'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 the mapping of 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.Comment: 8 pages, 6 figure

Drag force on a sphere moving towards an anisotropic super-hydrophobic plane

We analyze theoretically a high-speed drainage of liquid films squeezed between a hydrophilic sphere and a textured super-hydrophobic plane, that contains trapped gas bubbles. A super-hydrophobic wall is characterized by parameters $L$ (texture characteristic length), $b_1$ and $b_2$ (local slip lengths at solid and gas areas), and $\phi_1$ and $\phi_2$ (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 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 case of a sphere moving towards 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 towards a smooth hydrophilic plane that is shifted down from the super-hydrophobic wall. The value of this shift is equal to the average of the eigenvalues of the slip-length tensor. By analyzing striped super-hydrophobic surfaces, we then compute the correction to the Taylor equation for an arbitrary gap. We show that at 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 super-hydrophobic surface.Comment: 9 pages, 5 figures, submitted to PRE; EPAPS file include

Effective slip-length tensor for a flow over weakly slipping stripes

We discuss the flow past a flat heterogeneous solid surface decorated by slipping stripes. The spatially varying slip length, $b(y)$, is assumed to be small compared to the scale of the heterogeneities, $L$, but finite. For such "weakly" slipping surfaces, earlier analyses have predicted that the effective slip length is simply given by the surface-averaged slip length, which implies that the effective slip-length tensor becomes isotropic. Here we show that a different scenario is expected if the local slip length has step-like jumps at the edges of slipping heterogeneities. In this case, the next-to-leading term in an expansion of the effective slip-length tensor in powers of ${max}\,(b(y)/L)$ becomes comparable to the leading-order term, but anisotropic, even at very small $b(y)/L$. This leads to an anisotropy of the effective slip, and to its significant reduction compared to the surface-averaged value. The asymptotic formulae are tested by numerical solutions and are in agreement with results of dissipative particle dynamics simulations.Comment: 11 pages, 4 figures, submitted to Phys. Rev.

Effective slippage on superhydrophobic trapezoidal grooves

We study the effective slippage on superhydrophobic grooves with trapezoidal cross-sections of various geometries (including the limiting cases of triangles and rectangular stripes), by using two complementary approaches. First, dissipative particle dynamics (DPD) simulations of a flow past such surfaces have been performed to validate an expression [E.S.Asmolov and O.I.Vinogradova, J. Fluid Mech. \textbf{706}, 108 (2012)] that relates the eigenvalues of the effective slip-length tensor for one-dimensional textures. Second, we propose theoretical estimates for the effective slip length and calculate it numerically by solving the Stokes equation based on a collocation method. The comparison between the two approaches shows that they are in excellent agreement. Our results demonstrate that the effective slippage depends strongly on the area-averaged slip, the amplitude of the roughness, and on the fraction of solid in contact with the liquid. To interpret these results, we analyze flow singularities near slipping heterogeneities, and demonstrate that they inhibit the effective slip and enhance the anisotropy of the flow. Finally, we propose some guidelines to design optimal one-dimensional superhydrophobic surfaces, motivated by potential applications in microfluidics.Comment: 11 pages, 8 figures, submitted to J. Chem. Phy

Limiting propulsion of ionic microswimmers

Self-propulsion of catalytic Janus swimmers in electrolyte solutions is induced by inhomogeneous ion release from their surface. Here, we consider the experimentally relevant cases of particles which emit only one type of ions (type I) or equal fluxes of cations and anions (type II). In the limit of a thin electrostatic diffuse layer we derive a nonlinear outer solution for the electric field and concentrations of active (i.e. released from the surface) and passive ionic species. We show that for swimmers of type I both the maximum ion flux and propulsion velocity are constrained. This suggests that the propulsion of Janus swimmers can be optimized by tuning the concentration of active ions

Effective hydrodynamic boundary conditions for microtextured surfaces

We report measurements of the hydrodynamic drag force acting on a smooth sphere falling down under gravity to a plane decorated with microscopic periodic grooves. Both surfaces are lyophilic, so that a liquid (silicone oil) invades the surface texture being in the Wenzel state. A significant decrease in the hydrodynamic resistance force as compared with that predicted for two smooth surfaces is observed. To quantify the effect of roughness we use the effective no-slip boundary condition, which is applied at the imaginary smooth homogeneous isotropic surface located at an intermediate position between top and bottom of grooves. Such an effective condition fully characterizes the force reduction measured with the real surface, and the location of this effective plane is related to geometric parameters of the texture by a simple analytical formula.Comment: 4 pages, submitted to Phys. Rev.

Flow past superhydrophobic surfaces with cosine variation in local slip length

Anisotropic super-hydrophobic surfaces have the potential to greatly reduce drag and enhance mixing phenomena in microfluidic devices. Recent work has focused mostly on cases of super-hydrophobic stripes. Here, we analyze a relevant situation of cosine variation of the local slip length. We derive approximate formulae for maximal (longitudinal) and minimal (transverse) directional effective slip lengths that are in good agreement with the exact numerical solution and lattice-Bolzmann simulations for any surface slip fraction. The cosine texture can provide a very large effective (forward) slip, but it was found to be less efficient in generating a transverse flow as compared to super-hydrophobic stripes.Comment: 8 pages, 6 figure