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We study the variation of the dielectric response of ionic aqueous solutionsas function of their ionic strength. The effect of salt on the dielectricconstant appears through the coupling between ions and dipolar water molecules.On a mean-field level, we account for any internal charge distribution ofparticles. The dipolar degrees of freedom are added to the ionic ones andresult in a generalization of the Poisson-Boltzmann (PB) equation called theDipolar PB (DPB). By looking at the DPB equation around a fixed point-like ion,a closed-form formula for the dielectric constant is obtained. We express thedielectric constant using the "hydration length" that characterizes thehydration shell of dipoles around ions, and thus the strength of the dielectricdecrement. The DPB equation is then examined for three additional cases:mixture of solvents, polarizable medium and ions of finite size. Employingfield-theoretical methods we expand the Gibbs free-energy to first order in aloop expansion and calculate self-consistently the dielectric constant. Forpure water, the dipolar fluctuations represent an important correction to themean-field value and good agreement with the water dielectric constant isobtained. For ionic solutions we predict analytically the dielectric decrementthat depends on the ionic strength in a non-linear way. Our prediction fitsrather well a large range of concentrations for different salts using only onefit parameter related to the size of ions and dipoles. A linear dependence ofthe dielectric constant on the salt concentration is observed at low salinity,and a noticeable deviation from linearity can be seen for ionic strength above1\,M, in agreement with experiments.

Dipolar Poisson-Boltzmann Approach to Ionic Solutions: A Mean Field and Loop Expansion Analysis