Collapse or Swelling Dynamics of Homopolymer Rings: Self-consistent Hartree approach

TL;DR

This paper models the collapse and swelling dynamics of homopolymer rings using a self-consistent Hartree approach, deriving equations of motion and analyzing the process through numerical solutions and scaling arguments.

Contribution

It introduces a systematic derivation of the monomer correlation dynamics for homopolymer rings using the Martin-Siggia-Rose technique and Hartree approximation, providing insights into collapse and swelling behaviors.

Findings

Collapse time scales as t* ~ g/τ for segment g.

Numerical results align with scaling arguments.

Swelling modeled as homogeneous expansion neglecting topological effects.

Abstract

We investigate by the use of the Martin - Siggia - Rose generating functional technique and the self - consistent Hartree approximation, the dynamics of the ring homopolymer collapse (swelling) following an instantaneous change into a poor (good) solvent conditions.The equation of motion for the time dependent monomer - to - monomer correlation function is systematically derived. It is argued that for describing of the coarse - graining process (which neglects the capillary instability and the coalescence of ``pearls'') the Rouse mode representation is very helpful, so that the resulting equations of motion can be simply solved numerically. In the case of the collapse this solution is analyzed in the framework of the hierarchically crumpled fractal picture, with crumples of successively growing scale along the chain. The presented numerical results are in line with the corresponding simple scaling argumentation which in particular shows that the characteristic collapse time of a segment of length $g$ scales as $t^* \sim \zeta_0 g/\tau$ (where $\zeta_0$ is a bare friction coefficient and $\tau$ is a depth of quench). In contrast to the collapse the globule swelling can be seen (in the case that topological effects are neglected) as a homogeneous expansion of the globule interior. The swelling of each Rouse mode as well as gyration radius $R_g$ is discussed.