The Debye-Hückel approximation: its use in describing electroosmotic flow in micro- and nanochannels

In this work we consider the electroosmotic flow in a rectangular channel. We consider a mixture of water or other neutral solvent and a salt compound, such as sodium chloride, and other buffers for which the ionic species are entirely dissociated. Results are produced for the case where the channel height is much greater than the width of the electric double layer (EDL) (microchannel) and for the case where the channel height is of the order or slightly greater than the width of the EDL (nanochannel). At small cation, anion concentration differences the Debye-Hückel approximation is appropriate; at larger concentration differences, the Gouy-Chapman picture of the electric double emerges naturally. In the symmetric case, the velocity field and the potential are identical. We specifically focus in this paper on the limits of the Debye-Hückel approximation for a simplified version of a phosphate-buffered saline (PBS) mixture. The fluid is assumed to behave as a continuum and the volume flow rate is observed to vary linearly with channel height for electrically driven flow in contrast to pressure-driven flow which varies as height cubed. This means that very large pressure drops are required to drive flows in small channels. However, useful volume flow rates may be obtained at a very low driving voltage. In the course of the solution, we establish the relationship between the wall mole fractions of the electrolytes and the zeta potential. Multivalent electrolyte mixtures are also considered.