Solution
= Solution
An <amphiphile> has a solvent-affine part and a solvent-averse part within one molecule. In water these are commonly called <hydrophilic> head and <hydrophobic> tail. Two nonbiological examples are <sodium dodecyl sulfate>, with a twelve-carbon tail and an anionic sulfate head, and <cetyltrimethylammonium bromide>, with a sixteen-carbon tail and a cationic quaternary-ammonium head. Their <counterions> balance the ionic head-group charge; the head and tail do not have interchangeable interactions with water.
Aggregation reduces unfavorable exposure of <hydrocarbon> tails to water but loses molecular translational <entropy>. Interfacial <free energy>, head-group <solvation> and electrostatic repulsion, chain packing and <conformational entropy> determine the preferred aggregate. The <hydrophobic effect> is a solvent-mediated free-energy balance, rather than a literal attractive bond between tails.
The <amphiphile packing parameter>
$$
P=\frac{v}{a_0\ell_c}
$$
compares tail <volume> $v$, effective head-group <area> $a_0$ and maximum tail <length> $\ell_c$. A spherical core of radius $R$ obeys $v/a_0=R/3$, and a long cylindrical core obeys $v/a_0=R/2$. The condition $R\leq\ell_c$ gives the conventional spherical window $P\lesssim1/3$ and cylindrical window $1/3\lesssim P\lesssim1/2$. Less strongly curved bilayers become compatible for $1/2\lesssim P\lesssim1$, with planar packing near one; values above one can favor inverse <curvature> in an appropriate nonpolar surrounding medium. These windows are geometric guides rather than exact phase boundaries.
A larger hydrated or electrostatically repelling head increases $a_0$ and favors positive <curvature>. Screening of charged heads or a smaller head can lower $a_0$ and favor rods or sheets. Multiple tails increase the tail <volume> relative to head <area> and can favor bilayers. Tail <length>, branching, <temperature> and <solvent> selectivity alter both geometry and interaction <free energy>, so one cannot predict a unique aggregate from hydrophobicity alone.<Theory of self-assembly of hydrocarbon amphiphiles by Jacob N. Israelachvili and others>[1976, pp. 1525–1568: coupled thermodynamics, molecular packing constraints and globular, cylindrical and bilayer aggregates]{ref}
In the supplied chemical-potential expression, $\mu$ is the <equilibrium> <chemical potential> per <amphiphile>, $\mu_\alpha^0$ the standard <free energy> per <amphiphile> within an aggregate of size $\alpha$, and $x_\alpha$ the dimensionless molecular fraction belonging to that size class. The <aggregate molecular fraction versus aggregate number fraction> distinction explains the logarithm: the aggregate count fraction is $x_\alpha/\alpha$, and dividing its translational <chemical potential> by $\alpha$ gives a per-molecule value. Thus
$$
\alpha\frac{\mu-\mu_\alpha^0}{k_BT}=\ln(x_\alpha/\alpha),
\qquad
\boxed{x_\alpha=\alpha\exp\!\left[\frac{\alpha(\mu-\mu_\alpha^0)}{k_BT}\right].}
$$
Here $k_B$ is the <Boltzmann constant>, $T$ absolute <temperature>, and $\alpha$ a positive integer. The logarithm acts on a dimensionless ideal activity; a concentrated nonideal solution would require <activity coefficients>.
<equilibrium> exchange of <amphiphiles> gives the same per-molecule $\mu$ for every size. For monomers, $\mu=\mu_1^0+k_BT\ln x_1$. Substitution yields the <finite-micelle aggregation-number distribution>
$$
\boxed{x_\alpha=\alpha x_1^\alpha
\exp\!\left[\frac{\alpha(\mu_1^0-\mu_\alpha^0)}{k_BT}\right],\qquad
x_{\rm tot}=\sum_{\alpha\ge1}x_\alpha.}
$$
This formula by itself does not prescribe a preferred <micelle> size: the size dependence of $\mu_\alpha^0$ must include the competing packing and interaction terms. If their balance favors a finite size $\alpha_*$, a monomer population and a peak around that size can coexist. Dividing by $\alpha$ would instead plot the distribution of aggregate counts.
At very low $x_{\rm tot}$, the factors $x_1^\alpha$ suppress large aggregates and most molecules remain monomers. As $x_1$ rises, a favorable aggregate class grows extremely rapidly because of this high <power>. The <critical micelle concentration from standard chemical potentials> has the large-size estimate
$$
x_{1,*}\simeq\exp\!\left[\frac{\mu_{\alpha_*}^0-\mu_1^0}{k_BT}\right].
$$
More precisely, the illustrative crossover condition $x_{\alpha_*}=x_1$ gives
$$
\ln x_{1,*}=
\frac{\alpha_*(\mu_{\alpha_*}^0-\mu_1^0)/(k_BT)-\ln\alpha_*}{\alpha_*-1}.
$$
Added material above the crossover enters aggregates predominantly. Monomer <concentration> then increases slowly rather than being exactly pinned: its logarithm enters $\mu$, while aggregate populations respond with a large factor $\alpha_*$. For finite <micelles> this is a sharp association crossover, not an absolute prohibition on aggregates below one exact <concentration>.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2015/paper-1-aggregate-distribution.png]
{title=Original illustrative molecular-fraction distributions over aggregation number below and above a micellization crossover}
The drawing uses a declared illustrative standard-free-energy model favoring aggregates near $\alpha=40$. It is not an inferred distribution or critical <concentration> for either named <surfactant>. The separate monomer peak is at $\alpha=1$; a distribution of molecular fractions weights each aggregate by its number of molecules.
For the general physics of a nearly planar <fluid> sheet, let $h(x,y)$ be a small-slope displacement, $\kappa$ <membrane bending rigidity> and $\sigma$ mechanical tension. At fixed topology and symmetric reference <curvature>, the quadratic <Helfrich energy> is
$$
F[h]=\frac12\int d^2x\,[\kappa(\nabla^2h)^2+\sigma|\nabla h|^2].
$$
The bending term is the small-slope curvature-elastic <energy>; additional spontaneous-curvature or boundary terms require extra information.<Elastic Properties of Lipid Bilayers: Theory and Possible Experiments by Wolfgang Helfrich>[Section II.C, p. 695, equation (12): curvature-elastic energy and coefficients with units of energy]{ref}
For Fourier amplitudes $h_{\mathbf q}=A^{-1}\int_Ah(\mathbf x)e^{-i\mathbf q\cdot\mathbf x}d^2x$, equipartition gives the <membrane fluctuation spectrum under tension>,
$$
\boxed{\langle|h_{\mathbf q}|^2\rangle=
\frac{k_BT}{A(\kappa q^4+\sigma q^2)}.}
$$
Higher <temperature> increases fluctuations, while greater <membrane bending rigidity> or tension suppresses them. Short-wavelength modes are predominantly bending-controlled; longer wavelengths can be tension-controlled. Finite sheet size, supports, <confinement> and imposed <area> set <boundary conditions> and <infrared cutoffs>; continuum elasticity also needs a short-distance cutoff. <solvent> <viscosity> affects relaxation rates, although it does not enter this <equilibrium> amplitude at fixed elastic parameters.
\b[TODO: user-deferred biology. The requested importance of these fluctuations for bodily function is deferred.] The preceding result covers only the independent continuum and thermal physics.