5 Phase-Field-Crystal Model: A Tool for Probing Atoms in TSV
129
4. R. Evans, The nature of the liquid-vapour interface and other topics in the statistical mechanics
of non-uniform, classical fluids. Adv. Phys. 28, 143–200 (1979)
5. M. Greenwood, N. Provatas, J. Rottler, Free energy functionals for efficient phase field crystal
modeling of structural phase transformations. Phys. Rev. Lett. 105, 045702 (2010)
6. M. Greenwood, J. Rottler, N. Provatas, Phase-field-crystal methodology for modeling of structural transformations. Phys. Rev. E 83, 031601 (2011)
7. N. Provatas, K. Elder, Phase-Field Methods in Material Science and Engineering (WILEYVCH Verlag GmbH & Co, KGaA, Germany, 2010)
8. P. Chaikin, T. Lubensky, Principles of condensed matter physics. Cambridge University Press,
Cambridge, Great Britain (1995)
9. O. Eder, E. Erdpresser, B. Kunscht et al., The structure factor of liquid copper at 1393 K and
1833 K. J Phys. F: Metal Phys. 10, 183–195 (1980)
10. D. Belashchenko, Y. Zhuravlev, Application of the embedded-atom method to liquid copper.
Inorg. Mater. 44, 939–945 (2008)
11. G. Strang, Computational Science and Engineering (Wellesley-Cambridge Press, USA, 2007)
12. K. Wu, A. Adland, A. Karma, Phase-field-crystal model for fcc ordering. Phys. Rev. E 81,
061601 (2010)
13. E. Asadi, M.A. Zaeem, Quantitative phase-field crystal modeling of solid-liquid interfaces for
FCC metals. Comput. Mater. Sci 127, 236–243 (2017)
14. V. Sears, S. Shelley, Debye-Waller factor for elemental crystals. Acta Cryst. 47, 441–446 (1991)
15. J. Berry, N. Provatas, J. Rottler et al., Defect stability in phase-field crystal models: stacking
faults and partial dislocations. Phys. Rev. B 86, 224112 (2012)
16. Y.M. Jin, A.G. Khachaturyan, Atomic density function theory and modeling of microstructure
evolution at the atomic scale. J. Appl. Phys. 100, 013519 (2006)
17. P. Stefanovic, M. Haataja, N. Provatas et al., Phase-field crystals with elastic interactions. Phys.
Rev. Lett. 96, 225504 (2006)
18. P. Stefanovic, M. Haataja, N. Provatas et al., Phase field crystal study of deformation and
plasticity in nanocrystalline materials. Phys. Rev. E 80, 046107 (2009)
19. V. Heinonen, C. Achim, K. Elder et al., Phase-field-crystal models and mechanical equilibrium.
Phys. Rev. E 89, 032411 (2014)
20. M. Greenwood, N. Ofori-Opoku, J. Rottler et al., Modeling structural transformations in binary
alloys with phase field crystals. Phys. Rev. B 84, 064104 (2011)
21. N. Ofori-Opoku, V. Fallah, M. Greenwood et al., Multicomponent phase-field crystal model
for structural transformations in metal alloys. Phys. Rev. B 87, 134105 (2013)
22. E. Schwalbach, J. Warren, K. Wu et al., Phase-field crystal model with a vapor phase. Phys.
Rev. E 88, 023306 (2013)
23. G. Kocher, N. Provatas, New density functional approach for solid-liquid-vapor transitions in
pure materials. Phys. Rev. Lett. 114, 155501 (2015)
24. D. Taha, S. Mkhonta, K. Elder et al., Grain boundary structures and collective dynamics of
inversion domains in binary two-dimensional materials. Phys. Rev. Lett. 118, 255501 (2017)
25. P. Hirvonen, V. Heinonen, H. Dong et al., Phase field crystal model for heterostructures. Phys.
Rev. B 100, 165412 (2019)
26. T. Hirouchi, T. Takaki, Y. Tomita, Development of numerical scheme for phase field crystal
deformation simulation. Comput. Mater. Sci. 44, 1192–1197 (2009)
27. P. Chan, N. Goldenfeld, Nonlinear elasticity of the phase-field crystal model from the renormalization group. Phys. Rev. E 80, 065105 (2009)
28. M. Salvalaglio, A. Voigt, K. Elder, Closing the gap between atomic-scale lattice deformations
and continuum elasticity. NPJ Comput. Mater. 5, 48 (2019)
29. J. Berry, M. Grant, K. Elder, Diffusive atomistic dynamics of edge dislocations in two dimensions. Phys. Rev. E 73, 031609 (2006)
30. P. Chan, G. Tsekenis, J. Dantzig et al., Plasticity and dislocation dynamics in a phase field
crystal model. Phys. Rev. Lett 105, 015502 (2010)
31. J. Nye, Physical Properties of Crystals Their Representation by Sensors and Matrices (Oxford
University Press Inc., New York, 2006)
129
4. R. Evans, The nature of the liquid-vapour interface and other topics in the statistical mechanics
of non-uniform, classical fluids. Adv. Phys. 28, 143–200 (1979)
5. M. Greenwood, N. Provatas, J. Rottler, Free energy functionals for efficient phase field crystal
modeling of structural phase transformations. Phys. Rev. Lett. 105, 045702 (2010)
6. M. Greenwood, J. Rottler, N. Provatas, Phase-field-crystal methodology for modeling of structural transformations. Phys. Rev. E 83, 031601 (2011)
7. N. Provatas, K. Elder, Phase-Field Methods in Material Science and Engineering (WILEYVCH Verlag GmbH & Co, KGaA, Germany, 2010)
8. P. Chaikin, T. Lubensky, Principles of condensed matter physics. Cambridge University Press,
Cambridge, Great Britain (1995)
9. O. Eder, E. Erdpresser, B. Kunscht et al., The structure factor of liquid copper at 1393 K and
1833 K. J Phys. F: Metal Phys. 10, 183–195 (1980)
10. D. Belashchenko, Y. Zhuravlev, Application of the embedded-atom method to liquid copper.
Inorg. Mater. 44, 939–945 (2008)
11. G. Strang, Computational Science and Engineering (Wellesley-Cambridge Press, USA, 2007)
12. K. Wu, A. Adland, A. Karma, Phase-field-crystal model for fcc ordering. Phys. Rev. E 81,
061601 (2010)
13. E. Asadi, M.A. Zaeem, Quantitative phase-field crystal modeling of solid-liquid interfaces for
FCC metals. Comput. Mater. Sci 127, 236–243 (2017)
14. V. Sears, S. Shelley, Debye-Waller factor for elemental crystals. Acta Cryst. 47, 441–446 (1991)
15. J. Berry, N. Provatas, J. Rottler et al., Defect stability in phase-field crystal models: stacking
faults and partial dislocations. Phys. Rev. B 86, 224112 (2012)
16. Y.M. Jin, A.G. Khachaturyan, Atomic density function theory and modeling of microstructure
evolution at the atomic scale. J. Appl. Phys. 100, 013519 (2006)
17. P. Stefanovic, M. Haataja, N. Provatas et al., Phase-field crystals with elastic interactions. Phys.
Rev. Lett. 96, 225504 (2006)
18. P. Stefanovic, M. Haataja, N. Provatas et al., Phase field crystal study of deformation and
plasticity in nanocrystalline materials. Phys. Rev. E 80, 046107 (2009)
19. V. Heinonen, C. Achim, K. Elder et al., Phase-field-crystal models and mechanical equilibrium.
Phys. Rev. E 89, 032411 (2014)
20. M. Greenwood, N. Ofori-Opoku, J. Rottler et al., Modeling structural transformations in binary
alloys with phase field crystals. Phys. Rev. B 84, 064104 (2011)
21. N. Ofori-Opoku, V. Fallah, M. Greenwood et al., Multicomponent phase-field crystal model
for structural transformations in metal alloys. Phys. Rev. B 87, 134105 (2013)
22. E. Schwalbach, J. Warren, K. Wu et al., Phase-field crystal model with a vapor phase. Phys.
Rev. E 88, 023306 (2013)
23. G. Kocher, N. Provatas, New density functional approach for solid-liquid-vapor transitions in
pure materials. Phys. Rev. Lett. 114, 155501 (2015)
24. D. Taha, S. Mkhonta, K. Elder et al., Grain boundary structures and collective dynamics of
inversion domains in binary two-dimensional materials. Phys. Rev. Lett. 118, 255501 (2017)
25. P. Hirvonen, V. Heinonen, H. Dong et al., Phase field crystal model for heterostructures. Phys.
Rev. B 100, 165412 (2019)
26. T. Hirouchi, T. Takaki, Y. Tomita, Development of numerical scheme for phase field crystal
deformation simulation. Comput. Mater. Sci. 44, 1192–1197 (2009)
27. P. Chan, N. Goldenfeld, Nonlinear elasticity of the phase-field crystal model from the renormalization group. Phys. Rev. E 80, 065105 (2009)
28. M. Salvalaglio, A. Voigt, K. Elder, Closing the gap between atomic-scale lattice deformations
and continuum elasticity. NPJ Comput. Mater. 5, 48 (2019)
29. J. Berry, M. Grant, K. Elder, Diffusive atomistic dynamics of edge dislocations in two dimensions. Phys. Rev. E 73, 031609 (2006)
30. P. Chan, G. Tsekenis, J. Dantzig et al., Plasticity and dislocation dynamics in a phase field
crystal model. Phys. Rev. Lett 105, 015502 (2010)
31. J. Nye, Physical Properties of Crystals Their Representation by Sensors and Matrices (Oxford
University Press Inc., New York, 2006)