Orientation Distribution and Electrophoretic Motions of Rod-like Particles in a Capillary

We consider motions of charged rod-like particles in a capillary under the action of externally applied electric field as a model problem for the electrophoresis through a capillary. Since the electrophoretic velocity through the capillary is dependent upon the particle orientation relative to the electric field, the probability distribution function for the particle orientation is determined from the Smoluchowski equation. In the statistical equation considered here, the random rotary Brownian potential is balanced with the potentials from both electrical and hydrodynamic origins. First, the dipole moment associated with the asymmetrical distribution of particle surface charges and that induced by the external electric field are calculated by utilizing slender body theory. The results are strictly valid under the conditions that the contribution from double-layer distortion is negligible, which is typical of charged macromolecules in aqueous media. The dipole moments are expressed in terms of the particle charge (or zeta potential) distribution, external field strength, and aspect ratio of the particle. The two distinct dipole moments incorporate into the Smoluchowski equation for the particle orientation distribution which in turn determines the electrophoretic mobility. Finally, the particle velocity in a capillary is simply given as superposition of the electrophoretic velocity and the electroosmotic velocity which is driven by the surface charges on the capillary wall.