Direct Numerical Simulations of Electrophoresis of Charged Colloids

We propose a numerical method to simulate electrohydrodynamic phenomena in charged colloidal dispersions. This method enables us to compute the time evolutions of colloidal particles, ions, and host fluids simultaneously by solving Newton, advection-diffusion, and Navier-Stokes equations so that the electrohydrodynamic couplings can be fully taken into account. The electrophoretic mobilities of charged spherical particles are calculated in several situations. The comparisons with approximation theories show quantitative agreements for dilute dispersions without any empirical parameters; however, our simulation predicts notable deviations in the case of dense dispersions.

Figure 1

Relationship between surface charge $|Z|e$ and dimensionless zeta potential $y$ (a). Our numerical data follow nicely on the analytic solution of the nonlinear PB equation [34] but deviates notably from the Debye-Hückel linearized theory. Dimensionless mobility $E_m$ of a single particle is plotted in (b) as a function of dimensionless zeta potential $y$. For comparison, results of Smoluchowski, Henry, and O’Brien-White for $\kappa a=0.5$ are plotted. The color contours in (c) and (d) represent the total ionic charge density $\sum _{\alpha }e{Z}_{\alpha }{C}_{\alpha }$ around a single particle for (c) $Z=-100$ and (d) $Z=-500$. The electric field is applied in the horizontal ($+x$) direction.

Figure 2

Snapshots of the electrophoresis of dense dispersions with (a) fcc, (b) bcc, and (c) random particle configurations. The color map represents the total ionic charge density $\sum _{\alpha }e{Z}_{\alpha }{C}_{\alpha }$ in a plane perpendicular to the $z$ axis. The electric field is applied in the $+x$ direction normal to (1,0,0) face for fcc and bcc. See movies in Ref. 35.

Figure 3

The mobility coefficient $f(\kappa a,\phi )$ as a function of the volume fraction $\phi$ in (a) $\kappa a=1$ and (b) $\kappa a=0.5$. The solid lines represent the approximation theory proposed by Ohshima [31]. The theory is confirmed to be accurate for ${\phi }_{\mathrm{eff}}\le 1$, but tends to deviate from our numerical results for ${\phi }_{\mathrm{eff}}>1$ where overlapping of the electric double layers becomes notable.