Past exam of the soft condensed matter course of the University of Cambridge / 2016 / Paper 1 / 1 / b / Solution

For an equilibrium contact angle measured through the liquid, Young's equation balances the tangential interfacial forces:The pure-ethanol measurement therefore fixes the solid–vapour surface energy asThe inferred surface energy is $40.7\mathrm{mJ}{\mathrm{m}}^{-2}$, equivalently $40.7\mathrm{mN}{\mathrm{m}}^{-1}$. The equality of these units follows because creating interfacial area costs energy.

For the equal-fraction mixture, the stated linear relations give ${\gamma }_{SL}=20.5\mathrm{mN}{\mathrm{m}}^{-1}$ and ${\gamma }_{LV}=28.5\mathrm{mN}{\mathrm{m}}^{-1}$. Holding the solid–vapour surface energy fixed,The small decrease in ${\gamma }_{SL}$ increases the numerator of Young's equation, but the increase in ${\gamma }_{LV}$ has the larger relative effect and reduces $\cos \theta$. Consequently the contact angle increases and the mixture wets the surface less well. This inference uses equilibrium wetting on an unchanged smooth solid; a changing surface composition or contact-angle hysteresis would require additional measurements.