Electrokinetic flows over inhomogeneously slipping surfaces

Recent measurements and theory suggest that some solid/liquid surfaces can exhibit slip, at times to a surprising degree. Proposed mechanisms for slip include both intrinsic molecular slip as well as “effective” slip over complex and multiphase interfaces (i.e., covered with “lubricating” nanobubbles or trapped gas pockets). Electrokinetic flow velocities can be significantly enhanced over slipping surfaces as the high shear rates within the double layer drive high slip velocities at the interface. It is not clear, however, that surfaces whose effective slip results from a macroscale average of microscopically inhomogeneous slip will exhibit the same effective slip in an electrokinetically driven system. As well, since the surface charge density is generally set by surface chemistry, one might reasonably expect the ζ potential over “slipping” regions to be different than that over nonslip surfaces. Here, we consider electrokinetic flows over inhomogeneously slipping and inhomogeneously charged surfaces that exhibit a macroscopic effective slip. Using the Lorentz reciprocal theorem for the Stokes flow, we derive general relations for the electrokinetic slip velocity and flow rates over effectively slipping surfaces in the thin double-layer limit. We place particular emphasis on surfaces that consist of discrete slip and no-slip regions, which are meant to model either geometrically rough superhydrophobic surfaces or surfaces with nanobubbles. We derive several surprising results: (i) electro-osmotic flows with uniformly charged (i.e., constant ζ) liquids over an arbitrarily slipping surface are enhanced by precisely the same amount as would be found by naively assuming the (macroscopic) effective slip length to apply homogeneously; (ii) surfaces whose “slip” regions are uncharged (as one might expect should be the case) show no enhancement due to slip, instead giving precisely the same electro-osmotic flow as though the surface were nonslip and homogeneously charged. Our results have clear implications for the practical implementation of electrokinetic effects over effective slip surfaces and give an additional “handle” to probe the properties of surfaces that exhibit effective slip. We briefly discuss the effects of inhomogeneous slip upon electrokinetic pumps, surface conduction, and streaming currents.