The thermodynamics of elastically stressed crystals

Solid State Physics - Advances in Research and Applications

Volumn 59, Issue C, 2004, Pages 1-201

  Voorhees, P W ^a   Johnson, William C ^b  

^a Northwestern University   (United States)

^b University of Virginia   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 27144516887     PISSN: 00811947     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S0081-1947(04)80003-1     Document Type: Chapter

References (155)

1. Lee J.K., Barnett D.M., and Aaronson H.I. Metall. Trans. 8A (1977) 973
2. Khachaturyan A.G. Theory of Structural Phase Transformations in Solids (1983), John Wiley, New York
3. Mura T. Micromechanics of Defects in Solids (1987), Kluwer Academic Publishers, Dordrecht, The Netherlands
4. Lund A.C., and Voorhees P.W. Acta mater. 50 (2002) 2585
5. Su C.H., and Voorhees P.W. Acta mater. 44 (1996) 1987
6. Shchukin V.A., and Bimberg D. Rev. Mod. Phys. 71 (1999) 1125
7. Cullis A.G., Robbins D.J., Pidduck A.J., and Smith P.W. J. Cryst. Grwth. 123 (1992) 333
8. Eaglesham D.J., and Cerullo M. Phys. Rev. Lett. 64 (1990) 1943
9. Guha S., Madhukar A., and Rajkumar K.C. Appl. Phys. Lett. 57 (1990) 2110
10. Mederios-Ribeiro G., Bratkovski A.M., Kamins T.I., Ohlberg D.A.A., and Williams R.S. Science 279 (1998) 353
11. Norman A.G., Ahrenkiel S.P., Moutinho H., Al-Jassim M.M., Mascrenhas A., Mirecki-Millunchick J., Lee S.R., Twesten R.D., Follstaedt S.M., Reno J.L., et al. Appl. Phys. Lett. 73 (1998) 1844
12. Mirecki-Millunchick J., Twesten R.D., Follstaedt D.M., Lee S.R., Jones E.D., Zhang Y., Ahrenkiel S.P., and Mascarenhas A. Appl. Phys. Lett. 70 (1997) 1402
13. Gurtin M.E. Arch. Rational. Mech. Anal. 131 (1995) 67
14. Herring C. The Physics of Powder Metallurgy (1950), McGraw Hill 143-179
15. Herring C. In: Gomer R., and Smith C. (Eds). Structure and Properties of Solid Surfaces (1952), University of Chicago Press, Chicago 5-81
16. Millins W.W. In: Aaronson H.I., Laughlin D.E., Sekerka R.F., and Wayman C.M. (Eds). Int. Conf. Solid-solid Phase Transformations (1981), TMS-AIME, Warrendale, PA 49-66
17. M. E. Gurtin and E. Fried, Rev. Appl. Mech. (in press).
18. Larché F.C., and Cahn J.W. Acta metall. 33 (1985) 331
19. We use θ for the temperature instead of T so as to avoid confusion with the Piola-Kirchhoff stress tensor.
20. Gibbs J.W. The Scientific Papers of J. Willard Gibbs Volume 1: Thermodynamics (1961), Dover Publications, Mineola, NY
21. Larché F.C., and Cahn J.W. Acta metall. 21 (1973) 1051
22. Johnson W.C. In: Aaronson H.I. (Ed). Lectures on the Theory of Phase Transformations (2000), TMS-AIME, Warrendale, PA 35-134
23. Malvern L.E. Introduction to the Mechanics of a Continuous Medium (1969), Prentice-Hall, Inc., Englewood Cliffs, New Jersey
24. ij =0 if i≠j.
25. i d S.
26. 3i,3 .
27. The use of δ u represents an imaginary displacement performed in such a way that the forces acting on the body are not changed.
28. Nye J.F. Physical Properties of Materials (1985), Clarendon Press, Oxford
29. Fung Y.C. A First Course in Continuum Mechanics (1969), Prentice Hall, New Jersey
30. Johnson W.C. Acta Mater. 49 (2001) 3463
31. A nonlinear dependence of the lattice parameter on composition is treated in the same way. However, the resulting equations for the diffusion potential will not decouple.
32. Eshelby J.D. In: Kanninen M.F., Alder W.F., Rosenfeld A.R., and Jaffee R.I. (Eds). Inelastic Behavior of Solids (1970), McGraw-Hill, New York 77
33. Gurtin M.E. In: Marsden J.E., and Sirovich L. (Eds). Configurational Forces as Basic Concepts of Continuum Physics. Applied Mathematical Sciences (2000), Springer Verlag
34. Noziéres P. In: Godréche C. (Ed). Solids Far From Equilibrium (1991), Cambridge University Press 1-154
35. The treatment of crystals for which an atom can occupy two sublattices was elucidated for the hydrostatic case.
36. Johnson W.C. Acta metall. mater. 77 (1994) 1581
37. Mullins W.W., and Sekerka R.F. J. Chem. Phys. 82 (1986) 5192
38. Of course, the internal energy density can depend on other variables as well. For example, in an ionic crystal, the internal energy is also a function of the electric displacement and the various atomic species can have different charge states.
39. Guyer J.E., and Voorhees P.W. Phys. Rev. Lett. 74 (1995) 4031
40. Malyshkin V.G., and Shchukin V.A. Semiconductors 27 (1994) 1062
41. Johnson W.C., and Voorhees P.W. Metall. Trans. A 16A (1985) 337
42. Barnett D., Wang G., and Nix W. Acta metall. 30 (1982) 2035
43. Cottrell A.H., and Jaswon M.A. Proc. Roy. Soc. A199 (1949) 104
44. Hirth J.P., and Lothe J. Theory of Dislocations (1982), John Wiley and Sons, New York
45. Eshelby J.D. Proc. Roy. Soc. A252 (1959) 561
46. Piller J., Miller M.K., and Brenner S.S. In: Andren H.O., and Norden H. (Eds). Proc. 29th Int. Field Emmisions Symp. Goteborg, Sweden (1982)
47. Note that the application of the uniaxial stress in the absence of the two inhomogeneities changes the diffusion potential, but does not lead to a change in solute composition.
48. Johnson W.C., and Hut J.Y. Metall. Trans. 34a (2003) 2819
49. Lupis C.H.P. Chemical Thermodynamics of Materials (1983), Elsevier, Amsterdam
50. Bai Q., Chu W., and Hsiao C. Scripta metall. 21 (1987) 613
51. Porter D.A., and Easterling K.E. Phase Transformations in Metals and Alloys (1992), Chapman and Hall, London
52. Gurtin M.E., and Murdoch I. Arch. Rational Mech. Anal. 57 (1975) 291
53. Cahn J.W. Acta metall. 28 (1980) 1333
54. Hillert M., and gren J. Acta mater. 50 (2002) 2429
55. Cahn J.W. Acta metall. 37 (1989) 773
56. Rottman C. Phys. Rev. B 38 (1988) 12031
57. Johnson W.C. Acta. Mater. 48 (2000) 433
58. 13 for a dynamical treatment of the interfacial stress
59. Leo P.H., and Sekerka R.F. Acta metall. 37 (1989) 3119
60. Alexander J.I.D., and Johnson W.C. J. Appl. Phys. 58 (1985) 816
61. Asaro R.J., and Tiller W.A. Metall. Trans. A 3 (1972) 1789
62. Grinfeld M.A. Sov. Phys. Dokl. 31 (1986) 831
63. Srolovitz D.J. Acta. Mater. 37 (1989) 621
64. Spencer B.J., Voorhees P.W., and Davis S.H. Phys. Rev. Lett. 67 (1991) 3696
65. Onuki A. J. Phys. Soc. Japan 60 (1991) 345
66. The conditions for stability when the lattice depends on vacancy concentration have been examined by Spencer et al.
67. Mullins W.W. J. Appl. Phys. 28 (1957) 333
68. Weatherburn C.E. Differential Geometry of Three Dimensions (1927), Cambridge University Press, New York
69. Gurtin M.E., and Voorhees P.W. J. Appl. Math. Phys. 41 (1990) 782
70. Barvosa-Carter W., Aziz M.J., Gray L.J., and Kaplan T. Phys. Rev. Lett. 81 (1998) 1445
71. Voorhees P.W. In: M.K.S., et al. (Ed). Interfaces in the 21st Century (2002), Imperial College Press 167
72. Spencer B.J., Voorhees P.W., and Davis S.H. J. Appl. Phys. 73 (1993) 4955
73. Guyer J.E., and Voorhees P.W. Phys. Rev. B 54 (1995-1996) 11710
74. Leonard F.C., and Desai R.C. Appl. Phys. Lett. 73 (1998) 208
75. Spencer B.J., Voorhees P.W., and Tersoff J. Phys. Rev. Lett. 84 (2000) 2449
76. Spencer B.J., Voorhees P.W., and Tersoff J. Phys. Rev. B 235318 (2001)
77. Leo P.H., and Sekerka R.F. Acta mater. 49 (2001) 1771
78. Cahn J.W. Acta metall. 9 (1961) 795
79. Larché F.C., and Cahn J.W. Acta metall. 26 (1978) 1579
80. Cermelli P., and Gurtin M.E. Arch. Rat. Mech. Anal. 127 (1994) 41
81. Voorhees P.W., and Johnson W.C. J. Chem. Phys. 84 (1986) 5108
82. Johnson W.C., and Voorhees P.W. Metall. Trans. 16A (1985) 337
83. Larché F.C., and Cahn J.W. Acta metall. 30 (1982) 1835
84. Larché F.C., and Cahn J.W. Acta metall. 40 (1992) 947
85. Cahn J.W., and Kobayashi R. Acta Metall. Mater. 43 (1995) 931
86. Kirkaldy J.S., and Young D.J. Diffusion in the Condensed State (1987), Institute of Metals, London
87. Voorhees P.W., McFadden G.B., and Johnson W.C. Acta metall. mater. 40 (1992) 2979
88. Su C.H., and Voorhees P.W. Acta mater. 44 (1996) 2001
89. Akaiwa N., Thornton K., and Voorhees P.W. J. Comput. Phys. 173 (2001) 61
90. Thornton K., Akaiwa N., and Voorhees P.W. Phys. Rev. Lett. 86 (2001) 1259
91. Leo P.H., Mullins W.W., Sekerka R.F., and Vinals J. Acta metall. et mater. 38 (1990) 1573
92. Leo P.H., and Sekerka R.F. Acta metall. 37 (1989) 3139
93. Thompson M.E., and Voorhees P.W. Acta metall. mater. 47 (1999) 983
94. Cahn J.W. Acta metall. 10 (1962) 179
95. Roytburd A.L. J. Appl. Phys. 83 (1998) 228
96. Roytburd A.L. J. Appl. Phys. 83 (1998) 239
97. Alpay S.P., and Roytburd A.L. J. Appl. Phys. 83 (1998) 4714
98. Johnson W.C., and Chiang C.-S. J. Appl. Phys. 64 (1988) 1155
99. Mbaye A.A., Wood D.M., and Zunger A. Phys. Rev. B 37 (1988) 3008
100. Williams R.O. Metall. Trans. 11A (1980) 247
101. Williams R.O. CALPHAD 8 (1984) 1
102. Roitburd A.L. Sov. Phys. Sol. State 25 (1983) 17
103. Roitburd A.L. Sov. Phys. Sol. St. 26 (1984) 1229
104. Johnson W.C., and Voorhees P.W. Metall. Mater. Trans. 18A (1987) 1213
105. Johnson W.C. In: Barbee F.S.T.W., and Greer L. (Eds). Mater. Res. Soc Symp. Proc. vol. 103 (1987), Materials Research Society 61
106. Roytburd A.L., and Slutsker J. Mater. Sci. Engr. A238 (1997) 23
107. Johnson W.C., and Mueller W.H. Acta metall. mater. 89 (1991) 1991
108. Griffiths R.B., and Wheeler J.W. Phys. Rev. A 2 (1970) 1047
109. Because only n thermodynamic fields are independent, one of the thermodynamic fields must be dependent.
110. v .
111. As an example, consider the temperature-pressure phase diagram of a single-component system. There are two independent thermodynamic fields (temperature and pressure) and the dependent field is the molar Gibbs free energy. In the two-dimensional space of temperature and pressure, single phases are found in areas (two dimensions), regions of two-phase coexistence along lines (one dimension), and three-phase coexistence at points.
112. Liu Z.-K., and Agren J. Acta Metall. 38 (1990) 561
113. A unique description of the system not only requires knowledge of which phases are present but also the relative amounts of each phase.
114. Hillert M. Int. Metal. Rev. 30 (1985) 45
115. Johnson W.C., and Chiang C.-S. J. Mater. Res. 4 (1989) 678
116. Roytburd A.L., and Yu Y. Ferroelectrics 144 (1993) 137
117. Roytburd A.L. Phase Trans. 45 (1993) 1
118. Hilliard J.E. Phase Transformations (1970), American Society for Metals, Metals Park, OH
119. Stringfellow G.B. J. Electron. Mater. 11 (1982) 903
120. Glas F. J. Appl. Phys. 62 (1987) 15
121. Neuberger M. Handbook of Electronic Materials vol. 2 (1971), IFI/Plenum, New York 66
122. Huh J.-Y., and Johnson W.C. Acta. Metall. Mater. 43 (1995) 1995
123. This behavior is completely analogous to that of unstressed ternary alloys. In a plot of temperature and the composition of just one of the alloy components, the critical temperature does not coincide with the maximum temperature of the miscibility gap.
124. The plane of the temperature-composition phase diagram corresponds to zero stress. But the individual phases are stressed, even when the applied stress vanishes, owing to the misfit strains.
125. Of course, as proven in a previous subsection, a phase rule does exist for this system. The perception that the phase rule is violated arises when the thermodynamic fields and densities are not properly identified.
126. Cahn J.W., and Larché F.C. Acta metall. 32 (1984) 1915
127. Huh J.-Y., Howe J.M., and Johnson W.C. Acta. Metall. Mater. 41 (1993) 2577
128. J.-Y. Huh and W. C. Johnson vol. 311, Materials Research Society (1993), 119.
129. By absolutely stable, it is meant that, for all permissible variations in the phase compositions and stress states, the system has the lowest energy. Linearly stable means the system has the lowest energy in some region around the equilibrium state. A linearly stable state sits in an energy well.
130. Cahn J.W., and Kalonji G. In: Aaronson H.I., Laughlin D.E., Sekerka R.F., and Wayman C.M. (Eds). Int. Conf. Solid-Solid Phase Transformations (1982), TMS-AIME, Warrendale, PA 3
131. For example, the yield stress of the two-phase system would still reflect the symmetry of the cubic matrix phase when the tetragonal precipitates assume all orientational variants equally. This would not be true if the axis of tetragonality of all precipitates were parallel.
132. The Curie group contains the symmetry operations of the external field.
133. Sirotin Y.I., and Shaskolskaya M.P. Fundamentals of Crystal Physics (1982), Mir, Moscow
134. Wulff G. Zeit. f. Kristallog. 34 (1901) 1901
135. Johnson W.C., and Voorhees P.W. Solid State Phen. 23 (1992) 87
136. Taylor J.E. Bull. Am. Math. Soc. 84 (1978) 197
137. Johnson W.C., and Cahn J.W. Acta metall. 32 (1984) 1925
138. Johnson W.C., Berkenpas M.B., and Laughlin D.E. Acta metall. 36 (1988) 3149
139. Ardell A.J., and Nicholson R.B. Acta metall. 14 (1966) 1295
140. Doi M., Miyazaki T., and Wakatsuki T. Matls. Sci. and Engr. 74 (1985) 139
141. Phillips V.A. Acta metall. 14 (1966) 271
142. Satoh S., and Johnson W.C. Metal. Trans. 23A (1992) 2761
143. Kostorz G. Physica 120B (1983) 387
144. Barnett D., Lee J., Aaronson H., and Russel K. Scripta metall. 8 (1974) 1447
145. Tien J.K., and Copley S.M. Metall. Trans. 2 (1971) 215
146. Berkenpas M.B., Johnson W.C., and Laughlin D.E. J. Mater. Res. 1 (1986) 635
147. Iooss G., and Joseph D.D. Elementary Stability and Bifurcation Theory (1980), Springer-Verlag, New York
148. Thompson M.E., Su C.S., and Voorhees P.W. Acta metall. mater. 42 (1994) 2107
149. Leo P.H., Lowengrub J.S., and Jou H.J. Acta mater. 46 (1998) 2113
150. Schmidt I., and Gross D. J. Mech. Phys. Sold. 45 (1997) 1521
151. Jou H.J., Leo P.H., and Lowengrub J.S. J. Comp. Phys. 131 (1997) 109
152. Mueller R., and Gross D. Z. Angew. Math. Mech. 80 (2000) S397
153. Mueller R., and Gross D. Comp. Mater. Sci. 16 (1999) 53
154. Zhu J.Z., Liu Z.K., Vaithyanathan V., and Chen L.-Q. Scripta Mater. 46 (2002) 401
155. Johnson W. J. Amer. Cer. Soc. 77 (1994) 1581