  Patterned surfaces with large effective slip lengths, such as
super-hydrophobic surfaces containing trapped gas bubbles, have the potential
to greatly enhance electrokinetic phenomena. Existing theories assume either
homogeneous flat surfaces or patterned surfaces with thin double layers
(compared to the texture correlation length) and thus predict simple
surface-averaged, isotropic flows (independent of orientation). By analyzing
electro-osmotic flows over striped slip-stick surfaces with arbitrary
double-layer thickness, we show that surface anisotropy generally leads to a
tensorial electro-osmotic mobility and subtle, nonlinear averaging of surface
properties. Interestingly, the electro-osmotic mobility tensor is not simply
related to the hydrodynamic slip tensor, except in special cases. Our results
imply that significantly enhanced electro-osmotic flows over super-hydrophobic
surfaces are possible, but only with charged liquid-gas interfaces.
