#### Document Type

Article

#### Date

8-18-2006

#### Embargo Period

2-4-2013

#### Keywords

Coalescence growth; Smoluchowski equations; Thermal equilibrium

#### Disciplines

Chemistry

#### Description/Abstract

The Smoluchowski equations, which describe coalescence growth, take into account combination reactions between a j-mer and a k-mer to form a (j+k)-mer, but not breakup of larger clusters to smaller ones. All combination reactions are assumed to be second order, with rate constants K _{jk}. The K _{jk} are said to scale if K _{λj,γk} =λ ^{μ}γ ^{μ}K _{jk} for j ≤ k. It can then be shown that, for large k, the number density or population of k-mers is given by Ak ^{a}e ^{-bk}, where A is a normalization constant (a function of a, b, and time), a=-(μ+ ν), and b ^{μ+ν-1} depends linearly on time. We prove this in a simple, transparent manner. We also discuss the origin of odd-even population oscillations for small k. A common scaling arises from the ballistic model, which assumes that the velocity of a k-mer is proportional to 1/ √m _{k} (Maxwell distribution), i.e., thermal equilibrium. This does not hold for the nascent distribution of clusters produced from monomers by reactive collisions. By direct calculation, invoking conservation of momentum in collisions, we show that, for this distribution, velocities are proportional to m _{k} ^{-0-.577}. This leads to μ+ν=0.090, intermediate between the ballistic (0.167) and diffusive (0.000) results. These results are discussed in light of the existence of systems in the experimental literature which apparently correspond to very negative values of μ+ν.

#### Recommended Citation

Goodisman, Jerry and Chaiken, J., "Scaling and the Smoluchowski Equations" (2006). *Chemistry Faculty Scholarship*. 27.

https://surface.syr.edu/che/27

#### Source

local input

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 3.0 License.

## Additional Information

Copyright 2006 The Journal of Chemical Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and The Journal of Chemical Physics.

The article may be found at http://dx.doi.org/10.1063/1.2218836