Simplified Kinetic Model of Flux-Driven Phase Transitions and Patterns Competition in Open 2D System
DOI:
https://doi.org/10.15407/ujpe70.4.271Keywords:
open system, flux-driven transformation, spinodal decomposition, pattern formation, rate-dependent phase diagram, hysteresisAbstract
The regular solution approximation has a successful history of applications in the thermodynamics and kinetics of decomposition in alloys, treated as closed systems. It provides a qualitatively proper description of all stages of spinodal and nucleation-mediated decomposition for alloys under homogeneous external conditions without external fluxes. In this article, the kinetic mean-field model for open (flux-driven) systems is extended by incorporating the divergence of in- and out-fluxes into the master equations for occupation probabilities. The closest experimental analog of this model is the pattern formation during the co-deposition of a binary alloy under frozen bulk diffusion, but with reasonable surface diffusion, where the deposition rate V serves as the main external parameter. However, some peculiarities of the model may also be useful for describing eutectic and off-eutectic crystallizations. Rate-dependent phase T − C diagrams are determined for the steady the states of such an open system. The ratedependent instability region is subdivided into three distinct steady-state morphologies: spots (“gepard”-like), layers (“zebra”-like) – labyrinth or lamellae, and mixed patterns (a combination of “gepard” and “zebra”). This morphology map depends on the initial conditions, revealing memory effects and hysteresis. This implies that, unlike the equilibrium state of a closed system, which acts as an attractor for the evolution paths, the steady states of flux-driven systems may not be attractors. Variations of the model, including Monte Carlo simulations, are also discussed.
References
1. M. Henkel, H. Hinrichsen, S. L¨ubeck. Non-Equilibrium Phase Transitions. Vol. 1. Absorbing Phase Transitions (Springer, 2008).
2. G. Martin, P. Bellon. Radiation effects in concentrated alloys and compounds: equilibrium and kinetics of driven systems. Comptes Rendus Physique 9 (3-4), 323 (2008).
https://doi.org/10.1016/j.crhy.2007.11.006
3. Y. Ashkenazy, N.Q. Vo, D. Schwen, R.S. Averback, P. Bellon. Shear induced chemical mixing in heterogeneous systems. Acta Materialia 60 (3), 984 (2012).
https://doi.org/10.1016/j.actamat.2011.11.014
4. B.B. Straumal, A.S. Gornakova, O.B. Fabrichnaya, M.J. Kriegel, A.A. Mazilkin, B. Baretzky, S.V. Dobatkin. Effective temperature of high pressure torsion in Zr-Nb alloys. High Temperature Materials and Processes 31 (4-5), 339 (2012).
https://doi.org/10.1515/htmp-2012-0057
5. S.G. Brush. History of the Lenz-Ising model. Rev. Mod. Phys. 39 (4), 883 (1967).
https://doi.org/10.1103/RevModPhys.39.883
6. G. Martin. Atomic mobility in CahnпїЅs diffusion model. Phys. Rev. B 41 (4), 2279 (1990).
https://doi.org/10.1103/PhysRevB.41.2279
7. Z. Erd'elyi, I.A. Szab'o, D.L. Beke. Interface sharpening instead of broadening by diffusion in ideal binary alloys. Phys. Rev. Lett. 89 (16), 165901 (2002).
https://doi.org/10.1103/PhysRevLett.89.165901
8. D.L. Beke, Z. Erd'elyi. Resolution of the diffusional paradox predicting infinitely fast kinetics on the nanoscale. Phys. Rev. B 73 (3), 035426 (2006).
https://doi.org/10.1103/PhysRevB.73.035426
9. Z. Erd'elyi, M. Pasichnyy, V. Bezpalchuk, J.J. Tom'an, B. Gajdics, A.M. Gusak. Stochastic kinetic mean field model. Computer Phys. Commun. 204, 31 (2016).
https://doi.org/10.1016/j.cpc.2016.03.003
10. T.V. Zaporozhets, A. Taranovskyy, G. J'ager, A.M. Gusak, Z. Erd'elyi, J.J. Tom'an. The effect of introducing stochasticity to kinetic mean-field calculations: Comparison with lattice kinetic Monte-Carlo in case of regular solid solutions. Comput. Mater. Sci. 171, 109251 (2020).
https://doi.org/10.1016/j.commatsci.2019.109251
11. B. Gajdics, J.J. Tom'an, H. Zapolsky, Z. Erd'elyi, G. Demange. A multiscale procedure based on the stochastic kinetic mean field and the phase-field models for coarsening. J. Appl. Phys. 126 (6), 065106 (2019).
https://doi.org/10.1063/1.5099676
12. A. Gusak, T. Zaporozhets, N. Storozhuk. Phase competition in solid-state reactive diffusion revisited - Stochastic kinetic mean-field approach. J. Chem. Phys. 150 (17), 174109 (2019).
https://doi.org/10.1063/1.5086046
13. A. Titova, H. Zapolsky, A. Gusak. Memory effects during co-deposition of binary alloys. Scripta Materialia 241, 115897 (2024).
https://doi.org/10.1016/j.scriptamat.2023.115897
14. Y. Lu, B. Derby, H. Sriram, K. Kadirvel, C. Wang, X. Liu, Y. Wang. Microstructure development and morphological transition during deposition of immiscible alloy films. Acta Materialia 220, 117313 (2021).
https://doi.org/10.1016/j.actamat.2021.117313
15. M. Powers, J.A. Stewart, R. Dingreville, B.K. Derby, A. Misra. Compositionally-driven formation mechanism of hierarchical morphologies in co-deposited immiscible alloy thin films. Nanomaterials 11 (10), 2635 (2021).
https://doi.org/10.3390/nano11102635
16. S. Akamatsu, S. Bottin-Rousseau, M. ¸Serefo˘glu, V.T. Witusiewicz, U. Hecht, M. Plapp. In situ experiments in microgravity and phase-field simulations of the lamellar-torod transition during eutectic growth. Comptes Rendus. M'ecanique 351 (S2), 1 (2023).
https://doi.org/10.5802/crmeca.142
17. S. Bottin-Rousseau, V.T. Witusiewicz, U. Hecht, J. Fernandez, A. Laveron-Simavilla, S. Akamatsu. Coexistence of rod-like and lamellar eutectic growth patterns. Scripta Materialia 207, 114314 (2022).
https://doi.org/10.1016/j.scriptamat.2021.114314
18. M. ¸Serefo˘glu, S. Bottin-Rousseau, S. Akamatsu. Lamellarod pattern transition and confinement effects during eutectic growth. Acta Materialia 242, 118425 (2023).
https://doi.org/10.1016/j.actamat.2022.118425
19. L. R'atkai, G.I. T'oth, L. K¨ornyei, T. Pusztai, L. Gr'an'asy. Phase-field modeling of eutectic structures on the nanoscale: The effect of anisotropy. J. Mater. Sci. 52, 5544 (2017).
https://doi.org/10.1007/s10853-017-0853-8
20. A. Gusak, A. Titova. Eutectic crystallization and melting in sharp concentration gradients. J. Chem. Phys. 158 (16), (2023).
https://doi.org/10.1063/5.0147658
21. M. Atzmon, D.A. Kessler, D.J. Srolovitz. Phase separation during film growth. J. Appl. Phys. 72 (2), 442 (1992).
https://doi.org/10.1063/1.351872
22. J.W. Cahn, J.E. Hilliard. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28 (2), 258 (1958).
https://doi.org/10.1063/1.1744102
23. A.A. Vlasov. Theory of Many Particles (Gostekhizdat, 1950).
24. A.G. Khachaturyan. Theory of Structural Transformations in Solids (Courier Corporation, 2013).
25. A.S. Shirinyan, A.M. Gusak. Phase diagrams of decomposing nanoalloys. Philosophical Magazine 84 (6), 579 (2004).
https://doi.org/10.1080/14786430310001635431
26. A.S. Shirinyan, A.M. Gusak, M. Wautelet. Phase diagram versus diagram of solubility: What is the difference for nanosystems? Acta Mater. 53 (19), 5025 (2005).
https://doi.org/10.1016/j.actamat.2005.07.014
27. A.M. Gusak, A.O. Kovalchuk, B.B. Straumal. Interrelation of depletion and segregation in decomposition of nanoparticles. Philosophical Magazine 93 (14), 1677 (2013).
Downloads
Published
Issue
Section
License
Copyright Agreement
License to Publish the Paper
Kyiv, Ukraine
The corresponding author and the co-authors (hereon referred to as the Author(s)) of the paper being submitted to the Ukrainian Journal of Physics (hereon referred to as the Paper) from one side and the Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, represented by its Director (hereon referred to as the Publisher) from the other side have come to the following Agreement:
1. Subject of the Agreement.
The Author(s) grant(s) the Publisher the free non-exclusive right to use the Paper (of scientific, technical, or any other content) according to the terms and conditions defined by this Agreement.
2. The ways of using the Paper.
2.1. The Author(s) grant(s) the Publisher the right to use the Paper as follows.
2.1.1. To publish the Paper in the Ukrainian Journal of Physics (hereon referred to as the Journal) in original language and translated into English (the copy of the Paper approved by the Author(s) and the Publisher and accepted for publication is a constitutive part of this License Agreement).
2.1.2. To edit, adapt, and correct the Paper by approval of the Author(s).
2.1.3. To translate the Paper in the case when the Paper is written in a language different from that adopted in the Journal.
2.2. If the Author(s) has(ve) an intent to use the Paper in any other way, e.g., to publish the translated version of the Paper (except for the case defined by Section 2.1.3 of this Agreement), to post the full Paper or any its part on the web, to publish the Paper in any other editions, to include the Paper or any its part in other collections, anthologies, encyclopaedias, etc., the Author(s) should get a written permission from the Publisher.
3. License territory.
The Author(s) grant(s) the Publisher the right to use the Paper as regulated by sections 2.1.1–2.1.3 of this Agreement on the territory of Ukraine and to distribute the Paper as indispensable part of the Journal on the territory of Ukraine and other countries by means of subscription, sales, and free transfer to a third party.
4. Duration.
4.1. This Agreement is valid starting from the date of signature and acts for the entire period of the existence of the Journal.
5. Loyalty.
5.1. The Author(s) warrant(s) the Publisher that:
– he/she is the true author (co-author) of the Paper;
– copyright on the Paper was not transferred to any other party;
– the Paper has never been published before and will not be published in any other media before it is published by the Publisher (see also section 2.2);
– the Author(s) do(es) not violate any intellectual property right of other parties. If the Paper includes some materials of other parties, except for citations whose length is regulated by the scientific, informational, or critical character of the Paper, the use of such materials is in compliance with the regulations of the international law and the law of Ukraine.
6. Requisites and signatures of the Parties.
Publisher: Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine.
Address: Ukraine, Kyiv, Metrolohichna Str. 14-b.
Author: Electronic signature on behalf and with endorsement of all co-authors.
.png){kind=small}
