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

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.