Periodic Systems and Patterns (Phase-Field-Crystals and Swift-Hohenberg equation)
- P. K. Galenko and K. R. Elder, Marginal stability analysis of the phase field crystal model in one spatial dimension. Physical Review B, 83 064113 (2011).
- V. Lebedev, A. Sysoeva, and P. Galenko, Unconditionally gradient-stable computational schemes in problems of fast phase transitions. Physical Review E, 83 026705 (2011).
- P. Galenko, D. Danilov, and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics. Physical Review E, 79 051110 (2009).
- D.V. Alexandrov, P.K. Galenko, D.M. Herlach, Selection criterion for the growing dendritic tip in a non-isothermal binary system under forced convective flow. J. Crystal Growth, 312 2122 (2010).
- R. Lengsdorf, P. Galenko, D.M. Herlach, Measurement of dendritic growth on Al-Ni alloys in reduced gravity. Proceedings 19th ESA Symposium on European Rocket and Baloon Programmes and Related Research, Bad Reickenhall, 7-11 June 2009.
- S. Reutzel, H. Hartmann, R. Lengsdorf, P. Galenko, D.M. Herlach, Solidification of intermetallic Ni-Al alloy melts under reduced gravity conditions during parabolic flight experiments: Promising results for MSL-EML onboard the ISS. Proceedings of 59th International Astronautical Congress (IAC). Glasgow, Scotland, September 2008. (Glasgow: IAC, 2008), pp. 593-598.
- S. Reutzel, H. Hartmann, P.K. Galenko, and D. M. Herlach, Change of the kinetics of solidification and microstructure formation induced by convection in the Ni-Al system. Applied Physics Letters 91 (2007) 041913.
- P. Galenko, D. Herlach, G. Phanikumar, and O. Funke, Phase-field modeling of dendritic growth in droplets processed by electromagnetic levitation. In: Computational Modeling and Simulation of Materials III. Editor: P. Vincenzini (Faenza, Italy: Techna Group Publishers, 2004) p. 565-574.
- P. Galenko, D. Herlach, O. Funke, and G. Phanikumar, Phase-field modeling of dendritic solidification: Verification for the theoretical predictions with latest experimental data. In: Solidification and Crystallization. Editor: D.M. Herlach (Weinheim: Wiley-VCH, 2004) p. 52-60.
- P.K. Galenko, O. Funke O., J. Wang, and D.M. Herlach, Kinetics of dendritic growth under the influence of convective flow in solidification of undercooled droplets. Materials Science and Engineering A 375-377 (2004) 488-492.
- W. Löser, J. Fransaer, L. Granasy, D.M. Herlach, R. Hermann, D. Holland-Moritz, M. Krivilev, M. Kolbe, T. Volkmann, B. Gehrmann, J. Lindemann, E. Adar, S. Zhang, Nucleation and Phase Selection in Magnetic Alloys, In: Microgravity Application Programme - Successful Teaming of Science and Technology, Ed. By A. Wilson, ESA Publications Division, ESTEC, Noordwijk, The Netherlands 2005, pp. 62-71.
- W. Löser, R. Hermann, T. G. Woodcock, J. Fransaer, M. Krivilyov, L. Granasy, T. Pusztai, G. Toth, D.M. Herlach, D. Holland-Moritz, M. Kolbe, T. Volkmann, Nucleation and Phase Selection in Undercooled Melts: Magnetic Alloys of Industrial Relevance (MAGNEPHAS). Journal of the Japan Society of Microgravity Application, 25(3) (2008) 495-500.
- Y. Detandt , M. Krivilyov, Y. Salhi, D. Vanden Abeele, J. Fransaer, Direct numerical simulation of Taylor-Couette flows in the fully turbulent regime, In "Computational Fluid Dynamics 2006", eds. H. Deconinck, E. Dick, ISBN: 978-3-540-92778-5, Springer (2009), pp. 433-438.
- M. Krivilyov, J. Fransaer, Numerical simulation of unsteady flow inside an impulsively started liquid drop, In "Computational Fluid Dynamics 2006", eds. H. Deconinck, E. Dick, ISBN: 978-3-540-92778-5, Springer (2009), pp. 649-654.
- H. Hartmann, D. Holland-Moritz, P.K. Galenko, D.M. Herlach, Evidence of the transition from ordered to disordered growth during rapid solidification of an intermetallic phase. Europhysics Letters, 87(4) 40007 (2009).
- S. Reutzel, H. Hartmann, P. Galenko, H. Assadi, and D. Herlach, Non-equilibrium solidification of intermetallic compounds in Ni-Al systems. In: Frontiers in Solidification Science, ed. J. Hoyt, M. Plapp, G. Faivre, Sh. Liu (TMS: USA, 2007) p. 61-65.
A phase transformation in which both phases have equivalent symmetry but differ only in composition is well-known as spinodal decomposition (see Fig. 1). This transformation has been theoretically described by Cahn and Hilliard [1, 2].
Figure 1. Three-dimensional evolution of spinodally decomposed liquid. The modelling is provided using a "hyperbolic" model for phase separation [3, 4]. Snapshots for the decomposed (blue) phase are shown for various computational times.
Few advancements were made for strongly non-equilibrium phase separation. Binder et al.  generalized the linearized Cahn-Hilliard theory to the case of the existence of a slowly relaxing variable. Their calculations show that the instability of the system is not of the standard diffusive type, but rather it is controlled by the relaxation of the slow structural variable.
Recently, Cahn-Hillard theory has been modified by taking into account the relaxation of diffusion flux to its local steady state [3, 4]. The flux is considered as an independent thermodynamic variable in consistency with the extended irreversible thermodynamics. As a result, a partial differential equation of a hyperbolic type for phase separation with diffusion has been derived that can be called "a hyperbolic model for spinodal decomposition". Theoretically, this model can predict spinodal decomposition for short periods of time, large characteristic velocities of the process, large concentration gradients, or deep supercoolings at the earliest stages of decomposition. A comparative analysis for both Cahn-Hilliard's parabolic model and the hyperbolic model (modified Cahn-Hilliard) of spinodal decomposition are given in Refs. [6-9]. As a test for the hyperbolic model, its predictions are compared with experimental data in Fig. 2.
Figure 2. Dependence of the amplification rate ω/k2 upon square of the wave number k2 (solid line) given by the hyperbolic model  in comparison with scattering data of visible light (points) as described by Ref. . Experimental points were obtained on phase-separated SiO2-12 wt.%Na2O glass at the temperature 803 K.
- J.W. Cahn and J.E. Hilliard, J. Chem. Phys. 28, 258 (1958).
- J.W. Cahn, Acta Metall. 9, 795 (1961).
- P. Galenko, Phys. Lett. A 287, 190 (2001).
- P. Galenko and D. Jou, Phys. Rev. E 71, 046125-1-13 (2005).
- K. Binder, H.L. Frisch and J. Jäckle, J. Chem. Phys. 85, 1505 (1986).
- P. Galenko and V. Lebedev, Analysis of dispersion relation in spinodal decomposition of a binary system Philosophical Magazine Letters 87(11) 821-827 (2007).
- P. Galenko and V. Lebedev, Experimental test for hyperbolic model of spinodal decomposition in a binary system Letters to Journal of Experimental and Theoretical Physics 86(7) 527-529 (2007).
- P. Galenko and V. Lebedev, Non-equilibrium effects in in spinodal decomposition of a binary system Physics Letters A 327(7) 985-989 (2008).
- P. Galenko and V. Lebedev, Local nonequilibrium effect on spinodal decomposition in a binary system The International Journal of Thermodynamics (2008) 11(1) 21-28 (2008).
- N.S. Andreev, G.G. Boiko and N.A. Bokov, J. Non-Cryst. Solids 5, 41 (1970).
- D. Kharchenko, I. Lysenko and P. K. Galenko, Fluctuation effects on pattern selection in the hyperbolic model of phase decomposition. In: Stochastic Differential Equations. Editor: N. Halidias (Nova Science, New York, 2011), pp. 97-127.
- P.K. Galenko, D. Kharchenko, I. Lysenko, Stochastic generalization for a hyperbolic model of spinodal decomposition. Physica A, 389(17) 3443-3455 (2009).
- P. Galenko and D. Jou, Kinetic contribution to the fast spinodal decomposition controlled by diffusion. Physica A, 388(15-16) 3113-3123 (2009).
- N. Lecoq, H. Zapolsky, P. Galenko, Evolution of the structure factor in a hyperbolic model of spinodal decomposition. The European Physical Journal - Special Topics, 177 165-175 (2009).
- D. Kharchenko, P. Galenko, and V. Lebedev, Deterministic and stochastic phenomenological models in spinodal decomposition of a binary system. Progress in Physics of Metals [Uspekhi Fiziki Metallov], 10 27-102 (2009).
- M. Guerdane, F. Wendler, D. Danilov, H. Teichler, and B. Nestler, Crystal growth and melting in NiZr alloy: Linking phase-field modeling to molecular dynamics simulations. Physical Review B, 81(22) 224108 (2010).
- M. Selzer, B. Nestler, and D. Danilov, Influence of the phase diagram on the diffuse interface thickness and on the microstructure formation in a phase-field model for binary alloy. Mathematics and Computers in Simulation, 80(7) 1428-1437 (2010).
- D. Danilov, B. Nestler, M. Guerdane, H. Teichler, Bridging the gap between molecular dynamics simulations and phase-field modelling: dynamics of a [NixZr1−x]liquid–Zrcrystal solidification front. Journal of Physics D: Applied Physics, 42(1) 015310 (2009).
- B. Nestler, D. Danilov, A. Bracchi, and S. Schneider, A metallic glass composite: Phase-field simulations and experimental analysis of microstructure evolution. Materials Science and Engineering: A, 452-453 8-14 (2007).
- P.K. Galenko and D.M. Herlach, Fractals, morphological spectrum and complexity of interfacial patterns in non-equilibrium solidification. In: Complexux Mundi: Emergent Patterns in Nature. Edited by M.M. Novak (Singapore: World Scientific, 2006) p. 199-208.
- D. Danilov and B. Nestler, Phase-field modelling of solute trapping during rapid solidification of a Si-As alloy. Acta Materialia, 54(18) 4659-4664 (2006).
- D. Danilov and B. Nestler, Phase-field simulations of eutectic solidification using an adaptive finite element method. International Journal of Modern Physics B (IJMPB), 20 (7) 853-867 (2006).
- Y. Huang, A. Bracchi, T. Niermann, M. Seibt, D. Danilov, B. Nestler, S. Schneider, Dendritic microstructure in the metallic glass matrix composite Zr56Ti14Nb5Cu7Ni6Be12. Scripta Materialia, 53(1) 93-97 (2005).
- D. Danilov, B. Nestler, Phase-field simulations of solidification in binary and ternary systems using a finite element method. Journal of Crystal Growth, 275(1-2) e177-e182 (2005).
- B. Nestler, D. Danilov, P. Galenko, Crystal growth of pure substances: Phase-field simulations in comparison with analytical and experimental results. Journal of Computational Physics, 207(1) 221-239 (2005).
- D. Danilov, B. Nestler, Dendritic to Globular Morphology Transition in Ternary Alloy Solidification. Physical Review Letters, 93(21) 215501 (2004).
- M. Kolbe, X. Liu, T. Volkmann, R. Rostel, P. Galenko, G. Eggeler, B. Wei, D. Herlach, Interaction of solid ceramic particles with a dendritic solidification front. Materials Science and Engineering A, 375-377 524-527 (2004).
- P. K. Galenko, M. D. Krivilyov, S. V. Buzilov, Bifurcations in a sidebranch surface of a free-growing dendrite. Physical Review E, 55(1) 611-619. (1997).
- V.A. Zhuravlev and P.K. Galenko, Steady-state regimes of solidification in binary systems with heavy and light elements. Russian Metallurgy (Metally), 3 45. (1994).
- P.K. Galenko and O.V. Tolochko, The dynamics of dendritic structure formation of an amorphous structure in the nickel-zirconium system. Glass Physics and Chemistry, 19(2) 152 (1993).