Nanoparticle gel electrophoresis: Soft spheres in polyelectrolyte hydrogels under the Debye–Hückel approximation

[Display omitted] •Electrokinetic theory furnishes the soft-sphere electrophoretic mobility in polyelectrolyte gels.•Mobility is sensitive to the gel charge and permeability.•Mobility is sensitive to the core dielectric constant when gels are charged.•Matlab routines are available for parametric studies and interpretation of experiments. A mathematical model for electrophoresis of polyelectrolyte coated nanoparticles (soft spheres) in polyelectrolyte hydrogels is proposed, and evaluated by comparison to literature models for bare-sphere gel electrophoresis and free-solution electrophoresis. The utilities of approximations based on the bare-particle electrophoretic mobility, free-solution mobility, and electroosmotic flow in hydrogels are explored. Noteworthy are the influences of the particle–core dielectric constant and overlap of the polyelectrolyte shell. The present theory, which neglects ion-concentration and charge-density perturbations, indicates that the gel electrophoretic mobilities of metallic-core nanoparticles in polyelectrolyte gels can be qualitatively different than for their non-metallic counterparts. These insights will be beneficial for interpreting nanoparticle gel-electrophoresis data, optimizing electrophoretic separations, and engineering nanoparticles for technological applications.