AM  Vol.1 No.3 , September 2010
Solidification and Structuresation of Instability Zones
Abstract: Two mathematical crystallization models describing structure formations in instability zones are proposed and justified. The first model, based on a phase field system, describes crystallization processes in binary alloys. The second model, based on a modified Biot model of a porous medium and the convective Cahn–Hilliard model, governs oriented crystallization. Physical interpretation and numerical analysis are discussed.
Cite this paper: E. Lukashov and E. Radkevich, "Solidification and Structuresation of Instability Zones," Applied Mathematics, Vol. 1 No. 3, 2010, pp. 159-178. doi: 10.4236/am.2010.13021.

[1]   V. A. Avetisov, “p-Adic Description of Characteristic Relaxation in Complex System,” Journal of Physics A: Mathematical and General, Vol. 36, No. 15, 2003, pp. 4239-4246.

[2]   E. N. Kablov, “Cast Gas-Turbine Engine Blades (Alloys, Technology, Covering) [in Russian],” Moscow, 2001.

[3]   A. N. Kolmogorov, “On the Statistical Theory of Crystallization of Metals [in Russian],” Izv. Akad. Nauk SSSR, Ser. Mat., No. 3, 1937, pp. 355-359.

[4]   F. M. Shemyakin and P. F. Mikhalev, “Physic-Chemical Periodic Processes [in Russian],” Akad. Nauk SSSR, Moscow, 1938.

[5]   I. Z. Bezbakh, B. G. Zakharov and I. A. Prokhorov, “Ra-diographical Characterization of Microsegregation in Crystals [in Russian],” In: Proceedings of the 6th Inter-national Conference “Growth of Monocrystals and Heat-Mass Transfer”, Obninsk, Vol. 2, 2005, pp. 352- 361.

[6]   V. G. Danilov, G. A. Omel’yanov and E. V. Radkevich, “Hugoniot-Type Conditions and Weak Solutions to the phase Field System,” European Journal of Applied Mathematics, Vol. 10, 1999, pp. 55-77.

[7]   E. V. Radkevich, “Mathematical Aspects of Nonequilibrium Processes [in Russian],” Tamara Rozhkovskaya Publisher, Novosibirsk, 2007.

[8]   N. N. Yakovlev, E. A. Lukashev and E. V. Radkevich, “Problems of Reconstruction of the Process of Directional Solidification [in Russian],” Dokl. Akad. Nauk, Ross. Akad. Nauk, Vol. 421, No. 5, 2008, pp. 625-629; English translation: Doklady Physics, Technical Physics, Vol. 53, No. 8, 2008, pp. 442-446.

[9]   V. Visintin, “Models of Phase Transitions,” Birkhauser, Boston, 1996.

[10]   E. V. Radkevich, “The Gibbs--Thomson Effect and Exis-tence Conditions of Classical Solution for the Modified Stefan Problem,” In: Free Boundary Problems Involving Solids, Science and Technology, Harlow, 1993, pp. 135- 142.

[11]   O. A. Oleynik and E. V. Radkevich, “Second Order Equ-ations with Nonnegative Characteristic Form,” Am. Math. Soc., Providence, 1973.

[12]   A. A. Lacey and A. B. Tayler, “A Mushy Region in a Stefan Problem,” IMA Journal of Applied Mathematics, Vol. 30, No. 3, 1983, pp. 303-313.

[13]   M. A. Biot, “Mechanics of Deformation and Acoustic Propagation in Porous Media,” Journal of Applied Physics, Vol. 33, No. 4, 1962, pp. 1482-1498.

[14]   S. J. Watson, F. Otto, B. Y. Rubinstein and S. H. Davis, “Coarsening Dynamics for the Convective Cahn-Hilliard Equation,” University of Bonn, Preprint, 2003.

[15]   A. A. Golovin, S. H. Davis and A. A. Nepomnyashchy, “A Convective Cahn-Hilliard Model for the Formation of Facets and Carners in Crystal Growth,” Physical D, Vol. 118, 1998, pp. 202-230.

[16]   N. A. Zaitsev and Yu. G. Rykov, “Numerical Analysis of a Model Describing Metal Crystallization I. One-Dimensional Case,” Preprint, Keldysh Institute of Applied Physics RAS, No. 72, 2007.

[17]   W. Dreyer and B. Wagner, “Sharp-Interface Model for Eutectic Alloys. Part I. Concentration Dependent Surface Tension,” Preprint, 2003.

[18]   N. A. Zaitsev and Y. G. Rykov, “Numerical Analysis to a New Model of the Matel Solidification, 1-D Case,” Mathematical Simulation, 2010, to Appear.

[19]   N. P. Lyakishev and G. S. Burkhanov, “Metal Monocrystals [in Russian],” Eliz, Moscow, 2002.

[20]   P. E. Shalin, et al., “Monocrystals of Heat-Resistance Nickel Alloys [in Russian],” Mashinostroenie, Moscow, 1997.

[21]   J. W. Matthews and A. E. Blacslee, “Defects in Epitaxial Multilayers. I. Misfit Dislocations,” Journal of Crystal Growth, Vol. 27, No. 1, 1974, pp. 118-125.

[22]   V. N. Vigdorovich, A. E. Vol’pyan and G. M. Kurdyumov, “Oriented Crystallization and Physic-Chemical Analysis [in Russian],” Chemistry, Moscow, 1976.