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Conclusions and Perspectives

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The Diffuse Interface Approach in Materials Science

Part of the book series: Lecture Notes in Physics ((LNPMGR,volume 73))

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Abstract

Concluding the text at this point can only result in a partial conclusion: Research in the field of diffuse interface modeling is still young and fully active. It is likely, that while I am writing, further progress of considerable impact is achieved. As discussed progress could basically be twofold, either concerning modeling related issues or numerical ones.

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References

  1. T. Abel, E. Brener, H. Müller-Krumbhaar: Phys. Rev. E 55, 7789 (1997)

    Article  ADS  Google Scholar 

  2. T. Abel: Phasenfeldmodelle für Wachstumsprozesse in Multikomponenten-und Multiphasensystemen. PhD Thesis, Research Center Jülich, Jülich (1998)

    Google Scholar 

  3. Handbook of Mathematical Funktions ed. by M. Abramowitz, I.A. Stegun, 10th ed. (Wiley, New York 1972)

    Google Scholar 

  4. G. Adam, O. Hittmair: Wärmetheorie, 3rd ed. (Vieweg, Braunschweig 1988)

    Google Scholar 

  5. S. Akamatsu, G. Favier, T. Ihle: Phys. Rev. E 51, 4751 (1995)

    ADS  Google Scholar 

  6. M. Albrecht, S. Christiansen, J. Michler, W. Dorsch, H.P. Strunk, P.O. Hansson, E. Bauser: Appl. Phys. Lett. 67, 1232 (1995)

    Article  ADS  Google Scholar 

  7. S.M. Allen, J.W. Cahn: Acta Metall. 27, 1085 (1979)

    Article  Google Scholar 

  8. F.J. Almgren, J.E. Taylor, L. Wand: SIAM J. Cont. Opt. 31, 386 (1993)

    Google Scholar 

  9. R. Almgren, W. S. Dai, V. Hakim: Phys. Rev. Lett. 71, 3461 (1993)

    Article  ADS  Google Scholar 

  10. R. Almgren: J. Comp. Phys. 106, 337 (1993)

    MATH  ADS  MathSciNet  Google Scholar 

  11. R. Almgren: SIAM J. Appl. Math. 59, 2086 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. D.M. Anderson, G.B. McFadden, A.A. Wheeler: Phys. Fluids 9, 1870 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. D.M. Anderson, G.B. McFadden, A.A. Wheeler: Ann. Rev. Fluid Mech. 30, 139 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  14. D.M. Anderson, G.B. McFadden, A.A. Wheeler: NISTIR 6237 (1998)

    Google Scholar 

  15. D.M. Anderson, G.B. McFadden, A.A. Wheeler: Physica D 135, 175 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. D.M. Anderson, G.B. McFadden, A.A. Wheeler: Physica D 144, 154 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. D.M. Anderson, G.B. McFadden, A.A. Wheeler: In Interfaces for the Twenty-First Century, ed. by M.K. Smith, M.J. Miksis, G.B. McFadden, G.P. Neitzel, D.R. Canright (Imperial College Press, London 2002), p. 131

    Google Scholar 

  18. D.M. Anderson, G.B. McFadden, A.A. Wheeler: Physica D 151, 305 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. R. Aris: Vector, Tensors, and the Basic Equations of Fluid Dynamics (Dover, New York 1962), p. 84

    Google Scholar 

  20. G.S. Bales, A. Zangwill: Phys. Rev. B 41, 5500 (1990)

    Article  ADS  Google Scholar 

  21. A.-L. Barabasi, H. E. Stanley: Fractal Concepts in Surface Growth, 1st ed. (Cambridge University Press, Cambridge 1995)

    MATH  Google Scholar 

  22. M. Barber, A. Barbieri, J.S. Langer: Phys. Rev. A 36, 3340 (1987)

    Article  ADS  Google Scholar 

  23. M.N. Barber, D. Singleton: J. Math. Soc. B 36, 325 (1994)

    MathSciNet  Google Scholar 

  24. W. Blanke (ed.): Thermophysikalische Stoffgröβen, 1st ed. (Springer-Verlag, Berlin 1989), p. 155

    Google Scholar 

  25. J. Bechhoefer: Directional Solidification at the Nematic-isotropic Interface. PhD Thesis, The University of Chicago, Chicago (1988)

    Google Scholar 

  26. C. Beckermann, H.-J. Diepers, I. Steinbach, A. Karma, X. Tong: J. Comp. Phys. 154, 468 (1999)

    Article  MATH  ADS  Google Scholar 

  27. M. Ben Amar, B. Moussalam: Phys. Rev. Lett. 60, 317 (1988)

    Article  ADS  Google Scholar 

  28. M. Ben Amar, P. Pelce: Phys. Rev. A 39, 4263 (1989)

    Article  ADS  Google Scholar 

  29. E. Ben-Jacob, N. Goldenfeld, J.S. Langer, G. Schön: Phys. Rev. A 29, 330 (1984)

    Article  ADS  Google Scholar 

  30. E. Ben-Jacob, N. Goldenfeld, J.S. Langer, G. Schön: Phys. Rev. Lett. 51, 1930 (1983)

    Article  ADS  Google Scholar 

  31. C.M. Bender, S.A. Orszag: Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York 1978)

    MATH  Google Scholar 

  32. K.W. Benz: private communication (Freiburg 2002)

    Google Scholar 

  33. J. J. Binney: The Theory of Critical Phenomena: An Introduction to the Renor-malization Group, 1st repr. with corr. (Clarendon, Oxford 1999)

    Google Scholar 

  34. P. Bouissou, B. Perrin, P. Tabeling: Phys. Rev. A 40, 509 (1989)

    Article  ADS  Google Scholar 

  35. W.J. Boettinger, J.A. Warren: Met. Trans. A 27, 657 (1996)

    Article  Google Scholar 

  36. W.J. Boettinger, J.A. Warren: J. Cryst. Growth 200, 583 (1999)

    Article  ADS  Google Scholar 

  37. A. Bösch: Phasenfeld-Modelle für Mehrkomponentensysteme. Diploma Thesis, Research Center Jülich, Jülich (1993)

    Google Scholar 

  38. A. Bösch, H. Müller-Krumbhaar, O. Shochet: Z. Phys. 97, 367 (1995)

    Article  Google Scholar 

  39. R.J. Braun, B.T. Murray: J. Cryst. Growth 174, 41 (1997)

    Article  ADS  Google Scholar 

  40. R.J. Braun, J.W. Cahn, G.B. McFadden, H.E. Rushmeier, A.A. Wheeler: Acta Mater. 46, 1 (1998)

    Article  Google Scholar 

  41. A. J. Bray: Adv. Phys. 32, 357 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  42. E. Brener, V. I. Melnikov: Adv. Phys. 40, 53 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  43. E. Brener: Phys. Rev. Lett. 71, 3653 (1993)

    Article  ADS  Google Scholar 

  44. E. Brener, D. Temkin: Phys. Rev. E 51, 351 (1995)

    Article  ADS  Google Scholar 

  45. A.N. Brooks, T.J.R. Hughes: Comp. Methods Appl. Mech. Eng. 32, 199 (1982)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  46. R. Brower, D. Kessler, J. Koplik, H. Levine: Phys. Rev. Lett. 51, 1111 (1983)

    Article  ADS  Google Scholar 

  47. W.K. Burton, N. Cabrera, F.C. Frank: Phil. Trans. Roy. Soc. 243, 299 (1951)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. G. Caginalp, P. Fife: Phys. Rev. B 33, 7792 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  49. G. Caginalp: Arch. Rat. Mech. Anal. 92, 205 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  50. G. Caginalp: SIAM J. Appl. Math. 48, 506 (1988)

    Article  MathSciNet  Google Scholar 

  51. C. Caginalp, J. Jones: App. Math. Lett. 4, 79 (1991)

    Article  MathSciNet  Google Scholar 

  52. G. Caginalp, X. Chen: In On The Evolution Of Phase Boundaries, ed. by M.E. Gurtin and G.B. McFadden (Springer, New York, 1992), p. 1

    Google Scholar 

  53. G. Caginalp, W. Xie: Phys. Rev. E 48, 1897 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  54. J.W. Cahn, J. E. Hilliard: J. Chem. Phys. 28, 258 (1958)

    Article  ADS  Google Scholar 

  55. J.W. Cahn, J.E. Hilliard: J. Chem Phys. 31, 688 (1959)

    Article  ADS  Google Scholar 

  56. J.W. Cahn: Acta Metall. 8, 554 (1960)

    Article  Google Scholar 

  57. J.W. Cahn: Acta Metall. 9, 795 (1961)

    Article  Google Scholar 

  58. J.W. Cahn, D.W. Hoffmann: Acta Met. 22, 1205 (1974)

    Article  Google Scholar 

  59. J.W. Cahn, W.C. Carter: Met. Trans. 27A, 1431 (1996)

    Google Scholar 

  60. J.W. Cahn, P. Fife, O. Penrose: Acta Mater. 45, 4397 (1997)

    Article  Google Scholar 

  61. R.W. Cahn, P. Haase: Physical Metallurgy, 4th ed. (North Holland, Amsterdam 1996)

    Google Scholar 

  62. R.W. Cahn, P. Haase, E.J. Kramer: Materials and Technology, 1st ed. (VCH, Weinheim 1991)

    Google Scholar 

  63. D. Caraballo: A Variational Scheme for the Evolution of Polycrystals by Curvature. PhD Thesis, Department of Mathematics, Princeton University (Princeton, 1997)

    Google Scholar 

  64. B. Caroli, C. Caroli, B. Roulet: In Solids Far From Equilibrium, ed. by G. Godrèche (Cambridge University Press, Cambridge 1992), p. 155

    Google Scholar 

  65. M. Carrard, M. Gremaud, M. Zimmermann, W. Kurz: Acta Metall. 40, 983 (1992)

    Article  Google Scholar 

  66. Y.C. Chang, T.Y. Hou, B. Merriman, S. Osher: J. Comput. Phys. 124, 449 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  67. Ch. Charach, P.C. Fife: SIAM J. Appl. Math. 58, 1826 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  68. Ch. Charach, P.C. Fife: Open Syst. Inf. Dyn. 5, 99 (1998)

    Article  MATH  Google Scholar 

  69. Ch. Charach, C.K. Chen, P.C. Fife: J. Stat. Phys. 95, 1141 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  70. S. Chen, B. Merriman, S. Osher, P. Smereka: J. Comput. Phys. 135, 8 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  71. S. Chen, B. Merriman, M. Kang, R.E. Caflisch, C. Ratsch, L.-T. Cheng, M. Gyure, R.P. Fedkiw, C. Anderson, S. Osher: J. Comput. Phys. 167, 475 (2001)

    Article  MATH  ADS  Google Scholar 

  72. D.L. Chopp: J. Comput. Phys. 162, 104 (2000)

    Article  MATH  ADS  Google Scholar 

  73. S. Clarke, D.D. Vvendensky: Phys. Rev. Lett. 58, 2235 (1987)

    Article  ADS  Google Scholar 

  74. J.B. Collins, H. Levine: Phys. Rev. B 31, 6119 (1985)

    Article  ADS  Google Scholar 

  75. M. Conti: Phys. Rev. E 56, 3197 (1997)

    Article  ADS  Google Scholar 

  76. M. Conti: Phys. Rev. E 55, 701 (1997)

    Article  ADS  Google Scholar 

  77. M. Conti: Phys. Rev. E 55, 765 (1997)

    Article  ADS  Google Scholar 

  78. R. Cook, D. Malkus, M. Plesha: Concepts and Applications of Finite Element Analysis, 3rd ed. (Wiley, New York 1989)

    MATH  Google Scholar 

  79. J. B. Collins, H. Levine: Phys. Rev. B 31, 6119 (1985)

    Article  ADS  Google Scholar 

  80. S.R. Coriell, D. Turnbull: Acta Metall. 30, 2135 (1982)

    Article  Google Scholar 

  81. S.R. Coriell, R.S. Sekerka: J. Cryst. Growth 61, 499 (1983)

    Article  ADS  Google Scholar 

  82. S.R. Coriell, R.F. Boisvert, G.B. McFadden, L.N. Brush, J.J. Favier: J. Cryst. Growth 140, 139 (1994)

    Article  ADS  Google Scholar 

  83. S. R. Coriell, G. B. McFadden, R. F. Sekerka and W. J. Boettinger: J. Cryst. Growth 191, 573 (1998)

    Article  ADS  Google Scholar 

  84. D.P. Corrigan, M.B. Koss, J.C. La Combe, K.D. de Jager, L.A. Tennenhouse, M.E. Glicksman: Phys. Rev. E 60, 7217 (1999)

    Article  ADS  Google Scholar 

  85. M.E. Curtin, D. Polignone, J. Vinals: Math. Models. Methods Appl. Sci. 6, 815 (1996)

    Article  MathSciNet  Google Scholar 

  86. A.G. Cullis, D.J. Robbins, A.J. Pidduck, P.W. Smith: Mater. Res. Soc. Symp. Proc. 280, 383 (1993)

    Google Scholar 

  87. A.G. Cullis: Mater. Res. Soc. Symp. Proc. 399, 303 (1996)

    Google Scholar 

  88. J.A. Dantzig, L.S. Chao: In Proc. 10th U.S. Nat. Cong. of Appl. Mech., ed. by J. Lamb (1986), p. 249

    Google Scholar 

  89. S.H. Davis: Theory of Solidification, 1st ed. (Cambridge University Press, Cambridge 2001)

    MATH  Google Scholar 

  90. S. de Cheveigne, C. Guthmann, M.M. Lebrun: J. Phys. Paris 47, 2095 (1986)

    Google Scholar 

  91. S.R. de Groot, P. Mazur: Non-Equilibrium Thermodynamics, 1st ed. (Dover, New York 1984)

    Google Scholar 

  92. P.R.B. Devloo: An H-P Adaptive Finite Element Method for Steady Compressible Flow. Ph.D Thesis, The University of Texas at Austin, Austin, TX (1987)

    Google Scholar 

  93. C. Domb, M.S. Green: Phase Transitions and Critical Phenomena, Vol. 5 (Academic Press, London 1976)

    Google Scholar 

  94. W. Dorsch, S. Christiansen, M. Albrecht, P.O. Hansson, E. Bauser, H.P. Strunk: Surf. Sci. 896, 331 (1994)

    Google Scholar 

  95. W. Dorsch, B. Steiner, M. Albrecht, H.P. Strunk, H. Wawra, G. Wagner: J. Cryst. Growth 183, 305 (1998)

    Article  ADS  Google Scholar 

  96. F. Drolet, K. R. Elder, Martin Grant, J. M. Kosterlitz: Phys. Rev. E 61, 6705 (2000)

    Article  ADS  Google Scholar 

  97. D.J. Eaglesham, M. Cerullo: Phys. Rev. Lett. 64, 1943 (1990)

    Article  ADS  Google Scholar 

  98. Ch. Eck, P. Knabner, S. Korotov: ‘A two-scale model for the computation of solid-liquid phase transitions with dendritic microstructure’. Preprint No. 278, Institute for Applied Mathematics, University Erlangen-Nürnberg (2001)

    Google Scholar 

  99. W. Eckhaus: Matched Asymptotic Expansions and Singular Perturbations (North Holland, Amsterdam 1973)

    MATH  Google Scholar 

  100. K. R. Elder, F. Drolet, J.M. Kosterlitz, M. Grant: Phys. Rev. Lett. 72, 677 (1994)

    Article  ADS  Google Scholar 

  101. K.R. Elder, J.D. Gunton, M. Grant: Phys. Rev. E 54, 6476 (1996)

    Article  ADS  Google Scholar 

  102. K.R. Elder, M. Grant, N. Provatas, J.M. Kosterlitz: Phys. Rev. E 64, 021604 (2001)

    Article  ADS  Google Scholar 

  103. The appendix of this chapter follows closely the appendices of Elder et al. M. Grant, N. Provatas, J.M. Kosterlitz: Phys. Rev. E 64, 021604 (2001) [102].

    Google Scholar 

  104. G. Ehrlich: Surface Sci. 63, 422 (1977)

    Article  ADS  Google Scholar 

  105. H.F. El-Nashar, H.A. Cerdeira: Physica A 283, 6 (2000)

    Article  ADS  Google Scholar 

  106. H. Emmerich, K. Kassner, C. Misbah, T. Ihle: J. Phys.: Condens. Matter 11, 9985 (1999)

    Article  ADS  Google Scholar 

  107. H. Emmerich, D. Schleussner, K. Kassner, T. Ihle: J. Phys.: Condens. Matter 11, 8981 (1999)

    Article  ADS  Google Scholar 

  108. H. Emmerich, K. Kassner, T. Ihle, A. Weiβ: J. Cryst. Growth 211, 43 (2000)

    Article  ADS  Google Scholar 

  109. H. Emmerich: Phys. Rev. B 65, 233406 (2002)

    Article  ADS  Google Scholar 

  110. H. Emmerich: accepted for publication in Continuum Mech. Therm.

    Google Scholar 

  111. H. Emmerich, M. Jurgk, R. da Silva de Siquieri: in preparation

    Google Scholar 

  112. M.A. Eshelman, V. Seetharaman, R. Trivedi: Acta Metall. 36, 1165 (1988)

    Article  Google Scholar 

  113. M. Fabbri, V.R. Voller: J. Comp. Phys. 130, 256 (1997)

    Article  MATH  ADS  Google Scholar 

  114. R. Fedkiw, T. Aslam, B. Merriman, S. Osher: J. Comput. Phys. 152, 457 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  115. P.C. Fife: In CBMS-NSF Regional Conference Series in Applied Mathematics 53 (SIAM, Philadelphia 1988), p. 11

    Google Scholar 

  116. P.C. Fife, O. Penrose: Electron. J. Diff. Equations 1, 1 (1995)

    MathSciNet  Google Scholar 

  117. A detailed investigation of the ‘classical’ problem of density change flow without generalized boundary conditions was carried out by T. Fischalek, Otto-von-Guericke University Magdeburg (2000). Results presented at the TMR network ‘Pattern, Noise and Chaos’ meeting Leiden, July 2000. The calculations presented in this section are based on this work.

    Google Scholar 

  118. M. Fisher: Rev. Mod. Phys. 46, 597 (1974)

    Article  ADS  Google Scholar 

  119. G. Fix: Research Notes in Mathematics 2, (1983)

    Google Scholar 

  120. G.J. Fix: In Free Boundary Problems: Theory and Applications, Vol. II, ed. by A. Fasano and M. Primicerio (Pitman, Boston 1983), p. 580

    Google Scholar 

  121. E. Fried, M.E. Gurtin: Physica D 72, 287 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  122. E. Fried, M.E. Gurtin: Physica D 91, 143 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  123. H. Gao: J. Mech. Phys. Solids 42, 741 (1994)

    Article  MATH  ADS  Google Scholar 

  124. W. Gebhardt, U. Krey: Phasenübergänge und kritische Phänomene, 1st ed. (Vieweg, Braunschweig 1980)

    Google Scholar 

  125. A. Ghazali, C. Misbah: Phys. Rev. A 46, 5026 (1992)

    Article  ADS  Google Scholar 

  126. J.W. Gibbs: T. Conn. Acad. 2, 382 (1873)

    Google Scholar 

  127. M.E. Glicksman: Mater. Sci. Eng. 65, 45 (1984)

    Article  ADS  Google Scholar 

  128. M.E. Glicksman, S.P. Marsh: In Handbook of Crystal Growth, Vol. 1b, ed. by D.T.J. Hurle (Elsevier, Amsterdam 1993), p. 1075

    Google Scholar 

  129. M.E. Glicksman, M.B. Koss, E.A. Winsa: Phys. Rev. Lett. 73, 573 (1994)

    Article  ADS  Google Scholar 

  130. M. E. Glicksman: Microgravity News, NASA 4, 4 (1997)

    Google Scholar 

  131. M. E. Glicksman: private communication (Aachen 2002)

    Google Scholar 

  132. R. Glowinski, T.-S. Pan, J. Periaux: Comput. Methods Appl. Mech. Eng. 111, 283 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  133. G.H. Godunov, V.S. Ryabenkii: Difference Schemes: An Introduction to the Underlying Theory (North Holland, Amsterdam 1987)

    MATH  Google Scholar 

  134. H. Goldstein: Klassische Mechanik, 11th ed. (Aula-Verlag, Wiesbaden 1978)

    Google Scholar 

  135. G.H. Golub, J.M. Ortega: Scientific Computing (Academic Press, Boston 1993)

    MATH  Google Scholar 

  136. P.M. Gresho, S.T. Chan, M.A. Christon, A.C. Hindmarsh: Int. J. Numer. Methods Fluids 21, 837 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  137. M. Griebel, T. Dornseifer, T. Neunhofer: Numerische Simulation in der Strömungsmechanik, 1st ed. (Vieweg, Braunschweig 1995)

    Google Scholar 

  138. B. Grossmann, K.R. Elder, M. Grant, J. M. Kosterlitz: Phys. Rev. Lett. 71, 3323 (1993)

    Article  ADS  Google Scholar 

  139. J.D. Gunton, M. San Miguel, P. Sahni: In Phase Transitions and Critical Phenomena, Vol.8, ed. by C. Domb, J.L. Lebowitz (Academic Press, London 1983), p. 267

    Google Scholar 

  140. I. Gyrmati: Non-Equilibrium Thermodynamics, 1st ed. (Springer, New York 1970)

    Google Scholar 

  141. Proceedings of the Workshop on Virtual Molecular Beam Epitaxy, ed. by M.F. Gyure, J.J. Zinck, Comp. Mater. Sci.6, 113 (1996)

    Google Scholar 

  142. R. Haase, H. Schoenert: Solid-Liquid Equilibrium, 1st ed. (Pergamon, Oxford 1969)

    Google Scholar 

  143. B.I. Halpering, P.C. Hohenberg, S. Ma: Phys. Rev. B 10, 139 (1974)

    Article  ADS  Google Scholar 

  144. S.C. Hardy: J. Cryst. Growth 69, 456 (1984)

    Article  ADS  Google Scholar 

  145. D.W. Hoffmann, J.W. Cahn: Surf. Sci. 31, 368 (1972)

    Article  ADS  Google Scholar 

  146. P.C. Hohenberg, B.I. Halperin: Rev. Mod. Phys. 49, 435 (1977)

    Article  ADS  Google Scholar 

  147. G. Horvay, J. W. Cahn: Acta Metall. 9, 695 (1961)

    Article  Google Scholar 

  148. T. Hou, Z. Li, S. Osher, H.-K. Zhao: J. Comput. Phys. 134, 236 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  149. S.-C. Huang, M. Glicksman: Acta Metall. 29, 1697 (1981)

    Google Scholar 

  150. T. Ihle, H. Müller-Krumbhaar: Phys. Rev. E 49, 2972 (1994)

    Article  ADS  Google Scholar 

  151. T. Ihle: Eur. Phys. J. B 16, 337 (2000)

    Article  ADS  Google Scholar 

  152. S.E. Ingle, F.H. Horne: J. Chem. Phys. 59, 5882 (1973)

    Article  ADS  Google Scholar 

  153. G. P. Ivantsov: Dokl. Akad. Nauk USSR 58, 1113 (1947)

    Google Scholar 

  154. D. Jasnow: Rep. Prog. Phys. 47, 1061 (1984)

    Article  ADS  Google Scholar 

  155. J.-H. Jeong, N. Goldenfeld, J. Dantzig: Phys. Rev. E 64, 041602 (2001)

    Article  ADS  Google Scholar 

  156. R.A. Johnson, D.M. Belk: AIAA J. 33, 2305 (1995)

    Article  MATH  ADS  Google Scholar 

  157. D. Juric, G. Tryggvason: J. Comput. Phys. 123, 127 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  158. L.V. Kale, S. Krishnan: In Parallel Programming using C++, ed. by G.V. Wilson, P. Lu (MIT Press, Massachusetts 1996), p. 175

    Google Scholar 

  159. N.S. Kalospiros, B.J. Edward, A.N. Beris: Int. J. Heat Mass Trans. 36, 1191 (1993)

    Article  MATH  Google Scholar 

  160. N.S. Kalospiros, R. Ocone, G. Astarita, J.H. Meldon: Ind. Eng. Chem. Res. 30, 851 (1991)

    Article  Google Scholar 

  161. S. Kaplun: In Fluid Mechanis and Singular Perturbations, ed. by P.A. Lagerstrom, L.N. Howard, C.S. Liu (Academic Press, New York 1967)

    Google Scholar 

  162. A. Karma, J.S. Langer: Phys. Rev. A 30, 3147 (1984)

    Article  ADS  Google Scholar 

  163. A. Karma, B.G. Kotliar: Phys. Rev. A 31, 3266 (1985)

    Article  ADS  Google Scholar 

  164. A. Karma, A. Sarkissian: Phys. Rev. E 47, 513 (1993)

    Article  ADS  Google Scholar 

  165. A. Karma: Phys. Rev. E 48, 3441 (1993)

    Article  ADS  Google Scholar 

  166. A. Karma: Phys. Rev. E 49, 2245 (1994)

    Article  ADS  Google Scholar 

  167. A. Karma, W. Rappel: Phys. Rev. Lett. 77, 4050 (1996)

    Article  ADS  Google Scholar 

  168. A. Karma, W.-J. Rappel: Phys. Rev. E 53, R3017 (1996)

    Article  ADS  Google Scholar 

  169. A. Karma, W.-J. Rappel: Phys. Rev. E 57, 4323 (1998)

    Article  MATH  ADS  Google Scholar 

  170. A. Karma, M. Plapp: Phys. Rev. Lett. 81, 4444 (1998)

    Article  ADS  Google Scholar 

  171. A. Karma: Phys. Rev. Lett. 87, 115701 (2001)

    Article  ADS  Google Scholar 

  172. K. Kassner, C. Misbah, J. Müller, J. Kappe, P. Kohlert: Phys. Rev. E 63, 036117 (2001)

    Article  ADS  Google Scholar 

  173. K. Kawasaki, T. Ohta: Prog. Theor. Phys. 67, 147 (1982)

    Article  ADS  Google Scholar 

  174. K. Kawasaki, T. Ohta: Prog. Theor. Phys. 68, 129 (1982)

    Article  ADS  Google Scholar 

  175. K. Kawasaki, T. Ohta: Physica 118A, 175 (1983)

    ADS  Google Scholar 

  176. K. Kawasaki, M.C. Yalabik, J.D. Gunton: Phys. Rev. A 17, 455 (1978)

    Article  ADS  Google Scholar 

  177. L.J.T.M. Kemper: J. Chem. Phys. 115, 6330 (2001)

    Article  ADS  Google Scholar 

  178. J. Kepler: ‘Strena seu de nive sexangula’, ed. by G. Tampach (Frankfurt 1611); translated in A New Year’s Gift or on the Six-Cornered Snowflake, ed. by C. Hardie (Clarendon, Oxford 1966), p. 74

    Google Scholar 

  179. D. Kessler, J. Koplik, H. Levine: Phys. Rev. A 30, 3161 (1984)

    Article  ADS  Google Scholar 

  180. D. Kessler, H. Levine: Europhys. Lett. 4, 215 (1987)

    Article  ADS  Google Scholar 

  181. D. Kessler, J. Koplik, H. Levine: Adv. Phys. 37, 255 (1988)

    Article  ADS  Google Scholar 

  182. Y.T. Kim, N. Provatas, N. Goldenfeld, J. Dantzig: Phys. Rev. E 59, R2546 (1999)

    Article  ADS  Google Scholar 

  183. Y.T. Kim, N. Goldenfeld, J. Dantzig: Phys. Rev. B 62, 2471 (2000)

    Article  ADS  Google Scholar 

  184. M.B. King: Phase Equilibrium Mixtures, 1st ed. (Pergamon Press, Oxford 1969)

    Google Scholar 

  185. J.S. Kirkaldy, D.J. Young: Diffusion in the Condensed State (The Institute of Metals, London 1987)

    Google Scholar 

  186. R. Kobayashi: Bull. Jpn. Soc. Ind. Appl. Math. 1, 22 (1991)

    Google Scholar 

  187. R. Kobayashi: Physica D 63, 410 (1993)

    Article  MATH  ADS  Google Scholar 

  188. R. Kobayashi, J.A. Warren, W.C. Carter: Physica D 140, 141 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  189. P. Kopczynski, W.-J. Rappel, A. Karma: Phys. Rev. Lett. 77, 3387 (1996)

    Article  ADS  Google Scholar 

  190. D.J. Korteweg: Arch. Neerl. Sci. Exactes Nat. Ser. II 6 (1901)

    Google Scholar 

  191. M. Kruskal, H. Segur: ‘Asymptotics Beyond all Orders in a Model of Dendritic Crystals’. In Aero. Res. Ass. of Princeton Tech. Memo (Princeton 1985)

    Google Scholar 

  192. R. Kupfermann: Morphology, Coexistence and Selection in interfacial Pattern Formation. PhD Thesis, Tel Aviv University, Tel Aviv (1991)

    Google Scholar 

  193. J.C. LaCombe, M.B. Koss, M.E. Glicksman: Phys. Rev. Lett. 83, 2997 (1999)

    Article  ADS  Google Scholar 

  194. J.C. LaCombe, M.B. Koss, J.E. Frei, C. Giummarra, A.O. Lupulescu, M.E. Glicksman: Phys. Rev. E 65, 031604 (2002)

    Article  ADS  Google Scholar 

  195. B. Lafaurie, S. Zaleski, G. Zanetti: J. Comp. Phys. 113, 134 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  196. P.A. Lagerstrom, D.A. Reinelt: SIAM J. Appl. Math. 44, 451 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  197. P.A. Lagerstrom: Matched Asymptotic Expansions, 1st ed. (Springer, New York 1988)

    MATH  Google Scholar 

  198. Landoldt-Börnstein, Vol. IV/5 D (Springer, New York 1993)

    Google Scholar 

  199. J.S. Langer: Rev. Mod. Phys. 52, 1 (1980)

    Article  ADS  Google Scholar 

  200. J. Langer: In Directions in Condensed Matter Physics, ed. by G. Grinstein, G. Mazenko (World Scientific, Singapore 1986), p. 164

    Google Scholar 

  201. J.S. Langer: Phys. Rev. A 36, 3350 (1987)

    Article  ADS  Google Scholar 

  202. J. Langer: In Chance and Matter, Les Houches Session XLVI, ed. by J. Souletie, J. Vannenimus, R. Stora (North Holland, Amsterdam, 1987), p. 629

    Google Scholar 

  203. Y.-W. Lee, R. Ananth, W.N. Gill: Chem. Eng. Comm. 153, 41 (1996)

    Article  Google Scholar 

  204. F.K. LeGoues, M.C. Reuter, J. Tersoff, H. Hammar, R.M. Tromp: Phys. Rev. Lett 73, 300 (1994)

    Article  ADS  Google Scholar 

  205. M.-A. Lemieux, G. Kotliar: Phys. Rev. A 36, 4975 (1987)

    Article  ADS  Google Scholar 

  206. R.J. LeVeque, Z. Li: SIAM J. Numer. Anal. 31, No. 4, 1019 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  207. F. Liu, H. Metiu: Phys. Rev. E 49, 2601 (1994)

    Article  ADS  Google Scholar 

  208. J. Lowengrub, J. Truskinovsky: Proc. Roy. Soc. Ser. A 454, 2617 (1998)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  209. S. Luckhaus: ‘Solidification of alloys and the Gibbs-Thomson law’. Preprint No. 335, Institute of Mathematics, University of Bonn (1993)

    Google Scholar 

  210. B.Y. Lyubov: Kinetic Theory of Phase Transitions (Metallurgia Publishers, Moskau 1978)

    Google Scholar 

  211. A. Madhukar, S. V. Ghaisas: CRC Crit. Rev. Solid State Mater. Sci. 14, 1 (1988)

    Article  Google Scholar 

  212. G.B. McFadden, S.R. Coriell: J. Cryst. Growth 74, 507 (1986)

    Article  ADS  Google Scholar 

  213. G.B. McFadden, A.A. Wheeler, R.J. Braun, S.R. Coriell, R.F. Sekerka: Phys. Rev. E 48, 2016 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  214. G.B. McFadden, A.A. Wheeler, D.M. Anderson: Physica D 144, 154 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  215. B. Merriman, J. Bence, S. Osher: J. Comput. Phys. 112, 334 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  216. H. Metui, Y.-T. Lu, Z.Y. Zhang: Science 255, 1088 (1992)

    Article  ADS  Google Scholar 

  217. X. Mikic, E.C. Morse: J. Comp. Phys. 61, 154 (1985)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  218. Y.-W. Mo, D.E. Savage, B.S. Schwartzentruber, M.G. Lagally: Phys. Rev. Lett 65 1020 (1990)

    Article  ADS  Google Scholar 

  219. P. Morse, H. Feshbach: Methods of Theoretical Physics, 1st ed. (McGraw-Hill, New York 1953)

    MATH  Google Scholar 

  220. J. Müller, M. Grant: Phys. Rev. Lett. 82, 1736 (1999)

    Article  ADS  Google Scholar 

  221. H. Müller-Krumbhaar, H. Emmerich, E. Brener, M. Hartmant: Phys. Rev. E 63, 026304 (2001)

    Article  ADS  Google Scholar 

  222. W.W. Mullins: In Metal Surfaces: Structure, Energetics, and Kinetics, ed. American Society for Metals (1962), p. 17

    Google Scholar 

  223. W.W. Mullins, R.F. Sekerka: J. Appl. Physics 35, 444 (1964)

    Article  ADS  Google Scholar 

  224. M. Muschol, D. Lui, H. Z. Cummins: Phys. Rev. A 64 (1992) 1038

    Article  ADS  Google Scholar 

  225. T. Ohta, D. Jasnow, K. Kawasaki: Phys. Rev. Lett. 49, 1223 (1982)

    Article  ADS  Google Scholar 

  226. S. Ono, S. Kondo: Encyclopaedia of Physics 10, Springer (1960)

    Google Scholar 

  227. L. Onsager: Phys. Rev. 37, 405 (1931)

    Article  MATH  ADS  Google Scholar 

  228. S. Osher, J.A. Sethian: J. Comput. Phys. 79, 12 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  229. N. Palle, J. A. Dantzig: Met. Trans. A 27A, 707 (1996)

    Google Scholar 

  230. R.B. Pember, J.B. Bell, P. Colella: J. Comput. Phys. 120, 278 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  231. O. Penrose, P.C. Fife: Physica D 43, 44 (1990)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  232. O. Penrose, P.C. Fife: Physica D 69, 107 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  233. C.S. Peskin: J. Comput. Phys. 25, 220 (1977)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  234. R. Peters, J.S. Langer: Phys. Rev. Lett. 56, 1948 (1986)

    Article  ADS  Google Scholar 

  235. O. Pierre-Louis, C. Misbah, Y. Saito, J. Krug, P. Politi: Phys. Rev. Lett. 80, 4221 (1998)

    Article  ADS  Google Scholar 

  236. Y. Pomeau, M. Ben Amar: In Solids far from Equilibrium, ed. by Godrèche (Cambridge University Press, Cambridge 1991), p. 365

    Google Scholar 

  237. I. Prigogine: Non-Equilibrium Statistical Mechanics, 2nd ed. (Wiley, New York 1966)

    Google Scholar 

  238. N. Provatas, E. Elder, M. Grant: Phys. Rev. B 53, 6263 (1996)

    Article  ADS  Google Scholar 

  239. N. Provatas, N. Goldenfeld, J. Dantzig: Phys. Rev. Lett. 80, 3308 (1998)

    Article  ADS  Google Scholar 

  240. N. Provatas, N. Goldenfeld, J. Dantzig: J. Comp. Phys. 148, 265 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  241. J.J. Quirk: ICASE Report No. 92-7, NASA Langley Research Center, Hampton, VA (1992)

    Google Scholar 

  242. C. Ratsch, P. Ruggerone, M. Scheffler: In Surface Diffusion: Atomistic and Collective Processes, ed. by M.C. Tringides (Plenum, New York 1997), p. 83

    Google Scholar 

  243. W.H.S. Rouse: Great Dialogues of Plato (transl.)-A Mentor Book, 1st reissue ed. (Signet Classic, New York 1999)

    Google Scholar 

  244. J. S. Rowlinson: J. Stat. Phys. 20, 197 (1979) (an english translation of the original Ph.D. thesis by van der Waals)

    Article  MathSciNet  ADS  Google Scholar 

  245. Y. Saito, G. Goldbeck-Wood, H. Müller-Krumbhaar: Phys. Rev. A 38, 2148 (1988)

    Article  ADS  Google Scholar 

  246. A. Schmidt: J. Comp. Phys. 125, 293 (1996)

    Article  MATH  ADS  Google Scholar 

  247. L.Z. Schlitz, S. Garimella: J. Appl. Phys. 76, 4863 (1994)

    Article  ADS  Google Scholar 

  248. D. Schwabe, A. Scharmann: Phys. Bl. 42, 352 (1986)

    Google Scholar 

  249. V. Seetharaman, M.A. Eshelman, R. Trivedi: Acta Metall. 36, 1175 (1988)

    Article  Google Scholar 

  250. R.F. Sekerka, Z. Bi: Physica A 261, 95 (1998)

    Article  Google Scholar 

  251. R.F. Sekerka: In Proceedings of the Microgravity Materials Science Conference (Huntsville, Alabama 2000), p. 533

    Google Scholar 

  252. R.F. Sekerka, Z. Bi: In Interfaces for the Twenty-First Century ed. by M.K. Smith, M.J. Miksis, G.B. McFadden, G.P. Neitzel, D.R. Canright (Imperial College Press, London 2002), p. 147

    Google Scholar 

  253. J.A. Sethian: Proc. Nat. Acad. Sci. 93, 1591 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  254. J.A. Sethian: Level Set Methods and Fast Marching Methods, 2nd ed. (Cambridge University Press, Cambridge 1999)

    MATH  Google Scholar 

  255. M. S. Shephard, P. L. Baehmann, K. R. Grice: Comm. App. Num. Meth. 4, 379 (1988)

    Article  MATH  Google Scholar 

  256. W. Shyy: Computational Modeling for Fluid Flow and Interfacial Transport (Elsevier, Amsterdam 1994)

    Google Scholar 

  257. W. Shyy, H.S. Udaykumar, M.M. Rao, R.W. Smith: Computational Fluid Dynamics with Moving Boundaries (Taylor & Francis, Washington, DC, 1996)

    Google Scholar 

  258. A.J. Simon, J. Bechhoefer, A. Libchaber: Phys. Rev. Lett. 61, 2574 (1988)

    Article  ADS  Google Scholar 

  259. J. Smith: J. Comp. Phys. 39, 112 (1981)

    Article  MATH  ADS  Google Scholar 

  260. K. Somboonsuk, J.T. Mason, R. Trivedi: Metall. Trans. 15A, 967 (1984)

    Google Scholar 

  261. B.J. Spencer, S.H. Davis, P.W. Voorhees: Phys. Rev. B 47, 9760 (1993)

    Article  ADS  Google Scholar 

  262. D.J. Srolovitz: Acta Metallogr. 37, 621 (1988)

    Article  Google Scholar 

  263. H.E. Stanley: Phase Transitions and Critical Phenomena, 1st ed. (Oxford Univ. Press, New York 1971)

    Google Scholar 

  264. J.L. Steger: In Proceedings of the Computational Fluid Dynamics Symposium on Aeropropulsion, NASA CP-3078 (Natl. Aeronautics & Space Admin., Washington, DC, 1991), p. 1

    Google Scholar 

  265. Z. Suo: Advances in Appl. Mechs. 33, 193 (1997)

    Article  Google Scholar 

  266. M. Sussman, P. Smereka, S. Osher: J. Comput. Phys. 114, 146 (1994)

    Article  MATH  ADS  Google Scholar 

  267. J.E. Taylor: Acta Met. 40, 1475 (1992)

    Article  Google Scholar 

  268. J.E. Taylor, J.W. Cahn: Acta Met. 42, 1045 (1994)

    Article  Google Scholar 

  269. J.E. Taylor, J.W. Cahn: J. Stat. Phys. 77, 183 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  270. D. E. Temkin: Dokl. Akad. Nauk SSSR 132, 1307 (1960)

    Google Scholar 

  271. D. ter Haar: Collected Papers of Landau (Gordon and Breach, Washington, DC, 1967)

    Google Scholar 

  272. R. Tönhardt, G. Amberg: J. Cryst. Growth 194, 406 (1998)

    Article  Google Scholar 

  273. J.C. Tolendano, P. Tolendano: The Landau Theory of Phase Transitions (World Scientific, Singapore 1987)

    Google Scholar 

  274. R. Trivedi: Metall. Trans. 15A, 977 (1984)

    MathSciNet  Google Scholar 

  275. R. Trivedi, J.A. Sekhan, V. Seetharaman: Metall. Trans. A20, 769 (1989)

    Google Scholar 

  276. J.Y. Tsao: Materials Fundamentals of Molecular Beam Epitaxy (Academic Press, Boston 1993)

    Google Scholar 

  277. H. S. Udaykumar, W. Shyy: Numer. Heat Transfer B 27, 127 (1995)

    Article  ADS  Google Scholar 

  278. H.S. Udaykumar, W. Shyy: Int. J. Heat Mass Transfer 38, 2057 (1995)

    Article  MATH  Google Scholar 

  279. H.S. Udaykumar, W. Shyy, M M. Rao: Int. J. Numer. Methods Fluids 22, 691 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  280. H.S. Udaykumar, H.-C. Kan, W. Shyy, R. Tran-Son-Tay: J. Comput. Phys. 137, 366 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  281. H.S. Udaykumar, R. Mittal, W. Shyy: J. Comput. Phys. 153, 535 (1999)

    Article  MATH  ADS  Google Scholar 

  282. A. Umantsev, A.L. Roitburd: Sov. Phys. 30 651 (1988)

    Google Scholar 

  283. A.P. Umantsev: J. Chem. Phys. 96, 606 (1992)

    Article  ADS  Google Scholar 

  284. A. Umantsev, S.H. Davis: Phys. Rev. A 45 7195 (1992)

    Article  ADS  Google Scholar 

  285. S.O. Unverdi, G. Tryggvason: J. Comput. Phys. 100, 25 (1992)

    Article  MATH  ADS  Google Scholar 

  286. V. Venkatakrshnan: AIAA J. 34, 533 (1996)

    Article  ADS  Google Scholar 

  287. D.D. Vvendensky, A. Zangwill, C.N. Luse, M.R. Wilby: Phys. Rev. E 48, 852 (1993)

    Article  ADS  Google Scholar 

  288. J. Villain: J. Phys. I 1, 19 (1991)

    Article  Google Scholar 

  289. S.-L. Wang, R.F. Sekerka, A.A. Wheeler, B.T. Murray, S.R. Coriell, R.B. Braun, G.B. McFadden: Physica D 69, 189 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  290. S.-L. Wang, R.F. Sekerka: Phys. Rev. E 53, 3760 (1996)

    Article  ADS  Google Scholar 

  291. J.A. Warren, W.J. Boettinger: Acta Metall. Mater. A 43, 689 (1995)

    Article  Google Scholar 

  292. J.A. Warren, B.T. Murray: Mod. Simul. Mater. Sci. Eng. 4, 215 (1996)

    Article  ADS  Google Scholar 

  293. J.D. Weeks, G.H. Gilmer: Adv. Chem. Phys. 40, 157 (1979)

    Article  Google Scholar 

  294. H. Wenzel: In Physikalische Grundlagen metallischer Werkstoffe, 13th IFF-Spring School at Research Center Juelich (1982)

    Google Scholar 

  295. A.A. Wheeler, W.J. Boettinger, G.B. McFadden: Phys. Rev. A 45, 7424 (1992)

    Article  ADS  Google Scholar 

  296. A.A. Wheeler, W.J. Boettinger, G.B. McFadden: Phys. Rev. E 47, 1893 (1993)

    Article  ADS  Google Scholar 

  297. A.A. Wheeler, B.T. Murray, R.J. Schaefer: Physica D 66, 243 (1993)

    Article  MATH  ADS  Google Scholar 

  298. A.A. Wheeler, G.B. McFadden, W.J. Boettinger: Proc. Royal Soc. London A 452, 495 (1996)

    Article  ADS  Google Scholar 

  299. D.J. Wollkind, R.N. Maurer: J. Cryst. Growth 42, 24 (1977)

    Article  ADS  Google Scholar 

  300. G. Wulff: Z. Kristallogr. Mineral 34, 449 (1901)

    Google Scholar 

  301. L. Zhang, S. Garimella: J. Appl. Phys. 74, 2494 (1993)

    Article  ADS  Google Scholar 

  302. H.K. Zhao, T. Chan, B. Merriman, S. Osher: J. Comput. Phys. 127, 179 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  303. O. C. Zienkiewicz, J. Z. Zhu: Int. J. Numer. Meth. Eng. 24, 337 (1987)

    Article  MATH  MathSciNet  Google Scholar 

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(2003). Conclusions and Perspectives. In: The Diffuse Interface Approach in Materials Science. Lecture Notes in Physics, vol 73. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36409-9_8

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