Electrophoresis of tightly fitting spheres along a circular cylinder of finite length

Abstract

Electrophoresis of a tightly fitting sphere of radius $a$ along the centreline of a liquid-filled circular cylinder of radius $R$ is studied for a gap width $h_0=R-a\ll a$ . We assume a Debye length $\kappa ^{-1}\ll h_0$ , so that surface conductivity is negligible for zeta potentials typically seen in experiments, and the Smoluchowski slip velocity is imposed as a boundary condition at the solid surfaces. The pressure difference between the front and rear of the sphere is determined. If the cylinder has finite length $L$ , this pressure difference causes an additional volumetric flow of liquid along the cylinder, increasing the electrophoretic velocity of the sphere, and an analytic prediction for this increase is found when $L\gg R$ . If $N$ identical, well-spaced spheres are present, the electrophoretic velocity of the spheres increases with $N$ , in agreement with the experiments of Misiunas & Keyser (Phys. Rev. Lett., vol. 122, 2019, 214501).