  In this study we use non-equilibrium thermodynamics to systematically derive
a phase-field model of a polyelectrolyte gel coupled to a hydrodynamic model
for a salt solution surrounding the gel. The governing equations for the gel
account for the free energy of the internal interfaces which form upon phase
separation, the nonlinear elasticity of the polyelectrolyte network, and
multi-component diffusive transport following a Stefan--Maxwell approach. The
time-dependent model describes the evolution of the gel across multiple time
and spatial scales and so is able to capture the large-scale solvent flux and
the emergence of long-time pattern formation in the system. We explore the
model for the case of a constrained gel undergoing uni-axial deformations.
Numerical simulations show that rapid changes in the gel volume occur once the
volume phase transition sets in, as well as the triggering of spinodal
decomposition that leads to strong inhomogeneities in the lateral stresses,
potentially leading to experimentally visible patterns.
