Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2020 / Paper 2 / Section A(iii) / 18

At a straight contact line, take forces per unit transverse length and measure the contact angle $\theta$ through the liquid. Solid–vapor surface tension pulls outward along the solid, solid–liquid surface tension pulls inward, and the liquid–vapor force has horizontal component ${\gamma }_{LV}\cos \theta$. Tangential equilibrium givesThis is Young's equation. The substrate supplies the normal reaction balancing the vertical capillary component. The original force sketch labels directions at a right-hand contact line. The result assumes smooth, chemically uniform, rigid surfaces in partial-wetting equilibrium, with negligible pinning and line tension.

Advance the projected contact line by $dx$ per unit transverse length without wetting the gaps. A fraction $f$ converts solid–vapor interface into solid–liquid interface, contributing $f({\gamma }_{SL}-{\gamma }_{SV})dx$. The remaining fraction creates liquid–vapor interface above trapped gas, costing $(1-f){\gamma }_{LV}dx$. The outer droplet surface contributes ${\gamma }_{LV}\cos {\theta }^{\ast }dx$. ThusSubstituting Young's equation givesThis is the ideal Cassie–Baxter state relation. Its $f$ is projected solid-contact fraction, not the roughness factor of Wenzel wetting. A suspended interface over the gaps is essential; invasion of the texture changes the area bookkeeping.

With $f=3/4$,Hydrophobicity requires ${\theta }^{\ast }>90^{\circ }$, henceEquality is the ${\theta }^{\ast }=90^{\circ }$ boundary. A smooth-solid angle between about $70.53^{\circ }$ and $90^{\circ }$ can therefore become apparently hydrophobic if the suspended state persists.

For general $0<f\le 1$, the criterion is $\cos \theta <1/f-1$. For $1/2<f\le 1$, the threshold is ${\cos }^{-1}(1/f-1)$, reaching $90^{\circ }$ at $f=1$. For $f=1/2$, all $\theta >0$ satisfy it, with $\theta =0$ marginal. For $f<1/2$, the ideal equation predicts hydrophobicity for every original angle; formally ${\theta }^{\ast }\to 180^{\circ }$ as $f\to 0$.

These conclusions require retention of gas beneath the liquid. On intrinsically wettable textures the Cassie–Baxter state can be metastable. Pressure, condensation or defects may trigger invasion; penetration tests and contact-angle hysteresis distinguish that failure from the ideal suspended state. Roughness alone does not guarantee stable water repellency.