Mixing, segregation, and collapse transitions of interacting copolymer rings

TL;DR

This work introduces a lattice dumbbell model for two interacting ring polymers with mutual contacts and intra-block attractions, forming a phase diagram in the $(\\beta_c,\\beta_m)$ plane that features segregated-expanded (SE), segregated-collapsed (SC), and mixed (M) phases. It combines rigorous results on the existence and shape of phase boundaries with extensive Monte Carlo simulations to locate transitions, quantify thermodynamic and geometric observables, and examine finite-size effects. A key contribution is the detailed analysis of topological entanglement across phases, showing that linking is enhanced in the mixed phase while knotting tends to rise in the segregated-collapsed regime, with specific markers computed via the two-variable Alexander polynomial and Dowker codes. The findings illuminate how mixing and collapse compete in diblock ring polymers and underscore the role of topology as a diagnostic of phase structure in polymeric systems, with potential relevance to catenanes and related materials."

Abstract

A system of two self and mutual interacting ring polymers, close together in space, can display several competing equilibrium phases and phase transitions. Using Monte Carlo simulations and combinatorial arguments on a corresponding lattice model, we determine three equilibrium phases, two in which the rings segregate in space and are either extended (the segregated-expanded phase) or compact (the segregated-collapsed phase). The third is a mixed phase where the rings interpenetrate. The corresponding phase boundaries are located numerically and their critical nature is discussed. Finally, by looking at the topological properties of the three phases, we show that the two rings are likely to be linked in the mixed phase and knotted in the segregated-collapsed phase.

Figures (16)

• Figure 1: A cubic lattice model of an $AB$-diblock catenane with an $A$-block (red) and a $B$-block (blue). We implement this model by sampling lattice polygons while keeping the two polygons near each other. This is done by having at least one vertex in the $A$-block a unit distance from a vertex in the $B$-block. A pair of such vertices are marked with the solid line in the figure. The two polygons can be linked (as in this figure) to model a catenane.
• Figure 2: A hypothetical sketch of the model's expected phase diagram. There are at least three possible phases of mixed(M) or expanded(E)-collapsed(C) copolymer. The segregated-expanded (SE), segregated-collapsed (SC) and mixed (M) phases are explored in this paper.
• Figure 3: Examples of equilibrium configurations of $2n=800$ sampled at $\beta_c=0,\beta_m=0$ (SE), $\beta_c=0.5,\beta_m=0$ (SC) and $\beta_c=0,\beta_m=0.75$ (M).
• Figure 4: Crossing the Mixed-SC boundary either by keeping fixed $\beta_c >\beta_c^*$ and (a) varying $\beta_m$ or by (b) keeping fixed $\beta_m>\beta_m^*$ and varying $\beta_c$.
• Figure 5: (a1-a2) The average number of self contacts scaled by length, $\langle k_c \rangle / 2n$, as a function of $\beta_c$. Different symbols colours correspond to different system sizes (see the legends). (b1-b2) Plots of the corresponding variance of $k_c$ scaled by the system's size, $Var(k_c)/2n$, as a function of $\beta_c$. (c1-c2) Corresponding plots of the average number of mutual contacts scaled by $2n$, $\langle k_m \rangle / 2n$, as a function of $\beta_c$. (d1-d2) Plots of the mean-squared radius of gyration of the single components, $\langle R_g^2\rangle$, scaled by $n^{2/3}$, as a function of $\beta_c$. The two columns correspond to simulations performed at the fixed values of $\beta_m=0$ and $0.30$ respectively.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2