A kinetic model of a polyelectrolyte gel undergoing phase separation

Abstract

In this study we propose a novel phase-field theory based on non-equilibrium thermodynamics that resolves both the macroscopic deformations and the internal structure of a polyelectrolyte gel immersed in an ionic solution. The governing equations for the gel account for its electro-chemical response, the nonlinear elasticity of its polyelectrolyte network, multi-component Stefan–Maxwell diffusion and the energy cost of internal interfaces that form upon phase separation. These equations are coupled to a hydrodynamic model for the surrounding ionic solution. The full time-dependent model describes the evolution of the gel-solution system across multiple time and spatial scales revealing the mechano-electro-chemical mechanisms that regulate phase separation of the gel, which results in the emergence of complex spatial patterns. The rich dynamics of this system are investigated for a constrained gel undergoing uni-axial deformations. We find that the regulation of phase separation in the gel-bath system is dependent on the competition between two physical length scales: the Debye and Kuhn lengths which characterise the thickness of electric double layers and diffuse interfaces, respectively. When the Kuhn length is much larger than the Debye length, the standard electroneutral assumption can be invoked. In this case, we show that large-scale solvent flux can result in the phase separation of the gel. Depending on the concentration of ions in the surrounding bath, swelling/deswelling of the gel occurs either via propagation of a front from the gel-bath interface or via front propagation combined with spinodal decomposition. In the limit where the Kuhn and Debye length are commensurate, our model predicts a novel mode of phase separation which results in the gel bulk organising into spatially localised stable charged domains that emanate from the Debye layer and propagate through the whole gel.