Communications in Mathematical Sciences

Volume 17 (2019)

Number 7

Variational approach to concentration dependent dielectrics with the Bruggeman model: Theory and numerics

Pages: 1949 – 1974



Xiang Ji (Department of Mathematics and Mathematical Center for Interdiscipline Research, Soochow University, Suzhou, Jiangsu, China)

Shenggao Zhou (Department of Mathematics and Mathematical Center for Interdiscipline Research, Soochow University, Suzhou, Jiangsu, China)


The structure of the electric double layer has long been described by the classical Poisson–Boltzmann (PB) theory, in which a uniform dielectric coefficient is often assumed. Experimental data and molecular simulations evidence that the effective dielectric coefficient decreases with local ionic concentrations. In this work, a variational approach is developed to incorporate nonlinear concentration-dependent dielectrics described by the Bruggeman equation in a modified PB theory. The Bruggeman equation, which takes into account contributions from both counterions and coions systematically, provides a closure to the modified PB theory. In addition to ionic size and valence, our theory introduces a new source of ion-specificity, i.e., the dielectric coefficient of hydrated ions, to the continuum modeling of electrostatics. Asymptotic analysis reveals the connection between the modified PB theory and previous linear decrement models, and derives a criterion for counterion saturation. Robust numerical methods with efficient acceleration techniques are proposed to solve the resulting coupled equations. Dielectric coefficients predicted by our theory show good agreement with the experimental data for homogeneous electrolytes. The dielectric decrement effect on the ionic structure of electric double layers is assessed in extensive numerical simulations. With ion-specific parameters, our theory predicts asymmetric camel-shape profiles of differential capacitance against applied potentials for electrolytes with low salinity, and asymmetric bell-shape profiles for electrolytes with high salinity. The impact of counterion saturation, due to steric effects or dielectric decrement, on the shape of differential capacitance profiles is demonstrated through analysis and numerical investigations. To further understand the effect of concentration-dependent dielectrics, the modified PB theory is also applied to study the distribution of counterions around charged cylinders with various dielectric coefficients.


concentration-dependent dielectrics, Bruggeman model, Poisson–Boltzmann theory, differential capacitance, contact value theorem

2010 Mathematics Subject Classification

35J20, 35J25, 35Q92, 49S05, 92C05

This work is supported by the grants NSFC 11601361 and 21773165, Natural Science Foundation of Jiangsu Province, China (BK20160302), and Soochow University Q410700415.

Received 20 October 2018

Accepted 1 July 2019

Published 6 January 2020