Advertisement

Journal of Statistical Physics

, Volume 95, Issue 5–6, pp 1429–1503 | Cite as

Modeling of Phase Separation in Alloys with Coherent Elastic Misfit

  • Peter Fratzl
  • Oliver Penrose
  • Joel L. Lebowitz
Article

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.

kinetics of phase separation quenched alloys elastic interactions sharp interface model diffuse interface models atomic lattice models 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 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. 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. 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. 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. 5.
    I. Gyöngy, private communication (1998).Google Scholar
  6. 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. 7.
    J. K. Lee, Elastic stress and microstructural evolution, Materials Transactions JIM 39:114–132 (1998).Google Scholar
  8. 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. 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. 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. 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. 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. 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. 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. Google Scholar
  15. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 37.
    F. R. N. Nabarro, Inclusions and inhomogeneities under stress, Phil. Mag. Lett. 73:45–49 (1996).Google Scholar
  38. 38.
    G. Sauthoff, Influences of stresses on precipitation, Jour. de Physique IV 6(C1):87–97 (1996).Google Scholar
  39. 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. 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. 41.
    J. Svoboda and P. Lukáč, Modeling of kinetics of directional coarsening in Ni-superalloys, Acta Mater. 44:2557–2565 (1996).Google Scholar
  42. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 53.
    J. K. Lee, Coherence strain analysis via a discrete atom method, Scripta Met. et Mat. 32:559–564 (1995).Google Scholar
  54. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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).Google Scholar
  69. 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. 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. 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).Google Scholar
  72. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 96.
    P. W. Voorhees, Ostwald ripening of two-phase mixtures, Ann Rev. Mat. Sci. 22:197–215 (1992).Google Scholar
  97. 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. 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. 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. 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. 101.
    L.-Q. Chen and A. G. Khachaturyan, Scripta Met. Mater. 25:61–66 (1991).Google Scholar
  102. 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. 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. 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. 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. 106.
    A. Onuki, Interface motion in two-phase solids with elastic misfits, J. Phys. Soc. Japan 60:345–348 (1991).Google Scholar
  107. 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. 108.
    O. Penrose, A mean-field equation for the dynamic Ising model, J. Stat. Phys. 63:975–986 (1991).Google Scholar
  109. 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. 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. 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. 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).Google Scholar
  113. 113.
    A. Cerri, B. Schönfeld, and G. Kostorz, Decomposition kinetics in Ni-Ti alloys, Phys. Rev. B 42:958–960 (1990).Google Scholar
  114. 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. 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. 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. 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. 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. 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. 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. 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. 122.
    Y. Enomoto and K. Kawasaki, Computer simulation of Ostwald ripening with elastic field interactions, Acta Metall. 37:1399–1406 (1989).Google Scholar
  123. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 135.
    A. Onuki, Long-range interactions through elastic fields in phase-separating solids, J. Phys. Soc. Japan 58:3069–3072 (1989).Google Scholar
  136. 136.
    R. L. Pego, Front migration in the nonlinear Cahn–Hilliard equation, Proc. Roy. Soc. A 422:261–278 (1989).Google Scholar
  137. 137.
    D. J. Srolovitz, On the stability of surfaces of stressed solids, Acta Metall. 37:621–625 (1989).Google Scholar
  138. 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. 139.
    I. M. Kaganova and A. L. Roitburd, Equilibrium between elastically interacting phases, Soviet Physics JETP 67:1173–1183 (1988).Google Scholar
  140. 140.
    K. Kawasaki and Y. Enomoto, Statistical theory of Ostwald ripening with elastic field interactions, Physica A 150:463–498 (1988).Google Scholar
  141. 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. 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. 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. 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. 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. 146.
    W. C. Johnson, Precipitate shape evolution under applied stress––thermodynamics and kinetics, Metall. Trans. A 18A:233–247 (1987).Google Scholar
  147. 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. 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. 149.
    S. Yoshida, M. Fukaya, and T. Miyazaki, J. Japan Inst. Metals 51:18 (1987).Google Scholar
  150. 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. 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. 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. 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. 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. 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. 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. 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. 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. 159.
    H. Furukawa, Adv. Phys. 34:703 (1985).Google Scholar
  160. 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. 161.
    P. W. Voorhees, The theory of Ostwald ripening, J. Stat. Phys. 38:231–252 (1985).Google Scholar
  162. 162.
    J. W. Cahn and F. C. Larcheé, A simple model for coherent equilibrium, Acta Met. 32:1915–1923 (1984).Google Scholar
  163. 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. 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. 165.
    W. C. Johnson, On the elastic stabilization of precipitates against coarsening under applied load, Acta Metall. 32:465–475 (1984).Google Scholar
  166. 166.
    W. C. Johnson and J. W. Cahn, Elastically induced shape bifurcations of inclusions, Acta Metall. 32:1925–1933 (1984).Google Scholar
  167. 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. 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).Google Scholar
  169. 169.
    W. C. Johnson, Elastic interaction of two precipitates subjected to an applied stress, Metall. Trans. A 14A:2219–2227 (1983).Google Scholar
  170. 170.
    A. G. Khachaturyan, Theory of Structural Transformations in Solids (Wiley, New York, 1983).Google Scholar
  171. 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)).Google Scholar
  172. 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. 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. 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. 175.
    T. Mura, Micromechanics of Defects in Solids (Martinus Nijhoff, the Hague, 1982).Google Scholar
  176. 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. 177.
    H. Zabel and H. Peisl, Coherent:α – α′ phase transition of hydrogen in niobium, Acta Met. 28:589–599 (1980).Google Scholar
  178. 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. 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. 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. 181.
    V. Perovic, G. R. Purdy, and L. M. Brown, Acta Met. 27:1075–1084 (1979).Google Scholar
  182. 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. 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. 184.
    F. C. Larché and J. W. Cahn, Thermochemical equilibrium of multiphase solids under stress, Acta Metall. 26:1579–1589 (1978).Google Scholar
  185. 185.
    J. K. Lee and W. C. Johnson, Phys. Status Solidi 46:267 (1978).Google Scholar
  186. 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. 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. 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. 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. 190.
    J. D. Eshelby, The elastic energy-momentum tensor, J. Elasticity 5:321–336 (1975).Google Scholar
  191. 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. 192.
    K. Binder, Kinetic Ising model study of phase separation in binary alloys, Z. Phys. 267:313–322 (1974).Google Scholar
  193. 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. 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. 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. 196.
    P. Y. Robin, Thermodynamic equilibrium across a coherent interface in a stressed crystal, Amer. Mineralogist 59:1286–1298 (1974).Google Scholar
  197. 197.
    A. G. Khachaturyan and V. N. Hairapetyan, Phys Status Solidi (B) 35:735 (1973) (cited by Khachaturyan).Google Scholar
  198. 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. 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. 200.
    M. E. Gurtin, The Linear Theory of Elasticity Handbuch der Physik VIa–2 1-295 (Springer-Verlag, Berlin, 1972).Google Scholar
  201. 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. 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. 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. 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. 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. 206.
    J. D. Eshelby, Inelastic Behaviour of Solids, M. F. Kanninen et al., eds. (McGraw-Hill, New York, 1969), p. 77.Google Scholar
  207. 207.
    A. G. Khachaturyan, Phys. Status Solidi 35:119 (1969).Google Scholar
  208. 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. 209.
    A. G. Khachaturyan and G. A. Shatalov, Sov. Phys. JETP 29:557 (1969).Google Scholar
  210. 210.
    C. P. Sullivan, B. J. Piearcey, and G. A. Webster, J. Inst. Metals 96:274 (1968).Google Scholar
  211. 211.
    A. Khachaturyan, Sov. Phys. Solid State 9:2040 (1968) (English translation of Fiz. Tverd. Tela 9:2595).Google Scholar
  212. 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. 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. 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. 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. 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. 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. 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. 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. 220.
    J. W. Cahn, On spinodal decomposition in cubic crystals, Acta Metall. 10:179–183 (1962).Google Scholar
  221. 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. 222.
    J. W. Cahn, On spinodal decomposition, Acta Metall. 9:795–801 (1961).Google Scholar
  223. 223.
    J. D. Eshelby, Elastic inclusions and inhomogeneities, Prog. Solid Mech. 2:89–140 (1961).Google Scholar
  224. 224.
    M. Hillert, A solid solution model for inhomogeneous systems, Acta Met. 9:525–535 (1961).Google Scholar
  225. 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. 226.
    C. Wagner, Theorie der Alterung von Niederschlägen durch Umlösung––Ostwald–Reifung, Z. Electrochem. 65:581–591.Google Scholar
  227. 227.
    J. D. Eshelby, The elastic field outside an ellipsoidal inclusion, Proc. Roy Soc. A 252:561–569 (1959).Google Scholar
  228. 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. 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. 230.
    W. W. Mullins, Theory of thermal grooving, J. Appl. Phys. 28:333–339 (1957).Google Scholar
  231. 231.
    J. D. Eshelby, The continuum theory of lattice defects, Solid State Physics 3:79–144 (1956).Google Scholar
  232. 232.
    M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford Univ. Press, Oxford, 1954).Google Scholar
  233. 233.
    J. D. Eshelby, The force on an elastic singularity, Phil. Trans. Roy. Soc. Lond. A 244:87–112 (1951).Google Scholar
  234. 234.
    K. Robinson, Elastic energy of an ellipsoidal inclusion in an infinite solid, J. Appl. Phys. 22:1045–1054 (1951).Google Scholar
  235. 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. 236.
    F Bitter, On impurities in metals, Phys Rev. 37:1527–1547 (1931).Google Scholar
  237. 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).Google Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • Peter Fratzl
    • 1
  • Oliver Penrose
    • 2
  • Joel L. Lebowitz
    • 3
  1. 1.Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, and University of LeobenLeobenAustria
  2. 2.Department of MathematicsHeriot-Watt UniversityRiccarton, EdinburghU.K
  3. 3.Departments of Mathematics and PhysicsRutgers University, Hill Center, Busch CampusNew Brunswick

Personalised recommendations