Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2018
Phase-field analysis for the kinetic transition in an ordered crystal structure growing from an u... more Phase-field analysis for the kinetic transition in an ordered crystal structure growing from an undercooled liquid is carried out. The results are interpreted on the basis of analytical and numerical solutions of equations describing the dynamics of the phase field, the long-range order parameter as well as the atomic diffusion within the crystal/liquid interface and in the bulk crystal. As an example, the growth of a binary A 50 B 50 crystal is described, and critical undercoolings at characteristic changes of growth velocity and the long-range order parameter are defined. For rapidly growing crystals, analogies and qualitative differences are found in comparison with known non-equilibrium effects, particularly solute trapping and disorder trapping. The results and model predictions are compared qualitatively with results of the theory of kinetic phase transitions (Chernov 1968 Sov. Phys. JETP 26 , 1182–1190) and with experimental data obtained for rapid dendritic solidification of...
Apart from the topic "materials" contributions on condensed matter and materials science predomin... more Apart from the topic "materials" contributions on condensed matter and materials science predominate on the whole. This is also a consequence of sophisticated methods used in this field. In Vol. 9 examples for this are articles on "Magnetic Materials", "Magnetic Ordering in Solids", "Magnetic Resonance Spectrometers", "Magnetostrictive Materials", "Magnets", "Mechanical and Elastic Waves in Solids", "Mechanical Properties of Solids". In Vol. 10 we find "Metal-Insulator Transitions", "Metallic Glasses", "Metals and Alloys". In Vol. 11 "Mössbauer Effect", "Nanophase Materials", "Neutron Diffraction" should be mentioned whereas in Vol. 12 "Optical Microscopy", "Optical Properties of Solids", and "Nonlinear Optics" belong to this category. Readers will certainly acknowledge a summary of contents of previous volumes as given for the first time at the end of Vol. 12.
A general solution of the problem about local nonequilibrium heat transfer with a moving heat sou... more A general solution of the problem about local nonequilibrium heat transfer with a moving heat source at the planar solidification front is obtained. The steady-state and nonsteady-state regimes of rapid solidification are analyzed. The self-consistent regimes of solidification front motion are found.
A model for free dendritic growth during rapid solidification, which is based on a modified solut... more A model for free dendritic growth during rapid solidification, which is based on a modified solute diffusion equation that treats local nonequilibrium in the bulk phases is presented. Taking into account the finite speed of solute diffusive propagation, the model predicts the steady-state dendrite tip velocity and the dendritic tip radius. A satisfactory agreement of the model predictions with experimental data in Cu-Ni alloy is found.
Fluid flow decisively influences the heat and mass transport during solidification of melts and c... more Fluid flow decisively influences the heat and mass transport during solidification of melts and consequently the properties of the as-solidified materials. In this work we present investigations on the influence of convection on the non-equilibrium solidification of different undercooled Al-Ni alloy melts. The dendrite growth velocities in Al-Ni melts containerlessly processed by electromagnetic levitation have been measured on Earth and under the conditions of reduced gravity. While under terrestrial conditions strong electromagnetic fields are required for levitation against gravity, the fields necessary to compensate disturbing accelerations under reduced gravity conditions are smaller by orders of magnitude. Consequently, convective fluid flow induced by electromagnetic stirring effects is significantly decreased in experiments performed under reduced gravity. Comparative experiments on congruently melting Al50Ni50 alloys revealed that the dendrite growth velocities measured at ...
We consider the modifi ed Cahn-Hilliard equation for phase separation suggested to account for sp... more We consider the modifi ed Cahn-Hilliard equation for phase separation suggested to account for spinodal decomposition in deeply supercooled binary alloy systems or glasses. This equation contains, as additional term, the second-order time derivative of the concentration multiplied by a positive coefficient $\Tau_d$ (time for relaxation). We consider a numerical approximation scheme based on Fourier spectral method and perform numerical analysis of the scheme. We present results of numerical simulations for three spatial dimensions, and examine the stability and convergence of the scheme.
The European Physical Journal Special Topics, 2020
Hyperbolic-type equations of both phase field and concentration arising from a phase-field model ... more Hyperbolic-type equations of both phase field and concentration arising from a phase-field model for fast phase transformations in binary dilute systems yield in the one-dimension moving frame of reference to the concentration-and phase field governing equations, respectively. These equations have been solved numerically and applied to the case of Si-0.1 at.% As binary alloy [P.K. Galenko et al., Phys. Rev. E 84 , 041143 (2011)]. In this paper, the coupling of the hodograph equation for the interface with the solute diffusion equation leads to an exact analytical solution of the one-point Cauchy problem of an ordinary differential equation in a parametric form. Application of this solution to the case of Si-0.1 at.% As gives (i) the same tendency of concentration variation along dimensionless spatial coordinate (ii) the same values of interface velocity with a very slight difference in the value of concentration for a given undercooling at the interface. Based on the results obtained, the established hodograph-equation confirms again its usefulness to predict, for instance, certain aspects of rapid solidification processes for binary alloys.
The European Physical Journal Special Topics, 2020
In the present work we report results of the model experiments that allow us to study effect of d... more In the present work we report results of the model experiments that allow us to study effect of different cooling rates of samples on relationship for amorphous/crystalline phase in freezed Pd-based samples. For the model experiments Pd 40 Cu 30 Ni 10 P 20 alloy samples were used which are typical bulk metal glasses with a relatively small temperature interval of liquid to glass transition. The main focus of the study was on the effect of cooling rate and glass transition temperature on the microstructure of samples, cooled from the liquid to (semi)amorphous states of the alloy. Ground based experiments were conducted using the multi-zone vacuum furnace MEP-01 situated in the institute of launch complexes (NIISK, Moscow). Composition, structure and thermodynamic properties of the samples were determined by electron probe microanalysis, X-ray diffraction, transmission electron microscopy and differential scanning calorimetry.
The mathematical modeling of crystal structures and their dynamics during the structural transiti... more The mathematical modeling of crystal structures and their dynamics during the structural transitions can be performed by the method of phase field crystal in the hyperbolic formulation (MPFC method). This method is suitable for a continual modeling of the atomic density field at diffusion time intervals. The unstable behavior of the solution near local energy minima is discussed. The authors propose the hypothesis of the formation of metastable structures during the relaxation of domain to the stable state. Due to the presence of the set of different possible structures on the system's transformation path, the pattern of the atomic density field could remain still when the system is in the vicinity of local free energy minima. In the absence of fluctuations the system can remain in that condition for arbitrarily long. We propose the simple stochastic extension of hyperbolic PFC-model to find the effect of fluctuations on the dynamics of atomic density field.
We consider a thermo-solutal growth of an anisotropic dendrite in a binary liquid with allowance ... more We consider a thermo-solutal growth of an anisotropic dendrite in a binary liquid with allowance for the convective heat and mass transfer mechanism at the dendrite surface. We obtain a new stability criterion for stable mode of dendritic evolution. Also in this work, a stable relation for the dendrite tip velocity and its tip diameter via the selection theory and undercooling balance is discussed. The undercooling balance and selection criterion allow us to obtain two nonlinear equations determining the crystal tip velocity and its tip diameter as functions of undercooling.
Physica A: Statistical Mechanics and its Applications, 2017
A method based on boundary integral approach to the propagation of curved phase interface in a bi... more A method based on boundary integral approach to the propagation of curved phase interface in a binary non-isothermal mixture is developed. Previously known equations and solutions for thermally controlled growth and needle-like dendrites follow from the obtained boundary integral equations as limiting cases.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2018
Phase-field analysis for the kinetic transition in an ordered crystal structure growing from an u... more Phase-field analysis for the kinetic transition in an ordered crystal structure growing from an undercooled liquid is carried out. The results are interpreted on the basis of analytical and numerical solutions of equations describing the dynamics of the phase field, the long-range order parameter as well as the atomic diffusion within the crystal/liquid interface and in the bulk crystal. As an example, the growth of a binary A 50 B 50 crystal is described, and critical undercoolings at characteristic changes of growth velocity and the long-range order parameter are defined. For rapidly growing crystals, analogies and qualitative differences are found in comparison with known non-equilibrium effects, particularly solute trapping and disorder trapping. The results and model predictions are compared qualitatively with results of the theory of kinetic phase transitions (Chernov 1968 Sov. Phys. JETP 26 , 1182–1190) and with experimental data obtained for rapid dendritic solidification of...
Apart from the topic "materials" contributions on condensed matter and materials science predomin... more Apart from the topic "materials" contributions on condensed matter and materials science predominate on the whole. This is also a consequence of sophisticated methods used in this field. In Vol. 9 examples for this are articles on "Magnetic Materials", "Magnetic Ordering in Solids", "Magnetic Resonance Spectrometers", "Magnetostrictive Materials", "Magnets", "Mechanical and Elastic Waves in Solids", "Mechanical Properties of Solids". In Vol. 10 we find "Metal-Insulator Transitions", "Metallic Glasses", "Metals and Alloys". In Vol. 11 "Mössbauer Effect", "Nanophase Materials", "Neutron Diffraction" should be mentioned whereas in Vol. 12 "Optical Microscopy", "Optical Properties of Solids", and "Nonlinear Optics" belong to this category. Readers will certainly acknowledge a summary of contents of previous volumes as given for the first time at the end of Vol. 12.
A general solution of the problem about local nonequilibrium heat transfer with a moving heat sou... more A general solution of the problem about local nonequilibrium heat transfer with a moving heat source at the planar solidification front is obtained. The steady-state and nonsteady-state regimes of rapid solidification are analyzed. The self-consistent regimes of solidification front motion are found.
A model for free dendritic growth during rapid solidification, which is based on a modified solut... more A model for free dendritic growth during rapid solidification, which is based on a modified solute diffusion equation that treats local nonequilibrium in the bulk phases is presented. Taking into account the finite speed of solute diffusive propagation, the model predicts the steady-state dendrite tip velocity and the dendritic tip radius. A satisfactory agreement of the model predictions with experimental data in Cu-Ni alloy is found.
Fluid flow decisively influences the heat and mass transport during solidification of melts and c... more Fluid flow decisively influences the heat and mass transport during solidification of melts and consequently the properties of the as-solidified materials. In this work we present investigations on the influence of convection on the non-equilibrium solidification of different undercooled Al-Ni alloy melts. The dendrite growth velocities in Al-Ni melts containerlessly processed by electromagnetic levitation have been measured on Earth and under the conditions of reduced gravity. While under terrestrial conditions strong electromagnetic fields are required for levitation against gravity, the fields necessary to compensate disturbing accelerations under reduced gravity conditions are smaller by orders of magnitude. Consequently, convective fluid flow induced by electromagnetic stirring effects is significantly decreased in experiments performed under reduced gravity. Comparative experiments on congruently melting Al50Ni50 alloys revealed that the dendrite growth velocities measured at ...
We consider the modifi ed Cahn-Hilliard equation for phase separation suggested to account for sp... more We consider the modifi ed Cahn-Hilliard equation for phase separation suggested to account for spinodal decomposition in deeply supercooled binary alloy systems or glasses. This equation contains, as additional term, the second-order time derivative of the concentration multiplied by a positive coefficient $\Tau_d$ (time for relaxation). We consider a numerical approximation scheme based on Fourier spectral method and perform numerical analysis of the scheme. We present results of numerical simulations for three spatial dimensions, and examine the stability and convergence of the scheme.
The European Physical Journal Special Topics, 2020
Hyperbolic-type equations of both phase field and concentration arising from a phase-field model ... more Hyperbolic-type equations of both phase field and concentration arising from a phase-field model for fast phase transformations in binary dilute systems yield in the one-dimension moving frame of reference to the concentration-and phase field governing equations, respectively. These equations have been solved numerically and applied to the case of Si-0.1 at.% As binary alloy [P.K. Galenko et al., Phys. Rev. E 84 , 041143 (2011)]. In this paper, the coupling of the hodograph equation for the interface with the solute diffusion equation leads to an exact analytical solution of the one-point Cauchy problem of an ordinary differential equation in a parametric form. Application of this solution to the case of Si-0.1 at.% As gives (i) the same tendency of concentration variation along dimensionless spatial coordinate (ii) the same values of interface velocity with a very slight difference in the value of concentration for a given undercooling at the interface. Based on the results obtained, the established hodograph-equation confirms again its usefulness to predict, for instance, certain aspects of rapid solidification processes for binary alloys.
The European Physical Journal Special Topics, 2020
In the present work we report results of the model experiments that allow us to study effect of d... more In the present work we report results of the model experiments that allow us to study effect of different cooling rates of samples on relationship for amorphous/crystalline phase in freezed Pd-based samples. For the model experiments Pd 40 Cu 30 Ni 10 P 20 alloy samples were used which are typical bulk metal glasses with a relatively small temperature interval of liquid to glass transition. The main focus of the study was on the effect of cooling rate and glass transition temperature on the microstructure of samples, cooled from the liquid to (semi)amorphous states of the alloy. Ground based experiments were conducted using the multi-zone vacuum furnace MEP-01 situated in the institute of launch complexes (NIISK, Moscow). Composition, structure and thermodynamic properties of the samples were determined by electron probe microanalysis, X-ray diffraction, transmission electron microscopy and differential scanning calorimetry.
The mathematical modeling of crystal structures and their dynamics during the structural transiti... more The mathematical modeling of crystal structures and their dynamics during the structural transitions can be performed by the method of phase field crystal in the hyperbolic formulation (MPFC method). This method is suitable for a continual modeling of the atomic density field at diffusion time intervals. The unstable behavior of the solution near local energy minima is discussed. The authors propose the hypothesis of the formation of metastable structures during the relaxation of domain to the stable state. Due to the presence of the set of different possible structures on the system's transformation path, the pattern of the atomic density field could remain still when the system is in the vicinity of local free energy minima. In the absence of fluctuations the system can remain in that condition for arbitrarily long. We propose the simple stochastic extension of hyperbolic PFC-model to find the effect of fluctuations on the dynamics of atomic density field.
We consider a thermo-solutal growth of an anisotropic dendrite in a binary liquid with allowance ... more We consider a thermo-solutal growth of an anisotropic dendrite in a binary liquid with allowance for the convective heat and mass transfer mechanism at the dendrite surface. We obtain a new stability criterion for stable mode of dendritic evolution. Also in this work, a stable relation for the dendrite tip velocity and its tip diameter via the selection theory and undercooling balance is discussed. The undercooling balance and selection criterion allow us to obtain two nonlinear equations determining the crystal tip velocity and its tip diameter as functions of undercooling.
Physica A: Statistical Mechanics and its Applications, 2017
A method based on boundary integral approach to the propagation of curved phase interface in a bi... more A method based on boundary integral approach to the propagation of curved phase interface in a binary non-isothermal mixture is developed. Previously known equations and solutions for thermally controlled growth and needle-like dendrites follow from the obtained boundary integral equations as limiting cases.
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Papers by Peter Galenko