Past exam of the soft condensed matter course of the University of Cambridge / 2013 / Paper 1 / 4 / Solution

Use the Monge representation $\mathbf{r}(x,y)=(x,y,h(x,y))$ for a single-valued surface without overhangs. Its area element is $dA=\sqrt{1+|\nabla h{|}^2}dxdy$; at small slopes, $dA\simeq [1+|\nabla h{|}^2/2]dxdy$ and twice the mean curvature is $2\mathcal{H}\simeq {\nabla }^{2}h$, up to orientation sign. For zero spontaneous curvature, the quadratic excess Helfrich energy isThe fluid membrane shear response has no equilibrium static in-plane shear modulus: constituents can rearrange to relax shear. Bending and isotropic tension still cost energy; viscous shear stresses can matter during motion. A Gaussian curvature term is omitted under the stated approximation; for fixed closed-surface topology its integral is constant, but boundaries and topology changes require care. The small-slope Monge expression describes weak deformations, not an entire fully wrapped sphere.^[ref]

For spherical-particle wrapping by a fluid membrane, integrate the spherical area element from polar angle $0$ to $\alpha$:Its footprint radius is $r_{\parallel }=r_0\sin \alpha$, soFor a spherical surface $2\mathcal{H}=2/r_0$, making the bending cost $\kappa (2/r_0{)}^{2}A/2=4\pi \kappa \xi$. adhesion contributes $-2\pi w{r}_0^2\xi$, and tension contributes $\sigma \Delta A$. ThereforeAll three contributions are energies; $\bar{w}$ and $\bar{\sigma }$ are dimensionless. This is the specified bound-patch model: deformation of the free membrane and the neck is omitted.

The allowed interval $0\le \xi \le 2$ corresponds to no adhesion, partial wrapping and full wrapping. At $\xi =2$, the cap covers a whole sphere and its footprint shrinks to zero. This is a closed neck in the geometric limit. It does not by itself prove membrane scission: creating a detached enclosure changes topology and involves neck physics absent from this energy.

For positive membrane tension, $E^{\prime \prime }/(\pi \kappa )=2\bar{\sigma }>0$. Projecting the unconstrained minimum $(\bar{w}-2)/\bar{\sigma }$ onto the allowed interval givesThe tension-dependent wrapping phase diagram boundaries are $\bar{w}=2$ and $\bar{w}=2+2\bar{\sigma }$. At zero membrane tension, the energy is linear: the minimum jumps from $0$ to $2$ as $\bar{w}$ crosses $2$, while all indentation values are degenerate at $\bar{w}=2$. Negative membrane tension is outside the stable-flat-membrane interpretation used here. Increasing membrane tension makes the excess area required for wrapping more expensive, broadening the partially wrapped region.

TODO: user-deferred biological application. The final request for biological relevance is intentionally left unsolved; all preceding generic geometry and membrane mechanics are included.