Universal Phase Diagram for Wetting on Mesoscale Roughness

The wetting properties of solid substrates with mesoscale (between van der Waals tails and the capillary length) random roughness are considered as a function of the microscopic contact angle of the wetting liquid and its partial pressure in the surrounding gas phase. It is shown that the well-known transition occurring at Wenzel’s angle is accompanied by a transition line at which a jump in the adsorbed liquid volume occurs. This should be present generally on surfaces bearing homogeneous, isotropic random roughness. While a similar abrupt filling transition has been reported before for certain idealized groove or trough geometries, it is identified here as a universal phenomenon. Its location can be analytically calculated under certain mild conditions.

Figure 1

(a) Top view of the sample, showing the wetted areas in grey, the bare substrate in white. The normal vector to $\partial \mathcal{W}$, $\mathbf{n}$, lies in the ($x$, $y$) plane. (b) Two sets of contour lines of $f$ at heights $z$ and $z+dz$. The hatched area between the lines is equal to $p(z)dz$.

Figure 2

Graphic construction for solving Eq. (12). The dashed line represents the left-hand side of Eq. (12).

Figure 3

The phase diagram for wetting on surfaces with homogeneous, isotropic roughness. The most prominent feature is the existence of a transition line $[H_f(\theta )]$, at which the adsorbed average film thickness [and thereby the liquid coverage, Eq. (13)] jumps discontinuously from zero to a finite value.