Ring polymers in two-dimensional melts double-fold around randomly branching "primitive shapes"

TL;DR

The paper tests the Khokhlov–Nechaev–Rubinstein scenario by reconstructing the primitive shapes of unknotted, non-concatenated ring polymers in two dimensions and treating these shapes as branched trees. Using a kinetic Monte Carlo lattice model with stored length on a triangular lattice, the authors define an algorithm to extract loop-free primitive shapes and analyze them with observables and distribution functions from 2D lattice-tree theory, including Redner–Des Cloizeaux forms. They find that the ring primitive shapes exhibit 2D branched-polymer statistics across stiffnesses, with consistent scaling exponents and universal distribution shapes, thereby providing direct numerical support for the tree-like folding picture. Dynamics, however, reveal faster relaxation than predicted by branched-polymer theory, suggesting unique 2D effects and motivating future work to extend the approach to three dimensions and to explore threading phenomena. Overall, the work delivers a robust numerical protocol for connecting ring conformations to branched-polymer physics and clarifies the static versus dynamic aspects of ring melts in 2D.

Abstract

Drawing inspiration from the concept of the "primitive path" of a linear chain in melt conditions, we introduce here a numerical protocol which allows us to detect, in an unambiguous manner, the "primitive shapes" of ring polymers in two-dimensional melts. Then, by analysing the conformational properties of these primitive shapes, we demonstrate that they conform to the statistics of two-dimensional branched polymers (or, trees) in the same melt conditions, in agreement with seminal theoretical work by Khokhlov, Nechaev and Rubinstein. Results for polymer dynamics in light of the branched nature of the rings are also presented and discussed.

Figures (14)

• Figure 1: (a) Classical illustration of the primitive path (dashed line) of a linear chain (solid line) in an entangled polymer melt. By representing the effects of the constraints exerted by neighboring chains as a network of uncrossable topological obstacles ($\times$), the primitive path is identified as the shortest path which the polymer chain can be contracted into, at fixed chain ends and without crossing the obstacles. (b) In a melt of unknotted and non-concatenated rings, the equivalent topological barriers induce each ring to double-fold around a randomly branching path.
• Figure 2: Two-dimensional illustration of the lattice polymer model and kinetic Monte Carlo moves. Monomers (filled dots) occupy the spatial positions of the regular triangular lattice (empty dots) and two nearest neighbor monomers are connected by a black line representing the polymer bond between them. For two nearest neighbor monomers occupying the same lattice site the bond that connects them (represented by an arc) represents a unit of "stored length". Lattice positions connected by the double arrows are examples of allowed MC moves: (i) a unit of stored length unfolding to a normal bond, (ii) a bond folding into a unit of stored length, (iii) a Rouse-like move. Lattice positions connected by the double arrows with the cross are examples of forbidden MC moves: (iv) a monomer moving to the left that would occupy a lattice position with two monomers already present, (v) two non-nearest neighbor monomers on the same lattice site violating the excluded volume constraint.
• Figure 3: (a) Illustration of the $2d$ triangular lattice (orange dots, with $6$ unit cells drawn explicitly) of unit step $=a$ where ring polymers are simulated, along with the dual honeycomb lattice (blue dots, with $1$ cell drawn explicitly) of unit step $=a/\sqrt{3}$ where the corresponding trees reside. (b) Snapshots of ring melt configurations for $N_{\rm ring} =1280$ and $\kappa_{\rm bend} = 0, 1, 1.5$ (from left to right, see legend). (c) Examples of single ring polymers (blue/green/red circular contours) isolated from their corresponding melts, along with their primitive tree-like backbones (black lines inside each circular contour). Notice that the trees contain a certain amount of loops. (d) Final, loop-less, primitive shapes obtained by randomly removing one bond in each of the former loops.
• Figure 4: Conformational properties of trees: observables (symbols) and asymptotic power-law behaviors (dashed lines). (a) $\langle L\rangle \sim \langle N_{\rm tree}\rangle^{\rho}$, mean path length as a function of the mean tree weight $\langle N_{\rm tree}\rangle$. (b) $\langle N_{\rm br}\rangle \sim \langle N_{\rm tree}\rangle^{\epsilon}$, mean branch weight as a function of the mean tree weight $\langle N_{\rm tree}\rangle$. (c) $\langle R_{\rm path}^2\rangle \sim \langle L\rangle^{2\nu_{\rm path}}$, mean-square end-to-end spatial distance of paths of length $=\langle L\rangle$. (d) $\langle R_g^2\rangle \sim \langle N_{\rm tree}\rangle^{2\nu}$, mean-square gyration radius as a function of the mean tree weight $\langle N_{\rm tree}\rangle$. In each panel the dashed lines express the interval of possible values for the exponent, lying between the minimum lower-bound and the maximum upper-bound for all $\kappa_{\rm bend}$ (see Table \ref{['tab:ExpSummary-Obs']}). (Insets) Universal scaling plots for the observables. Here, the value of the exponent used in each plot corresponds to the average of the estimated best values for individual $\kappa_{\rm bend}$ (see Table \ref{['tab:ExpSummary-Obs']}).
• Figure 5: Conformational properties of trees: $p_{\langle N_{\rm tree}\rangle}(\ell)$, distribution functions of linear paths of length $\ell$. The dashed line of each panel corresponds to the predicted RdC functional form for trees, Eqs. \ref{['eq:q_RdC_path']}-\ref{['eq:RdC_K']}, with the exponents $\theta_{\ell}$ and $t_{\ell}$ as in Table \ref{['tab:ExpSummary-PDFs']}. Data of different colors denote different mean tree weight $\langle N_{\rm tree}\rangle$ (see legend).