Langmuir 1991, 7, 1537-1541
1537
A Unidimensional Fokker-Planck Approximation in the Treatment of Nucleation in Gases E. Ruckenstein’ and B. Nowakowskit Department of Chemical Engineering, State University of New York at Buffalo, Amherst, New York 14260 Received October 31,1990. In Final Form: January 22, 1991 A unidimensional Fokker-Planck equation is used to calculatethe rate of evaporationof molecules from clusters. The obtained result is further employed to derive an expression for the rate of nucleation. This is compared with a previously derived expression based on a three-dimensionaltreatment as well as with the classical theory. The unidimensional and three-dimensionalapproaches lead to results that are very close to one another. For higher supersaturations, they are much greater than those provided by the classical theory.
Introduction The kinetics of evaporation of molecules from the small clusters of an emerging new phase is one of the main problems in the nucleation theory. The first attempt to address this question was in the framework of the socalled classical theory of nucleation, which was initiated by Volmer and Weber1 and Farkas2 and formulated in a more comprehensive form by Becker and D6ring.3 In order to calculate the rate of evaporation of molecules, the classical approach relies on the principle of detailed balance and on an hypothetical equilibrium distribution of clusters. The appearance of molecular clusters is explained in terms of fluctuations whose probabilities are determined by the excess free energy of the created nuclei. The surface free energy of these small clusters is calculated by using the surface tension of a bulk phase. The classical theory of nucleation has been improved by Kuhrt? Lothe and Pound! and R e h s It is commonly acknowledged that the use of macroscopic thermodynamics for small clusters constitutes the main drawback of the classical nucleation theory. Recently, a kinetic approach to the theory of nucleation was proposed,’* which avoided the use of macroscopic thermodynamics. In that approach, the rate of evaporation of molecules from a cluster was calculated on the basis of a kinetic equation governing the motion of the molecules escaping from the potential well generated around the cluster. The form of this kinetic equation depends on the strength of the interactions of a molecule with the medium and is therefore different in a liquid and a gas. The high dissipation of energy in liquids results in a fast relaxation to the equilibrium velocity distribution in each point of the system. As a result, the probability density for a molecule to have a certain position in the potential well can be calculated by using the Smoluchowski approximation to the Fokker-Planck equation.1° In gases, because
* To whom correspondence should be addressed. + On leave from Agricultural University, 02-528 Warsaw, Poland.
(1) Volmer, M.; Weber, A. 2.Phys. Chem. (Leipzig) 1926, 119, 227. (2) Farkae, L. 2.Phys. Chem. (Leipzig) 1927, 125, 236. (3) Becker, R.; Dbring, W. Ann. Phys. (Leipzig) 1935, 24, 719. (4) Kuhrt, F. 2.Phys. 1952, 131, 186, 206. (6) Lothe, J.; Pound, G.M. J . Chem. Phys. 1962,36, 2080. (6) Reise, H. Adu. Colloid Interface Sci. 1977, 7, 1. (7) Nmimhan, G.; Ruckenstein,E. J. Colloid Interface Sci. 1989,128, 549. (8) Ruckenstein, E.; Nowakoweki, B. J. Colloid Interface Sci. 1990, 137, 683. (9) Nowakowski, B.; Ruckenatein, E. J. Chem. Phys. 1991,94,1397. (10) Gardiner,C. W. Handbook ofStochasticMethods;Springer:New York, 1983.
of the relatively infrequent collisions,one can differentiate between fast changing phase variables, position and momentum, and the slowly changing energy. Consequently, by disregarding the fast relaxation processes, the evolution of the system can be described by a master equation in the energy spaceeg The most detailed description of the random motion of a molecule is provided by the Fokker-Planck equation (FPE). As already noted, the Smoluchowski equation is obtained from the FPE in the high damping limit.1° In this paper we examine the low damping limit of the FPE, which corresponds to the physical conditions encountered in gases. A unidimensional Fokker-Planck equation in position coordinate and velocity constitutes the starting point in the derivation of an expression for the rate of evaporation from clusters. The FPE for the phase space variables is transformed into a kinetic equation in the energy space that is valid in the long time scale approximation. In the subsequent sections, an expression for the nucleation rate is derived on the basis of the unidimensional treatment. Finally, the results obtained are compared with those obtained on the basis of our previous three-dimensional treatment9 and the classical theory. Kinetic Equation i n the Energy Space Let us consider a molecule in a potential well that is generated in the gaseous layer around a cluster by the attractive interactions with the molecules of the cluster. If the energy of a molecule is 0 > E > @O (where @O is the energy level at the bottom of the well), its motion is confined within the well, and the molecule can be regarded as bound to the cluster. The molecule can leave the well only if it acquires a sufficient amount of energy from the collisions with the molecules of the gaseous medium to overcome the potential energy of the well. Hence, the process of evaporation from a cluster can be related to the rate by which the molecules pass over the upper energetic boundary of the potential well, E = 0. The random motion of a molecule is governed by the Fokker-Planck equation (also often called the Kramers equation when it includes an external field) for the probability density function p(x,v,t)with respect to the position x and velocity v of a molecule, at time t , or by the corresponding equivalent dynamical Langevin equation.lOJ1 Let us consider a spherically symmetric potential @ ( r )in the well around a cluster assumed spherical, where ~~~~
(11) Riaken, H. The Fokker-Planck Equation; Springer: New York, 1989.
0743-7463/91/2407-1537$02.50/0 0 - 1991 American Chemical Societv
Ruckenstein and Nowakowski
1538 Langmuir, Vol. 7, No. 7, 1991 r is the distance to the center of the cluster. If the thickness of the well, A,, is small in comparison to the radius of the cluster, R, the molecules move effectively in the onedimensional potential, since the curvature effects, which are expected to be of the order of X,/R, are negligible. One can therefore reduce the Fokker-Planck equation to the one-dimensional form
a !& = - 2 up + -(.u at dr au
+ -m1- )d@ p + r kmT a 2 p dr av2
(1)
where p is the probability density for the molecule to be located at r with a velocity u at time t , y is the friction coefficient, m is the mass of the molecule, k is the Boltzmann constant, and T denotes the temperature in degrees Kelvin. Some qualitative considerations regarding the motion of a molecule in the potential well are useful in the development of approximate procedures for solving eq 1. These molecules perform a kind of anharmonic oscillation, with a characteristic period estimated as t , = A,(m/kT)l12. On the other hand, the molecules exchange energy with the medium and the strength of these interactions can be characterized by the correlation time t , = y-'. In gases, the interactions with the medium are rather weak, due to the relatively infrequent molecular collisions. Consequently, the correlation time for Brownian motion, t,, is long compared to the period of oscillations in the well, t,, and a molecule performs many oscillations in the well before its energy changes significantly. As a result, one can differentiate between the fast changing phase variables, position and velocity,and the slowly changing energy. This distinction becomes even more clear after the introduction in the FPE (1)of the dimensionless variables x = r/X,
(24
using eq 7, one arrives to the following Fokker-Planck equation for the probability density p(x,E,t)
Taking advantage of the very different time scales for the changes in position and energy, one can assume that the probability density for the position x relaxes rapidly in the fast time scale to its stationary solution p,(xlE), which contains as a parameter the slowly changing energy E , assumed approximately constant in the fast time scale. Therefore, the joint probability density function can be decomposed as p ( x J , t )= p,(xlE) P(E,t) (9) The conditional probability density p,(xIE) is obtained as a stationary solution of eq 8
a
-[(2(E ax -
p,(xlE)] = 0
(10)
from which one immediately obtains
where C(E)is a constant that can be determined from the normalization condition. Combining eqs 9 and 11, the joint probability density can be expressed as
where d ( E ) = e ( 2 ( E- @))'I2dr
(13)
R1 and R2 being the roots of the equation E = W R ) . Introducing eq 12 into the Fokker-Planck equation (8), one obtains
and 9 = @/kT
a 1 -P(E,t) = d'(E) (2(E- @))'I2at [(2(E- W)- 11P(E,t)
where
In terms of the new variables, eq 1 acquires the form
a
@ at = - - (ax up)
+ daud 'dx kp + A $
t ~
+$)p
(3)
where the parameter A = tw/t, (4) compares two time scales. Further, eq 3 is transformed from the variables (x,u) to ( x J ) , where E is the dimensionless energy, scaled with kT, defined by
E = ' / 2 u 2+ (5) In terms of the new variables, the probability density p ( x J , t ) is given by"
from which, using eq 5, one finally obtains (7)
Converting from the velocity u to the energy E in eq 3 and
- *))'I2
)
(2(E- V?))1'2P(E,t) (14) dE2 4'(E) A closed equation for the probability density with respect to energy P(E,t) can be obtained after integrating eq 14 with respect to the position of a molecule between R1 and Rz. This yields
'-(a2
(-
= A A( - l ) P ( E , t ) )+ -P(E,t) a at dE b'(E)
where j is the flux in the energy space. A similar equation for P(E,t) can be established from a master equation in the energy space. The latter procedure leads to the following diffusion type equation:12 (12) Keck, J.; Carrier, G. J. Chem. Phys. 1965,43, 2284.
Langmuir, Vol. 7, No. 7, 1991 1539
Nucleation in Gases at
a t’) D ( E ) P ( E )-aE P ( E )
(16)
Here Pep@) is the equilibrium distribution given by
P ( E ) = W E ) exp(-E)/Z (17) and D ( E )is the diffusion coefficient in the energy spacel3
with d the dimensionality of the space. In eqs 16 and 17, Q(E)is the number of states of energy E for a single molecule with the Hamiltonian H(x,u)
W E ) = $, d r du 6(E - H(x,u))
(19)
and 2 is the canonical partition function
Equation 19 yields the following specific result for a molecule with the one-dimensional Hamiltonian H(x,u) = ‘/2u2 * ( x )
+
Comparing eqs 13 and 21, one finds that Q(E)coincides with C#J’(E). Consequently,the kinetic equation for energy, eq 15, derived from the Fokker-Planck equation coincides (in the one-dimensional case) with the diffusion equation for energy, eq 16, with the diffusion coefficientD(E) given by eq 18.
Rate of Evaporation of Molecules From a Cluster The rate of evaporation is obtained from the stationary solution of eq 15. Assuming that each molecule reaching the upper boundary of the well (E = 0) separates from the cluster, the following absorption boundary condition can be written:
p(E)lE=O= 0 (22) The steady-state solution of eq 15 that satisfies condition 22 has the form
where j~ is the steady-state flux in the energy space. The rate of evaporation per molecule, a1= j E , is obtained from eq 23 by using the normalization condition JP(E) dE = 1. This yields
Returning to the unscaled variables, a1 is replaced by y d . The overall evaporation rate per cluster a is given by a = N,yal, where N , is the number of vapor molecules in the well of a cluster. If n, is the average number density of vapor molecules in the well, the total number of molecules in the well is N , = n&rR2A,. Consequently, the dissociation rate per cluster is given by the expression (13) Borkovec, M.; Berm, B. J. J. Chem. phy6. 1985,82,797.
Although the quantities A, and n, are not known, they do not appear in the equations used to calculate the final results.
The Rate of Growth of a Cluster The rate of change of the cluster size is the difference between the rates of evaporation CY and condensation /3 per cluster. In the gaseous systems, the rate of impingment of molecules on a spherical cluster is given by the formula of the kinetic theory of gases14 where o is the mean thermal velocity of the vapor molecules and nl is the number density of vapor molecules at a large distance from the cluster. The critical cluster size R. is defined by the condition of (unstable) equilibrium between evaporation and condensation, CY*= &, and is a function of the concentration of the condensing component in the gaseous phase. From this relation, one can calculate the concentration of vapor that coexists in unstable equilibrium with a critical cluster of radius R e . Using eqs 25 and 26, one obtains
The rate a1decreases with increasing cluster size because the potential well is deeper for large clusters. Consequently, higher concentrations are needed to ensure the unstable equilibrium with smaller critical clusters. Clusters decay if their size is smaller than the critical one and grow when they are greater. The saturation number density n, is the vapor concentration that coexists with a cluster of infinite size, hence with the bulk condensed phase. The supersaturation s is given by
s = al(R*)/al(m) (28) In the classical theory of nucleation, the supersaturation and the critical radius are related by the Kelvin equation (14) s = exp(&)
where p is the number density of the condensed phase and Q is the surface tension, which constitutes the basic thermodynamic parameter of the classical theory. Macroscopic thermodynamics is expected to be valid for large clusters. It is therefore expected that the present theory will coincide with the classical result in the limit of large clusters. Consequently, eqs 28 and 29 can be used to calculate the surface tension. One may note that its value does not depend on the quantities A, and n,.
The Rate of Nucleation The rates of evaporation a and condensation /3 are involved in the equations governing the concentrations ni(t) of clusters consisting of i molecules (14) dni/dt = /3i-lni-l- aini - Bini + ai+lni+l= Ii-l - Ii (30) where Ii is the net flux of clusters passing from i population to (i + 1) population (14)Abraham, F. F. Homogeneous Nucleation Theory; Academic Press: New York, 1974.
Ruckenstein and Nowakowski
1540 Langmuir, Vol. 7,No. 7, 1991
Zi = Bini - ai+lni+l
(31)
The rate of nucleation is given by the steady-state flux Z, given by the stationary solution of eq 30. Traditionally, the set of discrete equations (30) are transformed into a partial differentialequation for the continuous distribution function n(g,t). While various methods of such a conversion are possible,'s in this paper we employ the Zeldovich approach16because in contrast to other expansions, it satisfies automatically the detailed balance principle for the equilibrium distribution niw (that follows from the condition I p 0)
Since the present theory provides independently the values of a and 0, eq 32 is used to calculate by recurrence the equilibrium distribution njq. This yields
In the Zeldovich expansion, eq 32 is used to rewrite eq 30 in the form
Replacing the discrete variable i by the continuousvariable g, and expanding eq 34 around the midpoint g = i, one obtains, in the second-order approximation, the following equation for the distribution function n(g,t)
where the flux of clusters Z(g,t) in the continuum formulation has the form
The stationary solution of eqs 35 and 36 must satisfy the boundary conditions n(l)/nq(l) = 1 and n(g)/nw(g)
- 0 for g
(374
m
(37b)
Equation 37b is valid if the very large clusters are removed from the system. The stationary solution of eq 36, which satisfies the boundary conditions (37a,b), has the form
where the function h(g) is defined by eq 33. Equation 38 provides the steady-state rate of nucleation (the rate of dissociation is involved in eq 38 via the function h(g)). A similar result is obtained by using the modified Goodrich approach.ls The results of the present theory will be compared with the classical theory of nucleation, which yields the following equation for the nucleation rate
-50
1
-150
1
-200
' '' '' 0.0
0.5
I
1.0
/
I
1
1.5
2.0
In s
Figure 1. Nucleation rate vs supersaturationfor various values of 6 = tfkT.
where q is the diameter of a molecule of the condensing species.
Results and Discussion The calculationswere performed for a binary interaction potential 412 that combines the rigid core repulsion with the nonretarded dispersion attraction
(m
for rI2 < q
Here r12 is the distance between the centers of two interacting molecules and z is a measure of the strength of these interactions. The potential @ in a well around the cluster is calculated by summing the binary interaction (40) over the whole cluster. For a spherical amorphous cluster of radius R, witha uniform distribution of molecules within it, the integration yields
'
@(r)= --4rcpq3( -)3(L)3 for r > R + q (41) 3 r-R r+R For amorphous bodies composed of randomly packed spheres, the number density is given by pq3 = l.2.17 The rate of nucleation was calculated by using eq 38, with the rate of dissociation given by eq 25. Figure 1 presents the nucleation rate plotted versus the supersaturation ratio s for several values of the interaction parameter t/kT. It is of interest to compare the results of the present theory with those provided by other approaches to the nucleation theory. Previously, the authors have developed a kinetic theory of nucleation in gases,9 based on a diffusion-like equation (16) in the energy space for the rate of evaporation. In that treatment, a three-dimensional motion of the molecules in the well was considered. Since the radial directionis the only one along which the potential energy in a well varies, one may expect the present onedimensional treatment for the rate of evaporation to constitute a fairly accurate approximation of the more exact three-dimensional solution. The unidimensional approximation used here is particularly justified, when the thickness of the well is small compared to the radius of the cluster; in such a system a molecule moves virtually
(15) Nowakowski, B.;Ruckenstein, E. J . Colloid Interface Sci. 1991, 142,599.
(16) Zeldovich, J. B. J. Exp. Theor. Phys. (Ruasian) 1942, 12, 525.
(17) Kittel, C.Introduction to Solid State Physics; Wiley: New York, 1976.
Langmuir, Vol. 7, No. 7, 1991 1541
Nucleation in Gases
theory for large clusters (clusters for which R*/v > IO). One can notice that the one-dimensional and threedimensional approaches provide very close results. For small supersaturations, the one-dimensional approximation tapers more rapidly, but all three approaches converge to common values for large critical clusters, because macroscopic thermodynamics is expected to be valid in such cases.
0
@a
- 50
h
C
‘1.
-
1
C
-100
-150
0.0
1.0
0.5
1.5
In s
Figure 2. Comparison among the nucleation rates predicted by various approaches.
in the one-dimensional potential well along the radial direction, and its motion along the lateral and longitudinal coordinates is independent and separable. Figure 2 compares the nucleation rates provided by the present one-dimensional approach for the rate of evaporation and by the three-dimensional treatment.s The results obtained by using the classical theory of nucleation (eq 39) are also presented. As already mentioned, the surface tension u involved in the classical theory was calculated from the best fit of the slope of the Kelvin equation (eq 29) to the numerical results of the present
Conclusion A kinetic approach to the theory of nucleation in gases is presented, in which the rate of evaporation of molecules from clusters is calculated independently, i.e. without employing the detailed balance principle and macroscopic thermodynamics as the classical theory does. The motion of the evaporating molecules within the potential well around a cluster is described by using a Fokker-Planck equation. Assuming that the effective thickness of the well is much smaller than the radius of the cluster, the motion of a molecule in the well is treated in a onedimensional approximation, in which only the radial coordinate and velocity are relevant. The effective onedimensional Fokker-Planck equation leads, after averaging over a short time scale, to a kinetic equation in the energy space, which is shown to coincide with the diffusion equation in the energy space obtained by another formalism. The rate of nucleation has been found to provide values close to those predicted on the basis of a threedimensional description of the motion of the evaporating molecules.
