# Minimal energy packings and collapse of sticky tangent hard-sphere polymers

Minimal energy packings and collapse of sticky tangent hard-sphere polymers. Hoy R.S. and O'Hern C.S. (2010). Phys. Rev. Lett. 105, 068001.

Leon Furchtgott, APP 225. Fall 2010.

## Summary/Introduction

This paper is related to work investigating The Free Energy Landscape of Hard Sphere Clusters (experimental) and Minimal Energy Clusters of Hard Spheres with Short Ranged Attractions (theoretical). In these papers, Arkus et al. were interested in enumerating minimal energy packings (MEPs) for 10 sticky hard spheres and predicting and measuring the probabilities of different configurations.

This paper looks at packings of sticky tangent hard-sphere polymers, which can model polymer collapse, protein folding and protein interactions. This is particularly interesting and different from the case of sticky hard spheres because of the addition of covalent-bond and chain uncrossability constraints.

This paper looks at 2 overarching questions. (1) How do the probabilities for obtaining polymer MEPs differ from those for sticky hard-sphere MEPs? (2) How do the properties of single compact polymers depend on collapse dynamics?

(1) Polymer constraints reduce the number of MEP configurations. This is because of *dividing surfaces* which split the packings into disjoint regions.

(2) For slow temperature quenches, single chains undergo a sharp transition to crystallites with a close-packed core. As the quench rate increases, the core size decreases and the exterior becomes more disordered.

## MEPs for polymers

The authors use Arkus et al's geometric and graph-theory techniques to generate all of the macrostates and microstates for a given <math>N</math> number of spheres and <math>N_c</math> number of contacts. They start by enumerating adjacency matrices satisfying hard-sphere and minimal rigidity conditions, as well as the proper polymeric covalent-bond constraints.

Given a valid adjacency matrix, they solve the system of quadratic equations <math>|r_i - r_j|^2 = d^2_{ij}</math> for the sphere positions <math>r_i</math>.

They thus obtain a set of macrostates, each characterized by an adjacency matrix that is nonisomorphic, and a unique set of interparticle distances -- each macrostate is distinct and cannot be obtained through rotation or reflection from another macrostate.

Each macrostate corresponds to several microstates, which, for polymers, correspond to multiple possible paths through a given macrostate.

The authors then examine the fraction <math>f</math> of adjacency matrices with <math>N_c</math> contacts that satisfies hard-sphere constraints and minimal rigidity, and they compare this fraction for monomers <math>f_m</math>, linear polymers <math>f_p</math> and polymer rings <math>f_r</math>. They find that hard-sphere constraints are more difficult to satisfy for polymers than for monomers: <math>f_r < f_p < f_m</math>.

This is because of the occurrence of *dividing surfaces* in polymer packings. A dividing surface is a minimal subset of a connected cluster of contacting monomers that geometrically splits it into two. If monomers J and K are split by S, then any path that starts in J cannot cross S unless it passes through all of the monomers in S. This greatly reduces the number of possible arrangements. (See figure 1 in the paper -- *I could not find jpgs of the figures so I am just referring to them without reproducing them into the article. Help appreciated.*).

Entropy suppression via blocking is strongest for structures of both high and low symmetry, therefore polymer MEPs favor intermediate structural symmetry. This is unlike monomer MEPs, which favor low structural symmetry for entropic reasons.

## Collapse of Sticky Tangent Hard-Sphere Polymers

The authors use molecular dynamics (MD) simulations to look at polymer dynamics during temperature quenches, as the polymer is cooled at different rates.

The MD parameters are in the paper and are straightforward.

As Figure 2 in the paper shows, the polymer exhibits very different behaviors depending on the quench rate <math>\dot T</math>. For low <math>\dot T</math>, there is a sharp transition between the random-coil configuration and the ordered crystallite configurations, which occurs at <math>T_{melt} = 0.37 \epsilon / k_B</math>. Both the number of particles with 12 contacts (<math>N_{cp}</math>) and the total number of contacts <math>N_c</math> undergo a sharp transition at this point.

For high <math>\dot T</math>, there are glassy dynamics near <math>T_{melt}</math>, and <math>N_{cp}</math> and <math>N_{c}</math> remain low even when T approaches 0. For <math>k_B \dot T \tau / \epsilon < 10^{-7} </math> the system has the characteristics of a first-order transition. For <math>k_B \dot T \tau / \epsilon > 10^{-3} </math>, the systems do not form minimally rigid clusters even at T = 0.

Figure 3 in the paper shows the effects of the quench rates on the end states. For low quench rates, the end state is a crystallite with a large close-packed core and a fairly crystalline exterior. For high quench rates, there is a small close-packed core with a large disordered exterior. The size and disorder of the exterior surface increases with <math>\dot T</math>.