The Hierarchical Reference Theory as applied to square well fluids of variable range

TL;DR

This paper investigates the numerical properties of the Hierarchical Reference Theory applied to square well fluids with variable range, highlighting challenges in numerical stability and boundary conditions, especially near critical points.

Contribution

It provides an analysis of the numerical challenges and limitations of the Hierarchical Reference Theory for square well fluids, especially for narrow wells and near critical points.

Findings

Agreement with simulations improves for lambda > 2

Numerical stiffness hampers critical point calculations for narrow wells

Inappropriate boundary conditions can adversely affect results

Abstract

Continuing our investigation into the numerical properties of the Hierarchical Reference Theory, we study the square well fluid of range lambda from slightly above unity up to 3.6. After briefly touching upon the core condition and the related decoupling assumption necessary for numerical calculations, we shed some light on the way an inappropriate choice of the boundary condition imposed at high density may adversely affect the numerical results; we also discuss the problem of the partial differential equation becoming stiff for close-to-critical and sub-critical temperatures. While agreement of the theory's predictions with simulational and purely theoretical studies of the square well system is generally satisfactory for lambda greater than about 2, the combination of stiffness and the closure chosen is found to render the critical point numerically inaccessible in the current formulation of the theory for most of the systems with narrower wells. The mechanism responsible for some deficiencies is illuminated at least partially and allows us to conclude that the specific difficulties encountered for square wells are not likely to resurface for continuous potentials.