Critical micelle concentration ($c_{\mathrm{CMC}}$)

The critical micelle concentration is the approximate surfactant concentration above which additional dissolved amphiphile predominantly enters micelles rather than remaining as isolated molecules. It is a concentration crossover governed by equality of the molecular chemical potential in the two environments, not a claim that absolutely no aggregates exist below a sharp threshold. Temperature, ionic conditions and molecular structure affect its value.

With $\mu ={\mu }_1^0+k_{B}T\ln x_1$, large aggregates become abundant when monomer chemical potential approaches the favorable aggregate standard value ${\mu }_{\ast }^0$. This gives $x_{1,\ast }\simeq \exp [({\mu }_{\ast }^0-{\mu }_1^0)/(k_{B}T)]$ in the large-aggregation approximation. For one finite preferred size $n$, equal molecular populations $x_n=x_1$ instead give $\ln x_{1,\ast }=[n({\mu }_n^0-{\mu }_1^0)/(k_{B}T)-\ln n]/(n-1)$.

For equilibrium association of $q$ isolated amphiphiles into one micelle, equality of chemical potentials gives the displayed relation between dimensionless thermodynamic activities. Large aggregation number makes aggregate abundance rise rapidly with monomer activity. A finite-aggregate model gives a concentration crossover, rather than an exactly discontinuous thermodynamic transition.

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.

If $x_{\alpha }$ counts amphiphile molecules belonging to aggregates of size $\alpha$, the corresponding dimensionless aggregate count is $n_{\alpha }=x_{\alpha }/\alpha$. Its ideal translational chemical potential contains $\ln n_{\alpha }$. Confusing these two distributions drops a factor of $\alpha$ and changes both normalization and the chemical-potential interpretation.