  We study the classic problem of ion solvation within the continuum theory of
Dipolar-Poisson models. In this approach an ion is treated as a point charge
within a sea of point dipoles. Both the standard Dipolar-Poisson model as well
as the Dipolar-Poisson-Langevin model, which keeps the dipolar density fixed,
are non-convex functionals of the scalar electrostatic potential $\phi$.
Applying the Legendre transform approach introduced by A.C. Maggs [A.C. Maggs,
Europhys. Lett. 98, 16012 (2012)], the dual functionals of these models are
derived and are given by convex vector-field functionals of the dielectric
displacement and the polarization field. We show that the
Dipolar-Poisson-Langevin functional generalizes the harmonic polarization
functional used in the theory of Marcus for electron transfer rate to nonlinear
regimes and can be quantitatively parametrized by molecular dynamics
simulations for SPC/E-water.
