Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / Section D / 25 / b

In electrophoresis, an applied electric field drives a charged particle and its associated diffuse ionic atmosphere relative to the liquid. Electric forcing is balanced by viscous stresses, giving a steady velocity $\mathbf{v}={\mu }_e\mathbf{E}$ in the linear regime. The relevant electrokinetic potential is the zeta potential at the slipping plane, rather than necessarily the solid surface potential.

For a thin electric double layer, the surrounding layer develops an electro-osmotic flow and the Smoluchowski electrophoretic mobility is ${\mu }_e=ϵ\zeta /\eta$. The sign of $\zeta$ fixes the direction. This transport is different from dielectrophoresis, which can move a neutral polarizable particle in a nonuniform field.

Here $r\gg {\lambda }_D$, so the thin-double-layer result givesand therefore $v=3.13\times 10^{-5}$ m s${}^{-1}$, along the field for the positive zeta potential.

In the opposite limit $r\ll {\lambda }_D$, the Hückel electrophoretic mobility is ${\mu }_e=2ϵ\zeta /(3\eta )$, giving $2.08\times 10^{-8}$ m${}^2$ V${}^{-1}$ s${}^{-1}$, two-thirds of the thin-layer value at the same $\zeta$. Both asymptotic expressions are independent of radius at fixed zeta potential; the crossover depends on $r/{\lambda }_D$. These classical approximations neglect double-layer relaxation and strong-potential corrections.