Nanoparticle gel electrophoresis: bare charged spheres in polyelectrolyte hydrogels

Nanoparticle gel electrophoresis has recently emerged as an attractive means of separating and characterizing nanoparticles. Consequently, a theory that accounts for electroosmotic flow in the gel, and coupling of the nanoparticle and hydrogel electrostatics and hydrodynamics, is required, particularly for gels in which the mesh size is comparable to or smaller than the particle radii. Here, we present an electrokinetic model for charged, spherical colloidal particles undergoing electrophoresis in charged (polyelectrolyte) hydrogels: the gel-electrophoresis analogue of Henry's theory for electrophoresis in Newtonian electrolytes. We compare numerically exact solutions of the model with several independent asymptotic approximations, identifying regions in the parameter space where these approximations are accurate or break down. As previously assumed in the literature, Henry's formula, modified by the addition of a constant electroosmotic flow mobility, is accurate only for nanoparticles that are small compared to the hydrogel mesh size. We derived an exact analytical solution of the full model by judiciously modifying the theory of Allison et al. for uncharged gels, drawing on the superposition methodology of Doane et al. to account for hydrogel charge. This furnishes accurate and economical mobility predictions for the entire parameter space. The present model suggests that nanoparticle size separations (with diameters ≲40 nm) are optimal at low ionic strength, with a gel mesh size that is selected according to the particle charging mechanism. For weakly charged particles, optimal size separation is achieved when the Brinkman screening length is matched to the mean particle size.