Scale-free cluster-cluster aggregation during polymer collapse

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.