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This paper develops a quantitative method to analyze the influence of axial variations in the zeta potential on the electrophoretic separation in a microfluidic channel. The width of the microfluidic channel is assumed to be less than the Debye length, and the Debye length is much smaller than the channel height. A net zeta potential is induced at the channel wall, which decreases slowly along the channel. The induced zeta potential causes a transverse electric field, which is used to manipulate the motion of particles in the electric field. The velocity of the colloid in the electric field is calculated by the full Navier-Stokes equation. The ratio of the dispersion coefficient to that for a Newtonian fluid is used to characterize the influence of the axial variation of zeta potential on the electrophoretic separation. The axial variations in the zeta potential are found to notably modify the transverse distribution of the dispersion coefficient. The dependence of the dispersion coefficient on the ratio of the half height of the channel to the Debye length is also analysed. The combined electrical and hydrodynamic drag forces on a particle can be approximated by a second-order differential equation for a small-amplitude oscillatory perturbation of the zeta potential. The oscillatory solution of this equation is used to analyse the electrophoretic separation at a particle trajectory along the channel. The solution for the zeta potential for a sinusoidal variation of the zeta potential along the channel is expanded in a Fourier series, and the solution of the particle trajectory is analysed using the coefficients of the Fourier series for the zeta potential. The particle trajectory along the channel is found to be nearly parallel to the electric field. The results in this paper can be used to describe the electrophoretic separation as well as to optimize the design of microfluidic channels.
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