On the Fokker-Planck approximation in the kinetic equation of multicomponent classical nucleation theory
aa r X i v : . [ phy s i c s . c l a ss - ph ] F e b On the Fokker-Planck approximation in the kinetic equation ofmulticomponent classical nucleation theory
Yuri S. Djikaev, ∗ Eli Ruckenstein, † and Mark Swihart ‡ Department of Chemical and Biological Engineering, SUNY at Buffalo,Buffalo, New York 14260
Abstract
We study the conditions of validity of a Fokker-Planck equation with linear force coefficients asan approximation to the kinetic equation of nucleation in the theory of homogeneous isothermalmulticomponent condensation. Starting with a discrete equation of balance governing the tempo-ral evolution of the distribution function of an ensemble of multicomponent droplets and reducingit (by means of Taylor series expansions) to the differential form in the vicinity of the saddle pointof the free energy surface, we have identified the parameters whereof the smallness is necessaryfor the resulting kinetic equation to have the form of the Fokker-Planck equation with linear (indroplet variables) force coefficients. The “non-smallness” of these parameters results either in theappearance of the third or higher order partial derivatives of the distribution function in the kineticequation or in its force coefficients becoming non-linear functions of droplet variables, or both; thiswould render the conventional kinetic equation of multicomponent nucleation and its predictionsinaccurate. As a numerical illustration, we carried out calculations for isothermal condensationin five binary systems of various non-ideality at T = 293 .
15 K: butanol–hexanol, water–methanol,water–ethanol, water–1-propanol, water–1-butanol. Our results suggest that under typical exper-imental conditions, the kinetic equation of binary nucleation of classical nucleation theory mayrequire a two-fold modification and, hence, under such conditions the conventional expression forthe steady-state binary nucleation rate may not be adequate for the consistent comparison oftheoretical predictions with experimental data.. ∗ Corresponding author. E-mail: idjikaev@buffalo.edu † deceased ‡ E-mail: swihart@buffalo.edu Introduction
Nucleation is the initial stage of any homogeneous first order phase transition − that does not occuras spinodal decomposition. At the nucleation stage of condensation, which will be solely consideredhereafter for the sake of concreteness, the initial growth of nascent particles (droplets) of the liquidphase is due exclusively to fluctuations; the association of two molecules and the following associationof the third, fourth, and so on molecules is thermodynamically unfavorable (i.e., is accompanied bythe increase of the free energy of the system), but does occur owing to fluctuations. However, aftera droplet attains some critical size (and composition, in the case of multicomponent condensation),the incorporation of every supplementary molecule into the droplet becomes thermodynamicallyfavorable (i.e., is accompanied by the decrease of the free energy of the system), and the dropletgrows regularly and irreversibly. The free energy of formation of the critical droplet (often referredto as a “nucleus”) determines the height of the activation, or nucleation, barrier.The distribution function of an ensemble of droplets with respect to the independent variables ofstate of a droplet represents the object of main interest in any theory of homogeneous condensation. Inparticular, such a distribution of near-critical droplets determines the nucleation rate. The temporalevolution of the distribution of near-critical droplets is governed by the equation whereof the finite-differences form is often referred to as a “balance equation” whereas its differential form is called a“kinetic equation” of nucleation.In the case of isothermal nucleation (where the temperature of any single droplet is constant andequal to the temperature of the vapor-gas medium), the kinetic equation of nucleation is assumed tobe well approximated by the Fokker-Planck equation. In the case of non-isothermal nucleation, wherethe possibility of the deviation of the droplet temperature from that of the surrounding medium istaken into account, the Fokker-Planck approximation has been shown to be inadequate to describe2he evolution of the distribution function with respect to the droplet temperature. In this work, wewill not consider the latter case, but will discuss the applicability of the Fokker-Planck approximationto the kinetic equation of isothermal multicomponent nucleation. In the kinetic theory of homogeneous isothermal condensation, the equation governing the temporalevolution of the distribution of near-critical droplets with respect to the number of molecules in adroplet (or with respect to numbers of molecules of different components in a droplet) is conven-tionally considered to have the Fokker-Planck form. The accuracy of such an assumption for unarynucleation has been thoroughly examined by Kuni and Grinin. On the other hand, its accuracy in thecase of multicomponent nucleation has been hardly studied at all. We are aware only of two relevantpapers; one by Kuni et al. , who qualitatively outlined the general principles of the Fokker-Planckapproximation in a kinetic equation of nucleation, and the other by Kurasov, who qualitatively dis-cussed this issue in the case of non-isothermal binary nucleation. In this section, we will first brieflyoutline the results of Kuni and co-workers concerning this issue in unary nucleation (subsection 2.1)and then we will attempt to shed some light on the validity of the Fokker-Planck approximation inthe kinetic equation of homogeneous isothermal multicomponent nucleation (subsection 2.2). Consider an ensemble of one-component droplets within the metastable vapor (of the same compo-nent) at temperature T , and denote the number of molecules in a droplet by ν ; this will be the onlyvariable of state if nucleation is isothermal (i.e., the droplet temperature is constant and equal to T ).3he capillarity approximation, whereon the thermodynamics of classical nucleation theory (CNT) isbased, requires the liquid droplets to be sufficiently large, with ν ≫
1, of spherical shape, with sharpboundaries, and uniform density inside. The metastability of the one-component vapor is usuallycharacterized by the saturation ratio ζ = n/n ∞ , where n is the number density of vapor moleculesand n ∞ is the equilibrium number density of molecules the vapor that is saturated over its bulkliquid at the given temperature. Clearly, the vapor-to-liquid transition may occur only if ζ >
1; attoo large ζ ’s, it will occur as spinodal decomposition, otherwise it will proceed via nucleation.Denote the distribution function of droplets with respect ν at time t by g ( ν, t ). Assuming thatthe droplets exchange matter with the vapor via the absorption and emission of single molecules, thetemporal evolution of g ( ν, t ) is governed by the balance equation ∂g ( ν, t ) ∂t = − (cid:2) ( W + ( ν ) g ( ν, t ) − W − ( ν + 1) g ( ν + 1 , t )) − ( W + ( ν − g ( ν − , t ) − W − ( ν ) g ( ν, t )) (cid:3) , (1)where W + ( ν ) andW − ( ν ) are the numbers of molecules that a droplet ν absorbs and emits, respec-tively, per unit time. A differential equation governing the temporal evolution of g ( ν, t ) can beobtained from the discrete balance equation (1) through the Taylor series expansions of W − ( ν ± , W + ( ν ± g ( ν ± , t ) (on its RHS) with respect to the deviation of their arguments from ν .According to the classical thermodynamics, the equilibrium distribution function has the form g e ( ν ) = n exp[ − F ( ν )] , (2)where F ( ν ) is the free energy of formation of a droplet ν (in units k B T , with k B being the Boltzmannconstant). In the framework of CNT, F ( ν ) can be written , as F ( ν ) = − bν + aν / , (3)where b = ln ζ and a = 4 π (3 v l / π ) / ( σ/k B T ), with v l being the volume per molecule in the liquidphase and σ the droplet surface tension (assumed to be equal to the surface tension of bulk liquid).4hen condensation occurs via nucleation, the function F ( ν ) has a maximum at some ν c = (2 a/ b ) .A droplet with ν = ν c is called “nucleus”; the subscript “c” will be marking quantities for it.Defining the quantity ∆ ν c as 12! | F ′′ c | (∆ ν c ) = 1 , (4)where F ′′ = ∂ F/∂ν , Kuni and Grinin pointed out that the free energy of droplet formation F ( ν )and equilibrium distribution g e ( ν ) can be accurately represented as F ( ν ) ≃ F c + 12! F ′′ c ( ν − ν c ) , g e ( ν ) ≃ g e ( ν c ) exp[ − F ′′ c ( ν − ν c ) ] , (5)respectively, in the entire region ( | ν − ν c | . ∆ ν c ) of the substantial change of g e ( ν ) in the vicinity of ν c if ∆ ν c /ν c ≪ . (6)The relative inaccuracy of representations (5) within the near-critical region | ν − ν c | . ∆ ν c is of theorder O (∆ ν c /ν c ) (hereafter O ( x ) denotes a quantity of the order of x ).As clear from eq.(5), ∆ ν c represents the characteristic scale of the substantial change of theequilibrium distribution function g e ( ν ) in the vicinity of ν c . Moreover, Kuni and Grinin showedthat in that vicinity ∆ ν c also represents the characteristic scale of the substantial change of thesteady-state distribution function g s ( ν ) as well as of the distribution g ( ν, t ), so that1 g ( ν, t ) ∂g ( ν, t ) ∂ν ∼ g s ( ν ) dg s ( ν ) dν ∼ g e ( ν ) dg e ( ν ) dν ∼ ν c . (7)The absorption rate W + ( ν ) of a droplet (in eq.(1)) is determined from the gas-kinetic theory, − W + ( ν ) = 14 n ¯ v T A ( ν ) , (8)where ¯ v T = p k B T /πm is the mean thermal velocity of vapor molecules (of mass m ) and A ( ν ) =4 π (3 v l / π ) / ν / is the surface area of the droplet ν . On the other hand, the droplet emission rate5 − ( ν ) is determined through W + ( ν ) from the principle of detailed balance, stipulating that for theequilibrium distribution of droplets W − ( ν ) g e ( ν ) = W + ( ν − g e ( ν − W − ( ν ) = W + ( ν −
1) exp[ F ( ν ) − F ( ν − . (9)Carrying out the Taylor series expansions on the RHS of eq.(1) and taking into account eqs.(3)-(9), Kuni and Grinin showed that in order for the resulting differential equation in the near-criticalregion | ν − ν c | . ∆ ν c to be accurately approximated by the Fokker-Planck equation ∂g ( ν, t ) ∂t = − W + c ∂∂ν (cid:18) − F ′ ( ν ) − ∂∂ν (cid:19) g ( ν, t ) (10)with the drift/force coefficient F ′ ( ν ) = ∂F/∂ν a linear function of ν , the strong inequality1∆ ν c ≪ in addition to condition (6). The parameters ∆ ν c /ν c and 1 / ∆ ν c can be consideredto represent the small parameters of the macroscopic theory of condensation.Thus, in order for the Fokker-Planck approximation to be accurate enough in the kinetic equationof nucleation, there must exist some near-critical region whereof the half-width ∆ ν c , defined byconstraint (3), satisfies the following requirements:a) ∆ ν c is large enough so that it represents the characteristic scale of substantial change of theequilibrium distribution function in the vicinity of ν c .b) ∆ ν c is small enough so that the quadratic approximation (eq.(5)) for the free energy of formationis accurate enough in the entire near-critical vicinnity.c) ∆ ν c is much greater than the elementary change of the droplet variable; this requirement ensuresthat in the Taylor series expansions of the RHS of eq.(1) the terms with the third and higher orderderivatives of the distribution function g ( ν, t ) can be neglected compared to the term containing thesecond order derivative of g ( ν, t ). 6ote that ( in unary condensation theory only! ) the requirements b) and c) are expressed throughstrong inequalities (6) and (11), whereas the requirement a), expressed as the operator estimates ineq.(7), is automatically satisfied due to constraint (4) if the requirement b) is satisfied. Now, consider a metastable N -component vapor mixture at temperature T , within which liquiddroplets of an N -component solution form as a result of isothermal condensation via nucleation.Again, in the framework of the capillarity approximation ( whereon the thermodynamics of macro-scopic theory of multicomponent condensation is based) the droplets are treated as large sphericalparticles with sharp boundaries, uniform internal composition, density, etc..., and with the samesurface tension as that of bulk liquid solution of the same composition. − Let ν i ( i = 1 , ..., N ) be the number of molecules of component i in a droplet; Since the tempera-ture of droplets is constant (and equal to T ), the state of the droplet is completely determined by theset { ν } ≡ ( ν , ..., ν N ) which can be thus chosen as the independent variables of state of the droplet;according to the capillarity approximation, ν i ≫ i = 1 , ..., N ). The droplet chemical compositioncan be characterized by a set { χ } ≡ ( χ , ..., χ N ) of mole fractions χ i ≡ χ i ( { ν } ) = ν i /ν ( i = 1 , ..., N )(with ν = P i ν i the total number of molecules in the droplet), of which only n − P i χ i = 1. The metastability of the vapor mixture can be characterized by the set of sat-uration ratios ζ i = n i /n i ∞ ( i = 1 , ..., N ) of its component vapors, where n i is the partial numberdensity of molecules of vapor i and n i ∞ is the equilibrium number density of molecules of this vapor(that would be saturated over its own pure bulk liquid) at temperature T .Denote the distribution function of droplets with respect { ν } at time t by g ( { ν } , t ). Depending onthe convenience, any function f of variables ν , ..., ν N can be denoted by either f ( ν , ..., ν N ) or f ( { ν } )or f ( ν i , e ν i ), where the “complementary” variable e ν i would represent all but one of the variables of state7f a droplet, with the “excluded” variable being ν i . In this notation, e.g., g ( { ν } , t ) = g ( ν i , e ν i , t ) = g ( ν , ..., ν N , t ). If the droplets exchange matter with the vapor via absorption and emission of singlemolecules (as usually assumed in multicomponent CNT), the temporal evolution of the distribution g ( { ν } , t ) is governed by the balance equation ∂g ( ν, t ) ∂t = − N X i =1 (cid:2) ( W + i ( { ν } ) g ( { ν } , t ) − W − i ( ν i + 1 , e ν i ) g ( ν i + 1 , e ν i , t )) − ( W + i ( ν i − , e ν i ) g ( ν i − , e ν i , t ) − W − i ( { ν } ) g ( { ν } , t )) (cid:3) , (12)where W + i ( ν ) and W − i ( ν ) ( i = 1 , .., N ) are the numbers of molecules of component i that a droplet ν absorbs and emits, respectively, per unit time. A differential equation governing the temporalevolution of g ( { ν } , t ) can be obtained from the discrete balance equation (12) through the Taylorseries expansions of W − i ( ν i ± , e ν i ) , W + i ( ν i ± , e ν i ), and g ( ν i ± , e ν i , t ) (on its RHS) with respect tothe deviation of their arguments from ν i ( i = 1 , ..., N ).According to the classical thermodynamics, the equilibrium distribution function has the form g e ( { ν } ) = n f exp[ − F ( { ν } )] , (13)where n f is the normalizing factor and F ( { ν } ) is the free energy of formation of a droplet ν (in unitsof k B T ). It can be written in the form , , F ( { ν } ) = − X b i ν i + a ( { ν } )( X i ν i ) / , (14)where b i ≡ b i ( { ν } ) = ln[ ζ i /χ i f i ( { χ } )] ( i = 1 , .., N ), f i ( { χ } ) is the activity coefficient of component i in the droplet, and a ( { ν } ) = 4 π (3 v l / π ) / ( σ ( { ν } ) /k B T ), with v l being the volume per molecule inthe liquid phase and σ ( { ν } ) the droplet surface tension (assumed to be equal to the surface tensionof bulk liquid solution of the droplet composition { χ } ).The function F = F ( ν , .., ν N ) determines a free-energy surface in an ”N+1”-dimensional space.Under conditions when condensation occurs via nucleation, it has a shape of a hyperbolic paraboloid8“saddle-like” shape in three dimensions). Hereafter, quantities for the “saddle” point will be markedwith the subscript “c”. A droplet, whereof the variables ( ν , ..., ν N ) coincide with the coordinates ofthe saddle point, is called “nucleus”; these coordinates are determined as the solution of N simulta-neous equations F ′ i ( { ν }| c ≡ ∂F∂ν i (cid:12)(cid:12)(cid:12)(cid:12) c = 0 ( i = 1 , .., N ) . (15)where F ′ i = ∂F ( { ν } ) /∂ν i ( i = 1 , .., N ).Let us define the quadratic approximation (QA) region Ω ν in the space of variables { ν } as thevicinity of the saddle point within which F ( { ν } ) can be accurately approximated as a quadratic form F ≡ F ( { ν } ) = F c + 12 N X i,j =1 F ′′ ijc ∆ ν i ∆ ν j , (16)where F ′′ ij = ∂ F/∂ν i ∂ν j ( i, j = 1 , .., N ) and ∆ ν i ≡ ν i − ν ic ( i = 1 , .., N ). In this approximation, theequilibrium distribution can be represented as g e ( { ν } ) ≃ g e ( { ν c } ) exp[ − N X i,j =1 F ′′ ijc ∆ ν i ∆ ν j ] ( { ν } ∈ Ω ν )) (17)Approximation (16) is equivalent to neglecting the cubic and higher order terms in the Taylor se-ries expansion of F ( { ν } ) with respect to deviations ∆ ν i in the vicinity of the saddle point. Therefore,considering that 1 / ν , wherein it is acceptable,can be determined by the condition ǫ ( { ν } ) . , (18)where ǫ ( { ν } ) = | P Ni,j,k =1 b ijk (∆ ν i )(∆ ν j )(∆ ν k ) || P Ni,j =1 a ij (∆ ν i )(∆ ν j ) | (19)with a ij ≡ ∂ F ( { ν } ) ∂ν i ∂ν j (cid:12)(cid:12)(cid:12)(cid:12) c , b ijk = 13! ∂ F ( { ν } ) ∂ν i ∂ν j ∂ν k (cid:12)(cid:12)(cid:12)(cid:12) c ( i, j, k = 1 , .., N ) (20)9et us define the saddle-point (SP) region Ω ν in the space of variables { ν } as the minimal vicinityof the saddle point within which the equilibrium distribution g e ( { ν } changes substantially. Accordingto eq.(18), its boundary should thus satisfy the constraint (analogous to constraint (4) of the unarynucleation theory ) | ∆ ν T A∆ ν | ≡ | N X i,j =1 F ′′ ijc ( ν i − ν ic )( ν j − ν jc ) | = 1 , (21)where the matrix notation was introduced with a real symmetric N × N -matrix A = [ a ij ] ( i, j =1 , .., N ) and a real column-vector ∆ ν = [∆ ν i ] ( i = 1 , .., N ) of length N, the superscript “T” markingthe transpose of a matrix or vector.Since the matrix A is real and symmetric, it is orthogonally diagonalizable, according to thespectral theorem. Therefore, there exists a real orthogonal N × N -matrix P ≡ [ p µν ] ( µ, ν = 1 , .., N )(such that P − = P T ) diagonalizing the matrix A , so that the matrix D = P T AP is a real diagonal N × N matrix (hereafter the Greek subscripts µ, ... = 1 , .., N do not indicate the relation to thechemical components 1 , .., N in the system). In virtue of the spectral theorem, the columns ofthe matrix P are linearly independent orthonormal eigenvectors of A whereof the correspondingeigenvalues λ , .., λ N are the diagonal elements of D . When the free energy surface has the shapeof a hyperbolic paraboloid, one of these eigenvalues is negative (say, λ < A ) < { x } ≡ ( x , .., x N ) as x µ = N X i =1 p iµ ∆ ν i ( µ = 1 , .., N ) , (22)constituting a column-vector x ≡ [ x µ ] ( µ = 1 , .., N ) of length N . Since the difference F − F c doesnot depend on the choice of independent variables of state of a droplet, and ∆ ν T A∆ ν = x T Dx ,approximation (16) for F in variables { x } becomes F = F c + X µ λ µ x µ , (23)10nd approximation (17) for the equilibrium distribution transforms into an approximation for theequilibrium distribution q e ( { x } ) in variables { x } , q e ( { x } ) = n x exp[ − F c − X µ λ µ x µ ] , (24)with a new normalization factor n x .Thus, the quadratic form in eq.(21), determining the boundary of the SP region Ω ν in variables { ν } , becomes a diagonal quadratic form in variables { x } , so that the constraint | N X µ =1 λ µ x µ | = 1 ( λ < , λ µ > µ = 1)) (25)will determine the boundary of SP region Ω x in variables { x } ; this equation is significantly simplerthan eq.(21). Once the boundary of the SP region is determined in variables { x } , it can be also foundin variables { ν } via transformation (22).One can then evaluate the accuracy of approximation (16) within the SP region Ω ν by calculatingthe ratio ε ( { ν } ) for { ν } ∈ Ω ν . According to eq.(18), the boundaries of the QA region Ω ν , wherethis approximation is acceptably accurate, are determined by the equality ε ( { ν } ) = 1 / ν of substantial change of g ( { ν } , t ). The latter is required to smoothly transition intothe equilibrium distribution for sub-critical droplets and into the stationary distribution for super-critical ones. Therefore, the QA region Ω ν of approximation (16) (which is a must for the Fokker-Planck approximation in the kinetic equation of CNT) must cover the entire SP region Ω ν (i.e., itis necessary that Ω ν ∈ Ω ν ) in order for the kinetic equation in Ω ν to have the Fokker-Planck formin Ω ν . Therefore, the QA region Ω ν of approximation (16) (which is a must for the Fokker-Planckapproximation in the kinetic equation of CNT) must cover the entire SP region Ω ν , i.e., it is necessarythat Ω ν ∈ Ω ν (in order for the kinetic equation in Ω ν to have the Fokker-Planck form with its forcecoefficients being a linear functions of { ν } in Ω ν ).11nlike the unary nucleation theory, one cannot straightforwardly obtain the operator estimatesfor the derivatives ∂g ( { ν } , t ) /∂ν i in the Taylor series expansions of g ( e ν i , ν i ± , t ) on the RHS of thebalance eq.(12) because of the presence of mixed terms a ij ∆ ν i ∆ ν j ( i, j = 1 , .., N ) in the exponentialof eq.(17) for g e ( { ν } ). However, one can notice that the lower limits of the half-widths of the SPregion Ω x in variables { x } can be estimated to be ∆ x ≡ / p | λ | , ∆ x ≡ / √ λ , ..., ∆ xN ≡ / √ λ N along the axes x , x , ..., x N , respectively, so that1 q ( { x } , t ) ∂q ( { x } , t ) ∂x µ ∼ q s ( { x } ) ∂q s ( { x } ) ∂x µ ∼ q e ( { x } ) ∂q e ( { x } ) ∂x µ ∼ xµ ( µ = 1 , .., N ) . (26)Therefore, since ∂g ( { ν } , t ) ∂ν i = N X µ =1 ∂J g ( { x } , t ) ∂x µ ∂x µ ∂ν i , (where J is the Jacobian of transformation ∆ ν = Px ) and noticing that ∂x µ /∂ν i = p iµ , one canobtain estimates1 g ( { ν } , t ) ∂g ( { ν } , t ) ∂ν i ∼ g s ( { ν } ) ∂g s ( { ν } ) ∂ν i ∼ g e ( { ν } ) ∂g e ( { ν } ) ∂ν i . N X µ =1 p iµ q | λ µ | . ( i = 1 , .., N ) . (27)Expanding the procedure of Kuni and Grinin to multicomponent nucleation, performing theTaylor series expansions of W − i ( ν i ± , e ν i ) , W + i ( ν i ± , e ν i ), and g ( ν i ± , e ν i , t ) on the RHS of eq.(12),and taking into account eq.(27), one can show that in order for the resulting differential equation tobe accurately approximated by the conventional Fokker-Planck equation of multicomponent CNT ∂g ( { ν } , t ) ∂t = − N X i =1 W + ic ∂∂ν i (cid:18) − F ′ i ( { ν } ) − ∂∂ν i (cid:19) g ( { ν } , t ) (28)with F ′ i ( { ν } ) ( i = 1 , .., N ) being linear superpositions of ∆ ν i ( i = 1 , .., N ) in the entire SP regionΩ ν , the parameters ǫ max ≡ max ∀{ ν }∈ Ω ν ǫ ( { ν } ) , νi ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j =1 p ij q | λ j | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( i = 1 , .., N ) , (29)12ust fulfill the strong inequalities ǫ max ≪ , (30)1∆ νi ≪ i = 1 , .., N ) . (31)Thus, the parameters ǫ max and ∆ νi ( i = 1 , .., N ) represent the small parameters of the macroscopictheory of multicomponent nucleation. The violation of any one of constraints (30) or (31) will leadto the necessity of going beyond the framework of the conventional Fokker-Planck approximationusually adopted for the kinetic equation in the multicomponent CNT.If constraint (31) on the parameter 1 / ∆ νi ( i = 1 , .., N ) is not satisfied for some i , then thekinetic equation will contain the third and higher order partial derivatives of the distribution function g ( { ν } , t ) with respect to ν i . (This constraint can be referred to as the SP region constraint becauseit characterizes how smoothly the distribution function varies in the SP region.) An elegant method(based on the combination the Enskog-Chapman method and method of complete separation ofvariables) for the solution of such a non-Fokker-Planck kinetic equation was developed by Kuni andGrinin (see also references 12,13 for its application).On the other hand, if the parameter ǫ max does not satisfy constraint (30), then the QA regionΩ ν of quadratic approximation (16) for F ( { ν } ) does not cover the entire SP region Ω ν , and it willbe necessary to retain the cubic and maybe even higher order (in ∆ ν i ( i = 1 , .., N )) terms in theTaylor series expansion for F ( { ν } ). (This constraint can be referred to as the QA region constraintbecause it characterizes the extent of the QA region.) As a result, the first derivatives F ′ i in thekinetic equation (28) will not be linear superpositions of deviations ∆ ν i ( i = 1 , .., N ) (they will bequadratic at least, or even of higher orders), hence the force/drift coefficients of equation (28) will nolonger be linear functions of { ν } , i.e., the kinetic equation will differ from the Fokker-Planck equationof multicomponent CNT. We are not aware of any work that would concern the solution of such a13inetic equation in the theory of multicomponent nucleation. As a numerical illustration of the foregoing, we have carried out calculations for isothermal conden-sation in five binary systems:(a) butanol (component 1) – hexanol (component 2);(b) water (component 1) – methanol (component 2);(c) water (component 1) – ethanol (component 2);(d) water (component 1) – 1-propanol (component 2);(e) water (component 1) – 1-butanol (component 2);These systems were chosen as representatives for the nucleation of droplets of ideal (a) and increas-ingly nonideal (b)-(e) binary solutions whose physical and chemical properties, necessary for theevaluation of parameters ǫ max and ∆ νi ( i = 1 , .., N ) in eq.(29), are relatively well known and avail-able from various sources. For all the systems, the molecular volumes v and v of pure liquids wereobtained using the density data of Lide , and the mean molecular volume of solution in the dropletwas (for the purpose of rough evaluations) approximated as v = χv + (1 − χ ) v , with χ = χ . Allcalculations were carried out for the same system temperature T = 293 .
15 K. Although the satura-tion ratios ζ and ζ were different in different systems, they were always chosen so that the heightof the barrier at the saddle point was in the range from 30 to 50, which would ensure a noticeablenucleation rate (according to binary CNT , , , ).The surface tension of 1-butanol(1)–1-hexanol(2) solution (which can be roughly treated as ideal)was assumed to depend on χ (= χ ) as σ ( χ ) = χσ + (1 − χ ) σ , where σ and σ are the surfacetensions of pure liquid butanol and pure liquid hexanol, respectively: σ = 25 .
39 dyn/cm was obtained14y linear interpolation of data from
Lide and σ = 26 .
20 dyn/cm was taken from
Gallant . Theactivity coefficients of both butanol and hexanol in this solution were set equal to unity (ideal solutionapproximation).For the composition dependence of the surface tension of droplets in the systems (b)-(e) we usedan expression σ ( χ ) = a + b/ ( d − χ ) + c/ ( d − χ ) (32)(with χ = χ and the dimension of σ dyn/cm), where the set of parameters a, b, c, d was differentin each system. These parameters were determined with the help of Mathematica 12.1 by fittingexpression (32) to appropriate experimental data (of Vazquez et al. for the systems (b)-(d)) and ofTeitelbaum et al. for the system (e)):(b) a = 16 . , b = 8 . , c = − . × − , d = 1 . a = 19 . , b = 3 . , c = − . × − , d = 1 . a = 23 . , b = 0 . , c = − . × − , d = 1 . a = 24 . , b = − . , c = 0 . , d = 1 . f ( χ ) = A (1 + A χA (1 − χ ) ) , ln f ( χ ) = A (1 + A (1 − χ ) A χ ) . (33)where the the pairs of parameters A and A for the considered systems are provided in refs.20,21:(b) A = 0 . A = 0 . A = 0 . A =1 . A = 1 . A = 2 . A = 1 . A = 4 . ζ and ζ of vapor mixture components are indicated in the figure captions.15n each Figure, the panel a) shows the SP region Ω x and the QA region Ω x in variables { x } ,whereas the panel b) shows the SP region Ω ν and the QA region Ω ν in variables { ν } ; both Ω x and Ω ν are shown as grayish areas in these Figures. The solid curves indicate the borders of theSP regions, whereas the dashed ones indicate the boundaries of QA regions. In the panel a) of eachFigure, the thin dashed lines delineate the rectangular central part of the SP region Ω x of half-widths∆ x and ∆ x which were used in calculating the parameters 1 / ∆ ν and ∆ ν according to eq.(31). In thepanel b) of each Figure, the corresponding central part of the SP region Ω ν is also shown, delineatedby thin dashed lines forming now a parallelogram.As clear from all these Figures, in any of the systems studied the QA region does not extendto many parts of the SP region. Moreover, the QA region does not even cover the central parts ofthe SP region, failing to hold even on some segments of its sub-critical and super-critical borders, atwhich the boundary conditions to the kinetic equation (28) are imposed.Thus, for all the systems studied, the quadratic approximation (16) for F ( { ν } ) is not sufficientlyaccurate in the entire SP region, and it is necessary to retain the cubic and maybe even higher order(in ∆ ν i ( i = 1 , .., N )) terms in the Taylor series expansion for F ( { ν } ). As a result, the first derivatives F ′ i in the kinetic equation (28) will not be linear superpositions of deviations ∆ ν i ( i = 1 , .., N ) (theywill be bilinear at least, or even of higher orders), hence the drift coefficients of equation (28) will nolonger be linear functions of { ν } , i.e., the kinetic equation will differ from the Fokker-Planck equationof multicomponent CNT. Therefore, it would be inadequate to use the conventional expression forthe steady-state binary nucleation rate, obtained on the basis of approximation (16), in comparingtheoretical predictions with experimental data for the binary nucleation rate in these systems.One can also notice, that the relative fraction of the SP region which is not covered by the QAregion increases with increasing non-ideality of the solution in droplets, being the smallest in thehexanol–butanol system and largest in the water–butanol system. This fraction can be probably16 able Small parameters 1 / ∆ ν and 1 / ∆ ν of the applicability of the Fokker-Planck approximation in thekinetic equation of binary nucleation T = 293 .
15 KBinary system ζ ζ / ∆ ν / ∆ ν . . .
11 0 . . . .
17 0 . . . .
09 0 . .
35 1 .
91 0 .
001 0 . .
25 2 . .
09 0 . ε max which apparently never satisfies constraint (30)).We have also evaluated the parameters 1 / ∆ ν and 1 / ∆ ν in all systems (a)-(e). Constraint (31) onthese parameters is necessary for neglecting the terms with the third and higher order derivatives inthe Taylor series expansions on the RHS of the balance equation (12). As clear from the Table, thesmallness of these parameters under metastability conditions that we considered is fulfilled perfectlywell. However, they are very sensitive to saturation ratios ζ and ζ , so that their smallness at agiven pair of ζ , ζ does not by any means guarantee their smallness at different metastability of thevapor mixture. Similar caution must be exercised with respect to the parameter ε max .Thus, both constraints (30) and (31) must be verified at given T, ζ , ζ , and only if they hold, onecan use the conventional CNT expression for the binary nucleation rate J s in comparing theoreticalpredictions with experimental data. Otherwise, another, more adequate theoretical theoretical ex-pression for J s must be obtained by solving a properly modified kinetic equation (which may be ofeven of non-Fokker-Planck form). 17 Concluding remarks
The distribution function of an ensemble of new phase particles with respect to their independentvariables of state constitute an object of main interest in a kinetic theory of any first-order phase tran-sition. In particular, such a distribution of near-critical droplets determines the nucleation rate. Atthe stage of nucleation, the temporal evolution of this distribution is governed by a kinetic equation.For isothermal transitions, it is conventionally approximated by the Fokker-Planck equation with thedrift/force coefficients being linear functions of independent variables of state of new phase particles.The applicability of this approximation to the kinetic equation of nucleation in homogeneous unarycondensation has been thoroughly examined by Kuni and Grinin. In this work, we have attempted to shed some light on the conditions necessary for a Fokker-Planck equation with linear force coefficients to be an adequate approximation to the kinetic equationof nucleation in a macroscopic theory of isothermal homogeneous multicomponent condensation.Starting with a discrete equation of balance governing the temporal evolution of the distributionfunction of an ensemble of multicomponent droplets and reducing it (by means of Taylor seriesexpansions) to the differential form in the vicinity of the saddle point of the free energy surface, wehave obtained the constraints that must be fulfilled in order for the resulting kinetic equation to havethe form of the Fokker-Planck equation and for its force coefficients to be linear functions of dropletvariables; we have also identified the corresponding “small” parameters.If those constraints (which can be referred to as the saddle point (SP) region and quadraticapproximation (QA) region constraints) are not satisfies, then either there will be the third or higherorder partial derivatives of the distribution function in the kinetic equation (when the SP constraintdoes not hold) or the drift coefficients of the Fokker-Planck equation will become non-linear functionsof independent variables (when the QA constraint does not hold), or both. In any of these cases,18he conventional kinetic equation of multicomponent nucleation and its predictions would becomeinaccurate.As a numerical illustration of the foregoing, we have carried out calculations for isothermalcondensation in five binary systems (at T = 293 .
15 K and vapor mixture metastabilities typicalto experimental conditions): butanol–hexanol, water–methanol, water–ethanol, water–1-propanol,water–1-butanol. These systems were chosen as representatives for the nucleation of droplets ofideal (a) and increasingly nonideal (b)-(e) binary solutions. Our numerical results suggest thatat considered temperature T and saturation ratios ζ , ζ the SP constraint on the smoothness ofthe droplet distribution in the SP region is well fulfilled, which substantiates neglecting the thirdand higher order derivatives of the distribution function in the conventional kinetic equation, i.e.,its generic Fokker-Planck form. However, the QA constraint on the quadratic approximation inthe Taylor series expansion of the free energy of droplet formation in the saddle point region isnot satisfied; therefore, the drift coefficients in that generic Fokker-Planck equation are not linearfunctions of droplet variables. Hence, the kinetic equation of binary nucleation does not have theform adopted in the binary CNT, so that the conventional expressions , , for the steady-statebinary droplet distribution and binary nucleation rate will provide inaccurate predictions and maymarkedly differ from experimental data.Moreover, numerical calculations show that whether the constraints on the small parameters (ofthe binary CNT kinetic equation) are satisfied or not is quite sensitive to the saturation ratios ζ , ζ and this sensitivity increases with increasing non-ideality of the liquid solution in droplets. Therefore,it is necessary to obtain the steady-state solutions of the modified kinetic equation, going beyondthe framework of the Fokker-Planck equation of CNT due to the non-fulfillment of either the SPregion constraint (when the third or even higher order derivatives of the distribution function arepresent in the kinetic equation) or the QA region constraint (when the force coefficients in the generic19okker-Planck equation are not linear functions of droplet variables) or both. Clearly, such solutionsare needed for the consistency of the comparison of theoretical predictions and experimental dataobtained under above. This will be the object of our further research. Acknowledgements - This research project was started and partially done when Eli Ruckenstein, one of the authors,was still alive, but was completed only after he passed away.
References (1) D. Kaschiev,
Nucleation : basic theory with applications (Butterworth Heinemann, Oxford,Boston, 2000).(2) J.W.P. Schmelzer,
Nucleation Theory and Applications (Wiley-VCH Verlag GmbH, 2005)(2 a ) V.V Slezov, Kinetics of First-Order Phase Transitions (Wiley-VCH, Berlin, 2009).(3) E. Ruckenstein and G. Berim,
Kinetic theory of nucleation (CRC, New York, 2016).(4) A.P. Grinin and F.M. Kuni,
Vestnik Leningradskogo universiteta. Seriya Fizika, Khimiya (inRussian), 1982, , 10-14.(5) F.M. Kuni, A.P. Grinin, and A.K. Shchekin, Physica A , 1998, , 67-84.(6) V.B. Kurasov,
Physica A , 2000, , 219-255.(7) J. Lothe and G.M.J. Pound, in
Nucleation ; Zettlemoyer, A. C., Ed.; Marcel-Dekker: New York,1969.(8)
Reiss, H.
J. Chem. Phys. (1950), 840. 209) A.E. Kuchma and A.K. Shchekin, J. Chem. Phys. (2019), 054104.(10) R. G. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 2013).(11)
Grinin, A.P. and
Kuni, F.M.
Theor. Math. Phys. (1989), 968.(12) Y. S. Djikaev, F. M. Kuni and A. P. Grinin, J. Aerosol Sci. , 1999, , 265-277.(13) Y. S. Djikaev, J. Teichmann and M. Grmela, Physica A , 1999, , 322-342.(14)
CRC Handbook of Chemistry and Physics . 75th Edition; Lide, D. R., Ed.; CRC Press: BocaRaton, 1994-1995.(15) Stauffer, D. Kinetic theory of two-component (“hetero-molecular”) nucleation and condensation.
J. Aerosol Sci. , , 319-333.(16) Melikhov, A.A., Kurasov, V.B., Djikaev, Y.S., and
Kuni, F.M.
Soviet Phys. Techn.Phys. (1991), 14.(17) Gallant, R.W., Physical properties of hydrocarbons, XI, Miscellaneous alcohols, HydrocarbonProcess., (1967), 46, 133-139.(18) G.Vazquez, E.Alvarez, J.M.Navaza J.Chem.Eng.Data (1995), (3), 611-614.(19) B. Y. Teitelbaum, T. A. Gortalova, and E. E. Siderova, Zh. Fiz. Khim. 25, 911-919 (1951).(20) Gmehling, J., and U. Onken, Vapor-Liquid Equilibrium Data Collection, vol. 1, part 1, Dtsch.Ges. fiir Chem. Apparatewesen, Chem. Tech. und Biotechnol., Frankfurt, Germany, 1977.(21) Perry’s Chemical Engineers Handbook . Perry, R. H.; Green, D. W., Eds; McGraw Hill Compa-nies, 1999. 21 aptions to Figures 1 to 5 of the manuscript “On the Fokker-Planck approximation in the kineticequation of multicomponent classical nucleation theory ”
Figure 1. The saddle point (SP) region and quadratic approximation (QA) region of the space ofdroplet variables for binary nucleation in 1-butanol(1)–1-hexanol(2) vapor mixture at T = 293 .
15 K, ζ = 2 .
0, and ζ = 1 .
7. a) The SP region Ω x and the QA region Ω x in variables { x } . b) The SPregion Ω ν and the QA region Ω ν in variables { ν } . The solid curves indicate the borders of the SPregions, whereas the dashed ones indicate the boundaries of the QA regions. Both Ω x and Ω ν areshown as grayish areas. The thin dashed line segments delineate the central part of the SP region(see the text).Figure 2. The saddle point (SP) region and quadratic approximation (QA) region of the space ofdroplet variables for binary nucleation in water(1)–methanol(2) vapor mixture at T = 293 .
15 K, ζ = 1 .
9, and ζ = 1 .
1. a) The SP region Ω x and the QA region Ω x in variables { x } . b) The SPregion Ω ν and the QA region Ω ν in variables { ν } . The solid curves indicate the borders of the SPregions, whereas the dashed ones indicate the boundaries of the QA regions. Both Ω x and Ω ν areshown as grayish areas. The thin dashed line segments delineate the central part of the SP region(see the text).Figure 3. The saddle point (SP) region and quadratic approximation (QA) region of the spaceof droplet variables for binary nucleation in water(1)–ethanol(2) vapor mixture at T = 293 .
15 K, ζ = 1 .
7, and ζ = 0 .
9. a) The SP region Ω x and the QA region Ω x in variables { x } . b) The SPregion Ω ν and the QA region Ω ν in variables { ν } . The solid curves indicate the borders of the SPregions, whereas the dashed ones indicate the boundaries of the QA regions. Both Ω x and Ω ν are22hown as grayish areas. The thin dashed line segments delineate the central part of the SP region(see the text).Figure 4. The saddle point (SP) region and quadratic approximation (QA) region of the space ofdroplet variables for binary nucleation in water(1)–1-propanol(2) vapor mixture at T = 293 .
15 K, ζ = 1 .
35, and ζ = 1 .
91. a) The SP region Ω x and the QA region Ω x in variables { x } . b) The SPregion Ω ν and the QA region Ω ν in variables { ν } . The solid curves indicate the borders of the SPregions, whereas the dashed ones indicate the boundaries of the QA regions. Both Ω x and Ω ν areshown as grayish areas. The thin dashed line segments delineate the central part of the SP region(see the text).Figure 5. The saddle point (SP) region and quadratic approximation (QA) region of the space ofdroplet variables for binary nucleation in water(1)–1-butanol(2) vapor mixture at T = 293 .
15 K, ζ = 1 .
25, and ζ = 2 .
5. a) The SP region Ω x and the QA region Ω x in variables { x } . b) The SPregion Ω ν and the QA region Ω ν in variables { ν } . The solid curves indicate the borders of the SPregions, whereas the dashed ones indicate the boundaries of the QA regions. Both Ω x and Ω ν areshown as grayish areas. The thin dashed line segments delineate the central part of the SP region(see the text). 23) Region - - -
10 0 10 20 30 - - x x b) Saddle Point Region
40 50 60 70 80 90 100 1102030405060 ν ν (aa) (bb) Figure 1:24)
Region - -
10 0 10 20 - - x x b) Saddle Point Region ν ν Figure 2:25)
Region - -
20 0 20 40 - - x x b) Saddle Point Region
40 50 60 70 80 90 100510152025303540 ν ν Figure 3:26)
Region - - -
10 0 10 20 30 - - - x x b) Saddle Point Region
40 60 80 100 12020406080 ν ν Figure 4:27)
Region - - -
10 0 10 20 30 - - x x b) Saddle Point Region
20 30 40 50 60 70304050607080 ν ν2