Shapes of Semiflexible Polymer Rings

The shape of semiflexible polymer rings is studied over their whole range of flexibility. Investigating the joint distribution of asphericity and the nature of asphericity as well as their respective averages, we find two distinct shape regimes depending on the flexibility of the polymer. For a small perimeter to persistence length the fluctuating rings exhibit only planar, elliptical configurations. At higher flexibilities three-dimensional, crumpled structures arise. Analytic calculations confirm the qualitative behavior of the averaged shape parameters and the elliptical shape in the stiff regime.

Figure 1

The overall shape of polymer configurations depending on the asphericity $\Delta$ and the nature of asphericity $\Sigma$. Along the solid line the structures are planar; configurations beyond that line are excluded as they do not correspond to real structures. The diamond indicates the peak in the distribution of flexible open polymers. The sequences of circles and triangles denote the mean shapes of open and ring polymers at integer flexibilities starting from $L/l_p=0$ at $(0,2)$ and at $(\pi /3,1)$, respectively.

Figure 2

The distribution of asphericity $\Delta$ and the nature of asphericity $\Sigma$ at different levels of flexibility. For tight rings, $L/l_p<5$, planar conformations dominate, while the configurations become truly 3D beyond $L/l_p$ of the order of 10. The crosses indicate mean and variance of each single shape parameter deviating from the most probable state. Note the change of the color scaling as the distribution spreads out.

Figure 3

Monte Carlo simulation data for the mean asphericity $⟨\Delta ⟩$ and the mean nature of asphericity $⟨\Sigma ⟩$ versus increasing flexibility $L/l_p$. Both grow linearly with raising flexibility for tight rings (solid line). Then, $⟨\Sigma ⟩$ saturates at a prolate shape, while the asphericity decreases in a power law (dashed line). Error bars are of the size of the symbols.