 Finite-micelle aggregation-number distribution

Here $\mu$ is the common equilibrium chemical potential per amphiphile and ${\mu }_{\alpha }^0$ the standard free energy per amphiphile in an aggregate of size $\alpha$. A minimum of the aggregate free-energy balance at a finite size produces a micellar population around that size, in addition to monomers. Translational entropy leaves a small aggregate population below the nominal critical micelle concentration; a preferred finite size is not supplied by the mass-action equation alone.