Drag force on a sphere moving toward an anisotropic superhydrophobic plane

We analyze theoretically a high-speed drainage of liquid films squeezed between a hydrophilic sphere and a textured superhydrophobic plane that contains trapped gas bubbles. A superhydrophobic wall is characterized by parameters $L$ (texture characteristic length), ${b}_{1}$ and ${b}_{2}$ (local slip lengths at solid and gas areas), and ${\ensuremath{\varphi}}_{1}$ and ${\ensuremath{\varphi}}_{2}$ (fractions of solid and gas areas). Hydrodynamic properties of the plane are fully expressed in terms of the effective slip-length tensor with eigenvalues that depend on texture parameters and $H$ (local separation). The effect of effective slip is predicted to decrease the force as compared with what is expected for two hydrophilic surfaces and described by the Taylor equation. The presence of additional length scales, $L$, ${b}_{1}$, and ${b}_{2}$, implies that a film drainage can be much richer than in the case of a sphere moving toward a hydrophilic plane. For a large (compared to $L$) gap the reduction of the force is small, and for all textures the force is similar to expected when a sphere is moving toward a smooth hydrophilic plane that is shifted down from the superhydrophobic wall. The value of this shift is equal to the average of the eigenvalues of the slip-length tensor. By analyzing striped superhydrophobic surfaces, we then compute the correction to the Taylor equation for an arbitrary gap. We show that at a thinner gap the force reduction becomes more pronounced, and that it depends strongly on the fraction of the gas area and local slip lengths. For small separations we derive an exact equation, which relates a correction for effective slip to texture parameters. Our analysis provides a framework for interpreting recent force measurements in the presence of a superhydrophobic surface.