= Solution
An <amphiphile> contains a solvent-compatible head and a solvent-avoiding tail. In water, aggregation hides hydrocarbon tails while leaving heads exposed. This reduces the free-energy cost of tail–water contact, competing with loss of <translational entropy>, headgroup repulsion and chain packing. A <micelle> is a finite equilibrium aggregate, not necessarily a permanently bound particle.
Equality of molecular <chemical potentials> controls aggregation. For a large preferred aggregation number, $\mu_1=\mu_1^0+k_BT\ln x_1$ approaches an aggregate value $\mu_*^0$, giving
$$
\boxed{x_{\rm CMC}\simeq\exp[(\mu_*^0-\mu_1^0)/(k_BT)].}
$$
Above the <critical micelle concentration>, added <surfactant> mainly increases aggregate population while <monomer> activity changes relatively little. Finite aggregation numbers give a smooth crossover rather than an exact bulk phase transition. The <amphiphile packing parameter> $p=v/(a_0l_c)$ compares tail volume, preferred head area and chain length; small $p$ favours spherical <micelles>, while larger values favour less curved aggregates. These geometric criteria are approximate, not universal shape phase boundaries.<Theory of self-assembly of hydrocarbon amphiphiles by Jacob N. Israelachvili and others>[1976, packing/free-energy model of generic hydrocarbon surfactant aggregates]{ref}
Surface-tension or conductivity changes with <concentration> locate an operational CMC; scattering probes size and aggregation number. <temperature>, impurities and salt must be controlled, and independent methods help distinguish aggregation from precipitation. In an ionic <surfactant>, salt screens headgroup repulsion and can change both the CMC and aggregate shape. This discussion concerns generic synthetic <surfactants> only.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2014/paper-1-micelle.png]
{title=Original generic micelle schematic and an explicitly illustrative aggregation crossover}
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