Does the configurational entropy of polydisperse particles exist?

TL;DR

This paper addresses the divergence of configurational entropy in polydisperse systems, proposing a method to compute a finite, meaningful entropy that aligns with experimental observations of glass transitions.

Contribution

It introduces a novel approach to calculate finite configurational entropy in polydisperse systems by modeling them as effective multi-component systems, resolving longstanding theoretical issues.

Findings

Finite configurational entropy can be computed for polydisperse systems.

The method applies to various models with different interactions.

Results are consistent across multiple simulation cases.

Abstract

Classical particle systems characterized by continuous size polydispersity, such as colloidal materials, are not straightforwardly described using statistical mechanics, since fundamental issues may arise from particle distinguishability. Because the mixing entropy in such systems is divergent in the thermodynamic limit we show that the configurational entropy estimated from standard computational approaches to characterize glassy states also diverges. This reasoning would suggest that polydisperse materials cannot undergo a glass transition, in contradiction to experiments. We explain that this argument stems from the confusion between configurations in phase space and states defined by free energy minima, and propose a simple method to compute a finite and physically meaningful configurational entropy in continuously polydisperse systems. Physically, the proposed approach relies on an effective description of the system as an $M^*$-component system with a finite $M^*$, for which finite mixing and configurational entropies are obtained. We show how to directly determine $M^*$ from computer simulations in a range of glass-forming models with different size polydispersities, characterized by hard and soft interparticle interactions, and by additive and non-additive interactions. Our approach provides consistent results in all cases and demonstrates that the configurational entropy of polydisperse system exists, is finite, and can be quantitatively estimated.