1 Thermodynamics and kinetics of nucleation in binary solutions

Nikolay V. Alekseechkin

Akhiezer Institute for Theoretical Physics, National Science Centre “Kharkov Institute of Physics and Technology”, Akademicheskaya Street 1, Kharkov 61108, Ukraine Email: [email protected]

Keywords: Binary solution; phase equilibrium; binary nucleation; diffusion growth; surface layer.

A new approach that is a combination of classical thermodynamics and macroscopic kinetics is offered for studying the nucleation kinetics in condensed binary solutions. The theory covers the separation of liquid and solid solutions proceeding along the nucleation mechanism, as well as liquid-solid transformations, e.g., the crystallization of molten alloys. The cases of nucleation of both unary and binary precipitates are considered. Equations of equilibrium for a critical nucleus are derived and then employed in the macroscopic equations of nucleus growth; the steady state nucleation rate is calculated with the use of these equations. The present approach can be applied to the general case of non-ideal solution; the calculations are performed on the model of regular solution within the classical nucleation theory (CNT) approximation implying the bulk properties of a nucleus and constant surface tension. The way of extending the theory beyond the CNT approximation is shown in the framework of the finite- thickness layer method. From equations of equilibrium of a surface layer with coexisting bulk phases, equations for adsorption and the dependences of surface tension on temperature, radius, and composition are derived. Surface effects on the thermodynamics and kinetics of nucleation are discussed.

1. Introduction

Binary nucleation covers a wide class of processes of phase transformations which can be divided into three groups: (i) gas-liquid (or solid) transformations, (ii) liquid-gas transformations, and (iii) transformations within a condensed state. The first group includes the binary droplet nucleation in a mixture of vapors of two substances [1-4], whereas the second group involves the bubble nucleation in binary fluids [5-7]. The third group includes liquid-liquid (LL), solid-solid (SS), and liquid-solid (LS) transformations. LL and SS transformations are the separation of liquid and solid solutions. LS transformation is the crystallization of a liquid alloy as well as the precipitation of a dissolved substance from a supersaturated liquid solution. Such a division is due to the different physics of the nucleation process within these groups, i.e. different equations of equilibrium for a critical nucleus as well as growth 2 equations for a postcritical one. These equations are common for the processes of the third group, so that LL, SS, and LS binary nucleation is the subject of the present theory. As a particular case, the nucleation of one-component precipitates from a binary solution is also considered. Binary as well as multicomponent nucleation is described in the framework of the formalism of the multivariable theory of nucleation [8] which is a universal theory – it describes the nucleation processes in different systems according to the same algorithm [1, 9-11]. Taking into account any phenomenon leads to the appearance of the corresponding variable in the theory and thereby the accuracy of the process description is increased. E.g., the addition of droplet temperature to the theory of vapor condensation and the consideration of heat exchange between the droplet and vapor allow us to calculate nonisothermal effects in nucleation [1, 11]. Taking into account surface effects [12, 13] essentially advances the theory beyond the CNT approximation. The classical work by Reiss [14] on binary nucleation can be considered also as the first work on the multivariable theory of nucleation; the basic concepts of the latter where introduced therein – the flux of nuclei in the phase space and the saddle surface representing the work of nucleus formation. The physical picture of nucleation was shown as the flow of nuclei in the saddle surface “gorge” through its pass. In the multivariable theory, a new-phase nucleus is described by the set of variables {X i } one of which, X1 , is “unstable” ; it describes the nucleus size - X1 = V , the nucleus volume, X1 = R , the radius, or X1 = N the total number of particles. The remaining variables X 2 ,..., X n are “stable”, so that the work of nucleus formation W (X1, ... , X n ) is represented by a saddle surface in the n -dimensional space. In the vicinity of

∗ the saddle point {X i }, it can be expanded up to quadratic terms:

n ∗ ∗ W (X1, ... , X n ) = W∗ + )2/1( ∑ hik (X i − X i )( X k − X k ) (1) i,k =1 Reiss’ work being an extension of the Zeldovich-Frenkel [15, 16] one-dimensional theory to binary

(+) (−) case employs the microscopic kinetics; it operates with the probabilities wi and wi of attachment and detachment of each kind monomers. This approach has become traditional for the binary-nucleation theory [17-21]; in particular, its finite difference equations are convenient for numerical studies of nucleation [3, 4, 18, 19]. In contrast to it, the approach of macroscopic kinetics is used in the present theory, as in the previous works [1, 9-12]. The advantages of this approach from the physical and analytical points of view were shown in Ref. [1] by the example of binary droplet nucleation. In particular, it is natural to use the fractions xi = Ni / N as the variables of nucleus description [8, 22], rather than the numbers Ni ; the basic equations of the theory – the equations of equilibrium and growth equations – are formulated just in terms of xi , as shown below.

The basis of the offered approach is the equations of motion of a nucleus in the space {X i }. In the vicinity of the saddle point, they are linear [8]: 3 n ∗ X& i = −∑ zik (X k − X k ) , Z = BH / kT (2) k =1 where X& i ≡ dX i / dt ; B is the matrix of “diffusivities” in the Fokker-Planck equation for the distribution function of nuclei F(X1, ..., X n ;t ) . These equations are obtained from the conditions that the flux of nuclei ∂F Ji = −bij + X& i F (3) ∂X j is equal to zero for the equilibrium distribution function Fe (X1, ..., X n ) ~ exp( −W ({ X i}) / kT ) . As is known from the theory of the Fokker-Planck equation, ∆X i ∆t X& i = lim (4) ∆t →0 ∆t where the averaging over all possible changes ∆X i (with the corresponding probability) in the time ∆t is done. Thus, despite the fact that the actual motion of a nucleus in the vicinity of the saddle point is chaotic (Brownian), only the regular component of this motion remains after the procedure of averaging, i.e. the velocities X& i are macroscopic . An equation for the steady state nucleation rate was derived in Ref. [8]:

W kT − ∗ I = C h−1 κ e kT (5) 0 2π 11 1

−1 −1 where h11 is the matrix H element; κ1 is the negative eigenvalue of the matrix Z , and C0 is the normalizing factor of the one-dimensional equilibrium distribution function Fe (X1 ) - it is determined in the framework of statistical mechanical approach. This equation essentially corrects the preceding result by Trinkaus [23]; it is invariant with respect to the space dimensionality and gives the result of the Zeldovich-Frenkel theory for n = 1. Also, the derivation of this equation in Ref. [8] does not employ the simultaneous diagonalization of the matrices H and B , which is possible only if the matrix B is symmetric; therefore, Eq. (5) is applicable also to the cases, when the matrix B includes antisymmetric elements [10]. The work by Russell [24] on nucleation in condensed phases should be also mentioned. The nucleation of a binary precipitate of a fixed composition is studied therein within the “shell model”; the variables of the theory are N , number of A atoms in the nucleus and x , number of A atoms in the nucleus’ shell of nearest neighbors. In other words, the variable N relates to the nucleus, whereas x does not belong to it. As is seen from the above description of a multivariable theory, this model does not correspond to it. Therefore, Russell’s model and the corresponding two-dimensionality seem artificial; the actual two-dimensionality appears, when both the variables describe the nucleus - N is the total number of atoms and x is the composition. When the fluctuations of nucleus composition are allowed, the problem becomes two-dimensional. The concentration of A atoms in the nucleus’ shell of nearest 4 neighbors is indeed an important quantity, however, it is considered within the present approach in connection with nucleus growth. So, the LL, SS, and LS binary nucleation is studied here for the first time within the macroscopic approach. As is seen from the foregoing, the matrices H and Z are all we need to calculate the nucleation rate I . The consideration is carried out within the CNT approximation [12]: the nucleus properties are assumed the same as the properties of the bulk phase (the actual nucleus inhomogeneity and the surface effects are not taken into account). Accordingly, the surface tension σ is constant; with the same accuracy, the partial molecular volumes υi are also constant. This approximation is justified as the first step in constructing the theory of binary nucleation in a condensed state; the aim of the paper is to show how the offered approach works in the given case. It should be noted that just this approximation is employed in the most of works on binary nucleation. Nevertheless, the extension of the theory beyond the CNT approximation is also considered in Appendix, where the surface effects in binary nucleation are taken into account within the classical thermodynamics. Equations for adsorption and the dependences of surface tension on radius, temperature, and composition are derived. It is shown that all these dependences of surface tension are due to the nucleus inhomogeneity [12, 13]; in particular, the dependence of surface tension on composition is due to adsorption (the difference in the compositions of surface layer and bulk new phase) and therefore it makes no sense to consider this dependence within the CNT approximation, where a nucleus is homogeneous and there is no adsorption. The outline of the paper is as follows. The thermodynamics of nucleation is considered in Section 2. The equations of equilibrium of a critical nucleus with the mother phase are derived, from which the critical radius and composition can be found. Section 3 is the kinetic part of the work: the macroscopic equations of diffusion growth of a one- and two-component nucleus are obtained here; the results of Section 2 are essentially used in these equations. In Section 4, the matrix Z and the nucleation rate are calculated for all cases considering in Section 2 as well as the obtained results are discussed. The summary of results is given in Section 5. In Appendix, the thermodynamics of surface layer is considered and equations for adsorption and the mentioned dependences of surface tension are derived; also, surface effects on the thermodynamics and kinetics of binary nucleation are discussed.

2. Equations of equilibrium for a critical nucleus

The new phase is denoted by α , the mother phase is β . The nucleation process is considered at constant temperature T and pressure Pβ ; the solution is supersaturated with respect to composition xβ (the component A fraction). The Laplace equation and its differential form are essentially used below: 5

α β 2σ α P − P = ≡ PL , dP = dP L (6) R∗ in view of Pβ = const , dP β = 0 ; asterisk denotes the critical value (it is omitted in equations of equilibrium for brevity).

2.1. Nucleation of a one-component precipitate from binary solution

Let component A precipitates. The chemical potential of component A in a non-ideal solution is µ β (T, Pβ , xβ ) = µ β (T, Pβ ) + kT ln f β (T, P β , xβ )xβ (7) where xβ and f β (T, Pβ , xβ ) are the fraction and the activity of component A in the solution (the subscript A is omitted for brevity); the bar relates to pure component A. The condition of equilibrium µα (T, Pα ) = µ β (T, Pβ , xβ ) of the precipitate with the mother phase in the differential form

dµα = dµ β (8)

at constant T and Pβ has the following form, in view of Eq. (6):  β  α β β β  ∂µ  υ dP L = µ& dx , µ& ≡  β  (9) x  ∂ T ,P β Eq. (8) means that the state of the system “nucleus + mother phase” changes while maintaining the

β equilibrium between them, i.e. the critical radius R∗ is adjusted to composition x or vice versa. In other

β words, Eq. (9) is an equation for the dependence R∗ (x ) . Integrating it from PL = 0 ( R∗ = ∞ ) to the

β β β α current PL , we get µ (x ) − µ(x∞ ) = υ PL , from where, in view of Eq. (7),

υα P  β β β β β β  L  f (x )x = f (x∞ )x∞ exp   (10)  kT 

β where x∞ (T ) is the composition of solution over the planar interface. This equation gives the desired

β dependence R∗ (x ) :

1 α β β β − β 2υ σ  x f (x ) R∗ (x ) = ln β + ln β β  (11) kT  x∞ f (x∞ )

β β The quantity S = x /x∞ is the supersaturation ratio; R∗ = ∞ for S = 1. For a regular solution, f β (xβ ) = exp [ω β 1( − xβ )2 ] (12) where ω β is the characteristic parameter of the solution. Thus, Eq. (10) becomes

υα P  β β β 2  L  β β β 2 x exp []ω 1( − x ) = C exp   , C(T) ≡ x∞ (T)exp [ω 1( − x∞ (T)) ] (13)  kT  from where 6 1 α β − β 2υ σ  x β β 2 β 2  R∗ (x ) = ln β + ω []1( − x ) − 1( − x∞ )  (14) kT  x∞ 

The ideal solution approximation is obtained by putting ω β = 0 , or f β = 1; hence,

υα P  β β  L  x = x∞ exp   (15)  kT  The same equation holds for a dilute solution. This is the familiar Ostwald-Freindlich equation; it gives

1 α  β − β 2υ σ  x  R∗ (x ) = ln β  (16) kT  x∞  It should be noted that Eqs. (15) and (16) are the same as the corresponding equations for a droplet in vapor [11] with replacing xβ by the vapor pressure; Eq. (15) is an analogue of the Kelvin equation for the equilibrium vapor pressure over the droplet of radius R .

β β α In literature, the linearized form of Eq. (15), x = x∞ 1( + 2υ σ / R∗kT ) is often used [25], which is valid only for large nuclei (near the binodal), or small supersaturations. At the same time, if the surface tension is not too small, the nucleation begins with nanosized nuclei and Eq. (15) has to be employed. On the other hand, Eq. (15) with constant surface tension σ is not suitable for too small nuclei. It is seen that xβ increases with decreasing R , however, it has not to exceed unity; a stronger condition xβ << 1 must be satisfied for a dilute solution. This shortcoming of CNT Eq. (15) is easily corrected by using the

radius-dependent surface tension σ (R) β in it; σ is radius-dependent in Eq. (6) [26] and the above T ,P procedure of integrating Eq. (9) does not imply the constancy of σ . In this way, the classical Kelvin equation was extended to small radii [13]. The linear asymptotics σ (R) = K(T )R at R → 0 [27, 28]

β β β α ensures a finite (spinodal) value of x in this limit, xs = x∞ exp (2υ K / kT ). Vice versa, the constant K

β can be found from this equation, if the spinodal value xs is known.

2.1. Nucleation of a compound from binary solution

Let the compound A nBm precipitates from a non-ideal solution of components A and B:

β β β β β β β β β µ A (T, P , x ) = µA (T, P ) + kT ln f A (T, P , x )x

β β β β β β β β β µB (T, P , x ) = µB (T, P ) + kT ln fB (T, P , x )( 1− x ) (17) This is the chemical reaction

nA + mB = An Bm ≡ C (18) in the two-phase system. The condition of equilibrium for chemical potentials is obtained from Eq. (18) by replacing the symbols A, B, and C by the corresponding chemical potentials [29]:

α α β β µC (T, P ) = nµA + mµB (19) 7 α β β β The differential form of this equation, dµC = nd µA + md µB , at constant T , P and in view of Eq. (6) is

α β β β β υC dP L = nµ& A dx + mµ& B dx (20)

β α α α where the point denotes the derivative with respect to x and υC = nυA + mυB . Integration of this equation gives

α β β β β β β β β υC PL = n[µA (x ) − µ A (x∞ )]+ m[µB (x ) − µB (x∞ )] (21)

β where x∞ (T ) is the equilibrium composition of solution over the planar interface with the compound. Employing Eq. (17), we get

n m n m υα P  xβ  1− xβ    f β (xβ )   f β (xβ )   C L = ln      + ln  A   B   ≡η(xβ ) (22) kT  xβ   xβ   f β xβ   f β xβ   ∞  1− ∞    A ( ∞ )   B ( ∞ )   from where the critical radius of compound nucleus for the given solution composition is found as

α 2υ σ −1 R (xβ ) = []η(xβ ) (23) ∗ kT Eq. (22) is transformed to the following one:

α n m υ P  β n β m β β β β  C L  (x ) 1( − x ) [][]f A (x ) fB (x ) = C exp    kT 

β n β m β β n β β m C(T ) ≡ (x∞ ) 1( − x∞ ) [f A (x∞ )] [fB (x∞ )] (24)

β β For ideal or dilute solutions, f A = fB = 1, so that only the first summand in Eq. (22) remains. For a regular solution,

β β β β 2 β β β β 2 f A (x ) = exp [ω 1( − x ) ], fB (x ) = exp [ω (x ) ] (25) and Eq. (24) acquires the following explicit form:  α  β n β m β β 2 β 2 υC PL (x ) 1( − x ) exp {}ω []n 1( − x ) + m(x ) = C exp   ,  kT 

β n β m β β 2 β 2 C(T ) ≡ (x∞ ) 1( − x∞ ) exp {ω [n 1( − x∞ ) + m(x∞ ) ]} (26)

2.3. Nucleation of a two-component precipitate

Differently from the previous case, here the nucleus composition is not fixed: first, the critical

α nucleus composition depends on its radius, x∗ (R∗ ) ; second, the composition can fluctuate around the

α critical value x∗ , which makes the problem two-dimensional . The variance of this fluctuation will be given later. As in the previous cases, it makes sense to start with a non-ideal solution, since the activities are experimentally determined quantity. For the convenience of using the resulting equations of equilibrium in the subsequent kinetic equations, the symmetric form of thermodynamic equations with respect to both components is employed here; in particular, the fractions xA and xB are used: 8 β β β β β β β β β µi (T, P , xi ) = µi (T, P ) + kT ln fi (T, P , xi )xi

α α α α α α α α α µi (T, P , xi ) = µi (T, P ) + kT ln fi (T, P , xi )xi , i = A, B (27)

α α α β β β β The equation of equilibrium dµi (T, P , xi ) = dµi (T, P , xi ) at constant T , P and in view of Eq. (6) gives

 ∂µα   ∂µ β  α α α β β α  i  β  i  υi dP L + µ&i dx i − µ&i dx i = 0 , µ&i ≡  α  , µ&i ≡  β  (28) ∂x ∂x  i T ,Pα  i T ,P β

α α β α β This is a Pfaffian equation [1] for the vector field F = υi i + µ&i j − µ&i k in the space r = (PL , xi , xi ) . The

β β α necessary and sufficient condition of its integrability by one relation xi = xi (xi , PL ) is F(∇ × F) = 0 . We have

β α β α  ∂µ ∂µ   ∂µ ∂υα   ∂µ ∂υα  ∇ × F = − &i + &i i −  &i + i j +  &i − i  = 0  α β   β   α   ∂xi ∂xi   ∂PL ∂xi   ∂PL ∂xi  in view of the obvious equality to zero of the derivatives in the first two summands and

∂µα ∂  ∂µα  ∂υα &i  i  i = α   = α ∂PL ∂xi  ∂PL  ∂xi Thus, the vector field is potential, F = ∇U , and Eq. (28) has the form dU = 0 ; its solution is U = const , the integration constant is determined by the chosen initial condition.

α β For component A, we integrate Eq. (28) in the space ( PL , xA , xA ) along the broken line whose segments are parallel to the coordinate axes, starting from the point corresponding to the planar interface

α β β β with pure component A in phase α : ( PL = 0 , xA = 1, xA = xA,∞ ), where xA,∞ (T ) is the fraction of component A in phase β over the planar interface of pure A in phase α . So, the path of integration is as

α β β α follows: (i) PL changes from 0 to the current PL at xA = 1 and xA = xA,∞ ; (ii) xA changes from 1 to the

α β β β β β current xA at the given PL and xA = xA,∞ ; (iii) xA changes from xA,∞ to the current xA at the given PL

α and xA . As a result, one obtains

α α α α β β β β υA PL + [µ A (xA ) − µA ]− [µA (xA ) − µA (xA,∞ )]= 0 (29a)

α β For component B, we integrate Eq. (28) in the space ( PL , xB , xB ) in the similar way; the starting

α β β β point is ( PL = 0 , xB = 1, xB = xB,∞ ), where xB,∞ (T ) is the fraction of component B in phase β over the planar interface of pure B in phase α . As a result,

α α α α β β β β υB PL + [µB (xB ) − µB ]− [µB (xB ) − µB (xB,∞ )]= 0 (29b)

α α It should be emphasized that these equations contain the partial volumes υA and υB of pure components

α in phase α (which are therefore specific volumes υi ), even if these quantities are composition-

α dependent, as a consequence of the integration path (integration over PL is performed at xi = 1). With the use of Eq. (27), Eqs. (29a, b) become as follows: 9  β β β α f A (xA )xA α α α α υ A PL ln − ln f A (P , xA )xA =  f β xβ kT  A,∞ A,∞ (30) f β (xβ )xβ υα P ln B B B − ln f α (Pα , xα )xα = B L  β β B B B  fB,∞ xB,∞ kT

β β β β β β α β where f A,∞ ≡ f A (xA,∞ ) , fB,∞ ≡ fB (xB,∞ ) , and P = P + PL .

α α α 2 β β β 2 For a regular solution, fi = exp [ω 1( − xi ) ] and fi = exp [ω 1( − xi ) ], so that Eq. (30) has the following form:

  α  β β β 2 α υA PL α α 2 β β β 2  xA exp []ω 1( − xA ) = CAxA exp  + ω 1( − xA ) , CA (T ) ≡ xA,∞ exp []ω 1( − xA,∞ )   kT   (31)  α   β β β 2 α υB PL α α 2 β β β 2 xB exp []ω 1( − xB ) = CB xB exp  + ω 1( − xB ) , CB (T) ≡ xB,∞ exp []ω 1( − xB,∞ )   kT 

β β α Eqs. (30) and (31) give the desired dependence xi = xi (xi , PL ) .

β β β α β α For large amounts of phases α and β , the equality µA (T, P , xA ) = µA (T, P , xA ) for regular solutions is

β β β β β 2 α β α α α 2 β (α ) β (α ) µA (T, P ) + kT ln xA + Ω 1( − xA ) = µ A (T, P ) + kT ln xA + Ω 1( − xA ) , Ω ≡ kT ω (32)

β According to the definition of xA,∞ ,

β β β β β 2 α β µA (T, P ) + kT ln xA,∞ + Ω 1( − xA,∞ ) = µ A (T, P ) (33) Combining Eqs. (32) and (33), we get

β xA β β 2 β 2 α α α 2 ln β + ω []1( − xA ) − 1( − xA,∞ ) = ln xA + ω 1( − xA ) (34) xA,∞ which is the first Eq. (31) for the planar interface ( PL = 0 ), as it must.

β β α β α β If we assume µA (T, P ) = µA (T, P ) instead of Eq. (33) and put ω = ω ≡ ω , then we have

β β 2 α α 2 ln xA + ω 1( − xA ) = ln xA + ω 1( − xA ) (35) instead of Eq. (34); the same equations hold for component B. Hence, it is seen that the conditions

β α β β α β µi = µi are equivalent to xA,∞ = xB,∞ = 1 and CA = CB = 1. These conditions together with ω = ω are fulfilled for transformations within the same state of aggregation – the separation of liquid or solid solutions with the same structure of both the phases, where the phases differ from each other only by composition; just these conditions were employed by Prigogine and Defay [29] in the analytical description of this phenomenon. In the model of regular solution, the binodal and spinodal were calculated in Ref. [29] and thereby the metastable region between these curves was shown, where

β α nucleation occurs. On the other hand, generally µi ≠ µi for transformations between different state of aggregation, e.g. the crystallization of a melt (the equality holds only at the melting temperature of pure component i ). So, the form of Eq. (31) is the most general. 10 β α β The system of equations (30) determines the radius R∗ (x ) and composition x∗ (x ) of a critical

α β nucleus for the given composition of the mother phase. For deriving the dependence x∗ (x ) , we denote

α α xA ≡ x , xB = 1− x and neglect the dependence of fi on P ; then divide the first Eq. (30) by the second

α α one and denote γ ≡ υA /υB . After simple transformations, one obtains

γ xα f α (xα ) xβ f β (xβ ) [f β xβ ] A = C A , C(T) ≡ B,∞ B,∞ (36) 1( − xα )γ α α γ 1( − xβ )γ β β γ f β xβ []fB (x ) []fB (x ) A,∞ A,∞

α β This equation implicitly gives the desired dependence x∗ (x ) . The critical radius then can be found from any of Eqs. (30), say, the first:

−1 α  β β β  β 2συ A f A (x )x α α α β R∗ (x ) = ln β β − ln f A (x∗ )x∗ (x ) (37) kT  f A,∞ xA,∞  For a regular solution, Eq. (36) acquires the following form: xα xβ exp {}ωα []1( − xα )2 − γ (xα )2 = C exp {ω β []1( − xβ )2 − γ (xβ )2 }, 1( − xα )γ reg 1( − xβ )γ

β γ (xB,∞ ) β β 2 β 2 Creg (T ) ≡ β exp {}ω []γ 1( − xB,∞ ) − 1( − xA,∞ ) (38) xA,∞

2.4. Mother phase dilute with respect to A, nucleus dilute with respect to B

Differently form the previous case, the composition cannot be arbitrary here, so that the integration of Eq. (28) differs by choosing the initial state for component B. We have for the chemical potentials of both components in both the phases:

β β β β α α α α µA =ψ (P ,T) + kT ln xA , µA = µA (P ,T) + kT ln xA

β β β β α α α α µB = µB (P ,T) + kT ln xB , µB =ψ (P ,T ) + kT ln xB (39)

β α α β α (β ) where by condition xA , xB << 1 and xA , xB ~ 1; ψ is not a pure-component chemical potential.

Eq. (28) for component A is integrated in the same way, as above: from the initial state ( PL = 0 ,

α β β α β β xA = 1, xA = xA,∞ ) to the current state ( PL , xA , xA ); the quantity xA,∞ (T ) has the same meaning, as before. As a result, we obtain Eq. (29a) which has the following form, in view of Eq. (39):

β α xA α υA PL ln β − ln xA = (40a) xA,∞ kT

α β β α α For component B, we change the order of α and β in Eq. (28), υi dP L − µ&i dx i + µ&i dx i = 0 , and

β α α α choose the initial state ( PL = 0 , xB = 1, xB = xB,∞ ), where xB,∞ (T ) is the fraction of component B in phase α over the planar interface of pure B in phase β . The path of integration is as follows: (i) PL

β α α β β changes from 0 to the current PL at xB = 1 and xB = xB,∞ ; (ii) xB changes from 1 to the current xB at the 11 α α α α α β given PL and xB = xB,∞ ; (iii) xB changes from xB,∞ to the current xB at the given PL and xB . As a result, one obtains

~α β β β α α α α υB PL − [µB (xB ) − µB ]+ [µB (xB ) − µB (xB,∞ )]= 0 (40b)

~α where υB is the partial volume of component B in phase α upon dilution; in view of our basic assumption

α of composition independence of υi , it is equal to the specific volume υB . With account for Eq. (39), Eq. (40b) is as follows:

α α xB β υB PL ln α − ln xB = (40c) xB,∞ kT Finally,

 xβ υα P   xβ υα P  A α  A L  α  A L   β = xA exp    β = x exp    x  kT   x  kT   A,∞ ,  A,∞ (41) xα υα P  1− xα υα P  xβ = B exp  A L  1− xβ = exp  A L   B α    α    xB,∞  kT   xB,∞  kT  where xA ≡ x and xB = 1− x was put.

For the planar interface ( PL = 0 ),  β β α x / xA,∞ = x  β α α (42) 1− x = 1( − x /) xB,∞

β α These equations can be derived directly from Eq. (39) with the use of xA,∞ and xB,∞ definition, as it was done above for a regular solution. From Eq. (42), the compositions of two coexisting bulk dilute solutions are determined:

α α 1− xB,∞ α β β β α α α β x = β α ≈ 1− xB,∞ 1( − xA,∞ ) , x = xA,∞ 1( − xB,∞ ) , xB = xB,∞ 1( − xA,∞ ) (43) 1− xA,∞ xB,∞ up to quadratic terms.

α β In order to obtain the composition x∗ (x ) of critical nucleus, we take the logarithm of Eq. (41), then

α α divide the first equation by the second one and denote γ ≡ υA /υB . After transformations, one obtains xα xβ 1 α γ = C β γ , C(T ) ≡ β α γ (44) 1( − x ) 1( − x ) xA,∞ (xB,∞ )

α β The critical radius is then obtained from Eq. (41) with the use of x∗ (x ) , as before. The critical radius also can be obtained directly form the system of equations (41). We express xα from the first equation and substitute it to the second one; after transformations, one obtains

β α α α β xA,∞ exp( υA PL / kT [1) − xB,∞ exp( −υB PL / kT )] α α α x = β α α , ∆υ ≡ υA −υB (45a) 1− xA,∞ xB,∞ exp( ∆υ PL / kT ) Up to quadratic terms, 12 β β α α α x = xA,∞ exp( υA PL / kT [1) − xB,∞ exp( −υB PL / kT )] (45b)

β α from where R∗ (x ) is determined. If component B is not soluble in A, then xB,∞ = 0 and Eqs. (45a, b) convert to Eq. (15) for the precipitation of pure component A.

β β Equations for the above quantities x∞ (T ) , xA,∞ (T ) , xB,∞ (T ) , etc. are found from the equality of

β chemical potentials of the corresponding bulk phases. E.g., for xA,∞ (T ) in Eq. (29a), we have by

β β β α β definition µA (T, P , xA,∞ ) = µA (T, P ) . The differential form of this equation,

β β β α β β dµ A (T,P , xA,∞ ) = dµA (T,P ) at constant pressure P reads

(αβ ) β β β α β β qA (T ) (αβ ) β α − sA dT + µ& A dx A,∞ = −sA dT , µ& A dx A,∞ = dT , qA (T ) = T(sA − sA ) (46) T

(αβ ) where qA is the heat of transition “ β (solution) → α (pure A)” for A atom. From this equation, the

β dependence xA,∞ (T ) is determined. The examples of similar equations are given in Refs. [29, 30].

2.5. The work of nucleus formation

The work of a near-critical nucleus formation is given by Eq. (1). The work W∗ of critical nucleus formation is given by the familiar Gibbs equation 1 4π W = σA = σR2 (47) ∗ 3 ∗ 3 ∗ where the critical radius is given by the above equations. The second differential of the work (the second summand in Eq. (1)) was calculated in Ref. [1] for a binary droplet; this calculation is also valid for the present theory. The variables ( V , x ) – the nucleus

α volume and composition x ≡ xA - are used here as the variables of nucleus description in the two- dimensional problem. So, the matrix H (the coefficients hik ) is

 P∗  − L 0  3V  ∂µα   ∗  α  A  H = α , µ& A ≡   (48)  µ  x & A,∗  ∂ T ,Pα  0 N∗   1− x∗  In the CNT approximation (a homogeneous nucleus with bulk properties), the matrix H is diagonal, when the composition x is used as a variable of nucleus description; it is non-diagonal, if the variables

( N1 , N2 ) are employed [1]. It was mentioned above that the composition of a two-component precipitate can fluctuate around the critical value x∗ ; the element hxx of Eq. (48) just determines the variance of this fluctuation:

2 −1 (x − x∗ ) = kTh xx . The positiveness of the element hxx is ensured by the thermodynamic condition of 13 α stability µ& A > 0 . The thermodynamic limit hxx → ∞ corresponds to transition to the single component

(V ) -theory; accordingly, it is realized for pure A ( x∗ →1) or B ( x∗ → 0 ) component. In the latter case,

α α α we have for a dilute solution µA =ψ (T, P ) + kT ln x , µ& A,∗ = kT / x∗ → ∞ at x∗ → 0 . Transformations in solid state (SS) can create elastic stresses which affect the nucleation kinetics. The work of nucleus formation in Ref. [1] and Eq. (48) resulting from it do not take into account this effect and therefore they can be applied only to the cases, where it is not essential. The detailed analysis of this phenomenon and the overview of works on this topic are given by Christian [31].

3. Equations of nucleus growth

3.1. The growth of a precipitate from ideal or dilute solution

At first, we consider the growth of a one-component (A) precipitate from binary solution. The concentration cA = cx A - the number of A atoms in unit volume - is employed in this Section; c is the total number of atoms in unit volume. The flux of A atoms to the nucleus of radius R across the interface

2 A is j+ = 4πR ν Alc R , where ν A is the probability of jump across the interface per unit time, l is the mean

A length of jump, and cR is the concentration of A atoms near the interface; the similar quantity enters the

Russell model [24] mentioned above, however, for other purpose. The reverse flux j− is assumed to be the same, as in the equilibrium of the nucleus of radius R with the mother phase:

e e 2 A A j− = j− = j+ = 4πR lν Ace (R) , where ce (R ) is the equilibrium concentration of component A for the nucleus of radius R ; it is given by Eqs. (10), (24) and their particular cases derived above. So, the net flux is

2 A A N& A ≡ j = j+ − j− = 4πR lν A (cR − ce ) (49) On the other hand, the flux j can be found from the solution of stationary diffusion problem

A A A ∆cA (r) = 0 with boundary conditions cA (∞) = c0 and cA (R) = cR , where c0 is the given concentration of

A A A binary solution: cA (r) = c0 − (c0 − cR )R / r and

A A A j = 4πRD 0 (c0 − cR ) (50)

A where D0 is the diffusion coefficient of component A in the solution. Comparing this equation to Eq.

A (49), we find the quantity cR :

A A A A D0 c0 + lν ARc e cR = A (51) D0 + lν AR

A A A A For the critical nucleus, R = R∗ , we have by definition ce = c0 and hence cR = c0 , as it must; according

A A A to Eq. (15), ce decreases with increasing R , thus ce < c0 for R > R∗ . 14 A It should be noted that the boundary condition cA (R) = ce is often employed in literature [25]. However, there is a contradiction between kinetics and thermodynamics in this point: on the one hand,

A A there is the flux of A atoms towards the nucleus, in view of the above inequality ce < c0 ; on the other hand, the nucleus cannot grow under this condition according to thermodynamics – it is in equilibrium

A with the mother phase. For this reason, the quantity cR is used here which is self-consistently determined. Substituting Eq. (51) in Eq. (50), we get finally

A A A γ A lν AR N& A = 4πRD 0 ΓA (c0 − ce ) , ΓA ≡ , γ A ≡ A (52a) 1+ γ A D0 Obviously, the same analysis can be applied to B atoms in considering the growth of a two- component precipitate, and

B B B γ B lν B R N& B = 4πRD 0 ΓA (c0 − ce ) , ΓB ≡ , γ B ≡ B (52b) 1+ γ B D0

A B where ce (R, x ) and ce (R, x ) are given by Eqs. (30), (31), and (41). It should be emphasized that

A B A A ce (R, x) + ce (R, x) ≠ 1 for R ≠ R∗ ; the equality holds only for R = R∗ , when ce (R∗, x∗ ) = c0 and

B B ce (R∗, x∗ ) = c0 . Eqs. (52a, b) determine the kinetics of evolution of a two-component precipitate – the change in its size and composition.

3.2. The growth of a precipitate from non-ideal solution

According to the thermodynamic theory of diffusion based on linear non-equilibrium thermodynamics [31, 32], the driving force of a diffusion process is the gradient of chemical potential, rather than the concentration gradient. As a consequence, the dependence of the diffusion coefficient on concentration arises in a non-ideal solution [32, 33]:  ∂ln f (x) D(x) = D 1+ (53) 0  ∂ln x  where f (x ) is the activity employed above; the index A is omitted for brevity. For a regular solution, this gives

D(x) = D0 [1− 2ω x 1( − x)] (54) Stationary diffusion equation is 1 d  dx  dx k r 2 D(x) = 0 , D(x) = 1 (55a) r 2 dr  dr  dr r 2 Integration of this equation, in view of Eq. (54), gives  2  k D x − ω x2 + ω x3 = − 1 + k (55b) 0  3  r 2

The above boundary conditions x(∞) = x0 and x(R) = xR determine the integration constants: 15  2   2  k = D x − ω x2 + ω x3 , k = RD (x − x ) − ω ( x2 − x2 ) + ω ( x3 − x3 ) (55c) 2 0  0 0 3 0  1 0  0 R 0 R 3 0 R 

2 2 The desired flux of A atoms towards the nucleus, in view of Eq. (55a), is j = 4πR ck 1 / R = 4πck 1 ,  2  j = 4πRD c (x − x ) − ω ( x2 − x2 ) + ω ( x3 − x3 ) (56) 0  0 R 0 R 3 0 R  Returning to concentrations and comparing this equation to Eq. (49), we have ω 2 ω γ (c − c ) = (c − c ) − (c2 − c2 ) + (c3 − c3 ) (57a) R e 0 R c 0 R 3 c2 0 R

Denoting cR − ce ≡ z and c0 − ce ≡ y , we see that this equation implicitly determines the function z(y ) , if

2 2 2 2 we note that c0 − cR = (c0 − ce ) − (cR − ce ) = y − z , c0 − cR = y + 2ce y − z − 2ce z , etc. The full representation of Eq. (57a) after rearrangements is as follows:  2   2  2ωce 2ωce 2ωce ω  2 2 ω 3 2ωce 2ωce 2ωce ω  2 2 ω 3 (γ + )1 − + 2 z +  2 − z + 2 z = 1− + 2  y +  2 −  y + 2 y  c c   c c  3 c  c c   c c  3 c (57b) It will be seen later that only the first (linear) term of the expansion z(y) = z′ )0( y is sufficient for our purpose; y = 0 corresponds to the critical nucleus, ce = c0 = cR , hence, z )0( = 0 . The derivative z′(y) = dz / dy can be easily found by differentiating both sides of Eq. (57b) with respect to y . Its value at zero (with the restored index A) is

γ~ 2ω c A  c A  ′ A ~ e  e  z )0( = ~ , γ A ≡1− 1−  (57c) γ A + γ A c  c 

A A Substituting z(y) = z′ )0( y for (cR − ce ) in Eq. (49), we get finally after simple transformations ~ A~ A A ~ γ Aγ A N& A = 4πRD 0 ΓA (c0 − ce ) , ΓA ≡ ~ (58a) γ A + γ A

B Eq. (53) has a symmetric form for both components: DB (x) = D0 [1+ ∂ln fB (x /) ∂ln xB ], so that the above analysis can be applied to component B without any changes, and

γ γ~ 2ω cB  cB  B~ B B ~ B B ~ e  e  N& B = 4πRD 0 ΓB (c0 − ce ) , ΓB ≡ ~ , γ B ≡1− 1−  (58b) γ B + γ B c  c  Eqs. (58a, b) determine the kinetics of growth of a two-component precipitate from a regular solution. For ~ ~ ~ ~ an ideal solution, ω = 0 , γ A = γ B = 1 and ΓA = ΓA , ΓB = ΓB .

In conclusion, the analysis of coefficients γ i and Γi should be done. Representing the diffusion

i 2 (bulk ) (bulk ) coefficient as D0 = l ν i , where ν i is the frequency of i atom jumps in the bulk mother phase (it includes all necessary quantities such as the coordination number, the correlation factor, etc. [31]), we get

R ν i γ i = (bulk ) (59) l ν i 16 The two limiting cases are as follows. (i) The interface-controlled growth with respect to component i : γ i << 1, Γi = γ i , and

2 i i N& i = 4πR lν i (c0 − ce ) (60a) This case corresponds to fast diffusion in the bulk and slow kinetics at the interface; the latter determines the growth with respect to component i . (ii) The diffusion-limited growth with respect to component i :

γ i >> 1, Γi = 1, and

i A A N& i = 4πRD 0 (c0 − ce ) (60b) This case corresponds to fast kinetics at the interface and slow diffusion in the bulk; the slowest process determines the growth, as before. Just Eq. (60b) is usually employed in literature [25] for describing the growth from solution; the interfacial kinetics falls out from consideration. As is seen from above, the appearance of quantities γ i in

i the present theory is due to introducing the quantity cR and employing the boundary condition

A A cA (R) = cR instead of cA (R) = ce ; just γ i allow revealing these limiting cases. In view of natural condition R > Ra ≈ l 2/ , where Ra is the atomic radius, the interface-controlled growth requires

(bulk ) ν i << ν i . For a sufficiently small (nanosized) nucleus, R is few times l 2/ , so that the diffusion-

(bulk ) limited growth requires the inverse condition ν i >> ν i . Likely, some intermediate case occurs for an actual growth. ~ The similar analysis holds for the quantities with tilde. (i) The interface-controlled growth: γ i << γ i , ~ ~ ~ ~ Γi = γ i , and Eq. (58a) goes to Eq. (60a). (ii) The diffusion-limited growth: γ i >> γ i , Γi = γ i , and

i ~ i i N& i = 4πRD 0γ i (c0 − ce ) (60c) ~ As γ i < 1, the condition for case (i) is stronger, whereas the condition for case (ii) is weaker, than in the previous consideration. ~ Finally, it should be noted that always γ i > 0 , though it involves a negative term. The equation for spinodal curve T (x ) of a regular solution is [29] kT 2/ Ω = x 1( − x ) , or 2ω x 1( − x) = 1; it is the same for xA and xB . This equation for ω > 2 (which corresponds to T < Tc = Ω 2/ k , Tc is the critical temperature)

s s yields two spinodal values x1 = 1( − 1− /2 ω 2/) and x2 = 1( + 1− /2 ω 2/) . The composition of mother

s s s s phase can be x0 < x1 or x0 > x2 ; the region x1 < x0 < x2 is unstable. Obviously, the equilibrium

i s s ~ i i compositions xe are located in the allowed regions, xe < x1 and xe > x2 , so that γ i = 1− 2ω xe 1( − xe ) > 0.

3.3. Nucleus motion equations in the (V , x) -space

Eq. (2) in the two-dimensional (V , x ) -space has the following form: 17 V& = −z (V −V ) − z (x − x )  VV ∗ Vx ∗ (61)  x& = −zxV (V −V∗ ) − zxx (x − x∗ )

The equation for x& can be also represented as

x& = axV& − λxx (x − x∗ ) (62) From comparison, the matrix Z is

 zVV zVx  Z =   (63) ax zVV ax zVx + λxx 

The condition of symmetry of the matrix B = kT ZH −1 (Onsager’s reciprocal relation) results in equation

−1 −1 zVx hxx = ax zVV hVV from where

zVx hVV ax = (64) zVV hxx In the one-dimensional ( V )-theory, the equation of motion is

V& = −zVV (V −V∗ ) (65)

4. Results and discussion

4.1. Nucleation rate of a one-component precipitate

Calculations can be done for the general case of Eq. (10), however, they are given below by the example of explicit Eq. (13). According to Eq. (58a),

α α A~ A A V& =υ A N& A = 4πRυA D0 ΓAc(x0 − xe ) (66)

A where xe (R ) is given by Eq. (13),

υα P  A β A 2  A L  xe exp []ω 1( − xe ) = C exp   (67)  kT 

A whereas x0 is the current fraction of component A in the solution.

All we need for calculating the nucleation rate is the quantity zVV in Eq. (65), zVV = −(dV& / dV )∗ . As

A A A is seen from Eq. (66), this derivative is reduced to calculating the derivative (dx e / dV )∗ ( xe = x0 at

V = V∗ , so that the derivative of the multiplier in front of brackets does not contribute to zVV ). The following useful equalities are employed below:

 d 2σA   dP   dP dR  P∗   L L L  2  =   =   = − = hVV (68a)  dV ∗  dV ∗  dR dV ∗ 3V∗ according to Eq. (48), and therefore 18  ∂x A   ∂x A ∂P   ∂x A   e   e L   e    = = hVV (68b) V  P V   P   ∂ ∗  ∂ L ∂ ∗  ∂ L ∗

A To get from Eq. (67) the derivative dx e /dP L , we differentiate both sides of this equation with respect to PL :

α β A 2 A α υ A PL −ω 1( − xe ) dx e υA kT e = C e β A A (69a) dP L kT 1− 2ω xe 1( − xe )

A A Taking this derivative at V = V∗ , we substitute xe = x0 and employ Eq. (67):

 dx A  υα x A υα x A  e  A 0 A 0   = β A A = ∗ (69b) dP kT 1 2 x 1( x ) kT ~  L ∗ − ω 0 − 0 γ A In view of the equality ~ Γ∗ γ ∗ A = A ≡ Γ′ (69c) ~∗ ∗ ~∗ A γ A γ A + γ A we get finally 4πR D A (υα )2 Γ′c x A 1 z = ∗ 0 A A 0 h = b h , b = 4πR D A (υα )2 Γ′c x A (70) VV kT VV kT VV VV VV ∗ 0 A A 0 where bVV is the diffusivity of the Fokker-Planck equation in the ( V )-space. It should be recalled that ΓA′

A ~∗ A ∗ ∗ ∗ is a function of composition x0 , since γ A is a function of x0 ; ΓA′ = ΓA = γ A /( 1+ γ A ) for ideal solution,

β ~∗ ω = 0 and γ A = 1.

Substituting h11 ≡ hVV and κ1 = zVV in Eq. (5), we find the steady state nucleation rate:

W A 2 A W h − ∗ α − ∗ VV kT σ 2D0 (υA ) ΓA′c x0 kT I = C0 bVV e = C0 e (71a) 2πkT kT R∗ which is the one-dimensional Zel’dovich-Frenkel equation with the diffusivity calculated within the present approach, Eq. (70). C0 is the normalizing factor of the equilibrium distribution function

α ( N ) α Fe (V ) = Fe (N)dN / dV = Fe (N /) υA . So, if the ( N )-space is used, we put C0 = C0 /υ A and

α 2 α 2 bVV = (υA ) bNN , hVV = hNN /( υA ) .

∗ ~∗ ∗ ~∗ In the case of interfaced-controlled growth, γ A << γ A , we have ΓA′ = γ A /γ A ,

2 2 A W 4πR lν (υα ) c x σ (2 υα )2 lν c x A − ∗ b = ∗ A A 0 , I = C A A 0 e kT (71b) VV ~∗ 0 ~∗ γ A kT γ A and the preexponential factor does not include R∗ . The same is true for the droplet nucleation [20]: the

−2 2 Zeldovich factor is proportional to R∗ , whereas the diffusivity is proportional to R∗ ; both these factors cancel each other.

∗ ~∗ In the case of diffusion-limited growth γ A >> γ A , ΓA′ =1 in Eqs. (70), (71a) and the diffusivity is

~∗ proportional to R∗ ; as a result, R∗ appears in the denominator. On the other hand, Eq. (71b) includes γ A 19 ~∗ in the denominator (for a non-ideal solution). Formally, both R∗ and γ A vanish on the spinodal.

However, the values of R∗ are bounded by Ra from below; the formal thermodynamic understanding of spinodal as corresponding to R∗ = 0 must be also corrected for this condition. At the same time, the nucleation rate sharply increases at high supersaturations (small R∗ ), which is the well known fact. So, Eqs. (71a, b) are physically correct.

4.2. Nucleation rate of a compound

It will be seen later that we should differ the roles of components here; let A and B be solute and solvent, respectively, so that the growth of compound is determined by component A: ~ υα N& 4πRυα D AΓ c V& = c A = c 0 A (x A − x A ) (72) n n 0 e where the division by n means that n A atoms from their full flux contribute to the formation of one

α α α compound molecule of volume υC = nυA + mυB ; it is implied that the required m B atoms at once attach

A to the mentioned n A atoms in this process. The quantity xe (R ) is given by Eq. (26):

υα P  A A n A m β A 2 A 2  C L  y(xe ) ≡ (xe ) 1( − xe ) exp {}ω []n 1( − xe ) + m(xe ) = C exp   (73)  kT 

A Further algorithm is the same as above, Eqs. (65)-(71). Calculating the derivative dx e /dP L and

A A taking it at V = V∗ , we substitute xe = x0 and employ Eq. (73); as a result,

 dx A  υα x A 1( − x A ) υα x A 1( − x A ) υα  e  c 0 0 c 0 0 c A   = A A β A A = A A ∗ ≡ ϕ(x0 ) (74a) dP kT n 1( x ) mx 1 2 x 1( x ) kT n 1( x ) mx ~ kT  L ∗ [− 0 − 0 ][− ω 0 − 0 ] []− 0 − 0 γ A from where

A α 2 A A 1 4πR∗D0 (υc ) ΓA′c x0 1( − x0 ) zVV = bVV hVV , bVV = A A (74b) kT n n 1( − x0 ) − mx 0

A A A The condition of dx e / dP L > 0 and accordingly bVV > 0 is n 1( − x0 ) − mx 0 > 0 , from where n x A < ≡ x (75) 0 n + m c i.e. the fraction of component A in the solution must be less than in the compound; this inequality can be regarded as a criterion for component A to be solute. Eq. (73) for component B is obtained simply by

A B A B substituting xe = 1− xe ; obviously, here the equality xe + xe = 1 holds, differently from the case of

B A binary precipitate of variable composition. Hence, (dx e / dP L )∗ = −(dx e / dP L )∗ . This means that the flux of atoms B is opposite to the flux of atoms A, i.e. their excess near the compound takes place. The left-hand side of or Eq. (73) for components A and B is shown in Fig. 1. The curve is symmetric for n = m and

s s s asymmetric for n ≠ m . The maxima are located at spinodal values, if x1 < xc < x2 ; at xc and x2 , if 20 s s s A xc < x1 ; at x1 and xc , if xc > x2 . Eq. (74a) has singularities both on the spinodal and at x0 = xc ; the

A A solution composition x0 = xc can be called degenerate. The derivative (dx e / dP L )∗ is positive on the ascending branch of the curve and negative on the descending one; the situation is mirror-symmetric for

A component B. In other words, when (dx e / dP L )∗ < 0 , component B as a solute determines the growth of compound.

~∗ ~∗ ~∗ From this reasoning and in view of the equality γ A = γ B ≡ γ , we get

W  σ 2D A (υα )2 Γ′c x A 1( − x A ) − ∗ C 0 c A 0 0 e kT , A solute  0 kT nR n 1( − x A ) − mx A I ∗ 0 0 =  B 2 B B W (76) σ 2D (υα ) Γ′c x 1( − x ) − ∗ C 0 c B 0 0 e kT , B solute  0 B B  kT mR ∗ m 1( − x0 ) − nx 0

s The regions where A or B determines the nucleation rate are shown in Fig. 1a’-c’. In the case xc < x1 ,

A s there is an additional narrow region xc < x0 < x1 , where B determines the nucleation (Fig. 1c’); the same

s holds for A in the case xc > x2 . In the case of ideal solution, Fig. 1b”, only the quantity xc separates regions A and B.

4.3. Nucleation rate of a binary precipitate

An equation for V& in this case is

α α V& = υA N& A +υB N& B (77a) where N& i are given by Eqs. (58a, b):

α A~ A A α B~ B B V& = 4πRc [υA D0 ΓA (x0 − xe ) +υB D0 ΓB (x0 − xe )] (77b)

α α α If the dependence of υi on composition is taken into account, then the term (υ&A N A +υ&B NB ) must be

α α α α added to the RHS of Eq. (77a). In view of the equalities υ&A = (∂υA / ∂x)x& and υ&B = (∂υB / ∂x)x& , where

α x ≡ xA , we have  α α  α α α α ∂υ A ∂υB υ&A N A +υ&B NB = N[]υ&A x +υ&B 1( − x) = Nx&x + 1( − x)  = 0 (77c)  ∂x ∂x  since the expression in brackets is equal to zero, according to the familiar property of partial volumes [29]. So, the dependence of partial volumes on composition does not change basic Eq. (77a).

An equation for x& is obtained by differentiating x = N A / N as follows: 1− x x x& = N& A − N& B (78a) N N

4πRc α A~ A A α B~ B B x& = [ 1( − x)υA D0 ΓA (x0 − xe ) − xυB D0 ΓB (x0 − xe )] (78b) N 21 The equations of nucleus motion now have the form of Eq. (61) and our aim is to find an explicit

A B form of the matrix Z , Eq. (63). Equations for xe (R, x ) and xe (R, x ) in Eqs. (77b) and (78b) are given by Eq. (31):

  α  A β A 2 υA PL α 2  xe exp []ω 1( − xe ) = CAxexp  + ω 1( − x)    kT   (79)  α   B β B 2 υB PL α 2 xe exp []ω 1( − xe ) = CB 1( − x)exp  + ω x    kT 

A Starting from calculating zVV = −(dV& / dV )∗ , we have to compute the derivatives ∂xe /∂V

B and ∂xe /∂V . For this purpose, we use the known from analysis theorem on the differentiation of an

A A implicit function. The function xe (PL , x ) is given implicitly by the relation Y (xe , PL , x) = 0 , where

 α  A A β A 2 υA PL α 2 Y (xe ,PL , x) = xe exp []ω 1( − xe ) − CAxexp  + ω 1( − x)  (80a)  kT  According to the mentioned theorem,

A A ∂xe ∂Y / ∂PL ∂xe ∂Y / ∂x = − A , = − A (80b) ∂PL ∂Y / ∂xe ∂x ∂Y / ∂xe

i i i Calculating ∂xe /∂PL , we then take it at the saddle point (V∗, x∗ ) , i.e. substitute xe = x0 , as before, and employ Eq. (79); as a result,

 ∂xi  υα xi  e  i 0   = β i i , i = A, B (81) P kT 1 2 x 1( x )  ∂ L ∗ − ω 0 − 0 from where and in view of Eq. (68b) 4πR c z = ∗ [D A (υα )2 Γ′ x + DB (υα )2 Γ′ 1( − x )]h (82) VV kT 0 A A 0 0 B B 0 VV

~∗ ~∗ ~∗ β i i A B where Γi′ = Γi /γ , γ = 1− 2ω x0 1( − x0 ) , as before, and hereafter we put x0 ≡ x0 and x0 = 1− x0 in final equations. It is seen that Eq. (82) is a direct extension of Eq. (70) to binary case.

A B Further, we employ Eq. (77b) to calculate zVx = −(dV& / dx )∗ ; the derivatives ∂xe /∂x and ∂xe /∂x are computed according to the second Eq. (80b). Being taken at the saddle point with the use of Eq. (79), they have the following form:

 ∂x A  1− 2ωα x 1( − x ) x A  e  ∗ ∗ 0   = β A A (83a)  ∂x ∗ x∗ 1− 2ω x0 1( − x0 )

 ∂xB  1− 2ωα x 1( − x ) xB  e  ∗ ∗ 0   = − β B B (83b)  ∂x ∗ 1− x∗ 1− 2ω x0 1( − x0 ) As a result,  A α B α  α D0 υ A ΓA′ x0 D0 υB ΓB′ 1( − x0 ) zVx = 4πR∗c[]1− 2ω x∗ 1( − x∗ )  −  (84)  x∗ 1− x∗  22

Eq. (78b) is used to obtain the elements of the second row. To get zxV = −(∂x& / ∂V )∗ the derivatives

A B ∂xe /∂V and ∂xe /∂V found above are employed again:

4πR∗c A α B α V∗ zxV = {D0 υA ΓA′ 1( − x∗ )x0 − D0 υB ΓB′ x∗ 1( − x0 )}hVV , N∗ = α α (85) kTN ∗ υA x∗ +υB 1( − x∗ )

A B Similarly, the above derivatives ∂xe /∂x and ∂xe /∂x are employed to compute zxx = −(∂x& / ∂x)∗ , α  A B  4πR∗c[1− 2ω x∗ 1( − x∗ )] D0 ΓA′ 1( − x∗ )x0 D0 ΓB′ x∗ 1( − x0 ) zxx =  +  (86) N∗  x∗ 1− x∗ 

The element zxV was found above directly from basic Eq. (78b). On the other hand, its value is dictated by Eqs. (63) and (64): zxV = ax zVV = zVx hVV / hxx , as a consequence of Onsager’s principle. The element hxx , Eq. (48), for a regular solution is

α 1− 2ω x∗ 1( − x∗ ) hxx = N∗kT (87) x∗ 1( − x∗ ) With the use of this equation and Eq. (84), it is easy to see that the same Eq. (85) is obtained, i.e. the necessary condition of self-consistency of the theory is fulfilled - the thermodynamic and kinetic equations of the present approach are consistent with Onsager’s principle. With the use of notations

α α 4πR∗cυA ΓA′ A 4πR∗cυB ΓB′ B ξ A ≡ x0D0 , ξB ≡ 1( − x0 )D0 (88) N∗ N∗ as well as Eq. (87), the matrix Z acquires the form  N [ξ υα + ξ υα ]h [ξ 1( − x ) − ξ x ]h  1  ∗ A A B B VV A ∗ B ∗ xx  Z =  ξ ξ  h  (89) x x h A x 2 B x2 xx kT []ξ A 1( − ∗ ) − ξB ∗ VV  α 1( − ∗ ) + α ∗    υA υB  N∗ 

It should be recalled that R∗ and x∗ are functions of x0 . Such a representation of Z is convenient to easy obtain the tensor of diffusivities of the Fokker-Planck equation in the (V , x ) -space:

 z h−1 z h−1  B = kT ZH −1 = kT  VV VV Vx xx  (90a)  −1 −1   zxV hVV zxx hxx 

 N ξ υα + ξ υα ξ 1( − x ) − ξ x   ∗[ A A B B ] [ A ∗ B ∗ ]  B =  1 ξ ξ  (90b) x x A x 2 B x2 []ξ A 1( − ∗ ) − ξB ∗  α 1( − ∗ ) + α ∗   N∗ υA υB  It is symmetric, as it must; this fact again shows that the thermodynamic and kinetic equations used here

α result in the fulfillment of Onsager’s principle. If we change the definition of ξi as ξi′ = ξi N∗υi 3/ V∗kT , the matrix Z elements expressed in terms of ξi′ are the same as for binary droplet nucleation in a mixture of two vapors, Ref. [1], Eq. (66) therein. This fact shows the universality of the present approach: 23 although the physics is different in these two systems (different equations of equilibrium and different equations of growth), the kinetic matrix Z has the same general form.

The remaining point in this part is to get the kinetic parameter λxx ; according to Eq. (63),

λxx = zxx − ax zVx and ax = zxV / zVV . From these equations, one obtains

α 2 hxx (υ ) ξ AξB α α α λxx = α β α α , υ = x∗υA + 1( − x∗ )υB (91) N∗kT υAυA ξ AυA + ξBυB The matrix Z for the nucleation of a precipitate dilute with respect to component B from mother

A phase dilute with respect to A is derived by the same algorithm with one simplification - xe (R, x ) and

B xe (R, x) are explicit functions now, they are given by Eq. (41):

 υα P  A  A L   xe = CA xexp     kT  β α  , C ≡ x , C ≡ x (92) υα P  A A,∞ B B,∞  B  A L  xe = CB 1( − x)exp     kT 

So, their derivatives with respect to PL and x are calculated directly. Eqs. (52a, b) are substituted now in

Eqs. (77a) and (78a); the element hxx has the form kT hxx = N∗ (93) x∗ 1( − x∗ ) As a result of application of the above procedure, the same Eq. (89) is obtained for the matrix Z , where ξi are given by Eq. (88) with Γi , Eqs. (52a, b), instead of Γi′ . In the approximation of diffusion- limited growth, Γi = Γi′ = 1, so that the quantities ξi become identical for both the problems; the matrices

Z for the same R∗ differ by x∗ and hxx . Since x∗ is close to unity in Eq. (93), this hxx is much greater, than hxx given by Eq. (87) for a regular solution. The characteristic equation for the matrix Z is

2 κ − (Sp Z)κ + det Z = 0 , Sp Z = zVV + zxx = zVV + λxx + ax zVx ; detZ = zVV λxx (94) The negative root is

1 2 κ1 = {Sp Z − (Sp Z) − 4det Z} (95) 2 This quantity is substituted in Eq. (5) to get the steady state nucleation rate of a binary precipitate.

4.4. Kinetic limits

As is seen from the foregoing, the kinetics of binary nucleation in a condensed state within the CNT approximation is governed by the two parameters: zVV and λxx . The parameter zVV characterizes the rate of nucleus growth; in a one-dimensional problem, this is the relative change of the nucleus volume per 24 unit time. The parameter λxx characterizes the rate of composition relaxation at constant V ; according to Eq. (62), (x&)V = −λxx (x − x∗ ) . Similarly to the thermodynamic limit hxx → ∞ discussed above, kinetic limits are also possible; they are determined by limiting relations between the kinetic parameters zVV and

λxx .

2 The unary nucleation limit λxx → ∞ , or λxx >> zVV , ax zVx , leads to the conditions (Sp Z) >> det Z and Sp Z > 0 ; Eq. (95) has the following asymptotics in this case:

det Z zVV λxx κ1 = = = zVV (96) Sp Z zVV + ax zVx + λxx which is the one-dimensional result. Indeed, a large value of λxx means that any deviation of the nucleus composition from x∗ rapidly relaxes, i.e. the nucleus grows with constant composition x∗ . In other words, the problem becomes one-dimensional; the variable x falls out from consideration, only the variable V remains. Alongside with the case of compound nucleation, the given case also can be attributed to Russel’s model [24], although the physics in both these cases is different.

In the opposite limit λxx → 0 , or λxx << zVV , ax zVx , under the same condition Sp Z > 0 , Eq. (95) yields

det Z zVV κ1 = = λxx (97) Sp Z zVV + ax zVx These results agree with the general rule that the nucleation rate is determined by the slowest kinetic process in the system [1, 9-11]. Similarly to nonisothermal effect in droplet nucleation [1, 11], we can characterize the deviation from the one-dimensional nucleation kinetics by considering the ratio

I κ1 ε ≡ 1( D) = (98) I zVV

Eq. (91) for λxx allows us to determine some conditions for the above limits. It was mentioned above that hxx is large for a dilute precipitate ( x∗ ~ 1), which is the thermodynamic limit; however, this precipitate grows from the mother phase dilute with respect to component A, i.e. x0 << 1. If, according to

Eq. (88), ξ A << ξB , then λxx ~ hxx ξ A ~ x0 /( 1− x∗ ) , i.e. λxx is determined by the relation of two small

A B quantities which can be assumed of the same order of magnitude. This analysis implies that D0 ~D0 . If

A B D0 >> D0 and therefore ξ A ~ ξB , then λxx is of the same order of magnitude as hxx and hence also large;

A B so, the one-dimensional limit requires here the kinetic condition D0 >> D0 . This is only a qualitative reasoning; a more exact quantitative criterion can be obtained by numerical comparison of λxx with zVV and ax zVx . As was shown by the example of binary droplet nucleation [1], the two-dimensional process does not differ greatly from the one-dimensional one (ε ~ 1), even if λxx and zVV are of the same order 25 of magnitude; this means that the characteristic times of volume change and composition change are the same and the composition has a chance to adjust to the change in volume.

For a regular solution, the condition λxx → 0 can be ensured by ξi → 0 ; e.g., λxx ~ξ A , if ξ A << ξB .

According to Eq. (88), the latter condition can be fulfilled due to either x0 → 0 (the mother phase is

A B dilute with respect to component A) or D0 << D0 . This case is similar to the nonisothermal limit in droplet nucleation [11, 12]; we have ε → 0 , and the problem is essentially two-dimensional. In addition to the nucleation rate I , the matrices H and Z determine the steady state distribution function of nuclei Fst (V , x ) . It allows us to calculate the mean steady state enrichment of nuclei with respect to one of the components [1]. This effect is a manifestation of two-dimensionality; it is more pronounced for small values of λxx . The nucleus composition does not have time to relax and its systematic deviation from x∗ occurs in the steady state. The similar effect is the mean steady state overheat of droplets in nonisothermal condensation [1, 11, 12].

5. Conclusion

The universal approach combining classical thermodynamics and the macroscopic kinetics of nucleus growth was applied here to the calculation of nucleation rates of precipitates in condensed binary

α solutions. Accordingly, the nucleation rate is expressed via thermodynamic ( T , x0 , ω , etc.) and

i macroscopic kinetic ( D0 , Γi′ ) parameters only. The macroscopic equations of nucleus growth, Eqs. (58a, b), have a simple form – the change in the number of i atoms in unit time is proportional to the difference

i i (c0 − ce ) between the current concentration of i atoms in the mother phase and their equilibrium

i concentration ce (R, x ) over the nucleus of radius R and composition x (in the two-dimensional problem). This form is a result of application of the principle similar to detailed balancing in statistical mechanics: a nucleus loses atoms with the same frequency as in the state of equilibrium with the mother phase. In other words, the probability of losing an atom is determined by the nucleus state ( R,x ), rather

i than by the mother-phase state. While a nucleus grows ( R and x change), the quantities ce (R, x ) change according to thermodynamic Eqs. (79), (92), or other. The nucleus grows with respect to component i ,

i i i i while c0 > ce (R, x ) ; the growth stops, when ce (R, x ) becomes equal to c0 , and the nucleus dissolves with

i i respect to i at c0 < ce (R, x ) . So, these simple kinetic equations describe a complex evolution of a nucleus – the change both in size and composition. It should be noted that the results of the present (V , x ) - theory can be reformulated in terms of the variables (N1, N2 ) in the same way, as for a binary droplet in Ref. [1]. However, just the variable x is natural for solving the given problem for the following reasons: (i) equations of equilibrium for a critical 26 nucleus in Section 2 are formulated in terms of x ; (ii) equations of nucleus growth are also written in terms of x ; (iii) Eq. (62) for x& allows us to reveal the parameter λxx governing the nucleation kinetics. The work of a near-critical nucleus formation in the (V , x ) -theory is a quadratic form with diagonal matrix. The cases of both unary and binary precipitates were studied. As a particular case of binary precipitates, the nucleation of a precipitate of fixed composition (compound) was considered, which is the subject of Russell’ theory [24]; this problem is solved here as a one-dimensional one. However, the most significant result of the present approach is the kinetics of nucleation of binary precipitates of a variable composition from non-ideal solutions, which is a two-dimensional problem. It is shown that the theory is consistent with Onsager’s principle of symmetry of kinetic coefficients; also, the results are similar to those for a binary droplet nucleation [1], despite the fact that basic equations are different. The nucleation of precipitates of a fixed composition is also possible here as a one-dimensional limit of the theory at a large value of the kinetic parameter λxx ; the composition rapidly relaxes to its critical value, and a nucleus growth with composition x = x∗ in the CNT approximation (beyond this approximation, x relaxes to xeq (V ) - see Appendix).

Appendix: Surface effects in binary nucleation

1. Equations for the chemical potentials of bulk and surface phases

We consider the three-phase system consisting of new (bulk) phase α , mother (bulk) phase β and the surface layer between them which is phase σ (Fig. 2); the surface of tension [27, 34] is employed as a dividing surface. This finite-thickness layer method (which is an alternative to Gibbs’ one) was developed in detail for curved interfaces by Rusanov [27, 28]. The fundamental equations for all these phases are given in Ref. [12]; only some of them are needed for our purpose. Equation

σ σ ασ α βσ β σ σ ad σ + s dT −υ dP −υ dP + ∑ xi dµi = 0 (A1) i= A,B is an analogue of Gibbs’ adsorption equation; here a = A/ N σ , sσ = Sσ / N σ , υασ = V ασ / N σ ,

βσ βσ σ σ σ σ σ υ = V / N , xi = Ni / N and A is the nucleus surface area, S is the entropy of the surface layer,

N σ is the total number of particles in it, V σ = V ασ +V βσ . It is seen that these specific quantities are the mean values for the surface layer, by definition.

Hereafter we put xA ≡ x and xB = 1− x for a binary system, as well as denote

 ∂µα   ∂µα  α  A  α  B  µ& A ≡  α  , µ& B ≡  α  x x  ∂ T α ,Pα  ∂ T α ,Pα 27 σ σ α β σ and similarly for phases β and σ . An equation for the chemical potential µi (T , P , P ,σ , x ) of component i in the surface layer is [12]

σ σ σ ασ α βσ β σ σ dµi = −si dT +υi dP +υi dP − aidσ + µ&i dx (A2)

σ where si , etc. are the partial molecular quantities,

 σ    σ  ∂S   ∂A  si =  σ  , ai =  σ  , etc.  ∂Ni  σ α β σ  ∂Ni  σ α β σ T ,P ,P ,σ ,N j≠i T ,P ,P ,σ ,N j≠i For a bulk phase,

α α α α α α α dµi = −si dT +υi dP + µ&i dx (A3) and the same for phase β . The isothermal-isobaric Gibbs-Duhem equation and Eq. (A1) for constant T σ ,

Pα , Pβ , and σ result in the following relations:

α σ α x α σ x σ µ& B = − α µ& A , µ& B = − σ µ& A (A4) 1− x 1− x

α σ β In the state of full equilibrium, the equality µi = µi = µi holds [12]; its differential form

α σ β dµi = dµi = dµi (A5) is used below for deriving the needed relations, as well as equation

α β dP = dP + dP L (A6) Also T α = T σ = T β ≡ T , where T is the common temperature of the system in equilibrium. The complex consisting of phases α and σ being in equilibrium with each other (but generally not in equilibrium with phase β ) is the density fluctuation (DF) [35, 36] within phase β ; its volume is by

V βσ greater, than the nucleus volume V bounded by the surface of tension (Fig.2).

1. Equilibrium of bulk phases

α β Equation dµi = dµi for i = A and B results in the following system of equations:

µ β dx β = (sβ − sα )dT +υα dP α −υ β dP β + µα dx α & A A A A A & A  β β β α α α β β α α (A7) µ& B dx = (sB − sB )dT +υB dP −υB dP + µ& B dx

α α β β Expressing µ& B via µ& A and µ& B via µ& A according to Eq. (A4), we reduce this set to the following equation:

α β β α β β x − x α α ∆s(αβ )dT + (υ − ∆υ(αβ ) )dP −υ dP = α µ& A dx (A8) 1− x with

β α β α α α ∆s(αβ ) ≡ s − s − (x − x )( sA − sB )

β α β α α α ∆υ(αβ ) ≡ υ −υ − (x − x )( υA −υB ) 28 This equation gives the relationship between the composition of phase α and the system state parameters – temperature and pressures. For the flat interface, Pα = Pβ , it reduces to the equation derived by Van der Waals [37].

2. Relations between the compositions of coexisting phases

As is known, the composition of the nucleus surface layer differs from the composition of bulk phase α , i.e. adsorption takes place. The need to incorporate the phenomenon of adsorption into the nucleation theory was noted by Wilemski [38, 39]. The above fundamental equations for a surface layer together with the equations of equilibrium allow deriving equations for the surface composition xσ at different conditions. Below, relations between xσ and xα , as well as between xα and xβ , are derived. Eq. (A7) is employed for deriving the dependence xα (xβ ) . Substituting Eq. (A6) to this set of

α β β equations and then excluding dP L from it, we get an equation connecting x , x , T , and P :

α β α α α υ + (x − x )( υ A −υB ) β β α β α α β α β µ& A dx = []υB (sA − sA ) −υA (sB − sB ) dT 1− x

α α β α β β υ α α + []υAυB −υBυA dP + µ& A dx (A9) 1− xα Isothermal-isobaric dependences are of the most practical interest. Thus, one obtains from this equation

 dx α  1− xα  υα −υα  µ β 1− xα υα + (υα −υα )xβ  µ β   A B β α & A B A B & A  β  = β 1+ α ()x − x  α ≡ β  α  α (A10) dx x x  T ,P β 1−  υ  µ& A 1−  υ  µ& A

α α α α α where equation υ = υA x +υB 1( − x ) was utilized.

α α β β For ideal solutions, µ& A = kT / x and µ& A = kT / x , this equation acquires the form  dx α  xα 1( − xα ) υα + (υα −υα )xβ    B A B  β  = β β  α  (A11) dx x x  T ,P β 1( − )  υ 

σ α α σ For deriving the relation between x and x , we use dµi = dµi with account for Eqs. (A3) and (A2):  σ ασ α βσ β σ σ α α α α α − sA dT +υA dP +υA dP − aAdσ + µ& A dx = −sA dT +υA dP + µ& A dx  σ ασ α βσ β σ σ α α α α α (A12) − sB dT +υB dP +υB dP − aBdσ + µ& B dx = −sB dT +υB dP + µ& B dx

α σ Further, the following steps are done: (i) Eq. (A1) for dσ with dµi instead of dµi , as well as Eq. (A6)

α are substituted; (ii) Eq. (A4) for µi is employed; (iii) dP L is excluded from the resulting set of equations

σ σ α β and Eq. (A4) for µi is employed. As a result, an equation connecting x , x , T , and P is derived. The isothermal-isobaric equation is then obtained as follows: 29  dx σ  1− xσ a + (a − a )xα  µα 1− xσ  (a − a )  µα   B A B & A A B α σ & A  α  = α   σ ≡ α 1+ ()x − x σ (A13) dx x a x  a   T ,P β 1−   µ& A 1−   µ& A

σ σ where equation a = aAx + aB 1( − x ) was utilized. For ideal solutions in both phase α and surface layer,

 dx σ  xσ 1( − xσ ) a + (a − a )xα    B A B  α  = α α   (A14) dx x x a  T ,P β 1( − )   It is seen that Eqs. (A13) and (A10) as well as (A14) and (A11) are quite similar. In the

α α σ approximation of constant υi (not depending on x ) and constant ai (not depending on x ), Eqs. (A11) and (A14) are equations with separated variables which are easily integrated. Eq.(A13) allows us to determine the surface layer composition xσ for a given xα , i.e. to find the difference (xσ − xα ) of compositions of surface and bulk phases (adsorption). Of course, an equation for

(dx σ / dx β ) as well as other relations of interest can be derived in a similar way. T ,P β

3. Dependence of surface tension on the new-phase composition

Eq. (A1) is basic for determining the dependences of surface tension on different state parameters of

σ α β α (β ) α (β ) coexisting phases. We replace dµi by dµi and dµi in this equation and express µ& B via µ& A ,

β β according to Eq. (A4); then Eq. (A7) for µ& A dx and Eq. (A6) are employed. As a result, the following set of equations is obtained:

α σ βσ β x − x α α ad σ = ∆s(σα )dT − []∆υ(σα ) +υ dP L − ∆υ(σα )dP + α µ& A dx (A15) 1− x

β β β σ β α β ασ β σ α 1( − x )ad σ = [∆s(σβ ) 1( − x ) + (x − x )( sA − sA )]dT + [ 1( − x )υ + (x − x )υA ]dP L

β β σ α β β β σ α α + [− ∆υ(σβ ) 1( − x ) + (x − x )( υA −υA )]dP + (x − x )µ& A dx (A16) with

α σ α σ α α ∆s(σα ) ≡ s − s − (x − x )( sA − sB )

β σ β σ β β ∆s(σβ ) ≡ s − s − (x − x )( sA − sB )

α σ α σ α α ∆υ(σα ) ≡ υ −υ − (x − x )( υA −υB )

β σ β σ β β ∆υ(σβ ) ≡ υ −υ − (x − x )( υA −υB )

α β Excluding dP L from this set of equations, we get an equation connecting σ with x , T , and P . An equation for the isothermal-isobaric dependence σ (xα ) has the following form: T ,P β

υα (xβ − xσ ) +υασ (xα − xβ ) µα ad & A dx α ( σ )T ,P β = α β α α α (A17) υB + x (υA −υB ) 1− x 30 The similar equation obtained in Ref. [27] is reduced to Eq. (A17) after simple transformations. For integrating this equation, it has to be complemented by the functions xβ (xα ) and xσ (xα ) which T ,P β T ,P β are determined from Eqs. (A10) and (A13). Excluding dP β from Eqs. (A15) and (A16) and utilizing equation 2 2σ dP = dσ − dR (A18) L R R2 we get an equation connecting σ with xα , T , and R [27]

 2  ∆υ   ∆υ    βσ (ασ ) β  (ασ ) a + υ − υ  dσ = ∆s(αβ ) − ∆s(ασ ) dT  R  ∆υ(αβ )   ∆υ(αβ ) 

2σ  ∆υ   ∆υ  µα + υ βσ − (ασ ) υ β dR + (xβ − xα ) (ασ ) − (xσ − xα ) & A dx α (A19) 2     α R  ∆υ(αβ )   ∆υ(αβ ) 1− x

α where ∆υ(ασ ) = −∆υ(σα ) , ∆s(ασ ) = −∆s(σα ) . From here, an equation for σ (x )T ,R is obtained as follows:

 2  ∆υ   ∆υ  µα a + υ βσ − (ασ ) υ β (dσ ) = (xβ − xα ) (ασ ) − (xσ − xα ) & A dx α (A20)   T ,R   α  R  ∆υ(αβ )   ∆υ(αβ ) 1− x

α It is of interest to consider the dependence σ (x )T ,R following from the condition of the DF internal equilibrium only, i.e. from the condition of equilibrium between phases α and σ at a fixed state of phase

α σ β , (dµi = dµi )(β ) . It is easily derived from Eq. (A15) which is just an equation for the internal equilibrium. Employing Eq. (A18) and then putting dP β = 0 in Eq. (A15), we get

 βσ  βσ α σ (2 ∆υ(σα ) +υ ) (DF ) 2σ (∆υ(σα ) +υ ) x − x α α a + (dσ ) = ∆s(σα )dT + 2 dR + α µ& A dx (A21)  R  R 1− x from where

 βσ  α σ (2 ∆υ(σα ) +υ ) (DF ) x − x α α a + (dσ )T ,R = α µ& A dx (A22)  R  1− x For a liquid droplet ( α ) in vapor ( β ), Eqs. (A19) and (A20) are simplified due to the condition

β ∆υ(αβ ) ≈ υ >> ∆υ(ασ ) and go into Eqs. (A21) and (A22), respectively. So, for a liquid binary droplet in

α vapor, an equation for σ (x )T ,R derived from the condition of full equilibrium coincides with that derived from the condition of the DF internal equilibrium only. It is seen from Eq. (A22) that the surface tension depends on xα due to adsorption, or the difference

α σ σ α (DF ) in compositions x and x . For integrating this equation, the dependence x (x )T ,R is needed. It is obtained from Eq. (A12) which is just an internal equilibrium equation. Substituting dσ from Eq. (A21) in Eq. (A12), as well as utilizing Eq. (A18) and putting dP β = 0 , we get from any equation of this set 31 (DF )  (2 ∆υ +υ βσ ) dx σ  1− xσ a + (σα )   = {a + (a − a )xα R  dx α  xα B A B   T ,R 1−

α 2 α ασ α σ ασ ασ  µ& A + [](υ −υ ) − (x − x )( υ A −υB )  σ (A23) R  µ& A In the limit of planar interface, this equation goes into Eq. (A13). T in Eqs. (A21)-(A23) is the DF temperature: T ≡ T α = T σ .

4. Surface effects on the work of binary nucleus formation

The second differential of the work with the surface layer contribution was calculated in Ref. [12]; for a binary nucleus, it has the following form:

∗ 2 ∗ PL 2 α σ (d W )(β ) ≡ − (dV ) + H + H 3V∗

α α α α α α α α H = ∑{Ni∗ [ds i dT − dυi dP ]∗ + µ&i∗dx dN i } i= A,B

σ σ σ ασ α σ σ σ σ H = ∑{Ni∗ [ds i dT − dυi dP ]∗ + µ&i∗dx dN i }+ ∑ Ni∗da idσ (A24) i= A,B i= A,B where H σ is just the mentioned surface layer contribution. H α and H σ are the positive definite quadratic forms of stable variables for phases α and σ ; only H α enters this equation in the CNT approximation. The matrix of the quadratic form H α for an incompressible droplet was found in Ref. [1] as  µα   α & A∗  N∗ α 0 α 1− x H =  ∗  (A25)  Cα   0 V ∗   T β 

α α α α with N∗ = N A∗ + NB∗ ; CV ∗ is the heat capacity of the critical droplet. The same equation with the replacement of superscript α by σ holds for the expression in braces for H σ , Eq. (A24), whereas the last term is represented as follows:

 ∂σ  dR  ∂σ  α  ∂σ  dσ =   dV +  α  dx +   dT (A26)  ∂R T ,xα dV  ∂x R,T  ∂T R,xα

 σ  σ σ 2 dN i ∑ Ni da i = dA − ∑aidN i =  − ∑ ai  dV (A27) i= A,B i= A,B  R i= A,B dV 

∗ ∗ ∗ σ 2χ  ∂σ  2 2χ  ∂σ  α 2χ  ∂σ  ∑ Ni∗da idσ =   (dV ) +  α  dVdx +   dVdT (A28) i= A,B 3V∗  ∂R T ,xα R∗  ∂x R,T R∗  ∂T R,xα where 32 R  dN σ  ∗  i  χ ≡ 1− ∑ ai   2 i= A,B  dV ∗ Droplet temperature is an important variable; it allows us to take into account nonisothermal effects in condensation [11, 12]. However, the heat conductivity in condensed matter is much higher, than in a vapor; thus, the temperature is not required as a variable of nucleus description and omitted below (it falls out from consideration as a result of the corresponding kinetic limit). From the above equations, the matrix of the work of binary nucleus formation in a condensed state is

 1   ∂σ ∗  χ  ∂σ ∗  − P∗ − 2χ       L α  3V∗   ∂R T ,xα  R∗  ∂x R,T H =  2  (A29)  χ  ∂σ ∗ µα µσ  dx σ   α & A∗ σ & A∗      N∗ α + N∗ σ  α   α  R∗  ∂T R,x 1− x∗ 1− x∗  dx ∗ 

2 ∗ Eq. (A24) for (d W )(β ) was derived at a fixed state of phase β [12], which is marked by the subscript; it was assumed that the mother-phase state does not change upon the nucleus formation. Therefore, equations derived above from the condition of DF internal equilibrium at a fixed state of phase β are employed for determining matrix H elements. Specifically: (i) equations of Section 2 determines

α β σ R∗ and x∗ for a given state (supersaturation) of phase β ; (ii) at the given R∗ and T = T , x∗ is a

α σ α function of x∗ which is determined from Eq. (A23); the derivative dx /dx is taken from this equation with R = R∗ also. (iii) The derivatives of surface tension in Eq. (A29) are determined by Eq. (A21).

α ασ When the nucleus is so small that phase α is absent, N∗ → 0 , V = V , only the surface parameters remain in Eq. (A29); the set of variables ( V , xσ ) is more convenient in this case for considering the kinetics of nucleus evolution. The element hxx in these variables has the form

2 µα  dx α  µσ α & A∗   σ & A∗ hxx = N∗ α  σ  + N∗ σ 1− x∗  dx ∗ 1− x∗

α and the first summand vanishes together with N∗ . Thus, the matrix H becomes as follows:

   ∗   ∗  α    1 ∗ ∂σ χ ∂σ  dx   − PL − 2χ    α   σ   3V   ∂R  α  R  ∂x   dx   H =  ∗  T ,x  ∗ R,T ∗  (A30) χ  ∂σ ∗  dx α  µσ    σ &1∗   α   σ  N∗ σ  R x dx 1 x   ∗  ∂ R,T  ∗ − ∗  The factor χ acquires the form

R∗ ai χ ≡ 1− ∑ ασ (A31) 2 i= A,B υi From inequality

ασ 2 σ ασ 2 σ ai a aB (υ A ) x + aA (υB ) 1( − x ) ∑ ασ − ασ = ασ ασ ασ > 0 (A32) i= A,B υi υ υ A υB υ 33 it follows that

R ai Ra 3 1 ∑ ασ > ασ = and χ < − (A33) 2 i=A,B υi 2υ 2 2 i. e. χ < 0 , which is an important property [12]. Comparing Eq. (A30) to the CNT matrix H , Eq. (48), we see that the dependence of surface tension

α on radius changes the element hVV (the nucleation barrier curvature), whereas the dependence on x

σ α yields the off-diagonal elements hVx . The presence of the derivative dx /dx in Eqs. (A29) and (A30) shows that taking into account the adsorption phenomenon is naturally required, when the nucleation work is written with the surface term, whereas this is not the case for the CNT nucleation work, Eq. (48). The quadratic form with the matrix H , Eq. (A29), can be identically transformed as follows:

2 2 H (V , x) = hVV (V −V∗ ) + 2hVx (V −V∗ )( x − x∗ ) + hxx (x − x∗ )

det H 2 2 det H 2 2 ≡ (V −V∗ ) + hxx (x − xeq ) = (V −V∗ ) + hxx (x′ − x∗ ) (A34) hxx hxx

hVx hVx xeq (V ) = x∗ − (V −V∗ ) , x′ = x + (V −V∗ ) (A35) hxx hxx Such transformation with respect to arbitrary stable variables was used in Ref. [8] for normalizing the equilibrium distribution function. The quantity xeq is a solution of equation ∂H / ∂x = 0 , or ∂W / ∂x = 0 ; this fact together with the form of Eq. (A34) leads to the conclusion that xeq plays the role of equilibrium composition for a noncritical nucleus. So, xeq = x∗ for noncritical nuclei holds only in the CNT approximation, where the dependence of surface tension on composition is absent, hVx = 0 . While the critical nucleus composition fluctuates around x∗ , the noncritical nucleus composition fluctuates around xeq with the same rms. The same Eqs. (A34) and (A45) with replacement x → T hold for a unary droplet nucleation due to the dependence of surface tension on droplet temperature T [12]. The presence of the off-diagonal elements hVx does not allow us to get Eq. (64) for ax and hence to write Eq. (62). In order to solve this problem, we must go to the new variables, (V , x) → (V , x′ ), where the matrix H′ is diagonal. This procedure with respect to temperature is performed in Ref. [12]; so the resulting equations from this work can be employed here with replacement T → x . The transition matrix is

 1 0 V −V∗  V −V∗  C =   ,   = C  (A36) − hVx / hxx 1  x − x∗   x′ − x∗ 

The matrices H and Z are transformed according to equations H′ = CTHC , Z′ = C−1ZC and acquire the form 34  h   det H   z − Vx z z   0  VV h Vx Vx ′ ′  xx  H =  hxx  , Z = (A37)    det H hVx   0 hxx   2 zVx zVx + zxx   hxx hxx  As the matrix H′ is diagonal now, we can write

x&′ = a′xV& − λxx′ (x′ − x∗ ) (A38)

z′xV = a′x zVV′ , z′xx = λxx′ + a′x zVx′ (A39)

and equation for a′x has the form of Eq. (64),

zVx′ hVV′ a′x = (A40) zVV′ hxx′ Substituting the elements from Eq. (A37), we get

−1  h2  h  z h  Vx  Vx 0  0 0 Vx VV a′x = 1− 1− ax  ax , ax ≡ (A41)  hVV hxx  hVV  zVV hxx

Substituting the obtained a′x in Eq. (A38) and returning to x , according to Eq. (A35), we have

−1  h   h   Vx 0  0 Vx x& = 1− ax  ax − V& − λxx′ (x − xeq ) (A42)  hVV   hxx 

The quantity λxx′ is obtained from Eq. (A39): λxx′ = z′xx − a′x zVx′ . Taking z′xx and zVx′ from Eq. (A37), we get after simple transformations

−1  h  h   0 Vx  Vx 0  λxx′ ≡ λxx = zxx − ax − 1− ax  zVx (A43)  hxx  hVV 

This is the form of equations for x& and λxx in the case of composition-dependent surface tension. The

form of Eq. (A42) confirms the meaning of xeq as the equilibrium composition for the given V ; the

composition x relaxes to xeq , rather than to x∗ . In the CNT approximation, hVx = 0 , Eq. (A42) acquires

the form of Eq. (62), and λxx is transformed to its CNT value. Finally, it should be noted that the equations of equilibrium of Section 2 for a binary precipitate must be rederived for the case of composition-dependent surface tension. If they have the same form, but with

α σ = σ (R, x ) (as for a binary droplet [1]), nevertheless, the elements zik in Eqs. (A41)-(A43) are not the

α CNT elements given by Eq. (89); they include the terms with derivatives (∂σ / ∂R)∗ and (∂σ / ∂x )∗ . The surface parameters entering the above equations can be estimated within statistical mechanics, the density functional theory [40-43], or obtained from computer simulations.

References

[1] N. V. Alekseechkin, Thermodynamics and kinetics of binary nucleation in ideal-gas mixtures, J. 35 Chem. Phys. 143, (2015) 054502. [2] P. Mirabel, J. L. Katz, Binary homogeneous nucleation as a mechanism for the formation of aerosols, J. Chem. Phys. 60 (1974) 1138-1144. [3] B. E. Wyslouzil, G. Wilemski, Binary nucleation kinetics. II. Numerical solution of the birth– death equations, J. Chem. Phys. 103 (1995) 1137-1151. [4] B. E. Wyslouzil, G. Wilemski, Binary nucleation kinetics. III. Transient behavior and time lags, J. Chem. Phys. 105 (1996) 1090-1100. [5] V. Talanquer, D. W. Oxtoby, Nucleation of bubbles in binary fluids, J. Chem. Phys. 102 (1995) 2156-2164. [6] A. E. Kuchma, A. K. Shchekin, D. S. Martyukova, A. V. Savin, Dynamics of ensemble of gas bubbles with account of the Laplace pressure on the nucleation stage at degassing in a gas- liqiud mixture, Fluid Phase Equilibria 455 (2018) 63-69. [7] T. N ĕmec, Homogeneous bubble nucleation in binary systems of liquid solvent and dissolved gas, Chemical Physics 467 (2016) 26-37. [8] N. V. Alekseechkin, Multivariable kinetic theory of the first order phase transitions, J. Chem. Phys. 124 (2006) 124512. [9] N. V. Alekseechkin, Thermodynamics and kinetics of vapor bubbles nucleation in one- component liquids, J. Phys. Chem. B 116 (2012) 9445-9459. [10] N. V. Alekseechkin, Nucleation kinetics of vapor bubbles in a liquid with arbitrary viscosity, Eur. Phys. J. B 86: 401 (2013). [11] N. V. Alekseechkin, Multivariable theory of droplet nucleation in a single-component vapor, Physica A 412 (2014) 186-205. [12] N. V. Alekseechkin, Surface effects in droplet nucleation, J. Aerosol Sci. 116 (2018) 1-24. [13] N. V. Alekseechkin, Tolman’s length and limiting supersaturation of vapor, Chemical Physics 500 (2018) 19-25. [14] H. Reiss, The kinetics of phase transitions in binary systems, J. Chem. Phys. 18 (1950) 840-848. [15] Ya. B. 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B 27 (1983) 7372-7378. [24] K. C. Russell, Linked flux analysis of nucleation in condensed phases, Acta. Met. 16 (1968) 761-769. [25] E. M. Lifshits, L. P. Pitaevskii, Physical Kinetics, Nauka, Moscow, 1979. [26] R. C. Tolman, The effect of droplet size on surface tension, J. Chem. Phys. 17 (1949) 333-337. [27] A. I. Rusanov, Phase Equilibria and Surface Phenomena (in Russian), Khimiya, Leningrad, 1967. [28] A. I. Rusanov, Phasengleichgewichte und Grenzflächenerscheidungen, Akademie- Verlag, Berlin, 1978. [29] I. Prigogine, R. Defay, Chemical Thermodynamics, Longmans Green, New-York, 1954. [30] C. Wagner, Thermodynamics of Alloys, Addison-Wesley Press, Cambridge, 1952. [31] J. W. Christian, The Theory of Transformations in Metals and Alloys, Part I, Pergamon, Oxford, 1975. [32] B. S. Bokshtein, S. Z. Bokshtein, A. A. Zhukhovitskii, Thermodynamics and Kinetics of Diffusion in Solids (in Russian), Metallurgiya, Moscow, 1974. [33] L. S. Darken, Diffusion, mobility and their interrelation through free energy in binary metallic systems, Trans. AIME 175.1 (1948) 184-194. [34] S. Ono, S. Kondo, Molecular Theory of Surface Tension in Liquids, in Handbuch der Physik, Vol. 10, edited by S. Flügge, Springer, Berlin, 1960, p.134. [35] D. Kashchiev, Thermodynamically consistent description of the work to form a nucleus of any size, J. Chem. Phys. 118 (2003) 1837-1851. [36] D. Kashchiev, Multicomponent nucleation: Thermodynamically consistent description of the nucleation work, J. Chem. Phys. 120 (2004) 3749-3758. [37] J. D. v. d. Waals, Ph. Kohnstamm, Lehrbuch der Thermostatik. I. Allgemeine Thermostatik, Verlag von Johann Ambrosius Barth, Leipzig, 1927. [38] G. Wilemski, Composition of the critical nucleus in multicomponent vapor nucleation, J. Chem. Phys. 80 (1984) 1370-1372. [39] G. Wilemski, Revised classical binary nucleation theory for aqueous alcohol and acetone vapors, J. Phys. Chem. 91 (1987) 2492-2498. [40] R. M. 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38 a 1.4

1.2 A, B

1.0 n=1 m=1 ) 0.8 A e xc=0.5

y(x 0.6

0.4 s s A x x 0.2 x 1 2 A 0,1 x0,2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 xA e

a' 40

20 A ) A 0

(x 0 φ

B -20 xs s 1 x2

-40 0.0 0.2 0.4 0.6 0.8 1.0 xA 0

39

b 1.8 1.6 B A 1.4 1.2 )

A e 1.0 n=1 m=2

y(x 0.8 x =0.33 0.6 c 0.4 s x xs 0.2 1 2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 xA e

b' 40 30

20 A 10 ) A 0

(x 0 φ -10 B -20 s x xs -30 1 2

-40 0.0 0.2 0.4 0.6 0.8 1.0 XA 0

b'' 20 ideal solution 15 n=1 m=2

10 xc=0.33

) 5 A A 0 (x

φ 0

-5 B -10 x -15 c

-20 0.0 0.2 0.4 0.6 0.8 1.0 xA 0

40

c 5 n=1 m=5 x =0.167 4 c B A

3 ) A e

y(x 2

1 s x x xs c 1 2 0 0.0 0.2 0.4 0.6 0.8 1.0 xA e

c' 40

20 A ) A 0

x 0 φ( B B -20

x s xs c x1 2 -40 0.0 0.2 0.4 0.6 0.8 1.0 xA 0

A B s s Fig. 1. The LHS of Eq. (73), y(xe ) (solid) and y(xe )(dashed), for ω = 3, x1 = 21.0 , x2 = 79.0 , and

A different pairs (n,m ) shown in figures (a), (b) and (c). The corresponding derivative (dx e / dP L )∗ - the

A function ϕ(x0 ) in Eq. (74a) – is shown in figures (a’), (b’) and (c’). The symbols A and B show here the regions, where the corresponding component is a solute and determines the nucleation kinetics. Fig. (b’’)

A shows the function ϕ(x0 ) for ideal solution, ω = 0 . Fig. (a) also demonstrates graphical solving Eq. (73): the straight line represents the value of the RHS of Eq. (73) for some critical radius R∗ ; it intersects the

A A curve (LHS) at two points x 1,0 and x 2,0 corresponding to the given R∗ .

41

β

σ Rβ

Rα R α Vβσ Vασ

Fig. 2. Phases α , σ , and β . The density fluctuation (DF) is bounded by bold line; the surface of tension is shown by dashed line.