Dipolar Poisson-Boltzmann Equation: Ions and Dipoles Close to Charge Interfaces

We present an extension to the Poisson-Boltzmann model where the dipolar features of solvent molecules are taken explicitly into account. The formulation is derived at mean-field level and can be extended to any order in a systematic expansion. It is applied to a two-plate system with oppositely charged surfaces. The ion distribution and profiles in the dipolar order parameter are calculated and can result in a large correction to the interplate pressure.

Figure 1

The DPB rescaled dielectric constant ${ϵ}^{\mathrm{eff}}(z)/{ϵ}_{\mathrm{bulk}}$ and the dipole density $c_d(z)/c_d$ profiles between two oppositely charged plates at separation $d=20\AA$. The surface charge density is $\sigma =\mp e/50{\AA }^2$. The reservoir contains $1:1$ salt of concentration $c_s={10}^{-5}\mathrm{M}$ and dipoles of density $c_d=10\mathrm{M}$. The dielectric constant is rescaled with respect to its bulk value ${ϵ}_{\mathrm{bulk}}=18.2$. The profiles have a strong variation in the vicinity of the plates (up to 2 Å) and then saturate to a value that is somewhat higher than their bulk values.

Figure 2

The DPB calculated pressure $\Pi$ as a function of the interplate separation $10\le d\le 140\AA$ for two oppositely charged plates. All other system parameters are as in Fig. 1. We compare the DPB with the usual PB models by plotting their relative difference $({\Pi }_{\mathrm{DBP}}-{\Pi }_{\mathrm{PB}})/{\Pi }_{\mathrm{PB}}$. In the inset the DPB pressure is plotted in units of ${10}^{-2}{k}_{B}T/{\AA }^3$.