Semiflexible Polymer Confined to a Spherical Surface

We develop a formalism for describing the kinematics of a wormlike chain confined to the surface of a sphere that simultaneously satisfies the spherical confinement and the inextensibility of the chain contour. We use this formalism to study the statistical behavior of the wormlike chain on a spherical surface. In particular, we provide an exact, closed-form expression for the mean square end-to-end distance that is valid for any value of chain length $L$, persistence length $l_p$, and sphere radius $R$. We predict two qualitatively different behaviors for a long polymer depending on the ratio $R/l_p$. For $R/l_p>4$, the mean square end-to-end distance increases monotonically with the chain length, whereas for $R/l_p<4$, a damped oscillatory behavior is predicted.

Figure 1

The conformation of a polymer chain wrapping around a unit sphere with a constant curvature that orbits after ten revolutions around the sphere.

Figure 2

The mean square end-to-end distance of a polymer chain confined to a sphere with $R/l_p=40$.

Figure 3

The mean square end-to-end distance versus the polymer chain length for $R/l_p=1/10$ (solid curve), $R/l_p=1/4$ (dashed curve), $R/l_p=2/5$ (dash-dotted curve), and $R/l_p=11/20$ (dotted curve).