Exact Analytical Solution of the Flory-Huggins Model and Extensions to Multicomponent Systems

TL;DR

This paper derives an implicit analytical solution to the Flory-Huggins model for polymer solutions, enabling explicit phase diagram mapping and extending the approach to multicomponent systems, thus simplifying analysis of phase separation.

Contribution

It introduces an implicit substitution method for solving the Flory-Huggins theory analytically, including multicomponent and polydisperse systems, with strategies for complex mixtures.

Findings

Derived implicit analytical solutions for single-polymer systems.

Extended solutions to multicomponent polymer mixtures.

Proposed efficient methods for phase coexistence mapping in complex systems.

Abstract

The Flory-Huggins theory describes the phase separation of solutions containing polymers. Although it finds widespread application from polymer physics to materials science to biology, the concentrations that coexist in separate phases at equilibrium have not been determined analytically, and numerical techniques are required that restrict the theory's ease of application. In this work, we derive an implicit analytical solution to the Flory-Huggins theory of one polymer in a solvent by applying a procedure that we call the implicit substitution method. While the solutions are implicit and in the form of composite variables, they can be mapped explicitly to a phase diagram in composition space. We apply the same formalism to multicomponent polymeric systems, where we find analytical solutions for polydisperse mixtures of polymers of one type. Finally, while complete analytical solutions are not possible for arbitrary mixtures, we propose computationally efficient strategies to map out coexistence curves for systems with many components of different polymer types.