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The Calculus of Boundary Variations and the Dielectric Boundary Force in the Poisson–Boltzmann Theory for Molecular Solvation

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Abstract

In a continuum model of the solvation of charged molecules in an aqueous solvent, the classical Poisson–Boltzmann (PB) theory for the electrostatics of an ionic solution is generalized to include the solute point charges and the dielectric boundary that separates the high-dielectric solvent from the low-dielectric solutes. With such a setting, we construct an effective electrostatic free-energy functional of ionic concentrations. The functional admits a unique minimizer whose corresponding electrostatic potential is the unique solution to the boundary-value problem of the nonlinear dielectric boundary PB equation. The negative first variation of this minimum free energy with respect to variations of the dielectric boundary defines the normal component of the dielectric boundary force. Together with the solute–solvent interfacial tension and van der Waals interaction forces, such boundary force drives an underlying charged molecular system to a stable equilibrium, as described by a variational implicit-solvent model. We develop an \(L^2\)-theory for boundary variations and derive an explicit formula of the dielectric boundary force. Our results agree with a molecular-level prediction that the electrostatic force points from the high-dielectric aqueous solvent to the low-dielectric charged molecules. Our method of analysis is general as it does not rely on any variational principles.

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Acknowledgements

BL was supported in part by the US National Science Foundation through Grant DMS-1913144, the US National Institutes of Health through Grant R01GM132106, and a 2019–2020 George W. and Carol A. Lattimer Research Fellowship, Division of Physical Sciences, University of California, San Diego. ZZ was supported in part by the Natural Science Foundation of Zhejiang Province, China, through Grant No. LY21A010011. SZ was supported by National Natural Science Foundation of China through Grant No. 21773165, Natural Science Foundation of Jiangsu Province, China, through Grant No. BK20200098, and National Key R&D Program of China through Grant 2018YFB0204404.

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Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, MC 0112, La Jolla, CA, 92093-0112, USA

    Bo Li

  2. College of Science, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, China

    Zhengfang Zhang

  3. School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China

    Shenggao Zhou

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  1. Bo Li
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  2. Zhengfang Zhang
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Li, B., Zhang, Z. & Zhou, S. The Calculus of Boundary Variations and the Dielectric Boundary Force in the Poisson–Boltzmann Theory for Molecular Solvation. J Nonlinear Sci 31, 89 (2021). https://doi.org/10.1007/s00332-021-09749-7

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