Journal
SIAM JOURNAL ON APPLIED MATHEMATICS
Volume 78, Issue 6, Pages 2973-2995Publisher
SIAM PUBLICATIONS
DOI: 10.1137/18M1167644
Keywords
electrostatic free-energy functionals; Legendre transforms; variational principles
Categories
Funding
- Frieda Daum Urey Academic and Oceanids Memorial Fellowship, University of California, San Diego
- National Science Foundation [DMS-1620487]
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In the Poisson-Boltzmann (PB) theory, the electrostatic free-energy functional of all possible electrostatic potentials for an ionic solution is often formulated in such a way that the Euler-Lagrange equation of such a functional is exactly the PB equation. However, such a PB functional is concave downward and maximized at its critical point, making it inconsistent in many applications where a macroscopic free-energy functional is minimized. Maggs [Europhys. Lett., 98 (2012), 16012] proposed a Legendre transformed form of the electrostatic free-energy functional of all possible dielectric displacements. This new functional is convex and minimized at the displacement corresponding to the critical point of the PB functional, and the minimum value is exactly the equilibrium electrostatic free energy. In this work, we study mathematically the Legendre transformed electrostatic free-energy functionals and the related variational principles. We first prove that the PB functional and its Legendre transformed functional are equivalent. We then consider a phenomeno-logical electrostatic free-energy functional that includes a higher-order gradient term, proposed by Bazant, Storey, and Kornyshev [Phys. Rev. Lett., 106 (2011), 046102] to describe charge-charge correlations. For such a functional, we introduce the corresponding Legendre transformed functional and obtain the related equivalence. We further consider the case without ions. We show that the electrostatic energy functional is equivalent to a Legendre transformed energy functional with constraint, and we prove the convergence of the Legendre transform of a perturbed electrostatic energy functional. Finally, we apply the Legendre transform to the dielectric boundary electrostatic free energy in molecular solvation.
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