Past exam of the soft condensed matter course of the University of Cambridge / 2017 / 3

Molecular area from the adsorption slope. The supplied Gibbs adsorption equation identifies the surface excess number density $\Gamma =1/\Sigma$ throughThe logarithm is understood relative to a fixed concentration unit or reference concentration; only concentration ratios enter its differences. The pure-water point $c=0$ supplies a baseline surface tension, but cannot be included in $\ln c$.

An estimate near the later working concentration $0.055\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$ uses its two bracketing points:Taking room temperature as $298\mathrm{K}$,This corresponds to about $4.4$ adsorbed molecules per $\mathrm{n}{\mathrm{m}}^2$. The data are not perfectly linear in $\ln c$, so the inferred molecular area depends somewhat on the concentration range: a fit to all five positive-concentration points gives about $0.26\mathrm{n}{\mathrm{m}}^2$, while the higher-concentration region gives roughly $0.22\mathrm{n}{\mathrm{m}}^2$. An effective area of order $0.2$–$0.3\mathrm{n}{\mathrm{m}}^2$ is justified; a concentration-independent exact value is not.

Contact angle. Linear interpolation between $0.05$ and $0.06\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$ gives ${\gamma }_{LV}(0.055)=(48.5+45.2)/2=46.85\mathrm{mN}{\mathrm{m}}^{-1}$. For an ideal equilibrium contact angle measured through the liquid, Young's equation isThe supplied solid-vapour and solid-liquid values have difference $40\mathrm{mN}{\mathrm{m}}^{-1}$. HenceThis assumes clean homogeneous equilibrium interfaces, negligible contact-angle hysteresis, and the stated solid interfacial energies applicable to the solution.

Complete wetting. The spreading coefficient is $S={\gamma }_{SV}-{\gamma }_{SL}-{\gamma }_{LV}=40\mathrm{mN}{\mathrm{m}}^{-1}-{\gamma }_{LV}$. The onset of complete wetting occurs at $S=0$, or ${\gamma }_{LV}=40\mathrm{mN}{\mathrm{m}}^{-1}$, if the solid interfacial energies remain unchanged. The last measured value is still $42.0\mathrm{mN}{\mathrm{m}}^{-1}$, so this requires extrapolation beyond the table.

Continuing the last local slope in concentration, $(42.0-45.2)/(0.07-0.06)=-320\mathrm{mN}{\mathrm{m}}^{-1}$ per $\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$, givesContinuing a constant local Gibbs slope in $\ln c$ instead gives approximately $0.078\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$. Thus the data suggest a threshold near $0.08\mathrm{M}$ under continued-trend assumptions, rather than fixing one exactly. A surface-tension plateau caused by micellization, or a change in ${\gamma }_{SL}$, could prevent or shift this threshold; no such extra data are given.

Capillary rise. For a circular capillary of radius $a$, vertical surface-tension force $2\pi a{\gamma }_{LV}\cos \theta$ balances the weight $\rho g\pi a^{2}h$. Thus capillary action givesUsing $a=0.5\mathrm{mm}=5.0\times 10^{-4}\mathrm{m}$, $\rho =10^3\mathrm{kg}{\mathrm{m}}^{-3}$ and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$,This is the height above the reservoir level in the ideal static capillary approximation; meniscus-volume corrections and contact-angle hysteresis are neglected. The positive $\cos \theta$ gives a rise, rather than a depression.

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 $\varphi (\mathbf{r})$ be a dimensionless local composition and writeThe 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 $\mathrm{J}{\mathrm{m}}^{-3}$ and $\varphi$ is dimensionless, $K$ has units $\mathrm{J}{\mathrm{m}}^{-1}$. 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 ${\varphi }_{-},{\varphi }_{+}$ have a common-tangent construction for phase coexistence. Subtract this tangent from $f_b$ to define an excess grand-potential density $\Delta \omega (\varphi )$, zero at both bulk compositions. For a planar profile varying along $z$, minimizing the excess free energy givesMultiply by ${\varphi }^{\prime }$ and integrate using ${\varphi }^{\prime }\to 0$ in the bulk:It follows that the interfacial surface tension isThis connects a continuous composition profile to a positive interfacial energy per area.

Broadening near criticality. Close to a symmetric mean-field thermodynamic critical point, an order parameter $m$ measuring departure from the critical composition has bulk densityThe coexistence values are $m=\pm m_0$, $m_0=\sqrt{|r|/u}$. The scalar quartic domain wall profile isThe curvature at either bulk minimum is $f_b^{\prime \prime }(m_0)=2|r|$, 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: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.

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.