### 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.

Original language | English (US) |
---|---|

Article number | 074304 |

Journal | The Journal of Chemical Physics |

Volume | 125 |

Issue number | 7 |

DOIs | |

State | Published - 2006 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*The Journal of Chemical Physics*,

*125*(7), [074304]. https://doi.org/10.1063/1.2218836

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

Research output: Contribution to journal › Article

*The Journal of Chemical Physics*, vol. 125, no. 7, 074304. https://doi.org/10.1063/1.2218836

}

TY - JOUR

T1 - Scaling and the Smoluchowski equations

AU - Goodisman, J.

AU - Chaiken, Joseph

PY - 2006

Y1 - 2006

N2 - 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 μγ 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.

AB - 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 μγ 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.

UR - http://www.scopus.com/inward/record.url?scp=33747619633&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33747619633&partnerID=8YFLogxK

U2 - 10.1063/1.2218836

DO - 10.1063/1.2218836

M3 - Article

AN - SCOPUS:33747619633

VL - 125

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 7

M1 - 074304

ER -