432

ISSN 1990-7931, Russian Journal of Physical Chemistry B, 2009, Vol. 3, No. 3, pp. 432–435. © Pleiades Publishing, Ltd., 2009.Original Russian Text © S.Sh. Rekhviashvili, A.M. Karmokov, E.V. Kishtikova, 2009, published in Khimicheskaya Fizika, 2009, Vol. 28, No. 6, pp. 23–27.

1. INTRODUCTION

The structure and physical properties of the surfaceof solids and various physicochemical processes thatoccur on it are of great interest for practical applicationsand theory. Indeed, the surface plays a key role in thewear of machine parts under the action of friction, con-tact melting, soldering and welding of different materi-als, photo- and thermoelectronic emission, etc. Cur-rently, analytic equations that describe the compositionof the surface have been obtained on the basic of molec-ular and electronic statistical models. These modelslead to fairly complex formulas for calculations con-taining a large number of unknown parameters, whichcomplicates their experimental verification and use inpractice. Somewhat simpler analytic equations wereobtained for calculations of the mean surface layercomposition with the use of the thermodynamic charac-teristics of the volume phase.

This work is concerned with the development anduse of the well-known van der Waals approximation [1–3] for describing surface segregation in a binary solu-tion at the interface with the vacuum and consistent cal-culations of the concentration profile, surface energy,and adsorption of components.

2. THEORETICAL MODEL

Let us consider a solid solution of components Aand B. The sample is a seminfinite medium separatedfrom the vacuum by a plane separation surface. To char-acterize the inhomogeneity of a thin surface layer, weintroduce the concept of excess free energy for the

i

thcomponent. The excess free energy is a function of thecoordinate, concentration, and concentration gradient;that is,

∆

F

i

(

x

,

c

i

(

x

),

dc

i

(

x

)/

dx

)

, where

x

is the coordinatecounted from the surface inside the sample,

c

i

(

x

)

is theconcentration of component

i

, and

i

= A or B. At ther-modynamic equilibrium,

∆

F

i

reaches a minimum

everywhere along the

x

coordinate. We must thereforeconsider the functional

(1)

where the limits of integration are set arbitrarily. Func-tional (1) is minimized according to the equation

(2)

Note that, in similar theories (see [3]), the volume den-sity of excess free energy, energy per unit volume,rather than excess free energy itself is minimized.When the total volume of a substance is constant, theminimization of energy and volume density of energy,however, gives mathematically equivalent results. Topass to the volume density of excess free energy, Eq. (2)should be divided by the mean volume occupied by anatom in the solution. To make presentation more visualphysically, we consider excess free energy.

In the simplest case, we can assume that the

∆

F

function does not depend explicitly on

x

. In addition,this function should be scalar and can therefore beexpanded into a series in even powers of the

dc

i

/

dx

derivative. This is completely analogous to the selec-tion of the Lagrange function

L

= –

U

(

x

) +

m

v

2

/2

, whichdoes not depend on time explicitly, in classical mechan-ics [4]. With an accuracy to the second term of theexpansion, we have

(3)

where

∆

F

0

i

and

b

i

are the expansion coefficients. Thefirst term in (3) is the deviation of free energy close

J ∆Fi x ci x( )ci x( )d

xd---------------, ,⎝ ⎠

⎛ ⎞ x,d

x1

x2

∫=

∂ ∆Fi( )∂ci

-----------------ddx------

∂ ∆Fi( )∂ dci/dx( )-----------------------– 0.=

∆Fi ∆F0i–bi

2----

dci

dx-------⎝ ⎠

⎛ ⎞2

,+=

Segregation Surface in Binary Solid Solutions

S. Sh. Rekhviashvili, A. M. Karmokov, and E. V. Kishtikova

Kabardino-Balkar State University, ul. Chernyshevskogo 173, Nalchik, 360004 Russiae-mail: rsergo@mail.ru

Received October 18, 2007

Abstract

—Analytic equations describing the concentration profile, surface energy, and adsorption for a binarysolid solution with the emergence of one of the components to the surface (surface segregation) were obtainedusing a unique approach, a modified van der Waals approximation. The calculation results were compared withthe experimental secondary ion mass spectrometry data on Cu–Ge and Cu–Al metallic binary solutions; agree-ment was satisfactory.

DOI:

10.1134/S1990793109030166

KINETICS AND MECHANISM OF CHEMICAL REACTIONS, CATALYSIS

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B

Vol. 3

No. 3

2009

SEGREGATION SURFACE IN BINARY SOLID SOLUTIONS 433

to the surface from its value in the volume of a homo-geneous solution, and the second term is the “gradi-ent energy” depending on the local composition. Theminus sign of the first term takes it into account thatfree energy can be zero depending on the concentra-tion gradient value. Substituting (3) into (2) yields

(4)

Equation (4) is a nonlinear second-order differentialequation; it extrapolates the van der Waals approxima-tion [1–3] to the separation boundary between a binarysolid solution and the vacuum. A similar equationappears in the van der Waals theory for the liquid–vaporone-component two-phase system.

No general analytic solution to (4) can be obtained.Certain simplifications are therefore necessary. In amacroscopic solid-state phase, interatomic interactionenergy is almost independent of impurity concentra-tion; the

b

i

coefficient can therefore be considered con-stant. In addition, the

∆

F

0

i

(

c

)

function can be repre-sented in the form of a series truncated, as above, afterthe quadratic term,

(5)

where

a

i

denotes expansion coefficients and

is thevolume concentration of component

i

. The zeroth and firstexpansion terms are absent in (5), because it is assumed

that

∆

F

0

i

=

∆

= 0 at

c

i

=

. Equation (5) is valid at anot too large concentration deviations from the volumevalues. This is a natural condition, which was implied inthe derivation of (4). Substituting (5) into (4) yields

(6)

The general solution to (6) at

a

i

< 0 has the form

(7)

where

c

1, 2

are arbitrary constants. Solution (7) shouldsatisfy the boundary conditions

(8)

where

is the surface concentration of component

i

.The only solution that obeys conditions (8) is

(9)

It follows from (9) that the concentration gradient atpoint

x

= 0, which determines the slope of the concen-

∂ ∆F0i( )∂ci

-------------------–12---

∂bi

∂ci

-------dci

dx-------⎝ ⎠

⎛ ⎞2

+ bi

d2ci

dx2---------.=

∆F0i ci( )ai

2---- ci ci

v( )–( )2,=

civ( )

F0i' civ( )

d2ci

dx2---------

ai

bi

---- ci civ( )–( )+ 0.=

ci x( ) civ( ) c1 x/Li–( )exp c2 x/Li( )exp ,+ +=

Li bi/ai–( )1/2,=

ci 0( ) cis( ), ci +∞( ) ci

v( ),= =

cis( )

ci x( ) civ( ) ci

s( ) civ( )–( ) x/Li–( ).exp+=

tration profile, is –( – )/Li. The Li parametersplay the role of diffusion lengths for components A andB. If bi/ai < 0, we have usual smooth concentration pro-files. If bi/ai > 0, Eq. (6) has periodic solutions withperiods of (bi/ai)1/2, which means solution stratificationduring segregation. The condition for the occurrence ofstratification is ai = 0 (or Li ∞).

Equation (6) and its solution (9) allow adsorptionand surface energy to be determined. After the separa-tion of variables in the first integral in (6), we can write

(10)

The multiplication of (10) by n(ci – ), where n is thetotal number of solution atoms per unit volume, and theintegration of it in x and ci yields

(11)

Equation (11) determines the total adsorption of com-ponent i from the bulk onto the surface (or vice versa)and exactly coincides with the well-known Guggen-heim–Adam equation, in which the Li values depend onthe properties of the volume phase. Clearly, surfaceenergy should satisfy the Gibbs adsorption equation.Let us calculate the differential of the chemical poten-tial from (5),

(12)

and rewrite (10) taking (12) into account,

(13)

Multiplying (13) by n∆Fi yields the differential of sur-face energy,

(14)

The summation over components in (14) yields theGibbs equation,

(15)

The integration of (15) gives

cis( ) ci

v( )

dx Li

dci

ci civ( )–( )

-----------------------.–=

civ( )

Γi n ci civ( )–( ) xd

0

∞

∫ nLi cid

civ( )

cis( )

∫= =

= nLi cis( ) ci

v( )–( ).

dµi dd ∆F0i( )

dci

-------------------⎝ ⎠⎛ ⎞– aidci,–= =

dxLi

ai

----dµi

ci civ( )–( )

-----------------------.=

dσi n∆FidxnLi

ai

--------∆Fi µid

ci civ( )–( )

-----------------------= =

= nLi ci civ( )–( )dµi– Γidµi.–=

dσ dσi

i

∑ Γi µi.di

∑–= =

434

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B Vol. 3 No. 3 2009

REKHVIASHVILI et al.

(16)

where σ0 is the surface energy of the solution at =

and ∆σ is the change in the surface energy causedby segregation. Note that (16) can be obtained by firstintegrating excess free energy (3) taking into accountthe first integral in (6),

(17)

After the substitution of (9) into (17) and subsequentelementary integration, we again obtain (16), which isas it should be. It follows from (11) and (16) thatadsorption and surface energy are directly proportionalto lengths Li. This leads us to conclude that the temper-ature dependences of lengths Li are what is responsiblefor the temperature dependence of segregation alwaysobserved experimentally [5].

3. A COMPARISON WITH EXPERIMENT

Let us compare theory with some experimentalresults obtained in [5]. In that work, Cu + 10 at % Aland Cu + 6 at % Ge single crystalline solid solutions

σ σ0 ∆σ,–=

∆σ n2--- aiLi ci

s( ) civ( )–( )2

,i

∑–=

cis( )

civ( )

∆σ n ∆Fi xd

0

∞

∫i

∑ n ∆F0i–bi

2----

cidxd

------⎝ ⎠⎛ ⎞

2

+⎝ ⎠⎛ ⎞ xd

0

∞

∫i

∑= =

= n bi

cidxd

------⎝ ⎠⎛ ⎞

2

x.d

0

∞

∫i

∑

with the orientations (100), (110), and (111) were stud-ied by secondary ion mass spectrometry in a vacuum.Layer-by-layer surface etching with measuring the sec-ondary ion peaks of the 63Cu+, 27Al+, and 74Ge+ isotopeswas used to construct profiles of concentration changes.The accuracy of the reproduction of profiles in experi-ments was ±5% of the value measured. The experimen-tal data led the authors of [5] to draw the empirical con-clusion of the exponential character of changes in theconcentration of the component that experienced segre-gation.

The experimental data (symbols) and the results ofcalculations by (9) (solid lines) are shown in Figs. 1, 2.

The and Li parameters in (9) were determined fromthe experimental data by the method of least squares,

and the volume concentrations of the components were considered given. The concentrations of atoms nin the alloys were calculated in the additive approxima-tion. The parameters found this way were used to cal-culate component adsorption values Γi. The calculationresults are listed in the table, which shows that thealloys exhibited different orientation dependences of allthe calculated parameters. In our model, this is relatedto the special features of the orientation dependence ofbi coefficients, which are close to diffusion coefficientsin their physical meaning.

4. CONCLUSIONS

The shortcoming of the model described above isthe assumption that the composition of a binary solidsolution on the surface and in the bulk is known before-hand, as follows from boundary conditions (8). There

cis( )

civ( )

43210x, nm

20

15

10

5

ci, ‡t %

4321x, nm

0

25

20

15

5

ci, ‡t %

Fig. 1. Profile of the distribution of Ge in the surface layerof the Cu + 6 at % Ge single crystalline solution at differentorientations: (�) (111), (�) (100), and (�) (110).

Fig. 2. Profile of the distribution of Al in the surface layerof the Cu + 10 at % Al single crystalline solution at differentorientations: (�) (111), (�) (110), and (�) (100).

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B Vol. 3 No. 3 2009

SEGREGATION SURFACE IN BINARY SOLID SOLUTIONS 435

are, however, theories that allow these compositions tobe determined [6]. In addition, the ai and bi coefficientsin (6) should be calculated separately for a particularsolution model.

In this work, three important conclusions are drawn:(1) in considering surface segregation in terms of a uni-fied approach, equations for concentration profiles, sur-face energy, and component adsorption were obtained;(2) a comparison of calculations with experiment gavesatisfactory agreement; and (3) the use of the resultsobtained in practice allows us to avoid confusion whichcan arise when the thermodynamic characteristics ofsurface segregation are calculated in various approxi-mations.

REFERENCES

1. J. S. Rowlinson, J. Stat. Phys. 20, 197 (1979).

2. J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258(1958).

3. J. Rowlinson and B. Widom, Molecular Theory of Cap-illarity (Clarendon, Oxford, 1982; Mir, Moscow, 1986).

4. L. D. Landau and E. M. Lifshitz, Course of TheoreticalPhysics, Vol. 2: The Classical Theory of Fields (Nauka,Moscow, 1988; Pergamon, Oxford, 1975).

5. A. M. Karmokov, Doctoral Dissertation in MathematicalPhysics (Kabard.-Balk. Gos. Univ., Nal’chik, 2000).

6. B. Ya. Pines, Notes on Metallophysics (Khark. Gos.Univ., Kharkov, 1961) [in Russian].

Characteristics of binary metallic solutions

System (hkl) Li, nm , at % , at % Γi, 1018 m–2

Cu + 6 at % Ge,n = 8.04 × 1028 m–3

(111) 1.22 16.9 6 11

(110) 1.02 13.6 6 6.2

(100) 1.19 15.1 6 8.7

Cu + 10 at % Al,n = 8.15 × 1028 m–3

(111) 0.98 21.3 10 9

(110) 0.83 20.6 10 7.2

(100) 0.79 16.7 10 4.3

cis( ) ci

v( )