Scale-free cluster-cluster aggregation during polymer collapse

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 $κ$, 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 $κ$, we observe the power-law scalings $C_s(t)\sim t^z$ and $N_s(t)\sim t^{-w} s^{-τ}$, 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 $κ\ge 5$, the exponents show deviation from this relation, which grows continuously with $κ$. 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.

Figures (13)

• Figure 1: Collapse of a flexible polymer. Representative snapshots at different times and their corresponding contact maps illustrating the sequence of events during the collapse of a flexible polymer of length $N=2048$. The first frame shows the nucleation of small clusters or "pearl" along the polymer chain, indicated by development of monomer-monomer contacts (dark blue region). The second and third frames reflect growth of clusters with long (contour) distance contacts. The final frame represents the globular state with contacts spanning the whole chain.
• Figure 2: Cluster-cluster aggregation in flexible polymers. Time evolution of the average number of clusters $\langle n_c\rangle$ for three different chain lengths. Profiles of $\langle n_c\rangle$ show a plateau at early times (blue background) followed by a monotonic decay (green background), consistent with the pear-necklace phenomenological picture.
• Figure 3: Cluster growth in flexible polymers. (a) Growth of the average cluster size $\langle C_s(t)\rangle$ during the collapse of polymers of three different lengths $N$. The dashed line shows the consistency of the data with a power-law growth with an exponent $z=1.67$. (b) Plots of the scaling functions $Y(y)$, demonstrating the collapse of data for different $N$ through a finite-size scaling analysis providing an unambiguous estimate of $z$. In both (a) and (b) the data are shown on a double-log scale.
• Figure 4: Kinetics of clusters of fixed sizes in flexible polymers. Time dependence of the number $N_s$ of clusters of fixed size $s$ on a double-log scale, during the collapse. Plots for multiple $s$ across different $N$ are included. The abscissa is scaled with $N^{1/z}$ for a better visualization. $N_s(t)$ display both monotonic and bell-shaped profiles. The dashed line shows the consistency of the data with the power-law decay.
• Figure 5: Verification of dynamic scaling in a flexible polymer. Double-log plots of the scaling function $f(x)\equiv s^2N_s(t)$ against the scaling variable $x\equiv s/t^z$ at different times for a polymer of length $N=2048$. The dashed black line shows the consistency of the data with the power law $\sim x^2$ in the regime $x\ll1$. Collapse of data at different times confirms the presence of colloid-like dynamic scaling during flexible polymer collapse.