Abstract
A theory is developed to describe the nonstationarity growth of spherical crystals in supercooled melts and supersaturated solutions. The first two corrections to the fundamental contribution to the law of crystal growth rate are determined using differential series, the expansion of unknown functions in a small parameter, and the integral Laplace–Carson transform. These corrections are shown to substantially change the growth rate of spherical crystals in metastable liquids.
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REFERENCES
A. C. Zettlemoyer, Nucleation (Dekker, New York, 1969).
J. W. Mullin, Crystallization (Butterworths, London, 1972).
D. Herlach, P. Galenko, and D. Holland-Moritz, “Metastable Solids from Undercooled Melts (Elsevier, Amsterdam, 2007).
Yu. A. Buyevich and D. V. Alexandrov, “On the theory of evolution of particulate systems,” IOP Conf. Series: Mater. Sci. Eng. 192, 012001 (2017).
D. V. Alexandrov and A. P. Malygin, “Transient nucleation kinetics of crystal growth at the intermediate stage of bulk phase transitions,” J. Phys. A: Math. Theor. 46, 455101 (2013).
V. G. Dubrovskii, Nucleation Theory and Growth of Nanostructures (Springer, Berlin, 2014).
D. V. Alexandrov and I. G. Nizovtseva, “Nucleation and particle growth with fluctuating rates at the intermediate stage of phase transitions in metastable systems,” Proc. R. Soc. A 470, 20130647 (2014).
Yu. A. Buyevich and V. V. Mansurov, “Kinetics of the intermediate stage of phase transition in batch crystallization,” J. Cryst. Growth 104, 861–867 (1990).
D. V. Alexandrov and A. P. Malygin, “Nucleation kinetics and crystal growth with fluctuating rates at the intermediate stage of phase transitions,” Modelling Simul. Mater. Sci. Eng. 22, 015003 (2014).
D. V. Alexandrov, “On the theory of transient nucleation at the intermediate stage of phase transitions,” Phys. Lett. A 378, 1501–1504 (2014).
D. A. Barlow, “Theory of the intermediate stage of crystal growth with applications to insulin crystallization,” J. Cryst. Growth 470, 8–14 (2017).
I. M. Lifshitz and V. V. Slyozov, “The kinetics of precipitation from supersaturated solid solutions,” J. Phys. Chem. Solids. 19, 35–50 (1961).
E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Pergamon, Oxford, 1981).
J. R. Hunt, “Self-similar particle-size distributions during coagulation: theory and experimental verification,” J. Fluid Mech. 122, 169–185 (1982).
Y. Enomoto and A. Okada, “Effects of Brownian coagulation on droplet growth in a quenched fluid mixture,” J. Phys.: Condens. Matter 2, 4531–4535 (1990).
V. V. Mansurov and A. V. Alyab’eva, “The kinetics of enlargement of particles in sols with the simultaneous process of distillation and coagulation,” Kolloidn. Zh. 54, 3–6 (1992).
A. V. Alyab’eva, Yu. A. Buyevich, and V. V. Mansurov, “Evolution of a particulate assemblage due to coalescence combined with coagulation,” J. Phys. II France 4, 951–957 (1994).
S. Simons, “On steady-state solutions of the coagulation equation,” J. Phys. A: Math. Gen. 29, 1139–1140 (1996).
D. V. Alexandrov, “On the theory of Ostwald ripening: formation of the universal distribution,” J. Phys. A: Math. Theor. 48, 035103 (2015).
D. V. Alexandrov, “The large-time behaviour of coarsening of a particulate assemblage due to Ostwald ripening and coagulation,” Phil. Mag. Lett. 96, 355–360 (2016).
D. V. Alexandrov, “Kinetics of particle coarsening with allowance for Ostwald ripening and coagulation,” J. Phys.: Condens. Matter. 28, 035102 (2016).
D. V. Alexandrov, “On the theory of Ostwald ripening in the presence of different mass transfer mechanisms,” J. Phys. Chem. Solids 91, 48–54 (2016).
D. V. Alexandrov, “A transient distribution of particle assemblies at the concluding stage of phase transformations,” J. Mater. Sci. 52, 6987–6993 (2017).
D. V. Alexandrov, “Kinetics of diffusive decomposition in the case of several mass transfer mechanisms,” J. Cryst. Growth 457, 11–18 (2017).
Yu. A. Buyevich, V. V. Mansurov, and I. A. Natalukha, “Instability and unsteady processes of the bulk continuous crystallization: I. Linear stability analysis,” Chem. Eng. Sci. 46, 2573–2578 (1991).
D. V. Alexandrov, “Nucleation and crystal growth kinetics during solidification: the role of crystallite withdrawal rate and external heat and mass sources,” Chem. Eng. Sci. 117, 156–160 (2014).
E. V. Makoveeva and D. V. Alexandrov, “A complete analytical solution of the Fokker–Planck and balance equations for nucleation and growth of crystals,” Phil. Trans. R. Soc. A 376, 20170327 (2018).
Yu. A. Buyevich and I. A. Natalukha, “Unsteady processes of combined polymerization and crystallization in continuous apparatuses,” Chem. Eng. Sci. 49, 3241–3247 (1994).
D. V. Alexandrov, “Mathematical modelling of nucleation and growth of crystals with buoyancy effects,” Phil. Mag. Lett. 96, 132–141 (2016).
A. O. Ivanov and A. Yu. Zubarev, “Non-linear evolution of a system of elongated droplike aggregates in a metastable magnetic fluid,” Physica A 251, 348–367 (1998).
R. F. Strickland-Constable, Kinetics and Mechanisms of Crystallization (Academic Press, London, 1968).
P. Bennema, “Theory and experiment for crystal growth from solutions: implications for industrial crystallization,” in Industrial Crystallization (Plenum Press, London, 1976), pp. 91–112.
D. V. Alexandrov, “Nucleation and crystal growth in binary systems,” J. Phys. A: Math. Theor. 47, 125102 (2014).
B. Chalmers, Physical Metallurgy (Wiley, New York, 1959).
V. P. Skripov, Metastable Liquids (Wiley, New York, 1974).
A. A. Chernov, Modern Crystallography III (Springer, Berlin, 1984).
D. V. Alexandrov, “Solidification with a quasiequilibrium two-phase zone,” Acta Mater. 49, 759–764 (2001).
D. L. Aseev and D. V. Alexandrov, “Nonlinear dynamics for the solidification of binary melt with a nonequilibrium two-phase zone,” Phys. Dokl. 51, 291–295 (2006).
D. L. Aseev and D. V. Alexandrov, “Directional solidification of binary melts with a nonequilibrium mushy layer,” Int. J. Heat Mass Transfer 49, 4903–4909 (2006).
B. Ya. Lyubov, Theory of Solidification in Large Volume (Nauka, Moscow, 1975).
V. A. Ditkin and A. P. Prudnikov, Integral Transforms and Operational Calculus (Pergamon, New York, 1965).
J. S. Kirkaldy and D. J. Young, Diffusion in Condensed State (Institute of Metals, London, 1987).
V. V. Mansurov, “The nonlinear dynamics of solidification of a binary melt with a nonequilibrium mushy region,” Math. Comput. Modelling 14, 819–821 (1990).
D. A. Barlow, J. K. Baird, and C.-H. Su, “Theory of the von Weimarn rules governing the average size of crystals precipitated from a supersaturated solution,” J. Cryst. Growth 264, 417–423 (2004).
D. V. Alexandrov and A. P. Malygin, “Flow-induced morphological instability and solidification with the slurry and mushy layers in the presence of convection,” Int. J. Heat Mass Transfer 55, 3196–3204 (2012).
D. V. Alexandrov and A. P. Malygin, “Coupled convective and morphological instability of the inner core boundary of the Earth,” Phys. Earth Planet. Int. 189, 134–141 (2011).
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This work was supported by the Russian Science Foundation, project no. 18-19-00008.
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Translated by K. Shakhlevich
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Alexandrov, D.V., Alexandrova, I.V., Ivanov, A.A. et al. On the Theory of the Nonstationary Spherical Crystal Growth in Supercooled Melts and Supersaturated Solutions. Russ. Metall. 2019, 787–794 (2019). https://doi.org/10.1134/S0036029519080020
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DOI: https://doi.org/10.1134/S0036029519080020