Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 1

For a binary fluid mixture, the scalar compositional order parameter $\varphi (\mathbf{r})$ may be taken as the local concentration difference between its two molecular species. Its spatial integral is fixed by their separately fixed total amounts:Locally, $\varphi$ can change only through transport and therefore obeys a continuity equation. By contrast, $\mathbf{p}$ is the local polar order parameter measuring the mean tail-to-head orientation of the surfactant molecules. Individual molecules can rotate in place, so the integral of $\mathbf{p}$ need not be conserved.

The composition changes most rapidly across an interface, so $\nabla \varphi$ points along its normal. The termtherefore couples the surfactant's head-to-tail orientation to the interface normal. Minimization aligns $\mathbf{p}$ antiparallel to $\nabla \varphi$ when $\lambda >0$ and parallel when $\lambda <0$. The sign is fixed by which fluid component is called positive $\varphi$ and by which molecular end defines positive $\mathbf{p}$, together with the preferential affinity of the head and tail for the two components.

The terms involving $\mathbf{p}$ can be written by completing the square:Translation of the integration variable in the Gaussian functional integral over $\mathbf{p}$ contributes only a $\varphi$-independent determinant. The effective free energy is thereforewhereThis is the surfactant renormalization of the square-gradient coefficient.

Surfactants accumulate at an interface and orient their two chemically distinct ends toward their preferred fluid components. The relaxation found in part c lowers the coefficient of the gradient energy, and hence generally lowers the surface tension. If the reduction makes $\tilde{\kappa }<0$, the positive fourth-gradient term proportional to $\gamma$ can stabilize structure at a nonzero wavevector, producing the modulated correlations characteristic of a microemulsion.

Use ${\partial }_x\mapsto iq$ under the Fourier transform and the reality conditions $\varphi (-q)=\varphi (q{)}^{\ast }$ and $p(-q)=p(q{)}^{\ast }$. With $\Psi =(\varphi ,p)$, the quadratic free energy iswhereThe opposite imaginary off-diagonal entries make $G(q)$ a Hermitian matrix. Reversing the Fourier-sign convention reverses both of those signs without changing any correlator.

The covariance of a centered multivariate Gaussian distribution is the inverse of its quadratic kernel. Writing $A(q)=a+\kappa q^2+\gamma q^4$, one findsConsequently the composition static structure factor is

The other diagonal entry of the same inverse givesor, more revealingly,The first term is the uncoupled local orientational fluctuation. The second shows that every nonuniform composition fluctuation induces a correlated surfactant-polarization fluctuation. It vanishes at $q=0$, as the coupling contains a gradient, and is enhanced at wavevectors where the composition structure factor is large.