Free energy of singular sticky-sphere clusters

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 $N\le 19$, 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 $N=10$.

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Cited by in corpus (11)

• Sticky Brownian Motion and its Numerical Solution
• From Sticky-Hard-Sphere to Lennard-Jones-Type Clusters
• Physical interpretation of the partition function for colloidal clusters
• Simulating sticky particles: A Monte Carlo method to sample a stratification
• The Gregory-Newton Problem for Kissing Sticky Spheres
• Coupling between long ranged repulsions and short ranged attractions in a colloidal model of zero shear rate viscosity
• Functional convergence to the local time of a sticky diffusion
• From canyons to valleys: Numerically continuing sticky hard sphere clusters to the landscapes of smoother potentials
• Calculating the symmetry number of flexible sphere clusters
• Thermal Fluctuations of Singular Bar-Joint Mechanisms
• Almost-rigidity of frameworks