  The classical problem of two uniformly charged infinite planes in
electrochemical equilibrium with an infinite monovalent salt reservoir is
solved exactly at the mean-field nonlinear Poisson-Boltzmann (PB) level,
including an explicit expression of the associated nonlinear electrostatic
contribution to the semi-grand-canonical potential. A linearization of the
nonlinear functional is presented that leads to Debye-H\"uckel-like equations
agreeing asymptotically with the nonlinear PB results in the weak-coupling
(high-temperature) and counterionic ideal-gas limits. This linearization scheme
yields artifacts in the low-temperature, large-separation or high-surface
charge limits. In particular, the osmotic-pressure difference between the
interplane region and the salt reservoir becomes negative in the above limits,
in disagreement with the exact (at mean-field level) nonlinear PB solution. By
using explicitly gauge-invariant forms of the electrostatic potential we show
that these artifacts -- although thermodynamically consistent with quadratic
expansions of the nonlinear functional -- can be traced back to the
non-fulfillment of the underlying assumptions of the linearization. Explicit
comparison between the analytical expressions of the exact nonlinear solution
and the corresponding linearized equations allows us to show that the
linearized results are asymptotically exact in the weak-coupling and
counterionic ideal-gas limits, but always fail otherwise, predicting negative
osmotic-pressure differences.
