ISSN 1062-8738, Bulletin of the Russian Academy of Sciences: Physics, 2008, Vol. 72, No. 10, pp. 1347–1349. © Allerton Press, Inc., 2008.Original Russian Text © M.A. Shebzukhova, A.A. Shebzukhov, 2008, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2008, Vol. 72, No. 10, pp. 1424–1426.

1347

Interphase Segregation in Binary Systemswith Curved Boundaries

M. A. Shebzukhova and A. A. Shebzukhov

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

Abstract

—The isotherm equation has been obtained for the interphase layer composition at the boundary of adispersed spherical particle with a radius of curvature

r

and the dispersive medium in a binary system. The ten-sion surface, spaced from the equimolecular surface at a distance equal to the Tolman length for a flat boundary(

δ

) has been chosen as a dividing surface. The latter is the theory parameter. Particular cases have been consid-ered, where the pressure and composition in one of the bulk phases are fixed along with temperature. The lim-iting case corresponds to the known Ostwald–Freundlich formula for the solubility of a dispersed particle in amatrix.

DOI:

10.3103/S1062873808100110

Preferred concentration of individual components atphase boundaries occurs in multicomponent heteroge-neous systems, which is known as the phenomenon ofinterface or surface segregation [1, 2]. It plays animportant (in some cases decisive) role in many pro-cesses, including the destruction of materials in corro-sive media, formation of emission properties, etc.

Interface segregation is quantitatively characterizedby the atomic concentration of the components in a thintransition layer between the bulk phases. Within theknown thermodynamic method of a finite-thicknesslayer (the Van der Waals–Guggenheim–Rusanovmethod) [3], consistent thermodynamic theory of theinterface tension and segregation at planar bound-aries in binary and multicomponent systems wasconstructed [4, 5].

It is of interest to construct a similar theory forcurved phase boundaries. Currently, the importance ofthis problem is determined primarily by the great inter-est in nanoobjects. The interface layer thicknesses aregenerally in the nanoscale range, and the radius of cur-vature of a dispersed particle in a matrix (dispersivemedium) can also be in the same range.

In [6], the dependence of the interface tension at acurved boundary on the radius of curvature and compo-sition of the coexisting bulk phases and the transitionlayer between them under isothermal conditions wasconsidered for a binary system within the method offinite-thickness layer. The purpose of this study was toobtain the isotherm equation (

T

= const) of the interfacelayer (phase

σ

) composition at the boundary of a dis-persed particle with a radius of curvature

r

(phase

α

)and the dispersive medium (phase

β

) in a binary sys-tem.

In the thermodynamic equilibrium state of the sys-tem, the total differential of the thermodynamic poten-

tial of the entire system is zero, and the necessary con-dition for the existence of an extremum of the functionof many variables is the equality to zero of the corre-sponding first-order partial derivatives. This conditionleads to the known phase equilibrium conditions in aheterogeneous system, expressed in terms of the molarthermodynamic potentials of the

g

(

ξ

)

phases and theirpartial derivatives with respect to the concentrations(

∂

g

(

ξ

)

/

∂

x

(

ξ

)

) (

ξ

=

α

,

β

,

σ

). As a dividing surface, wechoose the Gibbs tension surface, which is a mechani-cal equivalent of the real curved discontinuity surface[3]. The interface tension, related to the tension surface,is equal to the work of formation of a surface unit [3].Thus, we have the known formula

(1)

where

p

(

α

)

and

p

(

β

)

are the pressures in the

α

and

β

phases and

r

is the tension surface radius.We will use the equilibrium conditions for the dis-

persed particle and interface layer and take into account

the functional dependences

g

(

ξ

)

and

∂

g

(

ξ

)

/

∂

. In thiscase, for the interface layer, it is also necessary to takeinto account (along with temperature

T

, pressures

p

(

α

)

and

p

(

β

)

, and composition

the dependences on theinterface tension

σ

. As a result, we have the followingequation for a binary system under isothermal condi-tions:

(2)

where

g

jj

is the second derivative of the molar thermo-dynamic potential with respect to the concentration of

2σr

------ p α( ) p β( ),–=

xiξ( )

xiσ( )

ωidσ υiασ( ) υi

α( )–( )d p α( ) υiβσ( )d p β( )+=

– x jσ( )g jj

σ( )dx jσ( ) x j

α( )g jjα( )dx j

α( ),+

1348

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES: PHYSICS

Vol. 72

No. 10

2008

SHEBZUKHOVA, SHEBZUKHOV

the

j

th component;

is the partial molar volume ofthe

i

th component in the

α

phase;

ω

i

is the partial molar

surface of the

i

th component; and

and are thepartial molar volumes of the

i

th component in the partsof the interphase layer that are adjacent to the Gibbstension surface from the sides of the

α

and

β

phases

respectively (

i

≠

j

;

i

,

j

= 1, 2). In this case,

= +

and

υ

(

σ

)

= +

, where

is theaverage partial molar volume of the

i

th component inthe interface layer and

υ

(

σ

)

is the average molar volumein the interface layer. The equilibrium condition for thebulk phases

α

and

β

, separated by a curved boundary,yields

(3)

Considering jointly the relations (1)–(3) and excludingthe pressures

p

(

α

)

and

p

(

β

)

from them, we obtain

(4)

where

ρ

i

= ( – )/( – )

.

Further consideration of the problem of interfacesegregation of the components in a binary medium atthe boundary of a dispersed particle with a radius

r

(

α

phase) located in a dispersive medium (

β

phase) willbe performed for the simplest case where the chemicalpotentials of the components in all phases (α, β, and σ)are proportional to the corresponding concentrations(µi ~ lnxi). In this case, all partial values in (4) coincidewith the corresponding values for the components (ωi ≈ω0i, υi ≈ υ0i, ≈ ) and gii = RT/[xi(1 – xi)][3]. Thefactor before d(2σ/r) in (4), assigned to surface area ω0i,

can be written using and and expressed in ter-mes of the Tolman length δ0i:

(5)

It can easily be seen that, at δ0ir–1 � 1, this factor isequal to the Tolman length.

Let us write expression (4) twice for i = 1, 2 (respec-tively, j = 2, 1), when µi ~ lnxi. After integrating over theparticle radius of curvature from infinity to r and

υiα( )

υiασ( ) υiβ

σ( )

υiσ( ) υiα

σ( )

υiβσ( ) υi

σ( )xiσ( ) υ j

σ( )x jσ( ) υi

σ( )

υiβ( )d p β( ) υi

α( )d p β( )–

= x jβ( )g jj

β( )dx jβ( ) x j

α( )g jjα( )dx j

α( ).–

ωidσ υiβσ( ) ρiυi

β( )–( )d2σr

------⎝ ⎠⎛ ⎞ ρix j

β( )g jjβ( )dx j

β( )= =

+ 1 ρi–( )x jα( )g jj

α( )dx jα( ) x j

σ( )g jjσ( )dx j

σ( ),–

υiσ( ) υi

σ( ) υiβ( ) υi

α( )

υiβσ( ) υ0iβ

σ( )

υ0iασ( ) υ0i

α( )

υ0iββ( ) ρ0iυ0i

β( )–( )ω0i1–

= υ0iασ( ) 1 ρ0i–( )υ0i

α( )–( )[ ]ω0i1––

= δ0i 1δ0i

r------

13---

δ0i

r------⎝ ⎠

⎛ ⎞2

+ + .

excluding σ, we obtain an expression for the interface

layer composition in the binary system:

(6)

(7)

where σ0i∞ is the interface tension at the flat boundarybetween the α and β phases of the ith component,

For a dispersed particle, at the boundary with vapor,ρ0i → 0 and expression (6) is significantly simplified.At xi � 1, the concentration of the first component inthe surface layer linearly changes with its concentrationin the particle volume with a specified value of r.

Let us consider the cases where not only tempera-ture but also pressure is fixed in one of the bulk phases.In practice, a constant pressure in the dispersivemedium can be most easily provided (the pressure inthe external medium p(β) = const). In this case, we havethe following expression from the equilibrium condi-tion for two bulk phases:

(8)

which yields the known Ostwald–Freundlich formula[3] for the increase in the solubility of small particles of

a pure material ( = 1) in a matrix

(9)

Here, is the concentration of the ith component inthe ξ phase when r = ∞. Under the conditions T = constand p(β) = const, we obtain the following formula for thecomposition of the interface layer at the boundary of asmall particle with a radius r in a matrix:

(10)

x1σ( )

x1σ( ) 1 x1

σ( )–( ) γ–kx1

α( ) 1 x1α( )–( ) γ–

=

× x1β( )/x1

α( )( )ρ01

1 x1α( )–( )/ 1 x1

β( )–( )[ ]ρ02γ–

,

k σ02∞d0 σ01∞–( )ω01 RT( ) 1–[ ],exp=

γ γ d0, d0 1 b01+( ) 1 b02+( ) 1– ,= =

σ0i σ0i∞ 1 b0i+( ) 1– , γ 0 ω01ω021– ,= =

b0i 2 υ0iβσ( ) ρ0iυ0i

β( )–( ) rω0i( ) 1–=

= 2δ0i

r--------- 1

δ0i

r------

13---

σ0i

r-------⎝ ⎠

⎛ ⎞2

+ + .

2σr

------υ0iα( ) RT

xiβ( )

xiα( )--------ln

xi∞β( )

xi∞α( )--------ln–

⎝ ⎠⎜ ⎟⎛ ⎞

,=

xiα( )

xiβ( ) r( ) xi∞

β( ) 2συ0iα( )

r RT⋅----------------⎝ ⎠

⎛ ⎞ .exp=

xi∞ξ( )

x1σ( ) 1 x1

σ( )–( ) γ–kx1

α( ) 1 x1α( )–( ) γ–

,=

k σ2d0 σ1–( )ω01 RT( ) 1–[ ],exp=

σi σ∞ RTω0i1– xi∞

σ( )/xi∞α( )( ),ln–=

BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES: PHYSICS Vol. 72 No. 10 2008

INTERPHASE SEGREGATION IN BINARY SYSTEMS 1349

where

The composition of the interface layer for the case p(β) =const can also be expressed in terms of the concentration

in the dispersive medium in the form (10), where

When, along with temperature (T = const), the pressureis fixed in the dispersive phase (p(α) = const), the com-position of the interface layer is expressed in terms of r

and the volume concentrations ( or ) in the

form (10), where c0i = 2ρ0i (rω0i)–1 and c0i = 2(ρ0i –

1) (rω0i)–1.

In summary, we will consider the segregation at theboundary of a particle of radius r, located in a disper-sive medium, with constant temperature and composi-tion of one of the bulk phases. In this case, dependencesof type (10) are also obtained; in these dependences, at

= const, we have

(11)

where

σ∞ is the interface tension at the flat boundary of the

phase β with a constant composition ( = const) andthe small particle (phase α) with a radius r. At T = const

and = const, the right-hand side of (11) containsthe concentration in the phase α and the difference (1–ρ0i) instead of ρ0t.

REFERENCES

1. Shebzukhov, A.A., Poverkhnost, 1983, no. 8, p. 13.2. Shebzukhov, A.A., Poverkhnost, 1983, no. 9, p. 31.3. Rusanov, A.I., Fazovye ravnovesiya i poverkhnostnye

yavleniya (Phase Equilibria and Surface Phenomena),Leningrad: Khimiya, 1967.

4. Shebzukhov, A.A. and Karachaev, A.M., Poverkhnost-nye yavleniya na granitsakh kondensirovannykh faz(Surface Phenomena at the Boundaries of CondensedPhases), Khokonov, Kh.B., Ed., Nal’chik: Izd-voKBGU, 1983, p. 23.

5. Shebzukhov, A.A. and Karachaev, A.M., Poverkhnost,1984, no. 5, p. 58.

6. Shebzukhova, M.A. and Shebzukhov, A.A., Izv. Ross.Akad. Nauk, Ser. Fiz., 2007, vol. 71, no. 5, p. 656.

d0 1 b01 c01+ +( ) 1 b02 c02+ +( ) 1– ,=

γ γ 0d0, c01 2ρ0iυ0iα( ) rω0i( ) 1– .= =

x1β( )

c0i 2 ρ0i 1–( )υ0iα( ) rω0i( ) 1–=

and σi σ∞ RTω0i1– xi∞

σ( )/xi∞β( )( ).ln–=

x1α( ) x1

β( )

υ0iβ( )

υ0iβ( )

x1α( )

x1σ( ) 1 x1

σ( )–( ) γ–k x1

β( )( )ρ01

1 x1β( )–( )

ρ02γ–,=

γ γ 0d0, d0 1 b01+( ) 1 b02+( ) 1– ,= =

k σ02∞d0 σ01∞–( )ω01 RT( ) 1–[ ],exp=

x1β( )

x1β( )