Chapter 9
The diffuse structure of the electrical double layer

When considering electrosmotic flows, Chapter 6 focuses on outer solutions, namely solutions for flow far from boundaries. In this limit, we describe electroosmosis using an effectiveslip boundary condition _wall = μ_EO. A 1D integral model of the surface shows that, if the fluid properties are assumed uniform and the electrical potential at the wall is different from the bulk by a factor of φ₀, then μ_EO is given by μ_EO = -εφ₀∕η. The inner distribution of velocity and electrical potential need not be determined.

In this chapter, we address the electrical double layer (also called the Debye layer) near a charged wall and evaluate the spatial variation of charge and potential in this double layer. This determines the equilibrium structure of the fluid boundary layer near a surface in an electroosmotically driven system and describes the spatial variation of velocity near the wall. In the process, we relate the Coulomb force (related to the total wall charge density) and the distribution of the Coulomb force (related to the Debye length) to the velocity distribution. The flows that result are the inner solutions of electroosmotic flow problems. In total, the equilibrium electrical double layer solution leads to predictions of fluid flow and current in electrically driven micro- and nanofluidic systems that do not perturb this equilibrium. As this chapter involves detailed discussion of ion concentrations, a review of the terminology and parameters found in Appendix B is recommended.

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