Abstract
We develop a phase-field model for the simulation of precipitate microstructure pattern formation in nickel-base superalloys. The model accounts for the local effects from inhomogeneous and anisotropic elastic deformations, which mainly result from the lattice misfit between the precipitates and matrix phase. Further, in each time-step, we consider the chemical driving force for precipitate ripening to instantaneously equilibrate to a homogeneous value, leading to conserved phase volumes. The model is applied to study the equilibrium shape of a 2D single γ′-particle embedded in the γ-matrix with varying lattice misfit and γ/γ′ interface energies. Further, we apply the method of moment invariants to quantify the resulting equilibrium shapes of precipitates, which turns out to be a size independent characterization of the particle shape. Resulting values for the 2D moment invariants of experimental as well as simulated particle shapes are discussed and compared. Considering ideally spherical particles, we find that large values for the γ/γ′-interface width lead to systematic deviations in the resulting moment invariants.
Graphical abstract
Similar content being viewed by others
References
R.C. Reed,The Superalloys Fundamentals and Applications, 1st edn. (Cambridge University Press, New York, Cambridge, UK, 2006)
E. Fleischmann, C.H. Konrad, J. Preußner, R. Völkl, E. Affeldt, U. Glatzel, Metall. Mater. Trans. A 46, 1125 (2015)
I. Steinbach, Modell. Simul. Mater. Sci. Eng. 17, 073001 (2009)
M. Asta, C. Beckermann, A. Karma, W. Kurz, R. Napolitano, M. Plapp, G. Purdy, M. Rappaz, R. Trivedi, Acta Mater. 57, 941 (2009)
Y. Wang, J. Li, Acta Mater. 58, 1212 (2010)
I. Steinbach, O. Shchyglo, Curr. Opin. Solid State Mater. Sci. 15, 87 (2011)
L. Nguyen, R. Shi, Y. Wang, M. De Graef, Acta Mater. 103, 322 (2016)
M. Cottura, Y. Le Bouar, B. Appolaire, A. Finel, Acta Mater. 94, 15 (2015)
A. Gaubert, M. Jouiad, J. Cormier, Y. Le Bouar, J. Ghighi, Acta Mater. 84, 237 (2015)
M.P. Gururajan, T.A. Abinandanan, Acta Mater. 55, 5015 (2007)
M. Fleck, F. Schleifer, M. Holzinger, U. Glatzel, Metall. Mater. Trans. A 49, 4146 (2018)
L.T. Mushongera, M. Fleck, J. Kundin, F. Querfurth, H. Emmerich, Adv. Eng. Mater. 17, 1149 (2015)
J. Goerler, I. Lopez-Galilea, L. Mujica Roncery, O. Shchyglo, W. Theisen, I. Steinbach, Acta Mater. 124, 151 (2017)
M.S. Bhaskar, Comput. Mater. Sci. 146, 102 (2018)
Y. Pang, Y. Li, X. Wu, W. Liu, Z. Hou, Int. J. Mater. Res. 106, 108 (2015)
A. Jokisaari, S. Naghavi, C. Wolverton, P.W. Voorhees, O. Heinonen, Acta Mater. 141, 273 (2017)
B. Bhadak, R. Sankarasubramanian, A. Choudhury, Metall. Mater. Trans. A 49, 5705 (2018)
S. Maitra, Proc. IEEE 67, 697 (1979)
M.K. Hu, IRE Trans. Inf. Theory 8, 179 (1962)
J. MacSleyne, J. Simmons, M. De Graef, Modell. Simul. Mater. Sci. Eng. 16, 045008 (2008)
J. MacSleyne, J. Simmons, M. De Graef, Acta Mater. 56, 427 (2008)
J. MacSleyne, M. Uchic, J. Simmons, M. De Graef, Acta Mater. 57, 6251 (2009)
P. Callahan, J. Simmons, M. De Graef, Modell. Simul. Mater. Sci. Eng. 21, 015003 (2012)
J. Van Sluytman, T. Pollock, Acta Mater. 60, 1771 (2012)
R. Rettig, N. Ritter, H. Helmer, S. Neumeier, R. Singer, Modell. Simul. Mater. Sci. Eng. 23, 035004 (2015)
A. Chowdhury, E. Kautz, B. Yener, D. Lewis, Comput. Mater. Sci. 123, 176 (2016)
S. Haas, A. Manzoni, F. Krieg, U. Glatzel, Entropy 21, 169 (2019)
M. Plapp, Phys. Rev. E 84, 031601 (2011)
K. Kassner, C. Misbah, J. Müller, J. Kappey, P. Kohlert, Phys. Rev. E 63, 036117 (2001)
M. Fleck, E.A. Brener, R. Spatschek, B. Eidel, Int. J. Mater. Res. 4, 462 (2010)
A. Durga, P. Wollants, N. Moelans, Modell. Simul. Mater. Sci. Eng. 21, 055018 (2013)
M. Fleck, L.T. Mushongera, D. Pilipenko, K. Ankit, H. Emmerich, Eur. Phys. J. Plus 126, 95 (2011)
M. Fleck, H. Federmann, E. Pogorelov, Comput. Mater. Sci. 153, 288 (2018)
B. Nestler, F. Wendler, M. Selzer, B. Stinner, H. Garcke, Phys. Rev. E 78, 011604 (2008)
P. Voorhees, G. McFadden, W. Johnson, Acta Metall. Mater. 40, 2979 (1992)
A. Finel, Y. Le Bouar, B. Dabas, B. Appolaire, Y. Yamada, T. Mohri, Phys. Rev. Lett. 121, 025501 (2018)
A. Bösch, H. Müller-Krumbhaar, O. Shochet, Z. Phys. B 97, 367 (1995)
K. Glasner, J. Comput. Phys. 174, 695 (2001)
M. Weiser, Appl. Numer. Math. 59, 1858 (2009)
J. Eiken, IOP Conf. Ser. 33, 012105 (2012)
U. Glatzel, M. Feller-Kniepmeier, Scr. Metall. 23, 1839 (1989)
T. Pollock, A. Argon, Acta Metall. Mater. 42, 1859 (1994)
M. Probst-Hein, A. Dlouhy, G. Eggeler, Acta Mater. 47, 2497 (1999)
J. Preußner, Y. Rudnik, R. Völkl, U. Glatzel, Z. Metallkd 96, 595 (2005)
A. Jokisaari, P. Voorhees, J. Guyer, J. Warren, O. Heinonen, Comput. Mater. Sci. 149, 336 (2018)
K. Thornton, N. Akaiwa, P. Voorhees, Acta Mater. 52, 1353 (2004)
R. Völkl, U. Glatzel, M. Feller-Kniepmeier, Acta Mater. 46, 4395 (1998)
A.J. Ardell, V. Ozolins, Nat. Mater. 4, 309 (2005)
B. Sonderegger, E. Kozeschnik, Metall. Mater. Trans. A 40, 499 (2009)
Author information
Authors and Affiliations
Contributions
All authors were involved in the discussion of the results and the preparation of the manuscript. All authors have read and approved the final manuscript.
Corresponding author
Additional information
Contribution to the Topical Issue “Multiscale Materials Modeling”, edited by Yoji Shibutani, Shigenobu Ogata, and Tomotsugu Shimokawa.
Rights and permissions
About this article
Cite this article
Holzinger, M., Schleifer, F., Glatzel, U. et al. Phase-field modeling of γ′-precipitate shapes in nickel-base superalloys and their classification by moment invariants. Eur. Phys. J. B 92, 208 (2019). https://doi.org/10.1140/epjb/e2019-100256-1
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2019-100256-1