Amphiphile

An amphiphile is a molecule containing distinct hydrophilic and hydrophobic regions. Amphiphilic and amphipathic describe this combination. Membrane-forming phospholipids are examples; their two regions give different parts of one molecule different affinities for water. The word describes a structure, not a guarantee of a particular aggregate shape. See IUPAC Gold Book: amphiphilic.

A micelle is a finite aggregate of amphiphiles in a solvent. In an ordinary aqueous micelle, hydrophobic tails occupy the interior and hydrophilic heads contact the surrounding water. Association can lower the Gibbs free energy by reducing unfavorable tail exposure, despite loss of molecular translational entropy. Head-group interactions and chain packing help select the aggregate size and shape. A micelle has a filled tail-rich core, unlike a lipid vesicle with a solvent-filled interior bounded by a lipid bilayer.

For a long cylindrical micelle with negligible end caps, incompressible tail volume $v$ and head-group area $a_0$ give $v/a_0=R/2$. The geometric requirement $R\le {\ell }_c$ bounds the amphiphile packing parameter by $1/2$. The conventional cylindrical range is approximately $1/3\lesssim v/(a_0{\ell }_c)\lesssim 1/2$; aggregate transitions also depend on interactions and chain conformations.

The number of constituent molecules in an aggregate. An observed population generally has a distribution of aggregation numbers.

In a sharp-core spherical micelle of aggregation number $n$, incompressible tail volume $v$ per molecule and effective head-group area $a_0$ give $nv=4\pi R^3/3$ and $n{a}_0=4\pi R^2$. Division gives the displayed radius and aggregation number. If the maximum tail extension is $l_c$, the model requires $R\le l_c$, equivalently the amphiphile packing parameter $v/(a_0{l}_c)\le 1/3$. Input dimensions violating that inequality cannot describe an exact spherical filled core without modifying the molecular geometry assumptions.

For an amphiphile, let $v$ be its hydrophobic-tail volume, $a_0$ its effective head-group area, and $\ell$ its maximum tail extension. The dimensionless packing parameter $P=v/(a_0\ell )$ estimates compatible aggregate curvature. For a spherical micelle of core radius $r$ containing $n$ molecules, $nv=4\pi r^3/3$ and $n{a}_0=4\pi r^2$, giving $v/a_0=r/3$. Since $r\le \ell$, spherical packing requires $P\le 1/3$. The analogous long-cylinder calculation gives $v/a_0=r/2$ and the limiting value $1/2$; a planar monolayer has $v/a_0$ equal to its thickness and limiting value one. Thus small $P$ favors spherical micelles, intermediate values favor cylindrical aggregates, and values near one favor lipid bilayers. This is a geometric guide, not a full thermodynamic prediction of every phase boundary.

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.