Nanoparticle gel electrophoresis: Bare charged spheres in polyelectrolyte hydrogels

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

“…Therefore, when |ζ| ≲ k B T /e, the red lines agree with the theory of Li and Hill [23], which was undertaken-as a simplifying approximation-without the ion-concentration perturbations, also adopting the Debye-Hückel linearization. Together, the left panels in figure 1 show that ion-concentration perturbations, i.e., the so-called polarization and relaxation processes-decrease the intrinsic particle mobility |M * − M * eo | when |ζ| ≳ 2k B T /e, similarly to polarization and relaxation for spheres in Newtonian electrolytes [22] and uncharged gels [16].…”

“…This is due to the non-uniform charge in the gel that is induced by the particle charge, since otherwise this effect would be perfectly accounted for by the Brinkman factor (which accounts for hydrodynamic coupling under the influence of a uniform charge [5]). Indeed, this effect vanishes when κa ≪ 1, and can be captured when κa ≳ 1 by the fully linearized model of Li and Hill [23]. As demonstrated by electrokinetic-sonic-amplitude (ESA) experiments and theory for NP-doped gels [15], particle dynamics are dominated by non-hydrodynamic (e.g., adhesive and steric) particle-gel interactions when ℓ/a ≲ 2, so electrophoresis calculations with ℓ/a ≲ 2 are unlikely to have practical merit for interpreting gel electrophoresis experiments (and so have not been presented here).…”

“…With the computational parameters (number of grid points, radial extent of the domain, and adaptive mesh prescription) adopted in this work, the drag coefficient could be calculated to within about four significant-figures of the Brinkman formula (see below) over the full range of ionic strengths I. Electrophoretic mobilities (without the ion-concentration perturbations) were also compared to independent calculations in the literature for charged and uncharged gels [23]. A test of the terms arising from the ion-concentration perturbations was undertaken by comparison with independently calculated mobilities for charged gels (without regulation), provided by Professor S. A. Allison (private communication), using a hybrid methodology based on the works of Allison et al [25] and Allison et al [16].…”

“…This is especially significant for highly-charged particles, but practically negligible for the most weakly charged particles. Thus, under the conditions for which the Debye-Hückel approximation is valid, i.e., |ζ| ≲ 2k B T /e, M * − M * eo can be predicted using the various approximations explored by Li and Hill [23].…”

“…Li and Hill [23] calculated the electrical force for arbitrary diffuse-layer thicknesses (the sum of the foregoing forces (ii) and (iii)), but still subject to the Debye-Hückel approximation, also neglecting the electrolyte-ion concentration disturbances (see also Li et al [24] and Allison et al [25] for the soft-sphere analogue). These calculations assessed limitations of the earlier electrokinetic models for charged gels, also addressing the phenomenological superposition methodology proposed by Doane et al [5].…”