Research Article Unsteady electroosmosis in a microchannel with Poisson–Boltzmann charge distribution The present study is concerned with unsteady electroosmotic flow (EOF) in a micro- channel with the electric charge distribution described by the Poisson–Boltzmann (PB) equation. The nonlinear PB equation is solved by a systematic perturbation with respect to the parameter l which measures the strength of the wall zeta potential relative to the thermal potential. In the small l limits (l51), we recover the linearized PB equation – the Debye–Hu¨ckel approximation. The solutions obtained by using only three terms in the perturbation series are shown to be accurate with errors o1% for l up to 2. The accurate solution to the PB equation is then used to solve the electrokinetic fluid transport equation for two types of unsteady flow: transient flow driven by a suddenly applied voltage and oscillatory flow driven by a time-harmonic voltage. The solution for the transient flow has important implications on EOF as an effective means for trans- porting electrolytes in microchannels with various electrokinetic widths. On the other hand, the solution for the oscillatory flow is shown to have important physical implica- tions on EOF in mixing electrolytes in terms of the amplitude and phase of the resulting time-harmonic EOF rate, which depends on the applied frequency and the electrokinetic width of the microchannel as well as on the parameter l. Keywords: Electroosmosis / Oscillatory EOF / Poisson–Boltzmann equation / Transient EOF DOI 10.1002/elps.201100181 1 Introduction The Poisson–Boltzmann (PB) equation is a highly nonlinear differential equation that describes the charge distribution of ions in an aqueous solution due to a charged solid surface. The nonlinearity of the equation is adequately measured by the parameter l which expresses the strength of the wall zeta potential relative to the thermal potential. The PB equation is important in numerous fields, especially in the electrostatic energy of molecules and electrokinetics [1, 2]. Most of the literatures are concerned with surfaces with low charge potentials (l51) in which the PB equation can be reduced to the simpler linear Debye–Hu¨ckel approximation (DHA). For higher charge potentials, the PB equation should be used for accuracy. Another important parameter is the electrokinetic width denoted by K, which measures the channel width relative to the thickness of the electric double layer (EDL) – Debye length. Steady EOF, a fully developed flow by neglecting the entrance effect, exhibits very much different efficiencies in transporting electrolytes at different Ks. It has been shown from the linear DHA [3, 4] that for large K we have a nearly plug flow of which the dimensionless flow rate Q is proportional to the cross-sectional area of the channel, while for small K the dimensionless flow rate Q is quadratic as K 2 in the leading-order behavior. In other words, a very fine partition of a microchannel would give a much smaller total flow rate since a finer channel makes K smaller, and the flow rate even smaller (scaled as K 2 ). The same issue will be examined in the present study for higher charge potentials. In order to understand the structure of the solutions with the physical parameters involved, it is very useful and helpful if we may obtain solutions to the PB equation in analytical forms. In the literature, there are very few analytic solutions to the PB equation though the equation has been solved numerically for a variety of problems. There is a closed-form solution for the single infinite plate and some complicated, infinite series solutions are available for the potential outside a circular cylinder [5, 6]. The authors of [7] suggested a patching of two regions, each described by an approximation of the PB equation. Subsequently, their Chien C. Chang 1,2 Chih-Yu Kuo 1 Chang-Yi Wang 1,3 1 Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan, ROC 2 Institute of Applied Mechanics and Taida Institute of Mathematical Sciences, National Taiwan University, Taipei, Taiwan, ROC 3 Department of Mathematics and Department of Mechanical Engineering, Michigan State University, East Lansing, MI, USA Received March 21, 2011 Revised August 7, 2011 Accepted August 16, 2011 Colour online: See the article online to view Figs. 1–8 in colour. Abbreviations: DHA, Debye–Hu ¨ ckel approximation; EDL, electric double layer; PB, Poisson–Boltzmann Correspondence: Dr. Chien C. Chang, Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan, ROC E-mail: mechang@iam.ntu.edu.tw Fax: 1886-2-23625238 & 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com Electrophoresis 2011, 32, 3341–3347 3341