Exact asymptotic solution of an aggregation model with a bell-shaped distribution

Area of Science:

• Physical Chemistry
• Chemical Physics
• Materials Science

Background:

• Irreversible aggregation is a fundamental process in various scientific fields.
• Understanding aggregate behavior requires detailed theoretical frameworks.
• Previous work established a general scaling theory for Type III reaction rates.

Purpose of the Study:

• To present a detailed analysis of the scaling theory for irreversible aggregation with K(k,l)=1/k+1/l reaction rates.
• To investigate the behavior of large aggregates in the scaling limit.
• To explore results beyond the scaling limit, including fixed cluster size and time behaviors.

Main Methods:

• Detailed analysis within the scaling limit framework.
• Mathematical modeling of cluster size distribution.
• Investigation of reaction rates K(k,l)=1/k+1/l.

Main Results:

• The cluster size distribution collapses to a function of size ratio in the scaling limit.
• Confirmed general results for Type III reaction rates where larger aggregates react faster.
• Discovered an unexpected subleading correction for large clusters and deviations beyond the scaling limit.

Conclusions:

• The scaling theory provides a detailed understanding of irreversible aggregation.
• Subleading corrections and deviations highlight complexities beyond simplified models.
• The study offers insights into aggregate behavior at fixed sizes and times.