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Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation

Publication ,  Journal Article
Dai, S; Li, B; Lu, J
Published in: Archive for Rational Mechanics and Analysis
January 1, 2018

We study a phase-field variational model for the solvation of charged molecules with an implicit solvent. The solvation free-energy functional of all phase fields consists of the surface energy, solute excluded volume and solute-solvent van der Waals dispersion energy, and electrostatic free energy. The surface energy is defined by the van der Waals–Cahn–Hilliard functional with squared gradient and a double-well potential. The electrostatic part of free energy is defined through the electrostatic potential governed by the Poisson–Boltzmann equation in which the dielectric coefficient is defined through the underlying phase field. We prove the continuity of the electrostatics—its potential, free energy, and dielectric boundary force—with respect to the perturbation of the dielectric boundary. We also prove the Γ -convergence of the phase-field free-energy functionals to their sharp-interface limit, and the equivalence of the convergence of total free energies to that of all individual parts of free energy. We finally prove the convergence of phase-field forces to their sharp-interface limit. Such forces are defined as the negative first variations of the free-energy functional; and arise from stress tensors. In particular, we obtain the force convergence for the van der Waals–Cahn–Hilliard functionals with minimal assumptions.

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Published In

Archive for Rational Mechanics and Analysis

DOI

EISSN

1432-0673

ISSN

0003-9527

Publication Date

January 1, 2018

Volume

227

Issue

1

Start / End Page

105 / 147

Related Subject Headings

  • General Physics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics
 

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Dai, S., Li, B., & Lu, J. (2018). Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation. Archive for Rational Mechanics and Analysis, 227(1), 105–147. https://doi.org/10.1007/s00205-017-1158-4
Dai, S., B. Li, and J. Lu. “Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation.” Archive for Rational Mechanics and Analysis 227, no. 1 (January 1, 2018): 105–47. https://doi.org/10.1007/s00205-017-1158-4.
Dai S, Li B, Lu J. Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation. Archive for Rational Mechanics and Analysis. 2018 Jan 1;227(1):105–47.
Dai, S., et al. “Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation.” Archive for Rational Mechanics and Analysis, vol. 227, no. 1, Jan. 2018, pp. 105–47. Scopus, doi:10.1007/s00205-017-1158-4.
Dai S, Li B, Lu J. Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation. Archive for Rational Mechanics and Analysis. 2018 Jan 1;227(1):105–147.
Journal cover image

Published In

Archive for Rational Mechanics and Analysis

DOI

EISSN

1432-0673

ISSN

0003-9527

Publication Date

January 1, 2018

Volume

227

Issue

1

Start / End Page

105 / 147

Related Subject Headings

  • General Physics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics