Drag force on a sphere moving toward an anisotropic superhydrophobic plane

Abstract

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), b1 and b2 (local slip lengths at solid and gas areas), and φ1 and φ2 (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, b1, and b2, 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.

Effective hydrodynamic boundary conditions for microtextured surfaces

“…Since at short separations we observe f * → 0, one can conclude that slippage (which would lead to f * → 1/4 [20]) obviously does not mimic roughness when h is small, by overestimating the drag force. The same remark concerns effective superhydrophobic slippage where f * → 2(4 − 3φ)/(8 + 9φ − 9φ 2 ) [21] and is equal to 0.3 for this particular sample. Therefore, our experimental results are now compared with theoretical calculations made for an effective smooth plane shifted down from the top of the corrugations, i.e., by assuming Table I Table I we present the experimental values of s for different samples, and curves calculated with Eq.…”

Measurement of slip length on superhydrophobic surfaces

“…In this experiment [31], the authors have assumed that the boundary slip is described by the Vinogradova equation shown above. More recently, Asmolov et al [34] showed that on textured superhydrophobic surfaces, the effective slip length is not independent of the distance in the drainage experiments. Asmolov et al [34] suggested a new expression to evaluate the slip length on textured superhydrophobic surfaces, and it will be useful to check such expressions in forthcoming AFM experiments.…”

Effective slippage on superhydrophobic trapezoidal grooves

“…1). As in most previous publications [7,12,15,20,29,[35][36][37], we model the superhydrophobic plate as a flat heterogeneous interface. In such a description, we neglect an additional mechanism for a dissipation connected with the meniscus curvature [15,38,39].…”