Elastic properties of polycatenane chains and ribbons

TL;DR

The paper addresses how the elastic properties of mechanically interlocked polycatenane chains and ribbons respond to combined force and torque. It employs Monte Carlo simulations of rigid rings to measure extension and twist under controlled pulling and twisting, across free, wall-tethered, and ribbon topologies, including gaps. The work reports robust size-scaling with chain length, a sequence of force-extension regimes that align with the Marko-Siggia wormlike-chain model at high force, and a pronounced, width-dependent increase in torsional stiffness, as well as nuanced effects of gaps. These insights advance understanding of topologically constrained polymers and provide benchmarks for future experiments and simulations with semi-flexible rings.

Abstract

Single-chain elasticity is of fundamental importance in polymer physics, as it underlies many of the unique properties of polymer systems. Recently, there has been interest in characterizing the elastic properties of catenanes, molecular architectures composed of linked molecular rings. To date most studies have focused on the force-extension behavior of polycatenane and catenane dimers. In this study, we employ Monte Carlo computer simulations to investigate the elastic properties of a collection of catenane chains. In addition to polycatenane, we also examine the properties of catenane ribbons constructed by connecting two or three polycatenane chains together with a variable number of side-link rings. After first characterizing the behavior of free polycatenane chains and catenane ribbons, we examine their mechanical response to both an elongational force and a torque applied to the end rings of the chain. We find that the stretching induced by the force is counterbalanced by increasing the torque, which tends to twist the chains and in so doing reduce the extension length. At low torque, the twist angle of the end rings of the chain varies linearly with torque, and the associated torsional spring constant, characterizing the resistance of the chain to twist with the applied torque, tends to increase with stretching force. Relative to polycatenane, ribbons tend to be more elongated at low force and less elongated at strong force. In addition, increasing the ribbon width dramatically increases the torsional stiffness of the chain. Finally, decreasing the degree of side-linking in ribbons tends to decrease slightly the extension length at moderate force and to increase the torsional stiffness for sufficiently large gaps.

Figures (9)

• Figure 1: Illustration of the catenated chains examined in this study. Each chain in the illustration has a length of $N$=41. (a) Polycatenane ($n_{\rm w}=1$). (b) Catenane ribbon of width $n_{\rm w}$=2. (c) Catenane ribbon of width $n_{\rm w}$=3. (d) Catenane ribbon of width $n_{\rm w}$=2 and a gap length of $g$=4. (e) Catenane ribbon of width $n_{\rm w}$=3 and a gap length of $g$=2.
• Figure 2: (a) Snapshot of a polycatenane chain of length $N=39$ with end rings tethered to two parallel hard walls. The end rings are bound to the walls in a perpendicular orientation. The extension length, $R_{{\rm e},x}$, i.e. the distance between the centers of the end rings along the $x$ axis, is labelled. (b) Illustration of the definition of the twist angle, $\phi$, of the end rings. For visual clarity, the end rings shown in the snapshot have the same $y$ and $z$ coordinates, though this is not a constraint imposed in the simulations.
• Figure 3: RMS radius of gyration of catenane chains vs chain length, $N$. Results are shown for chains of width $n_{\rm w}=1$, $n_{\rm w}=2$, and $n_{\rm w}=3$. The solid lines show fits to the power function $\bar{R}_{\rm g}=cN^\nu$. The fits yielded scaling exponents of $\nu=0.65\pm 0.01$ for $n_{\rm w}=1$, $\nu=0.66\pm 0.01$ for $n_{\rm w}=2$, and $\nu=0.696\pm 0.003$ for $n_{\rm w}=3$. The inset shows the tangent-tangent correlation function $C(s)$ for chains of length $N=39$ and width $n_{\rm w}=1$, $n_{\rm w}=2$ and $n_{\rm w}=3$.
• Figure 4: (a) Scaled mean extension length, $\bar{R}_{{\rm e},x}/R_{\rm max}$, vs force applied to the end rings for a polycatenane chain. No confining walls are present, and no torque is applied to the end rings. Results for various values of the chain length, $N$, are shown. The solid line shows the relation $f^{1}$, the scaling expected in the weak force limit for standard polymers absent any confinement. The inset shows the data for $N=159$ and $N=39$ plotted on a semi-log scale. The solid line is a fit of the $N=159$ data to the function $c_0+c_1\ln(f)$ in the range $1\leq f \leq 10$. (b) Orientational order parameters $Q_{\rm cm}$ and $Q_{\rm n}$ (defined in the text) and scaled RMS distance between linked rings $\bar{b}/b_{\rm max}$ vs $f$ for the case of $N=159$. The inset shows $\xi\equiv 1-\bar{b}/b_{\rm max}$ vs $f$ in the high-force limit of $f\geq 100$. The solid line is a fit to the data using a power function $\xi=cf^{-\alpha}$ with $\alpha=0.52\pm 0.03$ and $c=0.66\pm 0.09$.
• Figure 5: (a) Scaled mean extension length vs force for a polycatenane chain confined between two hard walls. No torque is applied to the end rings. Results for various values of the chain length are shown. The dashed line shows the scaling expected for standard polymers in the weak-force limit in the absence of confinement. (b) Scaled mean twist angle, $\langle\phi\rangle/(N-1)$, vs extension force for polycatenane chains of various lengths. In each case, the applied torque is $\tau$=10. The inset shows the data for the unscaled twist angle.