  The electric double layer (EDL) that forms at the interface between a
polyelectrolyte gel and a salt bath is studied using asymptotic and numerical
methods. Specifically, matched asymptotic expansions, based on the smallness of
the Debye length relative to the typical gel dimensions, are used to construct
solutions of the governing equations and derive electroneutral models with
consistent jump conditions across the gel-bath interface. A general approach
for solving the equations of incompressible nonlinear elasticity in a curved
boundary layer is developed and used to resolve the gel mechanics in the EDL. A
critical feature of the model is that it accounts for phase separation within
the gel, which gives rise to diffuse interfaces with a characteristic thickness
described by the Kuhn length. We show that the solutions of the electroneutral
model can only be asymptotically matched to the solutions in the EDL, in
general, when the Kuhn length greatly exceeds the Debye length. Conversely, if
the Debye length is similar to or larger than the Kuhn length, then the entire
gel can self-organise into periodic, electrically charged domains via phase
separation. The breakdown of electroneutrality demonstrates that the commonly
invoked electroneutral assumption must be used with caution, as it generally
only applies when the Debye length is much smaller than the Kuhn length.
