We develop a numerical scheme to compute the dynamic electrophoretic velocity of a uniformly charged, dielectric, spherical colloidal particle under an unsteady electric field, for, in principle, arbitrary imposed electric field strength, $\beta =a^{*}{e}^{*}{E}_{\infty }^{*}/\left(k_B^{*}{T}^{*}\right)$. Here, $a^{*}$ is the characteristic size of the particle, $E_{\infty }^{*}$ is the applied field strength, and the factor $k_B^{*}{T}^{*}/e^{*}$ corresponds to the thermal voltage. We focus our computations on Debye lengths comparable to the particle size; i.e., $1/\left({\kappa }^{*}{a}^{*}\right)=\mathcal{O}\left(1\right)$, where $1/{\kappa }^{*}$ is the Debye length, and moderately-charged particles, i.e., ${\sigma }^{*}=\mathcal{O}({\epsilon }^{*}{k}_B^{*}{T}^{*}/\left(e^{*}{a}^{*}\right))$, where ${\sigma }^{*}$ is the surface charge density on the colloid and ${\epsilon }^{*}$ is the permittivity of the medium. For a suddenly applied field, the initial growth of the electrophoretic mobility (i.e., ratio of particle speed to field strength) occurs on the momentum diffusion time and is independent of $\beta$ in the practically relevant case of large Schmidt number, $\mathrm{Sc}={\nu }^{*}/D^{*}\gg 1$, where ${\nu }^{*}$ is the kinematic viscosity and $D^{*}$ is the mean ion diffusion coefficient. Subsequently, the Debye cloud deforms on the ion diffusion timescale, leading to the mobility approaching its steady-state in a $\beta$-dependent manner. For the stopping problem, where the field is suddenly switched off, most of the change in the particle speed occurs on the momentum diffusion timescale, and the particle fully stops after the Debye cloud regains its equilibrium spherical shape. Under an oscillatory field, the field frequency has a large influence on both the amplitude and phase lag of the mobility, as it determines if the Debye cloud has sufficient time to deform within an oscillation cycle. The time-average flow about the particle resembles a stresslet.

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• Electrophoresis
• Ionic transport
• Colloids
• Stokes equations