Free energy of singular sticky-sphere clusters

Area of Science:

• Condensed Matter Physics
• Statistical Mechanics
• Materials Science

Background:

• Particle networks with spring-like interactions model diverse systems like colloids and metamaterials.
• Zero-frequency vibrational modes (singularities) are crucial for system behavior, deviating from harmonic approximations.
• Understanding these singularities is key to predicting emergent properties in particle systems.

Purpose of the Study:

• To develop a theory for sticky particle systems that accounts for singularities in the sticky limit.
• To investigate the emergence of crystalline order from small- to large-scale particle assemblies.
• To determine the factors governing the stability and properties of sticky particle clusters.

Main Methods:

• Directly accounting for singularities in the sticky limit of pairwise interactions.
• Calculating asymptotic values of partition functions for singular clusters.
• Analyzing the geometric characterization of rigidity (second-order rigidity).
• Computing partition functions for rigid clusters up to N=21 particles.

Main Results:

• The asymptotic partition function of singular clusters depends on potential depth and range.
• Hyperstatic clusters (with >3N-6 contacts) dominate the cluster landscape for N≤21.
• Singular and isostatic clusters are less frequent than hyperstatic ones, despite entropic advantages.
• Hyperstatic clusters resemble fragments of close-packed lattices.

Conclusions:

• The theory provides a framework for understanding sticky particle systems, including their emergent order.
• The prevalence of hyperstatic clusters explains the emergence of crystalline order in sticky-sphere systems.
• This work elucidates how local interactions lead to global order, even in small systems (N=10).