Effect of Ring Composition on the Statics and Dynamics of Block Copolyelectrolyte Catenanes

Abstract

We use Langevin simulations to study the effect of ring composition on the structure and dynamics of model polycatenanes with copolyelectrolyte rings, each made of one charged and one neutral block. Key observables have a nonmonotonic dependence on ring composition, including the radius of gyration, mechanical bond length, orientational correlations, and rotational relaxation times. Microscopic analysis shows that these nonmonotonicities arise from the competition between electrostatic repulsion, pulling rings apart, and topological constraints, enforcing the proximity of neighboring rings. By locking charged-neutral interfaces at the mechanically bonded regions, this interplay can induce a strong chemical orientational order along the catenane while also hindering the local relaxation dynamics. Chemical orientation defects, manifesting as neutral-neutral interfaces, can emerge too and migrate along the catenane via coupled reorientations of neighboring rings. Our results clarify how ring composition and mechanical bonds can define the properties of topological materials across different scales.

Figure 1

Typical configurations of the considered copolyelectrolyte linear catenanes at different ring compositions. The catenanes are made of n = 12 rings, each of m = 20 monomers, see sketch. The rings are diblock copolyelectrolytes, with one neutral block of m_neu monomers (blue) and a charged one of 20 – m_neu unit-charge monomers (red). Monovalent counterions (gray) ensure the overall charge neutrality of the system.

Figure 2

Gyration radius of the catenane backbone, R_g, as a function of ring composition, m_neu. The inset shows the dependence of the catenane’s gyration radius to end-to-end distance ratio, R_ee/R_g, on the ring composition.

Figure 3

Local metric properties of catenanes as a function of ring composition. The profiled observables are the average ring’s gyration radius, ; the average mechanical bond length, b (defined as the distance between the centers of mass of concatenated rings); and the average distance of minimum approach between monomers of two concatenated rings, d_min, see sketch in panel (a). The shaded band marks the interval 0 ≤ m_neu ≤ 5 where the b and d_min curves exhibit qualitatively distinct behavior compared to longer neutral blocks.

Figure 4

Internal orientational correlation of catenanes as a function of ring composition. The profiled observables involve scalar products of mechanical bond vectors, b̂, and charged-to-neutral vectors, v̂^cn; see sketch in panel (a). Panels (b–d) show the m_neu dependence of the orientational correlation of mechanical bonds and chemical orientation vectors for the same and consecutive rings. Data points represent the average scalar product value, while the shaded band indicates the Q1–Q3 interquartile range, spanning from the 25th (Q1) to the 75th (Q3) percentiles. Panel (e) shows the probability distributions for consecutive charged-to-neutral vectors at different m_neu values.

Figure 5

Orientational correlation defects. Panel (a) shows the m_neu dependence of the orientational correlation (average scalar product) of pairs of charged-to-neutral vectors, v⃗^cn, at various distances along the catenane backbone. Negative values of the average scalar products are due to chemical orientation defects such as the one illustrated in panel (b) for a typical configuration at m_neu = 5. The arrows sketched at the bottom represent the projections of the charged-to-neutral vectors along the end-to-end distance vector of the catenane. The defect separates oppositely oriented runs of the v⃗^cn projections.

Figure 6

Probability distribution of mechanically bonded monomers in a given ring. The heatmaps represent the joint probability distribution of the indices of the two monomers of the central ring in the catenane closest to each neighboring ring, as sketched on the left. The heatmaps are shown for three different values of m_neu, with the neutral (blue) and charged (red) character of the monomers indicated by the colored sidebars. The profile at the top is the marginalized (one-dimensional) probability distribution. The diagonal symmetry of the heatmaps, reflecting the equivalence of the two monomers, was not imposed on the data.

Figure 7

Average number of contacting pair types in the catenane as a function of ring composition. The curves show the m_neu dependence of the average number of neutral–neutral (N–N), charged–charged (C–C), and mixed (C–N or N–C) pairs of contacting monomers in the catenane. The contacts refer to the two closest monomers of concatenated rings. The average numbers of contact types sum to n – 1 = 11 at each m_neu value. The three colored thin curves represent mean-field-like approximations, see main text. For 2 ≤ m_neu ≤ 7, the N–N curves plateau at about the value of 1, marked by the dashed horizontal line.

Figure 8

Characteristic times of global and local relaxation modes as a function of ring composition. The curves show the m_neu dependence of the characteristic rotational times of the entire catenane and one of its rings. The former, , is the characteristic decay time of the orientational correlation function of the catenane end-to-end vector. The latter, is the characteristic decay time of the orientational correlation function of the ring’s diameter vectors, averaged over all diameters.

Figure 9

Dynamics of chemical orientation defects. The traces in panel (a) represent the time evolution of the orientational correlation (scalar product) of consecutive charged-to-neutral vectors, and one of the two corresponding mechanical bonds b̂_i. The data are for a stretch of a few consecutive rings at m_neu = 5. The configurations corresponding to the two selected times (dashed lines) are shown in panel (b) and illustrate the hopping of the defect across neighboring rings.