Electric-field-induced displacement of a charged spherical colloid embedded in an elastic Brinkman medium

Abstract

When an electric field is applied to an electrolyte-saturated polymer gel embedded with charged colloidal particles, the force that must be exerted by the hydrogel on each particle reflects a delicate balance of electrical, hydrodynamic and elastic stresses. This paper examines the displacement of a single charged spherical inclusion embedded in an uncharged hydrogel. We present numerically exact solutions of coupled electrokinetic transport and elastic-deformation equations, where the gel is treated as an incompressible, elastic Brinkman medium. This model problem demonstrates how the displacement depends on the particle size and charge, the electrolyte ionic strength, and Young's modulus of the polymer skeleton. The numerics are verified, in part, with an analytical (boundary-layer) theory valid when the Debye length is much smaller than the particle radius. Further, we identify a close connection between the displacement when a colloid is immobilized in a gel and its velocity when dispersed in a Newtonian electrolyte. Finally, we describe an experiment where nanometer-scale displacements might be accurately measured using back-focal-plane interferometry. The purpose of such an experiment is to probe physicochemical and rheological characteristics of hydrogel composites, possibly during gelation.

Steady electrical and micro-rheological response functions for uncharged colloidal inclusions in polyelectrolyte hydrogels

“…Boundary conditions for the polymer displacement are v = Z at r = a and v → 0 as r → ∞. Other boundary conditions (see Hill & Ostoja-Starzewski 2008) …”

“…Our continuum model for a polyelectrolyte hydrogel comprises three phases: a charged, soft, porous solid (polymer network), solvent (water) and ions (counterions and added salt) (Hill et al 2003;Hill & Ostoja-Starzewski 2008). The porous medium is modelled as a compressible linear elastic solid with a continuous uniform distribution of fixed charge, and the solvent as an incompressible Newtonian fluid.…”

“…Li et al (2004aLi et al ( ,b, 2006) developed a so-called multi-physic, multi-effect model that adopts the Poisson equation to handle electrostatics. Similarly, Hill & Ostoja-Starzewski's (2008) electrokinetic model for uncharged hydrogels with charged spherical inclusions extends the two-fluid model of Levine & Lubensky (2000, 2001a by including electrokinetic influences. Our work extends these electrokinetic models to polyelectrolyte hydrogels by including the electrostatic forces owing to fixed charge on the polymer.…”

“…Accordingly, few experiments have been reported (Mizuno et al 2000(Mizuno et al , 2001(Mizuno et al , 2004 and our understanding of the electric-field-induced displacement is poor. Hill & Ostoja-Starzewski (2008) undertook the first theoretical study of electric-field-induced particle displacement. They calculated the steady electricfield-induced response of a charged, spherical colloid embedded in incompressible, uncharged polymer gels, showing that sub-nanometre displacements prevail under typical experimental conditions.…”

Hydrogel charge regulation and electrolyte ion-concentration perturbations in nanoparticle gel electrophoresis

“…Sufficiently large or adhesive particles, which are immobile on the time scales of steady electrophoresis or diffusion experiments, have other measurable electrokinetic properties in hydrogels, including an ability to drive steady electroosmotic flow [7,8,9], undergo an electric-field-induced displacement [10,11,12], and to generate an electrokinetic sonic amplitude (ESA) [13,14,15].…”