= Solution
\b[Diffuse-interface <free energy>.] A sharp boundary assigns one mathematical surface to two uniform fluids. Molecular mixing instead changes composition over a finite distance. In <square-gradient theory of a liquid interface>, let $\phi(\mathbf r)$ be a dimensionless local composition and write
$$
F[\phi]=\int\left[f_b(\phi,T)+\frac K2|\nabla\phi|^2\right]dV,
\qquad K>0.
$$
The bulk free-energy density $f_b$ favors the equilibrium phase compositions, while the <gradient> term penalizes changing composition too abruptly. Since $f_b$ has units $\mathrm{J\,m^{-3}}$ and $\phi$ is dimensionless, $K$ has units $\mathrm{J\,m^{-1}}$. The theory balances bulk mixing cost against this smoothness cost, rather than imposing a zero-width surface from the start.
At coexistence the two compositions $\phi_-,\phi_+$ have a <common-tangent construction for phase coexistence>. Subtract this tangent from $f_b$ to define an excess grand-potential density $\Delta\omega(\phi)$, zero at both bulk compositions. For a planar profile varying along $z$, minimizing the excess <free energy> gives
$$
K\phi''=\frac{d\Delta\omega}{d\phi}.
$$
Multiply by $\phi'$ and integrate using $\phi'\to0$ in the bulk:
$$
\boxed{\frac K2(\phi')^2=\Delta\omega(\phi).}
$$
It follows that the interfacial <surface tension> is
$$
\gamma=\int_{-\infty}^{\infty}
\left[\Delta\omega+\frac K2(\phi')^2\right]dz
=\int_{-\infty}^{\infty}K(\phi')^2dz
=\int_{\phi_-}^{\phi_+}\sqrt{2K\Delta\omega(\phi)}\,d\phi.
$$
This connects a continuous composition profile to a positive interfacial energy per <area>.
\b[Broadening near criticality.] Close to a symmetric mean-field <thermodynamic critical point>, an <order parameter> $m$ measuring departure from the critical composition has bulk density
$$
f_b(m)=\frac r2m^2+\frac u4m^4,\qquad
r\propto T-T_c<0,\quad u>0.
$$
The coexistence values are $m=\pm m_0$, $m_0=\sqrt{|r|/u}$. The <scalar quartic domain wall> profile is
$$
\boxed{m(z)=m_0\tanh(z/w),\qquad
w=\sqrt{\frac{2K}{|r|}}.}
$$
The curvature at either bulk minimum is $f_b''(m_0)=2|r|$, so the bulk <correlation length> is $\xi=\sqrt{K/(2|r|)}$ and this convention gives $w=2\xi$. As $T\to T_c^-$, the bulk distinction between phases weakens, its free-energy curvature and barrier decrease, and composition can vary gradually over an increasing distance at little bulk cost. The <gradient> penalty then favors a wider profile:
$$
\boxed{w\propto(T_c-T)^{-1/2}\quad\text{in mean-field theory}.}
$$
The same model gives $\gamma=2\sqrt{2K}|r|^{3/2}/(3u)$, which vanishes at criticality. The exponents here are mean-field results, not universal exact values for every real mixture.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2017-q3b-diffuse-interface.png]
{title=Approaching criticality reduces the composition contrast and broadens the equilibrium diffuse interface}
The figure uses two illustrative parameter values in this quartic model, not experimental parameters supplied by the paper. In addition to the intrinsic gradient-theory width, thermal <capillary waves> move and roughen an interface. Their increasing <amplitude> as <surface tension> decreases can broaden the spatially averaged measured profile further. Thus a measured broad interface can include both intrinsic composition smoothing and long-wavelength interface-position fluctuations.
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