  We present two successive mean-field approximations for describing the
mechanical properties and the swelling equilibrium of polyelectrolyte gels in
contact with a salt solution. The first mean-field approximation reduces the
many-chain problem of a gel to a corresponding single chain problem. The second
mean-field step integrates out the degrees of freedom of the flexible chain and
the ions. It replaces the particle-based description of the polyelectrolyte
with suitable charge distributions and an effective elasticity term. These
simplifications result in a computationally very efficient Poisson-Boltzmann
cell-gel description. Despite their simplicity, the single chain cell-gel model
shows excellent and the PB model very good agreement with explicit molecular
dynamics simulations of the reference periodic monodisperse network model for
varying chain length, polymer charge fraction, and external reservoir salt
concentrations. Comparisons of our models to the Katchalsky model reveal that
our approach is superior for strongly charged chains and can also predict the
bulk moduli more accurately. We further discuss chain length polydispersity
effects, investigate changes in the solvent permittivity, and demonstrate the
robustness of our approach to parameter variations coming from several modeling
assumptions.
