Gas cushion model and hydrodynamic boundary conditions for superhydrophobic textures

Superhydrophobic Cassie textures with trapped gas bubbles reduce drag, by generating large effective slip, which is important for a variety of applications that involve a manipulation of liquids at the small scale. Here we discuss how the dissipation in the gas phase of textures modifies their friction properties. We propose an operator method, which allows us to map the flow in the gas subphase to a local slip boundary condition at the liquid-gas interface. The determined uniquely local slip length depends on the viscosity contrast and underlying topography, and can be immediately used to evaluate an effective slip of the texture. Besides superlubricating Cassie surfaces, our approach is valid for rough surfaces impregnated by a low-viscosity “lubricant,” and even for Wenzel textures, where a liquid follows the surface relief. These results provide a framework for the rational design of textured surfaces for numerous applications.

Figure 1

Sketch of the typical 1D SH surface represented by rectangular grooves rigorously studied here. However, some of our conclusions are general and apply for any 1D and 2D textured surfaces—pillars, holes, and lamellae.

Figure 2

(a) Longitudinal and (b) transverse local slip lengths computed with $\varphi =0.75,\mu /{\mu }_g=50$. Dashed lines show the predictions of Eq. (3).

Figure 3

(a) Polar coordinates (side view) used to evaluate the flow near the edge of the groove, and illustration (top view) of the local slip length behavior at the edge of 2D (b) pillars, (c) hollows, and (d) channels.

Figure 4

Rescaled (a) longitudinal and (b) transverse local slip length for deep and shallow grooves. Solid and dotted curves correspond to the Cassie, $\mu /{\mu }_g=50$, and Wenzel, $\mu /{\mu }_g=1$, states at $\varphi =0.1,0.5,0.9$ (these curves coincide for shallow grooves and are nearly overlapping for deep grooves). Dash-dotted lines show asymptotic solutions near the edges of the groove.

Figure 5

(a),(b) Longitudinal and (c),(d) transverse effective slip lengths for textures with (a),(c) shallow and (b),(d) deep grooves. From top to bottom, $\mu /{\mu }_g=50,5,1$. Exact theoretical results are shown by circles; analytical results [Eq. (11)] with local slip given by Eq. (12) are plotted by solid curves. Filled squares show earlier data for perfect slip [10, 12]; filled circles show earlier data for the Wentzel state [31].

Figure 6

Apparent local slip as a function of the depth of the groove for (a) longitudinal and (b) transverse directions. Symbols show the values obtained using Eq. (11); solid curves show predictions of Eq. (13).