Scaling and the Smoluchowski equations

J. Goodisman, Joseph Chaiken

Research output: Contribution to journalArticle

4 Citations (Scopus)

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,γkμγ μK 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 ae -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.

Original languageEnglish (US)
Article number074304
JournalThe Journal of Chemical Physics
Volume125
Issue number7
DOIs
StatePublished - 2006

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Ballistics
ballistics
scaling
collisions
Velocity distribution
Coalescence
coalescing
conservation
Rate constants
Conservation
Momentum
monomers
velocity distribution
Monomers
momentum
oscillations
Hot Temperature

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics

Cite this

Scaling and the Smoluchowski equations. / Goodisman, J.; Chaiken, Joseph.

In: The Journal of Chemical Physics, Vol. 125, No. 7, 074304, 2006.

Research output: Contribution to journalArticle

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