Abstract. We discuss how the wettability and roughness of a solid impacts its hydrodynamic properties. We see in particular that hydrophobic slippage can be dramatically affected by the presence of roughness. Owing to the development of refined methods for setting very well-controlled micro-or nanotextures on a solid, these effects are being exploited to induce novel hydrodynamic properties, such as giant interfacial slip, superfluidity, mixing, and low hydrodynamic drag, that could not be achieved without roughness.

Super-hydrophobic array of grooves containing trapped gas (stripes), have the potential to greatly reduce drag and enhance mixing phenomena in microfluidic devices. Recent work has focused on idealized cases of stick-perfect slip stripes. Here, we analyze the experimentally more relevant situation of a pressure-driven flow past striped slip-stick surfaces with arbitrary local slip at the gas sectors. We derive approximate formulas for maximal (longitudinal) and minimal (transverse) directional effective slip lengths, that are in a good agreement with the exact numerical solution for any surface slip fraction. By representing eigenvalues of the slip length-tensor, they allow us to obtain the effective slip for any orientation of stripes with respect to the mean flow. Our results imply that flow past stripes is controlled by the ratio of the local slip length to texture size. In case of a large (compared to the texture period) slip at the gas areas, surface anisotropy leads to a tensorial effective slip, by attaining the values predicted earlier for a perfect local slip. Both effective slip lengths and anisotropy of the flow decrease when local slip becomes of the order of texture period. In the case of small slip, we predict simple surface-averaged, isotropic flows (independent of orientation).

Surface conduction in presence of slip is characterized by full Dukhin number, which is given by [1]:where m = 2ε(k B T /ze) 2 /(ηD), and D is the ion diffusivity, which is assumed equal for both types of ions. The first term in (1) is the Dukhin number for the no-slip surface, Du b=0 , while the second one, Du b , is due to hydrodynamic slip. In the Debye-Hückel modelThis restriction (2) allows the simplification:The parameter m ≈ 100/z 2 , and κL ≫ 1 since EDL is thin. Whence Du b=0 ≈ 1/(z 2 κL) ≪ 1 provided the potential is low. For a "slippery part" of Du we evaluate01. Therefore for increasing surface charge (potential) and b/L, the conductivity of the diffuse layer can become comparable to the bulk, and surface condition must be considered. The Péclet number, P e = U L/D , in presence of slip can be evaluated asTypically, electroosmotic velocity is of order is of order micrometers per second for no-slip surfaces [2]. For nanoscale patterns L < 1 µm and typical ion diffusivities D ≈ 10 −6 cm 2 /s this gives P e b=0 < 0.01 ≪ 1. The slip implies a correction factor (1 + bκ) , which suggests that the convective ion transport can safely be neglected only for bκ < 10. Larger values of bκ should relax this standard approximation of small P e. ELECTRO-OSMOTIC VELOCITY IN EIGENDIRECTIONSLongitudinal stripes.-In this configuration only x−velocity component remains, and the Stokes equation takes the formWe expand surface charge density in a Fourier series, and the potential is thenwhere ξ n = κ 2 + λ 2 n , λ n = 2nπ/L , q = q 1 φ 1 + q 2 φ 2 is the mean surface charge, andThe general solution to (6) for u(y, z) has the formwhere U n and U are determined by the slip boundary conditions. Imposing them on (9) in a thin EDL limit yields a dual serieswhich can be solved exactly by using a technique [3] to obtain the thin-EDL electro-osmotic velocity:Transverse stripes.-Although an external pressure gradient is equal to zero, local pressure variations contribute into a non-zero term ∇p in the Stokes equation, so that the flow is essentially two-dimensional. We first introduce a stream function f (x, y)2 which obeys inhomogeneous biharmonic equation:Here u and v are x and y velocity components, correspondingly. The general periodic solution to (14) has the formE t q n κ 2 η + g n y e −λny cos λ n x + + E t ε κ 2 η ∂ψ ∂y (15)Here the potential ψ(x, y) has exactly the form (7) with z replaced by x. The dual series problem in a thin EDL limit can be written aswhere a n = g n + E t q n ηκ 2 (ξ n − λ n ).These dual series can be solved exactly to obtainWe emphasize that a comparison of Eqs. (12) and (18) indicates that the EO flow is generally anisotropic, so that our results do not support an earlier conclusion [4] that the electro-osmotic mobility tensor is isotropic in the thin EDL limit. This inconsistensy [4] (due to an erroneous expression for a transverse electro-osmotic velocity, where factor of 2 was lost) has been corrected for a case b 2 = ∞ in [5]. Sciences, 31 Leninsky Prospect, 119991 Moscow, Russia 3 ITMC and...