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).
Highlights
Analyses and computations of colloidal particle electrophoresis in a liquid-filled circular cylinder usually assume that the cylinder is infinitely long (Keh & Anderson 1985; Keh & Chiou 1996; Yariv & Brenner 2002, 2003; Hsu & Yeh 2007; Sherwood & Ghosal 2018)
The volumetric flow rate far from the particle is determined by the electroosmotic slip velocity created by the uniform electric field if the wall of the cylinder is charged
If several particles are present within a cylinder or channel of finite length, electrophoresis is faster than for a single particle (Misiunas & Keyser 2019)
Summary
Analyses and computations of colloidal particle electrophoresis in a liquid-filled circular cylinder usually assume that the cylinder is infinitely long (Keh & Anderson 1985; Keh & Chiou 1996; Yariv & Brenner 2002, 2003; Hsu & Yeh 2007; Sherwood & Ghosal 2018). Sherwood & Ghosal (2018) used a lubrication analysis to determine the electrophoretic velocity of a tightly fitting sphere inside an infinite cylinder They found a pressure difference Δp which was nominally O((R/h0)5/2), whereas the total force due to wall shear stresses was O((R/h0)3/2). A force balance on a cylindrical control volume containing the spherical particle was impossible in the limit h0/R → 0, unless the leading-order pressure difference Δp vanished, which required the electrophoretic velocity of the sphere to be USmol/2 This is the electrophoretic velocity in the limit h0/R → 0 determined analytically by Yariv & Brenner (2003) and numerically by Keh & Chiou (1996). The hydrodynamic problem is strongly dominated by the region of the narrow gap (more so than the electrostatic problem), and a solution in the narrow gap is all that we shall need
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