Impact of ion partitioning and double layer polarization on diffusiophoresis of a pH-regulated nanogel

Diffusiophoresis of a charge regulated spherical polyelectrolyte nanogel (PE) due to an externally imposed ionic concentration gradient is considered. The immobile charge density of the nanogel develops through the association/dissociation reactions of their inorganic functional groups. The nanogel is ion and fluid permeable with dielectric permittivity different from that of the surrounding electrolyte medium. This difference in dielectric permittivity creates an ion partitioning due to the difference in self energy of ions. The Nernst–Planck equation for ion transport and the Poisson equation (PNP) for the electric field are modified to take into account the ion partitioning effects. The diffusiophoresis mechanism is governed by the electrophoresis generated by the induced electric field and chemiphoresis develops due to the mitigation of counterions across the double layer of the nanogel. In addition, the convection dominated double layer polarization and the counterion condensation as well as the electroosmotic flow created by the gel immobile charge play a role in the diffusiophoresis. All these effects are incorporated through the modified PNP equations coupled with the Navier–Stokes equations. The governing equations in their full form are solved numerically through a control volume approach. Present computed solutions for the limiting cases are in good agreement with the existing solutions based on the first-order perturbation analysis. In order to illustrate the diffusiophoresis mechanism we have measured the induced electric field and effective charge density of the PE and analyzed its dependence on several electrokinetic parameters. The contribution due to chemiphoresis is low for PE compared to a rigid colloid. For a highly permeable PE the diffusiophoretic velocity increases and approaches a saturation for higher range of the PE fixed charge density. The ion partitioning effect depletes counterions in PE to manifests its diffusiophoretic velocity. The diffusiophoretic velocity of PE for pH > IEP is higher than the case for which pH < IEP.