Numerical Calculations of the Bias Factor of an Edge Dislocation …560267/FULLTEXT01.pdf · 2012. 10. 12. · kristallina material sv aller n ar de uts atts f or bestr alning. Kantdis-lokationens

Numerical Calculations of the

Bias Factor of an Edge

Dislocation

Karl Samuelsson

Reactor Physics Department

Royal Institute of Technology

Stockholm, Sweden

Abstract

The ability to predict the behavior of the physical and mechanical

properties of reactor components during operation is of great impor-

tance. It is believed that the edge dislocation’s tendency to preferably

attract and annihilate interstitial atoms over vacancies is one of the

reasons for the swelling of crystalline materials under irradiation. The

bias factor of an edge dislocation can be calculated by solving the dif-

fusion equation with a drift term if the interaction energy between

the dislocation and a mobile defect is known; however, an analytical

solution only exists for some idealized interaction energies. Due to the

development of faster and more powerful computers, it is possible to

map the potential energy landscape surrounding an edge dislocation.

Using the results from such calculations, the bias factor would have to

be calculated numerically. In this thesis, the possibility of numerically

calculating the bias factor of an edge dislocation using the Partial Dif-

ferential Equation Tool-boxTM in MATLAB has been investigated. In

order to have a reference value, in this method, an interaction energy

for which an analytical solution to the diffusion equation exists has

been used. This numerical method tends to overestimate the bias

factor as the interaction energy is increased.

Sammanfattning

Förmågan att kunna förutsäga hur reaktorkomponenters fysikaliska

och mekaniska egenskaper förändras under bestr̊alning är värdefull.

En kantdislokations benägenhet att attrahera och annihilera inter-

stitialer framför vakanser tros vara en av anledningarna till varför

kristallina material sväller när de utsätts för bestr̊alning. Kantdis-

lokationens biasfaktor (p̊a engelska bias factor) kan beräknas genom

att lösa diffusionsekvationen med en driftterm givet en interaktion-

senergi mellan kantdislokationen och en punktdefekt. En analytisk

lösning existerar emellertid endast för vissa idealiserade interaktion-

senergier. Tack vare utvecklingen av kraftfullare datorer är det möjligt

att kartlägga interaktionsenergin kring en kantdislokation. Ifall resul-

tat fr̊an s̊adana beräkningar skulle användas för att beräkna biasfak-

torn för en dislokation, skulle diffusionsekvationen behöva lösas nu-

meriskt. I denna avhandling har möjligheten att beräkna biasfaktorn

numeriskt för en kantdislokation genom att använda Partial Differ-

ential Equation Tool-boxTM i MATLAB undersökts. För att ha ett

referensvärde har en speciell interaktionsenergi betraktats, för vilken

det existerar en analytisk lösning till diffusionsekvationen. Metoden

som användes visar en tendens att överskatta biasfaktorn för högre

interaktionsenergier.

Acknowledgements

In this short section I would like to thank Nils Sandberg, who has been

my advisor during this work. Nils is also the one who introduced me

to the field of radiation material science. I would also like to thank

the people at the reactor physics department at KTH for providing

me with a helpful and rewarding work environment.

Contents

List of Figures vii

List of Tables ix

1 Introduction 1

1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Behavior of Materials Under Irradiation . . . . . . . . . . . . . . 2

1.3 Sinks and Point Defects . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 The Edge Dislocation . . . . . . . . . . . . . . . . . . . . . 4

1.3.2 The Burgers Vector . . . . . . . . . . . . . . . . . . . . . . 5

1.4 The Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 The Migration of Point Defects . . . . . . . . . . . . . . . . . . . 7

1.5.1 Sink Strength and Bias Factor . . . . . . . . . . . . . . . . 8

1.6 The Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . 10

1.7 The Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . 14

2 Method 17

2.1 Stating the Governing Equations . . . . . . . . . . . . . . . . . . 17

2.2 Geometry and Boundary Conditions . . . . . . . . . . . . . . . . 19

2.3 The Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 The Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 The Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5.1 The Triangular Mesh Resolution . . . . . . . . . . . . . . . 31

2.5.2 The Square Mesh Resolution . . . . . . . . . . . . . . . . . 31

2.6 Other Input Parameters . . . . . . . . . . . . . . . . . . . . . . . 31

v

CONTENTS

3 Results and Discussion 33

3.1 Calculating the Bias Factor . . . . . . . . . . . . . . . . . . . . . 33

3.2 The Triangular Mesh Resolution . . . . . . . . . . . . . . . . . . . 39

3.3 The Square Mesh Resolution . . . . . . . . . . . . . . . . . . . . . 41

4 Conclusions 43

Bibliography 45

vi

List of Figures

1.1 A vacancy and an interstitial atom. . . . . . . . . . . . . . . . . 2

1.2 An edge dislocation is created by inserting a half-plane into an

ideal lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 The Burgers vector. . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 The position of a point defect in polar coordinates with the dislo-

cation in origo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Size interaction energy. . . . . . . . . . . . . . . . . . . . . . . . . 12

1.6 Size interaction energy. . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Flowchart of the procedure of numerically calculating the bias factor. 18

2.2 The transition from 3-dimensional to 2-dimensional geometry of

an edge dislocation. . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Geometry of the solution space. . . . . . . . . . . . . . . . . . . . 22

2.4 Discretization of a geometry. . . . . . . . . . . . . . . . . . . . . . 23

2.5 Geometry of the discretized solution space. . . . . . . . . . . . . . 24

2.6 The procedure of approximating a solution to a differential equa-

tion using the finite element method. . . . . . . . . . . . . . . . . 28

2.7 Geometry in a square mesh. . . . . . . . . . . . . . . . . . . . . . 30

3.1 Results from geometry 1. . . . . . . . . . . . . . . . . . . . . . . . 34





3.6 Comparison between different triangular mesh sizes. . . . . . . . . 39

vii

LIST OF FIGURES

3.7 Comparison between different triangular mesh sizes. . . . . . . . . 40

3.8 Comparison between different square mesh sizes. . . . . . . . . . . 41

viii

List of Tables

2.1 A list of the different geometries used. . . . . . . . . . . . . . . . 20

2.2 Different triangle mesh resolutions. . . . . . . . . . . . . . . . . . 31

2.3 Different square mesh resolutions. . . . . . . . . . . . . . . . . . . 32

ix

LIST OF TABLES

x

1

Introduction

1.1 General Introduction

Ever since the birth of the first nuclear reactors in the 1940s, physicists have been

concerned about the impact on the properties of the materials in the reactor from

the high radiation levels that are inherently present during operation [1]. In the

case of nuclear reactors, the majority of this radiation is neutrons hailing from

atoms that undergo fission in the fuel. When a non-fissile crystalline material

is bombarded by neutrons, a collision between an atom and a neutron may, if

the neutron has enough kinetic energy, result in a displacement of the target

atom, i.e. the atom will be displaced from its original position in the lattice.

The displaced atom may, if enough energy was transferred to it, collide with and

displace another lattice atom, and this may continue in several steps, resulting

in a displacement cascade [2]. A displacement is equivalent to the creation of an

interstitial atom, an atom that is not positioned at a lattice site; and a vacancy,

the empty space where the displaced atom was previously positioned. This is

illustrated in figure 1.1 (it should be noted that in this work, all figures depicting

an atomic lattice shows the case of the simple cubic structure, which is chosen

to make the figures easier to understand). Interstitial atoms and vacancies are

known as point defects. Any crystalline material at a temperature above 0 K

will contain defects due to local fluctuations in the energy [2], however, the point

defect concentration is drastically increased during irradiation due to interaction

between radiation and lattice atoms. The consequences of the increased number

1

1. INTRODUCTION

vacancy

interstitial

Figure 1.1: A vacancy and an interstitial atom.

of point defects in the reactor material have been the subject of study from

both experimental and theoretical scientists for the last 70 years, and continue to

puzzle the scientific community in its quest for materials with greater radiation

resistance.

1.2 Behavior of Materials Under Irradiation

All crystalline material in a reactor, e.g. the fuel, the cladding, structural ma-

terial etc. undergoes physical changes during operation. These changes include

swelling, segregation, growing and phase change [2]. It is not not difficult to real-

ize that if a structural material in a reactor changes dimensions during operation,

it may pose a threat to the integrity and stability of that reactor. In addition

to the thermal creep that, due to the high temperatures and pressures present

in a reactor, in itself is a problematic behavior, irradiation creep may also occur

[3, 4]. This is something that, if not taken into account may further jeopardize

the safety of nuclear reactors. These phenomena all change the physical proper-

2

1.3 Sinks and Point Defects

ties of the material that is being irradiated, and changes in physical properties

might lead to changes in mechanical properties, such as decreased ductility [2].

If the structural material loses ductility it will become brittle, which is most un-

wanted during operation and especially in an accident scenario. It is thus is easy

to realize that the ability to predict the physical behavior of reactor components

during operation is of great value.

If materials for which radiation has little or no impact on the physical and me-

chanical properties were to be developed, nuclear power could potentially become

safer and more economically attractive [5]. It could also help the development of

the next generation of nuclear reactors, which are supposed to create less radioac-

tive waste. Radiation resistant materials could thus help improve nuclear energy

in the aspects where it is most commonly criticized: the safety aspect, the eco-

nomical aspect, and the environmental aspect. Perhaps it should be noted that

the development of the next generation of reactors does not have its motivation

from an economical standpoint since it is not expected to decrease the cost of

electricity [6], but rather from a safety and environmental related standpoint.

With this in mind, the potential positive impact on nuclear power combined

with scientific curiosity is the reason why efforts are made to expand and improve

the field of radiation material science.


Point defects in a material have a tendency to group together and form so-called

dislocations [7], and these dislocations have a tendency to attract free point de-

fects. Dislocations are also produced when the material is plastically deformed

[8]. When a point defect is included into a dislocation, it disappears and this

is called annihilation. Anything in a material that, when comes in contact with

mobile point defects annihilates them, is called a sink. Sinks can be dislocations,

but not necessarily; for example, voids, grain boundaries, and precipitates can

also act as sinks.

3

1. INTRODUCTION

1.3.1 The Edge Dislocation

In this work, the sink properties of the so-called edge dislocation are being studied.

An edge dislocation can be seen as the addition of one half-plane of atoms into

an atomic lattice, something that will lead to deformation of the atomic planes

around it. This could be visualized by taking a book, opening it, cutting out

half of a page, and closing the book. This causes the surrounding intact pages to

curve around the edge of the cut page. The distortion of the surrounding atoms

causes the mobile point defects to drift towards or away from the dislocation. In

figure 1.2 an edge dislocation is created by inserting a half-plane into an ideal

lattice.

Figure 1.2: An edge dislocation is created by inserting a half-plane into an ideal

lattice.

4


1.3.2 The Burgers Vector

A concept that is important when defining dislocations is the Burgers vector. It

is a measurement of how much and in what direction a lattice is deformed by a

dislocation. It is defined as follows [7]:

Imagine that a dislocation line is encircled by taking atom-to-atom steps (see

figure 1.3 to the left). Then imagine that the dislocation is removed, but that

the same “steps” are made (see figure 1.3 to the right). What previously was an

encirclement is no longer a closed circuit (known as a closure failure), and the

step that must be taken in order to re-close the circuit is the Burgers vector, b.

b

Figure 1.3: The Burgers vector.

5

1. INTRODUCTION

1.4 The Rate Equations

A simple way to describe the global concentration of interstitials and vacancies

in a material is the so-called chemical rate equations:

dCidt

= K0 −KivCiCv −KisCiCs, (1.1)

dCvdt

= K0 −KivCiCv −KvsCvCs (1.2)

where K0 is the production rate of point defects, Kiv is the interstitial-vacancy

recombination coefficient, Kis is the interstitial-sink reaction rate coefficient, and

Kvs the vacancy-sink reaction rate coefficient.

Point defects are produced either by interaction between atoms and radiation

(if the material is being irradiated), or by emission of point defects from sinks.

This rate of production will depend on the radiation levels and temperature, but

also on the concentration and size distribution of sinks.

A point defect is lost when it recombines with another opposing point defect,

or when it is absorbed by a sink. A point defect’s susceptibility to annihilation

at a sink depends on the type of sink. A sink that has a large attracting force

will absorb more point defects than one with a weak force. The force between a

sink and a point defect will not only depend on the kind of sink, but also on the

kind of point defect. In the case of the edge dislocation, which is the case being

studied in this work, the force between the dislocation and an interstitial atom is

larger than that between the dislocation and a vacancy. This means that an edge

dislocation will attract and absorb more interstitial atoms than vacancies, leaving

excess vacancies in the lattice, some of which will group together and form voids.

Another way to describe this would be to say that the interstitial-sink reaction

rate coefficient is larger than the vacancy-sink reaction rate coefficient.

This tendency for a dislocation to prefer to absorb interstitial atoms rather

than vacancies is believed to be one of the reasons for the swelling of irradiated

crystalline materials [9, 10].

6

1.5 The Migration of Point Defects

Here, it should be noted that if one tries to predict the swelling of nuclear fuel

under irradiation, the rate equations as written in equations 1.1 and 1.2 are not

sufficient. The swelling will depend on, among other things, the formation and

size distribution of voids in the material. The predictions of these values requires

the rate equations to be stated in a more sophisticated manner (see e.g. [11]).

In the rate equations, the loss of point defects due to sinks depends on the

sink concentration. One may instead want to study the properties of a certain

local sink, that is, regardless of the global sink concentration in the material.

For this reason, a property called the sink strength has been introduced. The

sink strength is a measurement of how well a specific sink attracts mobile point

defects. The concept of sink strength will be further explained in section 1.5.1,

but before that a few other concepts must be introduced.


The migration of mobile point defects in a crystalline material is governed by two

processes: diffusion and drift [12]. The diffusion term comes from the random

movement of the point defects in the material, and can be expressed by the

following equation:

j = −D∇Cα (1.3)

where j is the diffusion flux, D is the diffusion coefficient, and Cα is the concen-

tration of either interstitials or vacancies.

This equation is called Fick’s first law and states that there will be a flow of,

in this case, point defects in the direction where the point defect concentration

is lower. If the concentration is constant everywhere, ∇Cα will be zero and therewill be no net flow of point defects.

The second governing process, known as the drift term, arises from any dif-

ference in potential energy throughout the material. A change in the potential

7

1. INTRODUCTION

energy causes the point defect to drift towards the direction where the potential

energy is lower. This phenomenon can be compared with a ball rolling down a

hill, i.e. the direction where the potential energy is lower. In this work, it is

the stress field around a dislocation that causes the change in potential energy,

but the change can also be due to applied stress on the material, temperature

gradients or chemical potentials. Adding this drift term to the diffusion equation

above results in the following equation [9]:

j = −D∇Cα −DCα∇EkBT

=

{β ≡ 1

kBT

}= −D∇Cα − βDCα∇E (1.4)

where kB is Boltzmann’s constant and T is the absolute temperature. E is the

interaction energy between the point defect and the dislocation.

If one assumes steady-state point defect concentrations around the dislocation,

the following equation will describe the point defect flux:

dCαdt

= ∇ · j = 0 (1.5)

which, combined with equation 1.4 can be written as:

∇2(DCα) + β∇(DCα) · ∇E + βDCα∇2E = 0. (1.6)

1.5.1 Sink Strength and Bias Factor

If equation 1.4 is solved, and the flux of point defects is obtained, the total point

defect current, Jtot into a cylinder around the dislocation can be calculated by:

Jtot = −r0∫dSr0 r̂ · j. (1.7)

Here Sr0 is the dislocation surface, r̂ is the normal vector to the dislocation

surface, and r0 is the radius of the circle. This radius can be interpreted as the

distance at which a point defect is close enough to the dislocation to be absorbed.

The sink strength of a dislocation is defined by Wolfer [9] as the constant of

proportionality between the currents of point defects into a sink and the difference

8


in the diffusion potential function far from and at the sink. The diffusion potential

function, Φ, is defined as:

diffusion potential = Φ = DCαeE

kBT = DCαeβE. (1.8)

It should perhaps be noted that this potential does not have a meaningful physical

interpretation, but is merely a re-written form of DCα, which simplifies equation

1.4 into:

j = −e−βE∇Φ (1.9)

and equation 1.5 becomes:

∇2Φ = β(∇E) · ∇Φ. (1.10)

Thus, the sink strength, commonly called k2, can be written as:

k2 =Jtot

Φ∞ − Φ0. (1.11)

Wolfer continues to define the dislocation bias factor, Z, as the ratio between

sink strength with and without a stress field around the dislocation:

Z =k2

k20(1.12)

where k20 is also known as the ideal sink strength. In their paper [12], Wolfer

and Ashkin regard Z as “a factor which describes completely the effect of the

interaction between point defects and an edge dislocation on the steady-state defect

current. Thus we may appropriately call it the bias factor of an edge dislocation.”

The net bias, ∆B, is then defined as the ratio between the interstitial dislo-

cation bias factor and the vacancy dislocation bias factor minus one:

∆B =ZinterstitialZvacancy

− 1. (1.13)

Unfortunately, in this field there seems to be a lack in the standardization of the

nomenclature. What Wolfer calls net bias is sometimes referred to as the bias

9

1. INTRODUCTION

factor, and what Wolfer calls bias factor is sometimes called the sink capture

efficiency [13, 14]. Furthermore, in some cases the sink strength is denoted by a

Z [15]. This can easily lead to confusion.

1.6 The Analytical Solution

In the case of an edge dislocation, Z can not be calculated analytically if the

surrounding stress field is used in its entirety [12]. However, the majority [9], [16]

of the interaction between the edge dislocation and a point defect comes from the

so-called size interaction, which can be approximated as:

E(r, θ) = Esize = −(1 + ν)

(1− ν)µbV

3π

sin(θ)

r= A · sin(θ)

r(1.14)

where ν is Poisson’s ratio, µ is the shear modulus, b is the length of one Burgers

vector, and V is the relaxation volume. θ and r mark the position of the point

defect if the dislocation is in the origin (see figure 1.4). The resulting energy

landscape can be seen in figures 1.5 and 1.6. The relaxation volume can be seen

as the difference between the point defect volume before it is inserted into the

material, and the hole in the material into which the point defect is inserted

[10, 17]. This concept is probably easier to imagine if the material is seen as

an elastic medium instead of a lattice of atoms. In fact, the expression for the

size interaction is derived by replacing the atomic lattice with an isotropic elastic

medium with certain elastic moduli [16], which means that the size interaction

energy, as given by equation 1.14, is not exact in the real case of an atomic lattice.

If one only uses the size interaction in equation 1.4 , the steady-state solution

for j can be found and Z can be calculated.

Frank S. Ham describes in his paper [18] from 1959 a way to analytically

calculate the point defect current into an edge dislocation with the assumption

10


edge dislocation line

point defect

r

θ

y

x

Figure 1.4: The position of a point defect in polar coordinates with the dislocation

in origo.

that the size interaction is the only interaction between the point defect and the

dislocation. Ham defines the following function:

Ψ(r, θ) = CαeAβ sin(θ)/2r (1.15)

and puts it in equation 1.5 together with equation 1.4 , which results in:

∇2Ψ−(Aβ

2r2

2)2Ψ = 0 (1.16)

and has the solution:

einθ(αnIn

(Aβ

2r

)+ βnKn

(Aβ

2r

)). (1.17)

Here, n is an integer, αn and βn are constants, and In and Kn are modified Bessel

functions of the first and second kind.

11

1. INTRODUCTION

−15−10

−50

510

15

−15−10−5051015

−1.5

−1

−0.5

0

0.5

1

1.5

y [b]

x [b]

β⋅Esize

Figure 1.5: Size interaction energy.

Ham does not use this solution to calculate the so-called bias factor, instead

he uses it to calculate an “effective capture radius”, which he defines as the radius

a cylinder surrounding the dislocation line would have to have in order to absorb

point defects at the same rate, were the potential field around it zero. This result,

however, shall not be used in this work. The expression for the effective capture

radius derived by Ham is based on the following boundary conditions:

limr→0

Cα = 0, (1.18)

limr→0

r[∂Cα/∂r + βCα(∂E/∂r)] 6 0. (1.19)

12


−15−10−5051015

−15

−10

−5

0

5

10

15

β⋅Esize

x [b]

y [b]

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 1.6: Size interaction energy.

Wolfer and Ashkin [12] use a very similar approach, but with a slightly different

size interaction;

E(r, θ) = A · cos(θ)r

(1.20)

and a slightly different diffusion potential function, η :

η(r, θ) = DCeβE/2 (1.21)

together with their own boundary conditions:

η(r0) = D0C0e−Aβ cos θ/2r0 , (1.22)

13

1. INTRODUCTION

η(R) = D∞C∞eAβ cos θ/2R (1.23)

where R is half the average distance between two dislocations and r0 is the radius

of the dislocation.

Using this approach, Wolfer and Ashkin end up with an expression for the

dislocation bias factor that they define as in equation 1.12:

Zedge =ln(R/r0)

∞∑n=0

(2− δn0)(Kn(ξr0/R)In(ξr0/R)

− Kn(ξ)In(ξ)

) (1.24)where

ξ =

∣∣∣∣βA2r0∣∣∣∣ (1.25)

and A is a material-dependent constant (see equation 1.14).

1.7 The Numerical Solution

In order to solve the equation

j = −D∇Cα − βDCα∇E, (1.26)

one must first know the gradient of the potential field ∇(E), around the dis-location, which is not trivial. In recent years, due to the development of more

powerful computers, calculating the potential field has become possible. This is

done using a method called molecular statics. The idea behind this method of

calculating the interaction energy is to construct an ideal lattice consisting of a

large number of atoms, called a supercell [19], in a computer, where the electronic

configuration of every atom is known. Based on the interatomic interactions, all

atoms are iteratively moved to their respective “resting location”, i.e. where the

forces from the surrounding atoms cancel each other out. The total energy of the

system is then calculated. By knowing this energy as well as the total energy of a

cell consisting of an edge dislocation and a point defect located at a certain lattice

14

1.7 The Numerical Solution

site, it is possible to calculate the interaction energy between the dislocation and

the point defect (it is also necessary to know the total energies of the cells with

only a dislocation an only a point defect [20]). This procedure would have to be

repeated for all possible locations of the point defect in order to map the energy

landscape around the dislocation. A molecular statics model does not take into

account the thermal vibrations of the atoms, meaning that formally, the model

simulates the material at 0 K.

Using the molecular statics method will yield a discrete expression for the

energy surface [20, 21]. If one wants to solve the diffusion equation analytically,

one must first fit the potential to a continuous surface, and even then, it is not

necessarily possible to get an analytical solution, since it might not exist.

This suggests that a more convenient way to obtain the diffusion flux, and

ultimately the dislocation bias, is to use the discrete values of the potential energy

from molecular statics calculations and solve the diffusion equation numerically.

In this work, the possibility of using this approach is explored more closely by

calculating the bias factor of an edge dislocation using the Partial Differential

Equation Toolbox in MATLAB. The procedure of the calculation is described in

chapter 2.

15

1. INTRODUCTION

16

2

Method

As mentioned in the previous chapter, the goal of this work is to examine the pos-

sibility of numerically solving the steady-state diffusion equation with the drift

term, and using the solution to calculate the bias factor. In order to assess the

calculated bias factor using this method, it is compared with the bias factor that

is produced using the numerical solution of Ham’s method [18], which means that

only the size interaction is taken into account. The bias factor obtained using

Ham’s method is taken from [12]. The numerical solver used is the Partial Differ-

ential Equation Toolbox in MATLAB, and the calculation process is represented

by figure 2.1.

2.1 Stating the Governing Equations

The PDE solver used in the Partial Differential Equation Toolbox numerically

finds a u that solves the following differential equation:

−∇ · (c∇u) + au = f on Ω (2.1)

where c,a,f are user-defined functions and Ω is the user-defined geometry. As

stated in section 1.5.1, combining equation 1.4 with 1.5 and 1.8 results in the

equation:

17

2. METHOD

Create geometry with

boundary conditions

and discretize it

State the potential field on

the discretized geometry

State the diffusion

equation and solve

it on the geometry

using pdenonlin

State the diffusion

equation for the ideal

sink and solve it using

pdenonlin

Calculate the gradients

in the x- and y- directions

using pdegrad

Calculate the

point defect

currents and

sink strengths Calculate the bias factor, Z

E(x,y)

Φ(x,y)

Φ0(x,y)

Φ0(x,y), x yΦ0(x,y), Φ (x,y), Φ (x,y),

k2, k20

x y

Figure 2.1: Flowchart of the procedure of numerically calculating the bias factor.

∇2Φ = β(∇E) · ∇Φ.

If this is to be written on the form of equation 2.1 using the size interaction

energy, the parameters c, a and f become:

c = −1, (2.2)

18

2.2 Geometry and Boundary Conditions

f = Aβ

[ux

(−2xy

(x2 + y2)2

)+ uy

(x2 − y2

(x2 + y2)2

)], (2.3)

a = 0. (2.4)

In the case of the ideal sink, since there is no interaction between the dislocation

and the point defects, the parameters c0, a0 and f0 become:

c0 = −1, (2.5)

f0 = 0, (2.6)

a0 = 0. (2.7)

This differential equation is also known as Laplace’s equation.


The migration of point defects in a crystalline material is essentially a 3-dimensional

problem. However, in the case of an edge dislocation that absorbs point defects,

the problem can be regarded as 2-dimensional if one chooses to look at the dif-

fusion in a single atomic layer. This is only possible under the assumption that

that the edge dislocation line goes straight through the lattice. The transition

from 3-dimensional to 2-dimensional geometry is illustrated in figure 2.2.

The geometry on which the differential equation is solved in this work consists

of an annulus and can be seen in figure 2.3. The necessity for an inner radius comes

from the fact that the analytical expression for the size interaction (see equation

1.14) diverges as limr→0. This inner boundary is also regarded as where point

defects are absorbed by the dislocation. According to [9], the diffusion potential

function (see equation 1.8) is constant on the inner and outer boundary. In this

work the boundary conditions have been chosen to be as follows:

19

2. METHOD

Φ(r0) = 0 s−1, (2.8)

Φ(R) = 1 s−1. (2.9)

The Partial Differential Equation Toolbox has a GUI where the user can define

the geometry and boundary conditions. In order to better assess the usefulness

of the numerical method, a number of geometries with different inner and outer

radii are used. A typical geometry as seen in the Partial Differential Toolbox can

be found in figure 2.5 A list of all geometries used can be found in table 2.1.

r0 [b] R [b] R/r0

Geometry 1 2 200 100


Geometry 3 5 50 10



Table 2.1: A list of the different geometries used.

In every PDE solver that uses the finite element method, the domain where

the solution is calculated is divided, or discretized, into smaller parts. Once this is

done, assuming a 2-dimensional domain, the geometry will be defined by a mesh

consisting of triangles and nodes (the corners of the triangles), this is illustrated

in figure 2.4. The numerical solution of the equation is given at each node, which

means that the more triangles the geometry consists of, the higher the resolution.

Higher resolution generally corresponds to longer computation time.

20


Figure 2.2: The transition from 3-dimensional to 2-dimensional geometry of an

edge dislocation.

21

2. METHOD

r0

R

Ω

∂Ω

Figure 2.3: Geometry of the solution space.

22


trianglenode

Ω

discretization

∂Ω

Figure 2.4: Discretization of a geometry.

23

2. METHOD

−10 −5 0 5 10−10

−8

−6

−4

−2

0

2

4

6

8

10

x [b]

y [b]

Figure 2.5: Geometry of the discretized solution space. Note that not all of the

geometry is shown here, but only the area closest to the inner boundary.

24

2.3 The Solver

2.3 The Solver

The numerical solver in Partial Differential Toolbox uses the so-called finite el-

ement method (FEM) to numerically approximate the solution to differential

equations [22]. The procedure of solving a non-linear differential equation with

the finite element method can be summarized in figure 2.6, and is explained in

more detail in this section. The explanation for the procedure can be found in

[22].

In order to examine if u does indeed solve the differential equation:

−∇ · (c∇u) + au = f on Ω

one defines the function r:

r(u) = −∇ · (c(u)∇u) + a(u)u− f(u) (2.10)

which is equal to zero if u is the solution to the differential equation.

Multiplying r with an arbitrary test function v, and integrating it over the

domain Ω will, if u is the solution to the differential equation, result in:

∫Ω

−(∇ · (c(u)∇u))v + a(u)uv dx =∫

Ω

f(u)v dx. (2.11)

Using Green’s formula, the equation above can be re-written as:

∫Ω

(c(u)∇u) · ∇v + a(u)uv dx−∫∂Ω

n · (c(u)∇u)v ds =∫

Ω

f(u)v dx (2.12)

where n is the outward pointing unit normal vector. The boundary conditions

can be defined in the general form:

n · (c(u)∇u) + q(u)u = g(u) on ∂Ω. (2.13)

25

2. METHOD

Combining equation 2.12 with the boundary conditions defined for the geometry

yields the following expression:

∫Ω

(c(u)∇u) · ∇v + a(u)uv dx−∫∂Ω

(−q(u)u+ g(u))v ds =∫

Ω

f(u)v dx (2.14)

and if we want to find a solution to r(u) = 0, a u must be found that satisfies:

(∫Ω

(c(u)∇u) · ∇v + a(u)uv − f(u)v dx−∫∂Ω

(−q(u)u+ g(u))v ds)

= 0.

(2.15)

Up until this point, all functions are continuous, which is not the desired case if

one wants to solve the differential equation numerically. Hence, u and v must be

discretized. Let uh be a piecewise linear approximation of u, with as many pieces

as the geometry has nodes:

uh =Nn∑j=1

Ujφj and v = φi (2.16)

where Nn is the number of nodes that defines the discretized geometry, φj(x) is

1 at the j:th node and zero everywhere else, and U contains the values for uh at

every node.

Equation 2.15 can now be written in the discrete form:

Nn∑j=1

[∫Ω

Ujc∇φj · ∇φi + Uja · φj · φi dx−∫∂Ω

Ujq · φj · φi ds∫∂Ω

Ujqφiφj ds

]−

−∫

Ω

fφi dx−∫∂Ω

gφi ds = 0 for all indices i. (2.17)

If one introduces the following functions:

Ki,j =

∫Ω

c∇φj · ∇φi dx, (2.18)

Mi,j =

∫Ω

aφj · φi dx, (2.19)

26

2.3 The Solver

Qi,j =

∫∂Ω

qφj · φi ds, (2.20)

Fi =

∫Ω

fφi dx, (2.21)

Gi =

∫∂Ω

gφi ds (2.22)

equation 2.17 can be re-written as:

(K +M +Q)U − (F +G) = 0. (2.23)

Note that this is only true if u is the solution to the differential equation. The

residual of any solution vector U , regardless of whether or not it solves the dif-

ferential equation, is defined as

ρ(U) = (K +M +Q)U − (F +G). (2.24)

If the norm of ρ(U) is zero, then u is the solution to r(u) = 0.

The PDE solver needs an initial guess for U in order to start finding a solution.

If Un is the first guess, the solver calculates the residual vector for this proposed

solution, and if the residual norm is not close enough to zero, a better solution is

needed. An improved solution Un+1 can be found using the following procedure:(∂ρ(Un)

∂U

)(Un+1 − Un) = −αρ(Un) (2.25)

where α is known as the damping coefficient. If α is chosen to be small enough,

then:

∣∣∣∣ρ(Un+1)∣∣∣∣ < ||ρ(Un)|| (2.26)and Un+1 can be calculated as:

Un+1 = Un + αpn (2.27)

27

2. METHOD

where pn is called the descent direction and is defined as:

pn = ρ(Un) · 1(

∂ρ(Un)∂U

) . (2.28)The residual vector of Un+1 can now be calculated, and if necessary an improved

solution Un+2 can be obtained from Un+1. This iteration process can now be

performed until the residual norm is close enough to zero.

Calculate residual

vector ρ(Un)

Geometry and

boundary conditions

K,M,Q,F,G

Un

ρ(Un)

Calculate descent

direction pn Choose damping

coefficient α

Un + 1 = Un + α·pn

Guess U

Figure 2.6: The procedure of approximating a solution to a differential equation

using the finite element method.

2.4 The Gradients

Once an acceptable numerical solution has been obtained, the gradients of this

solution must be calculated in order to calculate the point defect current. The

28

2.4 The Gradients

Partial Differential Toolbox has a function (pdegrad) that, given a solution vec-

tor, returns the gradients of the solution in the x- and y-directions. Before the

point defect current can be calculated the gradient vectors are mapped onto a

quadratic mesh (see 2.7). This is done in order to make the flux calculations more

convenient: before the mapping, the gradients are given at each triangle center as

a vector with the same length as the number of triangles in the mesh. Since there

is no simple way to see which triangles are closest to the inner boundary (where

we want to calculate the flux), a transformation to a square mesh is performed.

This transformation is done by the MATLAB function tri2grid.

Once the gradients are mapped on the square mesh, the flux in the x- and

y-direction at every square can be calculated by:

jx(x, y) = −ux(x, y) · e−E(x,y), (2.29)

jy(x, y) = −uy(x, y) · e−E(x,y) (2.30)

and the total current of point defects is obtained by summing the flux at all

squares, nn (nearest neighbor), closest to the inner boundary:

Jtot = constant ·∑nn

jx · r̂x + jy · r̂y. (2.31)

Note that this is the point defect current multiplied by a constant that depends

on the inner radius and the resolution of the square mesh. However, this constant

is unimportant since the same constant shows up when calculating the ideal sink

current, causing the constant to cancel itself out when calculating the bias factor.

With the point defect currents being calculated, the sink strengths (remember:

one sink strength with and one without the interaction between the dislocation

and point defect) are simply calculated using the equation:

k2 =Jtot

Φ∞ − Φ0and the bias factor can finally be calculated by:

29

2. METHOD

Figure 2.7: Geometry in a square mesh. Notice that not all of the geometry

has been mapped onto the square mesh, but only the area closest to the inner

boundary.

Z =k2

k20.

2.5 The Resolution

The impact on the calculated bias factor, caused by changes in the resolution

has been investigated. Here, we talk about two different kinds of resolutions:

30

2.6 Other Input Parameters

the triangle mesh resolution and the square mesh resolution. The triangle mesh

resolution refers to the amount of triangles that defines the geometry on which the

differential equation is solved. The square mesh resolution refers to the number

of squares that the gradients are mapped onto.

All resolution comparisons have been made on the same geometry; the one

with an inner radius of 6 b and an outer radius of 600 b. This is Geometry 5 in

table 2.1.

2.5.1 The Triangular Mesh Resolution

The different triangle mesh resolutions used can be found in table 2.2. Each mesh

refinement splits every triangle into four new ones.

No. of triangles No. of nodes

Resolution 1 810 425




Resolution 5 207360 104000

Resolution 6 829440 415360

Table 2.2: Different triangle mesh resolutions used for the geometry with R = 600

b and r0 = 6 b (Geometry number 5 in table 2.1). Note that a higher resolution

index indicates a higher resolution.

2.5.2 The Square Mesh Resolution

The different square mesh resolutions used can be found in table 2.3.


In the calculations, different values for Aβ have been used, where A is a mea-

surement of the magnitude of the interaction between the point defect and the

31

2. METHOD

Mesh size Total no. of squares

Square mesh 1 20×20 400Square mesh 2 100×100 10000Square mesh 3 200×200 40000Square mesh 4 400×400 160000

Table 2.3: Different square mesh resolutions used for the geometry with R = 600

b and r0 = 6 b (Geometry number 5 in table 2.1). Note that a higher square mesh

index indicates a higher resolution.

dislocation (see equation 1.14), and β = 1kBT

. The different values used for Aβ

are: 1, 5, 10, 15, and 20 b (b being the length of one Burgers vector). These

values are typical values for some metals found in steels at half of their respective

melting temperatures [9].

32

3

Results and Discussion

Using the aforementioned method, numerical calculations of the bias factor of

an edge dislocation have been performed. All calculations have been performed

under the assumption that the size interaction is the only interaction present

between the point defects and the dislocation. The results of these calculations

have been compared to the expression for the bias factor that Wolfer and Ashkin

[12] derived using the method conceived by Ham [18]. The significance of the two

different resolutions mentioned in section 2.5 has also been investigated.

3.1 Calculating the Bias Factor

Results from the calculations of the bias factor for different geometries (see section

2.2) and interaction magnitudes (see section 2.6) are presented below. In all

cases, square mesh 2 (see table 2.3) has been used together with a triangle mesh

resolution approximately the same as resolution 6 in table 2.2 (the number of

triangles and nodes are not identical in the different geometries, but the meshes

have all been refined the same number of times). In all figures, the analytical

value refers to the expression found in equation 1.24.

As can be seen in figures 3.1 - 3.5, the numerical method used produces

bias factors that resemble their corresponding analytical values. However, as

the interaction magnitude increases, the method starts to overestimate the bias

factor. For all geometries, the largest overestimation is when |Aβ| is equal to 20 b.

33

3. RESULTS AND DISCUSSION

The largest of these overestimations is in the case of geometry 1 (see figure 3.1)

where the calculated bias factor is 4% larger than its corresponding analytical

value. This is the geometry that has the smallest inner radius, and thus the

largest interaction energy at the inner boundary.

1 5 10 15 201

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Comparison (R = 200b, r0 = 2b)

|A0β| [b]

Zedge

analytical valuenumerical calculation

Figure 3.1: Results from geometry 1. Note that the overestimation is larger for

larger interaction magnitudes.

34


1 5 10 15 201

1.05

1.1

1.15

1.2

1.25

|A0β| [b]

Zedge





35


1 5 10 15 201

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

|A0β| [b]

Zedge





36


1 5 10 15 201

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

|A0β| [b]

Zedge





37


1 5 10 15 201

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.15

|A0β| [b]

Zedge





38

3.2 The Triangular Mesh Resolution

3.2 The Triangular Mesh Resolution

The results from the triangular mesh resolution investigation are found in this

section. All calculations here have been performed using the square mesh 2 in

table 2.3.

0 5 10 15 201

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

|Aβ| [b]

Zedge

Comparison between different resolutions (R = 600b, r0 = 6b)

analytical valueresolution 1resolution 2resolution 3resolution 4resolution 5

Figure 3.6: Comparison between different triangular mesh sizes. A higher reso-

lution index corresponds to a higher resolution. The specifics of all triangle reso-

lutions can be found in table 2.2.

As seen in figures 3.6 and 3.6, the calculated bias factors seem to approach the

analytical values as the triangle mesh becomes finer. It is not entirely unexpected

that a better result is produced by increasing the number of nodes, since in the

39


MATLAB function pdegrad, the gradients are evaluated in each triangle center

by checking the value of the solution at each node [22]. This means that a finer

triangle mesh will produce a better approximation of the gradients.

0 5 10 15 201

1.05

1.1

1.15

|Aβ| [b]

Zedge

Comparison between different resolutions (R = 600b, r0 = 6b)

analytical valueresolution 5resolution 6

Figure 3.7: Comparison between different triangular mesh sizes. A higher reso-

lution index corresponds to a higher resolution. The specifics of all triangle reso-

lutions can be found in table 2.2.

40

3.3 The Square Mesh Resolution

3.3 The Square Mesh Resolution

In this section, results from the square mesh resolution investigation are pre-

sented. Here, triangle mesh resolution 6 (see table 2.2) has been used for all

calculations.

0 5 10 15 201

1.05

1.1

1.15

|Aβ| [b]

Zedge

Comparison between different square mesh resolutions (R = 600b, r0 = 6b)

analytical valuesquare mesh 1square mesh 2square mesh 3square mesh 4

Figure 3.8: Comparison between different square mesh sizes. A higher square

mesh index corresponds to a higher resolution. The specifics of all square mesh

resolutions can be found in table 2.3.

One interesting result from the mesh size investigation is that the bias fac-

tor does not approach the analytical value when the square mesh becomes finer

(given a fixed triangular mesh resolution), see figure 3.8. This tendency remains

unexplained, but could probably be avoided altogether if the square grid mapping

41


could be avoided. It should probably be noted that the square mesh resolution

that yields the best value (square mesh 2) contains roughly the same amount of

squares as the triangle mesh contains nodes. As mentioned in section 2.4, the

reason for the square mapping to begin with is the fact that there is no easy way

to know which triangle is closest to the boundary in the way that the triangle

mesh is defined in MATLAB. It is, however, still possible.

42

4

Conclusions

As previously stated, the purpose of this work was to investigate the possibility of

calculating the bias factor numerically given an interaction energy, e.g. calculated

from molecular statics (see [21] for example). This was done using an interaction

energy corresponding to a known analytical expression for the bias factor. The

results of this work indicate that, given a high enough refinement of the triangular

mesh, a value for the bias factor can be calculated. The value of the bias factor,

however, is overestimated for high interaction magnitudes. The overestimation is

reduced if the triangle mesh resolution is increased.

Perhaps the largest limitation connected to this method is the fact that the

Partial Differential Toolbox is not able to generate a triangular mesh on a ge-

ometry if the ratio between the outer and inner radius exceeds a certain value

(∼100). Part of the motivation for this work was to find a way to calculate thebias factor for an arbitrary 2-dimensional geometry, in addition to interaction

energy, and as long as this limitation remains, the method used in this work will

be unable to do so.

If this calculation was to be performed for an arbitrary interaction energy,

repeated calculations with increasing resolution would have to be carried out until

the bias factor stabilizes, i.e. does not change when the resolution is increased.

Some of the code would also have to be re-written in order to use a calculated

43

4. CONCLUSIONS

energy landscape as input. Furthermore, it would probably be worthwhile to find

a way to avoid the square mesh mapping used in this method.

Finally, it should be mentioned that there are probably other methods one

could use in order to calculate the bias factor. The question whether or not these

other methods are more useful than the one presented in this work (or if they even

exist) should be addressed , especially if results from this code, or one similar to

it, were to be used in any attempt to predict the behavior of irradiated materials.

This question, however, is not part of the scope of this work.

44

Bibliography

[1] E. P. Wigner, “Theoretical physics in the metallurgical laboratory of

Chicago,” Journal of Applied Physics, vol. 17, no. 11, pp. 857–863, 1946.

1

[2] G. S. Was, Fundamentals of Radiation Materials Science: Metals and Alloys.

Springer, 2007. 1, 2, 3

[3] L. Mansur, “Theory and experimental background on dimensional changes

in irradiated alloys,” Journal of Nuclear Materials, vol. 216, no. 0, pp. 97 –

123, 1994. 2

[4] K. Ehrlich, “Irradiation creep and interrelation with swelling in austenitic

stainless steels,” Journal of Nuclear Materials, vol. 100, no. 13, pp. 149 –

166, 1981. 2

[5] L. Hallstadius, S. Johnson, and E. Lahoda, “Cladding for high performance

fuel,” Progress in Nuclear Energy, vol. 57, no. 0, pp. 71 – 76, 2012. Nuclear

Materials: Selected articles from the E-MRS 2011 Spring Meeting. 3

[6] E. Schneider and W. C. Sailor, “Nuclear fission,” Science & Global Security,

vol. 14, no. 2-3, pp. 183–211, 2006. 3

[7] D. Hull and D. J. Bacon, Introduction to Dislocations, Fourth Edition.

Butterworth-Heinemann, 2001. 3, 5

[8] J. P. Hirth and J. Lothe, Theory of Dislocations. Krieger Pub Co, 1992. 3

[9] W. G. Wolfer, “The dislocation bias,” Journal of Computer-Aided Materials

Design, vol. 14, no. 3, pp. 403–417, 2007. 6, 8, 10, 19, 32

45

BIBLIOGRAPHY

[10] P. T. Heald and M. V. Speight, “Point defect behaviour in irradiated mate-

rials,” Acta Metallurgica, vol. 23, no. 11, pp. 1389–1399, 1975. 6, 10

[11] B. N. Singh, S. I. Golubov, H. Trinkaus, A. Serra, Y. N. Osetsky, and A. V.

Barashev, “Aspects of microstructure evolution under cascade damage con-

ditions,” Journal of Nuclear Materials, vol. 251, pp. 107–122, 1997. 7

[12] W. G. Wolfer and M. Ashkin, “Diffusion of vacancies and interstitials to edge

dislocations,” Journal of Applied Physics, vol. 47, no. 3, pp. 791–800, 1976.

7, 9, 10, 13, 17, 33

[13] I. W. Chen, “Anisotropic diffusion of point defects to edge dislocations,”

Journal of Nuclear Materials, vol. 125, no. 1, pp. 52–63, 1984. 10

[14] S. Golubov, A. Barashev, B. Singh, and R. Stoller, “Dislocation bias revis-

ited.” Fusion Materials Semiannual Progress Report for the Period Ending

June 30, 2010, DOE-ER-0313/48, Prepared by Oak Ridge National Labora-

tory for the U.S. Department of Energy. 10

[15] D. Mordehai and G. Martin, “Enhanced annealing of the dislocation network

under irradiation,” Physical Review B - Condensed Matter and Materials

Physics, vol. 84, no. 1, 2011. 10

[16] R. Bullough and R. C. Newman, “The kinetics of migration of point defects

to dislocations,” Reports on Progress in Physics, vol. 33, no. 1, pp. 101–148,

1970. 10

[17] E. J. Mittemeijer, Fundamentals of Materials Science: The Microstructure-

Property Relationship Using Metals as Model Systems (Graduate Texts in

Physicsna). Springer, 2011. 10

[18] F. S. Ham, “Stress-assisted precipitation on dislocations,” Journal of Applied

Physics, vol. 30, no. 6, pp. 915–926, 1959. 10, 17, 33

[19] C. Domain, C. S. Becquart, and J. Foct, “Ab initio study of foreign intersti-

tial atom (c, n) interactions with intrinsic point defects in α-Fe,” Phys. Rev.

B, vol. 69, p. 144112, Apr 2004. 14

46

BIBLIOGRAPHY

[20] E. Clouet, “The vacancy-edge dislocation interaction in fcc metals: A com-

parison between atomic simulations and elasticity theory,” Acta Materialia,

vol. 54, no. 13, pp. 3543–3552, 2006. 15

[21] E. Kuramoto, K. Ohsawa, and T. Tsutsumi, “Computer simulation of defects

interacting with a dislocation in Fe and Ni,” Journal of Nuclear Materials,

vol. 283-287, no. PART II, pp. 778–783, 2000. 15, 43

[22] “Partial Differential Equation ToolboxTM User’s Guide.” http://www.

mathworks.com/help/pdf_doc/pde/pde.pdf, June 2012. MathWorks, Inc.

25, 40

47

http://www.mathworks.com/help/pdf_doc/pde/pde.pdfhttp://www.mathworks.com/help/pdf_doc/pde/pde.pdf

List of FiguresList of Tables1 Introduction1.1 General Introduction1.2 Behavior of Materials Under Irradiation1.3 Sinks and Point Defects1.3.1 The Edge Dislocation1.3.2 The Burgers Vector

1.4 The Rate Equations1.5 The Migration of Point Defects1.5.1 Sink Strength and Bias Factor

1.6 The Analytical Solution1.7 The Numerical Solution

2 Method2.1 Stating the Governing Equations2.2 Geometry and Boundary Conditions2.3 The Solver2.4 The Gradients2.5 The Resolution2.5.1 The Triangular Mesh Resolution2.5.2 The Square Mesh Resolution


3 Results and Discussion3.1 Calculating the Bias Factor3.2 The Triangular Mesh Resolution3.3 The Square Mesh Resolution

4 ConclusionsBibliography

Documents

Numerical Calculations of the Bias Factor of an Edge Dislocation …560267/FULLTEXT01.pdf · 2012. 10. 12. · kristallina material sv aller n ar de uts atts f or bestr alning. Kantdis-lokationens