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.

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.

E. Lukashov and E. Radkevich, "Solidification and Structuresation of Instability Zones,"

References

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

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[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.

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[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.

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[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.

[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.