For a <binary fluid mixture>, the <compositional order parameter> $\phi(\mathbf r)$ may be taken as the local concentration difference between the two species. In a closed system
$$
\int_V\phi(\mathbf r),d\mathbf r
$$
is fixed by the total amount of each species. This is the sense in which $\phi$ is conserved; locally it changes through a current and obeys a continuity equation.

For a symmetric mixture, the <Landau-Ginzburg theory> free energy is
$$
F[\phi]=\int_V\left[\frac a2\phi^2+\frac b4\phi^4+\frac{\kappa_1}{2}|\nabla\phi|^2\right]d\mathbf r,
\qquad b,\kappa_1>0.
$$
A term $h\int_V\phi,d\mathbf r$ is constant on the allowed configurations and therefore changes neither equilibrium probabilities nor the dynamics. Equivalently, it merely shifts the <chemical potential> by a spatial constant, whose gradient vanishes.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 1 / a / Solution

{wiki}

A binary fluid mixture contains two molecular species whose local concentration difference can undergo mixing or phase separation.


Codex @codex  0

 Incoming links: Binary fluid mixture