Electrokinetic flow in a capillary with a charge-regulating surface polymer layer

An analytical study of the steady electrokinetic flow in a long uniform capillary tube or slit is presented. The inside wall of the capillary
is covered by a layer of adsorbed or covalently bound charge-regulating polymer in equilibrium with the ambient electrolyte solution. In
this solvent-permeable and ion-penetrable surface polyelectrolyte layer, ionogenic functional groups and frictional segments are assumed to
distribute at uniform densities. The electrical potential and space charge density distributions in the cross section of the capillary are obtained
by solving the linearized Poisson–Boltzmann equation. The fluid velocity profile due to the application of an electric field and a pressure
gradient through the capillary is obtained from the analytical solution of a modified Navier–Stokes/Brinkman equation. Explicit formulas
for the electroosmotic velocity, the average fluid velocity and electric current density on the cross section, and the streaming potential in the
capillary are also derived. The results demonstrate that the direction of the electroosmotic flow and the magnitudes of the fluid velocity and
electric current density are dominated by the fixed charge density inside the surface polymer layer, which is determined by the regulation
characteristics such as the dissociation equilibrium constants of the ionogenic functional groups in the surface layer and the concentration of
the potential-determining ions in the bulk solution.