Continuous demixing transition of binary liquids: finite-size scaling from the analysis of sub-systems

TL;DR

This paper introduces a novel finite-size scaling method using sub-systems within large simulations to analyze critical behavior in binary liquids, addressing challenges posed by diverging correlation lengths.

Contribution

It proposes and validates an alternative finite-size scaling approach based on sub-system analysis, improving critical point estimation in binary liquids.

Findings

Estimated the critical Binder cumulant as 0.201 ± 0.001

Quantified confluent corrections with an exponent of approximately 0.83

Validated the method with consistency across multiple scaling routes

Abstract

A binary liquid near its consolute point exhibits critical fluctuations of the local composition; the diverging correlation length has always challenged simulations. The method of choice for the calculation of critical points in the phase diagram is a scaling analysis of finite-size corrections, based on a sequence of widely different system sizes. Here, we discuss an alternative using cubic sub-systems of one large simulation as facilitated by modern, massively parallel hardware. We exemplify the method for a symmetric binary liquid at critical composition and compare different routes to the critical temperature: (1) fitting the critical divergences of the correlation length and the susceptibility encoded in the composition structure factor of the whole system, (2) testing data collapse and scaling of moments of the composition fluctuations in sub-volumes, and (3) applying the cumulant intersection criterion to the sub-systems. For the last route, two difficulties arise: sub-volumes are open systems with free boundary conditions, for which no precise estimate of the critical Binder cumulant $U_c$ is available. Second, the periodic boundaries of the simulation box interfere with the sub-volumes, which we resolve by a two-parameter finite-size scaling. The implied modification to the data analysis restores the common intersection point, and we estimate $U_c=0.201 \pm 0.001$, universal for cubic Ising-like systems with free boundaries. Confluent corrections to scaling, which arise for small sub-system sizes, are quantified at leading order and our data for the critical susceptibility are compatible with the universal correction exponent $\omega\approx 0.83$.