Exact results and mean field approximation for a model of molecular aggregation

TL;DR

This paper introduces a simple one-dimensional model of molecular aggregation that captures key features of micelle formation, providing exact solutions and comparing them with mean field approximations to improve understanding of cluster formation.

Contribution

It offers an exactly solvable model that incorporates molecular volume and attraction saturation, advancing the study of micelle formation and cluster phenomena.

Findings

Exact solutions reveal system behavior at various densities and temperatures.

Mean field approximation closely matches exact results, suggesting new modeling approaches.

Addresses controversy over internal degrees of freedom in cluster formation models.

Abstract

We present a simple one-dimensional model with molecular interactions favouring the formation of clusters with a defined optimal size. Increasing the density, at low temperature, the system goes from a nearly-ideal gas of independent molecules to a system with most of the molecules in optimal clusters, in a way that resembles the formation of micelles in a dilution of amphiphilic molecules, at the critical micellar concentration. Our model is simple enough to have an exact solution, but it contains some basic features of more realistic descriptions of amphiphilic systems: molecular excluded volume and molecular attractions which are saturated at the optimal cluster. The comparison between the exact results and the mean field density functional approximation suggests new approaches to study the more complex and realistic models of micelle formation; in particular it addresses the long-standing controversy surrounding separation of internal degrees of freedom in the formulation of cluster association phenomena.