Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2019 / Paper 2 / Section B / 23 / ii / Solution

For ideal spherical micelles, $v/a_0=r/3$ and $r\le {\ell }_c$, giving $P\le 1/3$. For a long cylindrical micelle, $v/a_0=r/2$, so cylinders can accommodate larger packing parameters while keeping $r\le {\ell }_c$: roughly $1/3<P\lesssim 1/2$ favours cylinders. Relatively small head area or greater tail volume reduces positive curvature.

For a planar bilayer leaflet of hydrophobic thickness $\ell$, $v/a_0\simeq \ell$, with $\ell \le {\ell }_c$. Thus $1/2\lesssim P\lesssim 1$ favours bilayer/vesicle structures, with nearly planar bilayers toward $P\simeq 1$. The two leaflets put hydrophobic tails together and hydrophilic heads toward water. For $P>1$, inverse structures become possible. These bounds are approximate packing guides; chain stretching, head interactions and aggregate free energy determine actual phase boundaries.