= 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.
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