T H E JOlJRNAL
O F
PHYSIC:AI, C H E M I S T R Y Registered in U.S. Patent Office
62 Copyright, 1979, by the American Chemical Society
VOLUME 83, NUMBER 8
APRIL 19, 1979
Cluster Size Distribution from Homogeneous Condensation. A Stochastic Modelt C. F. Wllcox, Jr., Steven RUSSO,and S. H. Bauer" Depatfment of Chemistry, Cornel1 University, Ithaca, New York 14853 (Received November 10, 1978)
A stochastic model, based on the random placement in space of completely absorbing sinks, is shown to lead to a log normal distribution of ultimate cluster sizes for a finite density medium. The resulting distribution function not only shows the expected dependence of the variance on the density of sinks (corresponding to control of the density of critical size nuclei by the level of supersaturation),but also has the same numerical values for the variance as were reported in the literature.
In our precleding note2 we demonstrated that the condensation of a liquid from its supersaturated vapor, under homogeneous conditions, can be satisfactorily modeled by a random-walk process. A stochastic model expressed by a simple alogarithm showed all the anticipated properties of a condensing system: (i) that a random-walk does lead to a steady state population N , ( n < nt), which fluctuates mildly about a mean value; (ii) that the attainment of steady state via this random walk for n < nt leads to a recognixable critical size nucleus ( n cv nt);(iii) that for n > nt a catastrophic process sets in, i.e., unidirectional cluster growth occurs, leading to condensation; (iv) a physical interpretation was provided for the extremely low mean occupati~onallevels for n > nt even though there was a large condeinsation flux (33). However, a theory of nucleation arid condensation (gas liquid or gas sollid) is incomplete unless it provides
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Don 13unker had a style of his own, in the way he lived, in the scientific problems he selected to work on, and in the methods he developed to obtain their solutions. He was intensely preoccupied with molwular dynamics. He boldly asked searching questions and then proceeded to surprise us with some of the answers he uncovered. Along with the many multidimensional potential functions which Don and his students formulatled, and the thousands of trajectories which they calculated, often exploiting minimum computer facilities, he also found time to study systems dynamics by programming a computer to simulate the wildly chaotic drift of systems toward equilibrium; Le., purely stochastic models.' It is in this spirit that WE dedicate the following brief note to Don's memory, and state again that his thought-provoking models of molecules in the process of interconversion will be studied by many chemists, ]present and future.
a prediction of the size distribution of the clusters thus generated. T o the various formulations which have been proposed, for example, the self-consistent kinetic theory by Bauer and F r ~ r i pone , ~ must add some feature with a random aspect that somehow discriminates between the various critical size nuclei, such that a distribution of cluster size results. This distribution may be modified further by coalescence of the clusters to generate a second distribution [in many cases this is the distribution observed], prior to the total collapse of the droplets or crystallites into a single ball which is the condition of maximum thermodynamic stability; the last step rarely occurs. The basis for the first distribution is obvious. Critical size nuclei appear randomly distributed in space. The size to which any nucleus can grow is determined by the volume of gas in its immediate vicinity from which it can draw the monomers. These diffuse toward the nuclleus, impelled by the density gradient established a t its boundary (condensation rates being greater than evaporation rates for n > nt). It follows that systems with high supersaturation levels rapidly generate many critical nuclei, so that their mean cluster size is small, while low supersaturation levels produce few nuclei and their mean cluster size is large. This has been experimentally verified many times; for example, for the case of Pb vapor by Frurip and Bauer4 (their Figure 11). One cannot represent this random aspect by differential equations; a stochastic aspect must be incorporated. Two mathematical models are described below. The first and more complete one has not yet been tested numerically. For both, space is divided
0022-3654/79/2083-0897$01.00/0 0 1979 American Chemical Society
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The Journal of Physical Chemistry, Vo/, 83, No. 8, 1979
into m X m X m cubical cells, which a t t = 0 are filled either with critical size nuclei or monomers. Model I (Not Programmed at This Time). By random number selection place i2 nuclei (on x,,y l , 2,; 1 5 i 5 R) within the cube m X m X m (coordinates of each cell: a,, b,, e,; 1 I j Im). Impose the condition that the separation between adjacent nuclei must be equal to or greater than one, three, or five cell lengths. Allow for diffusion (via a random walk) as follows. Arbitrarily select a cell a,, b,, c, for which the occupation number is cy/ < nt (i.e,, it is not a nucleus). By means of another random number determine to which adjacent cell a monomer will diffuse; the probability is 1:6 to a,+l, b,, e, [or a,, bi+l, e]; etc], say this one has an occupation number Let the probability be unity that the monomer remain in the new cell if = 0, 1 or if it happens to be a nucleus, that is if aJtl 1 nt. However, let the probability be l / ( ~ y ~ + ~if) 1 < altl < nt. Play this game until a statistically significant number of cells are emptied, so as to generate a meaningful cluster size distribution. While these weighting factors are arbitrary, the successive play for a large number of trials should lead to the desired distribution. Frisch and Collins5 derived boundary conditions for Frick’s differential equation for diffusion to a multiplicity of sinks, allowing for their mutual perturbation. Thus, the stochastic analogues of their concentration gradients, which drive the monomers toward the various sinks, are the assumed probabilities for the migrating monomers to remain in the new cells, which depend on the cell occupation numbers, a,+1. Model 11 (Tested). Again, place R nuclei in m3 cells on a random basis. Then, from each nucleus draw radii to all its nearest nuclei, and bisect each line with a plane normal to it. This operation generates an irregular polyhedron around each nucleus (Voronoi polyhedron6). Such polyhedra appear in the Bernal theory of liquids7 and in statistical mechanical analyses of microemulsions.8 We now introduce the basic assumption that the material within the entire volume of each Voronoi polyhedron condenses onto its central nucleus. The distribution in cluster sizes thus reflects the distribution in Voronoi polyhedra volumes. The required algorithm is to find, for each cell in the m X m X m lattice which is occupied by a monomer, the nearest nucleus to which it is moved. A count of the number of monomers accrued for each of the s1 nuclei gives the desired size distribution. This model has been explored numerically for the cases with m 3 / Q= 417 and 1707, to simulate different levels of supersaturation. In both cases 300 nuclei were placed randomly, disallowing multiple cell occupation. For each density three subcases were explored in which the minimum distance between cells which incorporate nuclei was set a t 1,3, and 5 cell units. The resulting distributions of cluster sizes for a cube containing 125 000 cells are shown in Figure la, c; the numbers of clusters that are within each size range are plotted vs. the mean cluster size. Similar results were obtained for the cube containing 512 000 cells. This is a purely geometrical model. It departs from reality in a number of respects. For example, no correction is included for the varying temperature gradients developed around adjacent nuclei, due to their condensing different amounts of monomer. Also, no account is taken of the fact that the critical size nuclei are not generated simultaneously, so that the concentration gradients that are developed are time as well as distance dependent. This is not serious since the time to reach steady state is very short compared with the growth p e r i ~ d .Finally, ~ ~ ~ the random-walk aspect of molecular diffusion is not specif-
C. F. Wilcox, S. Russo, and S. H. Bauer
c . minimum separation = 5
30
20
lot ! i 0
0
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3
separation
z
4ot
500
0
I500
IO00
C l u s t e r Size Figure 1. Bar graphs for cluster size distribution, model 11: fl = 300; m 3 = 125 000. The minimum separation between nuclei is as follows: (a) 1, (b) 3, (c) 5. I n (a), the smooth curve is that calculated from the function parameters evaluated in Figure 2a. Percentoqe IO00
I
I
20
I
I
I
I
50 I
I
I
90
I
I
1
1
200
A IOOOL
c
-
1
ax*---a-a-*< Ha+a-**a
minimum separation = 3
n
=
200
I O O O ~
a I
1
I
4.0
5.0
6.0
7.0
Probiis Figure 2. Accumulative size distributions for the three cases illustrated in Figure 1.
The Journal of Physical Chemistry, Vol. 83, No. 8, 1979 899
Cluster Size Distribution from Homogeneous Condensation
5
0
20
P e r c e n ta g e 50 0 0
90
~
~
TABLE I: Dependence of t h e Variance (u) a n d M e a n Size ( n )on M o d e l Parameters ~
m
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50 300 m3in = 417 m = 80 ~2 = 300 m31n = 1701
’I-3.0
4.0
Figure 3. Same as Figure 2 for
P r o5.0 bits
il =
6.0
300 and m 3 = 5‘12000.
ically included. However, the model can be justified on the baisis that the largest concentration gradient which any monomer exlperiences directs it toward its nearest nucleus; we neglect the very small number which happen to be exactly midway between nuclei. It is the simplest model which can ble calculated with a microcomputer in an acceptable time. (The cases with m3/8= 417 required about 10 h each; the cases with m3/Q= 1’706required about 40 h eachi.) The Voronoi cell volume distribution shown in Figure 7 of ref 7 was obtained by a different algorithm but it is equivalent to the distribution functions illustrated in Figure 1.
Comments Inspection of the bar graphs shows that by forcing a more uniform distribution of nuclei (for example, disallowing separations closer than five cells) one generates a narrower distribution of cluster sizes. Plots of the accumulative distributions on log/probability scales (Figures 2a-c and 3a- c ) give nearly straight lines. Granqvist and Buhrmang demonstrated by direct observation that ultrafine metarl particles, whether clean or oxygen coated,
= =
min separation between nuclei
0
(n) U
(n)
1
3
5
1.67 373
1.51 386
1.40 396
1.66 1528
1.63 1543
1.54 1569
followed log normal distributions in sizes. They showed that this is generally the case for a variety of materi. 1s as reported in the literature. Indeed, it is expected whein the final size distribution is a consequence of coalescence of clusters in which binary combinations occur as random events. Given any prior size distribution the central limit theorem states that the new distribution developed by summing random variables V , = nl + n2 + n3 ... nK approaches normal as K increases. However, in our model growth is equivalent to single monomer accretion. Nevertheless, the log normal distribution appears to be an excellent representation. The distributions measured by Granqvist and Buhrman showed standard deviations of sizes in the range of 1.36-1.60. Our numerical experiments give standard devications in the range 1.40-1.67 (Table I). Our results also show the deficiency of small clusters (i.e,, slight positive curvature) noted by Granqvist and Buhrman, although our model is purely geometric. The dependence of the distribution parameters on the level of supersaturation is illustrated by comparing Figures 2c and 3c [ m 3 / Q= 417 and 1707, respectively]. As expected, a t the higher density of nuclei the distribution of cluster sizes is noticeably narrower [a = exp(s1ope) = 1.40 and 1.54, respectively].
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Acknowledgment. We thank Professor B. Widom for his informative discussions. This work was supported by the National Science Foundation under Grant No. DMR-78-10307. References and Notes (1) D. L. Bunker, B. Garrett, T. Kleindienst, and G. S. Long, 111, Combust. Flame, 23, 373 (1974). (2) S. H. Bauer, C. F. Wilcox, Jr., and S. Russo, J . Phys. Chem., 82, 59 (1978). (3) S . H. Bauer and D. J. Frurip, J . Phys. Chem., 81, 1015 (1977). (4) D. J. Frurip and S. H. Bauer, J . Phys. Chem., 81, 1007 (1977). (5) H. L. Frisch and F. C. Collins, J . Chem. Phys., 20, 1797 (1952). (6) G. F. Voronoi, J. Reine Angew. Math., 134, 198 (1908). (7) J. L. Ftnney, Proc. R . SOC. London, Ser. A , 319, 479 (1970). (8) Y. Taimon and S. Prager, Nature (London), 267, 333 (1977). (9) C. J. Granqvist and R A. Buhrman, J . A p p . Phys., 47, 2200 (1976).
