Bragg-Williams Theory for Particles with a Size-Modulating Internal Degree of Freedom

The field of soft matter teems with molecules and aggregates of molecules that have internal size-modulating degrees of freedom. Proteins, peptides, microgels, polymers, micelles, and even some colloids can exist in multiple-often just two dominating-states with different effective sizes, where size can refer to the volume or to the cross-sectional area for particles residing on surfaces. The size-dependence of their accessible states renders the behavior of these particles pressure-sensitive. The Bragg-Williams model is among the most simple mean-field methods to translate the presence of inter-particle interactions into an approximate phase diagram. Here, we extend the Bragg-Williams model to account for the presence of particles that are immersed in a solvent and exist in two distinct states, one occupying a smaller and the other one a larger size. The basis of the extension is a lattice-sublattice approximation that we use to host the two size-differing states. Our model includes particle-solvent interactions that act as an effective surface tension between particles and solvent and are ignorant of the state in which the particles reside. We analyze how the energetic preference of the particles for one or the other state affects the phase diagrams. The possibility of a single phase-two phases-single phase sequence of phase transitions as a function of increasing temperature is demonstrated.