# Analytical modeling of micelle growth. 1. Chain-conformation free energy of binary mixed spherical, wormlike and lamellar micelles

## TL;DR

This paper develops an analytical thermodynamic model for the growth of mixed micelles, revealing how surfactant chain-length differences influence micelle stability and shape, with implications for detergency formulations.

## Contribution

It introduces a generalized free-energy model for multicomponent, polydisperse micelles, deriving explicit expressions for chain conformation and free energy.

## Key findings

- Mixing surfactants with different chain lengths is nonideal and synergistic.
- The model applies to various micelle shapes and surfactant headgroups.
- Analytical expressions can improve understanding of micelle growth in practical applications.

## Abstract

Hypotheses: A quantitative molecular-thermodynamic theory of the growth of giant wormlike micelles of nonionic surfactants can be developed on the basis of a generalized model, which includes the classical 'phase separation' and 'mass action' models as special cases. The generalized model describes spherocylindrical micelles, which are simultaneously multicomponent and polydisperse in size. Theory: By analytical minimization of the free-energy functional we derived explicit expressions for the chain-extension and chain-end distribution functions in the hydrocarbon core of mixed micelles from two surfactants of different chainlengths. Findings: The hydrocarbon core of a two-component micelle is divided in two regions, outer and inner, where the ends of the shorter and longer chains are located. The derived analytical expression for the chain-conformation free energy implies that the mixing of surfactants with different chainlengths is always nonideal and synergistic, i.e. it leads to decrease of the micellar free energy and to enhancement of micellization and micelle growth. The derived expressions are applicable to surfactants with different headgroups (nonionic, ionic, zwitterionic) and to micelles of different shapes (spherical, wormlike, lamellar). The results can be incorporated in a quantitative theory of the growth of giant mixed micelles in formulations with practical applications in detergency.

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Source: https://tomesphere.com/paper/1905.01940