Dynamic electrical response of hydrogel-loaded microchannels

Abstract

We compute the polymer-skeleton displacement and solvent velocity in long, hydrogel-filled, thin-slit channels subjected to an oscillatory, longitudinal electric field. This work is motivated by recent attempts to advance electrical microrheology as a diagnostic tool for probing the zeta-potential of colloidal particles immobilized in hydrogels. Within the standard, linearized electrokinetic model for elastic porous networks, we obtain an exact analytical solution, furnishing the skeleton displacement and fluid velocity. This reveals intricate spatial and temporal dynamics, reflecting balances between electrical, elastic, inertial, and viscous forces. Formulas are evaluated using multi-precision complex arithmetic, which is necessary to accommodate large length-scale disparities. Here, we explore the dynamics in open channels, for a small, but experimentally representative, set of dimensional parameters. Upon traversing inertial-viscous, inertial-elastic, and visco-elastic characteristic frequencies, charged hydrogels transit from low-frequency dynamics, where the solvent can be considered to migrate in a stationary skeleton, to high-frequency dynamics, where the skeleton translates within a practically stationary fluid. At the inertial-elastic frequency, which presents a primary resonance peak, O(nm) displacements are achieved under accessible experimental conditions. Therefore, in back-focal-plane interferometry experiments, which can register Ångström-level harmonic particle displacements in hydrogels, it will be prudent to account for the background spatial and temporal dynamics.