2313
J . Phys. Chem. 1992, 96, 2313-2316
A Kinetic Treatment of Heterogeneous Nucleation B. Nowakowskit and E. Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 (Received: August 26, 1991)
An expression is derived for the rate of evaporation of a molecule interacting with a spherical cap cluster and the solid on which the cluster is located. The motion of the evaporating molecule in the potential well generated by its interaction with the cluster and substrate is described by a diffusion equation in the energy space. The rate of heterogeneous nucleation, calculated in the steady-state approximation, is strongly dependent on the interactions with the solid substrate. A comparison between the kinetic and classical theories shows that the former provides much larger values than the latter for small critical clusters (large supersaturations) but tends to the latter for large critical clusters.
Introduction Nucleation in supersaturated vapors frequently occurs on the surface of the container, foreign particles, or other surface heterogeneities. The number of nucleation sites as well as their catalytic effectiveness is generally not known. For this reason, it is usually assumed that the surface of the foreign substrate is homogeneous and that the extent of wetting of the solid by the liquid determines the wetting angle of the clusters on the solid surface. The classical theory of heterogeneous nucleation was formulated in the same way as the traditional approach to the homogeneous process.l*2 It employs the detailed balance principle and a hypothetical equilibrium distribution of clusters to calculate the rate constant of evaporation from clusters in terms of the rate constant of condensation. The equilibrium distribution is calculated on the basis of macroscopic thermodynamics. In particular, the macroscopic surface tension is used to calculate the surface free energy of a cluster. The extension of macroscopic properties to the small clusters constitutes the most questionable feature of the classical theory. Macroscopic thermodynamics was avoided in the lately developed kinetic theory of homogeneous n ~ c l e a t i o n . ~In , ~ that approach, the rate of evaporation from a cluster was obtained independent of the rate of condensation on the basis of the solution of the equation governing the motion of a molecule interacting with the cluster. In this sense, the latter approach can be considered more kinetic than the conventional treatment. The kinetic theory was applied to liquid/solid3 and vapor/liquid4 phase transitions. In this paper we extend the formalism developed for homogeneous nucleation in gases4 to the heterogeneous nucleation on a flat, homogeneous surface. The properties of the surface are assumed uniform, in the sense that the solid/liquid uIsand solid/gas u interfacial tensions have constant values over the entire surface. T l e nuclei formed are assumed to have the shape of a spherical cap, with a wetting angle 0 at the edge of the nuclei given by the Young equation5 uls ulg COS e = IJsg (1)
+
where ulgis the liquid/gas interfacial tension. In the next section, the theory4 for the rate of evaporation of molecules from a cluster is extended to heterogeneous nucleation by taking into account the interactions between the evaporating molecule and the system consisting of the cluster and the supporting solid. The obtained results are used, in a subsequent section, to calculate the rate of nucleation from the steady-state solution of the equations describing the population of clusters. Finally, numerical results are presented for the rate of heterogeneous nucleation. A comparison with the predictions of the classical nucleation theory concludes the paper. *Author to whom correspondence should be addressed. 'Present address: Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland.
0022-365419212096-23 13$03.00/0
Rate of Evaporation from the Cluster In the layer over the free surface of the cluster a potential well is generated due to the attractive intermolecular interactions. The form of the pair interaction potential 4(rI2)is assumed to combine the London-van der Waals attraction with the rigid core repulsion 4(r1z) =
OD
r12
< IJ
(2)
= - e ( ~ r / r ~ ~ ) rI2 ~ 2u
where r l z is the distance between the centers of the molecules, e is an interaction constant, and u is the molecular diameter (assumed the same for all the species). The vapor molecule in the well interacts with the cluster as well as with the solid. Assuming pairwise additivity, the effective interaction potential 9(r) in the position r in the well is calculated by integrating over the interactions with the molecules of the cluster and with those of the solid substrate. The interaction constant with the cluster, e,, is assumed to be greater than that with the solid, e,. The integration over the volume of the semiinfinite solid yields the following expression for the interactions 9,of the vapor molecule in the well with the solid (3) where x is the distance of the molecule from the surface of the solid and p, is the number density of molecules in the solid. The volume integral over the spherical cap, involved in the calculation of the interaction potential 9,with the cluster, can be performed analytically for two coordinates; the resulting formula has a complicated form, and for this reason the integration with respect to the third variable was carried out numerically. The details are presented in the Appendix. As long as the motion of the vapor molecule is confined within the well, one can consider that it belongs to the cluster. In this case its energy E lies in the range Q0 C E < 0 , where is the potential energy at the bottom of the well. The energy of the molecule is a stochastic variable, because of the exchange of energy during the collisions with the molecules of the gaseous medium. Eventually, the molecule in the well can acquire a sufficient amount of energy to overcome the potential well, and hence to leave the well. Therefore, the rate of evaporation of molecules from the cluster is related to the rate by which their energy passes over the upper energetic boundary of the well, E = 0. In gases, the collisions with the medium are relatively infrequent. As a result, the energy of the molecules in the well changes slowly (1) Abraham, F. F. Homogeneous Nucleation Theory; Academic Press: New York, 1974. ( 2 ) Zettlemoyer, A . C., Ed. Nucleation; Dekker: New York, 1969. (3) Ruckenstein, E.; Nowakowski, B. J. Colloid Interface Sci. 1990, 137, 583. (4) Nowakowski, B.; Ruckenstein, E. J . Chem. Phys. 1991.94, 1397,8487. ( 5 ) de Gennes, P. G. Reu. Mod. Phys. 1985, 57, 827.
0 1992 American Chemical Society
2314 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992
in comparison with the fast changes in their impulses and positions. Consequently, the time evolution in the energy space is governed by the diffusion-like equation for the probability density P(E,t) of energy4.6
Here D(E) is the diffusion coefficient in the energy space, P,(E) is the equilibrium distribution of energy given by the equation
Nowakowski and Ruckenstein should break down for large clusters. The Growth of the Cluster The rate of change of the size of the cluster is determined by the difference between the rates of evaporation and condensation. In gaseous media, the latter rate is given by the number of impingements of the vapor molecules on the surface of the cluster ?r
0 -(1 - COS 6)R2nlD 2
(12)
where r and p are the position and impulse of the molecule, respectively. The flux j(E,t) in the energy space is defined by
where 0 is the mean thermal velocity of the vapor molecules and nl is the number density of the vapor at a large distance from the cluster. For a given value of nl, there exists a size, called critical cluster, for which the rates of evaporation and condensation are equal. Clusters smaller than the critical one decay (statistically), whereas those that are larger grow. Hence, the critical cluster is in unstable equilibrium with the surrounding vapor. Using eqs 11 and 12 one can obtain the following relation between the vapor number density nl and the radius R. of the critical cluster 4x n1 = n -a!l(R.)
Using eq 4, one can write
The number density in the vapor that coexists with an infinite cluster is the saturation number density, 4.The supersaturation s = n l / n , constitutes the thermodynamic driving force for nucleation. Using eq 13, it can be expressed as
0
P,(E) = Q(E) e x p ( - E / k T ) / l
*o
dE Q(E) exp(-E/kT)
(5)
and Q(E) is the number of states of energy E. For a single molecule with the Hamiltonian H(r,p), Q(E) is given by Q(E) = l d r dp G(E-H(r,p))
r
(6)
(7) The rate of evaporation is provided by the steady-state solution of eq 7, Le., by the time-independcnt flux j E Assuming that every molecule that reaches the upper energy of the well separates from the cluster, one obtains the absorption boundary condition, P(E) = 0 at E = 0. The stationary solution of eq 7, which satisfies this boundary condition, is given by
The normalization constraint, .fP(E) dE = 1, yields an expression for the rate of evaporation per molecule, jE= cy1,
where m is the mass of the molecule, is the friction coefficient and d is the dimensionality of the geometrical space in which the Brownian motion takes place (equal to 3 in the system considered). The overall evaporation rate per cluster a! is given by the product between a!l and the number of molecules in the potential well over the cluster. Denoting by n, the number density of the vapor molecules in the well, the total number of molecules in a well of thickness X is N, = 2 4 1 - cos B)R2Xn,,.The overall evaporation rate is therefore given by 241
- COS e ) m n ,
L?
(14)
s = exp(
k)
where ulgis the macroscopic surface tension of the liquid and p is its number density. The Kelvin equation plays an important role in the classical theory of nucleation. The macroscopic thermodynamics is expected to be valid for large clusters, hence eq 15 should be recovered in the present theory for large nuclei. Consequently, comparing eqs 14 and 15 for R * / u>> 1, one can calculate the macroscopic surface tension in terms of molecular parameters.
The Rate of Nucleation The evolution of the ensemble of nuclei can be expressed in terms of the net flux of clusters passing from those consisting of i molecules, ni, to those containing ( i 1) molecules, ni+l, ri Bini - at+lni+l (16) Zi vanishes at equilibrium, and eq 16 yields then the detailed balance condition. Since the kinetic theory of nucleation provides independent results for a! and 8, the equilibrium distribution can be explicitly obtained from the detailed balance equation in the form
For convenience, eq 16 is converted to a continuum form,which, using the detailed balance equation, can be written
(1 1)
Pa@) l o D ( E ; ; : ( E )
Explicit values for the quantities n, and X are not needed because they are not present in the final equations. One may note that the above treatment implicitly assumes equipartition over the degrees of freedom, an assumption that (6) Keck, J.; Carrier, G. J . Chem. Phys. 1965.43, 2284. (7) Borkovec, M.;Berne, B. J. J . Chem. Phys. 1985,82, 797.
s = a!yR*)/d(m)
In the macroscopic thermodynamics, the supersaturation is related to the critical radius via the Kelvin equation
+
For Brownian dynamics, D(E) is given by7
a!=
w D
The continuum variable g replaces in (18) the discrete variable i. The rate of nucleation, calculated as the stationary solution of eq 18, is given by
(8) Zeldovich, J. B. Acta Physicochim. USSR 1943, 18, 1. (9) Shizgal, B.; Barrett, J . C. J . Chem. Phys. 1989, 91, 6505.
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2315
Heterogeneous Nucleation 0
--
TABLE I: Values of the Surface Tension ulp2/kTfor Various Interaction Parameters e./kT %/kT alga2/k T
* 2951
1.791 2.589 & 3.357 k 4.182 f
2 4 6 8
-50
h
C
c?
L
15 86
Y
C
0
d
-100
-25 -150
0.3
0.0
0.6 In s
0.9
1.2
0
-75
-25 -100
0.00
0.50 0.75 1.00 In s Figure 2. Comparison between the rates of nucleation obtained on the basis of the kinetic theory for various values of 6 = c,/kT and for eJkT = 0.5 and c,/kT = 1 (-).
- KINETIC
(e..)
THEORY CIASSICAL THEORY
-100
0.00
0.50 0.75 1.00 In s Figure 1. Rate of nucleation In ( I / @ & against supersaturation (In s) for various values of &‘ = c,/kT and for (a, top) c,/kT = 0.5 and (b, theobottom) c,/kT = 1 provided by the kinetic (-) and classical 0.25
(sa-)
ries. The equilibrium distribution e(g) exhibits a very sharp minimum at the critical size gl. This allows to calculate the integral in (19) by the steepest descent method. One obtains finally
In the classical theory of nucleation the hypothetical equilibrium distribution is calculated in terms of the macroscopic surface free energy of clusters. Equation 19 then provides the following expression for the steady-state nucleation
where Y
=
UIg(
2 - 3 COS o + 6093 4
0.25
o
The rates of nucleation were calculated using both eqs 20 and 21. The surface tension ulgwhich is needed in the classical expression was calculated as indicated in the previous section. This procedure enabled us to compare the predictions of the kinetic theory of nucleation with those of the classical theory.
Results of Calculations and Discussion The numerical calculations were performed using for the interaction constant with the cluster the values t,/kT = 2, 4,6, and 8, and for the interaction constant with the solid the values e,/kT = 0.5 and 1. The diameters of the molecules of all the species involved were assumed equal, and the number densities of mol-
ecules in the cluster and in the solid were calculated using the relation pa3 = 1.2 (valid for a random packing of spheresL0). The wetting angle for the liquid forming the nucleus was taken 0 = */6. The rate of nucleation based on the kinetic theory of nucleation was calculated numerically using eq 20, and the predictions of the classical theory were obtained using eq 21. Figure 1, a and b, presents the results obtained in a In-ln scale, namely -In (I/@lnl)vs the logarithm of supersaturation, In s, for c,/kT equal to 0.5 and 1, respectively. The continuous lines are based on the kinetic theory and the dashed lines on the classical theory. The values of the surface tension used in the classical formula (21) were calculated on the basis of the results of kinetic theory (eq 14) for clusters greater than R*/u> 17. The results obtained have been fitted to the Kelvin equation (15), and the surface tension was calculated by the least-squares method. The values of al,u2/kT are listed in Table I. Figure 2 compares the results provided by the kinetic theory of nucleation for two interaction constants with the solid substrate, namely e,/kT = 0.5 and 1. The rate of nucleation is higher for eJkT = 1 than for c,/kT = 0.5, because of the stronger “catalytic effect” of the solid. The stronger the attraction of the molecules in the well by the solid, the greater the difficulty to leave the well and evaporate from the cluster. Hence, the critical cluster is smaller for higher values of cs, and as a result the nucleation rate is higher. The predictions of both theories coincide in the range of small supersaturations, corresponding to large critical clusters. The classical theory is expected to be valid for large clusters, because it is based on macroscopic thermodynamics which is asymptotically correct in this range. At the other end, large supersaturations, the present theory provides much higher rates of nucleation than the classical theory. High supersaturations lead to small critical clusters, to which the macroscopic thermodynamics is no longer applicable. The macroscopic surface tension exhibited by bulk, planar interfaces overestimates the surface energy of small clusters. Consequently, according to eq 21 the classical theory underpredicts the population of critical clusters and the rate of nucleation. Similar relations between the results of the classical and the kinetic theories of nucleation were also observed in the case of homogeneous n u c l e a t i ~ n . ~ . ~ (10) Kittel, C . Introduction to Solid State Physics; Wiley: New York, 1974.
2316 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992
The macroscopic thermodynamics on which the classical theory of nucleation is based was avoided in the kinetic approach. Another assumption of the classical theory that can be reconsidered is the use of the simple formula (12) of the kinetic theory of gases for the number of collisions of the vapor molecules with the cluster. Equation 12 does not include the interaction of the condensing molecule with the substrate and the cluster. Such an interaction was shown to have some effect in the case of spherical particles on which condensation 0ccurs.II Appendix The Interaction Potential between a Molecule and a Spherical Cap Cluster. Assuming pairwise additivity, the effective interaction potential acting on a molecule in the well can be calculated using the integral W ) = JPc4(lr V - r’l) dr’
(‘41)
where r is the position of the molecule, r’ is the position of a molecule in the cluster, and pc is the number density of molecules in the cluster (assumed uniform). To calculate the volume integral over the spherical cap of the cluster, we locate the origin of the coordinate system r‘ in the center of the sphere containing the cap, and switch to cylindrical coordinates, in which r‘ = ( p cos 4, p sin 4, z ) . Because of the cylindrical symmetry, the position of the molecule in the well can always be written in the form r = (r sin J,, 0, r cos J,), where J, is the azimuthal angle of the position r. The square of the distance between the molecule and the point of integration r’ is
Nowakowski and Ruckenstein
where a(p,z) = ( r cos
+ ( r sin +)2
= 2pr sin J, (A61 The denominator of the function under the integral in (A4) can be recast in the form 0 2 - b2 = [ p 2 + ( r cos - z ) ~ ( r sin J,)2]2 + 4(r cos J, - ~ ) ~sin( J,)2 r (A7) Except for the unphysical case when r = r’, the expression A7 is always different from zero (being the sum of squares). Hence, the function under the integral A4 has no singularities. The integration of (A4) can be further performed by changing to the variable q = p2. This yields b(p,z)
+
R
0,= -cCPca6Lcos dz (Qi(4) + Q2(4) + Qdq))I$ Qi(q) = 4
+
4(r cos J , - ~ ) ~sin( J,)2[q2 r 4(r cos J, - ~ ) ~sin( J,)2]1/2 r (‘49) Q2(q) = [q2
+
-2(r sin +)2 4(r cos J, - z)2(r sin +)213/2
The integration over the polar angle 4 can be easily performed, and one obtains
Q
+ - ~ ) ~sin( J,)2 r q2 + 4(r cos J, - z)2(rsin +)2
l1
+
2(r cos J, - ~ ) ~sin( J,)2 r
H.J . Chem. Phys. 1980, 73, 6284.
IX 3/2
(A1 1)
The functions Qi appear in the integral A8 for the two limits of 4
+ - z ) -~ ( r sin +)2 q2 = ( r cos J, - z ) + ~ R2 - z2 - ( r sin J,)2 q , = (r cos
~~
(1 1) Marlow, W.
(A 10)
Q3(q)= y2(r sin ~ ) ) ~ [sin ( rJ,)2 - ( r cos J, - z ) ~ ]X
d4 (A31
(A81
where the functions Qi have the form
( r cos J , - z)Z(r sin J,)2[q2 4(r cos$
[ ( r cos J, - z ) +~ p2 + ( r sin J,)2 -2pr sin J, cos 413
(A5)
and
(r - r’)2 = ( r cos J, - z ) + ~ p2 + (r sin J,)2 - 2pr cos 4 sin J, (‘42) Using eqs A2 and 2, the integral A1 over the spherical cap acquires the form
J, - z ) + ~ p2
The third integral was calculated numerically.
(‘412) (A13)
