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.

伸展态聚合物在低于塌缩转变温度的条件下经受骤冷处理时，会发生塌缩并形成球状体。该过程始于单体团簇（或称“珠状”聚集体）的形成。
新生的团簇相互融合，使得平均团簇尺寸Cs持续增长，最终形成单一的球状体。已知此类团簇聚集过程与液滴聚并(droplet coalescence)具有相似性，这表明该聚合物塌缩过程中的团簇聚集，与诸多粒子体系（如胶体自组装(colloidal self-assembly)）中常见的团簇-团簇聚集(cluster–cluster aggregation)存在显著相似性，而这类粒子体系通常表现出普适动态标度行为(universal dynamic scaling behavior)。
受这一相似性的启发，我们通过分子动力学模拟(Molecular Dynamics Simulations)，验证了弯曲刚度κ可变的聚合物在塌缩过程中此类动态标度行为的存在。我们通过团簇尺寸分布Ns(t)随时间t的演化，以及团簇平均尺寸Cs(t)的增长来探究该动力学过程。无论弯曲刚度κ的取值如何，我们均观测到两类幂律标度(power-law scalings)关系：Cs(t) ~ t^z 与 Ns(t) ~ t^(-w)s^(-τ)，其中仅团簇增长的标度指数z≈1.67具有普适性。重要的是，我们的结果证实Ns(t)符合如下形式的动态标度：Ns(t) ~ s^(-2)f(s/t^z)，这表明团簇增长具有无标度(scale-free)特性。
有趣的是，对于柔性及弱刚性聚合物，其动态指数满足w=2z的关系，这与扩散控制型(diffusion-controlled)粒子团簇-团簇聚集体系中观测到的结果一致。而当κ≥5时，指数关系会偏离该式，且偏离程度随κ的增大持续加剧。
我们将上述偏离的根源归结为：所形成团簇的局部结构存在差异，导致有效扩散系数(effective diffusion constant)的团簇尺寸依赖性发生变化。最后，我们还讨论了可直接观测聚合物塌缩过程中动态标度行为的潜在实验策略。