Abstract
The phase-field crystal (PFC) method is known as a relatively new continuum approach for describing the microstructural dynamics of materials during structural and phase transitions. In contrast to molecular dynamics (MD) or density functional theory (DFT) methods, PFC allows description processes on diffusion time scales while maintaining an atomistic spatial resolution which explains its attraction to scientists over the last 20 years. After certain development, PFC serves today as a solution of a wide variety of new and increasingly complex problems. As a result, dozens of various modifications of the original PFC model are already helping to study special features in material science. New possibilities and challenges arise when extended spatio-temporal correlations are introduced in PFC model. The current paper briefly systematizes and characterizes PFC model’s applications mainly focused on spatio-temporal correlations, systems with memory and models suitable for the description of non-equilibrium phase transformations.
Similar content being viewed by others
References
J. Bueno, I. Starodumov, H. Gomez, P. Galenko, D. Alexandrov, Comput. Mater. Sci. 111, 310 (2016)
R. Jalili, D. Esrafilzadeh, S.H. Aboutalebi, Y.M. Sabri, A.E. Kandjani, S.K. Bhargava, E. Della Gaspera, T.R. Gengenbach, A. Walker, Y. Chao et al., Nat. Commun. 9, 1 (2018)
J. Wang, F. Ma, M. Sun 7, 16801 (2017)
G. Cassabois, P. Valvin, B. Gil, Nat. Photonics 10, 262 (2016). arXiv:1512.02962
D.M. Herlach, P.K. Galenko, D. Holland-Moritz, Metastable solids from undercooled melts (Elsevier, 2007), ISBN 9780080468860
N. Provatas, K. Elder, Phase-Field Methods in Materials Science and Engineering (Wiley-VCH, 2010), ISBN 9783527407477
W.J. Boettinger, J.A. Warren, C. Beckermann, A. Karma, Ann. Rev. Mater. Res. 163–194 (2002)
I.G. Nizovtseva, N. Moelans, A.A. Ivanov, Eur. Phys. J. Spec. Top. 229, 453 (2020)
I.G. Nizovtseva, N. Moelans, K. Reuther, M. Rettenmayr, D.V. Alexandrov, Eur. Phys. J. Spec. Top. 229, 433 (2020)
L. Gránásy, G.I. Tóth, J.A. Warren, F. Podmaniczky, G. Tegze, L. Rátkai, T. Pusztai, Prog. Mater. Sci. 106, 100569 (2019)
P. Galenko, I. Nizovtseva, K. Reuther, M. Rettenmayr, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 376, 20170207 (2018)
P. Galenko, I. Nizovtseva, K. Reuther, M. Rettenmayr, J. Exp. Theor. Phys. 127, 107 (2018)
M. Berghoff, B. Nestler, Comput. Condens. Matter 4, 46 (2015)
N. Wang, G. Kocher, N. Provatas, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 376, (2018)
N. Ofori-Opoku, V. Fallah, M. Greenwood, S. Esmaeili, N. Provatas, Phys. Rev. B Condens. Matter Mater. Phys. 87, 134105 (2013). arXiv:1211.0003
A. Emdadi, M. Asle Zaeem, E. Asadi, Comput. Mater. Sci. 123, 139 (2016)
A. Jaatinen, T. Ala-Nissila, J. Phys. Condens. Matter 22, 205402 (2010)
V.E. Ankudinov, P.K. Galenko, N.V. Kropotin, M.D. Krivilyov, J. Exp. Theor. Phys. 122, 298 (2016)
H. Emmerich, H. Löwen, R. Wittkowski, T. Gruhn, G.I. Tóth, G. Tegze, L. Gránásy, Adv. Phys. 61, 665 (2012)
P. Galenko, D. Danilov, V. Lebedev, Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 79, 51110 (2009)
G. Tegze, L. Gránásy, G.I. Tóth, F. Podmaniczky, A. Jaatinen, T. Ala-Nissila, T. Pusztai, Phys. Rev. Lett. 103, 035702 (2009)
E. Asadi, M. Asle Zaeem, JOM 67, 186 (2015)
V. Ankudinov, K.R. Elder, P.K. Galenko, Phys. Rev. E 102, 062802 (2020)
K. Elder, M. Katakowski, M. Haataja, M. Grant, Phys. Rev. Lett. 88, 245701 (2002)
J. Berry, K. Elder, M. Grant, Phys. Rev. B 77, 224114 (2008)
M. Guerdane, M. Berghoff, Phys. Rev. B 97, 144105 (2018)
G.I. Tóth, G. Tegze, T. Pusztai, G. Tóth, L. Gránásy, J. Phys. Condens. Matter 22, 364101 (2010)
S. Tang, R. Backofen, J. Wang, Y. Zhou, A. Voigt, Y. Yu, J. Cryst. Growth 334, 146 (2011)
F. Podmaniczky, G. Tóth, G. Tegze, T. Pusztai, L. Gránásy, J. Cryst. Growth 457, 24 (2017)
J. Swift, P.C. Hohenberg, Phys. Rev. A 15, 319 (1977)
V. Ankudinov, P. Galenko, J. Cryst. Growth 539, 125608 (2020). arXiv:1911.08248
E. Asadi, M. Asle Zaeem, Comput. Mater. Sci. 105, 101 (2015)
E. Asadi, M. Asle Zaeem, Comput. Mater. Sci. 105, 110 (2015)
P.K. Galenko, H. Gomez, N.V. Kropotin, K.R. Elder, Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 88, 13310 (2013)
P. Stefanovic, Density Functional Modeling of Mechanical Properties and Phase Transformations in Manocrystalline Materials (McMaster University, 2008)
N. Faghihi, S. Mkhonta, K. Elder, M. Grant, Phys. Rev. E 100, 022128 (2019)
N. Faghihi, S. Mkhonta, K.R. Elder, M. Grant, Eur. Phys. J. B 91, (2018)
G. Kocher, N. Provatas, Phys. Rev. Lett. 114, 155501 (2015)
G. Kocher, N. Provatas, Phys. Rev. Mater. 3, 053804 (2019)
Z.L. Wang, Z. Liu, Z.F. Huang, W. Duan, Phys. Rev. Mater. 4, 103802 (2020)
E. Alster, K. Elder, P.W. Voorhees, Phys. Rev. Mater. 4, 013802 (2020)
R. Backofen, A. Rätz, A. Voigt, Philos. Mag. Lett. 87, 813 (2007)
K.R. Elder, N. Provatas, J. Berry, P. Stefanovic, M. Grant, Phys. Rev. B 75, 064107 (2007)
K. Elder, K. Thornton, J. Hoyt, Philos. Mag. 91, 151 (2011)
S. Muralidharan, M. Haataja, Phys. Rev. Lett. 105, 126101 (2010)
R. Backofen, A. Voigt, J. Phys. Condens. Matter 21, 464109 (2009)
L. Gránásy, G. Tegze, G.I. Tóth, T. Pusztai, Philos. Mag. 91, 123 (2011)
G. Tegze, L. Gránásy, G.I. Tóth, J.F. Douglas, T. Pusztai, Soft Matter 7, 1789 (2011)
K.A. Wu, A. Karma, Phys. Rev. B 76, 184107 (2007)
P.K. Galenko, D. Jou, Phys. Rep. 818, 1 (2019)
P. Galenko, H. Gomez, N. Kropotin, K. Elder, Phys. Rev. E 88, 013310 (2013)
P. Galenko, V. Ankudinov, I. Starodumov, Phase Field Crystals: Fast Interface Dynamics (De Gruyter, Berlin, Berlin, 2018), ISBN 9783110588095
P. Stefanovic, M. Haataja, N. Provatas, Phys. Rev. E 80, 046107 (2009)
P. Stefanovic, M. Haataja, N. Provatas, Phys. Rev. Lett. 96, 225504 (2006). arXiv:quant-ph/0601619
M. Conti, A. Giorgini, M. Grasselli, J. Math. Anal. Appl. 436, 1297 (2016)
C. Wang, S. Wise, SIAM J. Numer. Anal. 49, 945 (2011)
A. Baskaran, Z. Hu, J. Lowengrub, C. Wang, S. Wise, P. Zhou, J. Comput. Phys. 250, 270 (2013)
M. Grasselli, H. Wu, arXiv preprint arXiv:1312.2834 (2013)
R. Prieler, J. Hubert, D. Li, B. Verleye, R. Haberkern, H. Emmerich, J. Phys. Condens. Matter 21, 464110 (2009)
M.A. Choudhary, D. Li, H. Emmerich, H. Löwen, J. Phys. Condens. Matter 23, 265005 (2011)
H. Emmerich, L. Gránásy, H. Löwen, Eur. Phys. J. Plus 126, 1 (2011)
V. Ankudinov, I. Starodumov, N.P. Kryuchkov, E.V. Yakovlev, S.O. Yurchenko, P.K. Galenko, Math. Methods Appl. Sci. (2020)
D. Kharchenko, V. Kharchenko, I. Lysenko, Cent. Eur. J. Phys. 9, 698 (2011)
S. van Teeffelen, R. Backofen, A. Voigt, H. Löwen, Phys. Rev. E 79, 051404 (2009)
H. Löwen, J. Phys. Condens. Matter 22, 364105 (2010)
B.P. Athreya, N. Goldenfeld, J.A. Dantzig, Phys. Rev. E 74, 011601 (2006)
S. Majaniemi, N. Provatas, Phys. Rev. E 79, 011607 (2009)
K.A. Wu, A. Adland, A. Karma, Phys. Rev. E 81, 061601 (2010)
Z.F. Huang, K. Elder, Phys. Rev. B 81, 165421 (2010)
R. Spatschek, A. Karma, Phys. Rev. B 81, 214201 (2010)
D.H. Yeon, Z.F. Huang, K. Elder, K. Thornton, Philos. Mag. 90, 237 (2010)
K. Elder, Z.F. Huang, N. Provatas, Phys. Rev. E 81, 011602 (2010)
S. Praetorius, A. Voigt, J. Chem. Phys. 142, 154904 (2015)
M. Seymour, N. Provatas, Phys. Rev. B 93, 035447 (2016)
M. Greenwood, N. Ofori-Opoku, J. Rottler, N. Provatas, Phys. Rev. B 84, 064104 (2011)
K. Elder, M. Seymour, M. Lee, M. Hilke, N. Provatas, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 376, 20170211 (2018)
S. Brazovskii, S. Dmitriev, Zh. Eksp, Teor. Fiz 69, 979 (1975)
K. Elder, M. Grant, Phys. Rev. E 70, 051605 (2004)
K.R. Elder, M. Katakowski, M. Haataja, M. Grant, Phys. Rev. Lett. 88, 245701 (2002)
P. Tupper, M. Grant, EPL (Europhysics Letters) 81, 40007 (2008)
P.K. Galenko, R. Wonneberger, S. Koch, V. Ankudinov, E.V. Kharanzhevskiy, M. Rettenmayr, J. Cryst. Growth 532, 125411 (2020)
V. Ankudinov, I. Starodumov, P.K. Galenko, Math. Methods Appl. Sci. (2020)
A. Jaatinen, C.V. Achim, K.R. Elder, T. Ala, Phys. Rev. E 80, 1 (2009)
T. Ramakrishnan, M. Yussouff, Phys. Rev. B 19, 2775 (1979)
M. Oettel, S. Dorosz, M. Berghoff, B. Nestler, T. Schilling, Phys. Rev. E 86, 021404 (2012)
E. Kats, V. Lebedev, A. Muratov, Phys. Rep. 228, 1 (1993)
L.D. Landau, E.M. Lifshits, L.P. Pitaevskii, Statistical physics (Butterworth-Heinemann, c1980, 1996), ISBN 9780750633727
S.A. Brazovskii, Zhurnal Eksperimentalnoi I Teoreticheskoi Fiziki 68, 175 (1975)
P. Galenko, D. Jou, Phys. Rev. E 71, 046125 (2005)
M. Cheng, J. Kundin, D. Li, H. Emmerich, Philos. Mag. Lett. 92, 517 (2012)
N. Ofori-Opoku, J.A. Warren, P.W. Voorhees, Phys. Rev. Mater. 2, 083404 (2018). arXiv:1803.01407
M.A. Choudhary, J. Kundin, H. Emmerich, Philos. Mag. Lett. 92, 451 (2012)
A.J. Archer, J. Chem. Phys. 130, 014509 (2009). arXiv:0812.1113
R. Wittkowski, H. Löwen, Mol. Phys. 109, 2935 (2011)
A.J. Archer, J. Phys. Condens. Matter 17, 1405 (2005)
M. Rauscher, J. Phys. Condens. Matter 22, 364109 (2010)
D. Frenkel, Liquids, Freezing and the Glass Transition (Amsterdam, 1991)
M. Rex, H. Wensink, H. Löwen, Phys. Rev. E 76, 021403 (2007)
M. Salvalaglio, R. Backofen, A. Voigt, K.R. Elder, Phys. Rev. E 96, 023301 (2017)
S. Praetorius, M. Salvalaglio, A. Voigt, Model. Simul. Mater. Sci. Eng. 27, 044004 (2019)
D. Jou, P. Galenko, Phys. Rev. E 88, 042151 (2013)
D. Jou, P.K. Galenko, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 376, 20170203 (2018)
G. Kocher, N. Ofori-Opoku, N. Provatas, Phys. Rev. Lett. 117, 1 (2016)
N. Goldenfeld, B.P. Athreya, J.A. Dantzig, Phys. Rev. E 72, 020601 (2005)
B.P. Athreya, N. Goldenfeld, J.A. Dantzig, M. Greenwood, N. Provatas, Phys. Rev. E 76, 056706 (2007)
Y. Shiwa, Phys. Rev. E 79, 013601 (2009)
U. Marconi, P. Tarazona, J. Chem. Phys. 110, 8032 (1999). arXiv:quant-ph/9810403
I. Aranson, B. Malomed, L. Pismen, L. Tsimring, Phys. Rev. E Stat. Phys. Plasmas, Fluids, Relat. Interdiscip. Top. 62, R5 (2000)
H. Ohnogi, Y. Shiwa, Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 84, 051603 (2011)
I. Starodumov, V. Ankudinov, P. Galenko, MATEC Web Conf. 129, 02035 (2017)
A. Archer, M. Rauscher, J. Phys. A Math. Gen. 37, 9325 (2004). arXiv:quant-ph/0405603
Q. Bronchart, Y. Le Bouar, A. Finel, Phys. Rev. Lett. 100, 015702 (2008)
D. Jou, P. Galenko, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 376, 20170203 (2018)
Y. Guo, J. Wang, Z. Wang, S. Tang, Y. Zhou, Wuli Xuebao/Acta Physica Sinica 61, 146401 (2012)
Y.L. Wu, Eur. Phys. J. C 55(1), 159 (2017)
G. Morales, Phys. Rev. E 99, 062218 (2019)
V. Lisy, J. Tothova, J. Comput. Theor. Transp. 42, 365 (2013)
D. Jou, P. Galenko, Phys. A Stat. Mech. Appl. 366, 149 (2006)
P.K. Galenko, D. Kharchenko, I. Lysenko, Phys. A Stat. Mech. Appl. 389, 3443 (2010)
D. Kharchenko, I. Lysenko, P. Galenko, Stoch. Differ. Equ. (2011)
J. García-Ojalvo, J. Sancho, Phys. Rev. E 49, 2769 (1994)
I. Nizovtseva, P. Galenko, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 376, 20170202 (2018)
P. Galenko, F. Iunes Sanches, K. Elder, Phys. D 308, 1 (2015)
J. Ramos, E. Granato, S. Ying, C. Achim, K. Elder, T. Ala-Nissila, Phys. Rev. E 81, 011121 (2010)
P.Y. Chan, N. Goldenfeld, J. Dantzig, Phys. Rev. E 79, 035701 (2009)
M. Greenwood, J. Rottler, N. Provatas, Phys. Rev. E 83, 031601 (2011)
S. Mkhonta, K. Elder, Z.F. Huang, Phys. Rev. Lett. 111, 035501 (2013)
M. Greenwood, N. Provatas, J. Rottler, Phys. Rev. Lett. 105, 1 (2010). arXiv:1002.3185
M. Greenwood, N. Ofori-Opoku, J. Rottler, N. Provatas, Phys. Rev. B Condens. Matter Mater. Phys. 84, 1 (2011)
V. Fallah, N. Ofori-Opoku, J. Stolle, N. Provatas, S. Esmaeili, Acta Mater. 61, 3653 (2013)
N. Smith, N. Provatas, Phys. Rev. Mater. 1, 053407 (2017)
V.W. Chan, N. Pisutha-Arnond, K. Thornton, Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 91, 053305 (2015)
K.L. Baker, W.A. Curtin, Phys. Rev. B 91, 014103 (2015)
S.K. Mkhonta, K.R. Elder, Z.F. Huang, Phys. Rev. Lett. 111, 35501 (2013). arXiv:1306.6357
H. Zapolsky, G. Demange, 12, (2017)
G.I. Tóth, T. Pusztai, G. Tegze, G. Tóth, L. Gránásy, Phys. Rev. Lett. 107, 175702 (2011). arXiv:1109.2729
N. Pisutha-Arnond, V.W. Chan, M. Iyer, V. Gavini, K. Thornton, Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 87, 1 (2013)
S. Ghosh, Comput. Mater. Sci. 138, 384 (2017)
K. Elder, G. Rossi, P. Kanerva, F. Sanches, S. Ying, E. Granato, C. Achim, T. Ala-Nissila, Phys. Rev. Lett. 108, 226102 (2012)
V. Fallah, J. Stolle, N. Ofori-Opoku, S. Esmaeili, N. Provatas, Phys. Rev. B 86, 134122 (2012)
P. Hirvonen, M.M. Ervasti, Z. Fan, M. Jalalvand, M. Seymour, S.M.V. Allaei, N. Provatas, A. Harju, K.R. Elder, T. Ala-Nissila, Phys. Rev. B 94, 035414 (2016)
Funding
This work has been supported by the grant of the Russian Science Foundation, project 19-71-10044.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Starodumov, I., Ankudinov, V. & Nizovtseva, I. A review of continuous modeling of periodic pattern formation with modified phase-field crystal models. Eur. Phys. J. Spec. Top. 231, 1135–1145 (2022). https://doi.org/10.1140/epjs/s11734-022-00518-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjs/s11734-022-00518-5