Free energy of singular sticky-sphere clusters

TL;DR

This paper develops a theoretical framework to calculate the free energy of singular sticky-sphere clusters, revealing how order emerges in large systems despite the dominance of hyperstatic clusters.

Contribution

It introduces a method to compute the free energy of singular clusters in the sticky limit, accounting for divergences and linking cluster properties to system order.

Findings

Asymptotic free energy depends on potential depth and range.

Hyperstatic clusters dominate the cluster landscape.

Order emerges from the prevalence of close-packed lattice fragments.

Abstract

Networks of particles connected by springs model many condensed-matter systems, from colloids interacting with a short-range potential, to complex fluids near jamming, to self-assembled lattices, to origami-inspired materials. Under small thermal fluctuations the vibrational entropy of a ground state is given by the harmonic approximation if it has no zero-frequency vibrational modes, yet such singular modes are at the epicenter of many interesting behaviors in the systems above. We consider a system of $N$ spherical particles, and directly account for the singularities that arise in the sticky limit where the pairwise interaction is strong and short ranged. Although the contribution to the partition function from singular clusters diverges in the limit, its asymptotic value can be calculated and depends on only two parameters, characterizing the depth and range of the potential. The result holds for systems that are second-order rigid, a geometric characterization that describes all known ground-state (rigid) sticky clusters. To illustrate our theory we address the question of emergence: how does crystalline order arise in large systems when it is strongly disfavored in small ones? We calculate the partition functions of all known rigid clusters up to $N\leq 19$, and show the cluster landscape is dominated by hyperstatic clusters (those with more than $3N-6$ contacts) -- singular and isostatic clusters are far less frequent, despite their extra vibrational and configurational entropies. Since the most hyperstatic clusters are close to fragments of a close-packed lattice, this underlies the emergence of order in sticky-sphere systems, even those as small as $N=10$.