A dynamic model of polyelectrolyte gels

Abstract

We derive a model that couples mechanical and electrochemical effects of polyelectrolyte gels. The gel is assumed to be immersed in a fluid domain. As the gel swells and de-swells, the gel-fluid interface can move. Our model consists of a system of partial differential equations for mass and linear momentum balance of the polymer and fluid components of the gel, the Navier-Stokes equations in the surrounding fluid domain, and the Poisson-Nernst-Planck equations for the ionic concentrations on the whole domain. These are supplemented by a novel and general class of boundary conditions expressing mass and linear momentum balance across the moving gel-fluid interface. A salient feature of our model is that it satisfies a free energy dissipation identity, in accordance with the second law of thermodynamics. We also apply Onsager's variational principle to derive the dynamic equations. The linear stability calculation reveals some interesting features of mechanical gel and polyelectrolyte gel. Particularly, in a one-dimensional analysis of two ion species system, we find how the global exponential decay rate is associated with gel's and ions' intrinsic decay rates. Lastly, we present some simulation results of the one dimensional dynamic model, for which the asymptotic behavior matches with the theoratical calculations.