Skip to main content
SpringerLink
Log in

Modeling of Phase Separation in Alloys with Coherent Elastic Misfit

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

Elastic interactions arising from a difference of lattice spacing between two coherent phases can have a strong influence on the phase separation (coarsening) behavior of alloys. If the elastic moduli are different in the two phases, the elastic interactions may accelerate, slow down or even stop the phase separation process. If the material is elastically anisotropic, the precipitates can be shaped like plates or needles instead of spheres and can arrange themselves into highly correlated patterns. Tensions or compressions applied externally to the specimen may have a strong effect on the shapes and arrangement of the precipitates. In this paper, we review the main theoretical approaches that have been used to model these effects and we relate them to experimental observations. The theoretical approaches considered are (i) “macroscopic” models treating the two phases as elastic media separated by a sharp interface, (ii) “mesoscopic” models in which the concentration varies continuously across the interface, and (iii) “microscopic” models which use the positions of individual atoms.

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. Y. Le Bouar, A. Loiseau, and A. G. Khachaturyan, Origin of chessboard-like structures in decomposing alloys. Theoretical model and computer simulations, Acta Mater. 46:2777–2788 (1998).

    Google Scholar 

  2. L. Q. Chen and L. J. Shen, Applications of semi-implicit Fourier-spectral method to phase-field equations, Computer Physics Commun. 108:147–158 (1998).

    Google Scholar 

  3. M. E. Gurtin and P. W. Voorhees, On the effects of elastic stress on the motion of fully faceted interfaces, Acta Mater. 46:2103–2112 (1998).

    Google Scholar 

  4. J. E. Guyer and P. W. Voorhees, Morphological stability and compositional uniformity of alloy thin films, J. Crystal Growth 187:150–165 (1998).

    Google Scholar 

  5. I. Gyöngy, private communication (1998).

  6. T. Koyama and T. Miyazaki, Computer simulation of phase decomposition in two dimensions based on a discrete type non-linear diffusion equation, Materials Transactions JIM 39:169–178 (1998).

    Google Scholar 

  7. J. K. Lee, Elastic stress and microstructural evolution, Materials Transactions JIM 39:114–132 (1998).

    Google Scholar 

  8. P. H. Leo, J. S. Lowengrub and H. J. Jou, A diffuse interface model for microstructural evolution in elastically stressed solids, Acta Mater. 46:2113–2130 (1998).

    Google Scholar 

  9. F. Léonard and R. C. Desai, Alloy decomposition and surface instabilities in thin films, Phys. Rev. B 57:4805–4815 (1998).

    Google Scholar 

  10. D. Y. Li and L. Q. Chen, Morphological evolution of coherent multivariant Ti11Ni14 preciptiates in Ti-Ni alloys under an applied stress––a computer simulation study, Acta Mater. 46:639–649 (1998).

    Google Scholar 

  11. D. Y. Li and L. Q. Chen, Computer simulation of stress-oriented nucleation and growth of theta′ precipitates in Al-Cu alloys, Acta Mater. 46:2573–2585 (1998).

    Google Scholar 

  12. A. M. Mebed and T. Miyazaki, Metall. and Mater. Trans. A 29:739–749 (1998). Computer simulation and experimental investigation of the spinodal decomposition in the beta Ti-Cr binary alloy system.

    Google Scholar 

  13. R. Mueller and D. Gross, The 3-D simulation of equilibrium morphologies of precipitates, Comp. Mater. Sci. 11:35–44 (1998).

    Google Scholar 

  14. P. Nielaba, P. Fratzl, and J. L. Lebowitz, Growth of ordered domains in a computer model with lattice misfit: to appear in J. Stat. Phys.

  15. I. Schmidt, R. Mueller, and D. Gross, The effect of elastic inhomogeneity on equilibrium and stability of a two particle morphology, Mechanics of Materials 30:181–196 (1998).

    Google Scholar 

  16. Y. Wang, D. Banarjee, C. C. Su, and A. G. Khachaturyan, Field kinetic model and computer simulation of ordered intermetallics from FCC solid solutions, Acta Mater. 46:2983–3001 (1998).

    Google Scholar 

  17. J.-H. Cho and A. J. Ardell, Coarsening of Ni3-Si precipitates in binary Ni-Si alloys at intermediate and large volume fractions, Acta Mater. 45:1393–1400 (1997).

    Google Scholar 

  18. H. J. Jou, P. H. Leo, and J. S. Lowengrub, Microstructural evolution in inhomogeneous elastic media, J. Comp. Phys. 131:109–148 (1997).

    Google Scholar 

  19. C. A. Laberge, P. Fratzl, and J. L. Lebowitz, Microscopic model for directional coarsening of precipitates in alloys under external load, Acta Mater. 45:3949–3961 (1997).

    Google Scholar 

  20. J. K. Lee, Morphology of coherent precipitates via a discrete atom method, Mater. Sci and Eng. A 238:1–12 (1997).

    Google Scholar 

  21. J. K. Lee, Studying stress-induced morphological evolution with the discrete atom method, JOM––Journal of the Minerals, Metals and Materials Society 49:37 (1997).

    Google Scholar 

  22. A. Malik, B. Schönfeld, G. Kostorz, W. Bührer, and J. S. Pedersen, Early stages of decomposition in Al-Ag and Al-Cu, Z. Metallkde 88:625–629 (1997).

    Google Scholar 

  23. T. Ohashi, K. Hidaka, and S. Imano, Elastic stress in single-crystal Ni-base superalloys and the driving force for their microstructural evolution under high-temperature creep conditions, Acta Mater. 45:1801–1810 (1997).

    Google Scholar 

  24. O. Paris, M. Fährmann, E. Fä hrmann, T. M. Pollock, and P. Fratzl, Early stages of precipitate rafting in a single crystal Ni-Al-Mo alloy investigated by small-angle X-ray scattering and TEM, Acta Mater. 45:1085–1097 (1997).

    Google Scholar 

  25. I. Schmidt and D Gross, The equilibrium shape of an elastically inhomogeneous inclusion, J. Mech. Phys. Solids 45:1521–1549 (1997).

    Google Scholar 

  26. M. E. Thompson and P. W. Voorhees, Spinodal decomposition in elastically anisotropic homogeneous systems in the presence of an applied traction, Modelling and Simulation in Mater. Sci. and Eng. 5:223–243 (1997).

    Google Scholar 

  27. M. Véron and P. Bastie, Strain induced directional coarsening in nickel-based superalloys: investigation on kinetics using the small angle neutron scattering (SANS) technique, Acta Mater. 45:3277–3282 (1997).

    Google Scholar 

  28. P. Vyskocil, J. Skov Pedersen, G. Kostorz, and B. Schönfeld, Small-angle neutron scattering of precipitates in Ni-Ti alloys––I. Metastable states in poly-and single crystals, Acta Mater. 45:3311–3318 (1997).

    Google Scholar 

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

    Google Scholar 

  30. P. Fratzl and O. Penrose, Ising model for phase separation in alloys with anisotropic elastic interaction––II. A computer experiment, Acta Mater. 44:3227–3239 (1996).

    Google Scholar 

  31. J. E. Guyer and P. W. Voorhees, Morphological instability of alloys and thin films, Phys Rev. B 54:11710–11724 (1996).

    Google Scholar 

  32. W. Hort and W. C. Johnson, Effect of uniaxial stress on coarsening of precipitate clusters, Met. and Mater. Trans. 27A:1461–1476 (1996).

    Google Scholar 

  33. H. Ikeda and H. Matsuda, Computer simulation of phase decomposition process generating precipitates harder than matrix, Mater. Trans. JIM 37:1413–1421 (1996).

    Google Scholar 

  34. M. Kato, T. Fujii, and S. Onaka, Elastic strain energies of sphere, plate and needle inclusions, Mater. Sci and Eng. A––Structural Materials 211:95–103 (1996).

    Google Scholar 

  35. J. K. Lee, A study on coherent strain and precipitate morphology via a discrete atom method, Metall. and Mater. Trans. 27A:1449–1459 (1996).

    Google Scholar 

  36. F. R. N. Nabarro, C. M. Cress, and P. Kotschy, The thermodynamic driving force for rafting in superalloys, Acta Mater. 44:3189–3198 (1996).

    Google Scholar 

  37. F. R. N. Nabarro, Inclusions and inhomogeneities under stress, Phil. Mag. Lett. 73:45–49 (1996).

    Google Scholar 

  38. G. Sauthoff, Influences of stresses on precipitation, Jour. de Physique IV 6(C1):87–97 (1996).

    Google Scholar 

  39. C. H. Su and P. W. Voorhees, The dynamics of precipitate evolution in elastically stressed solids––I. Inverse coarsening, Acta Mater. 44:1987–2000 (1996).

    Google Scholar 

  40. C. H. Su and P. W. Voorhees, The dynamics of precipitate evolution in elastically stressed solids––II. Particle alignment, Acta Mater. 44:2001–2016 (1996).

    Google Scholar 

  41. J. Svoboda and P. Lukáč, Modeling of kinetics of directional coarsening in Ni-superalloys, Acta Mater. 44:2557–2565 (1996).

    Google Scholar 

  42. Y. Z. Wang, L. Q. Chen, and A. G. Khachaturyan, Three-dimensional dynamic simulation of the equilibrium shape of a coherent tetragonal precipitate in Mg-partially stabilized cubic ZrO2, J. Amer. Ceramic Soc. 79:987–991 (1996).

    Google Scholar 

  43. T. A. Abinandanan and W. C. Johnson, Development of spatial correlations during coarsening, Mat. Sci. Eng. B 32:169–176 (1995).

    Google Scholar 

  44. J. Y. Buffière and M. Ignat, A dislocation based criterion for the raft formation in nickelbased superalloy single crystals, Acta Metall. Mater. 43:1791–1797 (1995).

    Google Scholar 

  45. J. W. Cahn and R. Kobayashi, Exponentially rapid coarsening and buckling in coherently self-stressed thin plates, Acta Met. Mater. 43:931–944 (1995).

    Google Scholar 

  46. M. Fährmann, P. Fratzl, O. Paris, E. Fährmann, and W. C. Johnson, Influence of coherency stress on microstructural evolution in model Ni-Al-Mo alloys, Acta Metall. Mater. 43:1007 (1995).

    Google Scholar 

  47. P. Fratzl and O. Penrose, Ising model for phase separation in alloys with anisotropic elastic interactions––I. Theory, Acta Metall. Mater. 43:2921–2930 (1995).

    Google Scholar 

  48. V. Gröger, P. Fratzl, W. Pahl, O. Paris, G. Bischof, and G. Krexner, Phase boundary structure of γ′-particles in Cu-10 at % Be, Acta Metall. Mater. 43:1305–1311 (1995).

    Google Scholar 

  49. J. E. Guyer and P. W. Voorhees, Morphological stability of alloys and thin films, Phys. Rev. Letters 74:4031–4034 (1995).

    Google Scholar 

  50. A. G. Khachaturyan, S. Semenovskaya, and T. Tsakalakos, Elastic strain energy of inhomogeneous solids, Phys. Rev. B 52:15909–15919 (1995).

    Google Scholar 

  51. T. Koyama, T. Miyazaki, and A. E. Mebed, Computer simulations of phase decomposition in real alloy systems based on the modified Khachaturyan diffusion equation, Metal. Mater. Trans. A 26:2617–2623 (1995).

    Google Scholar 

  52. C. A. Laberge, P. Fratzl, and J. L. Lebowitz, Elastic effects on phase segregation in alloys with external stresses, Phys. Rev. Lett. 75:4448–4451 (1995).

    Google Scholar 

  53. J. K. Lee, Coherence strain analysis via a discrete atom method, Scripta Met. et Mat. 32:559–564 (1995).

    Google Scholar 

  54. O. Paris, M. Fährmann, and P. Fratzl, Breaking of rotational symmetry during decomposition of elastically anisotropic alloys, Phys. Rev. Lett. 75:3458–3461 (1995).

    Google Scholar 

  55. O. Paris, F. Langmayr, G. Vogl, and P. Fratzl, A possible criterion for slowing down of precipitate coarsening due to elastic misfit interactions, Z. Metallkd. 86:860 (1995).

    Google Scholar 

  56. I. Schmidt and D. Gross, A strategy for determining the equilibrium shape of an inclusion, Arch. Mech. 47:379–390 (1995).

    Google Scholar 

  57. A. D. Sequiera, H. A. Calderon, G. Kostorz, and J. S. Pedersen, Bimodal size distribution of γ′ precipitates in Ni-Al-Mo––I. Small-angle neutron scattering, Acta Metall. Mater. 43:3427–3439 (1995).

    Google Scholar 

  58. A. D. Sequiera, H. A. Calderon, G. Kostorz, and J. S. Pedersen, Bimodal size distribution of γ′ precipitates in Ni-Al-Mo––II. Transmission electron microscopy, Acta Metall. Mater. 43:3441–3451 (1995).

    Google Scholar 

  59. K.-I. Udoh, A. M. El Araby, Y. Tanaka, K. Hisatsune, K. Yasuda, G. van Tendeloo, and J. van Landuyt, Structural aspects of AuCu I or AuCu II and a cuboidal block configuration of f.c.c. disordered phase in Au-Cu-Pt and AuCu-Ag, Mater. Sci. Eng. A 203:154–164 (1995).

    Google Scholar 

  60. Y. Wang and A. G. Khachaturyan, Shape instablity during precipitate growth in coherent solids, Acta Metall. Mater. 43:1837–1857 (1995).

    Google Scholar 

  61. Y. Z. Wang, H. Y. Wang, L. Q. Chen, and A. G. Khachaturyan, Microstructural development of coherent tetragonal precipitates in magnesium-partially stabilized zirconia––a computer simulation, J. Amer. Ceramic Soc. 78:657–661 (1995).

    Google Scholar 

  62. Y. Wang and A. G. Khachaturyan, Microstructural evolution during the precipitation of ordered intermetallics in multiparticle coherent systems, Phil. Mag. A 72:1161–1171 (1995).

    Google Scholar 

  63. N. D. Alikakos, P. W. Bates, and X. F. Chen, Convergence of the Cahn–Hilliard equation to the Hele-Shaw model, Arch. Rat. Mech. Anal. 128:165–205 (1994).

    Google Scholar 

  64. D. J. Arrell and J. L. Vallés, Interfacial dislocation based criterion for the prediction of rafting behavior in superalloys, Scripta Metall. Mater. 30:149–53 (1994).

    Google Scholar 

  65. P. Fratzl and O. Paris, Internal oxidation of Cu-Fe. II. The morphology of oxide inclusions from the minimization of elastic misfit energy, Acta Metall. Mater. 42:2027–2033 (1994).

    Google Scholar 

  66. W. Hort and W. C. Johnson, Diffusional boundary conditions during coarsening of elastically interacting precipitates, Metall and Met. Trans. A 25:2695–2703 (1994).

    Google Scholar 

  67. F. Langmayr, P. Fratzl, and G. Vogl, Crossover from ω-phase to α-phase precipitation in bcc Ti-Mo, Phys. Rev. B 49:11759–11766 (1994).

    Google Scholar 

  68. G. Muralidharan, J. E. Epperson, M. Petri, and Haydn Chen, Coherency strains and coarsening in Ni-Al-Si alloys: an experimental study, Solid-Solid Phase Transformations, W. C. Johnson, J. M. Howe, D. E. Laughlin, and M. A. Soffa, eds. (The Minerals, Metals & Materials Society, 1994).

  69. O. Paris, P. Fratzl, F. Langmayr, G. Vogl, and H. G. Haubold, Internal oxidation of Cu-Fe. I. Small-angle X-ray scattering study of oxide precipitation, Acta Metall. Mater. 42:2019–2026 (1994).

    Google Scholar 

  70. T. M. Pollock and A. S. Argon, Directional coarsening in nickel-base single crystals with high volume fractions of coherent precipitates, Acta Metall. Mater. 42:1859–1874 (1994).

    Google Scholar 

  71. Y. Y. Qu, H. A. Calderon, and G. Kostorz, Coarsening of coherent γ′-precipitates in a Ni-Al-Mo alloy, Solid-Solid Phase Transformations, W. C. Johnson, J. M. Howe, D. E. Laughlin, and W. A. Soffa, eds. (The Minerals, Metals & Materials Society, 1994).

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

    Google Scholar 

  73. M. E. Thompson, C. S. Su, and P. W. Voorhees, The equilibrium shape of a misfitting precipitate, Acta Metall. 42:2107–2122 (1994).

    Google Scholar 

  74. J. L. Vallés and D. J. Arrell, Monte Carlo simulation of anisotropic coarsening in nickelbase superalloys, Acta Metall. Mater. 42:2999–3008 (1994).

    Google Scholar 

  75. T. A. Abinandanan and W. C. Johnson, Coarsening of elastically interacting coherent particles––I. Theoretical formulation, Acta Metall. Mater. 41:17–25 (1993).

    Google Scholar 

  76. T. A. Abinandanan and W. C. Johnson, Coarsening of elastically interacting coherent particles––II. Simulations of preferential coarsening and particle migrations, Acta Metall. Mater. 41:27–39 (1993).

    Google Scholar 

  77. H. Ikeda and H. Matsuda, Effects of differences of elastic moduli between constituents on spinodal decomposition processes, Mater. Trans. JIM 34:651–657 (1993).

    Google Scholar 

  78. P. H. Leo and H. J. Jou, Shape evolution of an initially circular precipitate growing by diffusion in an applied stress field, Acta Metall. Mater. 41:2271–2281 (1993).

    Google Scholar 

  79. A. Maheshwari and A. J. Ardell, Morphological evolution of coherent misfitting precipitates in anisotropic elastic media, Phys. Rev. Lett. 70:2305–2308 (1993).

    Google Scholar 

  80. T. Miyazaki and T. Koyama, Stability against coarsening in elastically constrained many-particle systems, Mater. Sci and Eng. A 169:159–65 (1993).

    Google Scholar 

  81. S. Nambu and A. Sato, Elastic effect on domain morphology and kinetics of spinodal decomposition in the tetragonal system, J. Amer. Ceramic Soc. 76:1978–84 (1993).

    Google Scholar 

  82. S. Socrate and D. M. Parks, Numerical determination of the elastic driving force for directional coarsening in Ni-superalloys, Acta Metall. Mater. 41:2185–2209 (1993).

    Google Scholar 

  83. B. J. Spencer, P. W. Vorhees, and S. H. Davis, Morphological instability in epitaxially strained dislocation-free solid films––linear stability theory, J. Appl. Phys. 73:4955–4970 (1993).

    Google Scholar 

  84. B. J. Spencer, S. H. Davis, and P. W. Voorhees, Morphological instability in epitaxially strained dislocation-free solid films––nonlinear evolution, Phys. Rev. B 47:9760–9777 (1993).

    Google Scholar 

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

    Google Scholar 

  86. M. Doi and T. Miyazaki, A new parameter for describing the structure bifurcation in two-phase alloys containing coherent precipitates, J. Mater. Sci. 27:6291–6298 (1992).

    Google Scholar 

  87. W. C. Johnson and P. W. Voorhees, Elastically-induced precipitate shape transitions in coherent solids, Solid State Phenomena 23–24:87–103 (1992).

    Google Scholar 

  88. W. C. Johnson and P. W. Voorhees. In Nonlinear Phenomena in Materials Science, G. Martin and L. Kubin, eds. (Trans Tech, Clauthal, Germany, 1992).

    Google Scholar 

  89. F. C. Larché and J. W. Cahn, Phase changes in a thin plate with nonlocal self-stress effects, Acta Met. Mater. 40:947–955 (1992).

    Google Scholar 

  90. M. McCormack, A. G. Khachaturyan, and J. W. Morris, Jr., A two-dimensional analysis of the evolution of precipitates in elastic media, Acta Met. Mat. 40:325–336 (1992).

    Google Scholar 

  91. M. McCormack and J. W. Morris, Jr., Strain-induced shape shange of cubic preciptiates in a cubic matrix with positive anisotropy, Acta Met. Mat. 40:2489–2945 (1992).

    Google Scholar 

  92. H. Nishimori and A. Onuki, Evolution of soft domains in two-phase alloys: shape changes, surface instability and network formation, Phys. Lett. A 162:323–326 (1992).

    Google Scholar 

  93. R. Shneck, R. Alter, A. Brokman, and M. P. Daniel, Fundamentals of the anisotropy of elastic interactions between dilating particles in a cubic material, Phil. Mag. A 65:797–814 (1992).

    Google Scholar 

  94. A. Takeuchi, T. Koyama, and T. Miyazaki, Computer simulation of the phase decomposition of Al-Zn and Fe-Mo alloys based on the nonlinear diffusion equation (in Japanese), J. Japan Institute of Metals 57:492–500 (1993), see also Mater. Sci. Eng. A 169:159–165 (1993).

    Google Scholar 

  95. P. W. Voorhees, G. B. McFadden, and W. C. Johnson, On the morphological development of second-phase particles in elastically stressed solids, Acta Metall. Mater. 40:2979–2992 (1992).

    Google Scholar 

  96. P. W. Voorhees, Ostwald ripening of two-phase mixtures, Ann Rev. Mat. Sci. 22:197–215 (1992).

    Google Scholar 

  97. Y. Wang, L. Q. Chen, and A. G. Khachaturyan, Particle translational motion and reverse coarsening phenomena in multiparticle systems induced by long-range elastic interactions, Phys. Rev. B 46:11194–11197 (1992).

    Google Scholar 

  98. K. Binder, Spinodal decomposition. In Materials science and technology, P. Haasen, ed., Vol. 5, VCH (Weinheim, New York, 1991), p. 405.

    Google Scholar 

  99. J. C. Chang and S. M. Allen, Elastic energy changes accompanying γ′ rafting in nickelbase superalloys, J. Mater. Res. 6:1843–1855 (1991).

    Google Scholar 

  100. L.-Q. Chen and A. G. Khachaturyan, Computer simulation of structural transformations during precipitation of an ordered intermetallic phase, Acta Met. Mater. 39:2533–2551 (1991).

    Google Scholar 

  101. L.-Q. Chen and A. G. Khachaturyan, Scripta Met. Mater. 25:61–66 (1991).

    Google Scholar 

  102. P. Fratzl, J. L. Lebowitz, O. Penrose, and J. Amar, Scaling functions, self similarity, and the morphology of phase separating systems, Phys. Rev. B 44:4794–4811 (1991).

    Google Scholar 

  103. C. Leroux, A. Loiseau, D. Broddin, and G. van Tendeloo, Electron microscopic study of the coherent two-phase mixtures Ll0+Ll1 in Co-Pt alloys, Phil. Mag. B 64:57–82 (1991).

    Google Scholar 

  104. H. Nishimori and A. Onuki, Freezing of domain growth in cubic solids with elastic misfit, J. Phys. Soc. Japan 60:1208–1211 (1991).

    Google Scholar 

  105. A. Onuki and H. Nishimori, On Eshelby's elastic interaction in two-phase solids, J. Phys. Soc. Japan 60:1–4 (1991).

    Google Scholar 

  106. A. Onuki, Interface motion in two-phase solids with elastic misfits, J. Phys. Soc. Japan 60:345–348 (1991).

    Google Scholar 

  107. A. Onuki and H. Nishimori, Anomalously slow domain growth due to a modulus inhomogeneity in phase-separating alloys, Phys. Rev. B 43:13649–13652 (1991).

    Google Scholar 

  108. O. Penrose, A mean-field equation for the dynamic Ising model, J. Stat. Phys. 63:975–986 (1991).

    Google Scholar 

  109. B. J. Spencer, P. W. Voorhees, and S. H. Davis, Morphological instability in epitaxially strained dislocation-free solid films, Phys. Rev. Lett. 67:3696–3699 (1991).

    Google Scholar 

  110. Y. Wang, L.-Q. Chen, and A. G. Khachaturyan, Shape evolution of a precipitate during strain-induced coarsening––a computer simulation, Scripta Met. et Mat. 25:1387–1392 (1991).

    Google Scholar 

  111. Y. Wang, L.-Q. Chen, and A. G. Khachaturyan, Strain-induced modulated structures in two-phase cubic alloys, Scripta Met. et Mat. 25:1969–1974 (1991).

    Google Scholar 

  112. H. Calderon and G. Kostorz, Lattice misfit and decomposition in Ni-Al-Mo alloys. In Neutron Scattering in Materials Science, S. M. Shapiro, S. C. Moss, and S. D. Jorgensen, eds., Mat. Res. Soc. Symp. Proc. 166, p. 255 (1990).

  113. A. Cerri, B. Schönfeld, and G. Kostorz, Decomposition kinetics in Ni-Ti alloys, Phys. Rev. B 42:958–960 (1990).

    Google Scholar 

  114. W. C. Johnson, T. A. Abinandanan, and P. W. Voorhees, The coarsening kinetics of two misfitting particles in an anisotropic crystal, Acta Metall. Mater. 38:1349–1367 (1990).

    Google Scholar 

  115. P. H. Leo, W. W. Mullins, R. F. Sekerka, and J. Viñals, Effect of elasticity on late stage coarsening, Acta Metal. Mater. 38:1573–1580 (1990).

    Google Scholar 

  116. H. Nishimori and A. Onuki, Pattern formation in phase separating alloys with cubic symmetry, Phys. Rev. B 42:980–983 (1990).

    Google Scholar 

  117. A. Onuki. In Formation, Dynamics and Statistics of Patterns, K. Kawasaki, M. Suzuki and A. Onuki, eds. (World Scientific, Sinagapore, 1990).

    Google Scholar 

  118. B. Caroli, C. Caroli, B. Roulet, and P. W. Voorhees, Effect of elastic stresses on the morphological stability of a solid sphere growing from a supersaturated melt, Acta Met. 37:257–268 (1989).

    Google Scholar 

  119. A. Chakrabarti, R. Torál, and J. D. Gunton, Late stages of spinodal decomposition in a three-dimensional model system, Phys. Rev. B 39:4386–4394 (1989).

    Google Scholar 

  120. J. G. Conley, M. E. Fine, and J. R. Weertmann, Effect of lattice disregistry variation on the late stage phase transformation behavior of precipitates in Ni-Al-Mo alloys, Acta Metall. 37:1251 (1989).

    Google Scholar 

  121. C. M. Elliott, pp. 35–74 of Mathematical Models for Phase Change Problems, J. Rodrigues, ed., Int. Ser. Num. Math. Vol. 88 (Birkhauser, Stuttgart, 1989).

    Google Scholar 

  122. Y. Enomoto and K. Kawasaki, Computer simulation of Ostwald ripening with elastic field interactions, Acta Metall. 37:1399–1406 (1989).

    Google Scholar 

  123. J. Gayda and D. J. Srolovitz, A Monte Carlo finite element model for strain energy controlled microstructural evolution: “Rafting” in superalloys, Acta Metall. 37:641–650 (1989).

    Google Scholar 

  124. U. Glatzel and M. Feller-Kriepmayer, Calculations of internal stresses in the γ/γ′ microstructure of a nickel-base superalloy with high volume fraction of the #$-phase, Scripta Metall. 23:1839–1844 (1989).

    Google Scholar 

  125. H. Hein, Nucleation, growth and coarsening in Ni-5.0 at % Al-5.8 at % Ti, Acta Metall. 37:2145 (1989).

    Google Scholar 

  126. W. C. Johnson, P. W. Voorhees, and D. E. Zupon, The effects of elastic stress on the kinetics of Ostwald ripening: the two-particle problem, Metall. Trans. 20A:1175 (1989).

    Google Scholar 

  127. I. M. Kaganova and A. L. Roitburd, An anisotropic crystalline inclusion in an isotropic matrix, Sov. Phys. Crystall. 34:650–653 (1989) (English translation of Kristalografia 34:1076–82 (1988)).

    Google Scholar 

  128. M. J. Kaufman, P. W. Voorhees, W. C. Johnson, and F. S. Biancaniello, An elastically induced morphological instability of a misfitting precipitate, Metall. Trans. A 20A:2171–2175 (1989).

    Google Scholar 

  129. V. J. Laraia, W. C. Johnson, and P. W. Voorhees, Kinetics of Ostwald ripening in stressed solids––the low volume-fraction limit, Scripta Met. 23:1749–1754 (1989).

    Google Scholar 

  130. P. H. Leo and R. F Sekerka, The effect of surface stress on crystal–melt and crystal–crystal equilibrium, Acta Metall. 37:3119–3138 (1989).

    Google Scholar 

  131. P. H. Leo and R. F Sekerka, The effect of elastic fields on the morphological stability of a precipitate grown from solid solution, Acta Metall. 37:3139–3149 (1989).

    Google Scholar 

  132. T. Miyazaki and M. Doi, Shape bifurcations in the coarsening of precipitates in elastically constrained systems, Mat. Sci. Eng. A 110:175–185 (1989).

    Google Scholar 

  133. W. W. Mullins and J. Viñals, Self-similarity and growth kinetics driven by surface free energy reduction, Acta Metall. 37:991–997 (1989).

    Google Scholar 

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

    Google Scholar 

  135. A. Onuki, Long-range interactions through elastic fields in phase-separating solids, J. Phys. Soc. Japan 58:3069–3072 (1989).

    Google Scholar 

  136. R. L. Pego, Front migration in the nonlinear Cahn–Hilliard equation, Proc. Roy. Soc. A 422:261–278 (1989).

    Google Scholar 

  137. D. J. Srolovitz, On the stability of surfaces of stressed solids, Acta Metall. 37:621–625 (1989).

    Google Scholar 

  138. W. C. Johnson, M. B. Berkenpas, and D. E. Laughlin, Precipitate shape transitions during coarsening under uniaxial stress, Acta Met. 36:3149–3162 (1988).

    Google Scholar 

  139. I. M. Kaganova and A. L. Roitburd, Equilibrium between elastically interacting phases, Soviet Physics JETP 67:1173–1183 (1988).

    Google Scholar 

  140. K. Kawasaki and Y. Enomoto, Statistical theory of Ostwald ripening with elastic field interactions, Physica A 150:463–498 (1988).

    Google Scholar 

  141. A. G. Khachaturyan, S. V. Semenovskaya, and J. W. Morris, Jr., Theoretical analysis of strain-induced shape changes in cubic precipitates during coarsening, Acta Metall. 36:1563–1572 (1988).

    Google Scholar 

  142. V. J. Laraia, W. C. Johnson, and P. W. Voorhees, Growth of a coherent precipitate from a supersaturated solution, J. Mater. Res. 3:257–266 (1988).

    Google Scholar 

  143. Y. Oono and S. Puri, Study of phase-separation dynamics by use of cell dynamical systems I. modelling, Phys. Rev. A 38:434–453 (1988).

    Google Scholar 

  144. C. Rottman, P. W. Voorhees, and W. C. Johnson, The Gibbs–Thomson equation for a spherical coherent precipitate with applications to nucleation, Scripta Met. 22:293–298 (1988).

    Google Scholar 

  145. P. W. Voorhees and W. C. Johnson, Development of spatial correlations during diffusional late-stage phase transformations in stressed solids, Phys. Rev. Lett. 61:2225–2228 (1988).

    Google Scholar 

  146. W. C. Johnson, Precipitate shape evolution under applied stress––thermodynamics and kinetics, Metall. Trans. A 18A:233–247 (1987).

    Google Scholar 

  147. W. C. Johnson and P. W. Voorhees, Elastic interaction and stability of misfitting cuboidal inhomogeneities, J. Appl. Phys. 61:1610–1619 (1987).

    Google Scholar 

  148. I. M. Kaganova and A. L. Roitburd, Equilibrium shape of an inclusion in a solid, Sov. Phys. Dokl. 32:925–7 (1987).

    Google Scholar 

  149. S. Yoshida, M. Fukaya, and T. Miyazaki, J. Japan Inst. Metals 51:18 (1987).

    Google Scholar 

  150. M. B. Berkenpas, W. C. Johnson, and D. E. Laughlin, The influence of applied stress on precipitate shape and stability, J. Mater. Res. 1:635–645 (1986).

    Google Scholar 

  151. M. A. Grinfeld, Construction of a physically linear theory of coherent phase transformations, Izv. A. N. SSSR Mekhanika Tverdogo Tela 21(5):79–91 (1986); English translation in Mechanics of Solids 21:84–96 (1986).

    Google Scholar 

  152. W. C. Johnson and J. J. D. Alexander, Interfacial conditions for thermomechanical equilibrium in two-phase crystals, J. Appl. Phys. 59:2735–2746 (1986).

    Google Scholar 

  153. P. W. Voorhees and W. C. Johnson, Interfacial equilibrium during a first-order phase transformation in solids, J. Chem. Phys. 84:5108–5121 (1986).

    Google Scholar 

  154. T. Miyazaki, K. Seki, M. Doi, and T. Kozakai, Stability bifurcation in the coarsening of precipitates in elastically constrained systems, Mater. Sci. Eng. 77:125–132 (1986).

    Google Scholar 

  155. W. W. Mullins, The statistical self-similarity hypothesis in grain growth and particle coarsening, J. Appl. Phys. 59:1341–1349 (1986).

    Google Scholar 

  156. A. L. Roitburd, Phase equilibrium in solids, Soviet Phys. Solid State 28:1716–1718 (1986). (English translation of Fiz. Tverd. Tela 28:3051–3054).

    Google Scholar 

  157. J. B. Walsh, An introduction to stochastic partial differential equations, École d'été de probabilites de St. Flour XIV, P. L. Henneguin, ed., Lecture Notes Math 1180 (Springer, Berlin, 1986), pp. 265–437.

    Google Scholar 

  158. M. Doi, T. Miyazaki, and T. Wakatsuki, The effects of elastic interaction energy on the γ′ precipitate morphology of continuously cooled nickel-base alloys, Mater. Sci. Eng. 74:139–145 (1985).

    Google Scholar 

  159. H. Furukawa, Adv. Phys. 34:703 (1985).

    Google Scholar 

  160. A. L. Roitburd, Thermodynamics of solid solution precipitation, Sov. Phys. Solid State 27:982–90 (1985) (English translation of Fiz. Tverd. Tela 27:982–90).

    Google Scholar 

  161. P. W. Voorhees, The theory of Ostwald ripening, J. Stat. Phys. 38:231–252 (1985).

    Google Scholar 

  162. J. W. Cahn and F. C. Larcheé, A simple model for coherent equilibrium, Acta Met. 32:1915–1923 (1984).

    Google Scholar 

  163. M. Doi, T. Miyazaki, and T. Wakatsuki, The effect of elastic interaction energy on the morphology of γ′ precipitates in nickel-based alloys, Mater. Sci and Eng. 67:247–253 (1984).

    Google Scholar 

  164. H. Furukawa, Dynamic scaling theory for phase-separating unmixing mixtures: growth rates of droplets and scaling properties of autocorrelation functions, Physica A 123:497–515 (1984).

    Google Scholar 

  165. W. C. Johnson, On the elastic stabilization of precipitates against coarsening under applied load, Acta Metall. 32:465–475 (1984).

    Google Scholar 

  166. W. C. Johnson and J. W. Cahn, Elastically induced shape bifurcations of inclusions, Acta Metall. 32:1925–1933 (1984).

    Google Scholar 

  167. J. D. Gunton, M. San Miguel, and P. S. Sahni, The dynamics of first order phase transformations, Phase Transitions and Critical Phenomena, C. Domb and J. L. Lebowitz, eds., Vol. 8 (Academic Press, New York, 1983), pp. 267–466.

    Google Scholar 

  168. J. D. Gunton and M. Droz, Introduction to the Theory of Metastable and Unstable States, Lecture Notes in Physics, No. 183 (Springer-Verlag, 1983).

  169. W. C. Johnson, Elastic interaction of two precipitates subjected to an applied stress, Metall. Trans. A 14A:2219–2227 (1983).

    Google Scholar 

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

    Google Scholar 

  171. J. W. Morris, A. G. Khachaturyan, and S. H. Wen, The elastic contribution to the thermodynamics of phase transitions in solids, Proceedings of the International Conference on Solid–Solid Phase Tranformations Held at Pittsburgh, Pa, August 1981 (cited by Khachaturyan(170)).

  172. J. W. Cahn and F. C. Larché, Surface stress and the chemical equilibrium of single crystals. II. Solid particles imbedded in a solid matrix, Acta Metall. 30:51–56 (1982).

    Google Scholar 

  173. F. C. Larché and J. W. Cahn, The effect of self-stress on diffusion in solids, Acta Metall. 30:1835–1845 (1982).

    Google Scholar 

  174. T. Miyazaki, H. Imamura, and T. Kozakai, The formation of “γ′ precipitate doublets” in Ni-Al alloys and their energetic stability, Mat. Sci Eng. 54:9–15 (1982).

    Google Scholar 

  175. T. Mura, Micromechanics of Defects in Solids (Martinus Nijhoff, the Hague, 1982).

    Google Scholar 

  176. S. Wen, E. Kostlan, M. Hong, A. Khachaturyan, and J. W. Morris, The preferred habit of a tetragonal inclusion in a cubic matrix, Acta Met. 29:1247–1254 (1981).

    Google Scholar 

  177. H. Zabel and H. Peisl, Coherent:α – α′ phase transition of hydrogen in niobium, Acta Met. 28:589–599 (1980).

    Google Scholar 

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

    Google Scholar 

  179. W. C. Johnson and J. K. Lee, Elastic interaction energy of two spherical precipitates in an anisotropic matrix, Metal. Trans. A 10A:1141–1149 (1979).

    Google Scholar 

  180. T. Miyazaki, K. Nakamura, and H. Mori, Experimental and theoretical investigations on morphological changes of γ-precipitates in Ni-Al single crystals during uniaxial stress annealing, J. Mater. Sci. 14:1827–1837 (1979).

    Google Scholar 

  181. V. Perovic, G. R. Purdy, and L. M. Brown, Acta Met. 27:1075–1084 (1979).

    Google Scholar 

  182. H. Zabel and H. Peisl, Sample-shape-dependent phase transion of hydrogen in niobium, Phys. Rev. Lett. 42:511–514 (1979).

    Google Scholar 

  183. F. C. Larché and J. W. Cahn, A nonlinear theory of thermochemical equilibrium of solids under stress, Acta Metall. 26:53-60 (1978).

    Google Scholar 

  184. F. C. Larché and J. W. Cahn, Thermochemical equilibrium of multiphase solids under stress, Acta Metall. 26:1579–1589 (1978).

    Google Scholar 

  185. J. K. Lee and W. C. Johnson, Phys. Status Solidi 46:267 (1978).

    Google Scholar 

  186. E. Seitz and D. de Fontaine, Elastic interaction energy calculations for Guinier–Preston zones in Al-Cu and Cu-Be, Acta Metall. 26:1671–1679 (1978).

    Google Scholar 

  187. J. K. Lee, D. M. Barnett, and H. I. Aaronson, The elastic strain energy of coherent ellipsoidal precipitates in anisotropic crystalline solids, Metall. Trans. 8A:963–973 (1977).

    Google Scholar 

  188. A. Pineau, Influence of uniaxial stress on the morphology of coherent precipitates during coarsening––Elastic energy considerations, Acta Metall. 24:559 (1976).

    Google Scholar 

  189. A. L. Roitburd and N. S. Kosenko, Elastic energy of a plate inclusion in an anisotropic elastic medium, Scripta Met. 11:1039–1043 (1977).

    Google Scholar 

  190. J. D. Eshelby, The elastic energy-momentum tensor, J. Elasticity 5:321–336 (1975).

    Google Scholar 

  191. M. E. Gurtin and I. Murdoch, A continuum theory of elastic material surfaces, Arch. Rat. Mech. Anal. 57:291–323 (1975).

    Google Scholar 

  192. K. Binder, Kinetic Ising model study of phase separation in binary alloys, Z. Phys. 267:313–322 (1974).

    Google Scholar 

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

    Google Scholar 

  194. W. G. Hoover, W. T. Ashurst, and R. J. Olness, Two-dimensional computer studies of crystal stability and fluid viscosity, J. Chem. Phys. 60:4043–4047 (1974).

    Google Scholar 

  195. A. G. Khachaturyan and V. N. Airapetyan, Spatially periodic distributions of new-phase inclusions caused by elastic distortions, Phys. Status Solidi (A) 26:61–70 (1974).

    Google Scholar 

  196. P. Y. Robin, Thermodynamic equilibrium across a coherent interface in a stressed crystal, Amer. Mineralogist 59:1286–1298 (1974).

    Google Scholar 

  197. A. G. Khachaturyan and V. N. Hairapetyan, Phys Status Solidi (B) 35:735 (1973) (cited by Khachaturyan).

    Google Scholar 

  198. F. C. Larché and J. W. Cahn, A linear theory of thermomechanical equilibrium of solids under stress, Acta Metall. 21:1051–1063 (1973).

    Google Scholar 

  199. D. De Fontaine, Analysis of clustering and ordering in multicomponent solid solutions I. stability criteria, J. Phys. Chem. Solids 33:297–310 (1972).

    Google Scholar 

  200. M. E. Gurtin, The Linear Theory of Elasticity Handbuch der Physik VIa–2 1-295 (Springer-Verlag, Berlin, 1972).

    Google Scholar 

  201. H. E. Cook and D. de Fontaine, On the elastic energy of solid solutions II. influence of the effective modulus on precipitation from solution and the order–disorder reaction, Acta Metall. 19:607–616 (1971).

    Google Scholar 

  202. J. K. Tien and S. M. Copley, The effect of uniaxial stress on the periodic morphology of coherent gamma prime precipitates in nickel-base superalloy crystals, Metall. Trans. 2:215 (1971).

    Google Scholar 

  203. H. E. Cook, D. de Fontaine, and J. E. Hilliard, A model for diffusion on cubic lattices and its application to the early stages of ordering, Acta Met. 17:765–773 (1969).

    Google Scholar 

  204. H. E. Cook, The kinetics of clustering and short-range order in stable solid solutions, J. Phys. Chem Solids 30:2427–2437 (1969).

    Google Scholar 

  205. H. E. Cook and D. DeFontaine, On the elastic free energy of solid solutions––I. Microscopic theory, Acta Metall. 17:915–924 (1969).

    Google Scholar 

  206. J. D. Eshelby, Inelastic Behaviour of Solids, M. F. Kanninen et al., eds. (McGraw-Hill, New York, 1969), p. 77.

    Google Scholar 

  207. A. G. Khachaturyan, Phys. Status Solidi 35:119 (1969).

    Google Scholar 

  208. A. G. Khachaturyan and G. A. Shatalov, Potential of elastic interaction of defects in a crystal, Sov. Phys. Solid State 11:118 (1969) (English translation of Fiz. Tverd. Tela 11(1):59–66).

    Google Scholar 

  209. A. G. Khachaturyan and G. A. Shatalov, Sov. Phys. JETP 29:557 (1969).

    Google Scholar 

  210. C. P. Sullivan, B. J. Piearcey, and G. A. Webster, J. Inst. Metals 96:274 (1968).

    Google Scholar 

  211. A. Khachaturyan, Sov. Phys. Solid State 9:2040 (1968) (English translation of Fiz. Tverd. Tela 9:2595).

    Google Scholar 

  212. A. L. Roitburd, Orientational and habit relationships between crystalline phases in solidstate transformations, Sov. Phys. Cryst. 12:499 (1968) (English translation of Kristallografi 12:567–574 (1967)).

    Google Scholar 

  213. G. A. Webster and C. P. Sullivan, Some effects of temperature cycling on the creep behaviour of a nickel-base alloy, J. Inst. Met. 95:138–142 (1967).

    Google Scholar 

  214. J. D. Eshelby, On the elastic interaction between inclusions, Acta Met. 14:1306–1309 (1966) (appendix to A. J. Ardell, R. B. Nicholson, and J. D. Eshelby (1966) On modulated structure of aged Ni-Al alloys, Acta Metall. 14:1295–1309).

    Google Scholar 

  215. A. G. Khachaturyan, Some questions concerning the theory of phase transitions in solids, Fiz. Tverd. Tela. 8:2709–2717 (1966). English translation in Sov. Phys. Solid. State 8:2163 (1966).

    Google Scholar 

  216. J. W. Cahn, (unpublished, circa 1964). Report 64-RL-356 M, General Electric Research laboratory, Schenectady, New York (cited by Thompson and Voorhees(26)).

    Google Scholar 

  217. A. G. Khachaturyan, Nonlinear integral equations and their application to the problem of orderable alloys (in Russian) Fiz. Tverd. Tela 5:26–35 (1963). English translation: Soviet Physics Solid State 5:16 (1963).

    Google Scholar 

  218. A. G. Khachaturyan, Nonlinear equations of integral type and their application to the study of the crystal symmetries of interstitial solutions (in Russian), Fiz. Tverd. Tela 5:750–758 (1963). English translation: Sov. Phys. Solid State 5:548–553.

    Google Scholar 

  219. W. W. Mullins and R. F. Sekerka, Morphological stability of a particle growing by diffusion or heat flow, J. Appl. Phys. 34:323 (1963).

    Google Scholar 

  220. J. W. Cahn, On spinodal decomposition in cubic crystals, Acta Metall. 10:179–183 (1962).

    Google Scholar 

  221. A. G. Khachaturyan, Application of the method of two-time Green's functions to the problem of ordering alloys (in Russian), Fiz. Met. Metallogr. 13:493–501 (1962). English translation: Physics of Metals and Metallography 13(4):12–20.

    Google Scholar 

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

    Google Scholar 

  223. J. D. Eshelby, Elastic inclusions and inhomogeneities, Prog. Solid Mech. 2:89–140 (1961).

    Google Scholar 

  224. M. Hillert, A solid solution model for inhomogeneous systems, Acta Met. 9:525–535 (1961).

    Google Scholar 

  225. I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids 19:35–50 (1961).

    Google Scholar 

  226. C. Wagner, Theorie der Alterung von Niederschlägen durch Umlösung––Ostwald–Reifung, Z. Electrochem. 65:581–591.

  227. J. D. Eshelby, The elastic field outside an ellipsoidal inclusion, Proc. Roy Soc. A 252:561–569 (1959).

    Google Scholar 

  228. L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, London, 1959). (English Translation by J B. Sykes and W. H. Reid of Teoriya Uprugosti, Izdat. “Nauka, ” Moscow).

    Google Scholar 

  229. J. D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. A 241:376–396 (1957).

    Google Scholar 

  230. W. W. Mullins, Theory of thermal grooving, J. Appl. Phys. 28:333–339 (1957).

    Google Scholar 

  231. J. D. Eshelby, The continuum theory of lattice defects, Solid State Physics 3:79–144 (1956).

    Google Scholar 

  232. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford Univ. Press, Oxford, 1954).

    Google Scholar 

  233. J. D. Eshelby, The force on an elastic singularity, Phil. Trans. Roy. Soc. Lond. A 244:87–112 (1951).

    Google Scholar 

  234. K. Robinson, Elastic energy of an ellipsoidal inclusion in an infinite solid, J. Appl. Phys. 22:1045–1054 (1951).

    Google Scholar 

  235. M. M. Crum, Private communication cited in F. R. N. Nabarro (1940); The strains produced by precipitation in alloys, Proc. Roy. Soc. A 125:519–538 (1940).

    Google Scholar 

  236. F Bitter, On impurities in metals, Phys Rev. 37:1527–1547 (1931).

    Google Scholar 

  237. J. D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density (in Dutch), Verhandel. Konink. Akad. Weten. Amsterdam (Sect. 1), Vol. 1, No. 8 (1893). English translation by J. S. Rowlinson, J. Stat. Phys. 20:197–244 (1979).

Download references

Author information

Authors and Affiliations

  1. Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, and University of Leoben, A-8700, Leoben, Austria

    Peter Fratzl

  2. Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, Scotland, U.K

    Oliver Penrose

  3. Departments of Mathematics and Physics, Rutgers University, Hill Center, Busch Campus, New Brunswick, 08903, New Jersey

    Joel L. Lebowitz

Authors
  1. Peter Fratzl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oliver Penrose
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Joel L. Lebowitz
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fratzl, P., Penrose, O. & Lebowitz, J.L. Modeling of Phase Separation in Alloys with Coherent Elastic Misfit. Journal of Statistical Physics 95, 1429–1503 (1999). https://doi.org/10.1023/A:1004587425006

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1004587425006

Search

Navigation