the free surface energy is

known, an equilibrium shape is interior enveloping sets of planes, tangential to the

polar diagram in its each point. If on the polar diagram there are singular minima,

there are facets certainly on the crystal equilibrium shape, which correspond to

these singular points, because the planes tangential to the polar diagram of the sur-

face energy with angles which are a little deviated from a singular orientation never

belong to the interior enveloping.

7.1.3 The Curie�Wulff Principle

The Curie�Woolf principle is correct for crystals with perfect facets. According to

the principle, the free surface energy for a crystal, which is in equilibrium with an

environment, should have minimum

Xi

γiωi 5 0; ð7:3Þ

where ωi is the area of the ith facet. Considering this principle, Curie and Woolf

came to the conclusion that the crystal equilibrium shape is connected with the ani-

sotropic interfacial tension in such a manner that distances of facets from crystal

center hi (Figure 7.4) are proportional to the surface energies of facets γi.


Polyhedron volume is V 5 ð1=3ÞPihiωi. The volume of a crystal is constant for

equilibrium shape (in fact it is volume of a critical nucleus); the deviation in volume

is mainly connected with the change in distances from faces to the center. Hence,

dV 5Xi

ωidhi 5 0: ð7:4Þ

Having written down full differential from volume formally and having consid-

ered (Eq. (7.4)), we find thatP

ihidωi 5 0. And the minimum from expression for

the surface energy (Eq. (7.3)) givesP

iγidωi 5 0. The last two equalities come out

right simultaneously, if hiBγi.Thus, the less the surface energy, the closer the facet to the center of the crystal

and the larger its area more in comparison with other facets. The free surface

energy is minimal for facets, which correspond to crystal planes densely packed by

atoms. These facets are basic in the crystal habitus. At the increasing temperature

(e.g., at movement upward along a liquidus line and approaching of the composi-

tion of binary melts into the pure component), additional facets arise, and their rela-

tive area is gradually increasing. Then roundish sites can form. Round crystals

grow usually from pure melts of metals, and faces appear at the lowered tempera-

ture and certain concentration of melts (Figure 7.5), connected by balance

conditions—a liquidus line on the phase diagram.

7.2 Atomic Structure of Crystal Surfaces

7.2.1 Lifetime and Diffusion of Adsorbed Atoms

Adsorption of gas molecules on the surfaces of solids was considered in Chapter 5.

If the crystal is in contact with own vapor, adsorption will also take place. Almost

all possible places are occupied in the main atom layer of the crystal face, which

corresponds to a certain crystal plane (the surface structure can be reconstructed, it

was described in Chapter 4). The atoms, which have joined from the own vapor,

have the small number of bonds, and consequently the time of their life at the sur-

faces is small. Dynamic balance between quantities of atoms, which are joining

and leaving the surface, amounts in the course of time under the equilibrium pres-

sure. As mentioned above, the number of atoms, which falls from gas on the unit

area of a surface, is proportional to the pressure, I1 5P=ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πmkT

p, where m is the

molecule mass, k is the Boltzmann constant, and T is the temperature.

Desorption (or evaporation) molecules will occur only from the occupied part of

the surface. Intensity of the stream, which is evaporating, is proportional to concen-

tration of adsorbed atoms ns, I25 fns, f is the frequency of desorption of adsorbed

atoms, it depends on frequency of their oscillations υ and Boltzmann’s factor from

value of the adsorption energy Ua, f5 ν exp(�ua/kT).

Time of life of adsorbed atoms on the surfaces τs can be determined having put

probability of their detachment to be equal unit


P2 5 ν � expð�ua=kTÞτs 5 1; ð7:5Þ

(the magnitude ν exp(�ua/kT) gives the probability of a detachment of the

adsorbed atom during one oscillations). From here, we find

τs 5 ð1=νÞ exp ðua=kTÞ5 τ0expðua=kTÞ; ð7:6Þ

where τ0 is the time of one oscillation.

The structure of the crystal surfaces is such that every ad-atom is placed over

deepening between atoms of the main atom layer. In such places, ad-atoms interact

with several atoms of the bottom layer and consequently, are in the potential holes.

It is necessary to overcome the potential barrier uD for jump of the ad-atom in the

next deepening. This barrier is less than energy ua. Therefore, ad-atoms carry out

Figure 7.5 Forms of crystal growth at weak supercoolings of binary melts [2,3]: (A and B)

silver, temperature T5 700 and 450 K; (C and D) copper, T5 700 and 480 K; and (E and F)

aluminum, T5 580 and 400 K.


many jumps, and they pass the rather large distances during their lifetime τs. Thelifetime of an atom in one potential hole

τD 5 ð1=νÞ exp ðuD=kTÞ5 τ0 exp ðuD=kTÞ: ð7:7Þ

Apply Einstein’s formula (,x2. 5 2DΔt) at first to the moving connected

with one jump: a25 2DsτD, Ds5 a2/2τD. Then we will find an average moving

during the time τs:

λ2

s2 5 2Dsτs 5 a2

τsτD

;λ2

s 5

ffiffiffiffiffiffiλ2

s2

q5 a

ffiffiffiffiffiffiτsτD

r5 a exp

ua 2 uD

2kT

� �: ð7:8Þ

Time of one oscillation τ0 is B10213 s, uaD0.4ΔHev. τsD1027 s for the crys-

tals of zinc at temperature 600 K. As uDD0.1ua, we have estimated that

λ2

sD103a at average temperatures. An average run increases and can reach

1025�1022 cm at reduction of the temperature.

7.2.2 Structure of Steps on the Crystal Surface

In the first 20 years of twentieth century, Frenkel [4] suggested an idea that thermal

fluctuations can influence the structure of steps, having caused formation of breaks,

the concept about which has been introduced by Kossel and Stansky. Figure 7.6

shows the outcomes of the Monte Carlo (MC) modeling at various temperatures

(ratios ΔH/kT). There are structures of the Kossel crystal surface (simple cubic lat-

tice) with a step given in initial conditions. Initially, the sharp direct step becomes

a broken line with time of simulation because of formation of breaks. There are

ad-atoms and holes on the surface, and there are clusters from ad-atoms.

At increase of the temperature, the surface ceases to be sharp and turns into a tran-

sition zone with several atomic layers. The values of ΔH/kT given in subscripts to

figures are typical for the crystal�melt interfaces. In the case of the crystal�vapor

boundary, these values are much more, and the surface structure is considerably

more ordered.

Let us analyze a step roughness according to Ref. [1]. As a measure of a step

roughness, we will accept the ratio (U2U0)/U0, where U0 is the potential energy

Figure 7.6 Phase boundary cross section in the plane with a step, σ5Δμ/kT5 0.05:

(A) ΔH/kT5 7.5; (B) ΔH/kT5 5.4; and (C) ΔH/kT5 4.5.


of the direct step without breaks and U2U0 is the change of energy at the expense

of formation of breaks; atoms at the breaks have less bonds of type s�s (one bond

less in the case of Kossel’s crystal). Designated through ε energy of the broken off

bond, γl is the step edge energy (counting per the unit of length), n is the total

number of atom places on the step (U05 nε for Kossel’s crystal). Let n1 is the

number of breaks upward (the step {10}), n� is the number of breaks downwards;

n0 is the number of places on the step without breaks. For the direct step,

n15 n�5 nbr. Therefore, 2nbr1 n05 n.

Entropy of the rough step is determined by Boltzmann’s formula

S5 k lnW ;

where a thermodynamic probability

W 5n!

n0!n1!n2!5

n!

n0!ðnbr!Þ2:

According to definition,

γl 5U � TS5 nε1 2n3ε� kT ½n lnðn=eÞ � n0 lnðn0=eÞ � 2nbr lnðnbr=eÞ�:ð7:9Þ

The Stirling formula is taken into account here: ln(n!)5 n ln n�n.

We search for equilibrium structure; for this purpose, we find the minimum of

the expression for γl:

@γl=@nbr 5 2ε� kT ½2n0 lnðn0=eÞ � 2 lnðnbr=eÞ�5 0

(it is taken into account here that @/@nbr5�2(@/@n0)). It follows from the last

equation:

nbr

n05 e2ε=kT 5 η and as 2n0 η1 n0 5 n; n0 5

n

2η1 1; nbr 5

nη2η1 1

:

The average distance between breaks

λ3 51

2nbr5

an

2nbr5

a

2

11 2ηη

5a

2ð21 expðε=kTÞÞ; ð7:10Þ

ΔHev/kTfD25�30, ε/kTfD4�5 for the crystal�vapor boundary. At temperate tem-

peratures λ35 (102 100)a, and at low temperatures λ35 (1042 106)a. In the case

of the crystal�melt boundary, the heat of phase transition is much less and the

roughness of steps is much higher. Rewrite the relation for γl as follows:

γl 5 nε1 2nbrε� kT ½n0 lnðn=n0Þ � 2nbr lnðn=nbrÞ�:


Taking into account expressions for ratios n0/n and nbr/n, we will gain

γl 5 nε� 2nbrkT lnð11 2ηÞ � n0kT lnð11 2ηÞ;

or finally

γl 5 nε� nkT lnð11 2ηÞ5 � nkT lnðηð11 2ηÞÞ: ð7:11Þ

At low temperatures, γl. 0 always because nε. 0. At the temperature

increases, the edge energy decreases, and γl turns to zero at the roughening tem-

perature Tr, at which η5 0.5. It is obvious that ε/kTr5 0.7 at η5 0.5 and

ΔH/kTr5 4.2. At this and the higher temperatures, any clusters can be formed at

the surface because the free energy of steps which restrict them is equal to zero.

Such interface will be rough in the atom scale and multilayer. However, when the

width of interface goes to infinity, separate existence of phases is impossible.

Analysis of the roughness of steps in the model, that supposes existing of breaks

with one or two atoms in width, gives the condition of the γl vanishing at

ε/kTr5 0.8 and, accordingly, the condition ΔH/kTr5 4.8.

7.2.3 Roughness of the Crystal Faces

In the previous subsection, we came to the conclusion that the crystal faces, which

border with their own vapor, are feebly rough: the number of adsorbed atoms and

holes on their surfaces is small, and steps contain few breaks. The heat of melting

of all matters is much less than the latent heat of vaporization. Undimensional

entropy of melting ΔHf/kTf is within 1�2 for metals, 3�4 for semiconductors,

and5 5�10 for organic substances. However, there are some organic substances

similar to metals (with relatively small ratio ΔHf/kTf), e.g., camphor, camphene,

and succinonitrile. Crystals of metals have as a rule roundish forms of growth in

their own melts (Figure 7.5), which testifies a significant roughness of the interface

in the atomic scale.

Jackson [5] noted that all substances, for which ΔHf/kTf. 2, have faceted forms

of growth. The analysis of shapes of crystals growing from binary melts [3] has

shown that transition from faceted to roundish forms of growth responded to the

value of the ratio ΔH/kTr5 1.356 0.05. However, a much greater test of the

roughness (ΔH/kTr5 4.2 or 4.8) has been gained in the previous subsection.

Therefore, many authors analyzed the roughness of crystal surfaces, and a great

number of investigations have been devoted to modeling of crystal growth in

atomic scale.

Burton et al. [6] have applied Onsager’s solution of the problem of ordering

and magnetization in the two-dimensional model of Ising’s ferromagnetic for the

surface of Kossel’s crystal. They have gained that ΔH/kTr5 5.1. They also consid-

ered surfaces with several layers and analyzed ordering in Bette’s approach.

However, the values of the dimensionless entropy ΔS/k5ΔH/kTr were all the

same, too high.


Temkin [7] had made a weighable contribution to the theory. He considered

Kossel’s crystal with the interphase boundary of any width using the

Bragg�Williams approach, that is, without consideration of ordering at definition

of the entropy (as it is made for the above step).

Minimization of expression for the free energy of such boundary has allowed us

to research how the parameter εsl influences the concentration distribution of condi-

tionally “solid” atoms in the perpendicular direction to the interface (its depen-

dence on the layer number). Later, other authors carried out such analysis, taking

into account the ordering in consecutive layers, and some authors fulfilled the MC

modeling of the crystal surface. In all these works, the placement of “solid” atoms

over “liquid” atoms was forbidden. Figure 7.7 shows the outcomes of calculations

gained for two values of parameter ξ5 4εsl/kT. Both profiles of concentration

respond to rough surfaces. Temkin [7] discovered in the zero approach that the cri-

terion of transition from rough to the smooth surface is ΔH/kTr5 3.6.

Jackson [5] offered the simple model of interface with one transition layer and

analyzed it. His results were used in many textbooks. Let the number of atom

places equal N in such layer, and Ns is the number of “solid” atoms. If all solid

atoms are disposed together as one islet, the surface will be ordered completely.

Let us write the equation for the modification of free energy after casual place-

ment of solid atoms on the boundary surface

ΔF5ΔU2 TΔS5Nszsεslð12 xÞ2 kT lnN!

Ns!ðN2NsÞ!; ð7:12Þ

where expression for entropy is written according to the Boltzmann formula (the

thermodynamic probability is the number of placements of “solid” and “liquid”

atoms all along N sites); x5Ns/N; (1�x) is the part of liquid atoms. We apply the

Stirling formula and divide the equation by NkT:

ΔF

NkT5

zsεslkT

xð12 xÞ1 x ln x1 ð12 xÞlnð12 xÞ: ð7:13Þ

0.8

xi

0.6ξ = 1.67 ξ = 0.5

0.4

0.2

0–5 0 5 i –5 0 5 i

Figure 7.7 The part of “solid” atoms plotted versus the layer number i for the (100)

interface of Kossel’s crystal: (K) the zero approximation [7], (¢) the first approximation

(quasi-chemical method) and (I) MC modeling [8].


Figure 7.8 shows the ðΔF=NkTÞ dependences on the ratio x at different values

of the parameter α5 ðzsεsl=kTÞ5 ξðΔH=kTÞ (ξ5 η/z is the package density degree

in planes, parallel to boundary surface, η and z are the coordination numbers in

the plane and in crystal volume accordingly; for the simple cubic lattice ξ5 2/3).

All curves have an extremum at x5 0.5. It is a minimum at small α and maximum

at large. There are two minima at large α, which correspond to two, either small

or large (closely to 1), values of x. Jackson offered such treatment of the men-

tioned curves. At large α, the free energy minimum is responded to the smooth

surface with small, or close to 1, content of “solid” atoms in the transition layer.

At small values of α there is only one minimum at x5 0.5, which corresponds

to rough surface—identical quantity of the “solid” and “liquid” atoms disposed

without order.

Equilibrium value of x corresponds to minimum of the free energy:

@ðΔF=NkTÞ@x

5αð12 2xÞ1 ln x2 lnð12 xÞ5 0.x5 0:5:

Transition from curves with maximum to curves with minimum happens through

the curve with the inflection point at x5 0.5. We will express the second deriva-

tive, substitute x by the value 0.5, and then find the value α5 2 corresponding to

such transition. Hence, the value ΔH/kTr5 3α/25 3 is obtained in the most simple

model. It is closer to experimental data than results of other theories. Most likely,

the supposition VslDVll at calculation of parameter εsl through heat of fusion is

incorrect. The actual value εsl is much more because of the reduced density at the

interface. This has led to a paradox of deterioration of outcomes at improving of

theories. As research, which used a modeling by the MC method, considered the

same suppositions, there has been no improvement of outcomes.

α = 1.250.5

0

–0.50 0.5 1

x

α = 0.25

α = 0.5

α = 2.5

1.0

ΔF/N

kT

Figure 7.8 The dependences of the free energy change of transition layer after disordering

of its structure (dispersion of “solid” atoms) on the concentration of “solid” atoms at the

different values of the parameter α.


7.2.4 Simulation of Crystal Growth Within the “Solid-on-Solid” Model

The equations for the transition probabilities, deduced by Leamy and Jackson [8],

consider only bonds in the plane of crystal face,

p1 5 ν0τ expΔμkT

� �; ð7:14Þ

p2 5 ν0τ exp 4εslkT

ði2 2Þ� �

; ð7:15Þ

where i is the number of neighbors in the plane xy, Δμ5μl�μs is the odds of

chemical potentials of coexistent phases. Many authors [8�11] used these equa-

tions for simulation of crystal surfaces.

Leamy and Jackson [8] researched the structure of the crystal�melt surface,

studying the disposition of solid atoms in consecutive atomic layers (the quantity

of atoms was such that they filled half of the model volume). As they found, the

equilibrium structure depends on the magnitude of the parameter α (see the previ-

ous section). The outcomes obtained are featured in Figure 7.7. They coincide with

outcomes of analytical calculations for the model solid-on-solid at not-too-small

values of the parameter α. Using these types of surface features in what is called

the solid-on-solid (SOS), Gilmer [10] researched by simulation the kinetics of crys-

tal growth, which was well featured by the analytical theory in that time.

Figure 7.9 shows the surface at a certain instant of the MC simulation. Here a large

enough number of two-dimensional nuclei formed under conditions, for which the

structure shown in Figure 7.9 has been obtained (ϕ5 4kT, Δμ5 2kT), but the sur-

face is still smooth at the atomic scale, because the number of separate adsorbed

atoms and holes is not yet very large. Increasing driving force results in growth of

the nucleation rate.

Studying growth kinetics of the two-component crystal was fulfilled in the same

work [10]. According to Ref. [10], results for the one-component system coincided

Figure 7.9 The map of the interface obtained by Gilmer [10], ϕ5 4kT, Δμ5 2kT.


with the theory for growth by the mechanism of two-dimensional nucleation.

Growth rates depending on thermodynamic driving force Δμ were studied. The

velocities obtained for the one-component system and for the system with the

impurity component (the second component) are compared in Figure 7.10. At small

positive values of Δμ, formation of clusters happened only close to impurity

atoms, and this process stipulates moderate growth in the interval of Δμ values, in

which the facet (100) is usually stationary in the absence of impurity atoms.

A threshold supersaturation was completely absent at growth of two-component

crystals at used relations between energies of atom interaction.

7.3 The Surface Kinetics

7.3.1 Movement of a Step

Surfaces of crystals can be smooth or rough in the atomic scale, depending on tem-

perature and enclosing environment, which encloses them. In the case of smooth

surfaces, the structure and shape of steps, which exist on the surface at the screw

dislocations or arise from the formation of nuclei, appreciably determine growth

kinetics by the level-by-level mechanisms: a dislocation mechanism or mechanism

of two-dimensional nucleation.

Frenkel [4], and afterward Burton et al. [6], have shown that unlike flat facets,

steps, which are the straight lines at temperature 0 K, undergo “fusion” with a rise

in temperature (long before the fusion point), that is, become rough; this “fusion”

consists of the formation of numerous breaks. The average distance between breaks

of steps can be calculated by Eq. (7.10). Estimates for typical matters and tempera-

tures yield high concentration of breaks. So the step can be considered as the con-

tinuous receiver for adsorbed atoms in the course of crystal growth.

The probability of direct condensation of atoms into step break is taken over as

small, considering crystal growth from vapor. It is usually supposed that growth

rate is determined by diffusion of adsorbed atoms to the step from some area of the

0.4pB=0

pA=pBexp(–6)

0.2

00 1 2 3 4 Δμ/kT

R/(pAd)+

+ +

+

Figure 7.10 The normed growth rate in the one-component system (o) and in the system

with addition of the second component (Δ) [10].


surface (breadth λs on either side of from the step, λs is the average length of diffu-

sion run of adsorbed atom during the lifetime on the surface). The analysis in such

a model results in the following expression for growth rate of direct step:

VN 5 2σν0λsexp 2ΔHev

kT

� �; ð7:16Þ

where σ5Δμ/kT is the relative supersaturation; ν0 is the frequency, close to vibra-

tion frequency of atoms in crystal lattices; and ΔHev is the heat of vaporization.

There is possibility of direct joining of molecules to the breaks at growth from

solutions or melts. Owing to the complexity of taking into account several pro-

cesses: an attachment in result of activation, diffusion to the step in the melt or

solution, volume and surface diffusion, it is accepted to describe the growth rate of

step by the simple equation

VN 5 βstσ; ð7:17Þ

where σ is the local surface supersaturation (σ5 ln(C/Ce) for solutions, and

σ5ΔHΔT=ðkT2e Þ for melts, Te is the equilibrium temperature); βst is the kinetic

coefficient of straight step. As near the protuberant step the equilibrium concentra-

tion or temperature should be more than a nearly straight line (by analogy to

Thomson’s formula for three-dimensional case), the growth rate of such step can

be expressed through growth rate of straight step [6]:

VðrÞ5VN 12r�r

� �; ð7:18Þ

where r� 5 aγst/(kTσ) is the radius of critical two-dimensional nucleus; γst is the

free boundary energy of step in height a counted per one molecule.

7.3.2 The Dislocation Mechanism of Growth

As Franc has shown, if the screw dislocation has the Burgers vector with a nonzero

component, which is normal to the surface, the intersection of such dislocation

with the surface will conduct to an appearance of step, which cannot disappear at

crystal growth. The step is anchored in the crosspoint of the dislocation with the

surface; however, the step can move and form a spiral in the process of attachment

of atoms or molecules and, accordingly, a certain protuberance in the form of cone

is formed on the crystal face. The spiral rounds until curvature at center will not

reach 1/r�, according to Ref. [6], then all spirals “rotate” with conserving its shape.

The stationary spiral takes over approximately the Archimedean shape; such spiral

in polar coordinates is featured by the equation r5 2r�θ. Thus, with each full gyra-

tion during the time ð4πr�=VN 5 2π=wÞ, the moving of the step is 4πr�, where w is

the angular velocity of gyration. Therefore, the normal growth rate of crystal will

be expressed in the form


V 5aω2π

5aVN

4πr�5

βkT4πγst

σ2: ð7:19Þ

The parabolic law of growth will take place, if distances between spiral steps

are large enough (in the case of small supersaturations). At magnification of super-

saturation and, accordingly, reduction of distances between steps, the growth rate

increases more slowly with supersaturation because of overlapping of diffusion

fields near the steps, and the growth law comes nearer to the linear. More exact cal-

culation of distances between steps [11] yields to the value 19r� instead of 4πr� in

the denominator of Eq. (7.19).

7.3.3 Two-Dimensional Nucleation Growth Mechanism

Two-dimensional nuclei, which are the disks or clusters restricted by direct ribs,

cause the deposition of new layers when expanding at the crystal surface.

Frequently appearing as round nuclei [1,11,12]

I5h

Ω

� �3=2

βσ1=2 exp 2πγ2stk2T2

eσ

� �; ð7:20Þ

where there is the work of formation of two-dimensional nucleus of the critical size

δG(n�) in the exponential index divided by kT: exp(�δG(n�)/(kT)); h is the height

of the step formed by two-dimensional nucleus, Ω is the volume per one atom; Teis the equilibrium temperature.

According to Ref. [1], the growth rate of the face with area l2 in the case of for-

mation of new layers by individual two-dimensional nuclei,

VI 5 hl2I5 hl2h

Ω

� �3=4

βσ1=2exp 2πγ2stk2T2

eσ

� �: ð7:21Þ

In the case when the growing face is coated by numerous nuclei (frequency of

nucleation is large), the new layer will arise when the lower layer is not completely

filled yet. There will be nucleation on the surface of nuclei formed earlier. Suppose

that some nucleus (supercritical) grows with constant velocity. Neglecting its small

initial size, we will consider that it coats the area πV2Nt2 in the instant t. The num-

ber of the new nuclei arising on its surface is defined by the equation

n1 5

ðt0

IπV2Nt2dt: ð7:22Þ

Determine the latency period τw, as mean time, at which the number of new

nuclei that have already formed is equal to 1. Then

ðτoη0

πIV2Nt2dt5 1; τw 5 ðIV2

NÞ21=3;


and the velocity of growth

VII 5h

τwDhðIV2

NÞ1=3 5ffiffiffiffiffih3

Ω

rβσ5=6 exp 2

πγ2st3k2T2

eσ

� �: ð7:23Þ

This equation is accepted to feature growth by the mechanism of the two-

dimensional nucleation at large supersaturations. Equation (7.22) is not exact actu-

ally, because it does not take into account dependence of the velocity of growth of

two-dimensional nuclei on their curvature (Eq. (7.18)). This dependence can be

taken easily into account at numerical calculation of the growth rate during com-

puter simulations [13]. The latency period for the new nucleus formation on the

previous nuclei (from lower layer) can be found from the condition

Xi0i51

Iπðri2r�Þ2 ΔtD1; τw5 i0Δt; ð7:24Þ

in which the integral (Eq. (7.22)) is substituted by the total. Here Δt is some small

interval of time—the timestep; (ri2 r�)2 is the area of the below disposed nucleus,

on which there can be formed the new nucleus during the ith time step.

7.3.4 Growth Rate by the Normal Growth Mechanism

According to the Wilson [14] and Frenkel [15] theory, the pure crystal grows from

slightly supercooled melt with the velocity [1]

V 5D

a

ΔμkT

5DΔHfΔT

akT2; ð7:25Þ

where D is the diffusivity for the substance transport through the interface; Δμ is

the difference of chemical potentials of two phases; ΔT5 Tf�T.

The growth theory by the normal mechanism has gained development thanks to

research by Borisovet al. [16], Temkin [17], and other authors. The linear depen-

dence of growth rate on the supersaturation at the crystallization front V5βSσS is

considered as the standard indication of acting of the normal mechanism. That,

however, was called into question, e.g., by Ovsienko and Alfintsev [18] (see in

Ref. [19]).

Equation (7.25) is correct for small supersaturations (or supercoolings). An

equation that is more common is written in exponential form [20]

ν5βð12 expð�Δμ=RTÞÞ: ð7:26Þ

Kinetic coefficient β is usually estimated through coefficient of self-diffusion

D� (β5 fD�/a, a is the distant equaled to the particle size, f is the constant, which

is roughly equal to 1). In this case, β must depend exponentially on the temperature


at large supercoolings, βBexp(�Q/RT), as it is expected the Arrhenius-type

temperature dependence of the self-diffusion coefficient in the noncrystalline phase

(as in crystals).

7.3.5 Role of Bulk Transport Processes During Crystal Growth

The given above dependences of the growth rates on the supercooling of melts or

supersaturation of solutions can yield the crystal growth description when it is

determined by the kinetics of the surface processes (growth in the kinetic mode—

coefficients of the heat transport and diffusion are large (or the crystal size is

small)). In this case, the surface temperature and concentration do not differ essen-

tially from such values in surrounding medium. If growth is spotted by both joining

of atoms and transport processes, the surface concentration and temperature will

be determined by interaction of the surface processes and volume processes of the

heat and mass transport (in the volume of noncrystalline phase). In this case, the

surface concentration and temperature have values which lay in the gap between

equilibrium values and values in the melt or the solution far from the crystal. And

finally, if the potential energy barrier for joining atoms to the surface of the grow-

ing crystal is very small, the growth rate will be restricted by the thermal conduc-

tion or diffusion (growth in the diffusion mode), and the concentration and

temperature at the surface of the growing crystal will differ a little from equilib-

rium values (Eq. (7.31)).

7.3.6 Application of MC Simulation Technique to Study Growthof Small Crystals

In Ref. [21], the crystal growth in the narrow channel (acyclic boundary conditions)

was studied by simulation in the framework of the KMC method. The crystal size

in the growth direction was not restricted (it was carried out by periodic removal of

completely filled atomic layers). It is known that the morphology of crystals is con-

nected with the structure of interface and the growth mechanism. Depending on the

degree of the atomic roughness of different faces, the shapes of crystals can be fac-

eted, half faceted, or roundish. The equilibrium shape of the crystal is the shape of

the critical nucleus (three-dimensional), and it was featured [1] earlier that only the

size of equilibrium crystals depends on supersaturation (if the free surface energy

does not vary). Growth shapes can differ from the equilibrium shape because of

kinetic roughness development [3,22,23].

Elwenspoek and Van-der-Eerden [24] made the estimation of critical supersatu-

ration, at which the interface becomes kinetically rough, from the condition, that

the free energy of formation of two-dimensional nuclei becomes equal to kT.

Ovrutsky and Rasin [21] developed a special algorithm for determination of

parameters of the greatest clusters upper crystalline planes (Figure 7.11), such as

perimeter na, numbers of atoms in the cluster with one, two, and three broken

bonds in the considered crystalline plane (N1, N2, N3). It allowed us to determine


the facet size nf and all characteristics of critical two-dimensional nuclei: n�, na,nb5N11N21N3.

Voronkov [25] proved that the size of the equilibrium facet is more than twice

the two-dimensional nucleus fluctuating on it. Thus, only one two-dimensional

nucleus can be formed on the equilibrium facet. The results of Ref. [21] confirm

that. The size of the two-dimensional nucleus in Figure 7.11 is less than the half

size of the facet (black) because the relative supersaturation is more than the

Gibbs�Thomson shift that is necessary for beginning of growth of the crystal with

such a size.

Figure 7.12 shows crystals with entropy of fusion ΔH/kT5 5.5 in the square

channel (803 80) and their average profiles (the aliphatic line). Figure 7.12A

shows the crystal, which is not growing practically at the supersaturation

σ5Δμ/kT5 0.068 (0,V/aν, 1027, V is the growth velocity, a is the lattice

constant). At such supersaturation, the crystal does not grow practically, thanks to

the Gibbs�Thomson effect. As one can see from the drawing, there is the flat

site on the averaged profile, that is, the facet exists. When the crystal grows

(Figure 7.12B, σ5 0.08), its shape looks more correct—the face is incremented,

Figure 7.11 Three upper atomic layers [21], ϕ5 0.053, the upper atomic layer (white) is

two-dimensional nucleus.

Figure 7.12 Sections of crystals in the square channel [21], ΔH/kT5 5.5, (A) for not

growing crystal (the relative supersaturation σ5 0.068) and (B) for the growing crystal

(σ5 0.08, V/aνD3.73 1025).


and radiuses of curvature of roundish sites decrease. In it outcomes of simulation

agree with experimental data.

The temperature of disordering of the surface structure has been determined in

Ref. [26] as a function of the size of the channel. The value ϕ/kTRD0.95 (ΔH/

kTD5.7) has been obtained for the crystal in the channel 403N, and the value

ϕ/kTR5 0.856 0.03 is found for infinite square the facet (periodic conditions).

Earlier, the temperature of structural transition TR was determined by other meth-

ods. Leamy and Gilmer [27] found the value ϕ/kTR5 0.78 for the infinite crystal

from the condition of the vanishing of the free edge energy of steps. Xiao et al.

[28] found the value ϕ/kTR5 0.81, at which there was the heat capacity maxima in

the system that was simulated. Influence of the size of crystals on disordering of

the surface structure was also studied in Ref. [21]. The value ϕ/kTRD0.92 was

found to correspond to very small facet—the net size was 203 20. The value

ϕ/kTR5 0.85 was determined for the 1003 100 crystal face.

Figure 7.13A and B [29] shows the growing crystal that has the screw disloca-

tion of double height (2 atomic layers) on its face. The step connected with the dis-

location twists in the spiral with formation of the dislocation hillock (Figure 7.13).

Differences in structures in drawings 7.13A and 7.13B are connected with the odds

in heat of the phase transition and, accordingly, in values of the roughness parame-

ter α. Figure 7.14 shows the dependencies of growth rate (in relative units) on rela-

tive supersaturation σ5Δμ/kT for two values of the roughness parameter in the

cases when there is present or absent the screw dislocation on the crystal surface.

In the second case, the supersaturation must be larger than a certain threshold value

for the beginning of growth.

Results of modeling of crystal growth from the mixed melts with use of the lat-

tice liquid model are obtained in Ref. [30]. As objects, the alloys of systems, in

which the forms and kinetics of growing bismuth crystal vary differently at modifi-

cation of the melt composition and equilibrium (liquidus) temperature [31], were

chosen. The tabular and the diagram data were used for calculation of energies of

atom interaction.

Figure 7.13 Development of the dislocation hillock at two values of the roughness

parameter, α5 12 (A) and α5 8 (B), Δμ/kT5 3 [29].


Figure 7.15 shows the structure of the crystal surfaces [30] obtained as a result

of modeling of the Bi crystal growth from melts of systems Bi�Sn, Bi�In. As can

be seen from Figure 7.15C, the crystalline clusters in the surface layer are larger

and they have more correct form in the case of system Bi�Sn (evaporation heat of

tin much more than of indium). Outcomes of the level-by-level analysis of distribu-

tion of components and vacancies in crystalline phase are given in Figure 7.16.

Data of Figure 7.16 confirm that the interface width decreases essentially at magni-

fication of concentration of the second component. The data of simulations as a

whole are in agreement with experimental data in questions of differences in forms

and kinetics of growth of bismuth crystals in these systems [31].

7.4 Formation of Thin Films

7.4.1 Atomic Mechanisms of the Film Formation

Formation of thin films usually passes through stages of nucleation and growth.

These stages can include adsorption, surface diffusion, and formation of chemical

compounds and other surface atomic processes [32�35]. The basic mechanisms of

crystal growth on the substrates are presented in Figure 7.17 (it is such as in review

[33]) in a sequence, which maps the modification of conditions of crystallization.

What mechanism will take place in the yielded system is determined by the relation

between surface energies of the substrate αs, of the precipitated material αcr and

the interface energy αcr,sb of the precipitated material with the substrate. The islet

or the Folmer�Weber mechanism (Figure 7.17D) is realized, if atoms deposited

on the substrate interact with each other more strongly than with the substrate

(αsb,αcr1αsb,cr). Thus, small three-dimensional nuclei are formed directly on

0.4R

/k+ a

0.2

00 1

Δμ/KBT

L/KBT = 12

L/KBT = 6

2 3

Figure 7.14 Comparison of growth rates for perfect layer by layer (open symbols) versus

spiral R/k1a growth (solid symbols) for two parameterized temperatures [29].


the substrate. Then they develop into three-dimensional islets of the deposited

phase and, after filling of gaps between them, form the continuous rough films.

The islet mechanism is observed for a lot of metal systems on the dielectric sub-

strates, including many metals on halides, alkali metals on black lead, on mica, etc.

Figure 7.15 The interface images [30]: (A�C) growth from melts Bi�Sn, 5, 10, and 15 at.%

Sn accordingly and (D) Bi1 15 at.% In (two-dimensional nuclei in the high layer are black).

C

0.8

0.6

0.4

0.2

08 10

(A)

12

44

1

2 3

14

1

0.8

0.6

0.4

0.2

08 10

(B)

12

4

1

2

3

14

1

0.8

0.6

0.4

0.2

08 10

(C)

Z12

4

1

2

3

14

Figure 7.16 Concentration distributions of components and fraction of crystalline phase

in consecutive atomic planes [30]: (A and B) for system Bi�Sn (10 and 15 at.% Sn

accordingly), (C) Bi1 15 at.% In; (1) fraction of liquid A-component (Bi), (2) B-component,

(3) vacancies, and (4) fraction of crystalline phase (Bi).


In the case of αsb.αcr1αsb,cr, two mechanisms are possible: level-by-level

(Franc and van der Merve) (Figure 7.17A) or intermediate between the level-by-

level and islet (Stransky and Krastanov) (Figure 7.17C). Atoms interact more

strongly with the substrate than with each other at the level-by-level mechanism

(Figure 7.17A); they completely cover the substrate and form the completely filled

first coat on it; then the second layer and next layers are formed. It is accepted to

speak about multilayer growth (Figure 7.17B), if islets can be formed up on the

incompletely filled previous layer. Such growth is usually described within the lim-

its of Kashchiev’s model [36] in the case of autoepitaxy αsb5αcr, αsb,cr5 0.

Hence, the autoepitaxial films usually grow layerwise (level by level).

Straight level-by-level growth is observed at high temperatures and the multi-

layer at low surface temperatures. After formation of the first or several first layers,

the following level-by-level growth becomes unprofitable, and islets are formed up

on the first layers in the case of growth by the Stransky�Krastanov mechanism

(Figure 7.17C). The heteroepitaxial systems with disagreement in the parameter of

the crystal lattice are an example of such mechanism [37]. Changes of growth

mechanism in the systems with the disagreement are explained by relaxation of

elastic stresses in the three-dimensional islets without dislocations. Because of such

relaxation, three-dimensional growth becomes more favorable in energy than two-

dimensional after reaching some thickness of the deposited film [38�40].

Spontaneous formation of the elastic-strained three-dimensional islets having no

dislocations now is widely applied for direct formation of quantum points in the

semiconducting heteroepitaxial systems [33,37,41]. Figure 7.18 shows diagram-

matic representation of different atomic processes, which respond to adsorption and

crystal growth on the substrates.

During deposition from the vapor with pressure P, the stream of atoms to the

surface I5P(2πmkT)21/2, where m is the atom mass, k is the Boltzmann constant,

and T is the temperature of the transpiration source. Either the molecular beam, the

transpiration radiant, or the diffusion of ions from the solution can ensure the flux

I (m2/s). At first, individual atoms precipitate on the substrate in quantity n1 and

θ < 1 ML

θ > 2 ML

(A) (B) (C) (D)

1 ML < θ < 2 ML

Figure 7.17 Schemes of the basic mechanisms of the film formations [33]: (θ maps the

thickness of the film in number of the filled layers—ML) (A) level-by-level (Franc and van

der Merve); (B) multilayer (Kashchiev’s model); (C) multilayer1 islets (Stransky and

Krastanov); and (D) islet (Folmer and Veber).


relative concentration n1/N0 (N0 is the number of atomic places on the surface).

Then these atoms diffuse on the surface until either repeated transpiration will take

place, either formation of two- or three-dimensional clusters, either joining of

atoms to existing clusters. Real surfaces are far from perfect because of the pres-

ence of protuberances, breaks of steps, dislocations, and pointwise imperfections,

which can influence essential adsorption, diffusion, and the nucleation especially, if

the energy barrier for nucleation on the perfect surface is high (small concentration

of nuclei of the critical size).

In the case of growth of thin films on the solid body surface, it is acceptable to

select the following basic stages (Figure 7.19): origin (nucleation) of islets, their

independent growth, Ostwald’s ripening, confluence (coalescence) of islets, and

three-dimensional growth of the film.

Ostwald’s ripening stage is the late stage of the phase transition. In the case of

thin films, it takes place only at enough feeble radiant of deposition or in modes with

the growth stopping. The physical essence of the process consists of the following.

During stages of independent growth of islets, the supersaturation determined by

the concentration of adsorbed atoms strongly decreases, if islets grow for the account

of joining of adsorbed atoms. New islets thus are not formed any more. The distribu-

tion of islets on sizes, which is generated at the stage of nucleation, moves thus

toward the size magnification; however, the critical size of nucleuses also increases.

Therefore, islets of the subcritical size break up and supercritically grow. At the

yielded stage in system, there is the special sort of interaction, which is transmitted

through the generalized diffusion field. In result, the large islets grow at the expense

of decay of more small. This process is termed as Ostwald’s ripening.

The step break Surface diffusion

Surface diffusionDeposition Evaporation

Adsorption

Bond formationnucleation Diffusion

Figure 7.18 Diagrammatic representation of the surface processes [33].

Figure 7.19 Diagrammatic representation of successive steps of film growth by the two-

dimensional mechanism: (A) origin of islets; (B) independent growth; (C) coalescence; and

(D) formation of the three-dimensional film.


After completion of the stage of independent growth of islets, their confluence

(differently—the coalescence) occurs, in outcome of which the continuous film is

formed on the surface. At first, the coalescence process has the character of pair

collision of islets. Then, after magnification of infill degree, the plural collisions of

three, four, and so on of islets start to happen.

There are two basic mechanisms of the coalescence: the liquid-dropwise coales-

cence and solid-phase center formation. In the case of the liquid-dropwise coales-

cence, islets, which merge, behave like fluid drops; that is, the islets of the large

size formed from two little islets have the round shape the same as two small islets

had. In the case of the solid-phase coalescence, the islets, which merge, behave like

crystals: at collision of islets their growth in the collision places is terminated, and

in other places is prolonged in the former mode. Osipov [42] researched the liquid-

dropwise coalescence of two-dimensional islets in result of their side growth within

the limits of kinetic model for the distribution function of islets on their sizes. In

this case, the pair coalescence responded to the Smoluchovski model [43].

7.4.2 Kinetics of Epitaxial Growth of Thin Films

The study of the kinetics of thin film growth is important because it determines the

quality of their surface. The knowledge of kinetics allows optimization of the epi-

taxial growth technique, in particular, at epitaxy by the method of molecular beam.

Selection of parameters, which ensures the certain (desirable) characteristics of

films, is termed an optimization. The example of optimization by the way of

modeling of the process is the epitaxy of gallium nitride (GaN) for making

instrument structures; it is described in work [44].

The kinetics of the film epitaxial growth had been analyzed by Ovrutsky and

Posylaeva [45] by computer calculations of total volume of the growing crystalliza-

tion centers in consecutive atomic layers within notions of theory of mass crystalli-

zation. Authors used the ideas of Mehl and Johnson [46] concerning elimination of

the fictitious part of the crystalline phase increase. Two-dimensional mass crystalli-

zation in each atomic layer was considered, taking into account the area of the film

in the previous layer at every timestep—the model with many levels (the analog of

the BET theory of multilayer adsorption; see Section 5.1.4).

Figure 7.20 shows a model of multilayer mass crystallization applied by Ovrutsky

and Posylaeva [45]. Contributions from fictitious nuclei and the intersection area of

nonfictitious centers were excepted at calculations of an increase of crystalline phase

in the certain time slice. These areas are shown in white in the drawing.

Magnification of the area of all crystalline centers in zth layer:

ΔQz;k 5 ðQz21;k21 2Qz;k21Þ � Qz21;k21 �Xk21

j51

IzðtjÞΔt � 2π � rk2j �Δrk2j; ð7:27Þ

where Qz,k is the part of the crystalline phase in the layer z in the instant tk; rk2j is

the radius of the crystallization center, which has arisen in the past, in the time


slice [tk2j21, tk2j]; Iz(tj) is the rate of generation of two-dimensional nuclei in the

instant tj (Zeldovich’s correction [47] on nonstationary was taken into account for

its calculation for small times).

Figure 7.21 shows some dependences of the part Qz of the crystalline phase in

consecutive atomic layers on the layer number z for cases of the epitaxy and the

autoepitaxy at certain infill degree of the first layer. The number of atomic layers,

which corresponds to a placing of the basic diminution of magnitude Qz in

Figure 7.21, is connected with a structure of the film surface. It depends on interac-

tion with the substrate (it is only three to five layers at the autoepitaxy) and on

supersaturation. It follows from shown data that the certain time is needed for

Figure 7.20 The scheme of the model of multilayer crystallization [45], a—the fictitious

nucleus.

1.0

0.8

0.6

0.4

0.2

0 10 20 30

Qz

Z

Figure 7.21 Filling with the crystalline phase in the consecutive layers [30]: (V) autoepitaxy,Q15 0.99, σ5 0.05; (3 ) epitaxy, Q15 0.67, σ5 0.15; (Δ) epitaxy, Q15 0.90, σ5 0.15;

and (x) epitaxy, Q15 0.99, σ5 0.15.


formation of the almost the continuous film at the epitaxy. The thickness of films,

which ensures the continuity in the first layers, depends on crystallization condi-

tions. It is practically impossible to gain the continuous film at rather low

supersaturations.

This approach to problems of the description of the kinetics of crystal growth

and epitaxy has the advantage that, basically, it is possible to consider particulari-

ties of the growth mechanism and different properties of the substrate. The method

allows computing the growth kinetics at any supersaturation and value of the main

parameters: kinetic coefficient of the direct step and the free boundary energy of

the step, connected with of the crystal surface roughness in atomic scale. It is possi-

ble to evaluate the characteristics of the film’s structure, depending on crystalliza-

tion conditions in the case of epitaxy.

The MC method of kinetic simulations was used for studying the epitaxial

growth. Plotz et al. [48] investigated the level-by-level growth of films using the

Lennard-Jones potential; they described types of growing islets. Much et al. [49]

simulated the heteroepitaxial growth. They found the critical thickness of the film,

at which the excess dislocations of disagreement of the film crystal lattice and

substrate lattice arise. Vladimirova et al. [50] investigated structure modifications

of the vicinal surfaces of the GaAs crystal with many steps during epitaxial growth.

Authors considered the velocity of atom deposition and the surface diffusion, and

also the Erih�Shvebel barrier for joining of atoms to steps of lower layers. They

determined existence of “wandering” steps, groupings of steps, and even forma-

tions of the protuberances, which testify to the instability of the surface shape.

Simulation of the epitaxy processes, sublimation, and annealing in a three-

dimensional surface layer of silicon was fulfilled by the KMC method in Ref. [51].

In initial conditions, the thin film was set as some monomolecular layers in the

plane (111). Additional energy barriers were entered into the model for different

microscopic situations: surface defects, atoms, impurities, sites with discrepancy of

lattices in case of the heteroepitaxy. The interaction energy of atoms with neigh-

bors from the first, second, and third coordination orbs was considered.

7.4.3 Formation of Films Through the Liquid Phase at Deposition

As it is known, the molecular beam deposition on the substrate occurs only under

the condition that its temperature is below some value Tcond, which is termed the

critical temperature of condensation. Depending on the nature of the substrate and

density of the molecular beam, i.e., deposition velocity, the temperature Tcond can

be both more or less than the melting point of massive samples Tf [34].

Different authors have established that the metal films obtained by condensation

on the inert amorphous substrates consist of the separate isolated islets at the begin-

ning of process of their formation. Hence, they are formed by the origin and growth

of nuclei of the condensed phase that is their growth occurred through the islet

stage. As it follows from phase diagrams for samples of small sizes, nuclei formed

during the initial stage should be in liquid state owing to the small size, that is, the

condensation should occur into the liquid phase by the mechanism vapor!fluid


(v!f). However, experimental research specifies the possibility for the separate

isolated particles to settle at once in the form of microcrystals, or go through a tran-

sit stage of liquid droplets, and then to crystallize.

Thus there are two following mechanisms of condensation at growth of the film

islets: (1) formation of islets of the crystalline phase directly from the vapor (the

mechanism vapor!crystal) and (2) condensation at first with origin of islets of

the liquid phase, which can crystallize afterwards (vapor!fluid). The question is,

under what conditions the deposition with formation of islets will occur into the

crystalline phase by the Folmer�Weber mechanism, and under what liquid phase

at temperatures which are below melting point for large samples of the researched

substance. It concerns substances with the low position of triple point on the P, T

diagram (substances with very small vapor pressure). It is natural to expect that

substances with the large vapor pressure (with the high position of triple point, e.g.,

Zn, Cd, Mg) will crystallize by the mechanism v!cr.

Numerous experimental observations of nucleation at deposition of different

matters on different substrates yielded confirmation of realization of mechanisms

v!cr and v!f. Palatnik et al. [52] showed in the series of experimental research

that the mechanism of metal condensation, v!cr or v!f, is determined by the

deposition conditions, first of all, by the temperature and the material of the sub-

strate. According to the outcomes of this research, deposition on the inert amor-

phous substrate at temperatures Ttr, T, Tf, in initial stage can be carried out by

origin of liquid particles (the mechanism v!f), and at T, Ttr—by formation of

particles of the crystalline phase (the mechanism v!cr). Thus the boundary tem-

perature Ttr exists (Ttr, Tf), which corresponds to transition from the mechanism

v!f to the mechanism v!cr.

Formation of liquid phases was observed in Ref. [34] by means of the electron

diffraction investigation in the course of condensation Sn, Pb, In, and Bi on amor-

phous lacquer substrates at temperatures Ttr, T, Tf (boundary temperature

Tg� 0.68Tf for Bi and Ttr� 0.69Tf for Sn and Pb). The liquid phase was main-

tained, if the temperature of the substrate was above Ttr. If condensation was termi-

nated to the crystallization beginning, the liquid phase exists longer, if the medial

thickness of the deposited metal was smaller or the temperature of the substrate

was higher. Crystallization of liquid occurred always at cooling of the substrate

below temperature Ttr. The researches of condensation of silver and copper [34]

have also shown that, depending on temperature of the substrate, the deposited

phase may be liquid at the condensation beginning. The diffusive rings, which cor-

respond to liquid silver, have been obtained at the diffraction research during depo-

sition at temperatures of the substrate lower than 873 K.

Enough clear explanation for understanding the temperature Tg dependence on

the substrate material does not exist. Also not clear is why in films of one metal,

the liquid phase formed at the temperatures Ttr, T, Tf exists a long time after ter-

minating deposition, and in films of others, e.g., lead, crystallization occurs during

considerably smaller time period under other identical conditions.

The physical nature of the boundary temperature Ttr was not completely clear

until now. On the initial stage, the nuclei of condensed phase are liquid because of


size dependence of the melting point. Growing droplets are crystallizing at the sizes

predetermined by dimensional dependence of the melting point at temperatures of

substrate T, Ttr. Particles of liquid phase exist in the supercooled state during con-

densation at T. Ttr. That is, the temperature Ttr responds to supercooling at crystal-

lization on the substrate and, of course, it should depend on the material of the

substrate. The dependence of the free energy change δG on the substrate material is

essential for the initial stage of islet formation when their sizes are small and their

melting point is decreased. The islets will be in liquid state at temperatures below

the melting point of massive samples roughly up to the size (the equation from

Ref. [34] is given here in very simplified form):

r53ΔσL

� Tf 2 T

Tf; ð7:28Þ

where Δσ is the change of the free surface energy at melting of a small round crys-

tal and L is the melting heat per unit of volume.

Melting point of the islet in the form of spherical cap with the radius r is deter-

mined by the same relation, as for free particle with the radius r in the case of the

independence of the melting point on the magnitude of wetting angle θ. Fromexperiments [34], it is known that particles with a radius of 2�3 nm are liquid at

temperatures T�Ttr. Below temperature Ttr they crystallize after reaching by islets

of the size predetermined by the dimensional dependence of the melting point, and

further condensation occurs into the crystalline phase by the mechanism v!cr

(Figure 7.22). Because of this dependence, formation of continuous polycrystalline

films is usually observed with magnification of the mass thickness. All this allows

us to determine simply enough the temperature interval of existence of the super-

cooled liquid, using the condensates obtained on the substrate with the gradient of

temperatures in its wide interval (0.5Tf, T, Tf).

The studying of the supercooling effect on the crystallization is important not

only for understanding of nucleation phenomena at the fluid�crystal phase transi-

tions but also for solution of practical problems connected with crystallization—

improvement of the microstructure and properties of stable or metastable phases.

There is the critical size of small crystals r5Rc, at which the temperature of their

melting is equal to the boundary value, Tr5 Ttr, and supercooling for their crystalli-

zation is equal to zero [34]. The value (Tf�T)/Tf is the relative supercooling for

crystallization of large particles. Its limited value in the large sample (Tf�Ttr)/Tf is

equal to B0.4. The magnitude (Tr�Ttr)/Tr decreases with reduction of particle size

and it is equal to zero at r5Rc. In the opinion of Turnbull [20], the droplets with

r,Rc, may appear at T, Tr, but the transition occurs already without supercool-

ing, and, hence, becomes continuous.

Thus, according to experimental data concerning melting and crystallization of

small particles of In, Sn, Pb, and Au, the magnitude Rc consists of 2�3 nm. In the

range of temperatures Ttr, T, Tr, entropy of the system gains one of two of its

possible values, which corresponds to stable states, depending on the previous his-

tory of the system (e.g., cooling of droplets or heating of the crystalline particle).


7.4.4 Kinetic Modeling of Film Deposition from a Gas Phase

Ovrutsky and Rasshchupkyna [53] have used the model, in which there is no artifi-

cial separation of atoms to “solid” or “liquid,” for kinetic simulation of the film

condensation. In this model, vacancies were being formed as a result of their diffu-

sion from the liquid�gas surface. Simulation was spent on the model lattice

(Figure 7.23) with the smaller step in all directions than interatomic distance (four

times less than the expected lattice constant a). Only nodes belonged to planes xy

(z5 0), yz (x5 0), and zx (y5 0) are shown in Figure 7.23. Atomic places corre-

sponding to the FCC lattice are designated by quadrates (&). If atom A, which is

considered, is in the knot [000] the neighbor atom can be placed, not more close,

than a/2. The nodes designated by daggers (†), are forbidden for jumps in them of

the neighbor atoms. As objects for investigations two substances, Pb and Ni, have

been chosen, for which the experimental data concerning the modification of the

condensation mechanism are described in Ref. [34].

Figure 7.22 Films of tin with different mass thickness on the carbon substrates [34], on the

left—deposition by the mechanism vapor�crystal takes place, T5 313 K; on the right—the

mechanism the vapor�liquid, T5 363 K.


The probability for an atom to deposit on the substrate to be joined to a drop of

the fluid or to a small crystal is connected with a stream of atoms I to the surface

(the streams of 0.1�250 atoms in 1 μs to one atomic place were considered):

Pcond 5 Iτmin: ð7:29Þ

Joining was allowed if the corresponding node of the lattice has a free access

(it was open from above) and other atoms did not occupy any of 26 nearest

(forbidden) nodes of the lattice.

Probabilities of evaporation (vanishing) of atoms depend on its environment (the

number of the neighbor atoms on different distances), and probabilities of atom

jump on their old and new environment. Continually on time all atoms were tested

concerning jumps or evaporation, and also all free nodes near atoms or at the sub-

strate (disposed not less than a/2) were being tested on the occupation possibility

by atoms from the stream.

The probability of jump was defined by the equation:

pjump 5 τν0 expð�ðEnew � EoldÞ=ðkTÞÞ; ð7:30Þ

where Enew and Eold are the interaction energies of the atom with all neighbor

atoms within the orb of acting of the pair potential for possible new and previous

locations of the atom; νo is the frequency of natural oscillations of atoms. If new

energy was less than previous, the jump was fulfilled. In the case of magnification

of energy the jump was fulfilled, if the generated random number was less than the

probability pjump. Calculation of energies was spent counting 176 positions of

34

3

A

x

y

z

04

34

10

a / 4

2

12

304

Figure 7.23 Schema of the model net.


possible positions of the neighbor atoms. As the number of different possible tran-

sitions of atom is very large, it is difficult to spend the normalizing of probabilities.

Therefore, the timestep was reduced so that its further reduction did not influence

outcomes of calculations.

Figure 7.24 shows the incipient state of lead condensation on the substrate at the

temperature T5 610 K. If only single atoms migrate on the surface of the substrate,

it is the surface gas. The latency period of nucleation of drops depends on the

stream I. This time is very large at small intensity of the stream, or at low value of

the coefficient bs of interaction with the substrate. Figure 7.25 shows the growing

drop of lead (the section with atoms from two nearest planes of the model grid).

One can see that contact angles of drops with the substrate are incrementing at the

magnification of the drop size.

Figure 7.26 shows the drops of lead deposited at the temperature T5 580 K. On

the serial of plottings, there are collisions of liquid islets, which result in their con-

fluence. The drop formed here apparently already exceeds the critical size. There is

the FCC crystalline phase in it (the small number of atoms designated by more

dark shade).

Values of temperatures Ttr of the condensation mechanism modification

obtained in Ref. [53] for the islet of films (from the vapor�crystal to the

vapor�liquid�crystal) Ttr � 0.77Tf for Pb and Ttr� 0.88Tf for Ni are larger than

Figure 7.24 Lead islets on the substrate.

Figure 7.25 Growing drop of lead [53], T5 610 K, I5 32.9 mk/s, t5 0, 0.125, 0.161, and

0.204 mks accordingly (the cell size is equal to the lattice parameter a).


the experimental values (0.69Tf for Pb and 0.66Tf for Ni [34]). It is most likely

connected with the simplified calculation of the driving force of crystallization

(the odds of chemical potentials of two phases) under the simplified equation,

Δμ5ΔHΔΤ/Tf, which does not consider the odds of heat capacities of phases.

7.5 Shapes of Crystal Growth and Their Stability

7.5.1 Shapes of the Free Crystal Growth

The question about equilibrium shapes of crystals was considered in Sections 7.1.2

and 7.1.3. The equilibrium shape of the crystal of the limited size is actually the

shape of the critical nucleus. Growth shapes differ from the equilibrium shapes.

Crystals grow in the constant shape (roundish, faceted, or protuberant shapes),

remaining similar to itself, only under certain conditions. In particular, the similar

growth is possible while the size of the crystal and the deviation from equilibrium

do not exceed certain values [1,54]. Otherwise, crystals gain so-called skeletal or

dendritic (tree-like) shapes. Figure 7.27 shows a loss of stability and formation of

dendritic shapes of cyclohexanol and camphor crystals, growing by the normal

mechanism at different supercoolings.

7.5.2 Stability of Spherical Crystals

Dendrites are formed in consequence of instability of the initial roundish shape

of the crystal in relation to casually arising perturbations. We will consider, accord-

ing to Ref. [1], the stability of the full sphere with radius r, growing from the solu-

tion. Concentration distribution in the solution near the spherical crystal at

equilibrium concentration on its surface Cρ5Ce is yielded by Eq. (3.53). If we

solve the equation of Laplace together with the condition that the growth rate is

proportional to the surface supersaturation V 5 dρ=dt5βðCρ 2CeÞ (β is the kinetic

coefficient) and condition of the mass balance at the interphase boundary,

Figure 7.26 Confluence of drops of lead and their further crystallization [53], T5 580 K.


V 5DðCN 2CρÞ=ðρðCcr 2CρÞÞ, the functions C(r), V(ρ), and Cρ(ρ) can be found.

For the case of the feeble solution, these functions are the following:

C2CN 5ðCρ 2CNÞρ

r

βCcrρD

1

11βCcrρ=D;

V 5βðCN 2CeÞ11βCcrρ=D

andCρ 2Ce

CN 2Ce

51

11βCcrρ=D:

ð7:31Þ

Suppose that the protuberance in height δ{ρ was on the crystal surface, hence,

curvature of the surface in protuberance vertex was incremented on BMδ2/ρ,where the number M is larger when the protuberance is more acute. The protuber-

ance vertex hits in the supersaturated solution with concentration Cs1 (@C/@r)ρδ,where Cs5Cρ and @C/@r is determined from Eq. (7.31) at r5 ρ. A concentration

increase on the vertex stimulates its subsequent magnification to macroscopic sizes

and converts the full sphere into the dendritic crystal, that is, conducts to instabil-

ity. On the other hand, equilibrium concentration C0 at the vertex of the protuber-

ance with large curvature is also higher than over the remaining surface of the

crystal, according to Eq. (7.2),

C0 5Ce 112ΩγkTρ

12ΩγkTρ2

Mδ� �

; ð7:32Þ

where Ce is the equilibrium concentration of the solution over the infinite flat sur-

face. Thus, the surface energy, incrementing value C0 (C0.Ce), reduces supersatu-

ration on the protuberance vertex (V 5 βðCρ 2C0Þ), and consequently, counteracts

its subsequent elongation. It means that the free surface energy supports shape

stability of the surface.

Figure 7.27 Consecutive stages of the instable growth and development of dendritic joints

(A�D) of cyclohexanol crystals [55]; (E�H) shapes of camphor crystals [56,57].


The criterion of stability should take into account the conditions in which “kine-

matic force” that extends the protuberance is compensated by the reduction of ther-

modynamic driving force (supersaturation decrease). In other words, the velocity of

vertex of the protuberance concerning the nonperturbed front should not be

positive:

ΔV5 β ð@C=@rÞδ2Ce

2ΩγkTρ2

Mδ� �

# 0: ð7:33Þ

Substituting in Eq. (7.33) the derivative @C/@r found from Eq. (7.32) and solving

the equation obtained relatively R, we will obtain the criterion of stability in the

form:

ρ,RcD0:5Mr � 11ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11 4D=ðMβr�Þ

p� �; ð7:34Þ

where r� is the radius of the critical nucleus. The exact analysis was fulfilled by

Mullins and Sekerka [58]; they considered perturbation in the form of different har-

monics of the spherical functions imposed on the initial spherical shape:

rðθ;ϕÞ5 ρ1 δYlmðθ;ϕÞ, where ρ is the initial radius of sphere, Ylm is the surface

harmonic, which is the solution of the Laplace equation. Local curvature of the sur-

face for such shape depends on θ and ϕ:

K51

R1

11

R2

;Kðθ;ϕÞ5 2

ρ12

δYlmρ2

� �2

δΛγlmρ2

:

According to Eq. (7.2), the protuberance equilibrium concentration is more than Ce:

clnðθ;ϕÞ5 c0 112σΩρkT

1 ðl1 2Þðl2 1ÞσΩδγlmkTρ2

� �: ð7:35Þ

Substance reflux into medium from these sites is the factor stabilizing the shape

of the surface. If

ρ,RcðlÞ5ðl1 1Þðl1 2Þ

21 1

� �r�; ð7:36Þ

the shape is absolutely stable—small perturbation disappears in course of time. As the

relative criterion of the stability loss, the condition may be accepted that the ratio

_δe=δ_ρ=ρ

5 ðl2 1Þ 12RcðlÞρ

� �= 12

r�

ρ

� �ð7:37Þ

exceeds unity. Then shape perturbations develop fast.


According to Ref. [58], at large velocities of the surface processes (βρ�/Dc1 or

βTρ�/αTc1 for the case of growth from the melt) the sphere is stable, only if its

radius does not exceed MRc, that at l5 2 will make 7Rc. Even at small supercool-

ings, which exclude the spontaneous nucleation in the melt but which ensure

noticeable growth rate, ρ� has a magnitude of order 1026�1024 cm,

i.e., RC� 10262 1023 cm. For example, for iron (Ω5 1.23 10223 cm3,

γ5 0.204 J/m2) at supercooling ΔT5 10 K, spherical crystal is stable only to the

size B2.53 1026 cm. The critical radius of the stability decreases with supercool-

ing magnification, B1/ΔT.

According to Ref. [1], the values calculated under Eq. (7.34) are close to values

provided by the theory of small perturbations [58] if we suppose that 2M5 21(l1 1)(l1 2), where l is the number of the harmonics of perturbation. The value

l5 2 responds to the transmutation of the full sphere into the ellipsoid, l5 3

responds to the symmetry of the tetrahedron, l5 4—the cube, and so on. If symme-

try of perturbation is more, then l is more, the shape of full sphere is especially

inconvertible against such perturbations. The physical reason, which predetermines

occurrence and development of perturbations of this or that symmetry, is anisotropy

of the growth kinetics. Symmetry of angular dependence of kinetic coefficient

slightly differs from spherical even for rough surfaces in the atomic scale.

Apparently, from Figure 7.27, in the initial stage of the shape stability loss, the

protuberances connected with anisotropy of growth rate start to outstrip other sites

of interface. Their subsequent promoting into the supersaturated solution leads to

formation of dendrites. Each trunk of the developed dendrite grows already irre-

spective of initial protuberances and has the shape close to the paraboloid, and the

velocity of its growth is determined by curvature on the paraboloid vertex. On the

certain distance from the vertex, the radius of curvature of the paraboloid surface

exceeds the critical value Rc, and this local surface also loses the stability—the side

branches of the dendrite develop there. The number of protuberances exceeds the

number connected with anisotropy of growth rate at sharp magnification of the

supercooling (Figure 7.27H).

7.5.3 Stability of Polyhedrons

It follows from experimental data that perfect faceted shapes of crystals take place

only at very small supersaturations. Figure 7.28 shows schemes of transformation

(A) (B)

O

θ

O′

(C)

E

(D)

Figure 7.28 Consecutive stages (A�C) of the skeletal crystal development [1].


of the perfect crystal into the skeleton [1]. At the first stage of transformation, there

are flexures of facets with the maximal angles at their centers. Then troughs are

formed at the centers of crystal faces, where impurities collect, and legible indica-

tions of the face presence become lost.

Skeletal growth is caused by inconstancy of supersaturation along the faces.

Supersaturation distribution over the face had been found by solution of the diffu-

sion equation with boundary conditions, which express constancy of the mass flux

to each point of facet. The concentration field was also researched experimentally

for two-dimensional case by studying of the interference lines during growth of

crystal from thin films of salt solutions [1]. The curves of constant concentration

were determined in Ref. [59] under the shape of field of liquid crystals, arisen

around the anthracene crystal growing from the binary organic melt anthracene—

cholesterylkapronate (ether, cholesteryl amyl formic acid).

Chernov [1] considered the case of growth of polyhedron from solution in

approaching spherically the symmetrical approximation of the diffusion field

around it. At such diffusion field, each facet is intersected by several lines of equal

concentrations (or isotherms—at growth from melt), and near the vertices, the lines

go, which correspond to the largest supersaturation or supercooling. According to

Eq. (7.31), dC/drBβ/D at the crystal surface. The difference of supersaturations at

the center of the facet and at the vertex is proportional to βL/D. Therefore, if thecrystal size is so small that βL/D{1 (pure kinetic mode), supersaturation on the

surface is practically constant, also it is equal to the supersaturation in the volume

of solution. The same should happen at ideal stirring of solutions for crystals of

any sizes. In this case, crystal faces are practically flat (Figure 7.28A).

Supersaturation over the central sites of faces becomes less than over the vertexes

and ribs with increasing of the crystal size. However, as shown in experiments, the

surface remains macroscopically flat. Hence, there should be a mechanism which

compensates inconstancy of supersaturation.

The reason, which supports the polyhedral shape of growing crystal, is anisot-

ropy of growth rate V(n), n is the normal line vector. Influence of the surface

energy is not considered in Ref. [1], because the sizes of inconvertible polyhedrons

are big enough (B1022 cm). The parameter θ5 d(ln β)/dp is taken over as the

measure of anisotropy of the growth rate, here p characterizes orientation (declina-

tion) of the surface. If the declination p and density of steps in the central (most

curved) part of the facet (Figure 7.28B), is larger than for sites near the vertices

and ribs the larger is the value of the local kinetic coefficient. At strong depen-

dence V(p), i.e., at θ. 1, it is enough of deviation of order of 1� from orientation

of the singular facet to conserve the stationary value of the velocity V for any point

of curved facet:

Vðp;C � C0Þ5βðpðxÞÞðCðxÞ � C0Þ5Const; ð7:38Þ

where kinetic coefficient βðpÞ5βðp5 0Þffiffiffiffiffiffiffiffiffiffiffiffiffi11 p2

p. Inconstancy of supersaturation

C(x)2C0 along the facet (Figure 7.33A) is compensated by inconstancy of local

orientation (density of steps), and consequently, kinetic coefficient—β5β(p(x)).


The value of product β(p1)(C12C0) is physically defined by the activity of centers,

due to which the generation of new growing layers takes place.

If supersaturation is more than the critical value that is necessary for origin of

two-dimensional nuclei on the facets near the vertices, the last will be sources of

growth layers. At smaller than the critical value of supersaturation, the main

sources of formation of new layers are dislocations, which have arisen in the course

of growth and go out on the surface in the neighborhood of vertices, where super-

saturation is the greatest. Considering the relative change of supersaturation along

the crystal face, and also modifications of the local declinations, which are neces-

sary for compensation of differences in supersaturations (Eq. (7.38)), it is easy to

receive odds of declinations at the face center p2 and near the vertex p1 (under con-

dition of constant velocity):

p2 2 p1 5ðC1 2C2ÞðC1 2C0Þθ

: ð7:39Þ

If the relative supersaturation is (C12C2)/(C12C0)D0.2, and θB10, then

p22 p1� 23 1022. Thus, the local declinations B1� are enough for compensating

of such inconstancy of supersaturation, as it was discussed earlier.

The more crystal size is in comparison with D/β, the more should be the local

declination of the face to compensate heterogeneity of supersaturation. Occurrence

of sags at the center of a facet in the case of large supersaturations conducts to the

subsequent deterioration of diffusion supply of this site and, hence, to even its

strong lag from vertices. The distortions produce an occurrence on the surface of

sites with large kinetic coefficient. However, its anisotropy promotes growth rate

magnification already at a little larger increase of the declination and magnification

of a number of steps; thus the growth rate falls. As a result of these modifications,

the distortion increases even more, and so on. Thus, reaching some critical values

of local declinations results in the avalanche loss of stability, to be exact, the

impossibility of polyhedron to grow like itself (Figure 7.28C). Figure 7.29 shows

curved shapes of silver and bismuth crystals during growth. The first two photos

show crystals in binary melts. Figure 7.29C shows the decanted crystal (after the

melt dumping).

Figure 7.29 Distortion of the shape of growing faceted crystals [31]: (A) silver; (B and C)

bismuth; and (C) the decanted crystal of bismuth.


The kinetic coefficient increases sharply with magnification of local declination

of the surface, that is, the density of steps, in each point while the density is small.

Overlapping of diffusion fields of steps is sufficiently strong at large declinations,

and the kinetic coefficient of the face depends a little on its orientation. Linear rela-

tion V(p) responds to this area. According to morphological and kinetic data [1],

the area of strong dependence V(p) propagates to values of local declinations

pB1022�1021. Therefore, it is possible to take the value B1� mentioned above as

the critical value of maximum declination of surface at the face center, pcrit, at

which the skeletal growth begins. Calculation shows that the declination pcrit is

reached at the face center, when the crystal size is

Lcrit 5 f ðθÞðpcr � p1ÞD=βðp1Þ: ð7:40Þ

It is also the maximum size of the stable crystal; the skeletal growth develops at

exceeding of Lcrit. For growth from the melt, the critical size is

Lkp 5 f ðθÞðpcr � p1ÞαT=βTðp1Þ; ð7:41Þ

where αT is the thermal diffusivity, βT is the kinetic coefficient for growth from

the melt, and function f(θ)� 2.5, if θ. 1, and f(θ)� 1, if θ# 1.

7.5.4 Numerical Calculations of Evolution of the Crystal Shapes

Analytical stability theories of shapes of crystal growth featured above have certain

restrictions—the method of small perturbation was applied at consideration of

roundish shapes, and the linear relation of growth rate on supersaturation on the

sites near the vertexes of crystals, where new growth layers arise, was considered

in the analysis of the stability of faceted shapes. Growth of the vertices of small

enough crystals is predetermined by formation of new two-dimensional nuclei, and

the exponential law of growth rate in dependence on the local supersaturation takes

place in this case.

The free crystal growth was simulated in Refs. [60�64] by numerical solving of

the equations of heat or mass transport, taking into account the conditions of heat

or mass balance on the interface and surface kinetics, according to the mechanism

of growth under consideration. Figure 7.30A shows the grid which was used at con-

sideration of two-dimensional problems of free crystal growth [63], and

Figure 7.30B shows the grid for numerical solution of the directional solidification

problems [65].

Application of the implicit difference schemes with flattening, for example, with

allocation of the crystallization heat as additives to the heat capacity of the nearest

cells of the net (the method of the phase field), is justified in many cases.

However, explicit determination of the phase boundary with calculation of coordi-

nates of the surface points dependent on time and curvature of boundary is more

rational for solution of the crystal growth problems for cases when local curvature


of the surface influences essentially the growth rate and the shape of mobile sur-

faces varies.

The diffusion equation written down in finite differences in cylindrical

coordinates:

ð@C=@tÞ5D½ð@2C=@r2Þ1 ð1=rÞð@C=@rÞ1 ð1=r2Þð@2C=@θ2Þ�; ð7:42Þ

was solved at consideration of two-dimensional diffusion problems. Corresponding

derivatives were expressed through values of concentrations in nodes of the curvi-

linear net (Figure 7.30A). The net for computing concentrations was constructed as

follows. Radial rays were made from the center of the crystal to the edge of the

bath through equal intervals on angles. For symmetry reasons, the sector which cor-

responds to the anisotropy of growth rate, for example, 45� in the case of the

square crystal (Figure 7.30A), has been considered. Intervals on the distance along

radial rays from the surface of the crystal to the bath edge were chosen not con-

stant, but such increase in the arithmetical progression, according to Eq. (3.46).

Derivatives on coordinate r were written the same as in Section 3.4.3. And deriva-

tives on the angle θ were found after definition by interpolation of concentrations

on the nearest radial rays in points with the coordinate r, which was equal to the

coordinate r of the node under consideration.

The sweep method on serial time intervals has been applied for calculation of

concentrations in all nodes of the net, which are set by indexes m, n (on the angle

and on distance along the radial ray), and also coordinates of the surface nodes and

values of surface concentrations. Local growth rates were found according to the

certain functions (deduced theoretically, or those from approximate experimental

data) of their dependences on magnitude of superficial supersaturation V(σs)(σs5 (Cs�Ce)/Ce). And new values of surface concentrations were found from the

condition of mass balance at the interface, written for points of the crystal surface,

after definition of the concentration field (except for surface concentrations).

In connection with modification of coordinates of the surface nodes, the net was

(A) ρ1

ρm

m- /.nm.n

m+/.n

m+ /.n-l R

(B)

Figure 7.30 Reconstructed grids for consideration of two-dimensional problems of crystal

growth: (A) for free growth [63] and (B) for directional solidification [65].


rebuilt after each cycle of calculation (each time step): coordinates of all nodes

were modified in appropriate ways and the concentrations for the new position of

nodes were recalculated through defined values of concentration gradients.

Considering growth by the normal mechanism, anisotropy of interfacial tension

and kinetic coefficient was set. Angular dependences of the free surface energy and

kinetic coefficient for the cubic crystal were set in Ref. [66] as follows:

γ5 γ0ð11 zγ cos 4θÞ; ð7:43Þ

βs 5β0ð11 zβ cos 4θÞ; ð7:44Þ

where zγ and zβ are the coefficients which set the anisotropy degree. The surface

energy was taken into account through the correction of Gibbs�Thomson to super-

saturation, 2σsr�/r. Accordingly, the growth rate is determined by the equation

Vs 5βsσsð1� r�=rÞ; ð7:45Þ

where βs is the anisotropic kinetic coefficient; r� is the radius of the critical

nucleus; and r is the local radius of curvature of the surface. In addition, in the

case of the anisotropic surface energy, the growth rate was found from Herring’s

equation (7.2):

Vn 5βsσs 12Ks

ΩkT

γ1@2γ@θ2

� �� ; ð7:46Þ

where Ks is the local surface curvature, and expression in large brackets of

Eq. (7.2) is equal to γ0 (12 15zγ cos 4θ). In the direction of the maxima of surface

energy (θ5 0) growth rate is maximum, as the Gibbs�Thomson shift to supersatu-

ration is minimum.

At first, the Laplace equation was considered for numerical solutions of the

problem of the dendrite shape formation [67]. Using the techniques described

above, authors of Refs. [61�63,66] took into account the kinetic coefficient and its

anisotropy. Figure 7.31A shows evolution of the crystal shape at the given initial

roundish anisotropic shape. If anisotropy γ and βs is set, protuberances will develop

faster in the matching directions. However, the influence of anisotropy of interfa-

cial tension remains essential while the crystal size is smaller than B100r�. It wasfound in Ref. [66] that the curvature radii of protrusion do not change practically in

the range of radius values (1000�1500)r� because of reducing the surface supersat-

uration and growth velocity. Modeling of growth of the anisotropic particle from

the melt is fulfilled in Ref. [63] by a similar procedure.

In a number of up-to-date works [68,69], the shape changes of crystals are

considered using the lattice method, based on the Boltzmann kinetic equations

(the variant of this method for the description of currents in the fluid was men-

tioned in Section 3.4.3). Use of very shallow lattices (nets) allows us to spot


precisely enough the position of boundary surface within the limits of model of the

phase field (see Section 3.4.2) and count local declination of the surface and its

curvature that is necessary for taking into account anisotropy of surface energy and

kinetic coefficient. It is clear that such modeling requires use of the modern power-

ful COMPUTERS merged in so-called clusters. The principal problem of the proce-

dures based on the Boltzmann kinetic equation is connected with difficulties of

choice of coefficients, which characterize dispersion of the chosen mass elements

in order that the outcomes of modeling correspond to the values of diffusivities

(thermal conductions, viscosities) known from experiments. Figure 7.32 shows for-

mation of dendritic shapes of crystals at two values of parameter of anisotropy zγon the net 2503 2503 250.

Ovrutsky [60,64] for the first time simulated the growth of faceted crystals by

numerical solving of the diffusion problem by taking into account surface kinetics

caused by two-dimensional nuclei formation. Figure 7.31B shows the calculated

consecutive profiles of crystals for the case of growth in the two-dimension bath,

and Figure 7.33A shows graphs of the surface supersaturation distribution over the

face of cubic crystal along different directions. Apparently, from these graphs,

the greatest supersaturation is near the vertex of cubic crystal, and the least is at

the face center. The existence of two critical sizes of instability for the first time

has been established in Refs. [60,64]. It follows from Figure 7.33B that the declina-

tion at the facet center reaches several degrees at size B1022 cm, and in the case

of large enough supercooling (curve 3), the face is close to definitive losses of sta-

bility. Growth rate falls with the subsequent increasing of the crystal size because

the surface supersaturation decreases, including vertices (transition from kinetic

growth to diffusion mode), and dependence of growth rate on supersaturation is

exponential in the case of growth by the mechanism of two-dimensional nuclei

formation. Therefore, the angles start decreasing. However, the supersaturation at

1

500 R*

10–2cm1

2

34

5

6

7

2

34

5(A) (B)

Figure 7.31 Evolution of the shape of crystal growth [63,64]: (A) zγ5 zβ5 0, (CN�Ce)/

Ce5 0.17, CN5 0.8, r15 r0(θ)5 300r� 1 19cos 4θ, Γ 05 γ0Ω/kT5 2.243 1028 cm,

D5 2.83 10250 cm2/c, t55 0.004 s; (B) Bi1 20 mass% Sn, β5 0.05 cm/s, γ/kT5 0.134,

V5βσs5/6exp(2K/σs), K5πγ/(3k2T2), t75 572 s.


the center of the faces reaches zero in certain time, and the shape becomes defini-

tively unstable. The criterion of instability from condition of vanishing of supersat-

uration at centers of faces has been offered for the first time by Cahn [70].

Ueta and Saito [71] studied free crystal growth using simulations by the KMC

method for the model of two-dimensional lattice gas with gradient of density.

Figure 7.34 shows the shape evolution of the crystal having the initial square shape

(the Jackson roughness parameter α (see Section 7.2.3) was responded to atom-

ically smooth surface).

0.6(A) (B)

0.5

0.4

2 3

16

ϕ 0 (g

rad)

4

23a 2a

1a

2

3

1

0.3

1 2 3 10–4 10–3 10–2 10–1 100 101

Z0 (cm)102

X (10–2cm)

Cs–

Ce

C∞–C

e

Figure 7.33 Concentration distributions over the face of cubic crystal (A) and the slope

angles at the centers of faces in dependence on their size (B) [64], (A)—(1) along diagonal,

(2) along rib, (3) from the face center to the rib middle; (B)—the system Bi1 20 mas.% Sn,

for 1�3 σN5 0.00756, 0.0153, and 0.023 accordingly (supercoolings relatively the liquidus

temperature ΔT5 2.4, 4.8, and 6 K).

TemperatureTemperature

0.0124

–0.00164

–0.0156

–0.0296

–0.0436

0.0159

–0.0434

–0.103

–0.162

–0.221

(A) (B)

Figure 7.32 Three-dimensional dendritic crystals growing from the melt [68]: (A) zγ5 0.05

and (B) zγ5 0.0081, the surface temperature is higher at vertices.


7.6 Development of Cellular Structure During DirectionalSolidification

7.6.1 Concentration (Diffusion) Supercooling

Directional crystallization is a basis of the diversified technological processes of

manufacture of single crystals or alloys with the anisotropic structure. Ivantsov [72]

for the first time studied the phenomenon described below; he termed it a diffusion

supercooling. Afterward, Rutter et al. [73] termed it a concentration supercooling.

Figure 7.35 shows the scheme of directional crystallization from binary melt.

Such crystallization is defined by exterior factors: by the motion of container with

crystal and melt concerning the furnace (usually with constant velocity V) and

by the temperature distribution. The idealized T(x) distribution is shown in

Figure 7.35A. The fluid temperature is above liquidus, and the temperature

increases with the distance from the front of crystallization.

We consider that the concentration of impurity at the crystallization front

Cs5Ce, and impurity distribution before front is stationary, that is, it moves with

constant velocity V (together with the interface). Therefore, the derivative @C/@tin diffusion equation, @C/@t5D(@2C/@x2), can be expressed from the condition:

@C/@t5�V(@C/@x). Hence:

D@2C

@x21V

@C

@x5 0: ð7:47Þ

300

250

200

150

Y

100

50

00 50 100 150 200

X250 300

Figure 7.34 Two-dimensional MC simulation of the crystal growth [71].


After integrating this equation and taken into account the boundary conditions:

C5Ce at x5Xfr 5Vt and VðCcr 2CeÞ5D@C

@x

� �Xfr;

we will obtain x.Xfr:

Cðx; tÞ5CN 2 ðCcr 2CeÞ exp 2Vðx2XfrÞ

D

� �: ð7:48Þ

Ccr5CN for stationary process. Considering that ðCcr=CeÞ5 kd, one can rewrite

the equilibrium distribution number of impurity (7.48) in the following form:

Cðx; tÞ5CN 1112 kd

kd� exp 2

Vðx2VtÞD

� �� : ð7:49Þ

Let us approximate a site of liquidus under consideration by linear relation:

Te 5 Tf 2mCðx; tÞ: ð7:50Þ

It follows from Eqs. (6.39) and (6.40) that equilibrium temperature of liquidus

depends on coordinate x:

TeðxÞ5 Tf 2mCN 1112 kd

kd� exp 2

Vðx2XfrÞD

� �� : ð7:51Þ

Owing to the exponential law can happen that equilibrium temperature Te(x)

will increase with distance from the crystallization front at first faster than the real

T

C Ce

S L

xfr T (X ) Te(X) X

X

X

Cc r= C∞

(A)

(B)

Figure 7.35 Distributions of the temperature (A) and impurity concentration (B) at the

directional crystallization.


temperature T(x) (Figure 7.35). As a result, some melt zone before the crystalliza-

tion front will be supercooled, T(x), Te(x). The supercooling, obviously, will not

be at large enough gradient of real temperature G5 dT/dx. The following relation

will be condition of absence of the concentration supercooling:

GT $@TeðxÞ@x

Xfr 5Vm � CN

D� 12 kd

kd:

�� ð7:52Þ

Thus, there is such critical velocity

Vc 5GTDkd

mð12 kdÞCN;

at the exceeding of which the concentration supercooling exists.

Presence of the concentration supercooling routinely results in disturbances of a

flat front of crystallization, in cellular structure occurrence (protuberances move

ahead into fluid), and in different crystal imperfections predetermined by it. At the

considerable expansion of zone of the concentration supercooling, the interface

structure becomes cellular dendritic, that is, there is a side ramifying of the pro-

tuberances growing into fluid. Figure 7.36 shows the images of the cell shapes,

received in Ref. [74] in a study of directional solidification.

7.6.2 The Basic Results of the Theory of Small Perturbations

The criterion of concentration supercooling (Eq. (7.52)) defines the limiting values

of the interface velocity Vc, at which the concentration supercooling will arise at

certain temperature gradient GT [73] or the value of GT, which is needed for full

depression of the concentration supercooling. Disturbances of the front can be

avoided by reducing velocity or raising the temperature gradient. In a number of

early works, including Papapetrou [75], qualitative conceptions on how the flat

front at the directional crystallization of binary alloy loses stability have been

offered.

Figure 7.36 Shapes of cells for different relative velocities [74]: (A) ν5V/Vc5 1.3,

λ5 55 mkm; (B) ν5 2.6, λ5 45 mkm; and (C) ν5 9.5, λ5 45 mkm.


Mullins and Sekerka [76] for the first time theoretically considered losses of

stability of flat front concerning small periodic perturbations of the surface. They

supposed that all parameters in volume and on the surface are isotropic, a local

equilibrium on the interface takes place, and thermal and concentration fields are

stationary. Development or decay of the perturbation z5 δ(t)sin(2π/λ)x with small

amplitude δ(t), superimposed on a flat surface was explored. According to Refs.

[58,76], at violation of criterion (7.52), the value of the certain wavelength λ0(inertly stable wavelength) exists, for which the amplitude δ of the sinusoidal per-

turbation remains invariable:

λ0 5 2πγΩC0

RTGc

� �1=2

5 2πγΩDC0

VðC2CcrÞ

� �1=2

; ð7:53Þ

where γ is the free surface energy, Ω is the volume counting per one atom, R is the

gas constant, T is the temperature, Gc is the concentration gradient. The distortions

of the surface with the wavelength that is higher than λ0 are increasing during the

time, and those having the smaller wavelength—are decreasing.

The wavelength, which corresponds to perturbations with the fastest increasing

amplitude, is as follows:

λM 5ffiffiffi3

pλ0: ð7:54Þ

It is possible to construct the graph of dependence δ0/δ for each real case, similar

to that, which is shown in Figure 7.37.

For real values of parameters of growth, it appears that the area of lengths of

waves, at which δ0/δ. 0, in particular, wavelength λM, at which this magnitude

reaches maximum, has the same order, as a typical size of cells (B50 microns).

It was important to find correspondence between theoretically calculated

2π2π

2π 2π

λcλM

λ0 λ

1

2

δδ

3

Figure 7.37 The dependence of magnitude δ0/δ on λ21: (1) there is instability; (2) instability

exists in single point; and (3) the stability is conserved.


wavelength, at which the magnitude δ0/δ has maximum value under conditions of

instability, and sizes of cells known from experiments, in which the structure is

gained eventually by instable interface. The component of the Fourier series expan-

sion of any perturbation of a flat surface, which grows fastest, should lead finally

to the wavy or cellular structure with wavelength λM. However, this magnitude,

found theoretically (Eq. (7.54)), does not coincide with the size of cell obtained in

practice. Calculations are based on the supposition about smallness of the perturba-

tion amplitude (the first order on δ) that is completely admissible for stability

examination, but it does not allow receiving generally reliable information concern-

ing a finite size of cell.

For the case of the infinite kinetic coefficient and isotropic interfacial tension,

Noel et al. [77] have found expression for definition of critical wavelength λc, inrelation to which the surface is most instable, and the gamut of lengths of waves at

critical velocity, Vc, of the temperature field motion (Figure 7.37, curve 2) is tight-

ened into a point:

λc 5 2π½2γΩD= kGTVcð Þ�1=3: ð7:55Þ

Developing the Mullins�Sekerka theory, Langer [78] considered some simple

cases of instability of the interface, which resulted in the formation of the cellular

structure at crystallization (cases of pure substances with flat or spherical boundary

surfaces and the case of dilute solutions).

In a number of research studies, the linear analysis of stability has been fulfilled

taking into account wider spectrum of conditions. Influence of convective flows on

interface stability was explored in Ref. [79]. It has been shown that convection

essentially influences the stability. Sriranganathan et al. [80] studied the parameters

of crystallization influence on the loss of stability; the dependences mL, k, and σ on

the concentration were taken into account and also σ on the temperature.

According to Wheeler [81], periodical changes of the velocity of move result in

stabilization of the interface.

The linear theory (small perturbations) cannot precisely predict even develop-

ment of the distorted surface to cellular structure. The terms of higher order, which

are neglected for linearity deriving, become comparable in magnitude with those

which are kept, because of exponential magnification of the perturbation amplitude.

Hence, the yielded problem is essentially nonlinear.

For the first time, Coriell and Sekerka [82,83] analyzed the nonlinear problem

for a two-dimensional system. Within the limits of the nonlinear approach, the

influence of anisotropy of crystal properties on the morphology of the solidification

front was also explored in Ref. [84]. The author set anisotropy of surface energy in

the form: γBγ(11 zγ(12 cos 4θ)), where zγ is the anisotropy parameter and θ is

the angle between directions of singular (ideal) and vicinal (real) faces, was consid-

ered. It was found in a result of numerical modeling that wavelength λ varies to

10% at replacement of the value zγ5 0 to zγ5 0.2. However, small enough ampli-

tudes of distortions were all the same under consideration in nonlinear theories.


Obtaining the cellular structure with big protuberances or troughs, as it is observed

in experiment, has not been reached in theory.

The problem of cellular structure development at the considerable violation of

the stability criterion (Eq. (7.52)) is developed much more feebly than the problem

of stability in principle. Tiller et al. [73] calculated the shape of stable cell

(two-dimensional and hexagonal) in melt, which is not agitated. He yielded the

approximate equations for breadth (λ) and depths (d) of cells, and expression for

estimation of the overfall of concentration on the cell boundary. It follows from his

equations that the cell breadth decreases with increase in G and V. Weeks and van

Saarloos [85] studied area of stability of waves with lengths depending on the core

monitoring parameter

ν5VmLδC0

GTD5

V

Vc

; ð7:56Þ

where δC0 is the difference of concentrations of impurity in melt and crystal at sur-

face temperature, which corresponds to conditions of steady growth, Vc is the criti-

cal velocity. This parameter is included into the equation for length of inertly

stable wave [66]:

λ0 5 2πυlDd0υ21

� �1=2

; ð7:57Þ

where lD5D/V is the diffusion length; d0 is the chemical capillary length:

d0 5γ0Tsμ

ΔHρmlδC0

;

where γ0 is the average value of the free surface energy; μ is the molar mass; ΔH

is the latent heat of fusion; ρ is the density of the main component; ml is the slope

of the liquidus line; δC0 is the odds of impurity concentrations in the melt and in

crystal at the temperature, which corresponds to the interface position at stationary

growth.

The major trouble from the cellular structure is in its influence on the distribu-

tion of a dissolved matter—the cellular microsegregation; and this defines inhomo-

geneity of concentration in the obtained crystals. Flemings et al. [73] studied the

cellular microsegregation both experimentally and theoretically.

There is a number of experimental data concerning of loss of the interface

stability in the metal or organic (metallic) systems, which are characterized by

the normal mechanism of growth. However, their interpretation in many respects

is complicated because of inaccuracy in determination of thermal conditions.

Apparently, the results obtained in Refs. [86�88] for organic compounds, in Ref.

[89] for Al and Zn with impurities, in Ref. [90] for alloys Al�Ti and Al�Cr, in

Ref. [91] for alloys Bi�Sb are trusty. At loss of stability, the flat front is evolving


in according to magnitude of the major monitoring parameter G/(VC0); it is passing

through the different morphological stages [87,90,92]: flat front, two-dimensional

cells, the regular or hexagonal cells, perturbed cells, dendrites. On the basis of

these data, the scheme is constructed in Refs [1,93] (Figure 7.38), which shows

development of instability of the crystallization front.

As it was mentioned above, cellular structure formation is accompanied by cel-

lular microsegregation, which predetermines formation of dislocation structures

under the influence of concentration stresses. It was studied by the up-to-date

experimental methods of microanalysis, for example, for systems Fe�Ni [94],

Cu�Al [95], Sn�Pb [96].

7.6.3 Modeling Directional Solidification Using Finite DifferenceSchemes

Computer modeling of the directional solidification has been developed for the best

understanding of formation of cellular structure. The models of phase field have

been developed at first for study of solidification of the supercooled melt of pure

substance with diffusive interface [97�99], and since then, they were applied to

other processes of growth [100,101]. The phase field is such certain function of

time and coordinates, which is found during problem solution of the heat and mass

transport (see Section 3.4.2). In this case, the interface is defined as a set of contour

lines, which correspond to certain values of the phase field, for example, the tem-

perature is equal to the melting point value. Thus, it is possible that the problem

with the mobile boundary (Stefan’s problem) will be substituted effectively by the

simpler problem with the fixed boundary after introducing of one additional equa-

tion. Any differences between phases and their boundaries are not taken in consid-

eration, hence, all fields are considered as in the homogeneous phase from the

mathematical point of view. This approach allows carrying out calculations using

variables, which have saltuses at the interface positions. The method was applied

also for consideration of problem of the directional solidification.

Hunt and Lu [100,101] have developed a numerical model, having avoided

shortages of the supposition of the constant shape of the cell. The diffusion

equation for fluid phase was solved and the common balance of mass was taken

into account. Interface positions were calculated through definition of fusion

Figure 7.38 Surface evolution [1,93]: (A) pointwise dimples and “filaments,” grooves and

“fillets,” (B) parallel grooves, and (C) the hexagonal grid of grooves.


points, taking into account concentrations and the interfacial tension influence.

Interaction between cells was not considered, and any principle of selection was

not applied. As a result of calculations, the shapes and sizes of stable cells

were defined. Comparison with outcomes of experiment [102] for the system

suksinonitril10.075 mass% acetone is shown in Figure 7.39A. Apparently from

the drawing, the maxima and minima of the cell size distributions, and also average

values of sizes, which were calculated in Ref. [101] and measured in experiments

[102], correlate well.

Figure 7.39B shows the dimensionless shapes of cells (in terms of half of their

breadth) for three rates of solidification elected in experiments. The calculated

shapes of interface are in good correspondence with the typical shape of the cell

[100], which is characterized by magnitude of radius of the cell vertex that give

estimation of half of the breadth of the cell (93, 58, and 46 microns, accordingly,

for velocities 2.0, 2.5, and 3.0 microns/s).

Payment for exact enough computing of the interface positions was necessity of

use of very shallow nets. Therefore, either two-dimensional problems or three-

dimensional problems for very small volumes were usually considered. The simple

procedure taking exactly into account the boundary conditions at the moving, vari-

able in the shape, sharp phase boundaries was applied in Ref. [65]. It consists in

restriction of incremental values of surface concentrations at the first stage of cal-

culations. It has allowed us to consider the three-dimensional problems for real

scales of shaping of cellular microstructures. Calculations were made for the sys-

tem succinonitrile�acetone (SCN�ACE, 0.2 mol% ACE), which is the convenient

model material owing to the transparence and its structure similarity after solidifi-

cation to the structure of metal alloys. Besides, the results of experimental

researches are known [100,101] for this system.

300(A) (B)

250

λ(µ

m)

Experiments:

Therory: Hunt and LuTheor. λmax Theor. λmin

λmax

λav

λmin200

150

100

50

0100 101

V (μm/s)102

1.0

0.8

0.6

0.4

v = 3.0 μm/s

r′

v = 2.5 μm/s

v = 2.0 μm/s

0.2

0.0

–0.2

–0.4

–0.6

–0.8

–1.0–2.5 –2.0 –1.5 –1.0 –0.5

X′0.0 0.5

Figure 7.39 The average (K), maximum (¢), and minimum (£) sizes of cells with

wavelength λ, found from experiment [102]: (A) in comparison with the calculations of

Hunt and Lu [101] (----- theor. λmax, �� theor. λmin) and shapes of the interface cells

(B) calculated in Ref. [102] in the framework of the Hunt and Lu model [101] for three

velocities 2.0, 2.5, and 3.0 microns/s.


The solutions obtained in Ref. [65], reproduce the basic phenomena, which were

observed in experiments: occurrence of fine-meshed structure near to local pertur-

bation of the surface, the competition in growth of different cells and development

of more or less stationary structure with large enough cells.

Figure 7.40 shows consecutive profiles of the crystallization front for cases of

initial conditions with small periodic perturbations on the interface. As the velocity

of perturbation development (magnification of amplitude of cells) increases in dur-

ing the time, it is convenient to consider exponential component ω of the amplitude

dependences on time:

ξðk; tÞ5 ξ0 1 ξkexpðikrÞexpðωtÞ;

where k is the wave vector, jkj5 2π/λ.It was found in Ref. [65] that the perturbations developed in the certain gamut

of wavelengths. There are maxima in graphs ω(λ), which is biased toward smaller

wavelengths with magnification ν (Figure 7.41). At rather low value ν (ν5 1.5)

(Figure 7.41A, curve 3) outcomes of modeling differ a little from the values calcu-

lated according to the Mullins and Sekerka theory. The velocities of perturbation

development exceed essentially theoretical values at large velocity of pulling

sec307.251 ys= 4.261 λ0

0 0.8

(A) (B) (C)

1.6 2.4 3.2 0 0.8 1.6 2.4 3.2λ0

0 1 2 3 4

sec307.251 ys= 3.67 λ0

sec266.711 ys= 3.182 λ0

Figure 7.40 Consecutive profiles of the surface at the anisotropic kinetic coefficient [65],

ν5 2.5: (A) zβ5 0.4; (B) zβ5�0.4; (C) zβ5 0.4, zγ5 0.04, ϕcr5 22.5�.


(ν5 3, Figure 7.41A, curves 1 and 2). It is quite clear, as the analytical theory is

developed only for the case of very small amplitudes of disturbances.

In two-dimensional (isotropic) case, the maximum of the ω(λ) dependence

responds to the value λD0.9λcD1.6λ0D5.8λ0theor at ν5 1.5 (values λ0 were

defined from graphs, similarly figured on Figure 7.41), and λD0.55λcD3.5λ0D6.7

λ0theor at ν5 3. In a three-dimensional case, the maximum takes place at

λD0.5λcD1.8λ0D3.6λ0theor if ν5 1.5, and λD0.4λcD3.1λ0D5.0λ0theor if ν5 3.

Ovrutsky and Rasshchupkyna [65] found values λ0 (the minimum wavelength

that can develop in a two-dimensional case) are as follows: λ05 0.56λc at ν5 1.5

and λ05 0.16λc at ν5 3. They exceed in two to three times the value λ0theor(2.763 1025 m and 1.383 1025 m, accordingly). In the three-dimensional case,

the values λ0 obtained by simulations (λ05 0.28λc at ν5 1.5 and λ05 0.13λc at

ν5 3) exceed outcomes of the theory of small perturbation in 1.5�2 times.

Figure 7.41B presents the velocities of perturbation development at the anisotro-

pic kinetic coefficient. By marker it is designated the limiting values of wave-

lengths in a two-dimensional case, behind which there are bifurcations for the

dependence designated by the marker ¢. Shown values are obtained for two orien-

tations of crystal (also, two-dimensional and three-dimensional cases, ν5 3).

Perturbations will develop faster if a kinetic coefficient has maximum value in

the pulling direction. As it follows from Figure 7.41A, anisotropy of the kinetic

coefficient (curve 1) or interfacial tension (curve 2) differently influence the

dependences ω(λ).Losert et al. [103] studied the cellular structure development at different initial

perturbation of the surface by the method of phase field in diffusion mode (the infi-

nite kinetic coefficient). They considered feeble anisotropies of surface energy

(1%). Authors obtained specific enough shapes of cells (Figure 7.42) at superposi-

tion of the initial perturbations, which were set by two wave functions of different

amplitudes.

0.15

(A) (B)

1

2

3

ω (S–1)

ω (S–1)

0.1

0.05

00 1 2

0.4

0.2

00 0.95 1.0

λ/λcλ/λc

Figure 7.41 The exponential components ω of the perturbation velocity in dependence

on their wavelength [65]: (A) (2D), λc5 1.683 1024 m: 1�ν5 3, zγ5 0, zβ5 0.4,

β5 0.15 m/s; 2�ν5 3, zβ5 0, zγ5 0.06, β5 0.5 m/s; 3�ν5 1.5, zβ5 0, zγ5 0.06,

β5 0.5 m/s. (B) β5 0.15 m/s, zγ5 0, λc5 1.683 1024 m, λ0theor5 1.383 1025 m: 3D:

Δ2 ν5 3, zβ5 0.4; 2D: ¢2 ν5 3, zβ5 0.4.


7.6.4 Kinetic Modeling of Directional Solidification by the MC Method

Ovrutsky and Rasshchupkyna [104] studied by the kinetic MC simulations the

development of disturbance waves in shapes of the crystallization front during

directional solidification. Figure 7.43 shows how at the directional solidification

impurity atoms (dark gray) accumulate in front of the growing crystal. And

Figure 7.44 shows sections of the simulated samples near to crystallization front.

Ovrutsky and Rasshchupkyna [104] have compared the concentration distribu-

tion, which corresponds to Figure 7.43, with the analytical solution for the one-

dimensional diffusion problem of the directional solidification (Eq. (7.49)) and

have determined the diffusivity for model (DD1.23 1025 cm2/s). It has allowed

them to spend comparison of the cellular structures development data with out-

comes of theory of small perturbation [76] and to come to the conclusion that per-

turbation with large enough lengths of waves (in the atomic scale) develops

according to theoretical predictions.

Simulation has discovered the existence of a lower limit of lengths of waves for

which surface perturbation can develop with formation of the inconvertible cellular

structure: B30 interatomic distances; perturbations of the smaller size are destroyed

by fluctuations of growth. Stability of flat interface at the very large sample

Figure 7.42 Cellular structure development at two wave perturbations on the initial surface,

the succinonitrile�coumarin system, ν5 5, anisotropy zγ5 0.01 [103].

Figure 7.43 Fragment of section of the simulated sample.


movement velocities predicted by the theory [76] does not prove to be true simula-

tion. Local fluctuations of growth rate result in to random disturbances of the shape

of the surface, and structure of the crystal formed under such conditions is very

imperfect.

References

[1] A.A Chernov, Modern Crystallography. Volume 3: Crystal Growth, Springer, Berlin,

1984.

[2] A.M. Ovrutsky, Rep. AN USSR. Metals 4 (1978) 111�116 (in Russian).


[4] Ya.I. Frenkel, J. Exp. Theor. Phys. 16 (1) (1946) 39�51.

[5] K.A. Jackson., Liquid Metals and Solidification, American Society for Metals,

Cleveland, OH, 1958, pp. 174�186.

[6] W.K. Burton, N. Cabrera, F.C. Frank, Phil. Trans. R. Soc. 243 (1951) 299.

[7] D.E Temkin, Crystal Growth, vol. 5, Nauka, Moscow, 1965, pp. 89�93 (in Russian).

[8] H.J. Leamy, K.A. Jackson, J. Appl. Phys. 42 (5) (1971) 2121.

[9] J.P. van der Eerden, C. van Leeuwen, P. Bennema, et al., J. Appl. Phys. 48 (6) (1977)

2124.

[10] G.H. Gilmer, J. Cryst. Growth 42 (1977) 3.

[11] J.P. Hirth, G.M. Pound., Condensation and Evaporation—Nucleation and Growth

Kinetics, Pergamon Press, Oxford, 1963.

[12] V.V. Voronkov, Crystal Growth, Properties, and Applications, vol. 9, Springer-Verlag,

Berlin, 1983, pp. 76�111.


[14] Y.A Wilson, Philos. Mag. 50 (1900) 238.

[15] Ya.I. Frenkel, Phys. Z. Sowjetunion 1 (1932) 498.

Figure 7.44 Evolution of perturbations of the interface: (A) initial perturbation with

wavelength λ5 25 interatomic distances; (B) their evolution; and (C) growing perturbation

(λ5 50 interatomic distances, growth rate V5 5.43 106 of interatomic distances for 1 s).



























[16] A.G. Borisov, O.P Fedorov, V.V. Maslov, Sov. Phys. Crystallogr. 36 (5) (1991) 1267.

[17] D.E Temkin, Sov. Phys. Crystallogr. 28 (2) (1983) 240.

[18] D.E Ovsienko, G.A. Alfitsev, Phys. Metals Phys. Metall. 20 (3) (1965) 401.

[19] O.P. Fedorov, Crystal Growth Processes: Kinetics, Shape Formation, Defects, Naukova

Dumka, Kiev, 2010. 207 pp. (in Russian).

[20] D. Turnbull, Thermodynamics in Physical Metallurgy, American Society for Metals,

Cleveland, OH, 1952.

[21] A.M. Ovrutsky, I.G. Rasin, Trans. JWRI 30 (2001) 239.

[22] E. Miller, J. Cryst. Growth 42 (1977) 357.

[23] M. Elwenspoek, P. Bennema, J.P. van der Eerden, J. Cryst. Growth 83 (1987) 297.

[24] M. Elwenspoek, J.P. Van-der-Eerden, J. Phys. A. Math. Gen. 20 (1987) 669.

[25] V.V. Voronkov, Crystal Growth, vol. 9, Nauka, Moscow, 1972, pp. 257�263.

[26] P.J.C.M. van Hoof, W.J.P. van-Erckevort, M. Schoutsen, et al., J. Cryst. Growth 183

(4) (1998) 641.

[27] H.J. Leamy, G.H. Gilmer, J. Cryst. Growth 24 (1974) 499.

[28] R.F. Xiao, J. Iwan, D. Alexander, F. Rosenberger, Phys. Rev. A 43 (1991) 2977.

[29] J.D Weeks, G.H. Gilmer, Adv. Chem. Phys. 40 (1979) 157.

[30] A.M. Ovrutsky, M.S. Rasshchupkyna, A.A. Rozhko, J. Surf. Invest. (1) (2006) 85.

[31] A.M. Ovrutsky, I.V Salli, Growth and Defects of Metal Crystals, Naukova Dumka,

Kiev, 1972, pp. 95�103 (in Russian).

[32] LS. Palatnik, I.I Papirov., Epitaxial Films, Nauka, Moscow, 1971 (in Russian).

[33] J.A. Venables, G.D.T. Spiller, M. Hanbucken, Prog. Phys. 47 (1984) 399.

[34] N.T. Gladkih, S.V. Dukarov, S.V. Krishtal, Surface Phenomenon and Phase Transition

in Condensed Films, KhNU, Kharkov, 2004. (in Russian).

[35] I.V. Markov, Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal

Growth and Epitaxy, World Scient. Publ. Comp., Inc., 2003.

[36] D. Kashchiev, J. Cryst. Growth 40 (1977) 29.

[37] D. Bimberg, M. Grundmann, N.N. Ledentsov., Quantum Dot Heterostructures, Wiley,

New York, NY, 1999.

[38] J.H. van der Merwe, J. Appl. Phys. 41 (11) (1970) 4725.

[39] V.A. Shchukin, N.N. Ledentsov, P.S. Kop’ev, Phys. Rev. Lett. 75 (1995) 2968.

[40] I. Daruka, A.L. Barabasi, Phys. Rev. Lett. 79 (1997) 3708.

[41] S.A. Kukushkin, A.V. Osipov, Prog. Surf. Sci. 51 (1) (1996) 1.

[42] A.V. Osipov, Thin Solid Films 231 (1995) 173.

[43] M. Smoluchovski, Ann. Phys. 21 (1906) 759.

[44] S.Yu. Karpov, A.S. Segal, D.V. Zimina, Mater. Res. Soc. Symp. Proc. 743 (2003)

L3.40, 1.

[45] A.M. Ovrutsky, A. Posylaeva, Crystallogr. Rep. 47 (2002) 1.

[46] W.A. Johnson, R.F. Mehl, Trans. Am. Inst. Min. Metal. Petro. Eng. 135 (1939) 416.

[47] Ya.B. Zeldovich, N.N. Semenov, J. Exp. Theor. Phys. 12 (1940) 565.

[48] W.M. Plotz, K. Hingerl, H. Sitter, Phys. Rev. B 45 (2) (1992) 12122.

[49] F. Much, M. Ahr, M. Biehl, W. Kinzel, Europhys. Lett. 56 (2001) 791.

[50] M. Vladimirova, A. Pimpinelli, A. Videcoq, J. Cryst. Growth 220 (4) (2000) 631.

[51] A.V. Zverev, I.G. Neizvestny, N.L. Shvarts, et al., Phys. Eng. Semiconductors 35 (9)

(2001) 1067.

[52] L.S. Palatnik, M.Ya. Fuks, V.M. Kosevich, Mechanisms of Formation and

Substructure of Condensed Films, Nauka, Moscow, 1972 (in Russian).

[53] A.M. Ovrutsky, M.S. Rasshchupkyna, DNU Peports, Series Phys. Radio Electron. 15

(2) (2008) 93.






















































[54] Yu.F. Komnik, Physics of Metal Films. Dimensional and Structure Effects, Atomizdat,

Moscow, 1979.

[55] D.E. Ovsienko., Nucleation and Growth of Crystals from Melts, Naukova Dumka,

Kiev, 1994 (in Russian).

[56] A.M. Ovrutsky, V.V. Podolinsky, Metallofizika, 53, Naukova Dumka, Kiev, 1973.

p. 87 (in Russian).

[57] V.V. Podolinsky, J. Cryst. Growth 44 (1979) 511.

[58] W.W. Mullins, R.F. Sekerka, J. Appl. Phys. 34, (2) (1963) 323.




[62] D.E. Ovsienko, A.M. Ovrutsky, O.P. Fedorov, J. Exp. Theor. Phys. 100 (1991) 939.

[63] A.M. Ovrutsky, Rep. AN USSR 4 (1980) 80 (in Russian).

[64] A.M. Ovrutsky, J. Cryst. Growth 116 (1992) 158.

[65] A.M. Ovrutsky, M.S. Rasshchupkyna, Crystallogr. Rep. 53 (7) (2008) 1208.

[66] A.M. Ovrutsky, J. Exp. Theor. Phys. 99 (1991) 250.

[67] D.A. Kessler, H. Levine, Phys. Rev. A 39 (1989) 3041.

[68] I. Rasin, S. Succi, W. Miller, Phys. Rev. E 72 (6) (2005) 066705.

[69] I. Rasin, S. Succi, W. Miller, J. Comp. Phys. 206 (2) (2005) 453.

[70] J.W. Cahn (Ed.), Crystal Growth, Pergamon Press, New York, NY, 1967,

pp. 681�690.

[71] T. Ueta, Y. Saito, Research of Pattern formation, KTK Scientific Publisher, Tokyo,

1994, pp. 131�154.

[72] G.P. Ivantsov, Proc. USSR Acad. Sci. 81 (2) (1951) 179�182.

[73] W.A. Tiller, K.A. Jackson, J.W. Rutter, et al., Acta. Met. 1 (1953) 428.

[74] P. Kurowski, C. Guthmann, S. de Cheveigne, Phys. Rev. A 42 (1990) 7368.

[75] A.Z. Papapetrou, Crystallography 92 (1935) 89.

[76] W.W. Mullins, R.F. Sekerka, J. Appl. Phys. 35 (2) (1964) 444.

[77] N. Noel, H. Jamgotchian, B. Billia, J. Cryst. Growth 181 (1997) 117.

[78] J.S. Langer, Rev. Mod. Phys. 52 (1980) 1.

[79] S.R. Coriell, R.F. Sekerka, J. Cryst. Growth 61 (1983) 499.

[80] R. Sriranganathan, D.J. Woolkind, D.B. Oulton, J. Cryst. Growth 62 (2) (1983) 265.

[81] A.A. Wheeler, J. Cryst. Growth 67 (1) (1984) 8.

[82] S.R. Coriell, R.F. Sekerka, J. Cryst. Growth 34 (2) (1976) 157.

[83] S.R. Coriell, G.B Mc Fadden, R.F. Secerka, Annu. Rev. Mater. Sci. 15 (1985) 119.

[84] Alai Karma, Phys. Rev. Lett. 57 (1987) 858.

[85] J.D. Weeks, W. van Saarloos, M. Grant, J. Cryst. Growth 112 (1991) 244.

[86] E.L Zhivolup, O.P Fedorov, D.E. Ovsienko, Metallofizika, 13, Naukova Dumka, Kiev,

1991. p. 81 (in Russian).

[87] V. Nemoshkalenko, E. Zhivolub, E. Bersudsky, O. Fedorov, Space Research in

Ukraine 2000�2002, Naukova Dumka, Kiev, 2002.

[88] O.P Fedorov, A.P Shpak, E.L Zhivolub, O.B. Shuleshova, Crystallogr. Rep. 50 (6)

(2005) 1107.

[89] O.P. Fedorov, E.L. Zhivolub, Crystallogr. Rep. 43 (5) (1998) 877.

[90] K. Shibata, T. Sato, G. Ohiro, J. Cryst. Growth 44 (4) (1978) 419.

[91] H. Jamgotchian, B. Billia, L. Capella, J. Cryst. Growth 62 (3) (1983) 539.

[92] H. Biloni, R. Dibella, G.F. Bolling, Trans. Met. Soc. AIME 239 (1967) 2012.

[93] T. Takahaski, A. Kamio, N.A. Trung, J. Cryst. Growth 24 (1974) 477.

[94] Y. Migata, T. Suzuki, J.I. Uno, Metallur. Trans. A 16 (1985) 1799.























































[95] H.A. Palacio, M. Solari, H. Biloni, J. Cryst. Growth 73 (1985) 369.

[96] M.C. Flemings., Solidification Processing, McGraw-Hill, New York, NY, 1974.

[97] G.J. Fix., A. Fasano, M. Primicerio., Free Boundary Problems: Theory and

Applications, Pitman, Boston, 1983.

[98] J.S. Langer, G. Grinstein, G. Mazenko, Directions in Condensed Matter Physics,

World Scientific, Philadelphia, PA, 1986.

[99] J.B. Collins, H. Levine, Phys. Rev. B 31 (1985) 6119.

[100] S.Z. Lu, J.D. Hunt, J. Cryst. Growth 123 (1-2) (1992) 17.

[101] J.D. Hunt, S. Lu, Metall. Mater. Trans. A 27 (1996) 611.

[102] B. Kauerauf, G. Zimmerman, L. Murmann, et al., J. Cryst. Growth 193 (4) (1998)

701.

[103] W. Losert, D.A. Stillman, H.Z. Cummins, et al., Phys. Rev. E 58 (1998) 7492.

[104] A.M. Ovrutsky, M.S. Rasshchupkyna, Mater. Sci. Eng. A 495 (2008) 292.

[105] P. Bennema, J.P. van der Eerden, I. Sunagawa, Morphology of Crystals,

TERRAPUB, Tokyo, 1987, pp. 1�75.


















8 Modern Simulations by theMolecular Dynamics Method

8.1 Cluster Structure of Supercooled Liquids and Glasses

8.1.1 Amorphous and Nanocrystalline Materials

In the 1980s, a new field of research appeared in materials science: metallic

glasses. These materials are usually obtained from fast solidification, with cooling

rates as high as 106 K/s, of metal�nonmetal alloys or from mechanical alloying.

The topology of the resultant product is similar to the one found in the liquid state

of the same material, that is, with short-order spatial atomic correlations (around

5 A). Because of that, these materials are also referred to as amorphous materials.

Their importance lies in their excellent magnetic properties (if in their constituents

there are ferromagnetic elements [1,2]) and mechanical properties (mainly with

Al-based glasses [3,4]).

Glasses can be defined as noncrystalline solids. In the course of their prepara-

tion, a process commonly denoted as glass transition takes place. When a liquid is

cooled down with sufficiently high rates, crystallization may occur to a very lim-

ited degree or be completely absent down at temperatures corresponding to very

high viscosities η$ η(Tg)D10134 1012 Pa � s, where Tg is defined as the glass tran-

sition temperature. Below this temperature, the viscosity is so high that large-scale

atomic rearrangements in the system are no longer possible within the timescale of

the experiment, and the structure freezes, i.e., the structural rearrangements

required to retain the liquid in the appropriate metastable equilibrium state cannot

follow the changes of temperature any more. This process of freezing of the struc-

ture of a supercooled liquid is commonly denoted as glass transition and, as a

result, the system is transformed into a glass.

The main disadvantage that confines the use of these materials in the industry, is

their thermal instability, because the supercooled state is metastable. Any contribu-

tion of thermal energy is able to activate the crystallization processes. In many

cases, the primary crystalline phase that appears does not completely transform the

material: it can be followed by the precipitation of a second phase at higher tem-

peratures. Thus, the study of the materials obtained from the heat treatment of

metallic glasses was initially devoted to prevent its crystallization or to delay it to

higher temperatures. From these studies, it was discovered that these partially crys-

tallized materials had properties as good or better as their amorphous precursors,



http://dx.doi.org/10.1016/B978-0-12-420143-9.00008-9

and with the advantage of being thermally stable. The reason for the good proper-

ties of these kind of composite materials lies in the size of the crystals that grow

in the amorphous matrix and that are usually of nanometric scale (typically

between 20 and 50 nm [5]. These partially crystallized matters with appropriated

compositions are often the materials with valuable properties: the increased ductil-

ity, increased flow stress and fracture strength and super plasticity [6,7], as well as

hard and soft magnetic properties: low coercivity, high saturation magnetization,

reduced high-frequency losses, and stress-induced anisotropy [8,9]. In general,

composite materials are multiphase mixtures that are technologically important

because they can be produced with a wide variety of components to obtain desired

sets.

8.1.2 Techniques for Local Structure Analysis of Simulated Models

Satisfactory understanding of the structural properties of the stable and supercooled

melts is a fundamental problem because it is believed that the local structure of

melts has a strong influence on the nucleation mechanism [10].

Traditional techniques for the structure analysis are the following: calculations

of radial pair distribution functions (RPDF) and its Fourier transformation (the

structure factor S(q)), bond-angle distribution (N(θ)), coordination numbers, statis-

tics of Voronoi’s polyhedrons, and Delone’s simplexes [11]. These techniques are

generally accepted and they give the common structure characteristics of liquid and

amorphous materials.

The method of structure relaxation (SR) allows defining the structure of amor-

phous materials after statistical relaxation at the zero temperature. The particle

displacements are calculated at T5 0 depending on a potential energy profile

only. In the equilibrium state, resultant forces acting on every particle are equal to

zero. Depending on which algorithm is used, particles are shifted in series or all

simultaneously in the directions of resultant forces (Section 6.2.1). When studying

the cluster structure of simulated models, the problem appears connected with suffi-

ciently large thermal oscillations, which cause additional distortions of a local

order. The structure relaxation to zero temperature (in Kelvins) is not a good deci-

sion for this problem because the strongly different structure may correspond to

low temperatures. An effective way to solve this problem consists of averaging of

positions of atoms during the time of order 0.2 ps [12]. After that, snapshots of the

model sections give sure evidence of the real cluster structure.

Figure 8.1 gives the example of the Voronoi statistics for models of aluminum

and iron in liquid state [13]. However, the selection of polyhedrons into groups,

which may be considered as distorted basic elements of the structure, requires addi-

tional arguments. Therefore, several techniques were elaborated for calculation of

additional characteristics for the cluster selection.

Steinhardt et al. [14] measured both local and extended orientational symmetries

in computer-generated models of dense liquids and glasses. They introduced the

orientational order parameter W6 to demonstrate a short-range icosahedral ordering.

Their analysis starts by associating a set of spherical harmonics with every bond


joining an atom to its near neighbors. With a bond whose midpoint is at ~r , theyassociate the set of numbers,

Qlmð~rÞ � Ylmðθð~rÞ;φð~rÞÞ; ð8:1Þ

where the {Ylm (θ, φ)} are spherical harmonics, and θð~rÞ and φð~rÞ are the polar

angles of the bond measured with respect to some reference coordinate system

(they depend on type of the considered cluster, see examples in Figure 8.2).

dN/N (%)Al T= 943 K

Fe T= 1823 K

3

2

1

3

N 3 0110200 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 2 0 1 1 1 0 0 1 1 1 1 2 2 1 1 1 2 2 1 1 2 2 3 1 3 1 2 2 33 1 2 3 4 3 3 4 5 1 2 3 4 3 4 5 6 3 4 5 3 5 4 5 3 4 5 2 3 4 5 2 3 3 4 5 2 3 4 6 3 4 1 3 3 5 3 3 3108 8 8 8 9 6 6 6 7 7 7 7 4 4 4 4 5 5 5 6 2 3 3 8 5 5 6 6 6 6 7 7 4 4 4 5 5 2 2 3 3 4 4 4 3 4 2 31 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 1 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 2 2 3 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2

N 4N 5N 6N 7N 8

N 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 11 0 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 5 5 5 1 2 2 2 2 2 3 3 3 3 3 3 41212101010 8 8 8 8 8 8 9 6 6 6 6 6 6 6 7 7 7 7 4 4 4 4 4 5 5 5 5 6 6 3 3 4 8 5 5 5 6 6 3 4 4 4 4 5 20 2 2 3 4 0 1 2 3 4 5 2 0 1 2 3 4 5 6 1 2 3 4 2 3 4 5 6 2 3 4 5 2 3 4 5 2 3 2 3 4 2 3 3 2 3 4 5 2 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 2 2 1 1 2 1 0 0 0 1 1 1 1 1 1 2 100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

N 4N 5N 6N 7N 8

2

1

Figure 8.1 Statistics of the Voronoi polyhedrons for models of liquid aluminum and

iron [13]. Dark (red) columns respond to distorted icosahedrons; light (green) column—

fcc clusters; columns with horizontal hatching (blue)—bcc clusters.

fcc cluster hcp cluster Icosahedron

Figure 8.2 Different clusters of 13 atoms occurring in liquids near the melting temperature [14].

247Modern Simulations by the Molecular Dynamics Method

The spherical harmonics {Ylm (θ, φ)} for a given value of l and m (jmj, l) form a

(2l 11)-dimensional representation of the rotational group SO(3).

The orientational order parameters fQlmð~rÞg allow us to determine the range of

orientational order in various systems. In numerical studies, authors more often

considered averaged quantities [14],

Qlm � Qlmð~rÞ� �

;

of the order parameters. The first nonzero averages (other than Q00) occur for l5 4

in samples with cubic symmetry and for l5 6 in icosahedrally oriented systems.

Because the Qlm’s for a given l can be scrambled drastically by changing to a

rotated coordinate system, rotationally invariant combinations are considered.

Averaged bond order parameters were calculated by summation over all bonds

in the sample,

Qlm 51

Nb

Xbonds

Qlmð~rÞ: ð8:2Þ

Steinhardt et al. [14] found that all reasonable definitions lead to Qlm which van-

ish in isotropic liquids for l. 0, and which are nonzero in cubic solids for l$ 4.

The quantity Q00 (corresponding to the constant spherical harmonic γ00 5 1=ffiffiffiffiffiffi4π

pis always nonzero, and scales with the average coordination number associated

with a particular convention for assigning neighbors. After analyzing different char-

acteristics of the cluster ordering and symmetry, Steinhardt et al. [14] found that

the most effective parameter for distinguishing of clusters is the W6 invariant

(l5 6) which is connected with the values Qlm as follows:

W6 5 336ð6!Þ3=2ffiffiffiffiffiffiffi

19!p ð2 5Q3

60 1 6Q60jQ64j2ÞðQ2

6012jQ64j2Þ3=2: ð8:3Þ

It has a nontrivial extremum for:

Q260

jQ66 4j25

2

7

corresponding to:

jWcubic

6 j5 4ffiffiffiffiffiffiffiffiffiffiffiffiffi92378

p 5 0:013161 ð8:4Þ

(W6 52 0:013161 for fcc clusters and W6 510:013161 for bcc clusters).

The invariant W6 is easily minimized in an “icosahedral” subspace spanned by

Q60 and Q66 5, with all other Q6m5 0. Extreme of Eq. (8.3) occurs when the


characteristic of icosahedral clusters is satisfied to the relation for nondistorted

icosahedrons:

Q2

60 511

7jQ66 5j2:

The icosahedral value of W6 is W6 520:169754 [14]. And W6 520:012442for the hcp clusters.

To obtain a more detailed three-dimensional description of the local atomic con-

figuration, Honeycutt and Andersen [15] used the common-neighbor analysis, in

which the first two peaks of the pair-correlation function are decomposed. This

method is able to characterize the local environment surrounding each atomic pair

that contributes to the peaks of g(r), in terms of the number and properties of com-

mon nearest neighbors of the pair under consideration. The relative abundance of

selected pairs, averaged over the height inherent configurations, is analyzed. This

technique is named the cluster-type index method (CTIM). A set of four indices is

associated with each pair: (i) the first index denoted to what peak of g(r) the pair

under consideration belongs, i.e., the root pair; (ii) the second index represented the

number of near neighbors shared by the root pair; (iii) the third index was for the

number of nearest-neighbor bonds among the shared neighbors; (iv) a fourth index

was used to distinguish configurations with the same first three indices but with a

different topology. Using this method, authors distinguished between various local

structures such as fcc, hcp, bcc, and icosahedral environments. For example, they

considered that four bonded pairs are represented in a bulk fcc crystal: 1421, 2101,

2211, and 2441; the 1441, 1661, 2101, 2211, and 2441 are typical pairs of the bulk

bcc crystals; the local order built on a 13-atom icosahedron leads to the occurrence

of 1551, 1321, and 2331 pairs.

Ganesh and Widom [16,17] employed a simplified form of the

Honeycutt�Andersen analysis, counting the number of common neighbors shared

by a pair of near-neighbor atoms. They assign a set of three indices to each

18 bond. The first index is 1 if the root pair is bonded (separation less than or equal

to cutoff radius). The second index is the number of near-neighbor atoms common

to the root pair, and the third index gives the number of near-neighbor bonds

between these common neighbors. This identifies the number of atoms surrounding

the near-neighbor bond and usually equals the number of edge-sharing tetrahedra

whose common edge is the near-neighbor bond.

Tian et al. [18] analyzed the structures of nanoparticles by the CTIM. They had

defined the basic cluster as the structure composed of a core atom and its surround-

ing neighbor atoms. A set of six integers represents a basic cluster. The first integer

equals the total number of near-neighbors of the center atom, and the following

five integers represent the numbers of the 1441, 1551, 1661, 1421, and 1422 bond

types, respectively, expressed by using Honeycutt and Andersen index method

[15], by these bond types the surrounding atoms are connected with the center one

of the basic cluster.


Ovrutsky and Prokhoda [19] used the procedure for visually determining of the

atom number and coordinate of small cell in which it was placed. Putting the cursor

in the image of an atom (in the model section in which it was brightest), the values

of his number and coordinate of small cell were writing in the message windows. It

allowed determining the local RPDF (LRPDF). Visualization of the model and the

determination of LRPDF were often fulfilled after averaging in a definite time of

atom coordinates. A special procedure allowed determination of the number of

nearest neighbors and angles with nearest neighbors for the every atom. Atoms

were considered as centers of icosahedrons, if they had 12 nearest neighbors and

30 angles by a size in the interval 636 4� (exact value is 63.43�). It was possibleto determine the time of life of such surrounding by marking these atoms with new

color after the certain time (and writing this information), if mentioned characteris-

tics were the same. For the fcc or hcp surrounding it must be 24 angles in 606 4�

and 12 angles in 906 4�. An atom was considered as the center of the bcc ordered

group if it has 14(81 6) neighbors in the area of the first RPDF peak, 12 angles in

the interval 706 4�, and 12 angles in the interval 906 4�.

8.1.3 Cluster Structure of Supercooled Liquids and Glasses

Satisfactory understanding of the structural properties of stable and supercooled

melts is a fundamental problem because it is believed that the local structure of

melts has a strong influence on the nucleation mechanism [10].

Turnbull [20�22] established that metallic liquids could be supercooled if het-

erogeneous nucleation can be reduced or avoided. Later, Frank [23] hypothesized

that the supercooling of liquid metals might be due to frustrated packing of icosa-

hedral clusters. Icosahedral clustering of 12 atoms near a sphere is energetically

preferred to crystalline (e.g., fcc, hcp, or bcc) packings for the Lennard-Jones (L-J)

pair potentials. The icosahedron is favorable because it is made up entirely of four-

atom tetrahedra, the densest-packed cluster possible. Local icosahedral order cannot

be propagated throughout space without introducing defects. Many experiments

display indirect evidence of icosahedral structures in supercooled liquid metals, see

for instance, Refs. [24�26]. These results substantiated the presence of defective

icosahedra units. Computer simulations provide more direct evidence of the real

liquid structure. At first, common RPDF were analyzing only. Then detailed ana-

lyzing of the surroundings of atoms (type of the Voronoi polyhedral, the angles

between bonds and positions of atoms relatively nearest to them bond of two

atoms) were taken into account for determination type of the local ordering in

groups of atoms.

Many studies of Lennard-Jones systems have tested Frank’s hypothesis. Hoare

[27] found that for clusters ranging between 2 and 64 atoms at least three types of

“polytetrahedral” noncrystalline structures exist, with a higher binding energy than

hcp or fcc structures with the same number of atoms. Jakse and Pasturel [28] ascer-

tained the occurrence of a local order more complex than the icosahedral one.

Using CTIM for the Lennard-Jones (L-J) pair potentials, Honeycutt and Andersen

[15] defined an indication that the liquid is characterized by an ordering that is


more complex than the one found in the 13-atom icosahedron. They found the

crossover cluster size between icosahedral and crystallographic ordering around a

cluster size of 5000 atoms.

Analyzing the results of simulations, Steinhardt et al. [14] found that the short-

range order of the liquid state is dominated by icosahedral and distorted icosahedral

inherent structures because the 1551 and 1541 bonded pairs are preponderant.

However, although the 2331 pairs are relatively numerous, the absence of the 1321

pairs and the high value of the 2101 ones is a strong indication that the liquid

is characterized by a ordering that is more complex than the one found in the

13-atom icosahedron.

Ganesh and Widom [16] used first-principle molecular dynamics (MD) simula-

tions to generate realistic atomic configurations, providing structural detail not

directly available from experiment, based on interatomic forces that are more reli-

able than conventional simulations. They analyzed liquid copper and iron and some

alloys, for which there were known experimental results available for comparison,

to quantify the degree of local icosahedral and polytetrahedral order. The g(r) func-

tion obtained for their models was in a good agreement with the XRD experimental

data. Figure 8.3 shows the results of cluster ordering analysis.

For Cu, the W6 distribution (Figure 8.3A) shows a strong asymmetry favoring

negative values with tails extending toward the ideal icosahedron value. Because

the histogram vanishes as W6 approaches its limiting negative value, one can see

that there are practically no perfectly symmetric, undistorted icosahedra present.

However, a significant fraction has the W6 values close to the icosahedral value.

For Fe, the W6 distribution (Figure 8.3B) also shows the strong asymmetry and,

moreover, additional maximum corresponding to the right icosahedron value. That

is, the quantity of icosahedrons in the supercooled Fe is larger than in copper. It is

clear from graphics given in Ref. [16] that the number of fivefold bonds in pure Fe

increases with undercooling. As Ganesh and Widom [16] noted, the icosahedron

with a coordination of 12 is one such Kasper-Polyhedron with no disclinations.

8

Freq

uenc

y

Freq

uenc

y

4

0

8

4

0–0.2

(A) (B)

–0.1 0

W6

fcc fccicos icos

0.1 –0.2 –0.1 0

W6

0.1

Figure 8.3 Simulated W6 distributions for liquid Cu (A) and Fe (B) [16,17],

(A) T5 1313 K; (B) T5 800 K. Ideal icosahedron and FCC values are indicated.


Adding disclinations to the icosahedron, one finds that each disclination increases

the W6 value by the same amount, irrespective of its sign.

A careful analysis of atomic configurations generated by MD simulations [29]

has proved able to rationalize experimental data on supercooled liquid copper.

They elucidate the role played by defective icosahedra through a precise identifica-

tion of the geometric environment of each atom and showed that the energy associ-

ated with defective icosahedra, embedded in the disordered system, lowers the

overall energy and stabilizes the supercooled metal, preventing crystallization.

A deeper insight into the presence and the nature of icosahedral units in the

atomic configurations was obtained from the common-neighbor analysis introduced

in Refs. [15,30]. Celino and Rosato [29] used a set of three indices likely described

in Ref. [16] (see above). To highlight the impact of defective icosahedra, Celino

and Rosato [29] labeled each atom by the number N555 of its 555-type nearest

neighbors. Accordingly, for an atom at the center of an icosahedron, N555 is

equal to 12 hereafter termed a perfect icosahedra, distortions notwithstanding.

In Table 8.1, they reported the percentage of atoms in the supercooled liquid with

N555 A[0,12]. Atoms involved in 421 pairs, reminiscent of crystalline fcc struc-

tures, are also found in the supercooled system, as well as those arranged in a bcc

fashion also given in Table 8.1. Perfect icosahedra are present both in the liquid

and supercooled systems: in the latter, only 0.26% of the atoms have N5555 12.

Interestingly, the inclusion of atoms for which N5555 6 in the counting of those

Table 8.1 First Column: Percentages of Atoms with a Selected Number of

Nearest Neighbors Nxxx and a given Symmetry xxx. Second and Third Columns:

the Icosahedral ICO Symmetry is Identified by Counting N555. Fourth and Fifth

Columns: fcc and bcc Symmetries are Identified by Counting N421 and

N4441N666 Nearest Neighbors, Respectively [29].

Nxxx ICO ICO fcc bcc

T5 1623 K T5 1313 K T5 1313 K T5 1313 K

0 30.32 20.9 43.62 33.58

1 24.67 22.27 24.7 23.94

2 17.57 18.4 16.94 17.25

3 11.33 13.63 9.08 11.34

4 7.19 9.77 3.39 6.59

5 3.92 5.85 1.54 3.55

6 2.53 4.26 0.53 1.97

7 1.1 2.03 0.12 1.04

8 0.93 1.86 0.06 0.39

9 0.14 0.29 0.01 0.14

10 0.2 0.48 , 0.01 0.11

11 , 0.01 , 0.01 , 0.01 0.04

12 0.11 0.26 , 0.01 0.02


responsible of defective icosahedra yields 9.15% for the supercooled liquid. Much

smaller values are obtained for fcc and bcc seeds.

Table 8.1 shows that a large fraction of icosahedral atoms exists below the melt-

ing point. Their higher stability with respect to fcc or bcc atoms may prevent the

formation of large units reflecting fcc or bcc arrangements, ultimately leading to

crystallization. Clustering of icosahedral regions shown in Figure 8.4 leads to a

growth of the icosahedral seeds which will systematically increase their size from a

few tens to a few hundreds of atoms [31]. However, it is commonly accepted that

the relative stability of perfect and isolated icosahedral clusters with respect to fcc-

based ones is a decreasing function of their size. Therefore, as soon as the number

of atoms forming the icosahedral cluster overcomes a critical threshold, the fcc-

based clusters become energetically favored [29].

The fundamental problem of atom structure reconstruction at glass transition is

a subject of much research over many years. Evteev et al. [32,33] proposed the

structure model of the glass transition for pure metal, which shows the principal

difference between structures of metal crystals and metal glasses based on the

results of iron simulations. In their opinion, which is based on the simulation

results for iron with the Pak-Doyama paired potential of interatomic interaction,

the structural stabilization of the amorphous phase of pure iron during solidification

from melt is ensured by the formation of a percolation cluster from mutually

penetrating and contacting icosahedrons with atoms at vertices and centers.

Icosahedrons that contact at vertices or planes have one or three common atoms

correspondingly. Evteev et al. [33] found that changes of thermodynamic

Figure 8.4 The snapshot of the supercooled liquid where atoms with N555, 6 are

removed [29].


characteristics of their model during percolation cluster formation have some indi-

cations of the first-type phase transition. Figure 8.5 shows the cluster structure of

the model at different temperatures. Evteev et al. [32,33] determined the structures

after statistical relaxation of the model at T5 0 K. The percolation cluster appears

at the temperature 1260 K, which corresponds roughly to the glass transition tem-

perature. At a drop in a temperature, the number of icosahedral clusters increases

(Figure 8.6) that favors the percolation cluster formation.

Figure 8.7 shows the waiting periods for forming of the icosahedron percolation

cluster at different temperatures (1240�900 K) and for forming of the crystalliza-

tion center of the bcc phase. It was found that percolation cluster appears

and grows up at the temperatures less than the critical temperature Tg� 1180 K.

105

90

75

60

45

30

15

0 15 30 45 60 75 90 105(A) (B) (C)

105

90

75

60

45

30

15

0 15 30 45 60 75 90 105X (Å) X (Å)

Y (Å) Y (Å)105

90

75

60

45

30

15

0 15 30 45 60 75 90 105X (Å)

Y (Å)Y (Å)

Figure 8.5 Projections of the largest cluster from contacted icosahedrons on the model

border plane (after the relaxation at the zero temperature) [33], for a,b,c the temperature

T5 1460, 1260, and 1180 K.

N1

(%)

8

7

6

5

0 300 600 900 1200 1500 1800 2100T (K)

Figure 8.6 The number of atoms in the icosahedron centers in dependence on temperature

at the cooling rate 4.43 1012 K/s [32].


The smallest waiting time for appearing of the crystallization center responds to

this temperature. At higher temperatures (T. Tg), such waiting time becomes

higher and the numbers of icosahedrons decreases. The stable percolation cluster

do not form at T. Tg. At temperatures lower than Tg, the percolation cluster appear

before then the crystallization center. The annealing time that responds to forming

of the percolation cluster was smaller than 1.53 10211 s at studied temperatures (in

the range from 900 to 1180 K).

The number of icosahedrons and the lag for crystallization tc become higher at

reduction of the temperature. The part of atoms belonging to the fractal cluster

increased with time; it reached 39% at T5 1180 K and B60% at T5 900 K when

the crystallization center appears. The fractal cluster grows up yet some time after

crystallization center appearance. This means, in the opinion of Polukhin and

Vatolin [34], that crystallization centers grow at the expense of atoms, which do

not belong to the icosahedral clusters; such clusters prevent the crystallization.

Ovrutsky et al. [12,35] observed directly clusters in sections of simulated

models of supercooled pure aluminum and Al�Ni (10 at.% Ni) alloy. For better

visualization of structures, the coordinates of atoms were averaged during 1 ps.

Figure 8.8 shows the model sections and the snapshots of the same sections

(the brightness of images reduced with a distance from the chosen section) with

featuring only atoms, which enter into clusters. They found that the majority of

ordered atom groups contact one with others and form prolonged clusters

(Figure 8.8B). But large compact icosahedron clusters were also present in the

models of Al [12] and alloy Al�Ni [35] (Figure 8.8C and D). The number of dis-

torted icosahedral clusters (the angle deviations from right angles between bonds

are 6 4�) was larger than the number of clusters of other symmetries (fcc or bcc,

with the same angle dispersion).

1400

1200

1000

T (

K)

TgUndercooled liquid

Glass

Crystal

800

6001 10 100

t (10–11s)

Figure 8.7 The isothermal kinetic diagram [33], rhombuses—the waiting time for

crystallization centers, tc; pentagons—the waiting time for the stable percolation cluster

forming; dotted line is the result of theoretical calculation of the lag tc.


Figure 8.9 shows dispositions of neighbor atoms relative of centers ordered

groups and local distribution functions (n(r) instead g(r)). These functions have

characteristic indications of local ordering. For comparison, Figure 8.10 shows the

RPDF [32] obtained for the all atoms placed in the icosahedrons center after statis-

tical relaxation of the model at T5 0 K.

As stated before, the glassy state is unstable; therefore, when a glass is annealed

below Tg to the glass transformation range, it may undergo a transformation to a

more stable glass in which there is a readjustment of the structure reducing its vol-

ume. This process is named structural relaxation. Notwithstanding that the change

in the density during this process is small (usually less than 1%), there could be

important changes in the viscosity or ductility of glasses and also in their magnetic,

Figure 8.8 Section of the Al-10Ni alloy and images of clusters (B) [35] T5 650 K,

Ni atoms are dark (blue), central atoms of icosahedral clusters are dark (red); (C) the

crystallization center is from right, (D) the upper icosahedral cluster from (C).


electric, elastic, and diffusion properties [37]. It is also possible that glasses relax

to their supercooled liquid state; this process is named super relaxation [38].

It is clear that the numbers of different clusters and energy of systems as in a

hole should have changes during structural relaxation (at the temperature under

consideration). However, detailed investigations of the relaxation kinetics are

still rare.

Ovrutsky and Prokhoda [19] studied changes with time the LRPDFs of iron in

the places where the crystallization center appear in feature (states before crystalli-

zation center appears). Data concerning local structural changes at annealing of

amorphous iron are mapped in Figure 8.11. By the data in Figure 8.11, precrystal-

lizing reorganization in the field of nuclei formation happens approximately for

1503 10212 s. The first maximum becomes unsymmetrical (Figure 8.11C), that is

connected with the beginning of demarcation of the first and second coordination

orbs. The right component of the following doubled maximum becomes higher

than the left one (the type of the environment of atoms becomes on the average

closer to the bcc structure, than to fcc), and some new maximums are accurately

displayed. This LRPDF averaged on time answers to appearing of a crystallization

center with imperfect structure. Its structure becomes more perfect (Figure 8.11E,

G,I) during growth (increasing of its size).

n(r)

0 0.2 0.4 0.6 r (nm) 0 0.2 0.4 0.6 r (nm) 0 0.2 0.4 0.6 r (nm) 0 0.2 0.4 0.6 r (nm)

n(r) n(r) n(r)

(A) (B) (C) (D)

Figure 8.9 The n(r) distributions and relative positions of atoms (after averaging of atom

coordinates) [35,36], (A and C) icosahedral clusters (Al, T5 400 K and Fe, T5 800 K; (B) the

fcc ordered group in the center (Al, T5 400 K); (D) the bcc ordered group in Fe; T5 800 K.

14

g(r)

12

8

4

02 3 4 5 6 7 8 r (Å)

Figure 8.10 RPDF for atoms in the centers of icosahedrons, T5 1100 K [32].


For other chosen area (Figure 8.11B,D,F,H,J) structural modifications happen in

the same sequence, but later. LRPDFs (Figure 8.11), averaged during several peri-

ods of oscillations, allow defining nuclei occurrence even earlier than it becomes

appreciable visually.

Ovrutsky et al. [12,35] determined the lifetimes of icosahedron clusters, τl, inthe pure aluminum, pure iron and Al�Ni allow models by the means of periodic

repainting of the clusters under condition of saving corresponding parameters for

their determination. Then the dependences on time were found for quantities of

icosahedrons, which continued to exist. Values τl, calculated from equation: N(t)5N0exp (�t/τl) are shown by the dashed line in Figure 8.12. It was found that the

lifetimes of icosahedrons in the binary system are roughly the same as in the case

of pure aluminum. Unlike pure aluminum, these lifetimes are essentially less than

r (Å)

4

(A) (C) (E) (G) (I)

(B) (D) (F) (H) (J)

321

321

0 4 8 0 4 8 12 0 4 8 12 0 4 8 12 0 4 8 12

g(r)

g(r)

Figure 8.11 Local averaged on time RPDF, obtained during annealing of amorphous iron;

A, C, E, G, I—in the field of occurrence of the first nanocrystal; B, D, F, H, J—in other area

removed from the first nuclei, T5 950 K.

160

(A) (B)

120

80

40

0

160

120

80

40

0400 500

t, τ (10–12 s) t, τ (10–12 s)

600 700T (K) T (K)

800 600 800 1000 1200 1400

Figure 8.12 Latencies, t, of appearance of the first crystallization center and lifetimes of

icosahedrons depending on the temperature [12,35], (A) data for aluminum; (B) for iron; �,

lifetimes of icosahedrons; o, latencies of nucleation.


latencies (τn) of the crystallization center’s appearance, even at rather low

temperatures.

Relaxation phenomena at the beginning of annealing (after rapid quenching

from high temperatures, which were above the melting point) were studied for the

binary system. Figure 8.13 shows averaged energies of atom interaction plotted

versus time for two temperatures. Figure 8.14 shows such dependences for numbers

of clusters. Data shown in Figure 8.13 and Figure 8.14 correlate each other between

them. Relaxation times are roughly the same; and they increase when temperature

decreases. Ovrutsky et al. [12,35] assumed that equation:

NðtÞ5Nmaxð1� expð�t=τÞÞ ð8:5Þ

gives a time dependence of numbers of clusters. The best fit of simulation data (lines

in Figure 8.14) was obtained for the following values of τ: τ5 40 ps at T5 500 K

100–0.885

–0.890

–0.895

The

inte

ract

ion

ener

gy (

eV)

–0.900

–0.905

–0.910

–0.915

–0.920

200 300 400 500

Time (ps)

Figure 8.13 Averaged energy of atom interaction in the bulk of the alloy Al90Ni10 plotted

versus time; e, T5 500 K; x, T5 650 K.

400(A) (B)

350

300

250

200

150

The

num

ber

of c

lust

ers 320

28024020016012080400

0 20 40 60 80 100 120 140 160

The

num

ber

of c

lust

ers

100

50

00 100 200

Annealing time (ps) Annealing time (ps)300 400 500

Figure 8.14 Changes of numbers of clusters during relaxation annealing of the Al90Ni10alloy; (A) T5 500 K; (B) T5 650 K; K, fcc-ordered groups, x, icosahedrons, full lines arecalculated by Eq. (8.5).


and τ5 15 ps at T5 650 K in the case of the icosahedron clusters, τ5 15 ps at

T5 500 K and τ5 5 ps at T5 650 K in the case of the fcc clusters. It is clear that

τ and τl (see above) are the same values. The numbers of the fcc clusters are less

than the numbers of the icosahedron clusters as their lifetime is less.

As icosahedron configurations of atoms provide the least energy of packing in

comparison with other clusters, an increase of their number leads to lowering of

the system energy overall. Therefore, one could conclude that a relaxation time of

the system at the certain temperature may be determined as the time at the end

of which the energy and accordingly the number of the icosahedron clusters

become approximately permanent.

8.2 Nucleation Kinetics

8.2.1 The Main Classical Equation for the Nucleation Kinetics

Thermodynamical aspect of nucleation was considered in Section 2.6. The classical

theory operates with such concepts as free surface energy, the thermodynamical

driving force (Δμ5μl�μcr is the difference of the chemical potentials of liquid

and crystalline phases), the work of nucleus formation and the radius of critical

nucleus.

Kinetics of nucleation was researched in a lot of works, both experimentally and

theoretically. Moreover, in many books, for instance [39�41], this research is

described skillfully. The nucleation frequency, i.e., the number of stable nuclei

formed in the system in unit time, depends on the number of clusters that become

greater than the critical size after the incorporation of new atoms by a fluctuation.

Kinetic nucleation theory (KNT) assumes that, even in the later stages of the trans-

formation, there is a stationary distribution of clusters of a new phase embedded in

the matrix and, hence, there is a constant flux of clusters with radii greater than the

critical radius. This flux corresponds to the nucleation frequency; it can be calcu-

lated from the assumption of this stationary state and from the probability of exis-

tence of a cluster with a certain number of atoms in equilibrium.

KNT originated from the consideration of a chain reaction of clusters of varying

size [42�44]. A general formulation for KNT is based on the expression for change

with time of the distribution function of clusters c(n,t) (cluster concentration):

@cn@t

5 ðω1n21cn21 1ω2

n11cn11Þ2 ðω1n21cn 1ω1

n cnÞ ð8:6Þ

where ω1n21 and ω1

n are the average number of events that per unit time one particle

is attached and the number of particles in a cluster is increased from (n2 1) to n

and n to (n1 1), respectively; ω2n and ω2

n11 are the average number of events for a

cluster to release one particle per unit time.

Zeldovich [45] and Frenkel [46]) deduced equations analogous to Eq. (8.6)

taking n as a continuous variable. Using the principle of detailed balance


(i.e., all elementary reactions must have equal forward and reverse rates at equilib-

rium) with the relation ω1(n)c0(n)5ω�(n1 1)c0(n1 1), where c0(n) is the equilib-

rium distribution function; the dissolution frequency ω�(n) can be eliminated from

the equations. This produces a set of partial differential equations for cluster size

evolution in time, and instantaneous fluxes:

@cðn; tÞ@t

5@

@nω1c0ðnÞ @

@n

cðn; tÞc0ðnÞ

� �� 52

@Iðn; tÞ@n

; ð8:7Þ

Iðn; tÞ52ω1cðnÞ @

@n

cðn; tÞc0ðnÞ

� �: ð8:8Þ

At equilibrium, the size distribution of clusters coincides with the equilibrium

distribution function, c(n)5 c0(n), and it can be shown that:

c0ðnÞ5N expð�δGðnÞ=kTÞ; ð8:9Þ

where δGi is the Gibbs free energy change associated with formation of the cluster

of n particles, and N is the total number of clusters. Figure 8.15 shows two size dis-

tributions of nuclei� the equilibrium one (curve 1) and the steady state distribution

(curve 2), which is more real. The curve 2 corresponds to the case when nuclei

with the size that is larger r� (the critical size) are moved out the system because

they become the crystallization centers.

For the steady state, one can integrate the partial differential equations (8.8)

over all values of n. The resulting approximate solutions for the rate of nucleation:

Iðn�Þ5Nω1Z expð2δGðn�Þ=kTÞ; ð8:10Þ

where Z is the Zeldovich factor, it is approximately equal (δG(n�)/(2πkTn�))1/2 andis generally on the order of 0.05 [39]. This factor effectively corrects equilibrium

classical nucleation rates for steady state situations.

n*

12

C0(

n), C

(n)

Figure 8.15 The size distributions of nuclei; 1, the equilibrium distribution; 2, the steady

state distribution.


The question is how to determine the value ω1? Kelton et al. [40] expressed ω1

through macroscopical diffusion coefficient D (diffusion coefficient for atoms

crossing the interface between the matrix and the nucleus). However, the diffusion

coefficient does not always correlate with the velocity of growth. It is better to

express ω1 through kinetic coefficient βs for the growth velocity, which can be

determined in simulation experiments.

Voronkov [47] considered the change of nuclei size (the number of atoms in

clusters) as Markov’s process, for which the averaged values of magnitudes

ðn0 2 nÞ and ðn02nÞ2 are proportional to the time period:

ðn0 2 nÞ5AðnÞðt0 � tÞ; ðn02nÞ2 5 2BðnÞðt0 � tÞ; ð8:11Þ

where n is the initial size (the number of atoms) of nucleus; A(n) has a meaning of

the mean velocity of size change; B(n) is the diffusion coefficient for chaostical

motions of the nucleus size n in the space of sizes. The change of the size distribu-

tion is defined by the equation of the Fokker�Plank type, which coincides with

Eq. (8.7). The known expression for the rate I(n�) [46]:

Iðn�Þ5BðnÞ c0ðn�Þν

ffiffiffin

p ; ð8:12Þ

is valid in more common concept, where ν5 jn�n�jis the half maximum width of

the dependence δG(n) at the height δG(n�)�kT (Figure 8.16); and the diffusion

coefficient B(n) is connected with the growth velocity V(n)[47]:

BðnÞ52VðnÞkTðdGðnÞ=dnÞ21; ð8:13Þ

V(n)5 (dn/dt) is the rate of change of the nucleus size (the number n of

molecules).

n* n1

kT

V

0

δG(n)

δG(n*)

n

Figure 8.16 Schematic representation of the Gibbs energy changes associated with a

precipitate of size n.


As V(n)5�βs4πr2(dG(n)/dn)/(kTΩ) (with the Gibbs�Thomson shift,

V(r)5βsΔμ/kT for a large crystal), where Ω is the volume per one molecule and

Δμ is the thermodynamic driving force,

BðnÞ5βsð4π=ΩÞ1=3ð3nÞ2=3: ð8:14Þ

For spherical nuclei (homogeneous nucleation) with the number n5 (4/3)πr3/Ωof molecules, the work of their formation is:

δGðnÞ52nΔμ1 4πr2γ5 � nΔμ1 ð4πÞ1=3γað3nÞ2=3; ð8:15Þ

where γ is the free surface energy per surface unit and γa5 γΩ2/3—per one mole-

cule site on the nucleus surface. The first derivative of δG(n) is:

dðδGðnÞÞdn

52Δμ12ð4π=3Þ1=3γan21=352Δμð12 ðn � =nÞ1=3Þ52Δμð12 ðr � =rÞÞ:

From minimization of the first derivative, n� 5 ð4π=3Þð2γa=ΔμÞ3 and δG(n�)5(16π/3) γa

3/Δμ2. The second derivative at critical size:

d2ðδGðnÞÞdn2

� �n�52ð2=3Þð4π=3Þ1=3γan�24=3=Δμ: ð8:16Þ

Taking Taylor from δG(n) close n� (Figure 8.16), we will obtain equation for ν:

ν5 22kT=d2ðδGðnÞÞ

dn2

� �n�

� �1=2

5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi210=3πkTγ3a

q=Δμ2:

Then based on Eq. (8.12), the rate of nucleation will take a form:

I5 2βsΩ243

ffiffiffiffiffiffiγakT

rexp 2

16

3

πγ3akTΔμ2

� �: ð8:17Þ

One other aspect of steady state nucleation is the concept of induction time, or

time from the initial state until a steady state nucleation process has evolved. This

is the period needed for the system to reorganize itself and adopt the steady state

condition. During this period I5 Istexp(�τ/t). Estimate the induction time, τ, fromthe condition that it is the time of the cluster diffusion in the space of sizes from

size n5 1 to size n� (Eq. (8.11)):

τ5 ðn�Þ2=ð2BðnÞÞ5 32πΩ1=3

9βs

γ3aΔμ4

: ð8:18Þ


The time required to establish steady-state nucleation in the system is commonly

denoted as the time lag in nucleation. Numerical estimations of τ give its order

of 1024�10210 s depending on characteristics of the substance. Using para-

meters for pure metals at large supercooling (Δμ/kT5 0.5, γa/kTD0.5ΔH/

kTD0.5, βsD100 m/s), we have from Eq. (8.18) that τD253 10�12 s (see below

values of βs).

Actually, it has been suggested that a vitrification of some compositions

occurs because the cooling time is shorter than the transient time. According to

Ref. [48], the transient period can be estimated using the expression τ5 r�2/(π2D) for the quenched metallic alloy. According to Ref. [49], for melts and

solutions, τ is proportional to the viscosity η of the medium, τBη(Tf/(Tf�T)); Tfis the melting point. In condensation from vapor, this time is sufficiently small,

so that it is generally negligible. However, such is not the case in liquids, glasses,

and solids.

8.2.2 The Dependences of the Surface Tension on the Temperature andRadius of Nuclei

The most basic assumption of classical nucleation theory (CNT) is in that

the nucleus surface energy, γ, is a macroscopic property having a value equaled to

that for a planar interface, γN. Therefore, the size dependence of surface energy is

usually neglected when analyzing experimental data. CNT tests involving several

silicate glasses that nucleate homogeneously have demonstrated that experimental

crystal nucleation rates are much higher than theoretical ones [50].

It is difficult to evaluate the γ(T) dependence. It is taken at the first

approximation that γ is the certain fraction of the enthalpy of fusion [51�55].

Recent theories are based on the concept of the interface not being a geometrical

dividing surface but a diffuse layer. The practical result is an expression the γ(T)dependence [50]:

γðTÞ=γ Tfð Þ5 0:481 0:52T=Tf ; ð8:19Þ

which apparently fits a number of experimental data on homogeneous nucleation

rate. However, it is very important to know how the crystal-melt surface tension

depends on the nucleus radius. The size dependence was taken into consideration

in Ref. [56]. Freitasa, Galdez Costaa, Cabral, and Gomes [50] also Fokin, and

Zanotto [57] considered influence on nucleation of both dependences of the surface

tension—on the temperature and on the radius.

The surface energy of the nuclei is usually measured by fitting nucleation rate

data to the theory. In this case, one obtains the surface energy of the critical nuclei

with size r� as a function of temperature.

However, two different factors determine r�(T) dependence: the temperature

dependence of γ for a planar interface and its size dependence. Freitasa et al. [50]

examined the temperature dependence of macroscopic surface energy, decoupling


it from the size-dependent part. Several approximated equations had been chosen

to describe the curvature dependence of the crystal-melt surface energy, γ5 γ(r):

γðrÞ5 γN11 2δ=r

; ð8:20Þ

γðrÞ5 γNð12 2δ=rÞ; ð8:21Þ

γðrÞ5 γNð122δ=rÞ2; ð8:22Þ

where Tolman’s parameter, δ, characterizes the width of the interfacial area

between the coexisting phases (whose order is of atomic dimensions). Schmelzer

et al. [58] had showed that the Tolman’s (Eq. (8.20)), Vogelsberger’s (Eq. (8.21)),

and Rasmussen’s (Eq. (8.22)) expressions can be applied only to rcδ, rc4δ, andrc3δ, respectively.

Freitasa et al. [50] demonstrated an influence of the size and temperature depen-

dence of the surface tension by calculations of the work of critical cluster formation

δG(n) as a function of the nucleus size for different approximations of the γ(r)dependence (Figure 8.17); the typical values of the parameters are given in the cap-

tion to Figure 8.17. Also, they had calculated the γ(r,T) dependences, using experi-

mental rates of nucleation for several silicate glasses. The majority of their

calculation results give evidence that the γ(T) values decrease with temperature

whereas, according Spaepen’s theory [59], the surface tension of the crystal-melt

interface should decrease with lowering of the temperature.

2.0 × 10–18

1234

1.5 × 10–18

1.0 × 10–18

5.0 × 10

–10

1.0 × 10

–9

r (m)1.

5 × 10–9

2.0 × 10

–9

5.0 × 10–19

0.0

0.0

W (

J)

Figure 8.17 Calculated free energy changing of a model system due to the formation

of nucleus as a function of size [50]; 1—γ is size independent; 2—γ is size dependent

(Eq. (8.20)); 3—Eq. (8.21); 4—Eq. (8.22). The following parameters were used:

ΔGV 5ΔμNV5 4.083 108 J/m3, γN5 0.262 J/m2, δ5 3 � 10�10 m.


8.2.3 Critical Radii and Waiting Times: Results of Simulations for PureElements

The method of MD allows studying the peculiarities of crystallization in atomic

scale, in conditions that are impracticable in experiments. As a rule, periodic

boundary conditions are used in a simulation. Their use significantly increases the

degree of ordering in models, and it is known that the certain difficulties appear in

this case for organization of simulations at constant N,P,T [60]. Simulations at a

constant volume cannot be used for investigation of phase transitions, as a density

in the model is set artificially as a rule. That significantly influences the simulation

results; continuity of materials is violated sometimes if a density is set incorrectly

[34,61]. Models with a free surface are considered more often now [62,63]. The

models that contain many thousands of particles and free surfaces of phases are

real enough, as some problems related with using of periodic boundary conditions

are taken off in this case [19]. The size effects, such as size influence on the melt-

ing heat and temperature of melting of small nanocrystals, their structure, surface

tension, frequency spectrum of atomic oscillations, are the focus of many investiga-

tions and monographs, for instance Refs. [64�68].

The number of works devoted to the test of CNT is not sufficient. Bai and Li

[69] studied the relation between the critical nucleus size r� and the supercooling

ΔT for the L-J liquid (with the L-J potential parameters ε and σ). Figure 8.18

shows the dependence r�(ΔT) obtained for two system sizes.

A key result of CNT for the dependence r�(ΔT) is r� 5 2γsl/ΔμD(2γslTf/Lv)/ΔT [70,71]. Here Lv is the latent heat of fusion at melting temperature per unit

volume. By fitting the equation r� 5 k/ΔT for data shown in Figure 8.18, Bai and

Li [69] obtained the best fit at the value of k5 0.364. They related this value of k

0.02

14

12

10

8

6

4

2

0.04 0.06

N = 108.000

N = 32.000

0.08ΔT *, ε/k

r*, σ

0.10 0.12 0.14

Figure 8.18 The relation between the critical nucleus size r� and the supercooling ΔT [69].

The smallest stable solid sphere contains 43 atoms. A numerical fit for r� 5 k/ΔT is shown

by the solid line.


to the interfacial energy, k 5 22γslTf/Lv. Using the value Lv5 1.021ε/σ3 obtained

from melting of the pure L-J crystal and Tf5 0.618 � ε/k, Bai and Li [69]

had obtained γsl5 0.301ε/σ2, that is roughly 3.58 mJ/m2. This value is in excel-

lent agreement with Turnbull’s estimate (,0.329ε/σ2) [20], as well as other

calculations.

However, other results were obtained in Ref. [12] for pure aluminum

(Figure 8.19). At large supercooling in the amorphous state, the critical radius

ceases to reduce, i.e., the classical dependence does not obtain confirmation for

large supercooling in the case of metals. Ovrutsky and Prokhoda [12] determined

the equilibrium temperature for nucleus arising spontaneously by periodically

changing the temperature with a decrease of the interval of changes.

The rate of nucleation was in focus of a few recent studies on modeling

[72�75]. Simulation data are important for checking the temperature dependence

of the nucleation rate [73]. Simulation data for Al [73] (they were obtained for

NVT ensemble and for the potential from Ref. [76]) showed a good correlation, in

the opinion of the authors, with calculations based on the equations for time lag to

steady state of nucleation rate, τ, given in Refs. [77,78]. The simulated nucleation

time (the waiting time, tw, of initial decrease in energy, which was due to the

appearance of the new phase [73]), was always larger than τ at temperatures 500

and 550 K for all investigated sizes of systems (from 6000 to 106 atoms). The data

spread was large enough: at T5 500 K, the average simulated nucleation times

were 926 36 and 556 7 ps for the system composed of 16,384 and 1,000,188

atoms, respectively (Figure 8.20). These results demonstrate that, in order to obtain

accurate simulated nucleation times, the system size used must be large enough to

support the growth of multiple critical nuclei. It is important to point out that the

system-size dependence observed is due to the increased probability of critical

0 100 200 300

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

Rad

ii of

cry

tical

nuc

lei (

nm)

0.2

Undercooling (K)400 500 600

Figure 8.19 The critical radii of crystalline Al nuclei versus supercooling [12].


nucleus forming when the system size is larger, and it is not an artifact due to

periodic boundary conditions [73].

Minimum of the curve tw(ΔT) obtained in Ref. [12] (Figure 8.12A responded to

supercooling ΔT5 400 K) The averaged result for the nucleation time determined

for the system with 6000 atoms at the temperature of T5 500 K (five slightly dif-

ferent initial conditions) was tw5 40 ps, that is two times less than the value, 80 ps,

obtained in Ref. [73]. In opinion of Ovrutsky and Prokhoda [12], one of the reasons

of such disparity was the difference in the methods of nucleation time determina-

tion. Ovrutsky and Prokhoda [12] detected the appearance of the first crystalliza-

tions center studying the images of models from the past (the data from saved files)

and with help of LRPDFs calculations.

Evteev et al. [33] studied simultaneously percolation cluster formation and

nucleation of the crystalline phase in the iron model (Figure 8.7). The density in

their model was set in initial conditions and the model was cooled at constant vol-

ume (the NVT ensemble). Their results for minimum time lag is tw5 100 ps at

1200 K. The hatched curve in Figure 8.7 shows the temperature dependence of the

latency period for forming the first nucleus, which was calculated under the usual

equation for intensity I of nucleating (tcB1/I):

ttheor 5 ðNω1n�ZÞ21expδGðn�ÞkT

� �; ð8:23Þ

where N is the number of atoms in system. It has a minimum 10 ps at 950 K.

Evteev et al. [33] consider that great values of the latency periods tw (small inten-

sity of nucleation) in comparison with the theoretical values ttheor obtained for the

model caused exactly by that the percolation cluster from icosahedrons hinders

nucleation. There are also many of icosahedron cluster in amorphous state of pure

aluminum [12,73] and pure iron [36]. However, the time lags are smaller (to better

coincide with the theory). Nevertheless, the opinion [33], that the fractal cluster

200

100

50

104 105

System size (number of atoms)

T = 500 k

Nuc

leat

ion

time

(ps)

106

Figure 8.20 Nucleation time for Al at 500 K for different system [73]. For all system sizes,

the lowest nucleation time observed is close to the predicted transient times (dashed lines).

For smaller system sizes, a large scatter in the nucleation times was observed.


from icosahedrons is fundamental basis of the structural organization of the solid

amorphous state of pure metals, which essentially distinguishes them from melts,

has a sufficient reason.

8.3 Imperfect Structures of Small Crystallization Centers

8.3.1 Local Distribution Functions for Crystals of Different Size

Small nanocrystals do have not such strong ordering as the large crystal has.

Imperfect structures of nanocrystals have been studied experimentally [64]. And it

is known that the structure of crystallization centers is become gradually more per-

fect during annealing of amorphous alloys [79]. Results of simulating of the crys-

tallization of amorphous iron and Fe�B alloys [19] showed that small (,400

atoms) nanocrystals of iron have a highly imperfect variable structure. Small nano-

crystals are characterized by weak coordination of motion of neighboring atoms.

Spectrums of atom oscillations in small nanocrystals are more similar to those that

take place in amorphous phase (Figure 6.7). Short wavelengths of oscillations are

observed more frequently in the models with small nanocrystals, and the compo-

nents of velocities of nearest atoms had very often the opposite signs. Ovrutsky and

Prokhoda [19] offered the parameter βimp of noncoordination of oscillations that is

equal to the ratio of pairs of atoms with a different sign of the velocity components

to total number of pairs. This parameter was equal for amorphous state and small

nanocrystals (D350 atoms) of iron (βimp5 0.56 0.01 at T5 930 K) and βimp was

equal 0.456 0.01 for comparatively large nanocrystals (D2000 atoms). In the

opinion of authors, this is connected, obviously, with the short lifetimes of phonons

because of many transition boundaries between nanocrystals and clusters of amor-

phous phase or between nanocrystals with different structures and mutual

orientation.

Figures 8.21 and 8.22 show a change with time of LRPDFs determined for

atoms located in the center part of growing crystallization centers with diameter d

at simulation of pure aluminum and for model of iron. Atom coordinates averaged

during certain times were used for determination of LRPDF. These graphs demon-

strate that the structure of small CCs is in fact very imperfect and it becomes more

regular during growth.

8.3.2 Calculations of the Macroscopic Thermodynamic Driving Force

In Refs. [12,36], values of melting heats for the certain temperatures obtained by

analyzing of the simulation results were used to calculate relative supersaturations

σT5ΔμT/RT. The melting heats for the models of Al and Fe (Figure 8.23, curves

1) were determined as difference in potential energies per one atom in the non-

crystalline phase and in the single crystal after relaxation of such specimens

during 5�10 ps. Figure 8.24 shows the dependences of specific heats of two phases


(molar heat capacities Cl and Ccr) and entropy of fusion on the temperature.

The specific heats were determined as derivatives on temperature from enthalpies,

i.e., full energies. One can see that there are maxima on the temperature curves of

the heat capacity of noncrystalline phase, and it is connected with the glass transi-

tion. The structural transformation—disordering of the structure during heating,

leads to an increase of internal potential energy and absorption of heat.

0

1

(A) (B) (C)

2 4 6 8 0 2 4 6 8 0 2 4 6 8

g(r)

r (Å)

Figure 8.22 LRPDFs for the Fe growing crystal, T5 800 K; A,B,C—for the crystal size

(diameter) is equal 0.7, 0.9, and 1.8 nm correspondingly.

0

1

(A) (B) (C)

2 4 6 8 0 2 4 6 8 0 2 4 6 8

g(r)

r (Å)

Figure 8.21 LRPDFs for the Al growing crystal, T5 400 K; A,B,C—for the crystal size

(diameter) is equal 0.7, 1.1, and 2 nm correspondingly.


It is known that the entropy of fusion at the melting temperature ΔSm5ΔHm/Tm.

The differences in entropies at other temperatures were calculated from the equation:

ΔS5ΔSm 2

ðTmT

C1 2Ccr

TdT : ð8:24Þ

Data for difference in values of chemical potentials (ΔμT5ΔH�TΔS) are

given by lines with number 2 in Figure 8.23. They were calculated using values of

(A) (B)

40

30

20

10

0

40

30

20

10

00 0 300 600 900 1200 1500 1800200

3

2

11

2

3

400 600 800 1000

Cμ, ΔSμ (J/(mol K)) Cμ, ΔSμ (J/(mol K))

T (K) T (K)

Figure 8.24 Temperature dependences of molar specific heats and entropies of melting [12];

(A) aluminum, (B) iron; 1, 2—molar specific heats of amorphous and crystalline phases; 3—

the entropy of the phase transition.

10,000

(A) (B)

16,000

12,000

8000

4000

0

8000

6000

4000

2000

00 300 600 900 1200 1500 1800

22

3 3

4 4

11

ΔHT, ΔμT (J/mole) ΔHT, ΔμT (J/mole)

T (K) T (K)0 200 400 600 800 1000

Figure 8.23 The temperature dependence of melting heats and differences of chemical

potentials; (A) Al; (B) Fe; 1—molar melting heats; 2—thermodynamic driving force

calculated by equation ΔμT5ΔH2TΔS, 3—calculated by Spaepen’s equation ΔμT52ΔHf(1�T/Tf)T/(T1 Tf) [18], 4—new approximation: ΔμT5 0.5ΔHf(1�T/Tf)(11T/Tf).


ΔH from Figure 8.23 and ΔS from Figure 8.24. Ovrutsky and Prokhoda [12,36]

pointed that the errors of ΔS determination connected with the error of enthalpy

difference and the error of the Cl�Ccr difference were roughly 10%. The errors of

ΔμT determinations somewhat larger as ΔμT5ΔH�TΔS.

8.3.3 The Size-Dependent Thermodynamic Driving Force

Averaged potential energies of atom interaction depend on the size of CCs. After

their determination, Ovrutsky and Prokhoda [12,36] calculated the melting heats of

aluminum and iron crystals with different sizes at different temperatures. Squares in

Figure 8.25 show the results obtained. For calculation of the Δμr(r) dependences,

Ovrutsky and Prokhoda [12,36] took data of the ΔH(r) dependences and carried

out estimations of ΔS(r) (difference in entropies), taking the specific heat closer to

liquid for very small crystals. Calculated Δμ(r) values are plotted in the same

figure (curves 2). The dependence Δμr5ΔμT(1�r0/r) fits satisfactorily calculated

Δμ(r) values (curves 3). Thus, the theory of nuclei formation at large supercoolings

must take into account the dependence Δμ(r) and γ(r) dependence also.

8.3.4 Sizes of Critical Nuclei at Large Supercoolings

Examine how the dependences Δμ(r) and γ(r) (surface tension) can influence the

critical radii of nuclei. Consider that Δμr5ΔμT(1�r0/r), where the ΔμT(T) corre-

sponds to curves 1 in Figure 8.23, and γr5 γN(1�(r0/r)2) (the last dependence is

chosen for simplicity). Then the work of the crystal nucleus formation:

δGðrÞ524

3Ωπr3ΔμT 12

r0

r

1 4πr2γN 12

r20r2

� �; ð8:25Þ

5500

(A) (B)

10,000

9000

8000

7000

6000

5000

4000

3000

5000

4500

4000

1

1

2 23

33500

3000

2500

20000 0.5 1 1.5

r (nm)2 2.5 3 0 0.5 1 1.5

r (nm)2 2.5 3

ΔH(r), Δμ(r) (J/(mol K)) ΔH(r), Δμ(r) (J/(mol K))

Figure 8.25 The melting heats and Δμ differences versus the size of small crystallization

centers; (A) Al, T5 400 K; (B) Fe, T5 800 K; 1—ΔH(r); 2—Δμr5ΔH(r) �TΔS(r);

3—Δμr5ΔμT(1�r0/r).


where γN is the free surface energy for the interface between large crystal

and noncrystalline phase. It follows from the condition d(δG(r))/dr5 0 that the

critical radius:

r� 52γNΩΔμT

12

3r0: ð8:26Þ

The value of γN5 0.043 J/m2 at T5 400 K was obtained for aluminum from

data in Figure 8.19 in accordance with this equation. And γ(r�)5 γN(1�(r0/r�)2)5

0.023 J/m2 at this temperature, according to the assumption about the γ(r) depen-dence. Taking into account the dependences Δμ(r) and γ(r), one can obtain more

real results for the γN(T) and γ(r) dependences. By such a way, Ovrutsky and

Prokhoda [12] obtained that the values of γN do not increase when the temperature

decreases; and that corresponds to the theoretical result of Spaepen [59].

The dimension effect must influence the nucleation. Very small CCs cannot

appear at large supercoolings as the width of interface boundary is large with

respect to their size. As it follows from observations of simulated structures, nuclei

with a radius smaller than 0.35 nm disappear as a rule at very high supercoolings

(T, 500 K) as a result of rebuilding into a structure of some cluster of amorphous

phase. There are small and large enough clusters, sometimes up to 1�1.5 nm.

Large clusters are usually centered by the icosahedral cell and they contain the

curved atomic rows (Figure 8.8). Numbers of atoms having the icosahedral envi-

ronment become higher with a decrease of temperature and they are roughly two to

three times larger than numbers of atoms with the fcc environment. The cause of

that is the smaller energy of the icosahedral environment. Therefore, very small

nuclei of the crystal phase cannot grow if icosahedrons are near them; and they dis-

appear with the time.

The main indication of the amorphous state is the low rate of nucleation. Data

of Figure 8.12 indicate that the amorphous state exists for a long time at tempera-

tures below 300 K. Data of Figure 8.24 show that the main structure changes in the

noncrystalline phase of Al take place in the temperature interval from 450 to

700 K; and maximum of the specific heat of noncrystalline phase responds to tem-

perature TD550 K. The icosahedron lifetime is on the average B20 ps at

T5 550 K; it is less than the nucleation time (B30 ps). They exist B80 ps (far

from the crystallization center) at T5 500 K that is larger than the lag for nucle-

ation (B40 ps). This temperature may be assumed as glass-transition temperature

Tg for the model of aluminum investigated in Ref. [12].

8.4 Crystal Growth Kinetics in MD Models

8.4.1 On Mechanism and Kinetics of Growth of Metal Crystals

Studying snapshots of the model (such as in Figure 8.26), Ovrutsky et al. [12,35]

concluded that the growth mechanism from the amorphous phase consist in


collective displacements of atoms. Crystals influence essentially on the structure of

the amorphous phase near the interfaces. The front of crystallization is moving

after the ordering of some area. The new crystal with other orientation or the

twinned crystal is growing if the local structural coincidence is not full [12]. The

ordering in some places can be destroyed after the neighbor area is attached to

the crystal. Therefore, the crystal-melt interface is not plane in the atomic scale.

Near some clusters, the interface stops are relatively long. It is clear that the theory

for such growth cannot be simple.

For the quantitative description of the growth kinetics by the normal mechanism,

the Wilson�Frenkel law [80,81] is usually applied in the form (7.26). Considering

the classical theory of chemical reactions, Turnbull [82] described the growth

kinetics as difference of atom fluxes from melt to the crystal, I1, and from the crys-

tal to the melt I2:

I1 5 ν0 expð2Q1=kTÞ; ð8:27aÞ

I2 5 ν0 expð2Q2=kTÞ; ð8:27bÞ

where ν0 is the frequency of atom diffusion jumps through the interface; Q1 is the

activation energy for transitions of atoms from the crystal to the melt, Q2 is

the activation energy for transitions of atoms from the melt to the crystal. Under

the assumption that Q2�Q15Δμ, one can write down for the velocity of growth:

v5 aν0 expð2Q1=kTÞð12 expð2Δμ=kTÞÞ; ð8:28Þ

Figure 8.26 The sections of the model with growing crystal of iron (periodical conditions

along one direction) [36], the face [100]; (A)�(C) the temperature T5 600 K, the time of

annealing t5 2.5, 20 and 50 ps; (D) T5 500 K, t5 35 ps.


where a is an interatomic distance. Note that the expression ν0 exp(� Q1/kT) can be

expressed through the diffusion coefficient D (if we assume that Q1 is the diffusion

activation energy). The diffusion coefficient in the liquid can be written as [83]:

D5 ð1=6Þλ2ν0 expð�Q=kTÞ;

where λ is the averaged diffusion jump distance in the liquid. Thus, the kinetics

coefficient in Eq. (7.26) is β5 6f1aD/λ2. Jackson [83] entered prefactor f1 in this

expression for β, which takes into account the fact that growth only takes place at

repeatable step sites on the surface. For metals, which are very rough in atomic

scale interfaces with melts, this prefactor must be close to 1. Known experimental

dependences of the crystals velocities on the supercooling are in the book by

Fedorov [84]. He came to a conclusion that experimental values of the kinetics

coefficients are noticeably smaller than those which can be evaluated in the frame

of the Wilson�Frenkel model with assumption that aDλ at the melting tempera-

ture for which the typical values of diffusion coefficients are roughly equal

to (3�5)3 10�9 m2/s. If we assume that λ«a for better coincidence of the theoreti-

cal and experimental results, very small values of activation energy for diffusion

should be expected.

The Wilson�Frenkel model does not fit the results of simulations for the

Lennard-Jones liquid [85], for Cu and Ni [86], and for gold [87]. From these

papers, the values of β differ from the ratio 6D/a (the diffusion coefficients were

determined in these works by means of determination of the mean-square displace-

ments of atoms). Moreover, some experimental data for pure metals [85] allow

arguing that their crystallization is not thermally activated.

Broughton et al. [85] connected the kinetic coefficient with the average thermal

velocity (3kT/m)1/2 to explain the results of simulations of crystal growth in the L-J

liquid:

v5 f2ð3kT=mÞ1=2ð1� expð�Δμ=kTÞÞ; ð8:30Þ

where prefactor f2 is a constant of order 1. Later, Mikheev and Chernov [88] devel-

oped the density functional theory of freezing for the case of crystal-melt interface,

which is rather diffuse on the atomic scale. They proposed estimation, v5 (kT/m)1/2

ΔT/T, after simplifications of their results.

8.4.2 The Simulated Growth Velocities of Single Crystals

Figure 8.27 shows the simulated velocities for the case of L-J potential obtained in

the pioneer work [85]. The value f5 0.675 in Eq. (8.30) was chosen to give the

best fit with results of simulations. Figure 8.28 shows graphs with results of deter-

mination of growth velocities in large intervals of supercoolings for crystals of alu-

minum and iron [36]. Their modeling was fulfilled at boundary conditions featured

by Figure 8.26.


Studying the dependence ln(v(1-exp(�ΔμT/RT))�1) on 1/T (data of Figure 8.28),

Ovrutsky and Prokhoda [12,36] have determined the activation energy Q for the

surface kinetics. It is not constant; the value of Q is changing from 4500 to 400 K

at a decrease of temperature from 900 to 200 K in the case of aluminum, and in the

interval from 5500 to 500 K at the temperature change from 1400 to 350 K in the

case of iron. This confirms the point of view that conception of activated jumps of

atoms is not good approaching for the growth kinetics of metal crystals.

The dependences calculated according to Eq. 8.30 (full lines in Figure 8.28) fit

satisfactorily simulated data for the growth velocity only to ΔT5 100 K for Al and

160υ (m/s) υ (m/s)(A) (B)

350

300

250

200

150

100

50

0

120

80

40

00 200 400

ΔT (K) ΔT (K)

600 800 0 300 600 900 1200 1500

Figure 8.28 The velocities of single crystal growth plotted versus supercoolings [36]; (A) Al,

(B) Fe, lines are calculated according to Eq. (8.30): f5 0.35 for Al and f5 0.85 for Fe.

V (m/s)

0.4

0.2

0 0.2 0.4T/ε

Tf /ε

0.6

Figure 8.27 The velocity of crystal growth from the L-J liquid [85], ε is the energy

potential parameter; the dotted line is calculated according to Eq. (8.30). The full line

corresponds to the Wilson�Frenkel equation (8.29).


ΔT5 200 K for iron. Simulated curves passes maximum at ΔTD300 K and 400 K

for Al and Fe correspondingly, whereas the dependence (8.30) (with correct data

for ΔμT/RT) gives an increase of velocities up to ΔT5 650 K for Al and

ΔT5 1300 K for Fe. The simplified equation from Ref. [88] gives the dependence,

which increases only. It is obvious that the growth kinetics in a wide interval of

supercoolings cannot be described considering one certain mechanism only.

Celestiny and Debierre [87] had studied the anisotropy of growth velocity. They

simulated growth of pure gold in the model with gradient of temperature giving dif-

ferent initial orientations of the crystal; the kinetics of growth for the (111), (100),

and (110) faces was investigated. The authors found the linear low of growth up to

the supercooling of 100 K with kinetic coefficients close to thermal velocities in

accordance with (8.30). The kinetic coefficients in the dependence on v(ΔT) were

the following:

βT100 5 18:86 1:0 cm=s K;βT110 5 12:66 1:0 cm=s K;βT111 5 7:06 1:0 cm=s K:

The smallest value of βT111 had not obtained a satisfactory explanation.

Another interesting result consists in a large difference of kinetic coefficients

for solidification and melting (Figure 8.29). This difference was explained by dis-

tinctive features in growth and melting mechanisms, that is, additional time is

needed to build ordered structure in the case of growth.

The kinetics of isothermal crystallization and melting was studied for elemental

Ni in Ref. [89] employing MD simulations based on interatomic potentials of

the embedded-atom-method form. From these simulations, the authors calculated

the magnitude and crystalline anisotropy of the kinetic coefficient βT, defined as

υ (m

/s)

ΔT (K)

20

0

–20

–100 0 100 200

Figure 8.29 The velocity of growth and melting of the (111) crystal plane of the pure gold

model plotted versus supercooling [86].


the constant of proportionality between interface velocity and undercooling. They

obtain highly symmetric rates for crystallization and melting, from which they

extract the following values of βT for low index {100}, {110}, and {111} inter-

faces: βT1005 35.86 22, βT1105 25.56 1.6, and βT1115 24.16 4.0 cm/s K. The

author [89] discussed the results of this study in the context of previous MD simu-

lations for related systems, and kinetic models based upon transition-state and

density-functional theories. It is important that the values of the kinetic coefficients

obtained in result of simulations are closed to the value (βT5 20 cm/s K) obtained

in the experimental work [90], in which a containerless undercooling in an electro-

magnetic levitation device was used.

8.4.3 The Size Effect in Growth Velocity

Figure 8.30 shows the measured data of growth velocities of crystallization centers

(v5 dr/dt) plotted versus their size [12,36]. Growth velocities were also calculated

by the equation:

v5βσT ð1� r�=rÞ; ð8:31Þ

which takes into account the Gibbs�Thomson shift (Δσ5�σTr�/r). The dashed

lines show these results. The values of ΔμT were taken from Figure 8.23 (curves

2), kinetic coefficient β was determined from data for the comparatively large crys-

tal shown in Figure 8.26. It is clear from the Figure 8.30 that the dashed lines do

not fit well the growth velocities of small crystals.

The question of relation of the nuclei size and thickness δ of the physical inter-

face (a transition area between two phases) was considered in Ref. [59]. It was

shown that the Gibbs separating interface must be placed in the middle of the phys-

ical interface. Taking into account a decrease of ordering in small nanocrystals, it

υ (m/s) υ (m/s)

r (nm)

50

(A) (B)

60

50

40

30

20

10

0

40

30

20

10

00 0.5 1 1.5 2 2.5 3

r (nm)0 0.5 1 1.5 2 2.5 3

Figure 8.30 Dependences of growth rate of nuclei on their size [12,36], (A) aluminum

T5 400 K; (B) iron, T5 800 K; K—simulated data; dashed lines are calculated under the

Eq. (8.31), solid lines—the Eq. (8.32).


is quite reasonable to suppose that the minimum size of crystal nuclei (radius r0)

exists for every temperature, and Δμr5 0 for such size. Calculated Δμ(r) valuesplotted in Figure 8.25 were approximated by the dependence Δμr5ΔμT(1�r0/r).

This dependence (with r05 0.25 nm for Al and r05 0.35 nm for Fe) was taken into

account for calculation of the v(r) dependence given by solid line in Figure 8.30.

The following equation:

V 5βΔμT ð1� r0=rÞð12 r�=rÞ=kT ; ð8:32Þ

was used for calculations. The results calculated according to Eq. (8.32) (solid lines in

Figure 8.30) are in better coincidence with the simulated data. It means that

nanocrystals overcome comparatively slowly the initial stage of their existence.

The reasons for that are small effective supersaturation and the Gibbs�Thomson shift.

8.5 Recent MD Results on Crystallization from Alloy Melts

8.5.1 Growth of Disordered Solid Solutions from Alloy Melts. SoluteTrapping and Solute Drag Effects

The growth of some disordered solid solutions has been investigated using MD

simulations in several high-quality works [91�94]. Below, some fundamental phe-

nomena connected with the rapid solidification will be considered on the basis of

the paper [91] of several international groups of scientists and supplemental materi-

als to this paper [95]. Yang et al. [91] noted that substantial progress has been real-

ized over the past decade in the modeling of solidification under near-equilibrium

conditions; by contrast, predictive models for rapid solidification remain less devel-

oped, due in part to the need for a more detailed understanding of the nonequilib-

rium properties of CM interfaces.

For alloys solidifying at high velocities, the concentration of solute in the grow-

ing crystal is often found to be higher than its equilibrium value determined from

the phase diagram at the interface temperature [96]. The formation of a growing

solid with a solute composition beyond the equilibrium solubility limit through

rapid solidification is a process known as solute trapping [97�101]. The equilib-

rium distribution coefficient corresponds to equilibrium concentration of crystalline

and liquid phase according to the state diagram 2ke 5 xes=xel . At large supercooling

and velocity of growth xs . xes , it approaches xel and k5 k(v)5 xs/xl approaches 1.

Solute trapping plays a significant role in the processing of alloys using rapid

solidification techniques and it has been shown that trapping can also induce the

formation of metastable phases [102]. Additionally, the growth of a crystal with a

composition differing from that of its melt requires diffusion of solute across the

CM interface; the free-energy dissipation associated with this trans-interface diffu-

sion leads to a so-called solute drag effect that can significantly hinder the transfor-

mation rate [10,11].


Several models have been developed [98,99,101] to describe solute partitioning

as a function of interface velocity. These models differ in some significant details,

but each predicts a decreasing level of partitioning with increasing growth velocity

and each identifies a characteristic velocity at which significant trapping occurs.

For example, the theory for solute trapping developed by Galenko and Sobolev

[99] predicts a sharp transition to partitionless growth at a velocity dictated by the

atomic-scale relaxation processes in the bulk.

Yang et al. [91] considered a comparison of the k(V) results with known theories

for solute trapping: The continuous growth model (CGM) of Aziz and Kaplan [98]

is formulated by considering flux balances across a moving CM interface of

width λ. The theory predicts that appreciable trapping will occur when the interface

moves at a characteristic trapping velocity VD equal to the speed at which a solute

atom can traverse the interface: VD5D/λ, where D is the liquid diffusivity. In the

CGM, the k(V) function takes the following form (for dilute alloys):

kðVÞ5 ½ke 1 ðV 2VDÞ�=½11V=VD� ð8:33Þ

The local nonequilibrium model (LNM) of Galenko and Sobolev [99] is based

on a similar approach as the CGM but uses a generalized Fick’s law that accounts

for the finite relaxation time of the diffusion flux to its steady state.

The LNM yields:

kðVÞ5 ke½12 ðV=VLNMB Þ2�1 ðV=VLNM

D Þ12 ðV=VLNM

B Þ2 1 ðV=VLNMD Þ ð8:34Þ

for V less than the bulk liquid diffusion speed VLNMB (related to the relaxation time

for the diffusion flux), and k(V)5 1 for V .VLNMB .

The most recent theory for solute trapping due to Jackson et al. [101] is derived

using reaction rate theory to describe the rate of atom attachments to the active

sites of a sharp CM interface. For rough interfaces, the theory yields:

kðVÞ5 k1=ð11V=AÞe ; ð8:35Þ

where A is the characteristic velocities above which k(V) deviates strongly from ke.

To investigate the kinetic properties of alloy CM interfaces, Yang et al. [91]

considered two model systems. For the first, the interatomic interactions were taken

in the form of the truncated Lennard-Jones (L-J) potentials (Eq. (1.22)) for the

pair interactions (of type AA, BB, and AB) considered by Huitema et al. [103],

with the next parameters: σBB5 σAA; εBB5 0.540540εAA; σAB5 1.1σAA; εAB50.770265εAA. At r, 2.5σ, the potentials were multiplied by a cutoff function:

1.199 exp (0.25/(rij/σ�2.5))]. These potentials are shown in Figure 8.31.

Yang et al. [91] built the composition-temperature phase diagram for this system

(Figure 8.32) that features negligible solubility of the solute species (B) in the solid

(A), and thus displays a high degree of equilibrium solute partitioning.


They considered also an embedded-atom method model for Ni�Cu, with a

phase diagram [104] that displays extensive solubility. Compared to the L-J system,

Ni�Cu has a larger partition coefficient of ke 5 xes=xel 5 0:5, where xes and xel denote

equilibrium solidus and liquidus compositions, respectively.

For both the L-J and Ni�Cu systems, crystallization simulations began from

equilibrated two-phase solid�liquid simulation cells, with each of the bulk phases

prepared at their equilibrium phase-boundary compositions at a given temperature

and separated by CM interfaces oriented along {100} or {110}. For each equili-

brated system, they induced crystal growth by lowering the system temperature.

Several replicas with different initial velocity distributions were prepared, which

allowed evolving with independent trajectories. The positions of the CM interfaces

0.64

0.60

0.56

0.520.0 0.1

S + L

T ∗

L

0.2 0.3

Solute concentration (XB)

Figure 8.32 Calculated state diagram for the system with the L-J potentials [95].

1.0

0.5

AABBAB

0.0

–0.5

–1.02.01.5

rij (σAA)

Φ (

ε AA)

1.0 2.5

Figure 8.31 Binary L-J alloy potentials [95]. Solid (red) line: AA interaction. Dashed (blue)

line: BB interaction. Dotted (green) line: AB interaction.


during the growth simulations were monitored using a local structural order param-

eter. The density and composition were determined for all narrow layers parallel to

the moving interface. Therefore, Yang et al. [91] could obtain density and composi-

tion profiles along the oblong models (Figure 8.33). The solute composition (xB)

profile was defined as the ratio between the coarse-grained solute density profile

and the coarse-grained total density profile, xB(z)5 ρB(z)/ρ(z), where ρB(z) is the

solute density profile (Figure 8.34).

The top panel of Figure 8.34 shows a representative fine-scale density profile,

illustrating the diffuse nature of the CM interface. The bottom panel plots a

smoothed coarse-grained solute concentration profile from an L-J simulation with a

relatively low velocity (upper curve) and the highest velocity considered (lower

curve). For the slower V results, the peak in the concentration on the liquid side of

the interface reflects partitioning of the solute (xs, xl). For the high V case, the

concentration profile is nearly flat, indicating clear solute-trapping behavior.

3.0 Crystal Melt

xl

1.5

0.0

0.01 V ∗= 0.0708

V ∗

Z ∗

ρ ∗

V ∗= 0.0103

0.00–8 –4

XB

0 4 8

Figure 8.34 The averaged fine-grained profile for the total density across a CM interface

with a velocity V� 5 0.0103 in the model L-J alloy system (upper panel, and the smoothed,

coarse-grained profile of the solute mole fractions (lower panel) [91], the superscript� denotes L-J reduced units (sqrt(ε/m)�10�3); the solid and dashed lines in lower panel are

for V� 5 0.0103 and V� 5 0.0708, respectively)

Figure 8.33 A snapshot of the simulation cell of an MD trajectory for the equilibrium state

of the system used as a starting point for subsequent solidification simulations [95]. Blue and

red points represent solvent (A) and solute (B) atoms, respectively. There are two crystal-

melt interfaces in this periodic simulation cell.


Figure 8.35 from Ref. [91] plots MD results for the nonequilibrium partition

coefficient k(V)5 xs/xl. The interfacial region has two parts, on the liquid and solid

sides of the CM interface, and the solute composition in the former at a given time

t0 was compared with that in the latter at a later time t01Δt, where Δt is the time

required for the solid region to crystallize from the liquid. If the two compositions

are statistically equivalent, Yang et al. [91] concluded that the solidification is par-

titionless, i.e., k(V)5 1. Otherwise, k is computed taking xl as the peak composition

on the liquid side of the interface and xs as the average concentration of the solid

crystallized, as illustrated in Figure 8.34. In Figure 8.35, open circles and filled dia-

monds correspond to {110} and {100} interfaces, respectively, and show a statisti-

cally significant anisotropy.

In the lower panel of Figure 8.35, the solid and dashed lines fit the CGM and

Jackson models to all of the L-J MD data. Both theories fit well the data for the

lower velocities, where k 6¼ 1. However, the MD data at the highest V, which is

determined to be partitionless by the analysis of the MD data, are naturally under-

estimated by these theories, which predict k!1 as asymptotic behavior for V!1.

In the upper panel of Figure 8.35, the solid and dashed lines represent a fit of the

LNM model to the data for which k 6¼ 1. The fits of the {100} and {110} data pre-

dict a transition to partitionless solidification at a velocity VLNMB that is independent

of interface orientation, consistent with the LNM theory in which this parameter is

controlled by relaxation processes in the bulk liquid.

Thus, Yang et al. [91] found that the LNM of Galenko and Sobolev [99] gives a

best fit of the simulated results. The k(V) dependences for the system Cu�Ni are

only slightly different in cases of growth in {100} and {110} directions. The values

1.0

0.5k

k

0.0

1.0

0.5

0.00.00 0.02 0.04

V *

0.06

{100} MD{110} MD

{100} MD{110} MD{100} CGM{100} CGM{100} Jackson model{100} Jackson model

{100} LNM{110} LNM

Figure 8.35 MD calculated values of the partition coefficient (k) are plotted versus interface

velocity (V� in reduced units) [91], with open and filled symbols for {110} and {100}

interface orientations, respectively. Lines are fits of the MD data to available theories.


of the VLNMD parameters velocities are 1.4 and 1.6 m/s; and values of the VLNM

B

velocity are 15 and 21 m/s for the mentioned directions correspondingly. As men-

tioned above, k5 1 if V .VLNMB .

Yang et al. [91] have studied the dependences of growth velocities on the

driving force for interface migration ΔGm5ΔGchem1ΔGD in the form

ΔGm5 xeffΔμB1 (1�xeff)ΔμA proposed by Hillert [105], where ΔGchem is the

total chemical free-energy change due to solidification, and ΔGD is the free energy

dissipated due to solute drag; ΔμB denotes the difference between the chemical

potential of the solute species B in the solid versus the liquid phase, and similarly

for ΔμA. The effective composition xeff was taken in the form:

xeff 5 ð12 f Þxs 1 fUxl; ð8:36Þ

where the limit f5 0 corresponds to ΔGD5 0. In the version of the CGM theory

[98] that accounts for solute drag f5 1, while in the model of Jonsson and Agren

[106] it was assumed f5 1/2. According to diffuse-interface theories, the value f

depends on the nature of the variations of the solute concentration and diffusion

coefficient across the CM interface [105].

Yang et al. [91] plotted MD calculated velocities versus ΔGm at different values

of f. The dependences V(ΔGm) at f5 0 (without solute drag, ΔGm5ΔGchem) is

not linear, its slope increases with growth of ΔGchem in some intervals of values.

Thus, the MD calculated velocities indicates that the solute drag takes place at

comparatively small driving forces and is insignificant at large supersaturations.

The best-fit lines V(ΔGm) for the obtained velocities correspond to the value

f5 0.3 for the Ni�Cu system, and fD0.34 for the L-J system.

8.5.2 Crystallization of the Intermetallic Compound: Kinetics andDisorder Trapping

It is written clearly in the introduction to the paper [107] why crystallization of the

Al50Ni50 melt was investigated intensively last time: “Intermetallic NiAl (or nickel

aluminide) is an important compound because it exhibits attractive properties as

a result of its high energy of formation, low density, high-temperature corrosion

and oxidation resistance combined with a high yield strength [108�111]. It has a

B2-ordered crystal structure (CsCl prototype) which is retained up to the melting

temperature of about 1911 K [112]. However, its significant brittleness limits its

industrial application. It has been considered that refinement of the grain size to the

nanometer level could be a promising way to overcome this limitation because of

the enhancement of hardness and strength according to the Hall�Petch relation

[113,114]. A few experimental studies [115�117] have been performed to investi-

gate the velocity of growth and disorder trapping in undercooled NiAl alloy melts

as a function of undercooling.”

For the system Al�Ni, the quality potentials of interatomic interaction have

been elaborated by Mishin with coworkers [118,119], and this is also a reason


why many studies were fulfilled for this system using MD computer simulations

[107,120�124]. Levchenko et al. [120] have determined the thermophysical prop-

erties of an undercooled liquid Ni50Al50 alloy in the range of temperatures from

950 to 1550 K. In Ref. [121], the forming of the B2-NiAl ordered crystal structure

in the course of reaction in Al-coated Ni nanoparticle was studied. In Ref. [107],

particularities of near ordering in the undercooling melts and a structure relaxation

during isothermal annealing as changes in numbers of different Voronoi’s polyhe-

dron were studied. The waiting times for beginning of crystallization were also

determined. Kerrache et al. [122] have determined the kinetic coefficient β(β5 0.0025 m/(s K)) for the growth of the AlNi B2-phase studying the tempera-

ture dependence of the growth velocity. They considered that the classic

Wilson�Frenkel theory is unable to describe growth kinetics of this phase that a

leading role is played by the process of separation diffusion on the interface. Zhen

et al. [123] studied disorder trapping during crystallization of the B2-ordered NiAl

compound. They found that the majority of antisite defects are Ni atoms on the

Al sublattice, while the concentration of Al on the Ni sublattice is negligibly

small. The defect concentration was found to increase in an approximately linear

relationship with increasing the interface velocity. Zhen et al. [123] found also

that the growth velocities for the (100), (110), and (111) interface orientations are

very close to each other; from their results, one can find that the kinetic coeffi-

cient β is roughly 0.025 m/(s K). Ovrutsky and Prokhoda [124] studied the kinet-

ics of solidification of the NiAl melt in the wide interval of undercoolings using

different techniques of visualization of clusters, nuclei, and growing crystals.

Results obtained in the last two mentioned papers will be considered below in

more detail.

The first step before simulations is preparing samples for simulations. Initial

form of models in Ref. [124] was created in the auxiliary program. All atom coor-

dinates were passed in LAMMPS or in the CUDA program, developed in Ref.

[124], and their melting was accomplished (fully or partly). Two forms of samples

were prepared. Kinetics of nucleation was studied in spherical samples with a free

surface (nanodrops of 16,384 atoms). They were prepared in a result of fusion of

an ideal crystal given in initial conditions at a temperature above 2100 K over a

period of time .50 ps and quick (1�2 ps) cooling to the chosen temperature of

annealing (to obtain statistics for “one” temperature, the set temperatures for

annealing were different slightly, 0.1�0.2 K). To study growth kinetics, oblong

samples (of 32,768 atoms) in the form of cylinder with the B2-phase superstructure

were set in initial conditions (periodic boundary conditions along one axis were

applied). Then their larger part was transformed into amorphous state by the way

of many random displacements of atoms with further quick heating above melting

point and then quick cooling to the chosen temperature. The nanocrystals of

5�7 nm in size with initial interfaces of (100) and (110) crystal type were able to

grow at the constant temperature.

Three different crystallographic orientations were set up to prepare solid�liquid

interfaces oriented parallel to (100), (110), and (111) planes in Ref. [123]. The

dimensions of the cells varied according to crystalline orientation, but were


approximately 12a3 12a3 76a, where a is the lattice constant. The long directions

of the cells were chosen to be normal to the solid�liquid interface and the in-plane

dimensions were fixed to have periodic lengths dictated by the equilibrium lattice

constant (at zero pressure) for the crystal at the melting point. The dimensions

of the initial solid cells were (34.3 A3 34.3 A3 217.3 A), (33.2 A3 35.2 A

3 290.1 A), and (33.2 A3 35.9 A3 284.2 A) for the (100), (110), and (111) inter-

face orientations, respectively. The number of atoms in these simulation cells was

approximately 23,000, and varied slightly for the different orientations.

Zhen et al. [123] used the potentials elaborated in Ref. [118], which ensured the

melting point of the B2-phase Tm5 1520 K. Ovrutsky and Prokhoda [124] used

more up-to-date potentials from Ref. [119]. The melting point of an ideal crystal of

32,768 atoms with the structure of B2-phase was 17656 3 K when using

LAMMPS (for both cases—CPU and GPU) and 17606 3 K in the case of their

own CUDA program with all float variables. An inessential distinction from the

value determined in Ref. [119] (1780 K) was most likely connected with a differ-

ence in exploitable ensembles and different boundary conditions (the finite crystal

size in simulations).

For analyzing the results of simulations, Ovrutsky and Prokhoda [124] used the

program that enables viewing with the step in 1 A all sections of the model with

coordinates of atoms from saved files (the brightness of images of atoms placed

behind of chosen section decreases with a distance from the section; see

Figure 8.36). Thus, all the crystallization centers could be determined visually;

their structure was verified studying the graphs of local radial distribution functions

(LRPDF). Knowing the location of crystals, they studied changes in structure of

these areas in the past; the time of appearance of the crystallization center was

defined within a few picoseconds. Identification of clusters of different kinds was

fulfilled with the help of LAMMPS software or in results of procedure running,

which determines the number of nearest neighbors and angles with nearest neigh-

bors for every atom, and gives a color for atom visualization dependently on cluster

type [12].

To characterize the results, three types of analyses were performed in Ref. [123]

The first concerns the determination of interface velocity. This was done by track-

ing the interface position as a function of time through the use of a crystalline order

parameter.

The order parameter assigns a degree of crystallinity to an atom i and is defined

as φi 5 ð1=NÞPijrij 2 ridealij j2, where the sum extends over the N neighbors, rij is the

vector connecting sites i and j, and ridealij is the corresponding vector in an ideal

crystal. For bcc structures, the case of interest here, inclusion of first and second

neighbors (N5 14) had been found to be more effective in the identification of an

atom state—belonging to solid or liquid. The order parameter was averaged over

narrow bins oriented parallel to the solid�liquid interfaces, and analyzed as a func-

tion of distance along the interface normals. As the solid�liquid boundary is

crossed, a large jump in the value of the planar-averaged order parameter was

observed, as shown in Figure 8.37. In this figure, the liquid and solid regions have

high- and low-order parameter values, respectively.


32

24

16

Ord

er p

aram

eter

s

8

0–120 –80 –40 0

Z (Å)40 80 120

Figure 8.37 Calculated order parameter profile as a function of the coordinate of the atom

normal to the solid�liquid interface [123]. Points represent the calculated order parameter

for each atom.

Figure 8.36 The snapshots of the model section in two moments of time at the temperature

1000 K (in two and one color), [124]; (A) t5 235 ps; (B) t5 860 ps; the interfaces are

marked with white lines; in upper photos Ni atoms are blue.


Around the interface, the order parameter exhibits an abrupt change. To deter-

mine the interface position, Zhen et al. [123] used a hyperbolic tangent function to

fit the order parameter profile near each interfacial region. The function has the

form:

OPðzÞ5 c1 1 c2 tanhððz1 c3Þ=c4Þ ð8:37Þ

where c3 gives the interface position. In this figure, the blue solid and green dashed

lines display the fitting to the left and right interface, respectively. From plots of

interface position versus time, the interface velocity was obtained.

In spite of global thermostating (a Nose�Hoover thermostat), Zhen et al. [123]

plotted the temperature profile across the solid�liquid interface during solidifica-

tion, and they determined the values of the interface temperature, which were

somewhat larger than the temperatures in liquid (3�4 K higher than the value of

the thermostat temperature). Thus, Figure 8.38, which plots velocities versus the

temperature for cases of different orientations of crystals set in initial conditions,

describes the interface kinetics. As stated in Ref. [123] (and it is obvious), anisot-

ropy of the crystal growth velocity is small. Linear approximation of these data

gives the mentioned above value of the kinetic coefficient βD0.025 m/(s K).

To characterize the defect concentration in the crystal grown from the melt,

Zhen et al. [123] quenched to zero temperature snapshots from the growth simula-

tions using a steepest-descent algorithm to relax the atoms to their nearest local

minimum. The atoms in the crystalline regions can then be unambiguously assigned

to the individual (Ni and Al) sublattices of the ideal crystal structure. Defect con-

centrations were then calculated for each layer. For one layer, they first counted

the number of Ni (nNi) and Al (nAl) atoms at correct sites in this layer; then the

0(100)(110)(111)

–50

–100Vel

ocity

(cm

/s)

–1501480 1490 1500

Interface temperature (K)1510 1520

Figure 8.38 The crystal-melt interface velocity as a function of interface temperature for the

B2 NiAl system. Negative velocities correspond to solidification [123].


number of Ni (nantiNi ) and Al (nantiAl ) in antisites were counted by comparing to the

ideal crystal. If nidealNi and nidealAl are the number in the corresponding layer in a per-

fect crystal, the number of vacancies in Ni sites is nidealNi 2 nantiAl 2 nNi. Similarly, the

number of vacancies in Al sites is nidealAl 2 nantiNi 2 nAl. The total number of defects in

the system was calculated by adding the defects in each layer.

Figures 8.39 and 8.40 show snapshots during solidification along the (100)

direction for Al and Ni layers, respectively. Figure 8.39A�F correspond to times

of 22.0, 20.2, 0.0, 0.3, 1.0, and 4.0 ns, respectively, where the time reference

point corresponds to the such solid�liquid interface position defined by Eq. 8.37

that is close to the analysis section (window) of the model. One can see that dur-

ing growth, the ordering and local partitioning are occurring gradually. Some

atoms are required to diffuse out of the layer, and others into the layer to build

planes with the correct composition from a bulk melt with homogeneous concen-

tration. In Figure 8.39A, at 22.0 ns Ni and Al atoms are distributed approxi-

mately randomly in the bulk liquid. At 20.2 ns, this layer begins to solidify, and

at the same time, the concentration balance begins to change, with some Ni atoms

having moved out of the layer. When the solid�liquid interface is located pre-

cisely in this layer [snapshot (c)], there are many more Al atoms than Ni atoms.

From 0.3 to 1.0 ns [(d) and (e)], the number of Ni atoms continues to decrease.

Figure 8.39 Snapshots of an Al layer during the solidification for a (100) interface

orientation [123]. Al, blue filled circles; Ni, red open circles; (A) t522.0 ns, (B)

t520.2 ns, (C) t5 0.0 ns, (D) t5 0.3 ns, (E) t5 1.0 ns, and (F) t5 4 ns. t5 0 ns

corresponds to the time at which the solid�liquid interface is located in this layer.


Finally at 4.0 ns (f), when this layer is well inside the crystal, a few Ni atoms are

trapped in the Al layer. A similar solidification process for the Ni layer can be

seen in Figure 8.40.

Zhen et al. [123] had investigated the tendency for disorder trapping in the solid-

ifying crystal at different growth velocities. Both antisite defects and vacancies

were found, and almost all vacancies were located on the Ni sublattice. Figure 8.41

shows the concentration of defects in the B2 phase as a function of the interface

temperature. The error bars in the concentration of defects denote the estimated

uncertainties (standard errors) in the mean value. The dashed line is a least-squares

fit to a linear relation and illustrates the approximately linear increase in the con-

centration of defects as the interface temperature decreases. Considering the inter-

face velocity versus temperature shown in Figure 8.38, the concentration of

defects, i.e., the disorder trapping, should also be approximately a linear function

of interface velocity.

To determine the equilibrium vacancy concentrations, Zhen et al. [123] calcu-

lated the equilibrium point-defect concentrations at the melting temperature

(1520 K) concerned with applied potentials. In these calculations, they computed

the excess vibrational free energy of each point defect and used the values in a sta-

tistical�thermodynamic model given by Mishin et al. [118] to compute equilibrium

concentrations. The values obtained from this analysis are 0.65% and 0.33% for

vacancies on the Ni sublattice and Ni antisite defects on the Al sublattice, respec-

tively. The concentrations of the two other point defects are negligible: 0.0009%

and 0.0008% for Al-sublattice vacancies and Al antisites on the Ni sublattice,

Figure 8.40 The same as Figure 8.39 except for a Ni layer [123].


respectively. Zhen et al. [123] had noted that calculated the equilibrium concentra-

tions of vacancies on the Ni sublattice and Ni antisites on the Al sublattice are pre-

dicted to be dominant using the current embedded atom method (EAM) potentials

is consistent with known ab initio calculations. At the lowest temperature of

1483 K considered in the simulations, the concentrations of Ni vacancies and

Al-sublattice antisite defects are approximately 1.5 times larger than the equilib-

rium values at 1520 K (and are even larger than the equilibrium value at 1483 K).

Overall, the presence of defects at concentrations that are enhanced relative to the

equilibrium values, by an amount that increases with increasing interface velocity,

is completely consistent with the interpretation of defect formation by the disorder

trapping mechanism.

Ovrutsky and Prokhoda [124] have studied mechanisms and kinetics of forma-

tion and growth of crystallization centers with the structure of B2-phase in the

supercooled Al50Ni50 melts. After detecting of growing crystallization centers, they

studied changes of their structure during growth and changes of the structure of

supercooled models in the past before their formation in the places where they will

appear. The imperfect “noncritical” nucleus of finite size (B1 nm) is formed com-

paratively quickly (2�4 ps), after long annealing, in result of coordinated moving

of several atoms. Then its structure improves gradually without growth practically

(Figure 8.42) or it can disappear. Sometimes they see side-by-side two or three

1.6

1.4

1.2

1

0.8

0.6

0.08

0.06

0.04

Con

cent

ratio

n (%

)

0.02

0

1.6

1.4

1.2

1

0.8

0.61480 1490 1500

Interface temperature (K)

(100)(110)(111)

1510 1520

(A)

(B)

(C)

Figure 8.41 The concentration of defects of the B2 phase as a function of interface

temperature [123], (A) the vacancies, (B) Al at antisites, (C) Ni at antisites.


distorted bcc clusters as a nucleus; in spite of this, they did not see any bcc cluster

in this place 2 ps earlier. Sometimes the nucleus arises near the complex of several

other clusters, where parallel rows exist. If there are icosahedral clusters from every

side of the nucleus, it is not transformed into the crystallization center. Those

groups of bcc clusters are transformed into the crystallization centers of the B2-

phase, in which atoms of Ni and Al are mainly placed in right positions.

Otherwise, they disappear with time.

The degree of order in atom disposition varied at first gradually in the areas,

where crystallization centers (CCs) will be formed—the curved rows of Al or Ni

atoms are straightened slowly at first (Figure 8.42A). After straightening of the

rows and formation of planes, the nucleus begins to grow, very slowly at the begin-

ning, i.e., the crystallization center capable to the further growth (Figure 8.42C) is

already formed. However, transformation of the nucleus into the crystallization

center occurs practically without increase of its size. Thus, the crystallization center

is not formed by means of addition of atoms but by means of gradual improving of

the structure in some areas with a short-range order corresponding to the structure

of crystal phase.

Only CCs with superstructure ordering can grow at first. The structure in the

central part of CCs is improved during growth. Figure 8.43A and B show disposi-

tions of atoms in the central part of the growing small crystal, and Figure 8.43C

and D show LRPDFs corresponding to them. These images and LRPDFs give evi-

dence that the small crystal has the structure of B2-phase. One can see that the

LRPDF becomes sharper after increasing the crystal size.

Very small crystals grow very slowly because the Gibbs�Thomson shift to the

relative supersaturation is large for them and the thermodynamic driving force itself

(the difference in chemical potentials Δμ of amorphous and crystal phase) is small

for them because of the imperfect structure [122,124]. The growth velocity

increases quickly with the size. For the crystal shown in Figure 8.42, it increases

roughly three times (from the curve r(t)) at increasing of its size (diameter averaged

on different directions) from 1.5 to 2.5 nm.

If the supercooling is not small and the growth velocity becomes high enough

with increase of the crystal size, the new parts of the crystal have no good

Figure 8.42 Successive images of the model, which show the consecutive stages of

forming of the crystallization center [124]; C5 1000 K, times of annealing: (A) 77, (B) 87,

(C) 107 ps; atoms Al are light, Ni dark.


superstructure order as at the beginning of growth. Figure 8.44 shows growth of a

nanocrystal, which had an ideal structure at first, at two different temperatures. The

superstructure ordering is not perfect in the new parts of the growing crystal.

Whereas there is a limited number of defects at comparatively small supercooling

(Figure 8.44A, T5 1670 K), the ordering is bad at the large supercooling

(Figure 8.44B, T5 1100 K) and larger growth velocity. This is obviously connected

with the fact that the time of growth is small in comparison with the time that is

necessary for separation diffusion. There is the new CC at the bottom of

Figure 8.44B with the better ordering because its growth velocity, which is size

dependent, was smaller. The superstructure order is improving in nucleus shown in

Figure 8.42, which practically does not grow yet. Thus, simulations spent in Ref.

[124] confirmed the results of Ref. [123] concerning the dependence of the disorder

trapping on the growth velocity.

Figure 8.45A shows the waiting times for nucleation at different temperatures.

When lowering the temperature of annealing, nucleation becomes detectable begin-

ning from the temperature near 1200 K. The maximum rate of nucleation corre-

sponds to the temperature near 1050 K (the minimum waiting time is roughly

80 ps). The second nucleus appears in 20�40 ps after the first nucleus at such

supercoolings. The values for latencies are much smaller than the values from Ref.

[107], the factor is more than 10, because the system is larger with this factor than

g (r) g (r)

(A)

1 1

0 0.2 0.4 0.6 r (nm) 0 0.2 0.4 0.6 r (nm)

(C) (D)

(B)

Figure 8.43 Images of atoms from the central part of the growing crystallization center

(A,B) and LRPDFs from these areas (C,D) [124], T5 1200 K, time between pictures

Δt5 50 ps; (A) the crystal size (diameter) is 1.5 nm, (B) 2.5 nm.


the system studied in Ref. [107]. The curve of nucleation latencies is not symmet-

ric. It is connected with the collective movements of atoms remain to be possible at

large supercoolings—not jumps of atoms into vacancies as voids in the amorphous

phase that are much smaller than vacancies in crystals.

Ovrutsky and Prokhoda [124] found that the middle properties of the system, for

example an energy, became changed due to crystallization much later after the aris-

ing of the crystallization center, which grows very slowly at the beginning. At the

temperatures 1100 and 1000 K, the first CC arises (80�100 ps) close to the stage

of initial structure relaxation (quick decrease of the internal potential energy). Then

the curves of the energy have a small slope. The slope becomes larger due crystalli-

zation later (400 and 700 ps), when the growth velocity of CCs becomes suffi-

ciently large.

Figure 8.45B shows the temperature dependence of the growth velocity of

nanocrystals with a size of 5�7 nm, in oblong samples. Displacements of the inter-

face at the set temperatures (averaged in different directions) were measured in

photos of the model sections (excluding the initial period of relaxation of the amor-

phous phase. Due the periodical condition in one direction, the crystal, which is

placed at first from the right (Figures 8.36 and 8.44), quickly appear from the left;

its size is the sum of right and left parts. Photos for measurements of growth veloc-

ities (Figure 8.36) were made [124] in one color to see better interfaces. The inter-

faces have intricate contours because they move nonuniformly; they move forward

locally when appropriate ordering in some area of amorphous phase close to the

interface is formed. Not only different clusters, but new crystal nuclei with other

orientation hinder moving of the interface at large supercoolings, therefore, pro-

tuberances and cavities appear on the interphase surface. In the opinion of

Ovrutsky and Prokhoda [124], local structure changes in the amorphous phase near

the interface are possible when the coordinated displacements of atoms take place.

Figure 8.44 Sections of the models with growing nanocrystals of B2-phase, white dotted

lines indicate the initial interfaces, atoms of Ni are blue (dark) [124], (A) T5 1670 K, time

of annealing t5 875 ps; (B) T5 1100 K, t5 390 ps.


They write that the more obvious indications of such mechanism were obtained in

Refs. [12,36] for pure aluminum and iron. At large supercoolings (500�650 K), the

growth rate of the AlNi crystals reduces quickly, but it does not drop abruptly at

the temperatures smaller than 1000 K. This is the cause of asymmetry in the curve

for latencies shown in Figure 8.45A. The kinetics coefficient found in Ref. [124]

for comparatively small supercoolings (β5 0.0506 0.002 m/(s K)) does not contra-

dict essentially the data of Ref. [123] (B0,025 m/(s K)).

References

[1] K. Hono, K. Hiraga, Q. Wang, A. Inoue, T. Sakurai, Surf. Sci. 266 (1992) 35.

[2] G. Herzer, Phys. Scr. 49 (1993) 307.

[3] K. Nazako, Y. Kawamura, A.P. Tsai, A. Incue, Appl. Phys. Letl. 63 (1993) 2644.

tw (ps)

υ (m/s)

10,000

1000

100

10

20

10

0

–10

–20

–30

600

(A)

(B)

700 800 900 1000 1100 1200 T (K)

600 800 1000 1200 1400 1600 1800 T (K)

Figure 8.45 The temperature dependences of the waiting time for nucleation (A) and

growth velocity (B) [124], ’ and � are the latencies of the first nucleus forming; & are the

latencies for the second nucleus; � and V are the LAMMPS results.





[4] T. Benameur, A. Inoue, Mat. Sci. Forum 179�181 (1993) 813.

[5] M.E. Mc.Henry, M.A. Willard, D.E. Laughlin, Progr. Mat. Sci. 44 (1999) 291.

[6] Y.H. Kim, A. Inoue, T. Masumoto, Mater. Trans. J.I.M. 31 (1990) 747.

[7] I.T. Chen, Y. He, G.J. Shiflet, S.J. Poon, Scr. Metall. Mater. 25 (1991) 1421.

[8] K. Suzuki, N. Kataoka, A. Inoue, T. Masumoto, Mater. Trans. J.I.M. 32 (1991) 93.

[9] J.J. Croat, J.F. Herbst, R.W. Lee, F.E. Pinkerton, J. Appl. Phys. 55 (1984) 2078.

[10] D.R. Nelson, F. Spaepen, in: H. Ehrenreich, F. Seitz, D. Turnbull (Eds.), Solid State

Physics, vol. 42, Academic, New York, NY, 1989.

[11] R. Collins, Phase Transitions and Critical Phenomena, Academic, New York, NY,

1972.

[12] A.M. Ovrutsky, A.S. Prokhoda, J. Crystal Growth 314 (2011) 258.

[13] A.P Shpak, A.B. Melnik, Micro-Inhomogeneous Structure Unordered Metal Systems,

Academperiodika, Kiev, 2005 (in Russian).

[14] J. Steinhardt, D.R. Nelson, M. Ronchetti, Phys. Rev. B 28 (1983) 784.

[15] D. Honeycutt, H.C. Andersen, J. Phys. Chem. 91 (1987) 4950.

[16] P. Ganesh, M. Widom, arxiv.org/abs/cond-mat/0702263v1 [cond-mat.mtrl-sci] 10 Feb

2007.

[17] P. Ganesh, M. Widom, Phys. Rev. B 74 (2006) 134205.

[18] Z.A. Tian, R.S. Liu, H.R. Liu, C.X. Zheng, Z.Y. Hou, P. Peng, J. Non-Cryst. Solids

354 (2008) 3705.

[19] A.M. Ovrutsky, A.S. Prokhoda, Crystallogr. Rep. 54 (2009) 631.

[20] D. Turnbull, J. Appl. Phys. 21 (1950) 1022.

[21] D. Turnbull, J. Metals 188 (1950) 1144.

[22] D. Turnbull, R.E. Cech, J. Appl. Phys. 21 (1950) 804.

[23] F.C. Frank, R. Proc, Soc. London, Ser. A 215 (1952) 43.

[24] A. Di Cicco, A. Trapananti, S. Faggioni, A. Filipponi, Phys. Rev. Lett. 91 (2003)

135505.

[25] G.W. Lee, A.K. Gangopadhyay, K.F. Kelton, R.W. Hyers, T.J. Rathz, J.R. Rogers, D.

S. Robinson, Phys. Rev. Lett. 93 (2004) 037802.

[26] W.K. Luo, H.W. Sheng, F.M. Alamgir, J.M. Bai, J.H. He, E. Ma, Phys. Rev. Lett. 92

(2004) 145502.

[27] M.R. Hoare, J. Crystal Growth 17 (1972) 77.

[28] N. Jakse, A. Pasturel, Phys. Rev. Lett. 91 (19) (2003) 195501.

[29] M. Celino, V. Rosato, Phys. Rev. B 75 (2007) 174210.

[30] A.S. Clarke, H. Jonsson, Phys. Rev. E 47 (1993) 3975.

[31] F.H.M. Zetterling, M. Dzugutov, S.I. Simdyankin, J. Non-Cryst. Solids 39 (2001) 293.

[32] A.V. Evteev, A.T. Kosilov, E.V. Levchenko, J. Exp. Theor. Phys. 99 (3) (2004) 522

(in Russian).

[33] A.V. Evteev, A.T. Kosilov, E.V. Levchenko, O.B. Logachev, Phys. Solid State 48

(2006) 815 (in Russian).

[34] V.A. Polukhin, N.A. Vatolin, Simulation of Amorphous Metals, Nauka, Moscow, 1985

(in Russian).

[35] A.M. Ovrutsky, A.S. Prokhoda, V.F Bashev, Comput. Mater. Sci. 50 (6) (2011) 1937.

[36] A.M. Ovrutsky, A.S. Prokhoda, Metal. Phys. Adv. Technol. 33 (6) (2011) 707 (in

Russian).

[37] F.E. Luborsaky, Amorphous Metallic Alloys, Butterworth Monographs in Materials,

London, 1983.

[38] F. Faupel, W. Frank, M-P. Macht, V. Naundorf, K. Ratzke, H. Schober, S. Sharma, H.

Teichler, Rev. Mod. Phys. 75 (2003) 237.


















































[39] J.W. Christian, Transformations in Metals and Alloys, Part I, Oxford Pergamon,

Oxford, 1978.

[40] K.F. Kelton, H. Ehrenreigh, O. Turnbull, Solid State Physics, vol. 45, Academic, New

York, NY, 1991, pp. 75�177.

[41] D. Kashchiev, Nucleation: Basic Theory with Applications, Butterworth-Heinemann,

Oxford, 2000.

[42] M. Volmer, A.Z. Weber, Phys. Chem. 119 (1926) 277.

[43] R. Becker, W. Doring, Annalen der Physik 416 (1935) 719.

[44] R. Kaischew, I.N. Stranski, Z. Phys. Chem. B 26 (1935) 81.

[45] Y.B. Zeldovich, Acta Physicochim USSR 18 (1943) 1.

[46] Ya.I. Frenkel, Kinetic Theory of Liquids, Oxford University Press, Oxford, 1946.

[47] V.V. Voronkov, Soviet Phys. Crystallogr. 54 (2009) 631.

[48] L. Battezzati, S. Pozzovivo, P. Rizzi, in: H.S. Nalwa (Ed.), Encyclopedia of

Nanoscience and Nanotechnology, vol. 6, 2004, pp. 341�364.

[49] C. Birac, D. Lesueur, Phys. Status Solidi 36 (1976) 246.

[50] M.C. Freitasa, D.I. Galdez Costaa, A.A. Cabral, A.R. Gomes, J.M.R. Mercury, Mater.

Res. 12 (1) (2009) 101.

[51] N. Eustathopoulos, Int. Met. Rev. 28 (1983) 189.

[52] V.P. Skripov, Crystal Growth and Materials, North-Holland, Amsterdam, 1977, 327 p.

[53] Y.V. Naydich, V.M. Perevertaylo, N.F. Grigorenko, Capillary Phenomena in Processes

of Crystal Growth and Melting, Naukova Dumka, Kiev, 1983 (in Russian).

[54] J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity, Oxford University Press,

New York, NY, 1982.

[55] J. Israelachvili, Intermolecular and Surface Forces, second ed., Academic Press, San

Diego, CA, 1992.

[56] M.G. Weinberg, E.D. Zanato, S. Manrich, Phys. Chem. Glasses 33 (3) (1992) 99.

[57] M.V. Fokin, E.D. Zanotto, J. Non-Cryst. Solids 265 (2000) 105.

[58] J.W.P. Schmelzer, I. Gutzow, J. Schmelzer, J. Colloid Interface Sci. 178 (2) (1996) 657.

[59] F. Spaepen, Solid State Phys. 47 (1994) 1.

[60] D. Frenkel, B. Smit, Understanding Molecular Simulation. From Algorithms to

Applications, Academic Press, USA, UK, 2002.

[61] A.V. Evteev, A.T. Kosilov, A.V. Milenin, Phys. Solid State 43 (2001) 2187.

[62] S.C. Lee, N.M. Hwang, B.D. Yu, D.Y. Kim, J. Crystal Growth 223 (2001) 311.

[63] J.H. Shim, S.C. Lee, B.J. Lee, J.Y. Suh, J. Crystal Growth 250 (2003) 558.

[64] A.I. Gusev, Nanocrystal Materials: Production and Properties. Yekaterin-burg: UrO

RAN, 1998 (in Russian).

[65] N.T. Gladkih, S.V. Dukarov, S.V. Krishtal, Surface Phenomenon and Phase Transition

in Condensed Films, KHNU, Kharkov, 2004 (in Russian).

[66] W. Halperin, Rev. Modern Phys. 58 (1986) 353.

[67] A. Hori, J. Chem. Eng. Rev. 7 (1975) 28.

[68] K.F. Kelton, T.K. Croat, A.K. Gangopadhyay, L.Q. Xing, A.L. Greer, M. Weyland, J.

Non, Cryst. Solids 317 (2003) 71.

[69] Mo Li Xian-Ming Bai, J. Chem. Phys. 122 (2005) 224510.

[70] D. Turnbull, Solid State Phys. 3 (1956) 226.

[71] K.F. Kelton, A.L. Greer, Phys. Rev. B 38 (1988) 10089.

[72] M.I. Mendelev, M.J. Kramer, C.A. Becker, M. Asta, Philos. Mag. 88 (2008) 1723.

[73] R.S. Aga, J.R. Morris, J.J. Hoyt, M.I. Mendelev, Phys. Rev. Lett. 96 (2006) 245701.

[74] D. Moroni, P.R. ten Wolde, P.G. Bolhuis, Phys. Rev. Lett. 94 (2005) 235703.

[75] S. Izumi, S. Hara, T. Kumagai, S. Sakai, J. Crystal Growth 274 (2005) 47.

















































[76] M.I. Mendelev, D.J. Srolovitz, G.J. Ackland, S. Han, J. Mater. Res. 20 (2005) 208.

[77] H. Trinkaus, M.H. Yoo, Philos. Mag. A 55 (1987) 269.

[78] G. Shi, J.H. Seinfeld, K. Okuyama, Phys. Rev. A 41 (1990) 2101.

[79] K.F. Kelton, T.K. Croat, A.K. Gangopadhyay, L.Q. Xing, A.L. Greer, M. Weyland, J.

Non, Cryst. Solids 317 (2003) 71.

[80] Y.A. Wilson, Philos. Mag. 50 (1900) 238.

[81] Y.I. Frenkel, Phys. Z. Sowjetunion 1 (1932) 498.

[82] D. Turnbull, Principles of Solidification, Thermodynamics in Physical Metallurgy,

American Society for Metals, Cleveland, OH, 1950, pp. 282�306.

[83] K.A. Jackson, Interface Sci. 10 (2002) 159.

[84] O.P. Fedorov, Processes of Crystal Growth. Kinetics, Forms of Growth,

Imperfections, Naukova dumka, Kiev, 2010, 207 p. (in Russian).

[85] J.Q. Broughton, G.H. Gilmer, K.A. Jackson, Phys. Rev. Lett. 49 (1982) 1496.

[86] J.J. Hout, B. Sadigh, M. Asta, S.M. Foiles, Acta Mater. 47 (1999) 3181.

[87] F. Celestiny, J.M. Debierre, Phys. Rev. E 65 (2002) 041605.

[88] L.V. Mikheev, A.A. Chernov, J. Cryst. Growth 112 (1991) 591.

[89] D.Y. Sun, M. Asta, J.J. Hoyt, Phys. Rev. B 6 (2004) 024108.

[90] R. Willnecker, D.M. Herlach, B. Feuerbacher, Phys. Rev. Lett. 62 (23) (1989) 2707.

[91] Y. Yang, H. Humadi, D. Buta, B.B. Laird, D.Y. Sun, J.J. Hoyt, M. Asta, Phys. Rev.

Lett. 107 (2011) 25505.

[92] H.E.A. Huitema, B. Van Hengstum, J.P. van der Eerden, J. Chem. Phys. 111 (1999)

10248.

[93] F. Celestini, J.M. Debierre, Phys. Rev. B 62 (2000) 14006.

[94] Q. Yu, P. Clancy, J. Cryst. Growth 149 (1995) 45.

[95] http://link.aps.org/ supplemetal/10.1103/PhysRevLett.107.025505.

[96] M. Asta, et al., Acta Mater. 57 (2009) 941.

[97] J.A. Kittl, et al., Acta Mater. 48 (2000) 4797.

[98] M.J. Aziz, T. Kaplan, Acta Metall. 36 (1988) 2335.

[99] P. Galenko, S. Sobolev, Phys. Rev. E 55 (1997) 343.

[100] N.A. Ahmad, et al., Phys. Rev. E 58 (1998) 3436.

[101] K.A. Jackson, K.M Beatty, K.A. Gudgel, J. Cryst. Growth 271 (2004) 481.

[102] P.K. Galenko, D.M. Herlach, Phys. Rev. Lett. 96 (2006) 150602.

[103] H.E.A. Huitema, B. van Hengstum, J.P. van der Eerden, J. Chem. Phys. 111 (10)

(1999) 248.

[104] E.B. Webb, J.J. Hoyt, Acta Mater. 56 (2008) 1802.

[105] M. Hillert, Acta Mater. 47 (1999) 4481.

[106] B. Jonsson, J. Agren, J. Less-Common Met. 145 (1988) 153.

[107] E.V. Levchenko, A.V. Evteev, I.V. Belova, G.E. Murch, Acta Mater. 59 (2011) 6412.

[108] D.B. Miracle, Acta Metall. Mater. 41 (1993) 649.

[109] R.D. Noebe, R.R. Bowman, M.V. Nathal, Int. Mater. Rev. 38 (1993) 193.

[110] J.H. Westbrook, R.L. Fleischer, Intermetallic Compounds: Structural Applications,

vol. 4, Wiley, New York, NY, 2000.

[111] S.C. Deevi, D.G. Morris, V.K. Sikka, Mater. Sci. Eng. A 258 (1998) 124 (special

issue).

[112] P Nash, M.F. Singleton, J.L. Murray., in: P Nash (Ed.), Phase Diagrams of Binary

Nickel Alloys, vol. 1, ASM International, Metals Park, OH, 1991.

[113] R. Bohn, T. Haubold, R. Birringer, H. Gleiter, Scripta Met. 25 (1991) 6.

[114] M Fukumuto, M Yamasaki, M Nie, T Yasui, Quart J. Jpn. Weld Soc. 24 (2006) 87.



























http://link.aps.org/%20supplemetal/10.1103/PhysRevLett.107.025505

























[115] H. Hartmann, D. Holland-Moritz, P.K. Galenko, D.M. Herlach, Europhys. Lett. 87

(2009) 40007.

[116] S. Reutzel, H. Hartmann, P.K. Galenko, S. Schneider, D.M. Herlach, Appl. Phys.

Lett. 91 (2007) 041913.

[117] H. Assadi, S. Reutzel, D.M. Herlach, Acta Mater. 54 (2006) 2793.

[118] Y. Mishin, M.J. Mehl, D.A. Papaconstantopoulos, Phys. Rev. B 65 (2002) 224114.

[119] G.P. Purja Pun, Y. Mishin, Phil. Mag. 89 (2009) 3245.

[120] E.V. Levchenko, A.V. Evteev, D.R. Beck, I.V. Belova, G.E. Murch, Comput. Mat,

Science 50 (2010) 465.

[121] E.V. Levchenko, A.V. Evteev, D.P. Riley, I.V. Belova, G.E. Murch, Comput. Mat,

Science 47 (2010) 712.

[122] A. Kerrache, J. Horbach, K. Binder, Lett. Exploring Front. Phys. 81 (2008) 58001.

[123] X.Q. Zhen, Y. Yang, Y.F. Gao, J.J. Hoyt, M. Asta, D.Y. Sun, Phys. Rev. E 85 (2012)

041601.

[124] A.M. Ovrutsky, A.S. Prokhoda, Comput. Mat. Sci. (2013), 05.045, doi:org/10.1016/j.

commatsci.


















9 Computational Experimentsin Materials Science

9.1 Diffusion in Solids

As it is known, atoms seldom leave their balance position in solids. Nevertheless,

as it happens sometimes, the contacting matters can penetrate one another.

Penetration of one substance into another is termed an interdiffusion. As for self-

diffusion, it takes place if the concentration of atoms of radioactive isotopes of the

same substance, which was brought into some site of the sample, is flattening. Self-

diffusion of atoms in the crystal lattice is carried out by one of three mechanisms

from the point of view of classical theory:

1. If there is a vacancy in some knot of the crystal lattice, one of the neighbor atoms can ful-

fill the jump from its standing into the vacant knot. Then the vacancy will occupy the pre-

vious place of this atom. Such transitions are similar to the movement of vacancies.

Those atoms can jump into the vacancy, the kinetic energy of which is sufficient for per-

formance of local deforming of the crystal lattice.

2. If the energy of atom oscillations is large enough, the atom can jump from its place into

the space between lattice knots and turn into the interstitial atom. In this case, such a fluc-

tuation of energy is necessary that is considerably larger than for the jump into the vacant

knot, because the lattice deformation (and strain) will be much greater.

3. Lastly, the neighbor atoms can simply interchange by place. Thus, the final state does not

differ from the initial state, but the strain of the lattice during the jump will be much

more than for the first two termed mechanisms.

Calculations show that the contribution of jumps by the second and third

mechanisms to self-diffusion is small enough; the vacancy mechanism plays the

basic role. At the same time, diffusion of small impurity atoms, for example,

hydrogen in platinum or carbon in iron, occurs more often by the way of jumps

between the knots.

Frenkel has analyzed the self-diffusion in the volume-centered crystal lattice

(vcc). He supposed that diffusion is carried out by the way of jumps of atoms into

vacancies. Assume that in the two next atomic planes, which are apart d5 a/2 (a is

the lattice parameter), the concentrations of atoms of the radioactive isotope in

counting per unit area are different because of the existence of the gradient of con-

centration. The number of atoms per unit area in one plane is η(x), and in another

next plane is η(x1 a/2). The frequency of atom jumps from one plane to another is

spotted by the frequency of natural oscillations of atoms νo and geometrical factor



http://dx.doi.org/10.1016/B978-0-12-420143-9.00009-0

4 (the number of nearest knots on the next plane). It depends also on the probability

of existence of vacancies in the lattice knots, which is equal to the relative vacancy

concentration xv (xv5 exp(2 uv/kT), uv is the work of vacancy formation) and

probability of jumps, which is determined by the Boltzmann factor of work ua of

the crystal lattice distortion during displacements of atoms:

f 5 4no � expð�ua=kTÞ � xv 5 4νo � exp½�ðua 1 uvÞ=kT �: ð9:1Þ

Currents of atoms depend on the numbers of their jumps from the unit area of

the two next planes in one and opposite directions; they are proportional to the fre-

quency and surface concentrations:

I1 5 ηðxÞ � f 5 4νoηðxÞ � exp½�ðua 1 uvÞ=kT �;

I2 5 ηðx1 a=2Þ � f 5 4νoηðx1 a=2Þ � exp½�ðua 1 uvÞ=kT �:

The surface concentrations can be expressed through the volume concentrations,

multiplying them by the thickness of layers a/2, η(x)5 n(x) � (a/2). Thus, the total flux

I5 I1 2 I2 52a2νo � exp½�ðua 1 uvÞ=kT � � ðdn=dxÞ;

where dn/dx5 [n(x1 a/2)2 n(x)]/(a/2).

Comparing this equation with the diffusion equation (3.1) in the form

IM 5 dM=dS dt52DðdC=dxÞ;

where concentration C5mn (m is the mass of atoms, IM5 I �m), we will obtain the

relation for the diffusion coefficient:

D5 a2νo � exp½�ðua 1 uvÞ=kT �5D0 � expð2Q=RTÞ: ð9:2Þ

where Q is the activation energy and R is the gas constant.

Apparently, the diffusivity increases with temperature magnification under the

exponential law. Such dependence is confirmed experimentally. It is possible to

calculate the activation energy Q by Eq. (6.9) using the experimental data of mea-

suring of the diffusivity at different temperatures. Theoretical estimates of the Q

magnitude obtained as product of ua1 uv and the Avogadro number are in accor-

dance with experimental data. It means that jumps of atoms into the vacant knots

(moving of vacancies) give the main contribution to self-diffusion in crystals.

9.1.1 Model for Algorithm Construction

Impurity atoms or the radioactive atoms are in parallel planes; the distance between

them is equal to half of parameter d of the crystal lattice (one pixel on the monitor

screen). According to the program, atoms are disposed not in planes but in rows


with different coordinate y on the screen at the same coordinate x. The filling is

made up to half of the model length (and halves of screen) along x-axis, and the

zero number of particles and value of concentration is set for the other half.

All particles can jump to the right or to the left. Frequency of jumps is spotted

by Eq. (9.1). In this equation, the factor 4 is the geometrical factor for the face

(001) of the bcc crystal lattice. The chosen atom has four neighbor atomic places in

the neighbor crystal plane. Probability of vacancy placing in the neighbor position

is connected with the relative vacancy concentration xv5 nv/N (N is the total of

nodes in the crystal lattice), and the last is equal to Boltzmann’s factor from energy

of vacancy formation: xv5 exp(2 uv/kT). This magnitude is multiplied by 4, by the

frequency of atom oscillations νo5 1/τ0, and by the Boltzmann factor of the jump

activation energy: exp(2 ua/kT). We come to (1). Hence, one jump happens on the

average time Δt5 (1/f)5 τ0 exp(2 (ua1 uv)/kT)/4, where τ0 is the frequency

period. This time will be the timestep for our model. We will calculate it and keep

in mind that (ua1 uv)/kT5Q/RT, where Q is the diffusion activation energy deter-

mined from experimental data and R is the gas constant. Hence, all atoms will ful-

fill the jumps during timestep Δt. Direction of jumps will be spotted by the

random number (1 1 or 21).

The concentration (number of particles) will be supported by the stationary

value on the left boundary of the model. In all other points, it casually varies, but

on the average, in accordance to a course of the diffusion process, the concentration

of impurity atoms in each point x of the sample should vary according to Eq. (3.8).

If the ratio C(x,t)/C (the relative concentration) is known for the certain point of

the medium with coordinate x, it is possible to find the diffusivity, having found the

solution of the transcendental equation (3.8). The simplest way to solve Eq. (3.8)

consists of the search of sign change of the function, which is the difference of the

left and right parts of this equation. It is clear that for each value of the diffusion

coefficient, which will be tested, it is necessary to calculate numerically the integral

of errors. The corresponding procedure (TForm1.Button2Click) is described below.

In a result of the basic program work, the numbers of atoms for each coordinate

are calculated, the histogram of the concentration distribution is calculated and dis-

played periodically in the picture canvas, and the values of the relative concentra-

tion for five columns of the histogram are represented in this window. Besides, the

time, which is found as a product of the timestep with the number of the steps, and

mean-squared displacements of particles relative to their initial positions are also

represented in the form. It is easiest to spot the diffusivity by the Einstein’s equa-

tion through the mean-squared displacements (,x2. 5 2Dt).

The program interface (Figure 9.1) provides choosing of the chemical element

and temperature (offered values are shown in the window of type “Memo”). Pauses

in the work of the program are used for displaying the intermediate distributions of

concentration, for recording the interval limits for determination of the diffusion

coefficient according to Eq. (3.8), and for reading the values of the mean-squared

displacements of particles and the time of experiment.

Several basic procedures are shown below in short form: the procedureHistogram, which ensures the plotting of the concentration distribution on the

303Computational Experiments in Materials Science

coordinate x, the procedure TForm1.Timer1Timer, which fulfills the basic cycle of

atom jumps and their painting, and the procedure TForm1.Button2Click, the diffu-

sivity calculation according to Eq. (3.8).

Figure 9.1 The active controlling form of the program “Diffusion”.

Procedure Histogram;

{Procedure for showing of the particle distribution along the x-axis,}var j,i:integer; ss:string; conce:array [0..5] of real;

y:array[-1..641] of word; {array of quantity of particles for the every coordinate}nz:byte; {the width of columns in pixels }begin

for i:=1 to nn do {for all atoms} beginj:=x[i]+330{0}; {the atom positions on the screen}inc(y[j]); {the number of particles with a given coordinate x}end;

nz:=round(330/15); {the column width in pixels}for i:=0 to 640-nz do {the cycle of calculation of atoms distribution on the

coordinate cells - histogram columns}if i mod nz =0 then

for j:=1 to nz-1 do

begin

y[i]:=y[i]+y[i+j]; {recalculation of sums to the first coordinate of columns}y[i+j]:=0; {now y[i] is zero for other x-coordinates}end; j:=1;


for i:=329 to 640-nz do

if (i mod nz=0) and (j<6) then begin

conce[j]:=y[i]/ny; {calculation of relative concentrations}nz:=round(330/15);

maxx:=image1.Width; maxy:=image1.Height;

image1.Canvas.Brush.color:=clWhite;

image1.canvas.Rectangle(0,260,maxx,0);

image1.Canvas.Brush.color:=clYellow;

for i:=0 to 661 do

if y[i]>0 then

image1.canvas.Rectangle(i,260,i+nz-2,260-y[i]);

end;

procedure TForm1.Timer1Timer(Sender: TObject);

var j,i:integer; con, karu: real;

begin

with image1.Canvas do

for i:= 1 to nn do

begin

pixels[x[i]+330, round((maxy+1)/2 – {deleting of old images}

(maxy-ny+1)/2+round(i/15)+206)]:= clWhite;

if random(2)=1 then x[i]:=x[i]+1

else x[i]:=x[i]-1; {jumps of atoms }if x[i]<-330 then x[i]:=-326;

pixels[x[i]+330, round((maxy+1)/2-(maxy-

ny+1)/2+round(i/15)+206)]:= clRed; {imaging of atoms}end;

end;

procedure TForm1.Button2Click(Sender: TObject);

// The procedure for calculations of roots of the equation (3.8) for D, by searching//of changes of the function sign (“+” or “–“)

var k1,erf,k3,e,de,x:real;

i:word;

begin

if onoff='on' then begin

onoff:='off';

timer1.Enabled:=true;

Button1.caption:='divB'; end

else begin

onoff:='off';

timer1.Enabled:=false;

Button2.caption:='DifB'; end;

D1:=strtofloat(edit1.Text);

D2:=strtofloat(edit3.Text);

case ListBox1.Selected of

0:ik:=3; // choose of the column number

1:ik:=2; // equation (3.8) is rewritten in the form: (k1-erfc) =0


Apparently, from the text of procedure TForm1.Timer1Timer, atoms fulfill jumps

with probability 1. It means that the greatest possible timestep dt is chosen in the

program. It is spotted by the relation for probability of jumps

P5 dt � ν � expð�Q=RTÞ5 1;

where ν is the oscillation frequency of atoms. Each atom has a lot of tests during

this computer experiment (in every timestep). The number of steps spots the time

of the experiment, time.

9.1.2 Recommended Experiments

1. For chosen substance and parameters (Q and d), calculate the diffusion coefficients for

different temperatures, using values of mean-squared displacements of atoms

,x2. 5, x2. �(d/2)2. Let us estimate the error of diffusivity definition. Construct the

dependence graph ln D on 1/T. Approximate the written values by the linear relation.

Calculate the activation energy of diffusion in the relative units (Q/R) by the slope of the

dependence line (taking into account scales along axis). Compare the obtained value with

the tabular value noted in the window “Memo”.

2. For chosen substance and corresponding parameters (Q,d), calculate the diffusion coeffi-

cient at different temperatures, using the distribution of particles along the x-axis

figured by the histogram (printed values of the relative concentration C/C0). Enter the

searching boundaries of the diffusivity values into the windows Edit with inscriptionsD1 and D2 so that the expression (k1-erfc) in the shown procedure changed the sign in

these limits, for example, 102132 10211, either 102142 10212 or 10215�10213 m2/s.

Press the button with inscription Dif b. If the value calculated by the computer does

not coincide with one of inputted limits, you have obtained the required value according

to Eq. (3.8).

2:ik:=4; //function erfc is determined below;3:ik:=1; end;

c:= conce[ik]; x:=(ik+0.5)*nz*a/2;

k1:=2*c; k3:=x/2/sqrt(t);

d3:=(d1+d2)/2;

if d1<d2 then dd:=d1/100 else dd:=d2/100;

while d2-d1>dd do

begin

erf:=0; e:=k3/sqrt(d3); de:=e/100;

for i:=1 to 100 do erf:=erf+exp(-sqr(de*i))*de;

erf:=erf*2/sqrt(pi); erfc:=1-erf;

if (k1-erfc)>0 then d1:=d3 else d2:=d3;

d3:=(d1+d2)/2;

end ;

image1.Canvas.TextOut(500,290,' D3 ');. . . . . . . .end ;


9.2 Stefan’s Problem of Ice Growth

The analytical solution of this problem is considered in Section 3.3.2. It is unwieldy

enough and demands the same numerical analysis of the transcendental equations.

The heat conduction equation (3.38) in finite differences lays in the basis of the

program described below for the numerical solution of the problem of thermal

conductivity with the certain boundary conditions (e.g., T[1]5 210; T[im] 50; imis the maximum number in the list of points, in which the grid function is set).

Program Ice_Growing; {Pascal}uses graph,crt;

const L=334400*900; {heat of fusion, J/kg*900 kg/m^3}

al=1.165e-6; {k/(pcv) m*m/s 0.0000076;}k=2.21; {J/msK 0.14304;}xrange =3600*24; {one day for the x-axis} im=40;

type mas1=array[1..100] of real;

var T,B,x,time1,xfr,h1 :mas1;

i,j,k1,i1,cou,coumax,maxx,maxy,gd,gm:integer;

dx,dt,v,xf,time:real; f_time,f_x:text;

BEGIN ClrScr; {initial conditions}T[1]:=-10; T[im]:=0; j:=1; k1:=1; cou:=1;

for i:=2 to im-1 do T[i]:=-0.5; {the initial temperature}

xf:=0.001; {the initial thick of ice} dx:=(xf/(im-1));

dt:=0.5*(dx*dx)/al;

time:=0; B[1]:=-10;

while time<3600*24 do

begin

for i:=2 to im-1 do {solution of the parabolic equation}

begin {the new temperatures}B[i]:=T[i]+al*dt*(T[i+1]-2*T[i]+T[i-1])/(dx*dx)

+v*dt*(T[i+1]-T[i-1])/(2*dx)*i/im;

x[i]:=dx*i;

end ;

v:=(k/L)*(T[im]-T[im-1])/dx; {new rate of crystallization}

xf:=(xf+v*dt); {the new coordinate of the crystallization front}

time:=time+dt; dx:=xf/(im-1); {new distance between nodes}dt:=0.5*dx*dx/al;

for i:=2 to im-1 do T[i]:=B[i]; {rename}

if j=k1 then begin {writing of data for xfr(time) }time1[cou]:=time; xfr[cou]:=xf;

writeln('time= ',time1[cou],' xfr= ',xfr[cou]);

coumax:=cou; cou:=cou+1; k1:=k1+10000;

end; j:=j+1;

end; readkey;

for i:=2 to im do {writing data for T(x)}writeln('i=',i,' x= ',x[i],' T= ',T[i]);

readkey; gd:= 0; gm:= 0;


Only thermal conductivity through the solid phase (ice) is considered in the

equation of heat balance on the interface (3.39), and the melting heat is substituted

in counting per unit of volume—L5 Lρ � ρ. In the program, κ is the heat conductiv-

ity; al is the thermal diffusivity; im is the number of net nodes; dt is the timestep;xrange is the time of the crystallization process (24 h) and the maximal mark on

the x-axis at drawing); coumax is the number of records of the interface positions,

which are used in graph building.

According to calculations, the ice layer of approximately 10 cm is forming dur-

ing 24 h at the temperature at the ice surface 210�C. Actually, the growth rate of

ice is less at the air temperature T5210�C, as the surface temperature of ice is

higher. The exact solution requires evaluating of heat emission from the surface

ice�air. The heat emission depends essentially on the wind velocity.

As it follows from the calculated dependence xfr(t), the growth rate of ice

wanes faster at magnification of its thickness. It is related to the fact that the mag-

nitude of the temperature gradient, which determined the heat current, is decreasing

with the thickness increase.


1. Elaborate the procedure for imaging of reservoir with the variable ice layer on the water

surface.

2. Remake the program for C11 or Delphi. Ensure plotting of graphs of the dependencesxfr(t) and T(x) by means of the editor “Chart”, if you use the programming environ-

ment with Builder (for interface).

3. Try to solve the problem numerically, taking into account the heat emission from the sur-

face of ice with air in absence of wind.

9.3 Growth of a Spherical Crystals from a Binary Melt

The analytical solution of this problem was considered in Section 3.3.5. It was

obtained for the simplest case of the concentration constancy at the interface.

Actually, this concentration is not stationary. The surface concentration varies during

crystal growth so that the surface supersaturation becomes less σs5 (Cs2Ce)/Ce,

and the growth rate decreases V5 dρ/dt5βσs (β is the kinetic coefficient; Ce is the

equilibrium concentration for the certain temperature). Thus, transition from the

InitGraph(gd,gm,'');

maxx := getmaxx; maxy := getmaxy-40;

. . . . . . . . . . . {Drawing of the coordinate axes}MoveTo(trunc((time1[1]/xrange)*(maxx-30))+30,

maxy+20-trunc((xfr[1]/xf)*(maxy)));

for cou:=2 to coumax do {Drawing of graph}lineto(trunc((time1[cou]/xrange)*(maxx-30))+30,

maxy+20-trunc((xfr[cou]/xf)*(maxy)));

readln; CloseGraph;END.


kinetic mode of growth (Cs�CN, CN is the initial solution concentration in the

large bath) to the diffusion mode (Cs!Ce) takes place with time (see Section 7.5.2).

The basic problem of the numerical solution is an accomplishment of the bound-

ary condition of mass balance at the interface (3.54) for determination of values of

the surface concentrations. The timestep does not enter into this condition, so that a

divergence of computations takes place at inexactly setting of the initial surface

concentration. If the surface concentration jumps out of limits 0 to 1, the numerical

solution will become incorrect at once. To prevent dispersion, the special operator,

which limits changes of the surface concentrations at the beginning of simulation,

is applied in the program (C# programming language):

for (inti3 = 1; i3 <= 10; i3++){double delta = c_new[1] -c[1]; if(delta < 0) delta *= -1;

if (delta < filtr) break;else c_new[1] = 0.75 * c[1] + 0.25 * c_new[1];

}

A quality of the solution is checked by calculation of the integral mass balance

concerning of one of components: the amount s, which has gone out of melt, must

be equal to the amount that has entered into the crystal:

double S = 0; // mass S goes out from liquid, it is the sum by spherical layers // Vr1 is the mass that has entered to the crystal} for (int j = 1; j <= jmax - 1; j++) S = S + 4*3.1415*r[j]*r[j]*(r[j+1] - r[j])*(c01-(c[j]+c[j+1])*0.5);

double Vr1 = 4 * (ck - c01) * 3.1415 * r[1] * r[1] * r[1] /3;

Concentrations in the net nodes r[j] along coordinate r are calculated according

to Eq. (3.83) continually on each timestep (the sweep method). It was accepted that

the distant steps (the intervals on the distance between net nodes) are not constant—

they are incremented in the arithmetical progression according to Eq. (3.82). The

using System;using System.Collections.Generic;using System.Linq;using System.Text;

namespaceSphera // solution of the equation ρρρτ ∂

∂=∂∂=

∂∂ 2

2

2at special boundary conditions

{ using System;internalclass Program{

private const double D = 3E-05; //{cm^2/s}private const double c01 = 0.9;private const double ce = 0.8;private const int ck = 1;private const double bk = 0.1 * D/3e-8 *1e-3;private const double filtr = 0.0001;//bk is the kinetics coefficientprivate const int jmax = 16;private const double rb = 0.2;private const int tmax = 30000;

ΧΧΧ


private static void Main(string[] args){ double[] r = newdouble[0x11];

double[] c = newdouble[0x11];double[] c_new = newdouble[0x11];Console.WriteLine("Beginning");r[1] = 0.002;double t = 0; c[1] = 0.81; c[2] = 0.83; c[3] = 0.86;int nh2 = (jmax -1) * jmax; double a = 0.0002;double dr = (rb-r[1])*2 /nh2;for (int j = 4; j <= jmax; j++) c[j] = c01;

for (int j = 2; j <= jmax; j++) r[j] = r[j-1] + dr*(j-1); double v = bk * (c[1] -ce) / ce;double dt = a * dr * dr / D;

for (int k = 1; k <= 3; k++)for (int i1 = 1; i1 <= tmax; i1++){ //{main cycle in time}

if (i1 == 6000) a=a*2; if (i1 == 2000) a=a*2;if (i1 == 4000) a=a*2;double br = (rb -r[1]) * 2 / nh2; //{dr=(R_kuv-r[1])/nh-1;}dt = a * br * br / D;

for (int j = 2; j <= jmax - 1; j++){ //coordinates of grid nodesr[j] = r[j-1] + br*(j-1);double brj = br * j + br * (j -1); //(r[j+1]-r[j-1]);double Dcdr = (c[j + 1] - c[j - 1])/brj; //the first derivativedouble Dcdr2 =(c[j+1]-c[j])/(br*j); //the first derivative from the right

double Dcdr1=(c[j]-c[j-1])/(br*(j-1)); //the first derivative from the leftdouble D2cd2r = (Dcdr2 - Dcdr1) * 2 / brj; //the second derivativedouble drj =v*dt -v*dt*(j -1)*j / nh2; //{Change of r(j) in dt}c_new[j] = c[j] + dt*D*(D2cd2r + 2 * Dcdr/r[j]) + Dcdr*drj;} //new concentrations

c_new[1] = c[2] - v * (ck - c[1]) * br / D;

for (int i3 = 1; i3 <= 10; i3++){ double delta = c_new[1] - c[1]; if(delta <0) delta *= -1;

if (delta < filtr) break;else c_new[1] = 0.75 * c[1] + 0.25 * c_new[1];

}v =bk* (c_new[1] - ce)/ce; //{velocity of crystaL growth}r[1] = r[1] + v * dt;t = t + dt;

for (int j = 1; j <= jmax - 1; j++)c[j] = c_new[j]; c[jmax] = c_new[jmax -1];

if (i1 == 600) //rcr(t), c[1](t)Console.WriteLine("t = {0}, r_cr = {1}, c1 = {2}", t, r[1], c[1]);

if (i1 % 2000 == 0) //rcr(t), c[1](t)Console.WriteLine("t = {0}, r_cr = {1}, c1 = {2}", t, r[1], c[1]);

} // end on t Console.WriteLine(" ");double S = 0;

//balance: S is the mass that has leaved liquid, Vr1 is added to the crystalfor (int j = 1; j <= jmax - 1; j++)

S = S + 4*3.1415*r[j]*r[j]*(r[j+1]-r[j])* (c01-(c[j]+c[j+1])*0.5);double Vr1 = 4 * (ck -c01) * 3.1415 * r[1] * r[1] * r[1] /3;

Console.WriteLine("r1={0}, t={1}, S={2}", r[1], t, S);Console.WriteLine(" R={0}, Vr1={1}", rb, Vr1);Console.WriteLine(" ");for (int j = 1; j <= jmax; j++)//the concentration distributionConsole.WriteLine("j = {0}, r1 = {1}, c[j] = {2}", j,r[j],c[j]);Console.WriteLine("t= {0} ", t);Console.ReadKey();

}}

}


program listing in language C# (Console Application of C Sharp in Microsoft

Visual Studio) is given below.

According to Eq. (3.82), the nonuniform net is constructed from the crystal

“surface” r[1]50.002; (cm) to the edge of the bath rb50.2; (cm). The net is

reconstructed during crystal growth. Shifts of all nodes are spotted on each time-

step by the relation: drj 5 v*dt-v*dt*(j-1)*j/nh2; where v is the interface

velocity; corrections to the concentrations (in the equation for c_new[j]) are cal-

culated according to them. After calculating the surface concentration (the sepa-

rate equation for c_new[1]), it is compared to the previous value. Mixing of new

and old values is fulfilled, if a change of the value c_new[1] is more than the

magnitude of filtr.Outcomes of modeling in the form of two tables (as they are printed by the pro-

gram) are shown below. The first of them maps dependences of the size of the

spherical crystal and the surface concentration on the time. The second table maps

the concentration distribution (the dependence on coordinates of points in the liquid

part of bath). Between tables, there are two lines with the additional information;

values S and Vr1, which are printed for the terminating instant. Unlike the solution

described in Section 9.3, this model takes into account the surface kinetics by the

normal mechanism of growth (Section 7.3.4); at considered sizes, growth runs prac-

tically in the diffusion mode: the surface concentration is close to the equilibrium

value.

Dependences of the Crystal Size and Surface Concentration on the Time

t5 1.0997821875E-02 r_cr5 1.0374068505E-03 c15 8.0250906145E-01

t5 3.6664277988E-02 r_cr5 1.1129070205E-03 c15 8.0290934879E-01

t5 1.0986092785E-01 r_cr5 1.3721413490E-03 c15 8.0371090942E-01

t5 2.5560240158E-01 r_cr5 2.0079713415E-03 c15 8.0393818832E-01

t5 5.4392160675E-01 r_cr5 3.4232267540E-03 c15 8.0395849634E-01

t5 8.2813421787E-01 r_cr5 4.8053858563E-03 c15 8.0376886332E-01

t5 1.1084872027E1 00 r_cr5 6.0890878606E-03 c15 8.0355029479E-01

t5 1.3852979290E1 00 r_cr5 7.2728538749E-03 c15 8.0328428924E-01

t5 1.6588575711E1 00 r_cr5 8.3685182650E-03 c15 8.0316569971E-01

t5 1.9294145194E1 00 r_cr5 9.3890713484E-03 c15 8.0295689334E-01

t5 2.1971835173E1 00 r_cr5 1.0345321101E-02 c15 8.0279794504E-01

t5 2.4623439338E1 00 r_cr5 1.1247131790E-02 c15 8.0260157429E-01

t5 2.7250493576E1 00 r_cr5 1.2101798042E-02 c15 8.0240692040E-01

t5 2.9854353556E1 00 r_cr5 1.2914474355E-02 c15 8.0245895999E-01

t5 3.2436204918E1 00 r_cr5 1.3690716014E-02 c15 8.0233866595E-01

t5 3.4997072748E1 00 r_cr5 1.4435043739E-02 c15 8.0224048362E-01

r15 1.4435043739E-02 R5 2.0000000000E-01

s5 1.6148378771E-06 Vr15 1.5599198600E-06

The Concentration Distribution

[1]5 rj5 1.4435043739E-02 c[j]5 8.0224048362E-01

[2]5 rj5 1.5981064175E-02 c[j]5 8.3139905176E-01

[3]5 rj5 1.9073819399E-02 c[j]5 8.6670714649E-01

[4]5 rj5 2.3712952235E-02 c[j]5 8.8932060110E-01


[5]5 rj5 2.9898462683E-02 c[j]5 8.9788922078E-01

[6]5 rj5 3.7630350743E-02 c[j]5 8.9975562051E-01

[7]5 rj5 4.6908616415E-02 c[j]5 8.9998341264E-01

[8]5 rj5 5.7733259699E-02 c[j]5 8.9999931083E-01

[9]5 rj5 7.0104280594E-02 c[j]5 8.9999998122E-01

[10]5 rj5 8.4021679102E-02 c[j]5 9.0000000000E-01

[11]5 rj5 9.9485455222E-02 c[j]5 9.0000000000E-01

[12]5 rj5 1.1649560895E-01 c[j]5 9.0000000000E-01

[13]5 rj5 1.3505214030E-01 c[j]5 9.0000000000E-01

t5 3.4997072748E1 00 (in seconds)


1. Obtain solutions and construct graphs of dependences r1(t) and c(r[j]). Explain rea-

sons of noise in the concentration values. Why does the surface concentration decrease

with time?

2. Elaborate a variant of this program using Windows Forms Application in Microsoft

Visual Studio. Construct the graphs of the dependences r1t) and c(r[j]) by means of

the Chart editor.

9.4 Crystallization After Laser Processing ofa Metal Surface

The following program allows simulating the solidification of the hemisphere of

melt, which arose after heating of the surface site of the sample by the laser ray.

The heat conduction equation for the area of solid phase, which borders with the

melting zone, is considered in the case of spherical symmetry, @T@t 5αT @2T@r2 1 2

r@T@r

h i:

The solution may be found by the method of finite differences, in which

the derivatives are expressed through values of the temperature in the net nodes

(see expressions for new temperatures B[i] in the program). The repeated expres-

sion for B[i] in the program considers corrections dependent on shifts of nodes at

the net reconstruction in each timestep in connection with moving of the crystalli-

zation front.

The melting point is set at the melt-solid boundary and the room temperature is

set at the exterior (conditionally) hemispherical boundary in the solid phase.

Growth rate is found from the condition (3.39) of heat balance at the interface (see

expression for v in the program: v5xt*(T[2]-T[1])/(dR*L);). The following key

parameters are used in the program: g is the density; L is the melting heat in count-

ing per the unit of volume; xt is the heat conductivity; c is the specific heat; at is

the thermal diffusivity in the solid phase. Their values are the table data for alumi-

num (in SI units).


Program Laser; {Pascal} uses Crt, Graph; Const im=16; {number of nodes} var Gd, Gm, k,i,m,j:integer;

dt,dTr,c,Tpl,at,xt,g,L,time,dR,v:real; T,B,R,RR,tt:array[1..100] of real; Begin

time:=0; R[1]:=0.001; {the radius of the liquid zone} R[im]:=0.003; {the radius of surrounded solid} g:=2600;{kg/m**3} {the density of the matter} L:=430000*2600; {the heat of fusion, J/m**3} xt:=207; {the heat conductivity of Al, W/(m⋅K)} c:=1179; {the heat capacity J/(kg⋅K)}

at:=xt/(g*c); {the thermal diffusivity, m^^2/s} dR:=(R[im]-R[1])/(im-1); dt:=0.005*dR*dR/at;

Tpl:=660; {the melting temperature, oC} T[2]:=659.8; T[3]:=500; T[4]:=300; T[1]:=Tpl;

For m:=5 to im do T[m]:=20; dR:=(R[im]-R[1])/(im-1); For i:=2 to (im-1) do

R[i]:=R[i-1]+dR; B[1]:=Tpl;

v:=xt*(T[2]-T[1])/(dR*L); {the initial velocity from the heat balance} GD:=VGA; GM:=VGAHi; InitGraph(GD,GM,''); for k:=1 to 10 do beginFor j:=1 to 500 do begin For i:=2 to im-1 do begin dR:=(R[im]-R[1])/(im-1); R[i]:= R[im]-dR*(im-i);

dTr:=(T[i+1]-T[i-1])/(2*dR); {the first derivative} B[i]:=T[i]+dt*at*(((T[i+1]+T[i-1]-2*T[i])/

(dR*dR))+dTr*2/R[i]); {there is the second derivative} B[i]:=B[i]+v*dt*(im-i)/(im-1)*dTr; end;

v:=xt*(T[2]-T[1])/(dR*L); {the velocity of crystallization} R[1]:=R[1]+v*dt; time:=time+dt; delay(1000);end ; SetFillStyle(SolidFill,LightBlue); bar(0,200,420,420); {} SetFillStyle(SolidFill,Red); FillEllipse(200,200,round(R[1]*100000), round(R[1]*100000));

{fill circle drawing } SetColor(White); For i:=1 to 8 do if (i=1) or (i mod 2 =0) then begin Circle(200,200,round (R[i]*100000)); str(round(T[i]),ss); outtextxy(192,round (R[i]*100000)+204,ss); end;


SetFillStyle(SolidFill,Black); bar(0,0,450,200); {} SetFillStyle(SolidFill,WHITE); bar(0,200,420,100); RR[k]:=R[1]; tt[k]:=time; if R[1]<0.00001 then Break ; end ; CloseGraph; ClrScr; for k:=1 to 10 do writeln('R[1]=',RR[k],' time =',tt[k]); for i:=1 to 10 do writeln('i=',i,' T=',T[i]);Readkey; End.

Figure 9.2 shows the microbath with liquid (it is black) and some isotherms in

the solid for two instant of time: at the beginning and at the end of solidification

process, obtained according to outcomes of modeling. Isotherms are printed in the

cycle: For i: 5 1 to 8 do. Temperatures near isotherms are written in �C.It is quite possible to write the code of this program for any programming envir-

onments. There is a listing of the C11 program below that provides the most sim-

ple interface—with one Button for start, one Memo window for writing of results,

and with painting directly in the form:

Figure 9.2 Images of the microbath during solidification and isotherms in the solid phase,

(A) initial temperature field; (B) final temperature field.

#include <vcl.h>

#pragma hdrstop

#include "Unit1.h"

#pragma package(smart_init)

#pragma resource "*.dfm"

TForm1 *Form1;

//----------------------------------------------------

__fastcall TForm1::TForm1(TComponent*Owner):TForm(Owner)

{ }

//----------------------------------------------------

const im = 20; int Gd, Gm, k,i,m,j,kmax;

float dt,dTr,c,Tpl,at,xt,g,L,time1,dR,v;

float T[100]; float B[100]; float R[100];

float RR[100]; float tt[100]; AnsiString ss;


void __fastcall TForm1::Button1Click(TObject *Sender)

{ time=0;

R[1]=0.001; RR[1]=R[1]; //the initial radius of the liquid bath

R[im]=0.003; //the radius of surrounded solid

g=2600; //the closeness of matter in kg/m**3

L=430000*2600; //heat of fusion, J/m**3

xt=207; //the heat conductivity of Al, W/m/K

c=1179; //the heat capacity J/kg/grad

at=xt/(g*c); //the thermal diffusivity, m^^2/s

dR=(R[im]-R[1])/(im-1); // the step in distant

dt=0.005*dR*dR/at; // the timestep

Tpl=660; T[1]=Tpl; B[1]=Tpl; //temperature of fusion T[2]=659.8; T[3]=620; T[4]=540; T[5]=440;

for (m=6; m<=im; m++) T[m]=20;

for (i=2; i<=im-1; i++) R[i]=R[i-1]+dR;

v=xt*(T[2]-T[1])/(dR*L); //the initial velocity kmax=1; for (k=1; k<=20; k++)

{ for(j=1; j<=1000; j++)

{

for (i=2; i<=im-1; i++) // in the net nodes { dR=(R[im]-R[1])/(im-1);

R[i]= R[im]-dR*(im-i);

dTr=(T[i+1]-T[i-1])/(2*dR); //the first derivative B[i] = T[i] + dt*at*(((T[i+1]+T[i-1] - 2*T[i])/

(dR*dR)) + dTr*2/R[i]);

B[i]=B[i]+v*dt*(im-i)/(im-1)*dTr; //new temperatures }

for (i=2; i<=im; i++) T[i]=B[i]; //new temperatures v=xt*(T[2]-T[1])/(dR*L); //new velocity of the interface R[1] += v*dt; time += dt; //new size of liquid and new time

if (j==1) //drawing of the liquid bath and isotherms { int RKK= R[1]*100000;

Form1->Canvas->Brush->Color = clGray; //background Form1->Canvas->FillRect(Rect(0,200,450,450));

for (i=8; i>=1; i--)

{ int RK= R[i]*100000; //painting of the isotherms Form1->Canvas->RoundRect(200-RK,200-RK,200+RK,

200+RK,2*RK,2*RK);

if(i%2==0)

{ int TTT=T[i]; //values of the temperature for some isotherms ss=IntToStr(abs(TTT));

Form1->Canvas->TextOutA(190,RK+196,ss); }

} Form1->Canvas->Brush->Color = clRed;

Form1->Canvas->Ellipse(200-RKK, 200-RKK,200+RKK,

200+RKK); //image of liquid Form1->Canvas->Brush->Color = clLtGray;

Form1->Canvas->FillRect(Rect(0,0,450,200));

if (R[1]<=0.00001) break;



1. Substitute parameters for another substance in the program, for example, for iron or cop-

per (in the SI system).

2. Obtain solutions and construct graphs R1(t) and T(i). Compare times, for which the

microbath solidification occurs, in the case of aluminum and the substance for

consideration.

9.5 Directional Solidification

The theoretical analysis of the directional solidification is presented in

Section 7.6.1. The analytical solution has been obtained for the quasistationary con-

centration field that moves with the constant velocity. It is equal to the velocity of

pulling of the sample relatively the heating furnace. Actually, this problem is non-

stationary. The concentration field can come closer to the quasistationary field at

the certain time only in the case of the certain velocity of pulling. For this, the

velocity of the crystallization front will be equal to the velocity of propagation of

the diffusion field (diffusion zone). The exact solution of such problem is possible

only based on the diffusion equation dependent on time and, consequently, the

numerical solution is necessary. The program considered below ensures such a

solution. Extension of this program for two-dimensional and three-dimensional

cases (Section 7.6.3) allows describing evolution of the interface shapes—the cellu-

lar structure formation.

The program for solution of the one-dimensional problem of the concentration

field determination during the directional solidification contains the basic algorithm

for calculations of new concentrations continually on time. The program for the

solution of the one-dimensional problem of determination of concentration fields

during the directional solidification contains the basic algorithm for calculations of

new concentrations in every timestep. There are operators for determination of the

second derivatives, D2C, in all nodes of the net, new temperature at the crystalliza-

tion front, Tf, new concentration, ck, in the solid phase near the crystallization

RR[k]=R[1]; tt[k]=time; //data for R[k(t)] in the cycle for k if (R[1]<=0.00001) break; kmax +=1;

} //end in k

for (k=1; k < kmax; k++){ //data for R[k(t)] in Memo1 Memo1->Lines->Add(FloatToStrF(RR[k],ffGeneral,10,4));

Memo1->Lines->Add(FloatToStrF(tt[k],ffGeneral,10,4));

Memo1->Lines->Add(FloatToStrF(111,ffGeneral,10, 6));} }

for (i=1; i<=6; i++) // values of the temperature in six nodes near liquid Memo1->Lines->Add(FloatToStrF(T[i],ffGeneral, 10, 6));

Memo1->Lines->Add(FloatToStrF(111111,ffGeneral,10,6));

} /*end if j=1*/ Sleep(400);

} /* end in j=1000 */ if (R[1]<=0.00001) break;


front, and new concentration, cb[1], in the melt near the interface. The instanta-

neous velocity of growth is determined as a product of the kinetic coefficient and

surface relative supersaturation (v5βsσs, see Section 7.3.4), vf:5betha*(cb[1]-ce)/ce/bd, where cb[1] is the surface concentration (in liquid). The operator of cal-

culation of cb[1] and the small block for restriction of increments of the surface

concentration are placed further. For better accuracy, a calculation of cb[1] is made

twice—the second time after more precise definition of the growth velocity vf.New positions of the crystallization front, xf, and the orb radius, rb, of calcu-

lated area are then determined. Procedure Drawing for calculating and displaying of

the concentration distribution is periodically called. Procedure Theoretic for calcu-

lating of the concentration distribution according to Eq. (7.49) is called in the end

of program run, as well as in previous programs, intervals between nodes are

increased in the arithmetical progression; the net is reconstructed on each timestep,

and the corresponding member is present in the equation for determination of con-

centration (vf*dt*(c[i11]-c[i])/(x[i11]-x[i])). The main cs-file for pro-

gramming with Windows Form Application of the Visual Studio software is given

below.

using System;using System.Collections.Generic;using System.ComponentModel;using System.Data;using System.Drawing;using System.Linq;using System.Text;using System.Windows.Forms;

namespace WindowsFormsApplication3{ public partial class Form1 : Form

{const double scale = 2.98, imax = 17 /*the number of nodes*/,

dx = 0.04/*dx/bd*/, betha = 50.0 /*cm/s*/, D = 3e-5 /*cm^2/s*/, GT =/*232*/60 /*K/cm*/, nu = 2.5 /*V/Vcr*/, Pe = 0.3;

double[] C = new double[(int)imax + 2]; //Concentrations of A-componentdouble[] cb = new double[(int)imax + 2]; //New concentrations of

main componentdouble[] x = new double[(int)imax + 2];double V, dt, ko, ck, ce, bd, d2c, xf, vf, dc, time, ml,

kr, tf, vf0, y0, nh2, rb, T0, Te, dxx, cx, D_bez, cbb; // vf0 -velocity of moving; D_bez=D/bd/bd; T0=T_liqudus(C0); Te(C[x=x_front]

int i, j, k; string s, st;public void line(PaintEventArgs e, int x1, int y1, int x2, int y2){ e.Graphics.DrawLine(new Pen(Color.Black), x1, y1, x2, y2);}

public void line(PaintEventArgs e, double x1, double y1, double x2,doubley2){e.Graphics.DrawLine(new Pen(Color.Black), (int)x1, (int)y1,

(int)x2,int)y2);} //-----------------------------------------------

public void line(PaintEventArgs e, double x1, double y1, double x2, double y2, Color Col){e.Graphics.DrawLine(new Pen(Col), (int)x1, (int)y1,

(int)x2,(int)y2);} //-----------------------------------------


{st = d.ToString("0.00000"); // format of printing} //----------------------------------------public void Drawing(PaintEventArgs e){ int i = 1; double x1 = 0, x2 = 0, y1 = 0, y2 = 0;

x1 = 40 + Math.Round(xf * scale); y1 = 330;while (i <= imax)){x2 =40+Math.Round(x[i]*scale); y2=330- Math.Round((1-C[i])*300* 10);

line(e, x1, y1, x2, y2);x1 = x2; y1 = y2;i++; }

} //---------------------------------------public void Drawing(PaintEventArgs e, Color Col)

//of the concentration distribution{ int i = 1; double x1 = 0, x2 = 0, y1 = 0, y2 = 0;

x1 = 40 + Math.Round(xf * scale); y1 = 330;while (/*x1 < 640 && x2 < 640 && y1 < 480 && y2 < 480 && */(i < imax)){

x2=40 + Math.Round(x[i]*scale); y2 =330-Math.Round((1-C[i])*300* 10);line(e, x1, y1, x2, y2, Col); x1 = x2; y1 = y2;

public void str(double d, string s)

i++; } //-------------------------------------------public void Theoretic(){ for (int i = 1; i <= imax; i++)

C[i] =1 - 0.002*(1 + (1-ko)/ko*Math.Exp(-vf0*bd*(x[i] - xf)/D));} //--------------------------------------------public void outtextxy(PaintEventArgs c, int x, int y, string s){c.Graphics.DrawString(s, new Font("Arial", 10), new SolidBrush(Color.Black), (float)x, (float)y);} //---------------------------------------------public void outtextxy(PaintEventArgs c, double x, double y, string s){c.Graphics.DrawString(s, new Font("Arial", 10), new SolidBrush(Color.Black), (float)x, (float)y);} //-----------------------------------------------public Form1(){InitializeComponent();} //-----------------------------------------------private void pictureBox1_Click(object sender, EventArgs e){} //----------------------------------------------private void pictureBox1_Paint(object sender, PaintEventArgs e)

//MAIN{ ko = 0.1; time = 0;

line(e, 40, 330, 620, 330); line(e, 40, 330, 40, 30);line(e, 35, 180, 45, 180); line(e, 35, 30, 45, 30);line(e, 140, 335, 140, 330);line(e, 240, 335, 240, 330);line(e, 340, 335, 340, 330);line(e, 440, 335, 440, 330);line(e, 540, 335, 540, 330);outtextxy(e, 25, 10, "C_B"); outtextxy(e, 15, 33, "0.1");outtextxy(e, 10, 183, "0.05"); outtextxy(e, 138, 340, "2");outtextxy(e, 238, 340, "4"); outtextxy(e, 438, 340, "8");outtextxy(e, 338, 340, "6"); outtextxy(e, 538, 340, "10 bd");outtextxy(e, 620, 339, "x ");y0 = 0.0; xf = 0;nh2 = (imax - 2) * (imax - 1);rb = y0 + dx + nh2 * 0.5 * dx; //Calculating area in units bdT0 = 200.4; ml = 0.003;vf0 = nu * GT * D * ml / 0.02; //The velocity of pulling


bd = D / vf0; //The diffusion lengthD_bez = D / (bd * bd);str(betha, st); outtextxy(e, 60, 40, "betha(cm/s)=" + st);str(vf0, st); outtextxy(e, 60, 60, "vf0(cm/s)=" + st);str(bd, st); outtextxy(e, 60, 80, "bd(cm)=" + st);str(nu, st); outtextxy(e, 60, 120, "nu=" + st);str(GT, st); outtextxy(e, 60, 140, "GT(K/s)=" + st); //the temperature gradient

vf = vf0 / bd;dt = 0.2 * dx * dx / D_bez; time = 0;for (i = 2; i <= imax; i++)

C[i] = 0.998; C[1] = 0.998; //The initial concentration distribution

xf = 0; x[1] = xf; x[2] = xf + dx; // rb=dx(imax-1)*imax/2for (i = 3; i <= imax; i++)

x[i] = xf + dx + (rb - xf - dx) * (i - 2) * (i - 1) / nh2; //Theoretic; /*well initial distribution*/Drawing(e);for (k = 1; k <= 12; k++){ for (j = 1; j <= 30000; j++)

{ x[1] = xf; x[2] = xf + dx;for (int i = 3; i <= imax; i++)x[i] = xf + dx + (rb - xf - dx) * (i - 2) * (i - 1) / nh2;

for (int i = 2; i <= imax - 1; i++){dxx = x[i] - x[i - 1]; //The distant between nodescx = C[i] + (C[i + 1] - C[i]) / (x[i + 1] - x[i]) * dxx;

d2c = (cx - 2 * C[i] + C[i - 1]) / dxx / dxx; //The second deriviative

dc= D_bez*d2c*dt + vf*dt*C[i + 1] - C[i])/(x[i+1] - x[i]); cb[i] = C[i] + dc; //New concentrations}tf = T0 - GT*vf0 * time + GT*xf*bd; //The front temperaturetime = time + dt;ck = 1 - ko*(1 - C[1]); //The concentration in crystalce = 0.998 - (T0 - tf) * ml; //Equilibrium concentration (tf)vf = betha * (C[1] - ce) / ce / bd;double cb1 = cb[2] - vf * dx * (ck - C[1]) / D_bez;//New concentration near the interfacefor (int i = 1; i <= 20; i++)

{if (Math.Abs(cb1 - C[1]) < 2e-5) break; //Limitation of a changecb1 = 0.25 * cb1 + 0.75 * C[1]; }vf = betha * (cb1 - ce) / ce / bd; //more exact velocitycb[1] = cb[2] - vf * dx * (ck - cb1)/D_bez; //more exact new interface concentration for (int i = 1; i < 10; i++)

{if (Math.Abs(cb[1] - cb1) < 6e-6) break; //Limitation of a change

cb[1] = 0.25 * cb[1] + 0.75 * cb1; }vf = betha * (cb[1] - ce) / ce / bd; //more exact velocityxf = xf + vf * dt; /*New interface position*/ rb=rb +

vf*dt; for (int i = 1; i <= imax - 1; i++) C[i] = cb[i];

}if ((k == 1) || (k % 2 == 0))

{ //Information on time (min) and surface supersatirationstr(time / 60, st);Drawing(e);

outtextxy(e, 20 + Math.Round(xf *scale), 240-9*k, st+"min"); //"t="


Figure 9.3 shows the consecutive distributions of impurity concentration (of the

second component). This variant of the program provides mapping of all used para-

meters on the free part of the graph. Apparently, from the graph, the problem of

the directional solidification is essentially nonstationary, process of accumulation

of concentration (increase of its values) before the front can occur at its moving up

to several diffusional lengths (bd). Therefore, the cellular structure formation

begins on the nonstationary part. It leads to difficulties of comparison of the

approximate theory with the experiment.

Figure 9.3 Changes of the concentration distributions during the directional solidification,

the blue line at t5 70 s shows equilibrium concentrations Ce(T(x)); the red line gives the

theoretical distribution.

str((C[1] - ce) / ce, st);e.Graphics.FillRectangle(new SolidBrush(Color.Wheat), 59, 99, 230,

15);outtextxy(e, 60, 100, "(ce-c1)/ce=" + st);

}

if (k == 10) // Drawing of equilibrium concentrations ce(x){ for (int i = 1; i <= imax - 1; i++)

{cb[i] = C[i]; C[i] = C[1] + ml * GT * (x[i] - x[1]) * bd; }Drawing(e, Color.Blue);

for (inti = 1; i <= imax - 1; i++) C[i] = cb[i];}

}Theoretic(); // Calculation of the theoretical cB(x) distribution

// Drawing of the theoretical cB(x) distribution

Drawing(e, Color.Red); }



1. Obtain solutions to the problem of directional solidification for cases of different gradi-

ents of temperature, the velocity of the sample pulling, and different values of the kinetic

coefficient betha.2. Calculate the equilibrium temperatures in dependence on the coordinate x for final distri-

bution of the concentration—ensure calculating and showing the graph Te(x) together

with the graph C(x).

9.6 Ising’s Model

The theoretical basis for modeling of Ising’s ferromagnetic by the Monte Carlo

method was considered in Section 1.6. Application of the Metropolis technique or

thermostat technique quickly leads the system to the advantageous states from the

entropy-energy point of view (Figure 1.2). Therefore, the average of physical

values can be calculated using these configurations, which were realized by random

walks during modeling. It is clear that it should be a lot of such walks for the

guaranteed approach of the system to the equilibrium state and for improving

statistics.

Outcomes of computer modeling depend on the magnitude of energy of

exchange interaction J and temperature. We consider the two-dimensional ferro-

magnetic, which consists of the ordering domains at high enough values of ratio J/

kT. The ratio (U2U0)/U0, where U is the energy of the realized state and U0 is the

energy of completely ordered configuration, can be considered as the measure of

the structure ordering. If we take into account the influence of the external mag-

netic field in the equations for the transition probabilities (according to which the

new value of chosen degree of freedom will be accepted or rejected), the large

enough magnetization will correspond to the ordered state. Hence, it is possible to

determine the point of order vanishing (the Curie point) by the graph of depen-

dence of averaged magnetization per one spin on the temperature of the thermostat,

contacting to the model.

The basic elements of the program (the Pascal variant) for modeling of two-

dimensional system of spins are given below.

Program Ising; uses graph,crt;

label 0,1,2,3,4,5;

const n=50; {the number of spins n*n}

ever=1.2; {an influence of the external H-field}{b=J/kt}var m10,m:integer; c: Char;

type Tspin=array [1..n,1..n] of integer;

var w,en:array [1..6] of real;b:real;k:integer;

sm:string;

Procedure init (var sp:Tspin);

var x,y,k,r:integer; s:string;


begin

for x:=1 to n do

for y:=1 to n do {Initial spin directions} sp[x,y]:=random(2);

w[1]:=exp(4*b); w[2]:=exp(-4*b);

w[3]:=exp(2*b); w[4]:=exp(-2*b); {Gibbs weights of states} w[5]:=0.5; w[6]:=0.5; {b= J/kT}

en[1]:=-4*b; en[2]:=4*b; en[3]:=-2*b;

en[4]:=2*b; en[5]:=0; en[6]:=0;

end;

procedure showsp (var sp:Tspin; z:integer);

var x,y,i:integer;

begin

for x:=1 to n do for y:=1 to n do

putpixel (x+z,y,sp[x,y]*2); {mapping of spins} end;

procedure cheng (var sp:integer; var ver:real);

var i:real;

begin

if random<ver then begin {to change or not to change} if sp=1 then sp:=0 else sp:=1; end ;

end;

procedure chengm(var sp:Tspin); {analysis of the spin environment} var x,y,i:integer; ver,b:real;

begin

x:=random(n-2)+2; y:=random(n-2)+2;

i:=sp[x,y+1]+sp[x,y-1]+sp[x+1,y]+sp[x-1,y];

if sp[x,y]=0 then i:=4-i; case i of 0: ver:=w[1]/w[2]; 1: ver:=w[3]/w[4]; 2: ver:=1; 3: ver:=w[4]/w[3]; 4: ver:=w[2]/w[1]; end;

if sp[x,y]=1 then ver:=ver/ever

else ver:=ver*ever; cheng (sp[x,y],ver);

if x=2 then sp[n,y]:=sp[x,y] else if x=n-1 then sp[1,y]:=sp[x,y]; {boundary conditions} if y=2 then sp[x,n]:=sp[x,y] else

if y=n-1 then sp[x,1]:=sp[x,y];

end ; var i,Graphdriver,x,y,j,p:integer;

sp:Tspin; nnn,smax,max:string; nn,jm,xx,e,er: real;

xxx,yyy:array[1..5] of integer;

Mas:array[1..2,0..5] of real;

BEGIN randomize; i:=1; {main} Moveto(50,320); LineTo(50,100);

Moveto(50,320); LineTo(630,320);

for k:=1 to 6 do begin b:=0.2+(k-1)*0.5;

{pair energy changing} init (sp); xxx[1]:=50; yyy[1]:=320;

setfillstyle (0,0); setcolor(3); e:=0; J:=0;


for x:=2 to n-1 do {calculations of initial energy and magnetization}for y:=2 to n-1 do begin J:=J+sp[x,y];


if sp[x,y]=0 then i:=4-i;

case i of

0: er:=4*b; 1: er:=2*b; 2: er:=0;

3: er:=-2*b; 4: er:=-4*b; end ;

e:=e+er; end;

m:=0; repeat begin {main cycle}

begin J:=J-sp[x,y]; chengm(sp); if sp[x,y]=1 then J:=J+1 else J:=J-1; end;

if m mod (100)=0 then

begin {calculations of energy and magnetization every 100 steps } e:=0; J:=0;

for x:=2 to n-1 do for y:=2 to n-1 do

begin


if sp[x,y]=0 then i:=4-i;

if sp[x,y+1]=1 then J:=J+1 else J:=J-1;

case i of {counting of energy depending on the environment} 0: er:=4*b; 1: er:=2*b; 2: er:=0;

3: er:=-2*b; 4: er:=-4*b;

end; e:=e+er;

if sp[x,y]=1 then e:=e+0.1 else e:=e-0.1;

end ;

putpixel(50+round(m/100),200-round(e*40/sqr(n)),10);

{plot of the energy dependence on the number of tests} end; end ;

m:=m+1; until m>30000; {the number of steps}

str (b:4:2,sm); {Displaying of b=J/kT} outtextxy (360,92+10*(k-1),'b/kT='+sm);

str(e/sqr(n),:4:2,sm);

outtextxy(400,60+10*(k-1),'E/kT='+sm);Jm:=J/(n*n);

str(jm,sm); {Displaying of magnetization} outtextxy (320,30,'Magnetization='{+sm});

showsp(sp,160); end ; readkey;

End. From Delphi unit.pass:

Bind:= Bind0-(Myk-1)*1; //the induction of the magnet field B

if Bind<1 then Bind:=1; // Myk=1,2,3 . . . .

Em:= Bind/100; // the energy addition dependent on B

T:=T0+(Myk-1)*5*Myk; //a new temperature

b:=IkT0*T0/T; //J/kT in dependence on the temperature. . . . . . . . . . . . . . . . . . . . . . . .

enn:=1/b; en0:=(e0-e)/e0; //degree of orderingif (TRButton.Checked) then enn:=exp(-b); Series2.AddXY(enn,en0); //degree of orderingSeries3.AddXY(enn,Jm); // magnetization on enn


The window for spin mapping is at the left side of the active form (Figure 9.4);

the graph of the energy dependence on the number of steps of trials is placed at the

left and above. The graphs of the temperature dependence of magnetization and the

final relative energy ((E02E)/E0) are on the right side (the radiobutton “Tregim”

should be pressed); E/kT is the average energy per one spin (in units kT), J/kT is

the dimensionless exchange energy of spin interaction.


1. Consider attentively the program and find where the basic algorithm is placed. Determine

is it the thermostat algorithm or the Metropolis algorithm?

2. Explain how many of the Monte Carlo steps are necessary for our system to approach the

equilibrium state at the present value of the ratio J/kT� b.3. Select the value of the magnetic induction of external field, for which determination of

the magnetic transformation temperature will be the most precise.

4. Plot graphs of the dependence of magnetization on the magnetic induction at several

values of J/kT and analyze them. Use the possibilities of the “Chart” (the graph editor of

Delphi environment).

9.7 Adsorption

Physics of adsorption processes was considered in Chapter 5. The program consid-

ered below allows modeling the adsorption process, fulfilling its visualization and

constructing the graphs of isotherm or isobar of the adsorption.

Figure 9.4 The active controlling form of the program “ISING”.


The model for the computer analysis of adsorption is described in Section 5.1.4.

We have a substrate of certain area to which molecules are deposited. The site, in

which the deposition or evaporation of the molecule may happen, is chosen ran-

domly—a node of two-dimensional net with casual coordinates. The algorithm for

testing by the Monte Carlo method consists of trials of evaporation or deposition of

molecules, depending on whether the chosen site is occupied or free. We spot the

pressure P0, at which the dynamic equilibrium on numbers of deposited and evapo-

rated molecules takes place for each active center on the surface. If the pressure is

less than P0, the probability of molecule deposition will be rather small (the time

of deposition τdep of one molecule is large). Therefore, we consider the mean time

of the life of adsorbed molecule on the surface, τev, as the timestep. In every time-

step, the adsorbed molecule (if it exists) evaporates with the probability equal to 1.

If molecule is not present on this place, the deposition with probability τev/τdep is

possible.

If the pressure is more than P0, the time τdep will be rather small. The timestep

was chosen to be equal the mean time, during which one molecule deposits on the

one molecular site. Therefore, the deposition happens with the probability equaled

1; and evaporation with probability τdep/τev. On each timestep, the adsorbed mole-

cule evaporates, if random number (from 0 to 1) is less than this probability.

After a certain number of the Monte Carlo cycles, we find the part of occupied

places (magnitude θ). After a change of pressure, we observe the process and deter-

mine new value θ. Retrying the experiment for different pressures, we find the

adsorption isotherm and ensure its map by means of the editor Chart. The program

provides the modification of pressure or temperature after performance of the cer-

tain number of steps and buildup of graphs of the adsorption isotherm or isobar.

The text of program “Adsorby” (the variant in Pascal) with detailed explanations of

the important operators is given below.

Figure 9.5 shows the Delphi-program interface. The procedure “Show” paints

the adsorbed molecules on the screen (Image.Canvas). Hence, everyone can

observe a running of the adsorption process. Equilibrium between the numbers of

atoms, which have been attached to the substrate, and those which have come off

from it during the certain time, is established after a large enough number of trials.

Program Adsorb; uses crt, Graph;

const Xn=30;Yn=20; v=1e13; m=0.028/6e23; s=4e-20;

var site:array[0..100,0..100] of integer;

dukT,kT,du,dt,dt1,p,p1,rt,pp,pe,sq1:real;

x,y,i,j,k,xyz,xyz1,xyz2,index:integer;

Ni,W,a,b:word;

Q,lp:array[0..10] of real;

procedure Show; var x,y:integer;

begin a:=10; b:=10;

for x:=1 to Xn do

for y:=1 to Yn do begin

W:=site[x,y]+2; setfillstyle(1,W); bar(x*b,y*a,b*x+b,a*y+a);end;

end; BEGIN {main program}


Figure 9.5 The active controlling form of the program “Adsorb”.

ClrScr; Randomize;

dukT:=15; kT:=1.38e-23*300; pp:=1; {presure}sq1:=sqrt(m)*sqrt(6.283*kT);

pe:=v*exp(-dukT)*sq1/s; {the equilibrium pressure}for k:=1 to 6 do begin

pp:=pp*10; dt:=sq1/(pp*s); {the time of condensation}{ p:=v*exp(-dukT)*dt1;}

ni:=0; dt1:=exp(dukt)/v; {the time of evaporation}

p1:=dt/dt1; {probability of condensation if pp>pe else p1=1}

p:=dt1/dt; {probability of evaporation if pp<pe else p=1}for x:=0 to xn do

for y:=0 to yn do site[x,y]:=1; {init values}Gm:=0; Gd:=0; Initgraph (Gd,Gm,''); Show;

for j:=1 to 1000 do begin

x:=random(Xn+1); y:=random(Yn+1); {random site (place)}if pp>=pe then begin {condensation}if site[x,y]=1 then begin site[x,y]:=2;inc(ni);

goto 5;

end;

if (site[x,y]=2) and (p1>random) then {evaporation} begin site[x,y]:=1{site[x,y]-1}; dec(ni); end ;

if pp<pe then begin

if (site[x,y]=1) and (p>=rt) then begin

site[x,y]:=2; inc(Ni); goto 5; end ; {condensation}if site[x,y]=2 then begin dec(Ni);

site[x,y]:=1; {evaporation} end ; end;

end;

5: end; {end for j}


moveto(40,440);{. . . . . {drawing of coordinate axes)for k:=1 to 6 do {Graph}

lineto(40+k*70,440-round (Q[k]*400));

readkey; Closegraph;

for k:=1 to 6 do writeln('Q[k]=',Q[k],'P[k]=',lP[k]);

END.


1. Consider attentively the program and define where the basic algorithm is written. Explain

why probabilities of molecules’ deposition or evaporation are calculated differently in

cases of small and large pressures.

2. Define how many Monte Carlo steps are necessary for the system approach to the equilib-

rium state at present value of the ratio U/kT, where U is the binding energy of molecules

with the substrate.

3. Obtain the adsorption isotherms for different pressures.

4. Obtain the adsorption isobars for different temperatures.

5. Explore how magnitude U/kT influences the temperature of practically full desorption of

gas analyzing the adsorption isobars.

9.8 Determination of the Equilibrium Structure by theMonte Carlo Method

Application of the Monte Carlo method for searching of the equilibrium structures

was described in Section 6.2.1. Thus, as a rule, Metropolis’s algorithm is applied.

The interface of the corresponding program is shown in Figure 9.6.

The basic procedures of the program are given below.

Show; bar(380,380,400,400); readkey;

Q[k]:=ni/((Xn-1)*(Yn-1)); Lp[k]:=pp;

end; {end for k}

bar(0,0,640,480); setcolor(14);

line(40,50,40,440); line(40,440,600,440);

unit Monte-structure; //Delphi

interface

uses Windows, Messages, SysUtils, Variants, Classes, Graphics, Con-

trols, Forms, Dialogs, StdCtrls;

const NAtom =100; x0=15; y0=15; NAtomY=10; scale=5; r=3; e=121 ;

s=3.405 ;T=20;

type TAtom = record x,y,z:real; end;

type Form1 = class(TForm)

Button1: TButton;

Image1: TImage; Image2: TImage;

Label1: TLabel; Label2: TLabel; Label3: TLabel;

Label4: TLabel; Label5: TLabel;Label6: TLabel;

Label8: TLabel; Label9: TLabel; Label7: TLabel;

Edit1: TEdit; Edit4: TEdit; Button2: TButton;


Figure 9.6 The active controlling form of the program Monte-structure for searching of

the equilibrium structure by the Monte Carlo method.

Memo1: TMemo; Memo2: TMemo; Memo3: TMemo;

procedure Button1Click(Sender: TObject);

procedure FormCreate(Sender: TObject);

procedure Button2Click(Sender: TObject);

private {Private declarations}

public

end ;

var Form1: TForm1;

m10,m,n1,ny,ny_all:integer;

c: Char;sm:string;

Atoms:array[0..NAtom] of TAtom;

Eold,ver, Enew,dx,dy,dr: real;

y:array[1..28] of integer; gr:array[0..28] of real;

implementation {$R *.dfm}

Procedure init (); // setting of the initial coordinates of atomsvar i:integer;

begin

for i:=0 to NAtom-1 do begin

Atoms[i].x:=x0 +(i mod (10))*s;

Atoms[i].y:=Y0 +(i div (10))*s; end ;

end ;

Procedure CalcEnergy1(i:integer); {before change of atom position}var j:integer; dr:real;

begin Eold:=0;

for j:=0 to NAtom-1 do if j<>i then

begin

dr:=sqrt((atoms[i].x-atoms[j].x)*(atoms[i].x-


atoms[j].x)+(atoms[i].y-atoms[j].y)*(atoms[i].y-atoms[j].y));

Eold:=Eold+e*((exp(12*ln(s/dr))-exp(6*ln(s/dr))));

end ;

end ;Procedure CalcEnergy2(i:integer); {after change of atom position}

var j:integer;

begin enew:=0;

for j:=0 to natom-1 do if j<>i then begin

dr:=sqrt((atoms[i].x+dx-

atoms[j].x)*(atoms[i].x+dx-atoms[j].x)+(atoms[i].y+dy-

atoms[j].y)*(atoms[i].y+dy-atoms[j].y));

Enew:= Enew+e*(exp(12*ln(s/dr))-exp(6*ln(s/dr)));

end ;

end ;

Procedure Movement(); { change of atom position }var i:integer;

begin

i:=random(NAtom); CalcEnergy1(i);

dx:=(random(101)-50)/50*random*0.1;

dy:=(random(101)-50)/50*random*0.1;

CalcEnergy2(i);

if (Enew<Eold) then begin

atoms[i].x:= atoms[i].x+dx;

atoms[i].y:= atoms[i].y+dy ;

end else begin

ver:=exp((Eold-Enew)/T); if(random<ver)then begin

atoms[i].x:=atoms[i].x+dx; atoms[i].y:=atoms[i].y+dy;

end ; end ;

end ;

Procedure showsp(color:integer);

var x,y,i:integer; rect: TRect;

begin with for m1.image2.Canvas do begin

rect.Left:=0; rect.Top:=0;

rect.Bottom:=400;rect.Right:=400;

Brush.Color:=RGB(255,255,255); FillRect(rect);

Brush.Color:=RGB(55,55,55);

for i:=0 to 99 do

RoundRect(round(atoms[i].x*scale)-3, {images of atoms}round(atoms[i].y*scale)-3,round(atoms[i].x*scale)+

3,round(atoms[i].y*scale)+3,3,3);

end ;end ;

//--------------------

Procedure AxesDraw; . . . . . . .

Procedure histogram; //plotting of the pair distribution functionvar i1,i,j,iy:integer; scal,max:real;

begin

for i1:=1 to 28 do

for i:=0 to NAtom do



1. Closely consider the program and define the basic algorithm.

2. Define how many of the Monte Carlo steps are necessary for the approaching system to

the equilibrium state at the given potential of intermolecular interaction.

3. Obtain the images of the structure and radial pair distribution functions (RPDFs) for dif-

ferent temperatures.

4. Define the temperature, below which the ordered state with closely packed atoms is real-

ized. What kind of the crystal lattice is it?

for j:=0 to NAtom do

if j<>i then begin

dr:=sqrt((atoms[i].x–atoms[j].x)*(atoms[i].x+dx-atoms[j].x)

+(atoms[i].y-atoms[j].y)*(atoms[i].y- atoms[j].y));

ny_all:=ny_all+1; {the number of atoms with r<R}if round(dr*2)=i1 then

begin y[i1]:=y[i1]+1; ny:=ny+1; end; end;

for i1:=1 to 28 do //y[i1] is the number of atoms in the layergr[i1]:=(y[i1]/ny)*28*28/(2*i1);//R=28*0.5, r=i1*0.5, dr=0.5

gr[0]:=0; max:=0; scal:=160/4;

with m1.image1.Canvas do begin

Pen.Color:=clred; MoveTo(20,200);

for i1:=1 to 28 do

LineTo(20+i1*8,200-round(gr[i1]*scal));

end;

end;

Procedure TForm1.Button1Click(Sender: TObject);

Var i,j,color: integer; {display button}BEGIN

T:=Memo1.Lines.FloatToStrF(T,ffGeneral,5,2));

randomize;

showsp(0);for i:=0 to 10000 do movement();

AxesDraw; gistogramm; showsp(0);

end;

. . . . . . . . . . . . . . . .


Begin Application.Terminate; {pause}end ;

Procedure TForm1.FormCreate(Sender: TObject);

begin form1.label4.Caption :='n'+'x'+'n';

init(); showsp(0); {it shows parameters)Memo1.Lines.Add(FloatToStrF(T,ffGeneral,5,2));

Memo2.Lines.Add(FloatToStrF(e,ffGeneral,5,2));

Memo3.Lines.Add(FloatToStrF(s,ffGeneral,5,2));

end;

End.


9.9 Modeling of Crystal Growth by the Monte Carlo Method

9.9.1 The Crystal Growth Forms

It is known that the morphology of crystals is connected with the structure of inter-

face and mechanism of growth (see Section 7.2.3). The forms of crystals can be

facet, semifacet, and roundish, depending on the degree of atomic roughness of dif-

ferent facets. The equilibrium form of the crystal is the form of the critical nucleus

(three-dimensional), and it was considered [1] earlier, that only the size of equilib-

rium crystals depends on supersaturation (if the free surface energy does not vary).

Growth forms can differ from the equilibrium form because of kinetic roughening

development [2�4]. Authors of Refs. [5,6] estimated critical supersaturation at

which the facet becomes rough in the atomic scale, considering that the free energy

of formation of the two-dimensional nuclei becomes equal to kT.

9.9.2 Probabilities of Transitions

The standard method of the kinetic Monte Carlo modeling of Kossel’s crystal in

frameworks of the “solid-on-solid” model is applied here for modeling.

Probabilities of the atom attachment to the crystal and breakoff from it are defined

for the timestep, which is equal to mean time τ2, during which the atom with one

bond of the “solid-solid” (s-s) kind exists on the crystal surface. The probability of

atom breakoff is defined by the exponential factor from energy of its bond with the

neighbor “solid” atoms (s-atoms).

p� 5 νoτ� expð�2ϕn=kTÞ; ð9:3Þ

where ϕ5ΔH/6, ΔH is the melting heat per one atom, n is the number of the

nearest s-atoms, and νo is the vibration frequency.

The probability of joining of an atom to the crystal is connected with the odds

of chemical potentials of two phases, Δμ.

p1 5 νoτ2 exp½ðΔμ� 6ϕÞ=kT �: ð9:4Þ

9.9.3 Modeling of Growth of Kossel’s Crystal [6]

Consider the problem of the crystal growth in the narrow channel with ideal walls

without use of periodic boundary conditions. It is recommended to choose channels

with the quadratic (20�20, 40�40, 60�60, 80�80, or 100�100 atoms) or with rectan-

gular (40�160 atoms) section. The crystal size in growth direction is not restricted.

The special algorithm allows us to define parameters of all two-dimensional clus-

ters in high layers of the crystal and to write the parameters of the greatest of

them, such as perimeter na, numbers of atoms in clusters with one, two, or three

broken bonds (N1, N2, N3) in every upper crystalline plane. After the work of the


algorithm, it is possible to define facet size nf and all characteristics of the critical

two-dimensional nuclei: n�, na, nb5N11N21N3.

Figure 9.7 shows the front panel of the main form in program interface. It is

necessary to call the procedure Showf instead of Show in the basic program main_cr_u for featuring of the crystal sections perpendicular to the crystal surface in the

middle of the channels. The mentioned procedures are connected with the button

action—ButtonPaint. The procedure Show yields the crystal image from above on;

the colors of small squares corresponded to atoms depending on the number of the

atomic layer (see Figure 7.11). Procedure Showf ensures the rendering of the side

section of crystals with atoms in the form of cells of net. It shows the averaged pro-

files (Figure 9.8) by the thick line obtained, having obtained the average value of

more than 3000 sections (in other words, having done more than 109 Monte Carlo

steps). If the crystal grows, the change of number of atoms in it is taken into

account at the definition of the averaged profile.

Figure 9.8 shows the sections of crystals with entropy of fusion ΔH/kT5 5.5 in

the channel (160�40) and their average profiles (solid line). At the specified super-

saturation, σ5Δμ/kT5 0.068, the crystal practically does not grow (0,V/

aν0, 1027, V is the growth rate and a is the crystal lattice parameter) owing to an

acting of the Gibbs�Thomson shift to the supersaturation. Apparently, from

Figure 9.8, there is a flat site on the averaged profiles; it means that the facet exists.

Figure 9.8 Middle sections of not growing crystal in the rectangular channel, ΔH/kT5 5.5,

σ5 0.068, (A) along a long side; (B) along a short side.

Figure 9.7 The active controlling form of the program for modeling of growth of the Kossel

crystal.


When the crystal grows at small supersaturation (σ5 0.08), its form looks better

faceted—the facet becomes larger, and the radii of curvature of roundish sites

become less.

The face of this crystal (40�160) is essentially smaller than in the case of the

square channel (80�80; see Figure 7.12), despite the identical value of entropy of

fusion. Hence, the channel shape influences the form of the crystal-free surface.

Modification in the relative size of the facet in the growing crystal form is con-

nected with the anisotropy of the crystal growth rate; rounded sites become smaller

at increase of the supersaturation. When the facet becomes larger, it grows faster

due the increasing area, on which two-dimensional nuclei can be formed, increases.

The facet size, during stationary growth in the channel, is such that the velocity of

the facet growth coincides with the growth rate of rounded sites of the surface.

However, the kinetic roughness of the crystal face is increasing [2�4] at a subse-

quent increase in the supersaturation, and the face disappears at a certain high

supersaturation [5,6].


1. Explain the basic procedures of the program work.

2. Study the real structure of the crystal face and determine the average size of the fluctuat-

ing face and the statistical characteristics of two-dimensional nuclei. As shown by

Voronkov [7], the size of the equilibrium facet is twice more than the critical size of the

two-dimensional nucleus fluctuating on it. Thus, only one two-dimensional nucleus can

be formed on the equilibrium facet.

3. Set the time slice (for recording) of order 100 mks before start of the procedure

“Cluster”, and number of the data notations of order 500. As a result of automatic oper-

ation of the program after click of the button “Cluster” (in the form top, to the left of the

mark “�”), 500 values of the sizes (number of atoms) of the greatest cluster in several

top planes of the crystal will be recorded in the output file (find its path in the filemain_cr_u). The number of atoms in the cluster of topmost plane will give the facet

size, if there is no two-dimensional nucleus. Open this file in Excel by means of the spe-

cial insertion with the distributive sign “j” and construct the graph, similar to that shown

3500n

3000

2500

2000

1500

1000

500

03.2 3.3 3.4 3.5 3.6 3.7 t, 10–6 s

nf

n∗

Figure 9.9 Fluctuations of sizes of the greatest cluster and crystal face, ΔH/kT5 5.5,

σ5 0.068.


in Figure 9.9. Determine the size of the facet and the critical size of the two-dimensional

nucleus (the greatest cluster, which can disappear) by means of this graph.

4. If time allows, fulfill point 2 for several temperatures (values of the ratio ΔH/kT) and

construct the graph of the type shown in Figure 9.10.

The sizes of facets (nf) and critical nuclei (n�) depend on the supersaturation.

The curves for sizes of facets and sizes of nuclei (their product with 4), intersect at

such supersaturation, for which the dynamic equilibrium takes place (the crystal

does not grow). Thus, it is possible to determine the critical sizes of nuclei at dif-

ferent values of ΔH/kT. Then points of structural transition to the kinetic roughen-

ing (ϕ/kTtr) can be defined, as the points in the graph n�(ϕ/kT), in which the free

energy of formation of the critical nucleus (ΔF� 52n�Δμ1 naγa5 n�Δμ) is

equal to kT. In the last expression for ΔF�, γa is the free step energy in counting

per one atom and na is the number of atoms placed on the cluster edges.

9.10 The Method of Molecular Dynamics

The method of the molecular dynamics is described in Chapters 1 and 6. It consists

of numerical solution of Newton’s equations for all atoms—at the given instant,

the incremental values of coordinates and velocities of all particles are calculated

using discretizated Newton’s differential equations from their values at the previous

time moment. The basic elements of the program for model operation are enumer-

ated in Section 1.3.5. Examples of procedures for their implementation are given

below.

1800

1600

1400

1200

1000

800

600

4000.065 0.066 0.067 0.068

nf

4n∗

nf,4n∗

0.069 0.07

Δ H/kT = 5.3

Δ H/kT = 5.5

0.071 0.072 0.073kT

σ

Figure 9.10 The size of facets nf and critical nucleuses n� dependences on the

supersaturation.


9.10.1 Potentials and Forces

First of all, it is necessary to define the system model, which is the subject of

modeling. For the study of qualitative properties of the system of many particles,

the assumptions that dynamics is classical and molecules are chemically inert are

right enough. Suppose that the force of interaction of any two particles depends

only on the distance between them. Then the full potential energy U of the system

is defined by the sum of interaction energies of the pairs of atoms; in this case:

U5Vðr12Þ1Vðr13Þ1?1Vðr23Þ1?5XNi, j51

VðrijÞ ð9:5Þ

where V(rij) depends only on the absolute value of distance rij between particles

with indexes i and j. Equation (9.3) corresponds to “simple” fluids, which consist

of neutral atoms or molecules, the liquid argon in this case. It is known that argon

atoms feebly attract each other at comparatively large r; it is caused mainly by the

cross-polarization of atoms; the resulting force of attraction is termed the van der

Waals force. The Lennard-Jones potential describes well such interaction; it has the

following form:

VðrÞ5 εd

r

� �12

2 2d

r

� �6" #

: ð9:6Þ

Here, d is the value of distance at which the potential reaches minimum and ε is

the smallest energy (the depth of the potential well). This potential is short-range

and V(r) for r. 2.5σ, in fact, is equal to 0. The interaction potential between two

particles is calculated in the program by means of a function, which returns the

value of the interaction energy between two particles depending on the distance

between them, dr5squart(dx21dy2) (dx and dy are the differences of the particle

coordinates).

double potential_direct(double dx,double dy)

{ double d2,d6;

d2 = double(1.0/(dx*dx+dy*dy)); // divided by square of distance

d6 = double(d2*d2*d2); // obtaining of the 6th degree of expressiondouble potential = (potential_a*d6 - potential_b)*d6;

return potential;

}

The disposition of particles in the restricted volume may be random or in the

knots of some implicit grid. It is also necessary to choose the boundary conditions.

There is a possibility to choose one of two types of boundary conditions in the pro-

gram: elastic walls or periodic conditions. The elastic potential for boundaries is

chosen in the form of Hooke’s potential.


double border_potential(double x, double y)

// if the particle is outside the cell, the additive to potential is calculated {double p = 0; // internal variable for potential

if (x > cellsize_x) //if the particle has moved beyond the right edgep = p + border_k*(x-cellsize_x)*(x-cellsize_x)/2.0;

else if (x < 0) p=p+border_k*x*x/2.0; //if beyond the left edge

if (y > cellsize_y) //if beyond the upper edgep = p + border_k*(y-cellsize_y)*(y-cellsize_y)/2.0;

else if (y < 0) p = p + border_k*y*y/2.0;

return p; //the result, which the function returns}

For evaluations in case of periodic boundary conditions:

double potential_periodic(double dx,double dy)

// the interaction energy of the particle with mirror reflection of other particle { if (dx<0) dx=-dx; if (dy<0) dy=-dy;

//if the second particle at a distance of more than half of// linear size of the basic cell, its position is also reflected

if (dx > cellsize_x*0.5) dx = dx - cellsize_x;

if (dy > cellsize_y*0.5) dy = dy - cellsize_y;

double p= potential_direct(dx, dy);

return p;

}

For evaluating the particles’ motion and calculating the pressure produced by

them, calculations of forces are required. This is fulfilled by the procedures similar

in code to the functions evaluating the potentials given above. Using procedures

instead of functions as in the case of the potential energy calculations is caused by

the fact that the force is the vectorial magnitude, and it is necessary to count its

void force_direct(double dx, double dy, double &fx, double &fy)

// procedure returns the central force acting on the particle{ double d2,d6,f;

dddx=dx; dddy=dy;

d2 = 1.0/(dddx*dddx+dddy*dddy);

d6 = d2*d2*d2;

f = (12*potential_a*d6 - 6*potential_b)*d6*d2;

fx = f*dx;

fy = f*dy; // the force components have been calculated}

void border_force(double x, double y, double &fx, double &fy)

// if particle came off the cell, the turning force is calculated{ if (x > cellsize_x) fx = -border_k*(x-cellsize_x);

//returning the force, which is similar to Hooke’s Forceelse if (x < 0) fx = -border_k*x;


components and return two values, instead of one. The force is calculated as the

derivative of the energy.

9.10.2 Algorithms for Calculating Velocities and Coordinates

Algorithms for calculating velocities and coordinates sequentially in time are

described in Section 1.3.2. The velocity form of Verlet’s algorithm is applied most

void Step(Atom a[], Atom out_atoms[], double dt)

{ int i; double fx, fy; //Biman’s algorithm

Act atoms (a,accel_2); //calculates accel_2[i].vx and accel_2[i].vyfor (i=1;i<=N;i++)

{out_atoms[i].x = a[i].x + (a[i].vx+(4*accel_2[i].vx-

accel_1[i].vx)/6*dt)*dt;

out_atoms[i].y = a[i].y + (a[i].vy+(4*accel_2[i].vy-

accel_1[i].vy)/6*dt)*dt; //the new coordinates} Act atoms (out_atoms, acel_3);

for (i=1;i<=N;i++)

{out_atoms[i].vx = a[i].vx + (2*accel_3[i].vx+

5*accel_2[i].vx-accel_1[i].vx)/6*dt;

out_atoms[i].vy = a[i].vy + (2*accel_3[i].vy+

5*accel_2[i].vy-accel_1[i].vy)/6*dt; //the new velocities

accel_1[i].vx=accel_2[i].vx;

accel_1[i].vy=accel_2[i].vy; /the new old accelerations for the next dt} -----------------------

if (mode == Mode _Fixed) //Fixed, reflection from the border

{for (i=1;i<=N;i++) //and calculations of impulses for pressure

else fx = 0; // zero if particle is not beyond the boundaryif (y > cellsize_y) fy = -border_k*(y-cellsize_y);

// the same actions for y-coordinateelse if (y < 0) fy = -border_k*y; else fy = 0;

}

void force_periodic(double dx, double dy, double &fx, double &fy)

/* procedure returns the value of force acting on particle,from reflection of another particle (the other cell quarter)*/

{double ddx,ddy; // distances to the reflection of another particleif (dx<=0) ddx=-dx; else ddx=dx;

if (dy<=0) ddy=-dy; else ddy=dy;

if (ddx > cellsize_x*0.5) ddx = ddx - cellsize_x;

if (ddy > cellsize_y*0.5) ddy = ddy - cellsize_y;

force_direct(ddx, ddy, fx, fy); // calculation of forceif (dx < 0) fx = -fx; if (dy < 0) fy = -fy;

//sign of forces depends on the mutual placement of particles (the sign of dx)}


often in scientific works. The more exact Bimann’s algorithm (Eq. (1.45)) is used

here. The additional array of accelerations (accelerations are equal to forces because

the reduced mass is equal to 1) calculated on the previous timestep is used in the

procedure (three arrays: accel_1 [i], accel_2 [i], and accel_3 [i] are used in it):

The procedure (void) Step is preceded by procedure Act_atoms, which is called

for calculating of acceleration arrays: accel_1, accel_2, accel_3 (accel[i].vx iaccel[i].vy):

void Act atoms (Atom a[], Atom b[])

//procedure calculates arrays of data type of atom (forces=accelerations){ int i,j; double fx, fy;

for (i=1;i<=N;i++)

{ f[i].vx = 0; f[i].vy = 0; }

if (mode == Mode _Fixed) // fixed border, elastic walls{ for (i=1;i<=N-1;i++)

for (j=i+1;j<=N;j++) // summation of forces acting on particles{force_direct(a[i].x-a[j].x, a[i].y-a[j].y, fx, fy);

f[i].vx = f[i].vx + fx; f[i].vy = f[i].vy + fy;

//adding the corresponding contribution to the acceleration of the i-th particlef[j].vx = f[j].vx - fx; f[j].vy = f[j].vy - fy;

} //and immediately contribution to accelerating the j-th particle (m=1)

for (i=1;i<=N;i++) //calculation of the contribution of elastic walls{ border_force(a[i].x, a[i].y, fx, fy);

f[i].vx = f[i].vx + fx; f[i].vy = f[i].vy + fy; }

}

else // the case of periodic boundary conditions{ for (i=1;i<=N-1;i++)

for (j=i+1;j<=N;j++)

//call the procedure and an adding of the force components{force_periodic(a[i].x-a[j].x,a[i].y-a[j].y,fx,fy);

f[i].vx = f[i].vx + fx; f[i].vy = f[i].vy + fy;

f[j].vx = f[j].vx - fx; f[j].vy = f[j].vy - fy; }

}

}

{border_force(out_atoms[i].x,out_atoms[i].y,fx,fy);

if (fx>=0) impulse_x +=fx; else impulse_x -=fx;

if (fy>=0) impulse_y +=fy; else impulse_y -=fy;

} }

else //non Fixed – Periodic–entering of an atom from the other size

{for (i=1;i<=N;i++) //and calculations of impulses for pressure{if (out_atoms[i].x > cellsize_x)

{out_atoms[i].x = out_atoms[i].x - cellsize_x;

initial_atoms[i].x=initial_atoms[i].x-cellsize_x;

if (a[i].vx>=0) impulse_x += 2*a[i].vx;

else impulse_x -= 2*a[i].vx;

} /*and the same for other borders; coordinates of initial atoms[i] are needed for the mean-

squared displacement calculations*/ . . . . . . . }

}


9.10.3 Designations of Principal Constants and Variablesin the Program

The next parameters are used in the program for argon:

const N = 100; //total number of atoms

const cellsize_x = 2; // basic cell size

const cellsize_y = 2; // basic cell size

float potential_d=0.2; //distance to the minimum potential (0.3405 nm)

float potential_e = 100; //"depth" of the potential minimum, K

const border_k=2*potential_e*500; //for the elastic border potential

float x1=0; float x2=2.0; float y1=0; float y2=2.0; // border coordinatesfloat dt=0.00005; // the time step in dimensionless units

float initial_kinetic = 80; //the initial kinetic energy of atoms,

int mode; const Mode_Fixed=1; //the type of boundary condition

const Mode_Periodic=2; //the type of boundary conditionstruct Atom{double x,y; double vx,vy;};

// the type of record that contains 4 elementsstruct Energy{double u,k;}; //the type of record, contains 2 elements

Atom initial_atoms[N+1], atoms[N+1], //arrays of the Atom type;acel_1[N+1],acel_2[N+1],acel_3[N+1], temp_atoms[N+1];

Energy EA[N+1] //to record the kinetic, potential and total energy of particlesdouble initial_energy,impulse_x,impulse_y, total_time;

As the program provides application of dimensionless quantities, it is necessary to

provide relations between dimensional and dimensionless quantities or, in other words,

define the unit values for transforming dimensionless quantities into real ones. As the

distance d for minimum of the argon potential is 0.3405 nm and the corresponding

constant equals 0.2, the unit of length lun5 1.7025 nm. The kinetic energy is measured

in units of k � 1K (k is the Boltzmann constant). The dimensionless mass m5 1, hence,

mass unit is mAr5 6.69 � 10226 kg. Velocity υ5ffiffiffiffiffiffiffiffiffiffiffiffi2E=m

p; the dimensionless velocity

νdimless 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Edimless=mdimless

p; hence, υ=νdimless 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikB � 1K=mAr

p5 14:362 m=s. As

v/vdimless5 lun/tun (t is the time), tun5 lun/14.362 m/s5 1.1854 � 10210 s. These rela-

tions are considered for calculating the stationary values of the program:

mAr5 6.69e2 26; v_unit5 14.362; //sqrt(1.38e2 23/mAr);L_unit5 3.405e2 10/potential_d; t_unit5L_unit/v_unit;

The multiplier t_unit is introduced into the procedure make_steps to calculate

the variable total_time. The real time in seconds (s) is printed in the correspond-

ing window of the interface (panel Statistics).

9.10.4 Procedures for Calculations of the System Characteristics

In this section, procedures are considered which ensure calculations of system char-

acteristics, such as the kinetic, potential, and full energies of atoms; the energy


distributions of atoms; mean energies; and RPDF. This program contains the con-

venient procedure for buildup of distributions’ graphs. Buildup of corresponding

graphs also can be fulfilled by means of the editor Chart.

The following procedure prepares data for plotting of the energies distributions

of particles (for the kinetic, potential, and total energy) and defines the average

values of energies.

void calc_energy(Atom a[], Energy

EA[], double &kmin, double &kmax,

double &kave, double &umin, double

&umax, double &uave, double&emin,

double &emax, double &eave, double &eetotal)

{ int i,j; double e; eetotal = 0;

for (i=1;i<=N;i++) // kinetic energy, { EA[i].k =(a[i].vx*a[i].vx+a[i].vy*a[i].vy)*0.5;

EA[i].u = 0; // start for potential energy of each particle

eetotal = eetotal + EA[i].k; // the total energy, kinetic yet} for (i=1;i<=N-1;i++)


// calculation of potential energy of interaction of i-th and j-th particles{e = potential_direct(a[i].x-a[j].x, a[i].y-a[j].y);

EA[i].u = EA[i].u + e; EA[j].u = EA[j].u + e;eetotal = eetotal + e;} // adding of the energyif (mode == Mode _Fixed)

for (i=1;i<=N;i++)

//contribution to the potential energy due to interaction with elastic walls{e = border_potential(a[i].x, a[i].y);

EA[i].u = EA[i].u + e; eetotal = eetotal + e; }

else // the case of periodic boundary conditionsfor (i=1;i<=N-1;i++)


{e=potential_periodic(a[i].x-a[j].x,a[i].y-a[j].y);

EA[i].u = EA[i].u + e; EA[j].u = EA[j].u + e;


Procedure for RPDF calculation:

void calc_GR(double &gmin,double

&gmax,double &xmin,double&xmax)

{int i,j,k,nr; double d,d2,dr; int

intR=40; dr=potential_d/10; /*the distant interval between two orbs -- d_atom/10 */ int NCOUNT=0;

for (k=0;k<=100; k++) gr[k]=0;

//intial values zerofor (i=0;i<=N-1;i++)


{d2 = pow((atoms[i].x-atoms[j].x),2) + pow((atoms[i].y-

atoms[j].y),2);

d = sqrt(d2); //the distance between two atomsnr = floor(d/dr); // the number of spherical layer

if (nr>intR)continue; //intR*dr is the radius of area of calculation

gr[nr] += 1; //accumulation of data on each layers, dr in thickness

NCOUNT += 1;} // the sum of all gr[nr]for (int k = 1; k <= intR; k++)

gr[k]=gr[k]/(pow(k+1,3)-pow(k,3))*pow(intR,3)/NCOUNT;

eetotal = eetotal + e; } // adding of this contribution/* determination of maximum and minimum of energies for building of graphs*/kmin = 0; kmax = 0; kave = 0;

umin=0; umax=0; uave=0; emin=0; emax=0; eave=0;

for (i=1;i<=N;i++)

{e=EA[i].k; if (e> kmax) kmax= e; kave= kave + e;

e=EA[i].u; if (e< umin) umin= e; uave= uave + e;

e=EA[i].u+EA[i].k; if (e< emin) emin=e; eave=eave+e;

} // determination of average values of energies: kinetic, potential and totalkave = kave/N; uave = uave/N; eave = eave/N;

}


9.10.5 Statistics

The calculated values such as the average and maximum kinetic, potential,

and total energy are printed. The program tracks the numerical error in calcu-

lations of total energy and account stability also. Values of energies are given

in kelvins (K).

The pressure is calculated as the impulse flux through the lateral faces, P:5

impulse_x/cellsize_y/total_time;.The value of mean-squared displacements of particles from initial position is

given; the diffusivity, which is equaled to the quadrate of the displacements divided

by the total time, is also printed (the dimensionless values should be transformed

into the SI system).

The procedure (void), which calculates the mean-squared displacements of par-

ticles from their initial positions and their mean-squared velocities:

void calc_deltas(double &ave_dr2, double &ave_vx2,

double &ave_vy2)

{double dr2,vx2,vy2; int i; dr2 = 0; vx2 = 0; vy2 = 0;

for (i=0;i<=N;i++)

{dr2 = dr2 + pow((atoms[i].x-initial_atoms[i].x),2)+

pow((atoms[i].y-initial_atoms[i].y),2);

vx2 += atoms[i].vx*initial_atoms[i].vx;

vy2 += atoms[i].vy*initial_atoms[i].vy; }

} ave_dr2 = dr2/N; ave_vx2 = vx2/N; ave_vy2 = vy2/N;

xmin = 0; xmax = dr*intR; //xmax for graph is the size of the orb R;

gmin = 0; gmax = 0; // for the scale of graphfor (k=1;k<=intR; k++)

if (gr[k] > gmax) gmax = gr[k];} //the max value of gr[k]


The interface and procedures, which are started by buttons (Figure 9.11) are

described here:

1. The button 1 is named the button “Go”—see this button in the drawing. The button “Go”

click means the start-pause operation (the starting of the timer and stopping):

void __fastcall TForm1::StartButtonClick(TObject *Sender){if (mode == Mode_Fixed) PeriodicBorderSwitch->

Enabled=false;

else FixedBorderSwitch->Enabled=false;

Running=!Running;

if (Running) StartButton->Caption="Stop";

else StartButton->Caption="Go";

while (Running) {make_step(25, time_step);

Form1->UpdateStatistics(); Form1->RedrawImage(); }

Application->ProcessMessages(); }

3. Button 3 “Reset”—timer stopping, returning to the initial state of the system and timer

starting:

Figure 9.11 The active controlling form of the program “MD”.

void __fastcall TForm1::ResetButtonClick(TObject *Sender){ if (!Running)

{PeriodicBorderSwitch->Enabled=true;

FixedBorderSwitch->Enabled=true;}

if (Running)

{Running=false; reset_model(initial_x1,initial_x2,

initial_y1,initial_y2,initial_K);

Form1->RedrawImage(); Running=true; }

else {reset_model(initial_x1, initial_x2, initial_y1,

initial_y2, initial_K); Form1->RedrawImage(); }

Application->ProcessMessages();

}


4. Button 4 “Quit”: Program escaping on the button “Quit” pressing

Void __fastcall TForm:: QuitButtonClick (TObject *Sender) {if (Running) Running=!Running; . . . . . }

5. Buttons 5: Switching of modes: elastic walls or periodic conditions

void __fastcall TForm1::PeriodicBorderSwitchClick

(TObject *Sender)

{ if (Running)

{Running=false; mode= Mode_Periodic; Running=true;}

else { mode= Mode_Periodic; }

}

void __fastcall TForm1::FixedBorderSwClick(TObject Sender){if (Running)

{ Running=false; mode= Mode_Fixed; Running=true;}

else {mode=Mode_Fixed;

}

6. Button 6: A validation of the format of numbers introduced in the text fields after pressing

of the corresponding button “Apply”

7. Button 7 “Frame”: pressing fulfills the following steps (make_step) of modeling, whichcalls 25 times the void Step:

void __fastcall TForm1::FrameButtonClick(TObject Sender){if (mode==Mode_Fixed)

PeriodicBorderSwitch-> Enabled=false;

else FixedBorderSwitch->Enabled=false;

Running=false; StartButton->Caption="Go";

make_step(25, time_step); //a call of the void Step() 25 timesForm1->UpdateStatistics(); Form1->RedrawImage();

}

8. Updating statistics and making the corresponding notes in the form is ensured by opera-

tion of several procedures; one of them is the following:

void TForm1::UpdateStatistics()

{double kmin, kmax, kave, umin, umax, uave, emin, emax, eave,

etotal; double dr2, vx2, vy2;

calc_energy(atoms, EA, kmin, kmax, kave, umin, umax,

uave, emin, emax, eave, etotal);

calc_deltas(dr2, vx2, vy2);

TotalEnergyView->Caption=FloatToStrF(etotal,ffGeneral, 10,4);

AveKEnegryView->Caption = FloatToStrF(kave, ffGeneral, 6, 4);

. . . . . . . . . . . . . . . . . . . . . .

ErrorView->Caption=FloatToStrF((initial_energy-etotal)

/initial_energy, ffExponent, 1, 1);

DR2View->Caption = FloatToStrF(dr2, ffGeneral, 6, 4);


When creating the form (after start of the program), the system is positioned in

the initial state: initial values of coordinates and impulses and also values of the

program variables are taken from the main form.

The following procedure is responsible for the imaging of atoms. The color and

size of the background rectangle are obtained. The colors and positions of each

atom are set.

void TForm1::DrawAtoms(TCanvas *Canvas)

{ int i, x, y; double xscale, xshift, yscale, yshift;

xscale = (Canvas->ClipRect.Right-Canvas->

ClipRect.Left-20)/cellsize_x;

yscale = (Canvas->ClipRect.Bottom-Canvas->

ClipRect.Top-20)/cellsize_y;

xshift = Canvas->ClipRect.Left+10;

yshift = Canvas->ClipRect.Top+10;

Canvas->Brush->Color = RGB(255,255,255);

Canvas->Pen->Color = RGB(140,140,140);

Canvas->FillRect(Canvas->ClipRect);

Canvas->MoveTo(Canvas->ClipRect.Left+10,Canvas->

ClipRect.Top+10);

Canvas->LineTo(Canvas->ClipRect.Right-10, Canvas->

ClipRect.Top+10);

. . . . . . . . . . . . . . . . . . . . . . . . .

Color = RGB(255,255,255);

Canvas->Pen->Color = RGB(155,155,155);

for (i=1;i<=N;i++)

{ x = int(atoms[i].x * xscale + xshift);

y = int(atoms[i].y * yscale + yshift);

Canvas->RoundRect(x-2, y-2, x+3, y+3, 2, 2); }

}


1. Analyze the basic algorithm and its implementation. Justify connections of real physical

characteristics used for building the model and the dimensionless parameters, which are

applied in this program.

2. Find out how the time step influences the accuracy of calculations.

3. Define the basic characteristics of the studied system by analyzing the modeling results.

Convert the values of the pressure and diffusivity to SI units.

if (total_time > 0){DView->Caption = FloatTo

StrF(dr2/2.0/total_time, ffGeneral, 6, 4); //diffusivityXPressureView->Caption=FloatToStrF(impulse_x/

cellsize_y/total_time, ffGeneral, 6, 4); // pressureYPressureView->Caption = FloatToStrF (impulse_y/

cellsize_x/total_time, ffGeneral, 6, 4); } // pressure}


4. Study the temperature influence on the structure, that is, on the placement of atoms and

RPDF (consider the temperature from 10 to 100 K). Study and explain how boundary

conditions influence the structure.

5. Construct the void for the graph of the velocity distribution of particle.

6. Elaborate the void for definition of the velocity auto-correlation function and provide a

calculation of the self-diffusion coefficient using the autocorrelation function.

7. Develop the void for the system thermostatic control. Define the “melting” temperature

for the system that contacts with the thermostat.

9.11 Fractal Dimension and Renormalization

9.11.1 Definition of Fractal Dimension

One of the most interesting geometrical properties of objects is their form. Fractals

are applied for description of the nonregular forms in such different systems as tur-

bulence, coastal lines, mountain chains, and clouds. Mandelbrot’s book [8] contains

a discussion of fractal geometry; it contains a set of nice examples of histories

about the fractals obtained with computers.

Fractals, which exist in the real world, have a finite interval of sizes. Their phys-

ical definition is the following [9,10]: fractals are geometrical objects (lines, sur-

faces of the skew fields) which are very twisting, nonregular, and have the property

of self-similarity in the restricted scale. The up-to-date composites have the mani-

fold structure formed by several phases. At least one phase consists of clusters,

often nanodimensional. Clusters can be branched out, rarefied, or filamentary, that

is, the fractal is a geometrical object which fills space only partially.

Figure 9.12A,B shows the percolation clusters obtained during modeling of

vapor-phase adsorption at two pressures under condition of enough strong inter-

action between particles, which are deposited (the term “percolation” means the

connectivity between any elements, in this case, between cells which map the

atoms). These clusters develop from one center, and accordingly, there are connec-

tions between their elements. The cluster figured in Figure 9.12A is isolated.

Figure 9.12 Examples of percolation clusters. (A,B)The modeling of adsorption:

(A) P5 1400 Pa, (B) 3000 Pa; (C) purely mathematical generation on the square lattice

for the value of probability p5 0.5927.


The cluster shown in Figure 9.12C is obtained by modeling with the procedure of

the cell-like automatic machine without consideration of any physical content at

choice of probabilities p of the cell colors.

Purely mathematical generation of clusters has sense, as the geometrical nature

of the surface ordering has become clear after such simulations. It is found that

there is the threshold value of probability pc5 0.5927 of fillings of the chosen

cells. The clusters that form join, as a rule, that is, they are continuous in the space

if p. pc. Cells designated by points in Figure 9.12C are filled in such way that part

of these cells forms the continuous cluster, which joins the top and bottom sides of

the quadrate from 603 60 cells.

Fractal dimensionality df is applied as the quantitative measure of structure of

these objects with the nonregular form. For definition df, we will recollect at first

some simple concepts of usual Euclidean geometry. Consider a circular or spherical

object of mass M and radius R. The object can be continuous (the homogeneous

density), or can contain interstices, but in any case, we will suppose that its density

does not depend on its size. Hence, at the magnification of radius of the object

from R to 2R, its mass increases in R2 times, if the object is round, or in R3 times,

if it is spherical. This connection of mass and length can be written as

MðRÞBRd: ð9:7Þ

The number of “points” per unit area in each circle equally depends (mass—the

full number of points) on circle radius (d is the dimensionality of space). The object,

for which the relation (9.7) connects the mass and size, is termed “compact.” The

yielded relation means that if the linear size of a compact object increases in R time

at the invariable form, its mass increases in Rd time. This scale relation mass�size

is intimately connected with intuitive understanding of dimensionality, and it is

the useful generalization on dimensionality of more than 3. We will consider that if

the mass of M and size R satisfy the relation (9.7), the scale relation for the density

looks like

ρ5M=Rd: ð9:8Þ

The connection between the mass of an object and its characteristic size R can

be defined more generally than in Eq. (9.7). One statement of definition of the frac-

tal dimensionality is based on the relation

MðRÞBRdf : ð9:9Þ

The object is termed “fractal” if it satisfies to the relation (9.9) with the value

df, which is smaller than the space dimensionality d. If the relation (9.9) is fulfilled

for the object, its density will not be identical at all values R, and is scaled thus:

ρðRÞBM=RdBRdf2d: ð9:10Þ


As df, d, the density of the fractal object decreases with size magnification.

The yielded scale dependence of the density serves as the quantitative measure of a

rarefying or branching of the fractal objects. Another method of the description of

the fractals is based on the supposition that it contains the hollows of all sizes.

The percolation cluster shown in Figure 9.12C is an example of the casual or

statistical fractal, because the mass�size relation (9.7) is fulfilled only “on the

average” for it. This means that the average dependence M(R) is obtained on many

clusters and different initial points in the cluster. The relation (9.9) is fulfilled

not for any scales of length in all real physical systems, but within the limits of

restriction (top and bottom) of these scales. For example, the lower boundary of

length is stipulated by this or those microscopic distances, such as the lattice con-

stant or average distance between constituents of the object. The upper length is

routinely defined by the finite size of system. Presence of the specified limits com-

plicates the evaluation of the fractal dimensionality. For example, the line has

dimensionality 1. However, the line is figured as the long rectangle at large image

magnification.

The method of cells [11] is one of the most widespread ways of definition of the

fractal dimensionality. In this method, the object which is explored, is coated with

the net or lattice—the rectilinear segments in the case of line, quadrates with the

rib b in the case of the surface and cubes in the spatial case. The number of cells of

the net N(b), which belong to the object, is counted. Then more of the big net with

the new size of cells b15mb is chosen; its cells include m2 of primary cells. The

number of cells N(b1) belonged to the object (they have, at least, one point, the

cell, of the previous net, which was in the object) is also counted. The net is again

incremented and process of calculations of the magnitude N(bi) is retried. After

buildup of the bilogarithmic graph of the dependence of ln N(bi) on ln bi, the fractal

dimensionality is defined according to the equation

df 52Δðln NðbÞÞ=Δðln bÞ: ð9:11Þ

This equation will be applied below for the definition of the fractal dimensional-

ity using results of renormalization.

9.11.2 Fractal Dimensionality of Isolated Clusters

As the first task, it is recommended to calculate fractal dimensionality of the perco-

late clusters obtained by the Monte Carlo modeling of deposition of molecules

from the gas phase (see the application “Adsorption”). To obtain the convincing

proof of existence of the power dependence between ρ and R (see Eq. (9.10)) and

evaluate the fractal dimensionality with the comprehensible accuracy, the accuracy

of data not less than 1% from their magnitude is necessary. Fluctuations are large

enough during modeling of the small systems, and the data obtained should be

interpreted very cautiously.

The program for obtaining of clusters is given below in Pascal language (it

enters as one of procedures to the Delphi package). In this program, a consecutive


increasing of the area (in the form of quadrates), for which the density is calcu-

lated, is provided (procedure Show1 fulfills visualization of these quadrates).

Program Fractal; {P=2500 Pa, N_trial'=10000}

uses crt, Graph;

const Xn=33;Yn=33; v=1e13; m=0.028/6e23; s=4e-20;

var site:array[0..100,0..100] of integer;

dukT,kT,dt,dt1:real; //dukT=U/kT – binding energy with substratelp,p,p1,rt,pp,pe,sq1:real;

x,y,i,j,k,xyz,xyz1,xyz2,index,xn1,yn1,ii,jj,

N_trial,Gd,Gm:integer; Ni,W,a,b:word;

lnQ,Q,a,ln_a:array[0..10] of real; //density in the selected area

Procedure Show; //imaging of deposited atomsvar x,y:integer;

Begin

a:=10; b:=10;

for x:=1 to Xn do

for y:=1 to Yn do

begin

W:=site[x,y]+2; setfillstyle(1,W); bar(x*b,y*a,b*x+b,a*y+a);

end;

end;

Procedure Show1; //imaging of chosen area by another color

var x,y:integer; //xn1 – size of the chosen areabegin ClrScr;

a:=10; b:=10;

for x:=1 to Xn do

for y:=1 to Yn do

begin

if (x>round(Xn/2-xn1/2)) and (x<round(Xn/2+xn1/2))

and (site[x,y]=2)and (y>round(Xn/2-xn1/2)) and

(y<round(Xn/2+xn1/2)) then

W:=site[x,y]+3 else W:=site[x,y]+2;

setfillstyle(1,W); bar(x*b,y*a,b*x+b,a*y+a);

end;

end;

BEGIN

ClrScr; Randomize;

writeln('imput P, Pa'); Readln(Pp); //Pp– pressure

writeln('imput N_trial'); Readln(N_trial);// initial size of the chosen areadukT:=8; kT:=1.38e-23*300; sq1:=sqrt(m)*sqrt(6.283*kT);

pe:=v*exp(-dukT)*sq1/s; // Equilibrium pressure;dt:=sq1/(pp*s) // is the time of condensationof one atom in; v is the frequency of oscillations

Ni:=0; dt1:=exp(dukt)/v; //the time of evaporation

p1:=dt/dt1; p:=dt1/dt; {probabilities of deposition and evaporation of atoms}for x:=0 to xn do //initial surface without deposited atomsfor y:=0 to yn do

site[x,y]:=1; site[round(xn/2),round(xn/2)]:=2;

site[round(xn/2)+1,round(xn/2)]:=2;


9.11.3 Groups of Renormalization

The idea of study of some physical magnitudes in the neighborhood of critical

points at different scales of length was being applied not only to finite-dimensional

scaling but also was taken as a basis in the renorm-group method, which possibly

Gd=0; Gm=0; Initgraph (Gd,Gm''); Show;

for j:=1 to N_trial do // a cluster

begin

x:=random(Xn-1)+1; {random coordinates}y:=random(Yn-1)+1; rt:=random;

if (x>1) and (y>1) then begin

if pp>=pe then begin

if (site[x,y]=1) and ((site[x+1,y]=2) or

(site[x-1,y]=2) or (site [x,y-1]=2) or (site[x,y+1]=2))

then begin site[x,y]:=2; inc(ni);

goto 5; end;

if (site[x-1,y]+site[x+1,y]+site[x,y+1]+

site[x,y-1]>5) then p1:=p1*0.5;

if (site[x,y]=2) and (p1>rt) then

begin site[x,y]:=1; dec(ni);

end; end;

if pp<pe then begin

if (site[x,y]=1) and (p>=rt) and ((site[x+1,y]=2) or(site[x-

1,y]=2)or(site [x,y-1]=2)or (site[x,y+1]=2))

then begin

site[x,y]:=2; inc(Ni); goto 5; end;

if (site[x,y]=2) and (site[x-1,y]+site[x+1,y]+

site[x,y+1]+site[x,y-1]<7) then begin dec(Ni);

site[x,y]:=1; end; end;

5: end;

end; Show; Readkey;

xn1:=3; //3+5 is the initial size of the chosen areafor k:=1 to 5 do {the density determination in the selected areas}begin ClrScr;

xn1:=xn1+5; Q[k]:=0;

for x:=round(Xn/2-xn1/2) to round(Xn/2+xn1/2) do

for y:=round(Xn/2-xn1/2) to round(Xn/2+xn1/2) do

if site[x,y]=2 then Q[k]:=Q[k]+1;

Q[k]:=Q[k]/xn1/xn1; //densitylnQ[k]:=ln(Q[k]); a[k]:=xn1; ln_a[k]:=ln(xn1);Show1; Readkey; end;bar(380,380,400,400);. . . . . . . . . // plottingClosegraph;

for k:=1 to 5 do beginwriteln('Q= ',Q[k],'a= ',a[k]);writeln('lnQ= ',ln(lnQ[k]),'ln_a= ',ln(ln_a[k]));end;End.


was one of the major new methods of theoretical physics in the end of twentieth

century. Wilson suggested the first explanation of critical phenomena by means of

the renorm-group method in 1971. In 1981, he was awarded the Nobel Prize in

physics for the main contribution to working out of the renorm-group method. This

direct method of obtaining of critical indexes in junction with Monte Carlo techni-

ques often is the much more powerful tool than the usual Monte Carlo techniques.

If one sees the photo of the percolation configuration, which has been generated

on the square lattice for the value of filling probability p5 p0, pc, from the

increasing distance, it will be already impossible to see clusters, which consist of

one cell, and to distinguish the neighbor cells. Besides, narrow cross connections

between large clusters, in other words, those which connect large “spots,” will not

be distinguished in the photo from large distances. Hence, at p0, pc the photo

obtained from the large distance will be similar to the percolation configuration

generated for the value p1, p0. Besides, the length of connectivity ξðp1Þ of clusterswhich remained will be less than ξðp0Þ. If we depart further away, clusters in the

photo will seem as if they correspond to the value p5 p2, and p2, p1. In the end,

we are unable to distinguish any clusters, and the photo will seem as if it is made

for the trivial inconvertible point p5 0.

During examination of the photo of the configurations obtained with the value

of probability p0. pc, it would be possible to distinguish only small areas of unoc-

cupied cells. In the process of distance magnification, these hollows become less

noticeable in the photo and the configuration will look more and more homoge-

neous. Hence, the photo will look like the configuration generated for the value

p5 p1 with p1. p0. After considerable magnification of the distance, the photo

will seem as if it is obtained at the value p, which is equal to other trivial incon-

vertible point, p5 1.

If p05 pc, there are all scales of length, and this does not play a role in what

length we use for the system study. Therefore, the photo will look identical irre-

spective of distance, from which we consider it. In this sense, pc is the especial

nontrivial inconvertible point.

Consider the square net, which is divided into cells, the size of which in several

times is more than the size of primary cells of the net (Figure 9.13). If we look at

the net from the large distance, we will see that the knots are captured by the new

cell (all figured net), which forms the new superknot after renormalization.

Besides, the new net (with superknots) has the same symmetry, which the initial

Figure 9.13 Example of the cell with b5 4, used for the square lattice [8]. This net, which

contains b2knots, after renormalization will be turned into the single knot.


net had. However, a replacement of cells by new knots changes the scale length—

all distances decrease in b time, where b is the linear size of the cell. Thus, the

“renormalization” effect consists of replacement of several cells by the single

renormalized cell (knot) and rescaling of the length of connectivity for renorma-

lized lattices in b time. To conserve initial connectivity of lattice, we will consider

that the renormalized cell is engaged, if the initial group of the occupied cells con-

nects this cell.

Let us use, as a matter of convenience, the rule of vertical interlinking. The

effect of performance of the scale transformation of typical percolation configura-

tions for values p, which are more or less than pc, is illustrated accordingly in

Figure 9.14A,B. Here, initial configuration was renormalized three times with the

help of the program renorm, which transforms a four-cellular cell into one new

cell. In both cases, the transformations of renormalizing remove the system more

far from the state generated at pc. It is clear, that for p5 0.7, these transformations

revert the system to case with p5 1. If p5 0.5, there is the tendency for the system

to move to the state with p5 0.

For definition of parameters, which characterize the new configuration after

renormalization, it is admissible that each new cell is independent from the previ-

ous and is characterized only by the value p0—probability of that which is occu-

pied. As the transformation of renormalization is based on maintenance of the

basic property of percolation—binding together, we will consider the new knot

occupied, if it contains the joined cells, which “intersect” the initial knot. Hence, if

cells are engaged with probability p, the new knots are engaged with probability p0,where p0 is defined by the recursive relation or transformation of renormalization:

p0 5RðpÞ: ð9:12Þ

p = 0.7 p = 0.5L = 16

L = 4 L = 2 L = 4

L = 16L = 8 L = 8(A) (B)

Figure 9.14 Renormalization of configurations generated for two values of p: (A) p5 0.7

and (B) p5 0.5.


The function R(p) is the composite probability that the cells form the pairing

path. We will give an example, which makes clear the formal relation (9.12).

Figure 9.15 shows seven configurations which join, for the case of new cell with

b5 2. The probability p0 that the renormalizing cell will be occupied is equal to the

sum of probabilities of all possible variants:

p0 5RðpÞ5 p4 1 4p3ð1� pÞ1 2p2ð1� pÞ2: ð9:13Þ

The probability p0 of occupation of renormalizated cells differs from the proba-

bility p0 of occupation of initial cells. For example, assume that p05 0.5. After

realization of one renormalizing transformation, the value p0, obtained by

Eq. (9.13), is equal to p0 5 p15R(p05 0.5)5 0.44. If we fulfill the second transfor-

mation, we will obtain p25R(p1)5 0.35. It is easy to draw the deduction that the

subsequent application of transformation will approach system to the constant value

(point) p5 0. If p. pc, for instance, p5 0.7, we will hit the point p5 1 after conse-

cutive applying of transformations. For finding the nontrivial inconvertible points

corresponding to the critical threshold pc, it is necessary to find the value p that

satisfies to the equality p� 5R (p�). From the recursive relation (9.12), it follows

that the biquadratic equation solution concerning p� has two trivial inconvertible

points p� 5 0 and p� 5 1 and one nontrivial inconvertible point p� 5 0.61804, which

we will relate with pc. It is the calculated value p� for the case b5 2. Such value of

pc is close to the best-known estimation pc5 0.5927.

9.11.4 The Renorm-Group Calculation of the Fractal Dimensionality

The requirement of vertical interlinking at renormalization, which was mentioned

above, does not suit for the evaluation of fractal dimensionality. The rule of renor-

malization for use of Eq. (9.11) should provide the account of crates (new cells), in

which there is at least one cell of the previous net that belongs to the fractal object.

By such a rule, the single cells (e.g., the single adsorbed atoms), which do not enter

into the percolation clusters, will be counted. Therefore, the crates with one filling

Figure 9.15 Seven configurations with vertical junction, which will become a new occupy

cell with b5 2.


cell are considered in the program given below, but it is provided that the occupied

cell from the yielded crate contacts with at least to one occupied cell from the

neighbor crate (or on the vertical, or on the horizontal direction). In the operating

time of the program after “Start Fractal” button click (Figure 9.16), the number of

the occupied cells n is calculated on the primary net generated at p5 p1 and n0 isthe number of the occupied crates on the renormalized net with cells b’5mb. After

three renormalizing, fractal dimensionality is calculated under Eq. (9.11). The inter-

face of the program (Figure 9.16) allows us to introduce desirable values of the ini-

tial probability (p1) and to rewrite the outputted values n1 and n2 (after the first

renormalizing).

After “Start renorm” button click, renormalization of configurations is fulfilled

according to the requirement of the vertical interlinking of occupied cells. The inte-

ger variables j and i characterize its coordinates in the program.

The condition of the vertical interlinking with the upper cells:

if (matr[(2*i),(2*j)]*matr[(2*i),(2*j+1)]=1) OR

matr[(2*i+1),(2*j)]*matr[(2*i+1),(2*j+1)]=1)

then matr1[i,j]:=1;

Here, matr[i,j] is the array of coordinates of old net and the array matr1[i,j]gives indications of occupying of new cells after renormalization. Such renormali-

zation suits for the study of correlation of probabilities p and p1 and determination

of the critical value p1 (p15 pc).

Figure 9.16 Renormalizing of configurations for calculation of fractal dimensionality (the

button “Start fractal”).


The procedure for renormalization of configurations, which calculates the fractal

dimensionality (the button “Start fractal”) is given below:


begin

p1:=StrToFloat(Edit1.Text);pc:=StrToFloat(Edit2.Text);

d:=10; for i:=0 to 3 do n[i]:=0; randomize();

for i:=0 to 15 do

for j:=0 to 15 do

begin if Random < p1 then begin

Image1.Canvas.Brush.Color:=clBlack; matr[i,j]:=1;

end else begin // initial values of matr[i,j] equal 0 or 1Image1.Canvas.Brush.Color:=clWhite; matr[i,j]:=0;

end;

X1:=d*i; Y1:=d*j; X2:=X1+d; Y2:=Y1+d;

Image1.Canvas.Rectangle(X1, Y1, X2, Y2); //imaging end;

for i:=0 to 15 do

for j:=0 to 15 do

if matr[i,j]=1 then n[0]:=n[0]+1; // the number of black cellsd:=20;

for i:=0 to 7 do

for j:=0 to 7 do

begin //the first renormalizingif (matr[(2*i),(2*j)]*matr[(2*i-1),(2*j)]=1) or

(matr[(2*i+1),(2*j)]*matr[(2*i),(2*j-1)]=1)or

(matr[(2*i+1),(2*j)]*matr[(2*i+2),2*j]=1) or

(matr[(2*i),(2*j+1)]*matr[(2*i+1),(2*j+2)]=1)

then begin Image2.Canvas.Brush.Color:=clBlack;

matr1[i,j]:=1;

end else begin

Image2.Canvas.Brush.Color:=clWhite; matr1[i,j]:=0;

end;

X1:=d*i; Y1:=d*j; X2:=X1+d; Y2:=Y1+d;

Image2.Canvas.Rectangle(X1, Y1, X2, Y2); //imagingend;

for i:=0 to 7 do

for j:=0 to 7 do

if matr1[i,j]=1 then n[1]:=n[1]+1; // the number of black cellsd:=40;

for i:=0 to 3 do

for j:=0 to 3 do begin //the second renormalizingif (matr1[(2*i),(2*j)]*matr1[(2*i-1),(2*j)]=1) or

(matr1[(2*i+1),(2*j)]*matr1[(2*i),(2*j-1)]=1)or

(matr1[(2*i+1),(2*j)]*matr1[(2*i+2),2*j]=1) or

(matr1[(2*i),(2*j+1)]*matr1[(2*i+1),(2*j+2)]=1)

then begin

Image3.Canvas.Brush.Color:=clBlack; matr2[i,j]:=1;

end



1. Using the outcomes obtained after start of the procedure Fractal, define fractal

dimensionality according to Eq. (9.10) after buildup of graph of the dependence ln ρon ln R � (lnQ on ln_a in the Pascal program on page 350, a is the length of the rib

of the evolved quadrate). Fractal dimensionality depends on the magnitude of pressure

and probabilities p1 and p (program designations of probabilities of deposition and

breakoff of molecules), and also from the correction to p1, depending on the environ-

ment of the selected place.

2. Spend 10�20 starts of the procedure Fractal by the button “Start fractal” click for the

value of probability p1 chosen by you. Note values of fractal dimensionality; find the

average value and the mean-square error. Generate the percolation configuration with

p5 0.5927—the percolation threshold on the square lattices.

else begin


end;

X1:=d*i; Y1:=d*j; X2:=X1+d; Y2:=Y1+d;

Image3.Canvas.Rectangle(X1, Y1, X2, Y2); //imagingend;

for i:=0 to 3 do

for j:=0 to 3 do

if matr2[i,j]=1 then n[2]:=n[2]+1; // the number of black cellsd:=80;

for i:=0 to 1 do

for j:=0 to 1 do begin //the third renormalizingif (matr2[(2*i),(2*j)]*matr2[(2*i),(2*j+1)]=1) or

(matr2[(2*i+1),(2*j)]*matr2[(2*i+1),(2*j+1)]=1)or

(matr2[(2*i),(2*j)]*matr2[(2*i+1),(2*j)]=1) or

(matr2[(2*i),(2*j+1)]*matr2[(2*i+1),(2*j+1)]=1)

then begin

Image4.Canvas.Brush.Color:=clBlack; matr3[i,j]:=1;

end

else begin


end;

X1:=d*i; Y1:=d*j; X2:=X1+d; Y2:=Y1+d;

Image4.Canvas.Rectangle(X1, Y1, X2, Y2); //imaging end;

for i:=0 to 1 do

for j:=0 to 1 do

if matr3[i,j]=1 then n[3]:=n[3]+1;// the number of black cellsif(matr3[0,0]*matr3[0,1]=1)or matr3[1,0]*matr3[1,1]=1)

then Label2.Caption:='P= 1'else Label2.Caption:='P=0';

df:= ln(n[0]/n[3])/ln(8); //calculation of fractal dimensionLabel4.Caption:='df= '+FloatToStr(df); //displaying of df

Label8.Caption:='n1(16*16)= ' + IntToStr(n[0]);

Label10.Caption:='n2(8*8)= ' + IntToStr(n[1]);

End;


3. Note the deduced values of numbers n15 n[0] and n25 n[1]. Find root-mean-square

values ,n2. and ,n02.. Calculate fractal dimensionality for the case of one renom-

transformation and compare to the previous value.

4. Provide a graph of the dependence ln N (bi) from ln bi (see Eq. (9.11)) in the window pre-

pared by the tools of editor Chart (see Figure 9.16, the right part) and define fractal

dimensionality as the average tangent of the slope angle of the obtained line.

5. Start program Renorm by the button “Start Renom” click. Generate the percolation con-

figuration for p15 0.5927—the percolation threshold on the square nets. Note the termi-

nate value of probability p (0 or 1). Repeat clicking button “Start Renom” and note not

less than 20 finite values of the probability: 0 or 1.

6. Retry point 5 with another value of initial probability p1.Outcomes of evaluations according to tasks 5 and 6 uncover the geometrical nature of

the surface ordering.

9.12 Complex Analysis of Microstructures

Computer realization of methods used for estimation of structure parameters of

materials is the much-claimed technique for a long time [8,12]. Together with tradi-

tional characteristics such as the graininess, average aggregate size, the breadth of

boundaries between grains, and the distribution of grains on sizes, one new charac-

teristic is considered recently—fractal dimensionality [12,13]. Fractal dimensional-

ity shows how compactly the object fills the space in which it exists [9,14].

In connection with progress in the numeral photo technique, the up-to-date opti-

cal microscopes give the numeral images with full numerical information about a

color of every pixel. It opens many opportunities of the computer analysis of

images for calculation of characteristics of observable structures. Different compu-

tational methods of fractal dimensionality (the previous application see) are fea-

tured in the literature. In works [12,15,16], examinations on the determination of

correlations between values of fractal dimensionality and mechanical characteris-

tics of materials are carried out. The computerized process of definition of the

material structure was offered in Refs. [12,17,18]. However, from the description

of corresponding algorithms, it follows that in the majority, they go out of the pro-

cedures used earlier for the analysis of pictures of the old photo seal, imposing the

certain nets or drawing selective points. Thus, not all information noted in numeral

files was used.

Algorithms, which allow defining the wide spectrum of characteristics of micro-

structures, are practically instantaneously fulfilled by our program [19]. The pro-

gram is made by tools of Delphi. Therefore, start of the ready file with expansion

“.exe” does not demand installation of the additional programming environment on

the computer.

Figure 9.17 shows the program interface. At opening of the image file (with the

expansion “.bmp”) by the button click ( in Figure 9.17), the array map[x,y]

with the values of codes of color shades of the each picture point (pixel) is filled

and the image appears in the window “image”. After the button “picture” is

clicked, the image will be transformed to the four certain colors: black, green, light


blue and white (Figure 9.18B) which can correspond to different phases. The trans-

formed image comes up in the window “Paintbox”.

By means of “trackbars” (Figure 9.17, the trackbars are at the left), a correction

of the image reduced to four colors is spent (Figure 9.18) so that it is most like the

initial image (which is shown in the window “image”). It is essentially important

that the image in the window “Paintbox” is not mosaic, as it takes place at transfor-

mation of colors to a few quantity by the standard picture editors.

If after image correction there are separate merged sites between grains, which

are not parted by other color, it is possible to spend segments of straight lines by

means of the mouse, which will separate the grains by the white color. You bring

the cursor to the chosen place, press the mouse key, move the cursor, and let off

the key.

Further, it is necessary to choose in the window “Edit.Imap” one of the basic

shades (the integer value of Imap) that corresponds to structural component, which

should be analyzed. If phase composition varies along one of coordinates (e.g.,

Figure 9.18 Consecutive transformations of image of the object: (A) the initial image; (B)

the image reduced to four colors; (C) the modified image; (D) the two-colored image after

performance of the procedure Cluster; (E) the image after renormalization by the procedure

Fractal.

Figure 9.17 The program interface.


after chemical�thermal treatment of the specimen surface), it is recommended to

press the button “Y-distribution” and to call the procedure for plotting of the phase

amount dependence on the coordinate along axis perpendicular to the surface.

By means of the button “Start cluster” click, one of the basic procedures is put

in action, which allows defining sizes of all grains of one phase and total length of

all its boundaries. The procedure works with the array dmap[x,y], which values are

equal 1 or 0 at the beginning (1 if map[x,y]5 Imap, for instance, color Black in

Figure 9.18C). The function of the recursion fulfills the main work of the proce-

dure. It is called in two variants. At work of the first variant, the recursion repairs

separate inserts in grains or image imperfections, for example, tracks of scratches

that have arisen at preparation of the specimen surface. The serial call of the sec-

ond variant of the recursion function allows writing into the corresponding arrays

the sizes of all grains and length of their boundaries. After the “Start cluster” but-

ton click, the two-colored image (Figure 9.17 and Figure 9.18D) appears in the

window “Paintbox” in which only one phase, which is under consideration, is

shown by the chosen (red) color.

The main results of the program work are written in the lower zone of the inter-

face (Figure 9.17). There is the value of the average grain size (in the ratio to lat-

eral size of the drawing) and the value of the ratio of the number of pixels related

to the grain boundaries to all pixels, which fall into the grains of the certain phase.

The last magnitude is the characteristic of branching of boundaries of the phase

and it is the more adequate characteristic of the structure than the fractal

dimensionality (see below). Simultaneously, the graph of distribution of grains on

their sizes appears in the “Chart1” window (switching of page of the interface

from the picture window to the image window is necessary). Hence, the basic char-

acteristics of the structure are already defined. The characteristics of the structure

found thus are useful more than fractal dimensionality, as the last depends on mag-

nifying the image. The line, as it is known, has the fractal dimensionality equal to

1, and the incremented image of the line—2, because it is already a rectangle.

For definition of fractal dimensionality, it is necessary to press the “Fractal” but-

ton. After that, there will appear some rotatory images (the final image is shown in

Figure 9.18E) with the allowing ability reduced two times each time, but incremen-

ted to the size of the former image (the number of transformations is set in the edit

“Edit.k” window). Sequential groups of four pixels are considered for such trans-

formation. If the value of the array dmap[x,y] elements, at least of one of them, is

equal to 1, four pixels are replaced by one incremented cell (new pixel) with the

value dmap[x/2,y/2]51.Also, as well as in the previous section, it is possible to consider the requirement

of the connectivity for clusters (grains) without considering of the isolated points.

It is already possible at the performance of the procedure “Cluster”. The graph of

dependence Ln(n(dmap51)) on Ln(b), where b is the size of the net cells, appears

in the “Chart2” window on the picture page of the interface. Fractal dimensionality

is calculated under Eq. (9.7) given in the previous application:


Frac_dimens:=0.001*round((abs((ln(frac[kmax])-

ln(frac[1]))/ln(scale)))*1000);,

where scale5 2kmax, and its value is mapped in the lower zone of the interface

(value of fractal dimensionality for the given structure is equal to 1.496).

The program work on definition of fractal dimensionality was checked using the

images of the known generated fractals: Koch’s snowflake, the delta circuit of

Serpinsky, the blanket of Serpinsky. The discrepancy with known analytical values

did not exceed several percents. For obtaining the valid values at the recursion for

such objects as Koch’s snowflake and the figures of Serpinsky, it is necessary to

exclude the acting of the recursion function at the stage, when it removes the

scratches or spots (breaking up of the line in one place by means of the mouse is

enough in the case of Koch’s snowflake). Otherwise, empty triangles, circuits, or

quadrates will be filled with the chosen color and the higher value of the fractal

dimensionality will be obtained. The same problem concerns other hollow struc-

tures, for example, the images of sections of the nanotube.

As it is known, the image obtained by means of renom-transformation can be

the schematic characteristic of structure. Naturally, that it is necessary to press the

button “renorm”. Transformations are spent by the same scheme, as for fractal

dimensionality. However, a change of the element value of the array dmap[x/2,y/2]51 (by means of the auxiliary array) happens, if not less than two pixels from

four pixels correspond the yielded phase. If the quantities of those and other pixels

(2 and 2) are equal, an additional checkout is fulfilled, considering the value of thedmap[x,y] array elements in the points adjoining to yielded group of four pixels.

The final renom-image looks as though the photo of the structure was obtained

from the large distance (see examples from the previous application).


1. For several microstructures, define their basic characteristics: average grain size of the

certain phase, their branching (the roughness of boundaries), distribution of grains on

sizes, fractal dimensionality. Obtain the renormalized image.

2. Define the distribution of phase components depending on coordinates in the direction

normal to the surface for the specimens with the surface treatment or for the alloyed

materials. Determine average characteristics of grains of such structures.

9.13 How to Prepare Directives for Simulationswith LAMMPS

LAMMPS is the most popular code for simulations by the method of molecular

dynamics or by the Monte Carlo method. It is the perfect and correct program

within the limits of ability of these methods. LAMMPS is complex enough because

it contains a great number of files. Sometimes, it is more convenient to use your


own, more simple program, which can be quickly modified. However, it is impor-

tant to compare the results of simulations by your own program and the results

obtained with LAMMPS for some identical problem in checking the work of your

own program.

9.13.1 From Official LAMMPS Information

LAMMPS is a classical molecular dynamics code that models an ensemble of parti-

cles in a liquid, solid, or gaseous state. It can model atomic, polymeric, biological,

metallic, granular, and coarse-grained systems using a variety of force fields and


LAMMPS runs efficiently on a single-processor desktop or laptop machines, but

it is designed for parallel computers. It will run on any parallel machine that com-

piles C11 and supports the MPI message-passing library. LAMMPS is a freely

available, open-source code, distributed under the terms of the GNU Public License,

which means you can use or modify the code however you wish. LAMMPS was

originally developed under the US Department of Energy CRADA (Cooperative

Research and Development Agreement) between two DOE labs and three compa-

nies. It is distributed by Sandia National Labs (http://lammps.sandia.gov).

The common information on what systems may be simulated using LAMMPS is

given in the file “Section_intro.html” of documents in any version of LAMMPS.

Atom style commands include the list of particles, which can be considered, from

atoms and any molecules to finite-size spherical and ellipsoidal particles and hybrid

combinations of these.

Force fields are pointed by commands: pair style, bond style, angle style, dihedral

style, improper style, kspace style. All pairwise potentials are included in LAMMPS

as functions: Lennard-Jones, Buckingham, Morse, Born�Mayer�Huggins, Yukawa,

soft, hydrogen bond; or tabulated charged pairwise potentials: Coulombic and point-

dipole. All modern many-body potentials are in LAMMPS: EAM, Finnis/Sinclair

EAM, modified EAM (MEAM), embedded ion method (EIM), EDIP, ADP,

Stillinger�Weber, Tersoff, REBO, AIREBO, ReaxFF, COMB. Also, there are

coarse-grained potentials: DPD, GayBerne, REsquared, colloidal and DLVO meso-

scopic potentials: granular, Peridynamics, SPH, and many others.

Concrete atoms systems created in LAMMPS are pointed by commands: read_-

data (read in atom coords from files), lattice, create_atoms, delete_atoms, displa-

ce_atoms, replicate.

Fix commands give pointers for providing the ensemble type, constraints (ther-

mostatting options, pressure control, etc.), and boundary conditions (periodic or

walls of various kinds).

Run, run_style, or minimize commands give pointers about integrators. These

are most often the velocity-Verlet integrator for the MD simulations or the energy

minimization via conjugate gradient or steepest descent relaxation when using the

Monte Carlo method.

Dump and restart commands allow demanding to include in output files (the

dump files) information of various kinds that is the main goal of simulations: log


http://lammps.sandia.gov

file of thermodynamic info, text dump files of atom coordinates, velocities, other

per-atom quantities; binary restart files (for prolongation of calculations), per-atom

quantities (energy, stress, centro-symmetry parameter, CNA, etc); user-defined sys-

tem-wide (log file) or per-atom (dump file) calculations; spatial and time averaging

of per-atom quantities; time averaging of system-wide quantities; atom snapshots

in native, XYZ, XTC, DCD, CFG formats. For high-quality visualization, the

authors of LAMMPS recommend the following packages: VMD, AtomEye,

PyMol, Raster3d, RasMol.

LAMMPS requires as input a list of initial atom coordinates and types, molecu-

lar topology information, and force-field coefficients assigned to all atoms and

bonds.

For atomic systems LAMMPS provides a create_atoms command which places

atoms on solid-state lattices (fcc, bcc, user-defined, etc). Assigning small numbers

of force field coefficients can be done via the pair coeff, bond coeff, angle coeff

commands, and so on. For molecular systems or more complicated simulation

geometries, users typically use another code as a builder and convert its output to

LAMMPS input format, or write their own code to generate atom coordinate and

molecular topology for LAMMPS to read in.

There are now several freely available molecular dynamics codes, most of them

parallel, which can also be used in conjunction with LAMMPS to perform comple-

mentary modeling tasks: CHARMM, AMBER, NAMD, NWCHEM, DL_POLY,

Tinker. CHARMM, AMBER, NAMD, NWCHEM, and Tinker are designed pri-

marily for modeling biological molecules. CHARMM and AMBER use atom-

decomposition (replicated-data) strategies for parallelism; NAMD and NWCHEM

use spatial-decomposition approaches, similar to LAMMPS. Tinker is a serial code.

DL_POLY includes potentials for a variety of biological and nonbiological materi-

als; both a replicated-data and a spatial-decomposition version exist.

9.13.2 Some Package Command and Building of the Executable Filewith the GPU Package

If you use only CPU, there is no need to apply packages. You can download the

executable file from the LAMMPS website (http://lammps.sandia.gov/index.html),

and all necessary commands parameters, for writing the script file, will be loaded

from the pointed directory (in the command line) after you run the executable file.

However, the executable file for running with GPU must be compiled so that it can

take into account the architecture of your computer. For this purpose, the package

commands are applied. Currently, the following packages are used: GPU, USER-

CUDA, and USER-OMP. In this section, only using the GPU package will be con-

sidered. A certain version of CUDA software, which can work with your graphics

cards, have to be installed before using the packages. You can see more detailed

information in the Manual LAMMPS.pdf file.

The GPU style invokes options associated with the use of the GPU package.

The next settings specify the GPUs that will be used for simulation:


http://lammps.sandia.gov/index.html

package style args

style 5 gpu or cuda or omp

args 5 arguments specific to the style

gpu args 5 mode first last split keyword value

mode 5 force or force/neigh

first 5 ID of first GPU to be used on each node

last 5 ID of last GPU to be used on each node

split 5 fraction of particles assigned to the GPU

The split setting can be used for load balancing force calculation work between

CPU and GPU cores in GPU-enabled pair styles. If 0, split, 1.0, a fixed fraction

of particles is offloaded to the GPU, while force calculation for the other particles

occurs simultaneously on the CPU. If split, 0, the optimal fraction (based on CPU

and GPU timings) is calculated every 25 timesteps. If split5 1.0, all force calcula-

tions for GPU accelerated pair styles are performed on the GPU. In this case,

hybrid, bond, angle, dihedral, improper, and long-range calculations can be per-

formed on the CPU while the GPU is performing force calculations for the GPU-

enabled pair style. If all CPU force computations complete before the GPU,

LAMMPS will block until the GPU has finished before continuing the timestep.

The threads_per_atom keyword allows control of the number of GPU threads

used per-atom to perform the short range force calculation. By default, the value

will be chosen based on the pair style; however, the value can be set with this key-

word to fine-tune performance. For large cutoffs or with a small number of parti-

cles per GPU, increasing the value can improve performance. The number of

threads per atom must be a power of 2 and currently cannot be greater than 32:

keywords 5 threads_per_atom or cellsize

thr\eads_per_atom value 5 Nthreads

Nthreads 5 # of GPU threads used per atom

For performance reasons, you should not set Nthreads to more threads than there

are physical cores. Which combination of threads and MPI tasks gives the best per-

formance is difficult to predict and can depend on many components of your input.

The gpu/node keyword specifies the number N of GPUs to be used on each

node. The default value for N is 2—keywords 5 gpu/node or gpu/node/special or

timing or test or override/bpa:

gpu/node value 5 N

N 5 number of GPUs to be used per node

gpu/node/special values 5 N gpu1 .. gpuN

N 5 number of GPUs to be used per node

gpu1 .. gpuN 5 N IDs of the GPUs to use

The override/bpa keyword can be used to specify which mode is used for pair-

force evaluation. TpA 5 one thread per atom; BpA 5 one block per atom. If this

keyword is not used, a short test at the beginning of each run will determine which

method is more effective (the result of this test is part of the LAMMPS output).

override/bpa values 5 flag


flag 5 0 for TpA algorithm, 1 for BpA algorithm

The mode setting specifies where neighbor list calculations will be multi-

threaded as well. If mode is force, the neighbor list calculation is performed in

series. If the mode is force/neigh, a multithreaded neighbor list build is used. Using

the force/neigh setting is usually faster and should produce identical neighbor lists

at the expense of using some more memory (neighbor list pages are always allo-

cated for all threads at the same time and each thread works on its own pages). The

next example concerns the case of GPU package:

package gpu force 0 0 1.0

package gpu force 0 0 0.75

package gpu force/neigh 0 0 1.0

package gpu force/neigh 0 1 -1.0

If the “-sf gpu” (sf� suffix) command-line switch is used, then it is as if the

command “package gpu force/neigh 0 0 1” were invoked, to specify default settings

for the GPU package. If the command-line switch is not used, then no defaults are

set; one must specify the appropriate package command in the input script.

LAMMPS-specific code is in the GPU package. It makes calls to a generic GPU

library in the lib/gpu directory. This library provides NVIDIA support as well as

more general OpenCL support, so that the same functionality can eventually be

supported on a variety of GPU hardware.

As with other packages that include a separately compiled library, you need to

build the GPU library first, before building LAMMPS itself. Do the following,

using a Makefile in lib/gpu appropriate for your system. Compilation of this library

requires installing the CUDA GPU driver and CUDA TOOLKIT for your operating

system. Installation of the CUDA SDK is not necessary.

In addition to the LAMMPS library, the binary nvc_get_devices will also be built.

This can be used to query the names and properties of GPU devices on your system.

Example build process:

cd B/lammps/lib/gpu

emacs Makefile.linux

make -f Makefile.linux

./nvc_get_devices

cd ../../src

emacs ./MAKE/Makefile.linux

#make yes-asphere (#� this is not taken into account)

#make yes-kspace

make yes-gpu

make linux

If the file lib/libgpu.a. is produced, you can build LAMMPS with the GPU pack-

age installed:

cd B/lammps/src

make yes-gpu

make machine


When using GPUs, you are restricted to one physical GPU per LAMMPS pro-

cess, which is an MPI process running on a single core or processor. Multiple MPI

processes (CPU cores) can share a single GPU, and in many cases it will be more

efficient to run this way.

Additional input script requirements to run pair or PPPM styles with a gpu suf-

fix are the following:

To invoke specific styles from the GPU package, you can either append “gpu” to the style

name (e.g., pair_style lj/cut/gpu), or use the -suffix command-line switch, or use the suf-

fix command.

The newton pair setting must be off (setting the pairwise newton flag to off means that if

two interacting atoms are on different processors, both processors compute their interac-

tion and the resulting force information is not communicated).

The package gpu command must be used near the beginning of your script to control the

GPU selection and initialization settings. It also has an option to enable asynchronous

splitting of force computations between the CPUs and GPUs.

9.13.3 The LAMMPS Input Script

LAMMPS executes by reading commands from an input script (text file “in”), one

line at a time. When the input script ends, LAMMPS exits. Each command causes

LAMMPS to take some action. It may set an internal variable, read in a file, or run

a simulation. Most commands have default settings, which means you only need to

use the command if you wish to change the default.

In some cases, the ordering of commands in an input script is not important.

However, LAMMPS does not read your entire input script and then perform a sim-

ulation with all the settings. Rather, the input script is read one line at a time and

each command takes effect when it is read. Some commands are only valid when

they follow other commands. For example, you cannot set the temperature of a

group of atoms until the atoms have been defined. Sometimes command B will use

values that can be set by command A. This means command A must precede com-

mand B in the input script if it is to have the desired effect. For example, the

read_data command initializes the system by setting up the simulation box and

assigning atoms to processors. If default values are not desired, the processors and

boundary commands need to be used before read_data to tell LAMMPS how to

map processors to the simulation box.

Each nonblank line in the input script is treated as a command. LAMMPS com-

mands are case sensitive. Command names are lowercase, as are specified com-

mand arguments. Uppercase letters may be used in file names or user-chosen ID

strings.

If the last printable character on the line is an “&;” the command is assumed to

continue on the next line. The next line is concatenated to the previous line by

removing the “&” character and newline. This allows long commands to be contin-

ued across two or more lines.

All characters from the first “#” character onward are treated as comments and

are discarded.


The line is broken into “words” separated by whitespace (tabs, spaces).

Note that words can thus contain letters, digits, underscores, or punctuation

characters.

The first word is the command name. All successive words in the line are argu-

ments. If the argument is itself a command that requires a quoted argument, then

the double and single quotes can be nested in the usual manner. Only one of level

of nesting is allowed, but that should be sufficient for most use cases.

The “examples” directory in the LAMMPS distribution contains many sample

input scripts; the corresponding problems are discussed in the section “Examples”

and animated on the LAMMPS website.

A LAMMPS input script typically has four parts:

1. Initialization

2. Atom definition

3. Settings

4. Run a simulation

The last two parts can be repeated as many times as desired, that is, run a simu-

lation, change some settings, run some more, etc. Almost all the commands need

only be used if a nondefault value is desired.

1. Initialization

Set parameters that need to be defined before atoms are created or read-in from a file.

The relevant commands are units, dimension, newton, processors, boundary, atom_-

style, atom_modify.

If force-field parameters appear in the files that will be read, these commands tell

LAMMPS what kinds of force fields should be used: pair_style, bond_style, angle_style,

dihedral_style, improper_style.

2. Atom definition

There are three ways to define atoms in LAMMPS. Read them in from a data or

restart file via the read_data or read_restart commands (these files contain coordinates of

atoms and can contain molecular topology information), or create atoms on a lattice using

these commands: lattice, region, create_box, create_atoms.

3. Settings

Once atoms and molecular topology are defined, a variety of settings can be specified:

force field coefficients, simulation parameters, output options, etc. Force field coefficients are

set by these commands: pair_coeff, bond_coeff, angle_coeff, dihedral_coeff, improper_coeff,

kspace_style, dielectric, special_bonds. Various simulation parameters are set by these com-

mands: neighbor, neigh_modify, group, timestep, reset_timestep, run_style, min_style,

min_modify.

Fixes impose a variety of boundary conditions, time integration, and diagnostic

options. The fix command comes in many flavors: fix, fix_modify, unfix.

Various computations can be specified for execution during a simulation using the

compute, compute_modify, and variable commands.

Output options are set by the thermo, dump, and restart commands: dump, dump

image, dump_modify, restart, thermo, thermo_modify, thermo_style, undump,

write_restart.


4. Run a simulation

A molecular dynamics simulation is run using the run command. Energy minimization

(molecular statics) is performed using the minimize command. There are also command

for actions:

delete_atoms, delete_bonds, displace_atoms, change_box, minimize, neb prd, rerun,

run, temper

A full list of commands is in the Manual_LAMMPS.pdf file.

There are four basic kinds of LAMMPS output:

1. Thermodynamic output, which is a list of quantities printed every few timesteps to the

screen and logfile.

2. Dump files, which contain snapshots of atoms and various per-atom values and are

written at a specified frequency.

3. Certain fixes can output user-specified quantities to files: fix ave/time for time averag-

ing, fix ave/spatial for spatial averaging, and fix print for single-line output of

variables.

4. Restart files. You can point generate any number of dump files and fix output files,

depending on what dump and fix commands you specify.

The frequency and format of thermodynamic output is set by the thermo, ther-

mo_style, and thermo_modify commands. The thermo_style command also speci-

fies what values are calculated and written out. Predefined keywords can be

specified (e.g., press, etotal). Three additional kinds of keywords can also be speci-

fied (c_ID, f_ID, v_name), where a compute or fix or variable provides the value

to be output. In each case, the compute, fix, or variable must generate global values

for input to the thermo_style custom command.

Dump file output is specified by the dump and dump_modify commands. There

are several predefined formats (dump atom, dump xtc, etc.).

There is also a dump custom format where the user specifies what values are out-

puts with each atom. Predefined atom attributes can be specified (id, x, fx, etc.). Three

additional kinds of keywords can also be specified (c_ID, f_ID, v_name—compute

or fix or variable) provides the value to be output. In each case, the compute, fix, or

variable must generate per-atom values for input to the dump custom command.

There is also a dump local format where the user specifies what local values to

output. A predefined index keyword can be specified to enumerate the local values.

Two additional kinds of keywords can also be specified (c_ID, f_ID), where a com-

pute or fix or variable provides the values to be output. In each case, the compute

or fix must generate local values for input to the dump local command.

Several fixes take various quantities as input and can write output files: fix ave/

time, fix ave/spatial, fix ave/histo, fix ave/correlate, and fix print. The fix ave/time

command enables direct output to a file and/or time-averaging of global scalars or

vectors. The user specifies one or more quantities as input. The fix ave/spatial com-

mand enables direct output to a file of spatial-averaged per-atom quantities like

those output in dump files, within one-dimensional layers of the simulation box.

The per-atom quantities can be atom density (mass or number) or atom attributes

such as position, velocity, and force. They can also be per-atom quantities


calculated by a compute, by a fix, or by an atom-style variable. The spatial-

averaged output of this fix can also be used as input to other output commands.

The fix ave/histo command enables direct output to a file of histogrammed

quantities, which can be global or per-atom or local quantities. The histogram out-

put of this fix can also be used as input to other output commands. The fix ave/cor-

relate command enables direct output to a file of time-correlated quantities, which

can be global scalars. The correlation matrix output of this fix can also be used as

input to other output commands. The fix ave/atom command performs time-

averaging of per-atom vectors. The per-atom quantities can be atom attributes such

as position, velocity, force. They can also be per-atom quantities calculated by a

compute, by a fix, or by an atom-style variable. The time-averaged per-atom output

of this fix can be used as input to other output commands.

LAMMPS has several options for computing temperatures, any of which can be used

in thermostatting and barostatting. These compute commands calculate temperature,

and the compute pressure command calculates pressure. Thermostatting in LAMMPS is

performed by fixes, or in one case by a pair style. Four thermostatting fixes are currently

available: Nose�Hoover (nvt), Berendsen, Langevin, and direct rescaling (temp/

rescale), for example, fix temp/berendsen, fix langevin, fix temp/rescale. Fix nvt only

thermostats the translational velocity of particles. Only the nvt fixes perform time inte-

gration, meaning they update the velocities and positions of particles due to forces and

velocities respectively. The other thermostat fixes only adjust velocities; they do NOT

perform time integration updates. Thus, they should be used in conjunction with a con-

stant NVE integration fix such as these: fix nve, fix nve/sphere, fix nve/asphere.

Barostatting in LAMMPS is also performed by fixes. Two barosttating methods

are currently available: Nose�Hoover (npt and nph) and Berendsen: fix npt, fix

npt/sphere, fix npt/asphere, fix nph, fix press/berendsen.

The fix npt commands include a Nose�Hoover thermostat and barostat. Fix nph

is just a Nose�Hoover barostat; it does no thermostatting. Both fix nph and fix

press/bernendsen can be used in conjunction with any of the thermostatting fixes.

As with the thermostats, fix npt and fix nph only use translational motion of the

particles in computing T and P and performing thermo/barostatting. Fix npt/sphere

and fix npt/asphere thermo/barostat using not only translation velocities but also

rotational velocities for spherical and aspherical particles. As with the thermostats,

the Nose/Hoover methods (fix npt and fix nph) perform time integration. Fix press/

berendsen does NOT, so it should be used with one of the constant NVE fixes or

with one of the NVT fixes.

If one wants to view these values of T and P, one needs to specify them explic-

itly via a thermo_style custom command.

Below, the checked script for parallel computing of binary system (the Al�Ni

alloy, many thousands of atoms) with EAM potentials using GPU package is given

(comments to lines are given in lines marked by the sign #, the commands, which

are not in using now but can be included, are written with two sign # from the left).

newton off

package gpu force/neigh 0 0 1 threads_per_atom 2


#This specifies default settings for the GPU package (the threads_per_atom is chosen

#based on the pair style).

units metal

#This command sets the style of units used for a simulation.

Boundary s s p

#This sets the style of boundaries for the global simulation box in each dimension.

#The style p means the box is periodic; the styles f, s, and m mean the box is

nonperiodic.

atom_style atomic

read_data data.my

#Read in a data file that needs to run a simulation.

#The file can be ASCII text or a g-zipped text file (detected by a .gz suffix).

##lattice bcc 2.88

#Define a lattice for use by other commands; this does not contradict to read of

#coordinates from the data file.

##region box block 0 100 0 100 0 100

##create_box 1 box

##create_atoms 1 box

##set group all type 1

## These four commands are necessary if the data file is not included.

pair_style eam/alloy/gpu

pair_coeff � � Mishin-Ni-Al-2009.eam.alloy Ni Al

#The pair_coeff style specifies the pairwise force field coefficients for one or more pairs

#of atom types; style eam computes pairwise interactions for metals and metal alloys

#using embedded-atom method (EAM) potentials.

#There are several WWW sites that distribute and document EAM potentials stored in

#DYNAMO or other formats: http://www.ctcms.nist.gov/potentials;

#http://cst-www.nrl.navy.mil/ccm6/ap;

#http://enpub.fulton.asu.edu/cms/potentials/main/main.htm

##mass 1 58.71

##mass 2 26.9

#Molar masses are read from the data file with the EAM tabulated potentials.

velocityall create 1625.0 5812775

#These are the initial velocities of atoms for the temperature 1625 K using a random

#number generator

neighbor 2 bin

#This command sets parameters that affect the building of pairwise neighbor lists.

neigh_modify delay 5

#The delay setting means never build a new list until at least N steps after the previous

#build.

##compute 3 all pe/atom pair

##compute 4 all cna/atom 3.0

fix 1 all nvt temp 1625.0 1625.0 0.01

##fix 1 all nve

#These commands perform time integration on Nose�Hoover style non-Hamiltonian

#equations

#to generate positions and velocities sampled from the canonical nvt, npt, nph and nve

#ensembles. This updates the position and velocity for atoms in the group each

#timestep. The desired temperature at each timestep is a ramped value during the


http://www.ctcms.nist.gov/potentials

http://cst-www.nrl.navy.mil/

http://enpub.fulton.asu.edu/cms/potentials/main/main.htm

#run from Tstart to Tstop (T5Const in this example).

timestep 0.002

#Set the timestep size for subsequent molecular dynamics simulations in ps.

#Style custom allows you to specify a list of atom attributes to be written to the dump

#file for each atom. Possible attributes will appear in the order specified after every

#10000 timesteps.

run 100000

#Run or continue dynamics for a specified number (100000) of timesteps.

References

[1] A.A Chernov, Modern Crystallography Volume 3, Crystal Growth, Springer, Berlin,

1984.

[2] C.E. Miller, J. Crystal Growth 42 (1977) 357.

[3] M. Elwenspoek, P. Bennema, J.P. Van-der-Eerden, J. Crystal Growth 83 (1987) 297.

[4] A.M. Ovrutsky, Soviet Phys. Crystallogr. 30 (1985) 555.

[5] M. Elwenspoek, J.P Van-der-Eerden, J. Phys. A: Math. Gen. 20 (1987) 669.

[6] A.M. Ovrutsky, I.G. Rasin, Trans. JWRI 30 (2001) 239.

[7] V.V. Voronkov, Crystal Growth, vol. 9, Nauka, Moscow, 1972;257 p.

[8] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York, San

Francisco, 1983480 p.

[9] S.A. Saltykov, Stereometric Geometry, Metalurgiya, Moscow, 1976;270 p. (in Russian).

[10] V.I. Bolshakov, V.N. Volchuk, Ju.I. Dubrov, Fractal in Materials Technology, PGSA,

Dnepropetrovsk, 2005;253 p. (in Russian).

[11] R.M. Kronover, Fractals and Chaos in Dynamic Systems, (The lane with engl.)

POSTMARKER, Moscow, 2000;127 p.

[12] V.I. Bolshakov, V.N. Volchuk, J.I. Dubrov, Fractal in Materials Technology, PGSA,

Dnepropetrovsk, 2005;253 p.

[13] V.S. Ivanov, A.S Balankin, I.Z Bunin, A.A Oksogoev, Synergetics and Fractals in

Materials Technology, Science, Moscow, 1994;383 p.

[14] E. Feder, Fractals, Plenum, New York, NY, 1988;283 p.

[15] E. Hornbogen, Ztschr. Metallk. 78 (5) (1987) 352.

[16] G.A Malygin, Phys. Metals Metallurgical Sci. 68 (5) (1990) 22 (in Russian).

[17] S. Wolfram, Physica D. 10 (1984) 1�35.

[18] V.U Novikov, D.V Kozitsky, V.V. Jundunov, Mater. Technol. (1) (2001) 4 (in Russian).

[19] K.I. Dorogan, A.M Ovrutsky, Building, Metallurgical Science, Mechanical

Engineering, vol. 3, PGSA, Dnepropetrovsk, 2009;64 p. (in Russian).





























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