Anomalous Dynamics of Recalescence Front in Crystal Growth Processes: Theoretical Background
Abstract
:1. Introduction
2. Theoretical Modeling
2.1. The Model of Crystal Ensemble Nucleation
2.2. Analytical Solution
3. Numerical Example
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Stefan, J. Über einige Probleme der Theorie der Wärmeleitung. Sitzungsberichte Math.–Naturawissenschaftlichen Cl. Der Kais. Akad. Der Wiss. 1889, 98, 473–484. [Google Scholar]
- Stefan, J. Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Sitzungsberichte Math.–Naturawissenschaftlichen Cl. Der Kais. Akad. Der Wiss. 1889, 98, 965–983. [Google Scholar] [CrossRef] [Green Version]
- Meirmanov, A.M. The Stefan Problem; De Gruyter Expositions in Mathematics; De Gruyter: Berlin, Germany, 1992. [Google Scholar]
- Alexandrov, D.V.; Ivanov, A.A. The Stefan problem of solidification of ternary systems in the presence of moving phase transition regions. J. Exper. Theor. Phys. 2009, 108, 821–829. [Google Scholar] [CrossRef]
- Lee, D.; Alexandrov, D.V. Numerical modeling of one-dimensional binary solidification—The classical two-phase stefan problem. Int. J. Pure Appl. Math. 2010, 58, 381–416. [Google Scholar]
- Nash, G.E.; Glicksman, M.E. Capillary-limited steady-state dendritic growth: I. Theoretical development. Acta Metall. 1974, 22, 1283–1290. [Google Scholar] [CrossRef]
- Langer, J.S.; Turski, L.A. Studies in the theory of interfacial stability: I. Stationary symmetric model. Acta Metall. 1977, 25, 1113–1119. [Google Scholar] [CrossRef]
- Alexandrov, D.V.; Galenko, P.K. Boundary integral approach for propagating interfaces in a binary non-isothermal mixture. Physica A 2017, 469, 420–428. [Google Scholar] [CrossRef]
- Titova, E.A.; Alexandrov, D.V. The boundary integral equation for curved solid/liquid interfaces propagating into a binary liquid with convection. J. Phys. A Math. Theor. 2022, 55, 055701. [Google Scholar] [CrossRef]
- Pelcé, P. Dynamics of Curved Fronts; Academic Press: Boston, MA, USA, 1988. [Google Scholar]
- Brener, E.A.; Mel’nikov, V.I. Pattern selection in two-dimensional dendritic growth. Adv. Phys. 1991, 40, 53–97. [Google Scholar] [CrossRef]
- Galenko, P.K.; Alexandrov, D.V.; Titova, E.A. The boundary integral theory for slow and rapid curved solid/liquid interfaces propagating into binary systems. Phil. Trans. R. Soc. A 2018, 376, 20170218. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Toropova, L.V. Shape functions for dendrite tips of SCN and Si. Eur. Phys. J. Spec. Top. 2022, 231, 1129–1133. [Google Scholar] [CrossRef]
- Toropova, L.V.; Alexandrov, D.V.; Rettenmayr, M.; Liu, D. Microstructure and morphology of Si crystals grown in pure Si and Al-Si melts. J. Phys. Condens. Matter 2022, 34, 094002. [Google Scholar] [CrossRef]
- Alexandrov, D.V.; Galenko, P.K. A review on the theory of stable dendritic growth. Phil. Trans. R. Soc. A 2021, 379, 20200325. [Google Scholar] [CrossRef] [PubMed]
- Mansurov, V.V. The nonlinear dynamics of solidification of a binary melt with a nonequilibrium mushy region. Math. Comput. Model. 1990, 14, 819–821. [Google Scholar] [CrossRef] [Green Version]
- Alexandrov, D.V.; Ivanov, A.A.; Alexandrova, I.V. On the theory of bulk crystallization in the moving phase transition layer. J. Cryst. Growth 2020, 532, 125420. [Google Scholar] [CrossRef]
- Alexandrov, D.V.; Dubovoi, G.Y.; Malygin, A.P.; Nizovtseva, I.G.; Toropova, L.V. Solidification of ternary systems with a nonlinear phase diagram. Russ. Metall. (Met.) 2017, 2017, 127–135. [Google Scholar] [CrossRef]
- Makoveeva, E.V.; Alexandrov, D.V. Mathematical simulation of the crystal nucleation and growth at the intermediate stage of a phase transition. Russ. Metall. (Met.) 2018, 2018, 707–715. [Google Scholar] [CrossRef]
- Reinartz, M.; Kolbe, M.; Herlach, D.M.; Rettenmayr, M.; Toropova, L.V.; Alexandrov, D.V.; Galenko, P.K. Study on anomalous rapid solidification of Al-35 at%Ni in microgravity. JOM 2022, 74, 2420–2427. [Google Scholar] [CrossRef]
- Galenko, P.K.; Toropova, L.V.; Alexandrov, D.V.; Phanikumar, G.; Assadi, H.; Reinartz, M.; Paul, P.; Fang, Y.; Lippmann, S. Anomalous kinetics, patterns formation in recalescence, and final microstructure of rapidly solidified Al-rich Al-Ni alloys. Acta Mater. 2022, 241, 118384. [Google Scholar] [CrossRef]
- Skripov, V.P. Methastable Liquids; Wiley: New York, NY, USA, 1974. [Google Scholar]
- Buyevich, Y.A.; Goldobin, Y.M.; Yasnikov, G.P. Evolution of a particulate system governed by exchange with its environment. Int. J. Heat Mass Trans. 1994, 37, 3003–3014. [Google Scholar] [CrossRef]
- Kelton, K.F.; Greer, A.L. Nucleation in Condensed Matter: Applications in Materials and Biology; Elsevier: Amsterdam, The Netherlands, 2010. [Google Scholar]
- Alexandrova, I.V.; Alexandrov, D.V. Dynamics of particulate assemblages in metastable liquids: A test of theory with nucleation and growth kinetics. Phil. Trans. R. Soc. A 2020, 378, 20190245. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Makoveeva, E.V.; Alexandrov, D.V. On the theory of phase transformation process in a binary supercooled melt. Eur. Phys. J. Spec. Top. 2020, 229, 375–382. [Google Scholar] [CrossRef]
- Toropova, L.V.; Makoveeva, E.V.; Osipov, S.I.; Malygin, A.P.; Yang, Y.; Alexandrov, D.V. Nucleation and growth of an ensemble of crystals during the intermediate stage of a phase transition in metastable liquids. Crystals 2022, 12, 895. [Google Scholar] [CrossRef]
- Alexandrov, D.V.; Ivanov, A.A.; Nizovtseva, I.G.; Lippmann, S.; Alexandrova, I.V.; Makoveeva, E.V. Evolution of a polydisperse ensemble of spherical particles in a metastable medium with allowance for heat and mass exchange with the environment. Crystals 2022, 12, 949. [Google Scholar] [CrossRef]
- Strickland-Constable, R.F. Kinetics and Mechanisms of Crystallization; Academic Press: London, UK, 1968. [Google Scholar]
- Treivus, E.B. Kinetics of Growth and Dissolution of Crystals; Leningrad State University: Leningrad, Russia, 1979. [Google Scholar]
- Bennema, P. Industrial Crystallization; Plenum Press: New York, NY, USA, 1976. [Google Scholar]
- Alexandrova, I.V.; Ivanov, A.A.; Malygin, A.P.; Alexandrov, D.V.; Nikishina, M.A. Growth of spherical and ellipsoidal crystals in a metastable liquid. Eur. Phys. J. Spec. Top. 2022, 231, 1089–1100. [Google Scholar] [CrossRef]
- Alexandrov, D.V. Nucleation and evolution of spherical crystals with allowance for their unsteady-state growth rates. J. Phys. A Math. Theor. 2018, 51, 075102. [Google Scholar] [CrossRef] [Green Version]
- Alexandrov, D.V.; Alexandrova, I.V. On the theory of the unsteady-state growth of spherical crystals in metastable liquids. Phil. Trans. R. Soc. A 2019, 377, 20180209. [Google Scholar] [CrossRef] [Green Version]
- Alexandrov, D.V.; Nizovtseva, I.G.; Alexandrova, I.V. On the theory of nucleation and nonstationary evolution of a polydisperse ensemble of crystals. Int. J. Heat Mass Trans. 2019, 128, 46–53. [Google Scholar] [CrossRef]
- Alexandrov, D.V.; Malygin, A.P. Transient nucleation kinetics of crystal growth at the intermediate stage of bulk phase transitions. J. Phys. A Math. Theor. 2013, 46, 455101. [Google Scholar] [CrossRef]
- Fedoruk, M.V. Saddle-Point Method; Nauka: Moscow, Russia, 1977. [Google Scholar]
- Alexandrov, D.V. Nonlinear dynamics of polydisperse assemblages of particles evolving in metastable media. Eur. Phys. J. Spec. Top. 2020, 229, 383–404. [Google Scholar] [CrossRef]
- Lengsdorf, R.; Holland-Moritz, D.; Herlach, D.M. Anomalous dendrite growth in undercooled melts of Al–Ni alloys in relation to results obtained in reduced gravity. Scr. Mater. 2010, 62, 365–367. [Google Scholar] [CrossRef]
- Herlach, D.M.; Burggraf, S.; Reinartz, M.; Galenko, P.K.; Rettenmayr, M.; Gandin, C.-A.; Henein, H.; Mullis, A.; Ilbagi, A.; Valloton, J. Dendrite growth in undercooled Al-rich Al-Ni melts measured on Earth and in Space. Phys. Rev. Mater. 2019, 3, 073402-1-7. [Google Scholar] [CrossRef]
- Podmaniczky, F.; Gránásy, L. Molecular scale hydrodynamic theory of crystal nucleation and polycrystalline growth. J. Cryst. Growth 2022, 597, 126854. [Google Scholar] [CrossRef]
- McGinty, J.; Yazdanpanah, N.; Price, C.; ter Horst, J.H.; Sefcik, J. Nucleation and crystal growth in continuous crystallization. In The Handbook of Continuous Crystallization; The Royal Society of Chemistry: London, UK, 2020; pp. 1–50. [Google Scholar]
- Alexandrova, I.V.; Alexandrov, D.V.; Makoveeva, E.V. Ostwald ripening in the presence of simultaneous occurrence of various mass transfer mechanisms: An extension of the Lifshitz–Slyozov theory. Phil. Trans. R. Soc. A 2021, 379, 20200308. [Google Scholar] [CrossRef] [PubMed]
- Makoveeva, E.V.; Alexandrov, D.V. The influence of non-stationarity and interphase curvature on the growth dynamics of spherical crystals in a metastable liquid. Phil. Trans. R. Soc. A 2021, 379, 20200307. [Google Scholar] [CrossRef] [PubMed]
- Gránásy, L.; Tóth, G.I.; Warren, J.A.; Podmaniczky, F.; Tegze, G.; Rátkai, L.; Pusztai, T. Phase-field modeling of crystal nucleation in undercooled liquids—A review. Prog. Mater. Sci. 2019, 106, 100569. [Google Scholar] [CrossRef]
- Alexandrov, D.V. Dynamics of the phase transition boundary in the presence of nucleation and growth of crystals. J. Phys. A Math. Theor. 2017, 50, 345101. [Google Scholar] [CrossRef]
- Alexandrov, D.V.; Ivanov, A.A.; Alexandrova, I.V. Analytical solutions of mushy layer equations describing directional solidification in the presence of nucleation. Philos. Trans. R. Soc. A 2018, 376, 20170217. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Toropova, L.V.; Alexandrov, D.V. Dynamical law of the phase interface motion in the presence of crystals nucleation. Sci. Rep. 2022, 12, 10997. [Google Scholar] [CrossRef] [PubMed]
- Alexandrov, D.V.; Toropova, L.V. The role of incoming flow on crystallization of undercooled liquids with a two-phase layer. Sci. Rep. 2022, 12, 17857. [Google Scholar] [CrossRef] [PubMed]
- Makoveeva, E.V.; Alexandrov, D.V. Effects of external heat/mass sources and withdrawal rates of crystals from a metastable liquid on the evolution of particulate assemblages. Eur. Phys. J. Spec. Top. 2019, 228, 25–34. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Alexandrov, D.V.; Galenko, P.K.; Toropova, L.V. Anomalous Dynamics of Recalescence Front in Crystal Growth Processes: Theoretical Background. Crystals 2022, 12, 1686. https://doi.org/10.3390/cryst12121686
Alexandrov DV, Galenko PK, Toropova LV. Anomalous Dynamics of Recalescence Front in Crystal Growth Processes: Theoretical Background. Crystals. 2022; 12(12):1686. https://doi.org/10.3390/cryst12121686
Chicago/Turabian StyleAlexandrov, Dmitri V., Peter K. Galenko, and Liubov V. Toropova. 2022. "Anomalous Dynamics of Recalescence Front in Crystal Growth Processes: Theoretical Background" Crystals 12, no. 12: 1686. https://doi.org/10.3390/cryst12121686