The Journal of
Physical Chemistry
0 Copyright, 1987, by the American Chemical Society
VOLUME 91, NUMBER 1 JANUARY 1,1987
LETTERS Fluctuations in the Concentration of the Nucleus in Classical Nucleation Theory Howard Reiss* and Philippe Mirabel Institut de Chimie, Universite Louis Pasteur, Strasbourg, France (Received: September 22, 1986)
An analysis is performed which indicates that the relative fluctuation in the concentration of condensation nuclei in a supersaturated vapor, under conditions typical for the measurement of critical supersaturations, is of the order of IO6. Such a huge fluctuation must be taken into account in the development of a theory for the nucleation rate. It is also demonstrated that, in spite of this large fluctuation, the “maximum term method” yields the correct equilibrium distribution of clusters.
Introduction The classical theory of nucleation’ is essentially an equilibrium theory of rate in which the rate constants for the decay of clusters are obtained from the constants for the growth of clusters through the application of the principle of detailed balance. As a result the so-called “equilibrium distribution of clusters” enters the theory. Of course this equilibrium distribution must be “constrained” into equilibrium by disallowing clusters of size larger than the nucleus, since the nucleation process is taking place in a metastable phase. For convenience (and also to be concrete) we will restrict our considerations to metastable, supersaturated vapors involving condensation nuclei. It is weil-known that, under conditions typical for the measurement of nucleation rates, the actual concentration of nuclei s-l. Of is exceedingly small, usually of the order of course, what this means is that if 1 cm3 of the supersaturated vapor is observed for, say, 1 s, then during an aggregate time of IO-’* s a nucleus will be present in that cubic centimeter. If that nucleus is hit by molecules from the surrounding vapor at a nominal rate of lo’* s-l a nucleation rate of about 1 droplet cm-3 s-l will be observed. At such small concentrations the fluctuation in the concentration of the nucleus must be very large, and in a satisfactory theory (not necessarily an equilibrium one) there could
* Permanent address: Department of Chemisty and Biochemistry, UCLA, Los Angeles, CA 90024.
be serious consequences for the rate of nucleation and the classical approach might confronted with serious additional problems. In a sense, Lovett2 has drawn attention to this situation in the development of his so-called “hydrodynamic theory” of nucleation. He emphasizes the fact that clusters are continually growing and decaying in attempts to escape over the free energy barrier. Every once in awhile a cluster is successful and escapes before it decays. Viewed from the perspective of the “successful” cluster its development and escape is an exceedingly rapid “shotlike” phenomenon. Most of the time there is no cluster at the nucleus size. The problem is somewhat aggravated in the case of nucleation induced by ions or by molecules of a species different from those of the supersaturated vapor, e.g. in the nucleation of supersaturated monomer vapor by a polymer molecule. In these cases there may be only one ion or only one polymer molecule in the vapor, and at any given instant we do not have a distribution of clusters of various sizes. There is only one cluster, incorporating the foreign molecule, and at any given time it is of one or another size; clusters of varying size are not simultaneously present! In classical nucleation theory the so-called “equilibrium distribution” of clusters which specifies the “concentrations” of clusters of various sizes, (1) Abraham, F. F. Homogeneous Nucleation Theory; Academic: New York, 1974. (2) Lovett, R. J . Phys. Chem. 1980,84, 1483. J . Chem. Phys. 1984, 81, 6191.
0022-3654/87/2091-0001$01.50/0 0 1987 American Chemical Society
2 The Journal of Physical Chemistry. Vol. 91, No. 1, 1987 is usually arrived at by an application of the “law of mass action” in which the different sized clusters are viewed as different molecular species produced from one another by a chemical reaction. Alternatively the equilibrium distribution of clusters is arrived at through an appeal to the “maximum term method” of statistical mechanics, which should be identical with the law of mass action when it is valid, i.e. when fluctuations are small. The exact meaning of the law of mass action may have to be sharpened when “reactant” and “product” molecules are only alternately present in the system as would be the case when the clusters were forming on say one ion or one polymer. Furthermore, when the fluctuations in the concentrations of the species are relatively large, one has to be careful about the application of the maximum term method. In this paper we shall investigate the validity of the maximum term method for species whose concentrations are small enough to have a large relative fluctuation. Since the maximum term method is consistent with the law of mass action our analysis applies to that method as well. At the same time we will characterize both the average concentration of the nucleus as well as thefluctution of that concentration. The results derived below demonstrate that the average concentration, rigorously obtained without an application of the maximum term method, is identical with that obtained with the method, even in the face of an enormous relative fluctuation, and even when there can be only one cluster (one ion or one polymer) present at any given time. The reason for this lies in the fact that the clusters are considered to be components of an ideal gas mixture. However, the truly enormous fluctuations which can be realized must be carefully taken into account in the development of a theory for the rate of nucleation.
Derivation of the Distribution For convenience, we will examine the equilibrium distribution of clusters for the case in which the clusters develop on ions, allowing only one ion to a cluster. To be concrete we might think of ions in supersaturated water vapor, a system in which nucleation has often be in~estigated.~Then a cluster consists of an ion together with i water molecules. The partition function of such a cluster will be denoted by q,. Note that qo represents the partition function of a bare ion. We will denote by a the partition function of a free water molecule. Finally, we will deal with the case in which the vapor of volume Vcontains a constant number I of ions distributed among the various clusters. It is easiest to derive the distribution of clusters, using the grand ensemble. To this end we write the grand partition function, with T and Vfixed, for the system as follows:
Letters
Then it is an easy matter to invert the order of the two summations in eq 1, taking account of eq 3 and 4 so that the grand partition function may be expressed as
-
z = -1 cz!r I(X’qJ”1 y C -(aXY I!
X = pl/kT
(2)
where h l represents the chemical potential of a molecule of the supersaturated vapor, k is the Boltzmann constant, T i s the temperature, and where the mixture of clusters is treated as a mixture of ideal gases. M is the total number of water molecules in the system and the ni in curly brackets under the summation sign in eq 1 indicates that the sum goes over all sets of n, such that
Cni =I i
f=o
J!
The sum over f results in an exponential, and a little thought will indicate that the remaining sum is a simple multinomial expansion so that we may write, in place of eq 6 for the grand partition function, the following expression: (7) It should be remarked that the inversion of the order of summation, leading to eq 6, is possible because of the “open” nature of the grand ensemble. Also we should remember that the sum over i , in eq 7, extends only as far as the cluster of nucleus size, since in the supersaturated system we are constraining the vapor so as to disallow clusters of larger size. Now, inspection of eq 1 reveals that the average number of clusters of size i will given by ni.(M-Zl’”i)n(q51/ni!)
m
”” (“,I
-71
M=O
i
-7
(M -
(ni) =
cini)! i
I
L
T. V
Substituting eq 7 into this expression gives IVq, (ni) = -
C X’qj
(9)
j
In the similar manner the average value of n: is given by
[h”’]
Again, substituting eq 7 into this expression yields [””2
(n:) = I ( I - 1) -
CA’S’
in which
ni.
i
+I
- = CX’qj
The relative fluctuation in the concentration can be defined in the standard way as follows:
-Ani- (ni)
[ ( n ; ) - (ni)2]’/2 (12) (ni)
Substitution of eq 9 and 11 into eq 12 yields
(3)
i.e. the total number of clusters equals the total number of ions. We ignore clusters which do not contain ions, assuming that the supersaturation is low enough so that these are not important. It is also true, in eq 1, that
Cini Ikt
(4)
i
In general, we know that
( n i )
