Buckling of stiff polymer rings in weak spherical confinement

Confinement is a versatile and well-established tool to study the properties of polymers either to understand biological processes or to develop new nanobiomaterials. We investigate the conformations of a semiflexible polymer ring in weak spherical confinement imposed by an impenetrable shell. We develop an analytic argument for the dominating polymer trajectory depending on polymer flexibility considering elastic and entropic contributions. Monte Carlo simulations are performed to assess polymer ring conformations in probability densities and by the shape measures asphericity and nature of asphericity. Comparison of the analytic argument with the mean asphericity and the mean nature of asphericity confirm our reasoning to explain polymer ring conformations in the stiff regime, where elastic response prevails.

Figure 1

(Color online) Dominant shape of a stiff polymer ring without (a) and with (b) spherical confinement. (a) Without confinement the first order bending mode excited by thermal fluctuations induces a planar ellipse that exceeds along its major axis the radius of the corresponding rigid ring. (b) Enclosed by spherical confinement the otherwise planar ellipse is compressed and Euler buckles into a bananalike shape.

Figure 2

(Color online) Buckled rod of length $S$ with maximal height $h_0$. The distance between its hinged ends $2\tilde{R}$ determines the rod trajectory parameterized by $\vartheta \left(s\right)$, $s∊\left[0,S\right]$ and the maximal height $h_0$.

Figure 3

(Color online) Probability density and representative snapshots of polymer rings from Monte Carlo simulation data for the flexibilities $L/l_p=1$, 3 and 8. As indicated by the cartoons on top, the polymer configurations in the first column (A–C) are projected onto the plane spanned by the intermediate ${\Lambda }_2$ and the largest principal axis ${\Lambda }_1$. In the second column polymer configurations (D–F) are projected onto the plane spanned by the smallest ${\Lambda }_3$ and the largest principal axis ${\Lambda }_1$. The gradient in the density of states from high to low is color coded from bright (yellow) to dark (black), the absolute scale of the probability density halves starting from 0.0016 onward as the flexibility increases. The snapshots are chosen such that their asphericity matches the mean configuration of the observed ensemble.

Figure 4

(Color online) Monte Carlo simulation data for the mean asphericity $⟨\Delta ⟩$ and the mean nature of asphericity $⟨\Sigma ⟩$ versus flexibility $L/l_p$ for polymer rings of contour radius $R_c$, that are confined by impenetrable spheres of radii $R=1.0R_c$ to $R=\infty$. Relatively weak confinement already induces dramatic changes in the shape of polymer rings.

Figure 5

(Color online) Comparison of the mean asphericity $⟨\Delta ⟩$ and mean nature of asphericity $⟨\Sigma ⟩$ versus flexibility $L/l_p$ calculated from our analytical description in Eqs. (5, 11) (light colors) and from Monte Carlo simulation data (dark symbols) for polymer rings of contour radius $R_c$ inside spheres of radii $R=1.0R_c$ to $1.3R_c$ (lower red series). For reference data and analytical predictions for a free polymer ring are displayed in each diagram as well (upper blue series).