# Travelling-wave gel dipolophoresis of hydrophobic conducting colloids

## TL;DR

This paper presents a new theoretical model for the movement of hydrophobic colloids in a porous medium under electric fields.

## Contribution

A unified weak-field formulation for dipolophoresis in a Brinkman-hydrogel medium is introduced.

## Key findings

- Explicit expressions for dipolophoretic mobility are derived using Brinkman, Navier slip, and Debye length scales.
- A rigorous derivation of the Helmholtz–Smoluchowski slip velocity is provided for hydrophobic colloids in porous media.
- New solutions for mobility and electroosmotic velocity under sinusoidal excitation are presented.

## 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.