#  On the Fokker-Planck approximation in the kinetic equation of
multicomponent classical nucleation theory

Yuri S. Djikaev, Eli Ruckenstein, and Mark Swihart
Department of Chemical and Biological Engineering, SUNY at Buffalo,
Buffalo, New York 14260 Corresponding author. E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We examine the validity of the Fokker-Planck equation with linear force
coefficients as an approximation to the kinetic equation of nucleation in
homogeneous isothermal multicomponent condensation. Starting with a discrete
equation of balance governing the temporal evolution of the distribution
function of an ensemble of multicomponent droplets and reducing it (by means
of Taylor series expansions) to the differential form in the vicinity of the
saddle point of the free energy surface, we have identified the parameters
whereof the smallness is necessary for the resulting kinetic equation to have
the form of the Fokker-Planck equation with linear (in droplet variables)
force coefficients. The “non-smallness” of these parameters results either in
the appearance of the third or higher order partial derivatives of the
distribution function in the kinetic equation or in its force coefficients
becoming non-linear functions of droplet variables, or both; this would render
the conventional kinetic equation of multicomponent nucleation and its
predictions inaccurate. As a numerical illustration, we carried out
calculations for isothermal condensation in five binary systems of various
non-ideality at $T=293.15$ K: 1-butanol–1-hexanol, water–methanol,
water–ethanol, water–1-propanol, water–1-butanol. Our results suggest that
under typical experimental conditions the kinetic equation of binary
nucleation of classical nucleation theory may require a two-fold modification
and, hence, the conventional expression for the steady-state binary nucleation
rate may not be adequate for the consistent comparison of theoretical
predictions with experimental data.

##  1 Introduction

Nucleation is the initial stage of any homogeneous first order phase
transition1-3 that does not occur as spinodal decomposition. At the nucleation
stage of condensation, hereinafter the sole subject of consideration for the
sake of concreteness, the initial growth of nascent particles (droplets) of
the liquid phase is due exclusively to fluctuations; the association of two
molecules and the subsequent association of the third, fourth, and additional
molecules is thermodynamically unfavorable (i.e., is accompanied by an
increase in the free energy of the system), but does occur owing to
fluctuations. However, after a droplet attains some critical size (and
composition, in the case of multicomponent condensation), the incorporation of
each additional molecule becomes thermodynamically favorable (i.e., is
accompanied by a decrease in the free energy of the system), and the droplet
grows irreversibly. The free energy of formation of the critical droplet
(often referred to 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 of state of a droplet represents the main object of
interest in any theory of homogeneous condensation. In particular, such a
distribution of near-critical droplets determines the nucleation rate. The
temporal evolution 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 (in which the temperature of any single
droplet is constant and equal to the temperature of the surrounding vapor-gas
medium), the kinetic equation of nucleation is assumed to be well approximated
by the Fokker-Planck equation. In the case of non-isothermal nucleation, where
the possibility of the deviation of the droplet temperature from that of the
surrounding medium is taken into account, the Fokker-Planck approximation has
been shown to be inadequate to describe the evolution of the distribution
function with respect to the droplet temperature. Hereinafter, we do not
consider the nonisothermal case, but focus on the applicability of the Fokker-
Planck approximation to the kinetic equation of isothermal multicomponent
nucleation.

##  2 The Fokker-Planck approximation in the kinetic equation of homogeneous
isothermal nucleation

In the kinetic theory of homogeneous isothermal condensation, the equation
governing the temporal evolution of the distribution of near-critical droplets
with respect to the number of molecules in a droplet (or with respect to
numbers of molecules of different components in a droplet) is conventionally
considered to have the Fokker-Planck form. The accuracy of such an assumption
for unary nucleation has been thoroughly examined by Kuni and Grinin.4 On the
other hand, its accuracy in the case of multicomponent nucleation has been
studied very little. We are aware of only two relevant papers; one by Kuni et
al.,5 who qualitatively outlined the general principles of the Fokker-Planck
approximation in a kinetic equation of nucleation, and the other by Kurasov,6
who qualitatively discussed this issue in the case of nonisothermal binary
nucleation. In this section, we will first briefly outline the results of Kuni
and co-workers concerning this issue in unary nucleation (subsection 2.1) and
then attempt to shed some light on the validity of the Fokker-Planck
approximation in the kinetic equation of homogeneous isothermal multicomponent
nucleation (subsection 2.2).

###  2.1 Unary nucleation

Consider an ensemble of one-component droplets within the metastable vapor (of
the same component) at temperature $T$, and denote the number of molecules in
a droplet by $\nu$; this will be the only variable of state if nucleation is
isothermal (i.e., the droplet temperature is constant and equal to $T$). The
capillarity approximation,7 whereon the thermodynamics of classical nucleation
theory (CNT) is based, requires the liquid droplets to be sufficiently large,
with $\nu\gg 1$, and of spherical shape, with sharp boundaries, and uniform
density inside. The metastability of the one-component vapor is usually
characterized by the saturation ratio $\zeta=n/n_{1\infty}$, where $n$ is the
number density of vapor molecules and $n_{1\infty}$ is the number density of
molecules of the vapor in equilibrium with its bulk liquid at the system
temperature. Clearly, the vapor-to-liquid transition can occur only if
$\zeta>1$; at sufficiently large values of $\zeta$, it will occur as spinodal
decomposition, otherwise it will proceed via nucleation.

Denote the distribution function of droplets with respect $\nu$ at time $t$ by
$g(\nu,t)$. Assuming that the droplets exchange matter with the vapor via the
absorption and emission of single molecules, the temporal evolution of
$g(\nu,t)$ is governed by the balance equation

$\frac{\partial g(\nu,t)}{\partial
t}=-\left[(W^{+}(\nu)g(\nu,t)-W^{-}(\nu+1)g(\nu+1,t))-(W^{+}(\nu-1)g(\nu-1,t)-W^{-}(\nu)g(\nu,t))\right],$
(1)

where $W^{+}(\nu)andW^{-}(\nu)$ are the numbers of molecules that a droplet
$\nu$ absorbs and emits, respectively, per unit time. A differential equation
governing the temporal evolution of $g(\nu,t)$ can be obtained from the
discrete balance equation (1) through the Taylor series expansions of
$W^{-}(\nu\pm 1),W^{+}(\nu\pm 1)$, and $g(\nu\pm 1,t)$ (on its RHS) with
respect to the deviation of their arguments from $\nu$.

According to classical thermodynamics, the equilibrium distribution function
has the form

$g_{e}(\nu)=n\exp[-F(\nu)],$ (2)

where $F(\nu)$ is the free energy of formation of a droplet of size $\nu$ (in
units $k_{B}T$, $k_{B}$ being the Boltzmann constant). In the framework of
CNT, $F(\nu)$ can be written4,5 as

$F(\nu)=-b\nu+a\nu^{2/3},$ (3)

where $b=\ln\zeta$ and $a=4\pi(3v_{l}/4\pi)^{2/3}(\sigma/k_{B}T)$, with
$v_{l}$ being the volume per molecule in the liquid phase and $\sigma$ the
droplet surface tension (assumed to be equal to the surface tension of the
bulk liquid). When condensation occurs via nucleation, the function $F(\nu)$
has a maximum at some $\nu_{c}=(2a/3b)^{3}$. A droplet with $\nu=\nu_{c}$ is
called “nucleus”; the subscript “c” will mark quantities related thereto.

Defining the quantity $\Delta\nu_{c}$ by the equality

$\frac{1}{2}|F_{c}^{\prime\prime}|(\Delta\nu_{c})^{2}=1,$ (4)

where $F^{\prime\prime}=\partial^{2}F/\partial\nu^{2}$, Kuni and Grinin4
pointed out that the free energy of droplet formation $F(\nu)$ and equilibrium
distribution $g_{e}(\nu)$ can be accurately represented as

$F(\nu)\simeq
F_{c}+\frac{1}{2}F_{c}^{\prime\prime}(\nu-\nu_{c})^{2},\;\;\;g_{e}(\nu)\simeq
g_{e}(\nu_{c})\exp[-\frac{1}{2}F_{c}^{\prime\prime}(\nu-\nu_{c})^{2}],$ (5)

respectively, over the entire region $(|\nu-\nu_{c}|\lesssim\Delta\nu_{c})$ of
the substantial change of $g_{e}(\nu)$ in the vicinity of $\nu_{c}$ if

$\Delta\nu_{c}/\nu_{c}\ll 1.$ (6)

The relative inaccuracy of representations (5) within the near-critical region
$|\nu-\nu_{c}|\lesssim\Delta\nu_{c}$ is of the order of
$\Delta\nu_{c}/\nu_{c}$.

As clear from eq.(5), $\Delta\nu_{c}$ represents the characteristic scale of
the substantial change of the equilibrium distribution function $g_{e}(\nu)$
in the vicinity of $\nu_{c}$. Moreover, Kuni and Grinin4 showed that in that
vicinity $\Delta\nu_{c}$ also represents the characteristic scale of the
substantial change of the steady-state distribution function $g_{s}(\nu)$ as
well as of the distribution $g(\nu,t)$, so

$\frac{1}{g(\nu,t)}\frac{\partial
g(\nu,t)}{\partial\nu}\sim\frac{1}{g_{s}(\nu)}\frac{dg_{s}(\nu)}{d\nu}\sim\frac{1}{g_{e}(\nu)}\frac{dg_{e}(\nu)}{d\nu}\sim\frac{1}{\Delta\nu_{c}}.$
(7)

The absorption rate $W^{+}(\nu)$ of a droplet (in eq.(1)) is determined from
the gas-kinetic theory,1-5

$W^{+}(\nu)=\frac{1}{4}n\;\overline{v}_{T}A(\nu),$ (8)

where $\overline{v}_{T}=\sqrt{8k_{B}T/\pi m}$ is the mean thermal velocity of
vapor molecules (of mass $m$) and $A(\nu)=4\pi(3v_{l}/4\pi)^{2/3}\nu^{2/3}$ is
the surface area of the droplet. On the other hand, the droplet emission rate
$W^{-}(\nu)$ is determined through $W^{+}(\nu)$ from the principle of detailed
balance, stipulating that for the equilibrium distribution of droplets
$W^{-}(\nu)g_{e}(\nu)=W^{+}(\nu-1)g_{e}(\nu-1)$, so that, according to eq.(2),

$W^{-}(\nu)=W^{+}(\nu-1)\exp[F(\nu)-F(\nu-1)].$ (9)

Carrying out the Taylor series expansions on the RHS of eq.(1) and taking into
account eqs.(3)-(9), Kuni and Grinin4 showed that for the resulting
differential equation in the near-critical region
$|\nu-\nu_{c}|\lesssim\Delta\nu_{c}$ to be accurately approximated by the
Fokker-Planck equation

$\frac{\partial g(\nu,t)}{\partial
t}=-W^{+}_{c}\frac{\partial}{\partial\nu}\left(-F^{\prime}(\nu)-\frac{\partial}{\partial\nu}\right)g(\nu,t)$
(10)

with the drift/force coefficient $F^{\prime}(\nu)=\partial F/\partial\nu$ a
linear function of $\nu$, the strong inequality

$\frac{1}{\Delta\nu_{c}}\ll 1$ (11)

must be fulfilled in addition to condition (6). The parameters
$\Delta\nu_{c}/\nu_{c}$ and $1/\Delta\nu_{c}$ can be considered to represent
the small parameters of the macroscopic theory of condensation.

Thus, for the Fokker-Planck approximation to be suitable in the kinetic
equation of nucleation, there must exist some near-critical region whereof the
half-width $\Delta\nu_{c}$, defined by constraint (3), satisfies the following
requirements:
a) $\Delta\nu_{c}$ is large enough to represent the characteristic scale of
substantial change of the equilibrium distribution function in the vicinity of
$\nu_{c}$.
b) $\Delta\nu_{c}$ is small enough that the quadratic approximation (eq.(5))
for the free energy of formation is acceptable across the entire near-critical
vicinity.
c) $\Delta\nu_{c}$ is much greater than the elementary change of the droplet
variable; this requirement ensures that in the Taylor series expansions of the
RHS of eq.(1) the terms with the third and higher order derivatives of the
distribution function $g(\nu,t)$ can be neglected compared to the term
containing the second order derivative of $g(\nu,t)$.
Note that (in unary condensation theory only!) the requirements b) and c) are
expressed through strong inequalities (6) and (11), whereas the requirement
a), expressed as the operator estimates in eq.(7), is automatically satisfied
due to constraint (4) if the requirement b) is satisfied.

###  2.2 Multicomponent nucleation

Now, consider a metastable $N$-component vapor mixture at temperature $T$,
within which liquid droplets 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 macroscopic theory of
multicomponent condensation is based) the droplets are treated as spheres with
sharp boundaries, internal thermodynamic equilibrium, and with the surface
tension of the bulk liquid of the same composition.7-9

Let $\nu_{i}\;\;(i=1,...,N)$ be the number of molecules of component $i$ in a
droplet. Becasue the temperature of the droplet is constant (and equal to
$T$), the state of the droplet is completely determined by the set
$\\{\nu\\}\equiv(\nu_{1},...,\nu_{N})$ which can be thus chosen as the
independent variables of state of the droplet; the capillarity approximation
implies that $\nu_{i}\gg 1\;\;(i=1,...,N)$. The droplet chemical composition
can be characterized by a set $\\{\chi\\}\equiv(\chi_{1},...,\chi_{N})$ of
mole fractions $\chi_{i}\equiv\chi_{i}(\\{\nu\\})=\nu_{i}/\nu\;\;(i=1,...,N)$
(with $\nu=\sum_{i}\nu_{i}$ the total number of molecules in the droplet), of
which only $n-1$ are independent because $\sum_{i}\chi_{i}=1$. The
metastability of the vapor mixture can be characterized by the set of
saturation ratios $\zeta_{i}=n_{i}/n_{i\infty}\;\;(i=1,...,N)$ of its
component vapors, where $n_{i}$ is the partial number density of molecules of
vapor $i$ and $n_{i\infty}$ is the number density of molecules of vapor $i$ in
equilibrium with its pure bulk liquid at temperature $T$.

Denote the distribution function of droplets with respect $\\{\nu\\}$ at time
$t$ by $g(\\{\nu\\},t)$. Introduce the “complementary” variable
$\widetilde{\nu_{i}}$ to include all but one of the variables of state of a
droplet, with the “excluded” variable being $\nu_{i}$; any function $f$ of
variables $\nu_{1},...,\nu_{N}$ can be thus represented as either
$f(\nu_{1},...,\nu_{N})$ or $f(\\{\nu\\})$ or
$f(\nu_{i},\widetilde{\nu_{i}})$. For instance,
$g(\\{\nu\\},t)=g(\nu_{i},\widetilde{\nu_{i}},t)=g(\nu_{1},...,\nu_{N},t)$.

If the droplets exchange matter with the vapor via absorption and emission of
single molecules (as usually assumed in multicomponent CNT), the temporal
evolution of the distribution $g(\\{\nu\\},t)$ is governed by the balance
equation

$\displaystyle\frac{\partial g(\nu,t)}{\partial t}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{N}\left[(W_{i}^{+}(\\{\nu\\})g(\\{\nu\\},t)-W^{-}_{i}(\nu_{i}+1,\widetilde{\nu_{i}})g(\nu_{i}+1,\widetilde{\nu_{i}},t))\right.$
(12)
$\displaystyle\left.-(W^{+}_{i}(\nu_{i}-1,\widetilde{\nu_{i}})g(\nu_{i}-1,\widetilde{\nu_{i}},t)-W_{i}^{-}(\\{\nu\\})g(\\{\nu\\},t))\right],$

where $W^{+}_{i}(\nu)$ and $W^{-}_{i}(\nu)\;\;(i=1,..,N)$ are the numbers of
molecules of component $i$ that a droplet $\nu$ absorbs and emits,
respectively, per unit time. A differential equation governing the temporal
evolution of $g(\\{\nu\\},t)$ can be obtained from the discrete balance
equation (12) through the Taylor series expansions of $W_{i}^{-}(\nu_{i}\pm
1,\widetilde{\nu}_{i}),W_{i}^{+}(\nu_{i}\pm 1,\widetilde{\nu}_{i})$, and
$g(\nu_{i}\pm 1,\widetilde{\nu}_{i},t)$ (on its RHS) with respect to the
deviation of their arguments from $\nu_{i}\;\;(i=1,...,N)$.

According to classical thermodynamics, the equilibrium distribution function
has the form

$g_{e}(\\{\nu\\})=n_{\mbox{\tiny f}}\exp[-F(\\{\nu\\})],$ (13)

where $n_{\mbox{\tiny f}}$ is the normalization factor and $F(\\{\nu\\})$ is
the free energy of formation of a droplet $\nu$ (in units of $k_{B}T$). It can
be written in the form1,8,9

$F(\\{\nu\\})=-\sum b_{i}\nu_{i}+a(\\{\nu\\})(\sum_{i}\nu_{i})^{2/3},$ (14)

where $b_{i}\equiv
b_{i}(\\{\chi\\})=\ln[\zeta_{i}/\chi_{i}f_{i}(\\{\chi\\})]\;\;(i=1,..,N)$,
$f_{i}(\\{\chi\\})$ is the activity coefficient of component $i$ in the
droplet, and
$a(\\{\chi\\})=4\pi(3v_{l}/4\pi)^{2/3}(\sigma(\\{\chi\\})/k_{B}T)$, with
$v_{l}(\\{\chi\\}$ being the composition-dependent average volume per molecule
in the liquid phase and $\sigma(\\{\chi\\})$ the droplet surface tension
(assumed equal to the surface tension of a bulk liquid solution of droplet
composition $\\{\chi\\})$.

The function $F=F(\nu_{1},..,\nu_{N})$ determines a free-energy surface in an
(N+1)-dimensional space. Under conditions when condensation occurs via
nucleation, it has the form of a hyperbolic paraboloid (“saddle-like” shape in
three dimensions). A droplet, whereof the variables $(\nu_{1},...,\nu_{N})$
coincide with the coordinates of the saddle point, is called “nucleus”; these
coordinates are determined as the solution of $N$ simultaneous equations

$F^{\prime}_{i}(\\{\nu\\}|_{c}\equiv\left.\frac{\partial
F}{\partial\nu_{i}}\right|_{c}=0\;\;(i=1,..,N).$ (15)

where $F^{\prime}_{i}=\partial F(\\{\nu\\})/\partial\nu_{i}\;\;(i=1,..,N)$.
Quantities for the nucleus will be again marked with the subscript “c”.

Let us define the quadratic approximation (QA) region $\Omega_{2\nu}$ in the
space of variables $\\{\nu\\}$ as the vicinity of the saddle point within
which $F-F_{c}$ can be accurately approximated as a quadratic form

$F-F_{c}=\frac{1}{2}\sum_{i,j=1}^{N}F^{\prime\prime}_{ijc}\Delta\nu_{i}\Delta\nu_{j}\;\;\;(\\{\nu\\}\in\Omega_{2\nu}),$
(16)

where
$F^{\prime\prime}_{ij}=\partial^{2}F/\partial\nu_{i}\partial\nu_{j}\;\;(i,j=1,..,N)$
and $\Delta\nu_{i}\equiv\nu_{i}-\nu_{ic}\;\;(i=1,..,N)$. In this
approximation, the equilibrium distribution can be represented as

$g_{e}(\\{\nu\\})\simeq
g_{e}(\\{\nu_{c}\\})\exp[-\frac{1}{2}\sum_{i,j=1}^{N}F^{\prime\prime}_{ijc}\Delta\nu_{i}\Delta\nu_{j}]\;\;\;\;\;(\\{\nu\\}\in\Omega_{2\nu}).$
(17)

Approximation (16) is equivalent to neglecting the cubic and higher order
terms in the Taylor series expansion of $F(\\{\nu\\})$ with respect to
deviations $\Delta\nu_{i}$ in the vicinity of the saddle point. Therefore,
considering that $1/3$ is already much smaller than $1$, the QA region
$\Omega_{2\nu}$, wherein approximation (16) is acceptable, can be determined
by the condition

$\epsilon_{\mbox{\tiny 32}}(\\{\nu\\})\lesssim\frac{1}{3},$ (18)

where

$\epsilon_{\mbox{\tiny
32}}(\\{\nu\\})=\frac{|\sum_{i,j,k=1}^{N}b_{ijk}(\Delta\nu_{i})(\Delta\nu_{j})(\Delta\nu_{k})|}{|\sum_{i,j=1}^{N}a_{ij}(\Delta\nu_{i})(\Delta\nu_{j})|}$
(19)

with

$a_{ij}\equiv\left.\frac{1}{2!}\frac{\partial^{2}F(\\{\nu\\})}{\partial\nu_{i}\partial\nu_{j}}\right|_{c}\;,\;\;\;\;b_{ijk}=\left.\frac{1}{3!}\frac{\partial^{3}F(\\{\nu\\})}{\partial\nu_{i}\partial\nu_{j}\partial\nu_{k}}\right|_{c}\;\;\;(i,j,k=1,..,N)$
(20)

Let us define the saddle-point (SP) region $\Omega_{\nu}$ in the space of
variables $\\{\nu\\}$ as the minimal vicinity of the saddle point within which
the equilibrium distribution $g_{e}(\\{\nu\\}$ changes substantially.
According to eq.(17), its boundary should thus satisfy the constraint
(analogous to constraint (4) of the unary nucleation theory4)

${\bf|\Delta\nu}^{\mbox{\tiny T}}{\bf
A}{\bf\Delta\nu}|\equiv|\frac{1}{2}\sum_{i,j=1}^{N}F^{\prime\prime}_{ijc}(\nu_{i}-\nu_{ic})(\nu_{j}-\nu_{jc})|=1,$
(21)

where the matrix notation was introduced with a real symmetric $N\times
N$-matrix ${\bf A}=[a_{ij}]\;\;(i,j=1,..,N)$ and a real column-vector
${\bf\Delta\nu}=[\Delta\nu_{i}]\;\;(i=1,..,N)$ of length N, where ${\bf
A}=[a_{ij}]\;\;(i,j=1,..,N)$ is a real symmetric $N\times N$-matrix and
${\bf\Delta\nu}=[\Delta\nu_{i}]\;\;(i=1,..,N)$ is a real column-vector of
length N, the superscript “T” marking the transpose of a matrix or vector.

Because the matrix ${\bf A}$ is real and symmetric, it is orthogonally
diagonalizable, according to the spectral theorem.10 Therefore, there exists a
real orthogonal $N\times N$-matrix ${\bf
P}\equiv[p_{\alpha\gamma}]\;\;(\alpha,\gamma=1,..,N)$ (such that ${\bf
P}^{-1}={\bf P}^{T}$) diagonalizing the matrix ${\bf A}$, so the matrix ${\bf
D}={\bf P}^{\mbox{\tiny T}}{\bf A}{\bf P}$ is a real diagonal $N\times N$
matrix (hereafter the Greek subscripts $\alpha,\gamma=1,..,N$ do not indicate
the relation to the chemical components $1,..,N$ in the system). By virtue of
the spectral theorem,10 the columns of the matrix ${\bf P}$ are linearly
independent orthonormal eigenvectors of ${\bf A}$. The corresponding
eigenvalues $\lambda_{1},..,\lambda_{N}$ are the diagonal elements of ${\bf
D}$. When the free energy surface has the shape of a hyperbolic paraboloid,
one of these eigenvalues is negative (say, $\lambda_{1}<0$), while all others
are positive, hence $\det({\bf A})<0$.

Let us introduce the new variables $\\{x\\}\equiv(x_{1},..,x_{N})$ as

$x_{\alpha}=\sum_{i=1}^{N}p_{i\alpha}\Delta\nu_{i}\;\;\;(\alpha=1,..,N),$ (22)

comprising a column-vector ${\bf x}\equiv[x_{\alpha}]\;\;(\alpha=1,..,N)$ of
length $N$. Because the difference $F-F_{c}$ does not depend on the choice of
independent variables of state of a droplet, and ${\bf\Delta\nu}^{\mbox{\tiny
T}}{\bf A}{\bf\Delta\nu}={\bf x}^{\mbox{\tiny T}}{\bf D}{\bf x}$,
approximation (16) for $F$ in variables $\\{x\\}$ becomes

$F-F_{c}=\sum_{\alpha}\lambda_{\alpha}x_{\alpha}^{2}\;\;\;(\\{x\\}\in\Omega_{2x}),$
(23)

and approximation (17) for the equilibrium distribution transforms into an
approximation for the equilibrium distribution $q_{\mbox{\tiny e}}(\\{x\\})$
in variables $\\{x\\}$ (with a new normalization factor $n_{x}$):

$q_{\mbox{\tiny e}}(\\{{\bf x}\\})\simeq q_{\mbox{\tiny e}}(\\{{\bf
x_{c}}\\})\exp[-\sum_{\alpha}\lambda_{\alpha}x_{\alpha}^{2}]\;\;\;(\\{x\\}\in\Omega_{2x}).$
(24)

Thus, the quadratic form in eq.(21), determining the boundary of the SP region
$\Omega_{\nu}$ in variables $\\{\nu\\}$, becomes a diagonal quadratic form in
variables $\\{x\\}$. That only one eigenvalue of ${\bf A}$ is negative
($\lambda_{1}<0$), whereas all others are positive
($\lambda_{\alpha}>0\;\;(\alpha\neq 1)$), allows one to identify $x_{1}$ as
the single thermodynamically unstable variable and $x_{2},..,x_{N}$ as
thermodynamically stable ones. This allows one to impose standard boundary
condition on the multidimensional kinetic equation,11,12

$\displaystyle\frac{q(\\{{\bf x}\\},t)}{q_{e}(\\{{\bf x}\\})}$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1&\;\;(x_{1}\rightarrow-\infty\;\;\mbox{and}\;\;\sum_{\alpha}\lambda_{\alpha}x_{\alpha}^{2}<0),\\\
0&\;\;(x_{1}\rightarrow\infty\;\;\mbox{and}\;\;\sum_{\alpha}\lambda_{\alpha}x_{\alpha}^{2}<0),\end{array}\right.$
(27) $\displaystyle q(\\{{\bf x}\\},t)$ $\displaystyle\rightarrow$
$\displaystyle
0\;\;\;\;\mbox{as}\;\;\;\;\sum_{\alpha}\lambda_{\alpha}x_{\alpha}^{2}\rightarrow\infty.$

requiring that in variables $\\{x\\}$ the distribution function $q(\\{x\\},t)$
of small sub-critical droplets smoothly transition into the equilibrium
distribution, whereas for large super-critical droplets $q(\\{x\\},t)$
smoothly transition into stationary distribution (by convention, sub-critical
and super-critical droplets are assigned negative and positive values,
respectively, of the unstable variable $x_{1}$; in specific applications,
these signs depend on the coefficients of transformation (22), i.e., on the
orthogonal matrix ${\bf P}$).

These boundary conditions to eq.(12) are imposed on the boundary of the SP
region $\Omega_{\nu}$ of substantial change of $g(\\{\nu\\},t)$. In variables
$\\{\bf x\\}$, the constraint

$|\sum_{\alpha=1}^{N}\lambda_{\alpha}x_{\alpha}^{2}|=1\;\;\;(\lambda_{1}<0,\;\lambda_{\alpha}>0\;\;(\alpha\neq
1))$ (28)

will determine the boundary of SP region $\Omega_{x}$ in variables $\\{x\\}$;
this equation is simpler than eq.(21). Once the boundary of the SP region is
determined in variables $\\{x\\}$, it can be also found in variables
$\\{\nu\\}$ via transformation (22).

One can then evaluate the accuracy of approximation (16) within the SP region
$\Omega_{\nu}$ by calculating the ratio $\varepsilon_{\mbox{\tiny
32}}(\\{\nu\\})$ for $\\{\nu\\}\in\Omega_{\nu}$. According to eq.(18), the
boundaries of the QA region $\Omega_{2\nu}$, where this approximation is
acceptably accurate, are determined by the equality
$\varepsilon_{32}(\\{\nu\\})=1/3$.

Strictly speaking, approximation (16) is needed in the entire SP region
$\Omega_{\nu}$ (at boundaries of which conditions (25) are imposed) in order
for the kinetic equation of CNT therein to have the Fokker-Planck form with
its force coefficients being linear functions of $\\{\nu\\}$; in other words,
it is necessary that $\Omega_{\nu}\subseteq\Omega_{2\nu}$. However, even if
approximation (16) is not fulfilled in some (relatively small) part(s) of
$\Omega_{\nu}$, one can expect the Fokker-Planck equation with linear force
coefficients to be an acceptable approximation for the kinetic equation in the
entire $\Omega_{\nu}$ if the parameter

$\omega=\frac{\mu[\Omega_{\nu}\setminus(\Omega_{\nu}\cap\Omega_{2\nu})]}{\mu[\Omega_{\nu}]}$
(29)

(with $\mu[\Omega]$ denoting the measure of a set (region) $\Omega$) is
negligibly small. Clearly, the smaller $\omega$, the more accurate the kinetic
equation of CNT. One can roughly assume that this accuracy is sufficient if
approximation (16) fails only in 10% of $\Omega_{\nu}$ or less, and require
that

$\omega\lesssim 0.1.$ (30)

In the case of isothermal binary nucleation ($N=2$), the space of droplet
variables is two-dimensional, and the measure $\mu[\Omega]$ of any region
$\Omega$ therein is the surface area of $\Omega$; for isothermal ternary
nucleation ($N=3$), the space of droplet variables is three-dimensional, and
the measure $\mu[\Omega]$ of a region $\Omega$ therein is the volume of
$\Omega$.

In contrast to the unary nucleation theory, one cannot obtain the operator
estimates for the derivatives $\partial g(\\{\nu\\},t)/\partial\nu_{i}$ in the
Taylor series expansions of $g(\nu_{i}\pm 1,\widetilde{\nu}_{i},t)$ on the RHS
of the balance eq.(12) in a straightforward manner because of the presence of
mixed terms $a_{ij}\Delta\nu_{i}\Delta\nu_{j}\;\;\;(i,j=1,..,N)$ in the
exponential of eq.(17) for $g_{e}(\\{\nu\\})$. However, the lower limits of
the half-widths of the SP region $\Omega_{x}$ in variables $\\{{\bf x}\\}$ can
be estimated to be $\Delta^{x}_{1}\equiv
1/\sqrt{|\lambda_{1}|},\Delta^{x}_{2}\equiv
1/\sqrt{\lambda_{2}},...,\Delta^{x}_{N}\equiv 1/\sqrt{\lambda_{N}}$ along the
axes $x_{1},x_{2},...,x_{N}$, respectively, so

$\frac{1}{q(\\{x\\},t)}\frac{\partial q(\\{x\\},t)}{\partial
x_{\alpha}}\sim\frac{1}{q_{s}(\\{x\\})}\frac{\partial q_{s}(\\{x\\})}{\partial
x_{\alpha}}\sim\frac{1}{q_{e}(\\{x\\})}\frac{\partial q_{e}(\\{x\\})}{\partial
x_{\alpha}}\sim\frac{1}{\Delta^{x}_{\alpha}}\;\;(\alpha=1,..,N).$ (31)

Therefore, because

$\frac{\partial
g(\\{\nu\\},t)}{\partial\nu_{i}}=\sum_{\alpha=1}^{N}\frac{\partial
J\,g(\\{{\bf x}\\},t)}{\partial x_{\alpha}}\frac{\partial
x_{\alpha}}{\partial\nu_{i}},$

(where $J$ is the Jacobian of transformation ${\bf\Delta\nu}={\bf P}{\bf x}$)
and $\partial x_{\alpha}/\partial\nu_{i}=p_{i\alpha}$, one can obtain
estimates

$\frac{1}{g(\\{\nu\\},t)}\frac{\partial
g(\\{\nu\\},t)}{\partial\nu_{i}}\sim\frac{1}{g_{s}(\\{\nu\\})}\frac{\partial
g_{s}(\\{\nu\\})}{\partial\nu_{i}}\sim\frac{1}{g_{e}(\\{\nu\\})}\frac{\partial
g_{e}(\\{\nu\\})}{\partial\nu_{i}}\lesssim\sum_{\alpha=1}^{N}p_{i\alpha}\sqrt{|\lambda_{\alpha}|}.\;\;\;(i=1,..,N).$
(32)

Expanding the procedure of Kuni and Grinin4 to multicomponent nucleation,
performing the Taylor series expansions of $W_{i}^{-}(\nu_{i}\pm
1,\widetilde{\nu}_{i}),W_{i}^{+}(\nu_{i}\pm 1,\widetilde{\nu}_{i})$, and
$g(\nu_{i}\pm 1,\widetilde{\nu}_{i},t)$ on the RHS of eq.(12), and taking into
account eq.(30), one can show that for the resulting differential equation to
be accurately approximated by the conventional Fokker-Planck equation of
multicomponent CNT

$\frac{\partial g(\\{\nu\\},t)}{\partial
t}=-\sum_{i=1}^{N}W^{+}_{ic}\frac{\partial}{\partial\nu_{i}}\left(-F^{\prime}_{i}(\\{\nu\\})-\frac{\partial}{\partial\nu_{i}}\right)g(\\{\nu\\},t)$
(33)

with $F^{\prime}_{i}(\\{\nu\\})\;\;(i=1,..,N)$ being linear superpositions of
$\Delta\nu_{i}\;\;(i=1,..,N)$ in the SP region $\Omega_{\nu}$, the parameters

$\frac{1}{\Delta^{\nu}_{i}}\equiv\left|\sum_{\alpha=1}^{N}p_{i\alpha}\sqrt{|\lambda_{\alpha}|}\right|\;\;\;(i=1,..,N),$
(34)

must fulfill the strong inequalities

$\frac{1}{\Delta^{\nu}_{i}}\ll 1\;\;\;\;\;(i=1,..,N),$ (35)

in addition to the parameter $\omega$ satisfying constraint (28).

Thus, the parameters $\omega$ and $1/\Delta^{\nu}_{i}\;\;\;(i=1,..,N)$
represent the small parameters of the macroscopic theory of multicomponent
nucleation. The violation of any one of constraints (28) or (33) will
necessitate going beyond the framework of the conventional Fokker-Planck
equation with linear force coefficients usually adopted for the kinetic
equation in the multicomponent CNT.

If constraint (33) on the parameters $1/\Delta^{\nu}_{i}\;\;(i=1,..,N)$ is not
satisfied for some $i$, then the kinetic equation will include contributions
of the third and higher order partial derivatives of the distribution function
$g(\\{\nu\\},t)$ with respect to $\nu_{i}$. This constraint can be referred to
as the SP region constraint, because it 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 of
variables) for solving such a non-Fokker-Planck kinetic equation was developed
by Kuni and Grinin13 (see also references 14,15 for its applications).

On the other hand, if the parameter $\omega$ does not satisfy constraint (28),
then the QA region $\Omega_{2\nu}$ of quadratic approximation (16) for
$F(\\{\nu\\})$ does not cover a sufficiently large part of the SP region
$\Omega_{\nu}$ and it is necessary to retain the cubic and perhaps even higher
order (in $\Delta\nu_{i}\;\;(i=1,..,N)$) terms in the Taylor series expansion
for $F(\\{\nu\\})$. This constraint can be referred to as the QA region
constraint, because it characterizes the extent of the QA region. As a result,
the first derivatives $F^{\prime}_{i}$ in the kinetic equation (31) will not
be linear superpositions of deviations $\Delta\nu_{i}\;\;(i=1,..,N)$ (they
will be quadratic at least, or of even higher orders). Hence the force
coefficients of equation (31) will no longer be linear functions of
$\\{\nu\\}$, i.e., the kinetic equation will differ from the conventional
Fokker-Planck equation of multicomponent CNT. We are not aware of any work
addressing the solution of such a kinetic equation in the theory of
multicomponent nucleation.

##  3 Numerical evaluations

As a numerical illustration of the foregoing, we have carried out calculations
for isothermal condensation in five binary systems:
(a) 1-butanol (component 1) – 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 representative of the nucleation of droplets of
ideal (a) and increasingly nonideal (b)-(e) binary solutions whose physical
and chemical properties, necessary for the evaluation of parameters $\omega$
and $\Delta^{\nu}_{i}\;\;\;(i=1,..,N)$ in eqs.(27) and (32), are relatively
well known. For each system, the molecular volumes $v_{1}$ and $v_{2}$ of pure
liquids were obtained from the density data of Lide,16 and the mean molecular
volume of solution in the droplet was approximated as $v=\chi
v_{1}+(1-\chi)v_{2}$, with $\chi=\chi_{1}$. All calculations were carried out
for the same system temperature $T=293.15$ K. Although the saturation ratios
$\zeta_{1}$ and $\zeta_{2}$ were different in different systems, they were
always chosen so that the height of the nucleation barrier at the saddle point
was in the range from $30$ to $50$, which would ensure a greater than $1$
cm-3s-1 nucleation rate (according to binary CNT1,9,17,18).

The surface tension of 1-butanol(1)–1-hexanol(2) solution (which can be
considered as nearly ideal) was assumed to depend on $\chi(=\chi_{1})$ as
$\sigma(\chi)=\chi\sigma_{1}+(1-\chi)\sigma_{2},$ where $\sigma_{1}$ and
$\sigma_{2}$ are the surface tensions of pure liquid butanol and pure liquid
hexanol, respectively; $\sigma_{1}=25.39$ dyn/cm was obtained by linear
interpolation of data from Lide16 and $\sigma_{2}=26.20$ dyn/cm was taken from
Gallant.19 The activity coefficients of both butanol and hexanol in this
solution were set equal to unity (ideal solution approximation).

For the composition dependence of the surface tension in systems (b)-(e) we
used the expression

$\sigma(\chi)=a+b/(d-\chi)+c/(d-\chi)^{2}$ (36)

(with $\chi=\chi_{1}$ and the dimension of $\sigma$ dyn/cm), where a set of
parameters $a,b,c,d$ for each system. These parameters were determined with
the help of Mathematica 12.1 by fitting expression (34) to appropriate
experimental data (of Vazquez et al.20 for the systems (b)-(d)) and of
Teitelbaum et al.21 for the system (e)):
(b) $a=16.3343,b=8.85203,c=-1.85715\times 10^{-7},d=1.160977$
(water(1)–methanol(2));
(c) $a=19.6512,b=3.25232,c=-2.24934\times 10^{-8},d=1.0880297$
(water(1)–ethanol(2));
(d) $a=23.4678,b=0.43188,c=-2.53012\times 10^{-10},d=1.02205$
(water(1)–1-propanol(2));
(e) $a=24.5474,b=-0.0309657,c=0.00338759,d=1.00811323$
(water(1)–1-butanol(2)).

The composition dependence of the activity coefficients in systems (b)-(e) was
described by the van Laar equations

$\ln
f_{1}(\chi)=\frac{A_{12}}{(1+\frac{A_{12}\chi}{A_{21}(1-\chi)})^{2}},\,\,\ln
f_{2}(\chi)=\frac{A_{21}}{(1+\frac{A_{21}(1-\chi)}{A_{12}\chi})^{2}}.$ (37)

with pairs $A_{12}$ and $A_{21}$ from refs.22, 23:
(b) $A_{12}=0.5619$ and $A_{21}=0.8041$ (water(1)–methanol(2) solution);
(c) $A_{12}=0.9227$ and $A_{21}=1.6798$ (water(1)–ethanol(2) solution);
(d) $A_{12}=1.1572$ and $A_{21}=2.9095$ (water(1)–1-propanol(2) solution);
(e) $A_{12}=1.0996$ and $A_{21}=4.1760$ (water(1)–1-butanol(2) solution).

Some results of numerical calculations are presented in Figures 1-5.
Saturation ratios $\zeta_{1}$ and $\zeta_{2}$ of vapor mixture components are
indicated in the figure captions.

In each Figure, panel a) shows the SP region $\Omega_{x}$ and the QA region
$\Omega_{2x}$ in variables $\\{x\\}$, whereas panel b) shows the SP region
$\Omega_{\nu}$ and the QA region $\Omega_{2\nu}$ in variables $\\{\nu\\}$;
both $\Omega_{2x}$ and $\Omega_{2\nu}$ are shown as grayish areas in these
Figures. The solid curves indicate the borders of SP regions, whereas the
dashed ones indicate the boundaries of QA regions. In panel a) of each Figure,
the thin dashed lines delineate the rectangular central part $\Omega^{c}_{x}$
of the SP region $\Omega_{x}$ of half-widths $\Delta^{x}_{1}$ and
$\Delta^{x}_{2}$ which were used in calculating the parameters
$1/\Delta^{\nu}_{1}$ and $\Delta^{\nu}_{2}$ according to eq.(32). In panel b)
of each Figure, the corresponding central part $\Omega^{c}_{\nu}$ of the SP
region $\Omega_{\nu}$ is also shown, delineated by thin dashed lines forming a
parallelogram. The arrows show the direction of the growth of droplets at the
saddle point.

As evident from these Figures, in each system the QA region only partially
covers the SP region. Moreover, the QA region does not even cover the central
parts of the SP region; approximation (16) fails to hold even on some segments
of its sub-critical and super-critical borders, at which the stricter boundary
conditions to the kinetic equation (31) are imposed.

Thus, for all the systems studied, the quadratic approximation (16) for
$F(\\{\nu\\})$ is accurate not in the entire SP region, and it is necessary to
retain the cubic and perhaps even higher order (in
$\Delta\nu_{i}\;\;(i=1,..,N)$) terms in the Taylor series expansion for
$F(\\{\nu\\})$. As a result, the first derivatives $F^{\prime}_{i}$ in the
kinetic equation (31) will not be linear superpositions of deviations
$\Delta\nu_{i}\;\;(i=1,..,N)$ (they will be quadratic or even of higher
orders). Hence, the force coefficients of equation (31) will no longer be
linear functions of $\\{\nu\\}$, i.e., the kinetic equation will differ from
the conventional Fokker-Planck equation of multicomponent CNT. Therefore, the
conventional expression for the steady-state binary nucleation rate, obtained
on the basis of approximation (16), is not adequate for comparing theoretical
predictions with experimental data in these systems.

We have also evaluated the parameters $1/\Delta^{\nu}_{1}$,
$1/\Delta^{\nu}_{2}$, and $\omega$ in all systems (a)-(e). For simplicity,
$\omega$ was estimated from below by calculating the ratio

$\widetilde{\omega}=\left.\frac{\mu[\Omega_{\nu}\setminus(\Omega_{\nu}\cap\Omega_{2\nu})]}{\mu[\Omega_{\nu}]}\right|_{\nu\in\Omega^{c}_{\nu}}$
(38)

only within the central part $\Omega^{c}_{\nu}$ of the SP region
$\Omega_{\nu}$. As clear from panels a) of the Figures, calculating the RHS of
eq.(27) in increasingly larger enclosures (rectangles) will result in
increasingly larger results, because beginning from some large enough
enclosing enclosure the surface area of the region
$(\Omega_{\nu}\cap\Omega_{2\nu})$ will remain constant while the surface area
$\Omega_{\nu}$ will continue to increase asymptotically approaching its
limiting value, while the RHS of eq.(27) asymptotically approaches $\omega$
from below. Therefore, one can guarantee that $\omega>\widetilde{\omega}$, and
if $\widetilde{\omega}\gtrsim 0.1$, then constraint (28) will certainly not
hold. (Recall that the measures of any region $\Omega$ in variables
$\\{\nu\\}$ and $\\{x\\}$ are related as
$\mu[\Omega(\\{\nu\\})]=J\mu[\Omega(\\{x\\})]$).

Constraint (33) on the parameters $1/\Delta^{\nu}_{1}$ and
$1/\Delta^{\nu}_{2}$ is necessary for neglecting the terms with the third and
higher order derivatives in the Taylor series expansions on the RHS of the
balance equation (12) and thus ensuring the Fokker-Planck form of the kinetic
equation. As clear from the Table, the smallness of these parameters under
metastability conditions that we considered is fulfilled well. However, they
are sensitive to the saturation ratios $\zeta_{1}$ and $\zeta_{2}$, so their
smallness at given $\zeta_{1},\zeta_{2}$ does not guarantee their smallness at
different metastability of the vapor mixture.

On the other hand, constraint (28) on the parameter $\omega$ is necessary in
order to ensure that the force coefficients of the Fokker-Planck equation are
linear functions of droplet variables in the predominant part of
$\Omega_{\nu}$. As evident from the Table, under considered metastability
conditions this constraint is not satisfied in any of the systems studied. We
note again, however, that $\omega$ is quite sensitive to the saturation ratios
$\zeta_{1}$ and $\zeta_{2}$, so its smallness at a given pair of
$\zeta_{1},\zeta_{2}$ does not guarantee its smallness at different
metastability of the vapor mixture.

Thus, both constraints (28) and (33) must be verified at given
$T,\zeta_{1},\zeta_{2}$, and only if they hold, can one confidently use the
conventional CNT expression for the binary nucleation rate $J_{s}$ for
purposes of comparing theoretical predictions with experimental data.
Otherwise, another, more adequate theoretical expression for $J_{s}$ must be
obtained by solving a properly modified kinetic equation (which may be of non-
Fokker-Planck form).

Table: Small parameters $1/\Delta^{\nu}_{1}$, $1/\Delta^{\nu}_{2}$, and
$\widetilde{\omega}\;(<\omega)$ of the Fokker-Planck approximation with linear
force coefficients in the kinetic equation of binary nucleation at $T=293.15$
K.  Binary system $\zeta_{1}$ $\zeta_{2}$ $F_{c}$ $1/\Delta^{\nu}_{1}$
$1/\Delta^{\nu}_{2}$ $\widetilde{\omega}\;\;(\omega>\widetilde{\omega})$
1-butanol(1)–1-hexanol(2) $2.88$ $3.41$ $40.11$ $0.19$ $0.04$ $0.15$ $2.07$
$3.41$ $49.96$ $0.16$ $0.02$ $0.12$ $2.88$ $1.95$ $49.95$ $0.15$ $0.08$ $0.19$
$4.30$ $3.41$ $30.07$ $0.23$ $0.11$ $0.21$ $2.88$ $6.39$ $30.21$ $0.25$ $0.01$
$0.15$ water(1)–methanol(2) $1.45$ $0.81$ $40.14$ $0.01$ $0.09$ $0.51$ $1.29$
$0.81$ $50.14$ $0.07$ $0.04$ $0.14$ $1.45$ $0.68$ $50.11$ $0.07$ $0.06$ $0.15$
$1.71$ $0.81$ $30.21$ $0.12$ $0.08$ $0.19$ $1.45$ $1.01$ $30.17$ $0.11$ $0.05$
$0.18$ water(1)–ethanol(2) $1.52$ $0.9$ $39.81$ $0.07$ $0.08$ $0.21$ $1.39$
$0.90$ $50.22$ $0.06$ $0.05$ $0.19$ $1.52$ $0.69$ $49.93$ $0.05$ $0.09$ $0.18$
$1.71$ $0.90$ $30.14$ $0.09$ $0.12$ $0.24$ $1.52$ $1.21$ $30.07$ $0.09$ $0.06$
$0.23$ water(1)–1-propanol(2) $1.3$ $1.9$ $39.86$ $0.01$ $0.06$ $0.42$ $1.1$
$1.9$ $49.93$ $0.07$ $0.01$ $0.14$ $1.3$ $1.55$ $50.82$ $0.01$ $0.04$ $0.54$
$1.45$ $1.9$ $30.35$ $0.07$ $0.06$ $0.53$ $1.3$ $2.4$ $29.89$ $0.09$ $0.03$
$0.30$ water(1)–1-butanol(2) $1.32$ $2.6$ $40.07$ $0.09$ $0.02$ $0.28$ $1.02$
$2.6$ $50.21$ $0.10$ $0.01$ $0.13$ $1.32$ $2.16$ $50.03$ $0.02$ $0.06$ $0.32$
$1.48$ $2.95$ $30.45$ $0.01$ $0.09$ $0.74$ $1.32$ $3.37$ $30.25$ $0.12$ $0.03$
$0.27$

##  4 Concluding remarks

We have attempted to shed some light on the conditions necessary for the
Fokker-Planck equation with linear force coefficients to be an adequate
approximation to the kinetic equation of nucleation in a macroscopic theory of
isothermal homogeneous multicomponent condensation. Starting with a discrete
equation of balance, governing the temporal evolution of the distribution
function of an ensemble of multicomponent droplets, and reducing it (by means
of Taylor series expansions) to the differential form in the vicinity of the
saddle point of the free energy surface, we have obtained the constraints
necessary for the resulting kinetic equation to have the form of the Fokker-
Planck equation and for its force coefficients to be linear functions of
droplet variables; we have also identified the corresponding “small”
parameters.

If those (saddle point (SP) region and quadratic approximation (QA) region)
constraints are not satisfies, then either contributions from the third or
higher order partial derivatives of the distribution function in the kinetic
equation will be signinficant (when the SP constraint does not hold) or the
force coefficients of the Fokker-Planck equation will become non-linear
functions of droplet variables (when the QA constraint does not hold), or
both. In any of these cases, the conventional kinetic equation of
multicomponent nucleation and its predictions would become inaccurate.

As a numerical illustration, we have carried out calculations for isothermal
condensation in five binary systems (at $T=293.15$ K and vapor mixture
metastabilities typical of 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 of ideal
(a) and increasingly nonideal (b-e) binary solutions. Our results suggest that
the SP constraint on the smoothness of the droplet distribution in the SP
region is well fulfilled, which substantiates neglecting the third and 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 in the Taylor series expansion of the free energy
of droplet formation in the saddle point region is not satisfied; therefore,
the force coefficients in that generic Fokker-Planck equation are not linear
functions of droplet variables. Hence, the kinetic equation of binary
nucleation does not have the form adopted in the binary CNT, so the
conventional expressions1-3,8,9,17,18 for the steady-state distribution of
binary droplets and steady-state rate of binary nucleation can be of
insufficient accuracy for making comparisons with experimental data.

Moreover, numerical calculations show that whether the constraints on the
small parameters are satisfied or not is quite sensitive to the saturation
ratios $\zeta_{1},\zeta_{2}$ and this sensitivity increases with increasing
non-ideality of the liquid solution in droplets. Therefore, for many
conditions of likely interest, it is necessary to obtain the steady-state
solutions of the modified kinetic equation, going beyond the framework of the
Fokker-Planck equation of CNT due to the non-fulfillment of either the SP
region constraint (when the third or even higher order derivatives of the
distribution function are present in the kinetic equation) or the QA region
constraint (when the force coefficients in the generic Fokker-Planck equation
are not linear functions of droplet variables) or both. Clearly, such
solutions are needed for consistently comparing theoretical predictions and
experimental data obtained under conditions when the corresponding constraints
are not fulfilled. This will be an object of our further research.

#### Acknowledgements

This research project was initiated and partially completed when Eli
Ruckenstein, one of the authors, was still alive but was completed only after
he passed away. We thank Buzz I. Dzhikkaity for the numerical calculations of
the parameter $\widetilde{\omega}$ for all the systems in the Table.

#### Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

### References

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### Captions

to Figures 1 to 5 of the manuscript “On the Fokker-Planck approximation in the
kinetic equation of multicomponent classical nucleation theory”  by Y. S.
Djikaev, E. Ruckenstein, and Mark Swihart

####

Figure 1. The saddle point (SP) and quadratic approximation (QA) regions of
the space of droplet variables for binary nucleation in
1-butanol(1)–1-hexanol(2) vapor mixture at $T=293.15$ K, $\zeta_{1}=2.88$, and
$\zeta_{2}=3.49$. a) The SP region $\Omega_{x}$ and the QA region
$\Omega_{2x}$ in variables $\\{x\\}$. b) The SP region $\Omega_{\nu}$ and the
QA region $\Omega_{2\nu}$ in variables $\\{\nu\\}$. The dashed and solid
curves indicate the boundaries of the SP and QA regions, respectively. Both
$\Omega_{2x}$ and $\Omega_{2\nu}$ are shown as grayish areas. The thin-dashed
quadrilateral delineates the central part of the SP region (see the text);
vertices of the same color correspond to one same droplet in variables
$\\{x\\}$ (a) and $\\{\nu\\}$ (b). The arrow points from the saddle point (red
dot) toward super-critical droplets.
Figure 2. The saddle point (SP) and quadratic approximation (QA) regions of
the space of droplet variables for binary nucleation in water(1)–methanol(2)
vapor mixture at $T=293.15$ K, $\zeta_{1}=1.45$, and $\zeta_{2}=0.81$. a) The
SP region $\Omega_{x}$ and the QA region $\Omega_{2x}$ in variables $\\{x\\}$.
b) The SP region $\Omega_{\nu}$ and the QA region $\Omega_{2\nu}$ in variables
$\\{\nu\\}$. The dashed and solid curves indicate the boundaries of the SP and
QA regions, respectively. Both $\Omega_{2x}$ and $\Omega_{2\nu}$ are shown as
grayish areas. The thin-dashed quadrilateral delineates the central part of
the SP region (see the text); vertices of the same color correspond to one
same droplet in variables $\\{x\\}$ (a) and $\\{\nu\\}$ (b). The arrow points
from the saddle point (red dot) toward super-critical droplets.
Figure 3. The saddle point (SP) and quadratic approximation (QA) regions of
the space of droplet variables for binary nucleation in water(1)–ethanol(2)
vapor mixture at $T=293.15$ K, $\zeta_{1}=1.52$, and $\zeta_{2}=0.90$. a) The
SP region $\Omega_{x}$ and the QA region $\Omega_{2x}$ in variables $\\{x\\}$.
b) The SP region $\Omega_{\nu}$ and the QA region $\Omega_{2\nu}$ in variables
$\\{\nu\\}$. The dashed and solid curves indicate the boundaries of the SP and
QA regions, respectively. Both $\Omega_{2x}$ and $\Omega_{2\nu}$ are shown as
grayish areas. The thin-dashed quadrilateral delineates the central part of
the SP region (see the text); vertices of the same color correspond to one
same droplet in variables $\\{x\\}$ (a) and $\\{\nu\\}$ (b). The arrow points
from the saddle point (red dot) toward super-critical droplets.
Figure 4. The saddle point (SP) and quadratic approximation (QA) regions of
the space of droplet variables for binary nucleation in water(1)–1-propanol(2)
vapor mixture at $T=293.15$ K, $\zeta_{1}=1.30$, and $\zeta_{2}=1.90$. a) The
SP region $\Omega_{x}$ and the QA region $\Omega_{2x}$ in variables $\\{x\\}$.
b) The SP region $\Omega_{\nu}$ and the QA region $\Omega_{2\nu}$ in variables
$\\{\nu\\}$. The dashed and solid curves indicate the boundaries of the SP and
QA regions, respectively. Both $\Omega_{2x}$ and $\Omega_{2\nu}$ are shown as
grayish areas. The thin-dashed quadrilateral delineates the central part of
the SP region (see the text); vertices of the same color correspond to one
same droplet in variables $\\{x\\}$ (a) and $\\{\nu\\}$ (b). The arrow points
from the saddle point (red dot) toward super-critical droplets.
Figure 5. The saddle point (SP) and quadratic approximation (QA) regions of
the space of droplet variables for binary nucleation in water(1)–1-butanol(2)
vapor mixture at $T=293.15$ K, $\zeta_{1}=1.32$, and $\zeta_{2}=2.6$. a) The
SP region $\Omega_{x}$ and the QA region $\Omega_{2x}$ in variables $\\{x\\}$.
b) The SP region $\Omega_{\nu}$ and the QA region $\Omega_{2\nu}$ in variables
$\\{\nu\\}$. The dashed and solid curves indicate the boundaries of the SP and
QA regions, respectively. Both $\Omega_{2x}$ and $\Omega_{2\nu}$ are shown as
grayish areas. The thin-dashed quadrilateral delineates the central part of
the SP region (see the text); vertices of the same color correspond to one
same droplet in variables $\\{x\\}$ (a) and $\\{\nu\\}$ (b). The arrow points
from the saddle point (red dot) toward super-critical droplets.

<EMAIL_ADDRESS>\vspace{1cm}}\hfil\hskip
8.5359pt&\includegraphics[width=236.15796pt]{drPhysA_smallParam_f2Aa.eps}\\\\[28.45274pt]
\vspace{0.cm}\leavevmode\hbox{b) \vspace{1cm}}\hfil\hskip
8.5359pt&\vspace{0.0cm}\includegraphics[width=236.15796pt]{drPhysA_smallParam_f2Bb.eps}\\\\[8.5359pt]
\end{array}$ Figure 1:

<EMAIL_ADDRESS>\vspace{3cm}}\hfil\hskip
8.5359pt&\includegraphics[width=247.53888pt]{drPhysA_smallParam_f1Aa.eps}\\\\[54.06006pt]
\vspace{0cm}\leavevmode\hbox{b) \vspace{3cm}}\hfil\hskip
8.5359pt&\vspace{0.0cm}\includegraphics[width=247.53888pt]{drPhysA_smallParam_f1Bb.eps}\\\\[19.91684pt]
\end{array}$ Figure 2:

<EMAIL_ADDRESS>\vspace{3cm}}\hfil\hskip
8.5359pt&\includegraphics[width=247.53888pt]{drPhysA_smallParam_f4Aa.eps}\\\\[54.06006pt]
\vspace{0cm}\leavevmode\hbox{b) \vspace{3cm}}\hfil\hskip
8.5359pt&\vspace{0.0cm}\includegraphics[width=247.53888pt]{drPhysA_smallParam_f4Bb.eps}\\\\[19.91684pt]
\end{array}$ Figure 3:

<EMAIL_ADDRESS>\vspace{3cm}}\hfil\hskip
8.5359pt&\includegraphics[width=247.53888pt]{drPhysA_smallParam_f5Aa.eps}\\\\[54.06006pt]
\vspace{0cm}\leavevmode\hbox{b) \vspace{3cm}}\hfil\hskip
8.5359pt&\vspace{0.0cm}\includegraphics[width=247.53888pt]{drPhysA_smallParam_f5Bb.eps}\\\\[19.91684pt]
\end{array}$ Figure 4:

<EMAIL_ADDRESS>\vspace{3cm}}\hfil\hskip
8.5359pt&\includegraphics[width=247.53888pt]{drPhysA_smallParam_f3Aa.eps}\\\\[54.06006pt]
\vspace{0cm}\leavevmode\hbox{b) \vspace{3cm}}\hfil\hskip
8.5359pt&\vspace{0.0cm}\includegraphics[width=247.53888pt]{drPhysA_smallParam_f3Bb.eps}\\\\[19.91684pt]
\end{array}$ Figure 5: