Relation between positional specific heat and static relaxation length: Application to supercooled liquids

TL;DR

This paper links the positional specific heat to the static relaxation length in supercooled liquids, proposing a phenomenological model that predicts a power-law divergence at the Vogel-Fulcher temperature, aligning with numerical simulations.

Contribution

It introduces a phenomenological framework connecting positional specific heat with static relaxation length, providing a temperature dependence consistent with simulations and mean field theory.

Findings

Static relaxation length varies as (T - T0)^(-1) in supercooled liquids.

Phenomenological exponent matches simulation results, not mean field estimates.

Predicts a divergence of relaxation length at T0, consistent with glass transition theories.

Abstract

A general identification of the {\em positional specific heat} as the thermodynamic response function associated with the {\em static relaxation length} is proposed, and a phenomenological description for the thermal dependence of the static relaxation length in supercooled liquids is presented. Accordingly, through a phenomenological determination of positional specific heat of supercooled liquids, we arrive at the thermal variation of the static relaxation length $\xi$, which is found to vary in accordance with $\xi \sim (T-T_0)^{-\nu}$ in the quasi-equilibrium supercooled temperature regime, where $T_0$ is the Vogel-Fulcher temperature and exponent $\nu$ equals unity. This result to a certain degree agrees with that obtained from mean field theory of random-first-order transition, which suggests a power law temperature variation for $\xi$ with an apparent divergence at $T_0$. However, the phenomenological exponent $\nu = 1$, is higher than the corresponding mean field estimate (becoming exact in infinite dimensions), and in perfect agreement with the relaxation length exponent as obtained from the numerical simulations of the same models of structural glass in three spatial dimensions.