16.1 Electroosmosis with temporally-varying interfacial potential [AC electrokinetics top]

We showed in Section 6.3.2 that outer solutions for electroosmosis can be predicted (for thin double layers) by replacing the body force applied to the electrical double layer with an effective slip. In this case, the net charge density source terms in the Navier-Stokes and Poisson equations are ignored, and their effects are replaced by an effective electroosmotic slip velocity boundary condition:

where φ₀ can be a function of space and time.

We now consider double layer potentials induced by voltages applied at electrodes, which requires that we solve the dynamic equations to predict the potential drop across the electrical double layer (φ₀) as a function of the voltage applied at the electrodes.

The time required for a double layer to charge up and reach its equilibrium state is central to predicting the electrokinetic response of systems with unsteady potential boundary conditions. Analytically, our ability to model these systems hinges on the spatial discretization of the system and the accuracy of the models used to describe the system or its elements. As a first step, we begin by identifying characteristic frequencies using equivalent electrical circuit models and use double layer models to predict the differential double layer capacitance. The predicted double layer capacitance (and therefore the predicted system response) is a strong function of the double layer model—in fact, the study of double layer capacitance is usually a much more rigorous test of double layer models than, for example, the study of electroosmosis. Thus more detailed models of the double layer lead to better predictions in the equivalent circuit framework.

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