Scale-Free Cluster–Cluster Aggregation during Polymer Collapse

An extended polymer collapses to form a globule upon being 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 the growth of the average cluster size Cs, eventually leading to a single globule. The aggregation of the clusters is known to be analogous to droplet coalescence. This suggests a striking resemblance between such aggregation and cluster–cluster aggregation found in many particle systems, like colloidal self-assembly, typically characterized by a universal dynamic scaling behavior. Motivated by this similarity, we verify the presence of such dynamic scaling during the collapse of a polymer with varying bending stiffness κ, using molecular dynamics simulations. We probe the dynamics via the time (t) evolution of the size distribution of clusters Ns(t) and the growth of clusters Cs(t). Irrespective of κ, we observe the power-law scalings Cs(t) ∼ tz and Ns(t) ∼ t–ws–τ, of which only the cluster growth is universal with z ≈ 1.67. Importantly, our results indeed show that Ns(t) exhibits a dynamic scaling of the form Ns(t) ∼ s–2f(s/tz), indicative of 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 κ ≥ 5, the exponents show deviation from this relation, which grows continuously with κ. We identify differences in the local structures of the clusters formed, leading to variations in the cluster-size dependence of the effective diffusion constant, as the origin of the above deviation. We also discuss potential experimental strategies to directly visualize the observed dynamic scaling in a collapsing polymer.