Oscillatory response of charged droplets in hydrogels

Characterization of droplet-hydrogel interfaces is of crucial importance to engineer droplet-hydrogel composites for a variety of applications. In order to develop electrokinetic diagnostic tools for probing droplet-hydrogel interfaces, the displacement of a charged droplet embedded in a polyelectrolyte hydrogel exposed to an oscillating electric field is determined theoretically. The polyelectrolyte hydrogel is modeled as an incompressible, charged, porous, and elastic solid saturated with a salted Newtonian fluid. The droplet is considered an incompressible Newtonian fluid with no charges within the droplet. The droplet-hydrogel interface is modeled as a surface with the thickness of zero and the electrostatic potential ζ. The polymer is allowed to slide past the surface of the droplet, with the extent of the sliding quantified with the kinetic friction law, for which the tangential stress of the polymer is proportional to the relative velocity of the polymer and the droplet fluid. The standard regular perturbation method is used to obtain a leading-order solution in ζ-potential for the droplet harmonic displacement, neglecting inertial and temporal ion-concentration effects. It is found that the electrical response is modulated significantly by the polymer boundary condition at the droplet surface. At frequencies higher than a transitional frequency, the impact of the polymer boundary condition vanishes, and the response asymptotes to that with stick boundary condition for the polymer at the droplet surface. In addition, the oscillatory susceptibility of the droplet, defined as the ratio of the droplet displacement to the strength of an applied oscillatory non-electrical force, is determined theoretically; this theoretical study helps to evaluate the validity of the linear response theory used in interpretation of microrheological experiments done with droplets. It is observed that at frequencies less than the transitional frequency, the oscillatory susceptibility would indicate only slip boundary condition for the polymer at the inclusion surface unless the inclusion viscosity is infinity.