Skip to main content
SpringerLink
Log in

Mathematical and Numerical Aspects of a Phase-field Approach to Critical Nuclei Morphology in Solids

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We investigate a phase-field model for homogeneous nucleation and critical nucleus morphology in solids. We analyze the mathematical properties of a free energy functional that includes the long-range, anisotropic elastic interactions. We describe the numerical algorithms used to search for the saddle points of such a free energy functional based on a minimax technique and the Fourier spectral implementation. It is demonstrated that the phase-field model is mathematically well defined and is able to efficiently predict the critical nucleus morphology in elastically anisotropic solids without making a priori 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. Anderson, D., McFadden, G., Wheeler, A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1998)

    Article  MathSciNet  Google Scholar 

  2. Brener, E., Iordanskii, S., Marchenko, V.: Elastic effects on the kinetics of a phase transition. Phys. Rev. Lett. 82, 1506–1509 (1999)

    Article  Google Scholar 

  3. Brezzi, F., Rappaz, J., Raviart, P.: Finite dimensional approximation of nonlinear problems part I: branches of nonsingular solutions. Numer. Math. 36, 1–25 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cahn, J., Hilliard, J.: Free energy of a nonuniform system, III. Nucleation in a two-component incompressible fluid. J. Chem. Phys. 31, 688–699 (1959)

    Article  Google Scholar 

  5. Chen, L.Q.: Phase-field models for microstructure evolution. Annu. Rev. Mater. Sci. 32, 113–140 (2002)

    Article  Google Scholar 

  6. Chu, Y., Moran, B., Reid, A., Olson, G.: A model for nonclassical nucleation of solid-solid structural phase transformations. Metall. Mater. Trans. A 31, 1321–1331 (2000)

    Article  Google Scholar 

  7. Conti, S., Schweizer, B.: A sharp-interface limit for a two-well problem in geometrically linear elasticity. Arch. Ration Mech. Anal. 179, 413–452 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Du, Q., Gunzburger, M., Peterson, J.: Analysis and approximation of the Ginzburg-Landau Model of superconductivity. SIAM Rev. 34, 54–81 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Du, Q., Liu, C., Wang, X.: A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198, 450–468 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Du, Q., Zhu, L.: Analysis of a mixed finite element method for a phase field elastic bending energy model of vesicle membrane deformation. J. Comput. Math. 24, 265–280 (2006)

    MATH  MathSciNet  Google Scholar 

  11. E, W., Ren, W., Vanden-Eijnden, E.: String method for the study of rare events. Phys. Rev. B 66, 052301 (2002)

    Article  Google Scholar 

  12. Eshelby, J.: The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. Lond. A 252, 561–569 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  13. Evans, L.C.: Partial Differential Equations. Am. Math. Soc., Providence (1998)

    MATH  Google Scholar 

  14. Feng, W., Yu, P., Hu, S., Liu, Z., Du, Q., Chen, L.: Spectral implementation of an adaptive moving mesh method for phase-field equations. J. Comput. Phys. 220, 498–510 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Feng, X., Prohl, A.: Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math. Comput. 73, 541–567 (2004)

    MATH  MathSciNet  Google Scholar 

  16. Fischer, S., Karplus, M.: Conjugate peak refinement: an algorithm for finding reaction paths and accurate transition states in systems with many degrees of freedom. Chem. Phys. Lett. 194, 252–261 (1992)

    Article  Google Scholar 

  17. Gagne, C., Gould, H., Klein, W., Lookman, T., Saxena, A.: Simulations of spinodal nucleation in systems with elastic interactions. Phys. Rev. Lett. 95, 095701 (2005)

    Article  Google Scholar 

  18. Garcke, H., Kwak, D.: On asymptotic limits of Cahn-Hilliard systems with elastic misfit. In: Analysis, Modeling and Simulation of Multiscale Problems, pp. 87–111. Springer, Berlin (2006)

    Chapter  Google Scholar 

  19. Garcke, H., Rumpf, M., Weikard, U.: The Cahn-Hilliard equation with elasticity, finite element approximation and qualitative analysis. Interfaces Free Bound. 3, 101–118 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gránásy, L.: Diffuse interface theory of nucleation. J. Non-Cryst. Solids 162, 301–303 (1993)

    Article  Google Scholar 

  21. Henkelman, G., Jonsson, H.: A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J. Chem. Phys. 111, 7010–7022 (1999)

    Article  Google Scholar 

  22. Henkelman, G., Uberuaga, B., Jonsson, H.: A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys. 113, 9901–9904 (2000)

    Article  Google Scholar 

  23. Hu, S., Chen, L.: A phase-field model for evolving microstructures with strong elastic inhomogeneity. Acta Mater. 49, 1879–1890 (2001)

    Article  Google Scholar 

  24. Ionova, I., Carter, E.: Ridge method for finding saddle points on potential energy surfaces. J. Chem. Phys. 98, 6377–6386 (1993)

    Article  Google Scholar 

  25. Jerrad, R., Sternberg, P.: Critical points via Γ-convergence: general theory and applications. Preprint (2007)

  26. Khachaturyan, A.: Theory of Structural Transformations in Solids. Wiley, New York (1983)

    Google Scholar 

  27. Kohn, R., Reznikoff, M., Tonegawa, Y.: The sharp-interface limit of the action functional for Allen-Cahn in one space dimension. Calc. Var. Partial Differ. Equ. 25, 503–534 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Leo, P., Lowengrub, J., Jou, H.: A diffuse interface model for microstructure evolution in elastically stressed solids. Acta Mater. 61, 2113–2130 (1998)

    Article  Google Scholar 

  29. Li, X., Thornton, K., Nie, Q., Voorhees, P., Lowengrub, J.: Two- and three-dimensional equilibrium morphology of a misfitting particle and the Gibbs-Thomson effect. Acta Mater. 52, 5829–5843 (2004)

    Article  Google Scholar 

  30. Li, Y., Zhou, J.: A minimax method for finding multiple critical points and its applications to semilinear PDE. SIAM J. Sci. Comput. 23, 840–865 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. Moré, J., Munson, T.: Computing mountain passes and transition states. Math. Program. 100, 151–182 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  32. Onuki, A.: Ginzburg-Landau approach to elastic effects in the phase-separation of solids. J. Phys. Soc. Jpn. 58, 3065–3068 (1989)

    Article  Google Scholar 

  33. Poduri, R., Chen, L.: Non-classical nucleation theory of ordered intermetallic precipitates—application to the Al-Li alloy. Acta Mater. 44, 4253–4259 (1996)

    Article  Google Scholar 

  34. Rabinowitz, P.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence (1986)

    Google Scholar 

  35. Roy, A., Rickman, J., Gunton, J., Elder, K.: Simulation study of nucleation in a phase-field model with nonlocal interactions. Phys. Rev. E 57, 2610–2617 (1998)

    Article  Google Scholar 

  36. Sagui, C., Somoza, A.M., Desai, R.: Spinodal decomposition in an order-disorder phase-transition with elastic fields. Phys. Rev. E 50, 4865–4879 (1994)

    Article  Google Scholar 

  37. Wang, Y., Chen, L.Q., Khachaturyan, A.G.: Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap. Acta Mater. 41, 279–296 (1993)

    Article  Google Scholar 

  38. Wang, Y., Khachaturyan, A.: Three-dimensional field model and computer modeling of martensitic transformations. Acta Mater. 45, 759–773 (1997)

    Article  Google Scholar 

  39. Yu, P., Hu, S., Du, Q., Chen, L.: An iterative-perturbation scheme for treating inhomogeneous elasticity in phase-field models. J. Comput. Phys. 208, 34–50 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  40. Zhang, L., Chen, L., Du, Q.: Morphology of critical nuclei in solid state phase transformations. Phys. Rev. Lett. 98, 265703 (2007)

    Article  Google Scholar 

  41. Zhang, L., Chen, L., Du, Q.: Diffuse-interface description of strain-dominated morphology of critical nuclei in phase transformations. Acta Mater. (2008). doi:10.1016/j.actamat.2008.03.043

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Penn State University, PA, 16802, Centre County, USA

    Lei Zhang

  2. Department of Materials Science and Engineering, Penn State University, PA, 16802, Centre County, USA

    Long-Qing Chen

  3. Department of Mathematics and Department of Materials Science and Engineering, Penn State University, PA, 16802, Centre County, USA

    Qiang Du

Authors
  1. Lei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Long-Qing Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qiang Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiang Du.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, L., Chen, LQ. & Du, Q. Mathematical and Numerical Aspects of a Phase-field Approach to Critical Nuclei Morphology in Solids. J Sci Comput 37, 89–102 (2008). https://doi.org/10.1007/s10915-008-9207-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-008-9207-7

Keywords

Search

Navigation