= Square-gradient theory of a liquid interface
{title2=$F=\int[f_b(\phi)+(K/2)|\nabla\phi|^2]\,dV$}
A diffuse-interface <free energy> balances the bulk preference for coexisting compositions against a positive penalty for sharp composition gradients. For dimensionless composition $\phi$, $f_b$ is a bulk energy density and $K$ has units of energy per length. Subtract the bulk coexistence common tangent to define $\Delta\omega(\phi)$, with equal zero minima at the two compositions. A planar equilibrium profile obeys $K\phi''=\Delta\omega'(\phi)$ and $K(\phi')^2/2=\Delta\omega(\phi)$. Its <surface tension> is $\int K(\phi')^2dz$. As a <thermodynamic critical point> is approached, the composition contrast and bulk curvature vanish, the <correlation length> grows, and the intrinsic interface broadens. A <scalar quartic domain wall> gives the mean-field hyperbolic-tangent profile. Thermally excited <capillary waves> can broaden a measured profile further.
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