Scale-free cluster-cluster aggregation during polymer collapse

TL;DR

This study investigates the universal dynamic scaling behavior during polymer collapse, revealing scale-free cluster growth characterized by power-law dynamics and how stiffness influences the aggregation exponents.

Contribution

It demonstrates the presence of universal dynamic scaling in polymer collapse and explores how bending stiffness affects the scaling exponents and cluster structures.

Findings

Power-law scaling of cluster size and number distributions.

Universal cluster growth with exponent z ≈ 1.67.

Deviations in scaling relations for stiffer polymers.

Abstract

An extended polymer collapses to form a globule when subjected to a quench below the collapse transition temperature. The process begins with the formation of clusters of monomers or ``pearls''. The nascent clusters merge, resulting in growth of the average cluster size $C_s$, eventually leading to a single globule. The aggregation of the clusters are known to be analogous to droplet coalescence. This suggests a striking resemblance between such an aggregation and cluster-cluster aggregation found in many {particle systems}, like in colloidal self-assembly, typically characterized by a universal dynamic scaling behavior. Motivated by that, here, we verify the presence of such dynamic scaling during the collapse of a polymer with varying bending stiffness $\kappa$, using molecular dynamic simulations. We probe the dynamics via time evolution of the size distribution of clusters $N_s(t)$ and growth of $C_s(t)$. Irrespective of $\kappa$, we observe the power-law scalings $C_s(t)\sim t^z$ and $N_s(t)\sim t^{-w} s^{-\tau}$, of which only the cluster growth is universal with {$z\approx 1.67$.} Importantly, our results indeed show that $N_s(t)$ exhibits a dynamic scaling of the form $N_s(t)\sim s^{-2}f(s/t^z)$, indicative of a scale-free cluster growth. Interestingly, for flexible and weakly stiff polymers the dynamic exponents obey the relation $w=2z$, as also found in diffusion-controlled cluster-cluster aggregation of particles. For $\kappa \ge 5$, the exponents show deviation from this relation, which grows continuously with $\kappa$. We identify the differences in local structures of the clusters formed, leading to variations in cluster-size dependence of the effective diffusion constant to be the origin of the above deviation. We also discuss potential experimental strategies to directly visualize the observed dynamic scaling in a collapsing polymer.