A mathematical model for a copolymer in an emulsion

TL;DR

This paper develops a mathematical model for a copolymer in an emulsion, revealing a complex phase diagram with multiple localized and delocalized phases depending on percolation properties.

Contribution

It introduces a tractable mathematical framework for copolymer behavior in emulsions, analyzing phase transitions based on percolation regimes.

Findings

Single critical curve in supercritical regime separates phases

Three critical curves in subcritical regime define multiple phases

Identification of tricritical points where phases meet

Abstract

In this paper we review some recent results, obtained jointly with Stu Whittington, for a mathematical model describing a copolymer in an emulsion. The copolymer consists of hydrophobic and hydrophilic monomers, concatenated randomly with equal density. The emulsion consists of large blocks of oil and water, arranged in a percolation-type fashion. To make the model mathematically tractable, the copolymer is allowed to enter and exit a neighboring pair of blocks only at diagonally opposite corners. The energy of the copolymer in the emulsion is minus $\alpha$ times the number of hydrophobic monomers in oil minus $\beta$ times the number of hydrophilic monomers in water. Without loss of generality we may assume that the interaction parameters are restricted to the cone $\{(\alpha,\beta)\in \mathbb{R}^2\colon |\beta|\leq\alpha\}$. We show that the phase diagram has two regimes: (1) in the supercritical regime where the oil blocks percolate, there is a single critical curve in the cone separating a localized and a delocalized phase; (2) in the subcritical regime where the oil blocks do not percolate, there are three critical curves in the cone separating two localized phases and two delocalized phases, and meeting at two tricritical points. The different phases are characterized by different behavior of the copolymer inside the four neighboring pairs of blocks.