Past exam of the quantum condensed matter course of the University of Cambridge / 2006 / Section A / A1 / b / Solution

An amphiphile contains a solvent-compatible head and a solvent-avoiding tail. In water, aggregation can lower the Gibbs free energy by reducing exposed hydrocarbon area through the hydrophobic effect. This competes with loss of translational entropy, head-group repulsion and the cost of deforming the tails. Above the critical micelle concentration, added amphiphiles mainly enter aggregates rather than continuing to increase the isolated-molecule concentration proportionally.

The amphiphile packing parameter $P=v/(a_0{\ell }_c)$ compares tail volume $v$, preferred head area $a_0$ and tail length ${\ell }_c$. Rough geometric ranges favor spherical micelles for $P<1/3$, cylindrical aggregates for $1/3<P<1/2$, and lipid bilayers for $1/2<P<1$. These are organizing estimates rather than universal sharp phase boundaries. A bilayer can close into a vesicle to eliminate exposed edges; concentration, salt and temperature can change the preferred aggregate.

A simple experimental test varies concentration and measures surface tension: its slope against log concentration changes near the critical micelle concentration, while light scattering detects the appearance of larger aggregates. These observations distinguish the concentration threshold from an assumption that every molecule forms an aggregate at arbitrarily low concentration.