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Cahn–Hoffman ξ-Vector and Its Relation to Diffuse Interface Models of Phase Transitions

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In this paper we review two important theoretical areas to which J. W. Cahn has made major contributions: (i) The theory of the ξ-vector developed by Hoffman and Cahn, which provides an elegant setting for the description of the equilibrium shapes of sharp interfaces in the presence of anisotropic surface energy. (ii) Diffuse interface theories of phase transitions. We describe recent work which connects these two complementary facets of models of interfaces by the development of a generalized ξ-vector for diffuse interface models with anisotropic surface energy. We show that the generalized ξ-vector plays a central role in both the mathematical and physical aspects of a wide range of diffuse interface theories of interfaces with either anisotropic surface energy or attachment kinetics.

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REFERENCES

  1. D. W. Hoffman and J. W. Cahn, A vector thermodynamics for anisotropic surfaces. I. Fundamentals and application to plane surface junctions, Surf. Sci. 31:368–388 (1972).

    Google Scholar 

  2. J. W. Cahn and D. W. Hoffman, A vector thermodynamics for anisotropic surfaces. II. Curved and faceted surfaces, Acta Metall. Mater. 22:1205–1214 (1974).

    Google Scholar 

  3. G. Wulff, Z. Kristallog. 34:449 (1901).

    Google Scholar 

  4. Mathematical Crystallography (Oxford University Press, 1903).

  5. F. C. Frank, The geometrical thermodynamics of surfaces, in Metal Surfaces (American Society of Metals, Ohio, 1962), pp. 1–16.

    Google Scholar 

  6. W. W. Mullins, Solid surface morphologies governed by capillarity, in Metal Surfaces (American Society of Metals, Ohio, 1962), pp. 17–66.

    Google Scholar 

  7. P. G. de Gennes, Short range order effects in the isotropic phase of nematics and cholesterics, Mol. Cryst. Liq. Cryst. 12:193–214 (1971).

    Google Scholar 

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

    Google Scholar 

  9. J. W. Cahn, On spinodal decomposition, Acta Metall. 9:795–801 (1961).

    Google Scholar 

  10. J. W. Cahn and S. M. Allen, A microscopic theory for domain wall motion and its experimental verification in Fe-Al alloy domain growth kinetics, J. Phys.(Paris) Colloque C7, C7-51 (1977).

    Google Scholar 

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

    Google Scholar 

  12. R. J. Braun, J. W. Cahn, G. B. McFadden and A. A. Wheeler, Anisotropy of interfaces in an ordered alloy: a multiple-order-parameter model, Phil. Trans. Roy. Soc. London A 355:18331787 (1997).

    Google Scholar 

  13. J. S. Rowlinson and B. Widow, Molecular Theory of Capillarity (Clarendon, Oxford, 1989).

    Google Scholar 

  14. J. E. Taylor, Mean curvature and weighted mean curvature, Acta Metall. Mater. 40:1475–1485 (1992).

    Google Scholar 

  15. Lord Rayleigh. On the theory of surface forces.––II. Compressible fluids, Phil. Mag. 33:209–220 (1892).

    Google Scholar 

  16. J. D. van der Waals, Verhandel. Konink. Akad. Weten. Amsterdam (Sect. 1), 1(8) (Dutch); Transl. Ostwald, 1894, Thermodynamische theorie der kapillarität unter voraussetzung stetiger dichteänderung, Z. Phys. Chem. 13:657–725 (German); Transl. Anonymous, 1895, Arch. Néerl. 28:121–201 (French). Transl. J. S. Rowlinson, 1979, Translation of J. D. van der Waals's “The thermodynamic theory of capillarity under the hypothesis of a continuous density variation. ” J. Stat. Phys. 20:197–244 (From Dutch, German, French).

  17. B. I. Halperin, P. C. Hohenberg, and S.-K. Ma, Renormalization-group methods for critical dynamics: I. Recursion relations and effects of energy conservation, Phys. Rev. B 10:139–153 (1974).

    Google Scholar 

  18. J. S. Langer. Unpublished notes.

  19. J. S. Langer. Models of pattern formation in first-order phase transitions, Directions in Condensed Matter Physics, G. Grinstein and G. Mazenko, eds. (Philadelphia, World Scientific, 1986), pp. 165–186.

    Google Scholar 

  20. O. Penrose and P. C. Fife, Thermodynamically consistent models of the phase-field type for the kinetics of phase transitions, Physica D 43:44–62 (1990).

    Google Scholar 

  21. A. P. Umantsev, Thermodynamic stability of phases and transition kinetics under adiabatic conditions, J. Chem. Phys. 96:605–617 (1992).

    Google Scholar 

  22. S.-L. Wang, R. F. Sekerka, A. A. Wheeler, B. T. Murray, S. R. Coriell, R. J. Braun, and G. B. McFadden, Thermodynamically consistent phase-field models for solidification, Physica D 69:189–200, (1993).

    Google Scholar 

  23. P. C. Fife and O. Penrose, Interfacial dynamics for thermodynamically consistent phase-field models with nonconserved order parameter, Electronic J. Diff. equations 1:1–49 (1995).

    Google Scholar 

  24. C. Charach and P. C. Fife, On thermodynamically consistent schemes for phase-field equations. To appear Open Systems and Information Dynamics.

  25. S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (Dover, New York, 1984).

    Google Scholar 

  26. G. Caginalp and P. C. Fife. Phase-field methods for interfacial boundaries, Phys. Rev. B 33:7792–7794 (1986).

    Google Scholar 

  27. R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Physica D 63:410–423 (1993).

    Google Scholar 

  28. J. E. Taylor, Mean curvature and weighted mean curvature, Acta Metall. Mater. 40:1475–1485 (1992).

    Google Scholar 

  29. H. J. Diepers, C. Beckermann, and I. Steinbach, A phase-field method for alloy solidification with convection, in Solidification Processing 1997, J. Beech and H. Jones, eds., Proc. 4th Decennial Int. Conf. on Solid. Process. (University of Sheffield, Sheffield, 1997), pp. 426–430.

    Google Scholar 

  30. An Anisotropic Multi-phase-field Model: Interfaces and Junctions, Phys. Rev. E. 57:2602–2609 (1998).

  31. R. Kikuchi and J. W. Cahn, Theory of interphase and antiphase boundaries in fcc alloys, Acta Metall. 27:1337–1353 (1979).

    Google Scholar 

  32. J. W. Cahn and R. Kikuchi, Transition layer in a lattice-gas model of a solid-liquid interface, Phys. Rev. B 31:4300–4304 (1980).

    Google Scholar 

  33. S.-W. Lai, Theory of ordering dynamics for Cu3Au, Phys. Rev. B 41:9239–9256 (1990).

    Google Scholar 

  34. J. W. Cahn, S. Han and G. B. McFadden, J. Stat. Phys., this issue.

  35. G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equation, Phys. Rev. A 39:5887–5896 (1989).

    Google Scholar 

  36. N. Provatos, N. Goldenfeld and J. Dantzig, Efficient computation of dendritic microstructures using adaptive mesh refinement, Phys. Rev. Lett. 80:3308–3311 (1998).

    Google Scholar 

  37. G. Caginalp and P. C. Fife, Phase-field methods for interfacial boundaries, Phys. Rev. B 34:4940–4943 (1986).

    Google Scholar 

  38. G. Caginalp, The role of microscopic anisotropy in the macroscopic behaviour of a phase boundary, Ann. Phys. 172:136–155 (1986).

    Google Scholar 

  39. G. B. McFadden, A. A. Wheeler, R. J. Braun, S. R. Coriell and R. F. Sekerka, Phase-field models for anisotropic interfaces, Phys. Rev. E 48:2016–2024 (1993).

    Google Scholar 

  40. A. A. Wheeler and G. B. McFadden, A ξ-vector formulation of anisotropic phase-field models: 3D asymptotics, Eur. J. Appl. Maths. 7:367–381 (1996).

    Google Scholar 

  41. A. A. Wheeler and G. B. McFadden, On the notion of a ξ-vector and a stress tensor for a general class of anisotropic diffuse interface models, Proc. Roy. Soc. Lond. A 453:1611–1630 (1997).

    Google Scholar 

  42. P. C. Fife, private communication.

  43. R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume II (Interscience Publishers, New York, 1962).

    Google Scholar 

  44. D. Anderson, G. B. McFadden, and A. A. Wheeler, A phase-field model with convection. Preprint (1998).

  45. D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Ann. Rev. Fluid Mech. 30:139–165 (1998).

    Google Scholar 

  46. E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameters, Physica D 68:326–343 (1993).

    Google Scholar 

  47. E. Fried and M. E. Gurtin, Dynamic solid-solid phase transitions with phase characterized by an order parameter, Physica D 72:287–308 (1994).

    Google Scholar 

  48. M. E. Gurtin, The nature of configurational forces, Arch. Rat. Mech. Anal. 131:67–100 (1995).

    Google Scholar 

  49. Herbert Goldstein, Classical Mechanics (Addison–Wesley Publishing Company, 1980).

  50. D. Anderson, G. B. McFadden, and A. A. Wheeler, Unpublished notes (1998).

  51. D. J. Korteweg, Sur la forme que prennent les équations du mouvements des fluides si l'on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité, Arch. Néerl. Sci. Exactes Nat. Ser. II 6:1–24 (1901).

    Google Scholar 

  52. M. J. P. Musgrave, Crystal Acoustics; Introduction to the Study of Waves and Vibrations in Crystals (Holden-day, San Francisco, USA 1970).

    Google Scholar 

  53. F. C. Frank, in Growth and Perfection of Crystals, R. H. Doremus, B. W. Roberts, and D. Turnbull, eds. (Wiley, New York, 1958).

    Google Scholar 

  54. R. J. Braun, J. W. Cahn, G. B. McFadden, H. E. Rushmeier, and A. A. Wheeler, Theory of growth rates in the ordering of an FCC alloy, Acta Materr. 46:1–12 (1998).

    Google Scholar 

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  1. Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton, SO17 1BJ, U.K

    A. A. Wheeler

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Wheeler, A.A. Cahn–Hoffman ξ-Vector and Its Relation to Diffuse Interface Models of Phase Transitions. Journal of Statistical Physics 95, 1245–1280 (1999). https://doi.org/10.1023/A:1004575022280

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