  We present a derivation of generalized Poisson-Boltzmann equations starting
{from} classical theories of binary fluid mixtures, employing an approach based
on the Legendre transform as recently applied to the case of local descriptions
of the fluid free energy. Under specific symmetry assumptions, and in the
linearized regime, the Poisson-Boltzmann equation reduces to a phenomenological
equation introduced by (Bazant et al., 2011), whereby the structuring near the
surface is determined by bulk coefficients.
