 Solution

Let $A$ be projected area. Close-packed circular footprints cover $\varphi A$ and the exposed flat substrate covers $(1-\varphi )A$. A hemisphere's curved area is twice its circular footprint, so the actual wetted solid area isThe Wenzel roughness factor is therefore $r=A_{\mathrm{wet}}/A=1+\varphi =1.9069$, not $\varphi$ or $2$.

Using the Wenzel wetting relation,The nanostructure makes this surface more wettable, because planar PMMA already has $\theta <90^{\circ }$. Under complete Wenzel wetting, roughening amplifies the underlying hydrophilic character; it does not turn this material hydrophobic.

The calculation assumes identical chemistry on spheres and substrate, complete liquid penetration, ideal hemispheres, a macroscopic contact angle and negligible pinning/line-tension effects. Actual dried particles may have different contact geometry, deformation, contamination or trapped air. A hydrophobic coating with planar angle above $90^{\circ }$ would make roughness increase the angle instead. Designing a texture that retains air in a Cassie–Baxter state is another route, but its stability against liquid penetration must be checked.