On the Smoluchowski limit in Brownian coagulation of aerosols

On the Smoluchowski Limit in Brownian Coagulation of Aerosols In previous papers ( 1, 2), we have established a connection between the sticking probability and the van der Waals interparticle interactions for Brownian coagulation of aerosols. The proposed model ( 1) has been shown to reduce to the Smoluchowski expression in the continuum limit. The continuum limit is practically attained when the mean free path ~g of the molecules is much smaller than the radius a of the aerosol particles, hence when the Knudsen number Kn = Xg/a is very small. In a recent comment by Sceats (3), it has been claimed that our result is incorrect and that the continuum limit for the rate of aerosol coagulation is greater than the Smoluchowski limit. The purpose of this letter is (i) to demonstrate the validity of the Smoluchowski limit for vanishingly small Knudsen numbers and (ii) to point out that the retardation effect is responsible for such a behavior. The effect ofinterparticle forces on the coagulation coefficient of equal size spherical particles, in the absence of hydrodynamic interactions, can be accounted for via the stability ratio W defined as

F°~exp[4~(h)/kT] W = Za Jo (~a + - ~ dh,

where

a* = a/X.

One can easily see that the stability ratio W is a function of the dimensionless particle size a* and dimensionless Hamaker constant A* ( = A / k T ) . For sufficiently small particles, i.e., a* ~ 0, Eq. [3] reduces to the expression valid for the unretarded potential. On the other hand, for particles much larger than the characteristic wavelength X, i.e., a* ~ ~ , Eq. [3] leads to

dy w = f0~ (1 + y)~

where 4~(h) is the interaction potential between two partides of radius a whose centers are separated by a distance h + 2a, k is the Boltzmann constant, and Tis the absolute temperature. A simple, accurate, and convenient expression for the van der Waals retarded interaction potential is given by

[51

REFERENCES 1. Narsimhan, G., and Ruckenstein, E., J. Colloid Interface Sci. 104, 344 (1985). 2. Narsimhan, G., and Ruckenstein, E., J. Colloid Interface Sci. 107, 174 ( 1985 ). 3. Sceats, M. G., J. Colloid Interface Sci. 129, 105 (1989). 4. Gregory, J., J. Colloid Interface Sci. 83, 138 ( 1981 ). 5. Alam, M. K., AerosolSci. Technol. 6, 41 (1986).

(4) [2]

where 4~uis the unretarded potential, X is the characteristic wavelength for the internal molecular motion, and the factor within the parentheses accounts for the retardation effect. Introducing Eq. [2] in Eq. [1], one obtains

1.

It is relevant to note that the mean free path Xg at room temperature in air is of the order of 1000 A and that the characteristic wavelength X has the same order of magnitude. Consequently, Kn = Xffa is of the same order as 1/a*. This implies that Smoluchowski's result is indeed obtained in the limit of vanishingly small Knudsen numbers. Recent numerical calculations (5) on the effect of retarded van der Waals forces on aerosol coagulation are also in agreement with our result ( 1 ).

[1]

~ = ~ u [ 1 - 5'32h x ln( 1 + 5.~2h)]

[4]

GANESAN NARSIMHAN

Department of Agricultural Engineering Purdue University West Lafayette, lndiana 47907 ELI R U C K E N S T E I N

Department of Chemical Engineering State University of New York Buffalo, New York 14260

W = ~ o ( 1 +1 y)--------5 e x p[~bu(y) [ ~ T - - ( 1 - 10.64ya* 1

d

y

H

Received February 5, 1990; accepted May 8, 1990

294 0021-9797/90 $3.00 Copyright © 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.

Journal of Colloid and Interface Science, Vol. 138, No. 1, August 1990