Skip to main content

Advertisement

SpringerLink
Log in

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

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

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 \({\Gamma}\)-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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acerbi E., Bouchitté G.: A general class of phase transition models with weighted interface energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(6), 1111–1143 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Allen S.M., Cahn J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)

    Article  Google Scholar 

  3. Andelman, D.: Electrostatic properties of membranes: the Poisson–Boltzmann theory. In: Lipowsky, R., Sackmann, E. (eds.) Handbook of Biological Physics, vol. 1, pp. 603–642. Elsevier, Amsterdam 1995

  4. Berne B.J., Weeks J.D., Zhou R.: Dewetting and hydrophobic interaction in physical and biological systems. Annu. Rev. Phys. Chem. 60, 85–103 (2009)

    Article  ADS  Google Scholar 

  5. Cahn J.W., Hilliard J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)

    Google Scholar 

  6. Cai Q., Ye X., Luo R.: Dielectric pressure in continuum electrostatic solvation of biomolecules. Phys. Chem. Chem. Phys. 14, 15917–15925 (2012)

    Article  Google Scholar 

  7. Chandler D.: Interfaces and the driving force of hydrophobic assembly. Nature 437, 640–647 (2005)

    Article  ADS  Google Scholar 

  8. Che J., Dzubiella J., Li B., McCammon J.A.: Electrostatic free energy and its variations in implicit solvent models. J. Phys. Chem. B 112, 3058–3069 (2008)

    Article  Google Scholar 

  9. Cheng H.-B., Cheng L.-T., Li B.: Yukawa-field approximation of electrostatic free energy and dielectric boundary force. Nonlinearity 24, 3215–3236 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Cheng L.-T., Xie Y., Dzubiella J., McCammon J.A., Che J., Li B.: Coupling the level-set method with molecular mechanics for variational implicit solvation of nonpolar molecules. J. Chem. Theory Comput. 5, 257–266 (2009)

    Article  Google Scholar 

  11. Davis M.E., McCammon J.A.: Electrostatics in biomolecular structure and dynamics. Chem. Rev. 90, 509–521 (1990)

    Article  Google Scholar 

  12. Dzubiella J., Swanson J.M.J., McCammon J.A.: Coupling hydrophobicity, dispersion, and electrostatics in continuum solvent models. Phys. Rev. Lett. 96, 087802 (2006)

    Article  ADS  Google Scholar 

  13. Dzubiella J., Swanson J.M.J., McCammon J.A.: Coupling nonpolar and polar solvation free energies in implicit solvent models. J. Chem. Phys. 124, 084905 (2006)

    Article  ADS  Google Scholar 

  14. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton 1992

  15. Fonseca I., Morini M., Slastikov V.: Surfactants in foam stability: A phase-field model. Arch. Ration. Mech. Anal. 183(3), 411–456 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin 1983

  17. Giusti E.: Minimal Surfaces and Functions of Bounded Variation. Birkhauser, Boston (1984)

    Book  MATH  Google Scholar 

  18. Hasted J.B., Riston D.M., Collie C.H.: Dielectric properties of aqueous ionic solutions. parts I and II. J. Chem. Phys. 16, 1–21 (1948)

    Article  ADS  Google Scholar 

  19. Hutchinson J.E., Tonegawa Y.: Convergence of phase interfaces in the van der Waals–Cahn–Hilliard theory. Calc. Var. Partial Differ. Equ. 10, 49–84 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ilmanen T.: Convergence of the Allen–Cahn equation to Brakkes motion by mean curvature. J. Diff. Geom. 38, 417–461 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li B.: Minimization of electrostatic free energy and the Poisson–Boltzmann equation for molecular solvation with implicit solvent. SIAM J. Math. Anal. 40, 2536–2566 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li B., Cheng X.-L., Zhang Z.-F.: Dielectric boundary force in molecular solvation with the Poisson–Boltzmann free energy: a shape derivative approach. SIAM J. Appl. Math. 71, 2093–2111 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li B., Liu Y.: Diffused solute-solvent interface with Poisson–Boltzmann electrostatics: free-energy variation and sharp-interface limit. SIAM J. Appl. Math. 75, 2072–2092 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li B., Zhao Y.: Variational implicit solvation with solute molecular mechanics: from diffuse-interface to sharp-interface models. SIAM J. Appl. Math. 73(1), 1–23 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lum K., Chandler D., Weeks J.D.: Hydrophobicity at small and large length scales. J. Phys. Chem. B 103, 4570–4577 (1999)

    Article  Google Scholar 

  26. Mizuno M., Tonegawa Y.: Convergence of the Allen–Cahn equation with Neumann boundary conditions. SIAM J. Math. Anal. 47, 1906–1932 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Modica L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 123–142 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  28. Modica L., Mortola S.: Un esempio di \({\Gamma-}\) convergenza. Boll. Un. Mat. Ital. B 14(5), 285–299 (1977)

    MathSciNet  MATH  Google Scholar 

  29. Padilla P., Tonegawa Y.: On the convergence of stable phase transitions. Commun. Pure Appl. Math. LI, 551–579 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Röger M., Schätzle R.: On a modified conjuecture of De Giorgi. Math. Z. 254, 675–714 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rowlinson J.S.: Translation of J. D. van der Waals’ The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. J. Stat. Phys. 20, 197–244 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Sato N.: A simple proof of convergence of the Allen–Cahn equation to Brakkes motion by mean curvature. Indiana Univ. Math. J. 57, 1743–1751 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sharp K.A., Honig B.: Electrostatic interactions in macromolecules: theory and applications. Annu. Rev. Biophys. Chem., 19, 301–332 (1990)

    Article  Google Scholar 

  34. Sternberg P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101, 209–260 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sun H., Wen J., Zhao Y., Li B., McCammon J.A.: A self-consistent phase-field approach to implicit solvation of charged molecules with Poisson–Boltzmann electrostatics. J. Chem. Phys. 143, 243110 (2015)

    Article  ADS  Google Scholar 

  36. van der Waals, J.D.: Thermodynamische theorie der capillariteit in de onderstelling van continue dichtheidsverandering. Verhand. Kon. Akad. v Wetensch. Amst. Sect. 1 1893 (in Dutch)

  37. Wang Z., Che J., Cheng L.-T., Dzubiella J., Li B., McCammon J.A.: Level-set variational implicit solvation with the Coulomb-field approximation. J. Chem. Theory Comput. 8, 386–397 (2012)

    Article  Google Scholar 

  38. Xiao L., Cai Q., Ye X., Wang J., Luo R.: Electrostatic forces in the Poisson–Boltzmann systems. J. Chem. Phys. 139, 094106 (2013)

    Article  ADS  Google Scholar 

  39. Zhao Y., Kwan Y., Che J., Li B., McCammon J.A.: Phase-field approach to implicit solvation of biomolecules with Coulomb-field approximation. J. Chem. Phys. 139, 024111 (2013)

    Article  ADS  Google Scholar 

  40. Zhou H.X.: Macromolecular electrostatic energy within the nonlinear Poisson–Boltzmann equation. J. Chem. Phys. 100, 3152–3162 (1994)

    Article  ADS  Google Scholar 

  41. Zhou S., Cheng L.-T., Dzubiella J., Li B., McCammon J.A.: Variational implicit solvation with Poisson–Boltzmann theory. J. Chem. Theory Comput. 10(4), 1454–1467 (2014)

    Article  Google Scholar 

  42. Ziemer, W.P.: Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Springer, New York 2002

Download references

Author information

Author notes

  1. Shibin Dai

    Present address: Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, AL, 35487-0350, USA

Authors and Affiliations

  1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003, USA

    Shibin Dai

  2. Department of Mathematics and Quantitative Biology Graduate Program, University of California, San Diego, 9500 Gilman Drive, Mail Code: 0112, La Jolla, CA, 92093-0112, USA

    Bo Li

  3. Department of Mathematics, Department of Physics, and Department of Chemistry, Duke University, Box 90320, Durham, NC, 27708-0320, USA

    Jianfeng Lu

Authors
  1. Shibin Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianfeng Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bo Li.

Additional information

Communicated by I. Fonseca

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dai, S., Li, B. & Lu, J. Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation. Arch Rational Mech Anal 227, 105–147 (2018). https://doi.org/10.1007/s00205-017-1158-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-017-1158-4

Search

Navigation