Electrophoresis at large peclet numbers

Abstract

The theory of concentration polarization of the double layer (DL) of a spherical particle, developed by J.Th.G. Overboek, F. Booth, S. Dukhin, V. Shilov, R.W. O'Brien, R.J. Hunter and others, in a linear approximation with respect to the external electric field, is generalized as applied to the regime of large Peclet numbers which is realized in strong electric fields. In this regime, the concentration field arising outside DL in its polarization is established not only in the process of diffusion but also under the effect of convection. The theory of concentration polarization is developed in terms of the convective diffusion equation. As with the linear treatment, polarization of DL at large Peclet numbers causes a change in the Stern potential, the formation of a dipole moment and the long-range potential whose distribution is described by a harmonic function. The peculiarity of polarization at large Peclet numbers consists of the formation of the distribution within the diffusion layer, which is not described by a harmonic function, and of the respective deviation from the electrical neutrality. As the external electric field grows, the initial diffuse layer deviates strongly from spherical symmetry and electric neutrality, and the screening of the surface charge is provided not only by the diffuse atmosphere but also by the charge originating in the diffusion layer, i.e. a secondary DL forms. The effect of the electric field on the outer plate of the secondary DL gives rise to the additional mechanism of electro-osmotic slip which is considered to be the secondary electro-osmosis. In strong electric fields, a nonlinear additional term appears for the Smoluchowski formula of electrophoretic velocity caused by the variation ofζ-potential over the surface of the particle and by the secondary electro-osmotic slip. The quantitative theory of the weak nonlinear concentration polarization of DL is developed and a correction term for the Smoluchowski formula based on this theory has been derived. The theory has also been developed for the strong concentration polarization of DL at which the deviation from the Smoluchowski formula is substantial. In the theory, it is assumed that the thickness of the diffuse DL which expanded due to the decrease of concentration polarization, is small compared with the thickness of the diffusion layer. It is also assumed to be possible to neglect the tangential component of the polarization field as compared with the external one. The fulfilment of these assumptions is provided by the restrictions imposed on the value of relaxation criterion Rel. With growing criterion Rel, we can expect the initiation of nonlinear effects to begin at a lower value of the electric field, which is most essential. The effect of the growth of the polarization space charge accelerating the electrophoresis can turn out to be more essential than the effect of the tangential polarization field slowing it down. In this case, the thickness of the diffuse layer and that of the diffusion one is closer together and the difficulty in constructing the theory increases sharply.