Circular Polycatenanes: Supramolecular Structures with Topologically Tunable Properties

Polycatenanes, macrochains of topologically interlocked rings with unique physical properties have recently gained considerable interest in supramolecular chemistry, biology, and soft matter. Most of the work has been, so far, focused on linear chains and on their variety of conformational properties compared to standard polymers. Here we go beyond the linear case and show that, by circularizing such macrochains, one can exploit the topology of the local interlockings to store twist in the system, significantly altering its metric and local properties. Moreover, by properly defining the twist (Tw) and writhe (Wr) of these macrorings we show the validity of a relation equivalent to the Călugăreanu-White-Fuller theorem $Tw+Wr=\mathrm{const}$, originally proved for ribbonlike structures such as double stranded DNA. Our results suggest that circular polycatenanes with storable and tunable twist can form a new category of highly designable multiscale structures with potential applications in supramolecular chemistry and material science.

Figure 1

(a) 0-Twist reference configuration. In the center, we show the twist angles between consecutive normals. The dots inside the rings indicate that all their normals point in the direction $z>0$. (b) The circular polycatenane resulting from detaching ring $n$ from ring $n-1$ and closing it switching crossings. (c) The same configuration can be obtained by detaching ring $n$ from ring 1 and flipping it by half a turn before closing the catenane again. The $x$ indicates that the normal of ring $n$ now points below the plane. (d) A circular polycatenane obtained after four clockwise flips of the $n$th ring, followed by two counterclockwise flips. As explained in the text, all reference configurations can also be obtained by switching the links on even rings, allowing us to calculate the topological invariant $n_{tw}=n_{+}-n_{-}$. The configuration (d) is equivalent to a catenane with $n_{+}=2$ red rings and no yellow rings, as explained in the Supplemental Material [40]. (e) The stiffness of the rings allows us to map them on their centers ${\mathbf{R}}_i$ and normal vectors ${\widehat{\mathbf{N}}}_i$. Notice how the vectors ${\widehat{\mathbf{N}}}_i$ need not to be perpendicular to the backbone vectors ${\mathbf{T}}_i$.

Figure 2

Typical configurations for $n=20,300$ rings with $n_{tw}=0$ (left column) and $n_{tw}=0.5n$ (right column). The latter are visibly more crumpled. The scale of the snapshot is preserved at fixed values of $n$. The color map highlights the sequence of elementary rings along the backbone.

Figure 3

(a) $\mathrm{\Delta }{C}_n(d,n_{tw}/n)/C_n(d,0)$ as a function of the index distance $d$ along the backbone, for six different values of $n_{tw}/n$. (b) Squared radius of gyration as a function of $n$ and $n_{tw}/n$. $R_g$ is in units of $D=m/\pi$. (c) Scaling of the absolute value of the writhe $|Wr|$ with $n$, for different values of $n_{tw}/n$. (d) $⟨Tw+Wr⟩$ for different values of $n_{tw}$ and $n$. Results for $n=20$ include different aspect ratios $p$. The error bars correspond to the standard deviation of the reported values.