The Journal of Physical Chemistry, Vol. 82, No. 12, 1978
Mobility in Transverse Field
1417
Electrophoresis of Highly Charged Colloidal Cylinders in Univalent Salt Solutions. 1. Mobility in Transverse Field Dirk Stigter Western Regional Research Center, Agricultural Research Service, U.S. Depatfment of Agriculture, Albany, California 947 IO (Received January 30, 1978) Publication costs assisted by the U.S. Department of Agriculture
This paper deals with the theory of electrophoresis of a highly charged cylinder in a transverse electric field, neglecting end effects. The cylinder is suspended in a univalent salt solution, and the assumptions allow for polarization of the Gouy ionic atmosphere (relaxation effect). In solving the electrohydrodynamic problem the strategy is followed as developed by others for the colloidal sphere. It is possible to determine the fluid flow field completely from the boundary conditions, without explicit formulation of the hydrostatic pressure and without application of the balance of forces on the cylinder. The final result is a set of coupled integral equations which is solved numerically by an iteration method. In practice the treatment is important for randomly oriented cylinders. For this reason numerical results are expressed in terms of the reduced mobility of randomly oriented cylinders, with a 2:1 average between perpendicular and parallel orientation in the external field, as discussed further in the following paper.
Introduction Electrophoretic theory has a long history in colloid science. The methods used in this paper rest on a hierarchy of important developments tracing from von Smoluchowski,’ Huckel,2Henry,:j O ~ e r b e e kand , ~ Booth5 to Wiersema et al.617 The earlier treatments,l including that by Henry of the ~ y l i n d e rneglect ,~ the relaxation of the ionic atmosphere of the colloid and, hence, are valid mainly for low {potentials. The more recent paper^^-^ deal with the relaxation effect of the colloidal sphere. The present treatment of the cylinder includes the relaxation effect and is largely a modification of the work by Wiersema6 on spherical particles. The uniformly charged cylinder with a Gouy ionic atmosphere was introduced already in a previous paper8 in which some electrophoretic results for double helical DNA were reinterpreted. In this and the following paper a more detailed account is given of the electrophoretic theory, with further applications. The theory of cylindrical particles differs from that of spheres in that there is an orientational problem. In general, experimental values of the electrophoretic velocity U are average results. Instantaneous values for a single cylinder depend on its orientation with respect to the external field X. The mobility of the cylinder in perpendicular orientation is expressed as
vi lx = f i D t I 8 ~
(1.I) where P is the average potential in the shear surface of the cylinder, D is the dielectric constant, and 17 the viscosity of the surrounding solution. The numerical factor f l depends on the ratio of the radius a of the cylinder and the thickness of the ionic atmosphere, that is, the Debye length 1 / ~ It. varies between f L = 1 for KU 0 and f L = 2 for KU m. For intermediate values of KU Henry’s theory3 for f l is valid only when the ionic atmosphere retains its cylindrical symmetry during electrophoresis, that is, in the case of low potentials, e[/’lhT
