Travelling-wave gel dipolophoresis of hydrophobic conducting colloids

Abstract

A unified 'weak-field' formulation is provided for calculating the combined nonlinear effect of dielectrophoresis and the induced-charge electrophoresis (dipolophoresis) of polarized rigid hydrophobic spherical colloids freely suspended in an electrolyte-saturated Brinkman-hydrogel (porous) medium under a general (direct or alternating currents) non-uniform electric forcing. Explicit expressions for the modified total dipolophoretic mobility of a conducting (metallic) spherical colloid are given in terms of the Brinkman (Darcy), Navier slip, and Debye (screening) length scales. Also presented is a rigorous derivation of the Helmholtz-Smoluchowski slip velocity in terms of these three length scales, including the induced electroosmotic flow field around a hydrophobic rigid colloid embedded in a Brinkman medium that is forced by an arbitrary (non-uniform) ambient electric field. The available solutions for a free (non-porous) electrolyte solution under a uniform forcing and no-slip surface are obtained as limiting cases. For the purpose of illustration, we present and analyse some newly explicit solutions for the mobility and the associated induced-charge electroosmotic velocity field of a slipping colloid set in an effective (hydrogel) porous medium, which is exposed to an ambient 'sinusoidal' travelling-wave excitation depending on frequency and wave number.

Fig. 1

Contours of the a stream function and b corresponding velocity vector following Eq. (66) and for $\alpha =1$, where $w\left(\alpha ,\beta ,{\lambda }_0\right)$ was taken as 1/72 for the contour levels plotted in a. Only one quarter of the flow field plane is plotted as the particle is stationary in the short wave limit and hence other quarters are mirror images

Fig. 2

Variation of the colloid’s dimensionless mobility $U_1$ with the excitation’s dimensionless RC frequency $\mathrm{\Omega }$ for three dimensionless wave numbers k’s following Eq. (41), and a $\left(\alpha ,\beta \right)=\left(0,0\right)$, b $\left(\alpha ,\beta \right)=\left(1,0\right)$, c $\left(\alpha ,\beta \right)=\left(0,1\right)$, and d $\left(\alpha ,\beta \right)=\left(1,1\right)$, where ${\lambda }_0=1$