Why do ultrasoft repulsive particles cluster and crystallize? Analytical results from density functional theory

TL;DR

This paper uses density functional theory to analytically explain why ultrasoft repulsive particles form clusters and crystals, revealing universal properties and linking microscopic interactions to macroscopic phase behavior.

Contribution

It provides an analytical framework connecting the Fourier transform of pair potentials to cluster formation and crystallization in ultrasoft particle systems.

Findings

Stable crystals have a density-independent lattice constant set by the potential's Fourier minimum.

All cluster crystals exhibit a universal Lindemann ratio of 0.189 at freezing.

Clusters emerge due to negative Fourier components of the interaction potential.

Abstract

We demonstrate the accuracy of the hypernetted chain closure and of the mean-field approximation for the calculation of the fluid-state properties of systems interacting by means of bounded and positive-definite pair potentials with oscillating Fourier transforms. Subsequently, we prove the validity of a bilinear, random-phase density functional for arbitrary inhomogeneous phases of the same systems. On the basis of this functional, we calculate analytically the freezing parameters of the latter. We demonstrate explicitly that the stable crystals feature a lattice constant that is independent of density and whose value is dictated by the position of the negative minimum of the Fourier transform of the pair potential. This property is equivalent with the existence of clusters, whose population scales proportionally to the density. We establish that regardless of the form of the interaction potential and of the location on the freezing line, all cluster crystals have a universal Lindemann ratio L = 0.189 at freezing. We further make an explicit link between the aforementioned density functional and the harmonic theory of crystals. This allows us to establish an equivalence between the emergence of clusters and the existence of negative Fourier components of the interaction potential. Finally, we make a connection between the class of models at hand and the system of infinite-dimensional hard spheres, when the limits of interaction steepness and space dimension are both taken to infinity in a particularly described fashion.