Statistical mechanics of Roskilde liquids: Configurational adiabats, specific heat contours and density dependence of the scaling exponent

TL;DR

This paper derives exact thermodynamic relations for Roskilde liquids, linking configurational specific heat behavior along adiabats to the density dependence of the scaling exponent, supported by computer simulations of Lennard-Jones and Girifalco fluids.

Contribution

It introduces a fluctuation formula for the slope of $C_V$-contours and clarifies the relation between the scaling exponent $oldsymbol{ extgamma}$ and the pair potential, extending isomorph theory.

Findings

The sign of $doldsymbol{ extgamma}/d ho$ influences $C_V$ contour slopes.

$oldsymbol{ extgamma}$ and $R$ are key parameters in characterizing Roskilde liquids.

Self-diffusion is more invariant along adiabats than $C_V$-contours.

Abstract

We derive exact results for the rate of change of thermodynamic quantities, in particular the configurational specific heat at constant volume, $C_V$, along configurational adiabats (curves of constant excess entropy $S_{\textrm{ex}}$). Such curves are designated isomorphs for so-called Roskilde liquids, in view of the invariance of various structural and dynamical quantities along them. Their slope in a double logarithmic representation of the density-temperature phase diagram, $\gamma$ can be interpreted as one third of an effective inverse power-law potential exponent. We show that in liquids where $\gamma$ increases (decreases) with density, the contours of $C_V$ have smaller (larger) slope than configurational adiabats. We clarify also the connection between $\gamma$ and the pair potential. A fluctuation formula for the slope of the $C_V$-contours is derived. The theoretical results are supported with data from computer simulations of two systems, the Lennard-Jones fluid and the Girifalco fluid. The sign of $d\gamma/d\rho$ is thus a third key parameter in characterizing Roskilde liquids, after $\gamma$ and the virial-potential energy correlation coefficient $R$. To go beyond isomorph theory we compare invariance of a dynamical quantity, the self-diffusion coefficient along adiabats and $C_V$-contours, finding it more invariant along adiabats.