The smooth cut-off Hierarchical Reference Theory of fluids

TL;DR

This paper presents a comprehensive formulation of the Hierarchical Reference Theory (HRT) with a smooth cut-off, applying it to a hard core Yukawa fluid to analyze phase behavior, critical properties, and compare with renormalization group results.

Contribution

It introduces a self-consistent derivation of the smooth cut-off HRT and explores its application to fluid phase equilibria, highlighting its advantages over traditional methods.

Findings

HRT yields convex free energy with flat isotherms in two-phase region.

HRT can directly determine fluid-fluid phase coexistence without Maxwell construction.

The theory's predictions align well with Monte Carlo simulations for various Yukawa ranges.

Abstract

We provide a comprehensive presentation of the Hierarchical Reference Theory (HRT) in the smooth cut-off formulation. A simple and self-consistent derivation of the hierarchy of differential equations is supplemented by a comparison with the known sharp cut-off HRT. Then, the theory is applied to a hard core Yukawa fluid (HCYF): a closure, based on a mean spherical approximation ansatz, is studied in detail and its intriguing relationship to the self consistent Ornstein-Zernike approximation is discussed. The asymptotic properties, close to the critical point are investigated and compared to the renormalization group results both above and below the critical temperature. The HRT free energy is always a convex function of the density, leading to flat isotherms in the two-phase region with a finite compressibility at coexistence. This makes HRT the sole liquid-state theory able to obtain directly fluid-fluid phase equilibrium without resorting to the Maxwell construction. The way the mean field free energy is modified due to the inclusion of density fluctuations suggests how to identify the spinodal curve. Thermodynamic properties and correlation functions of the HCYF are investigated for three values of the inverse Yukawa range: z=1.8, z=4 and z=7 where Monte Carlo simulations are available. The stability of the liquid-vapor critical point with respect to freezing is also studied.