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.