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Boundary conditions for the solid-liquid interface of the solidifying pure melt have been derived. In the derivation the model of Gibbs interface is used. The boundary conditions include both the state quantities of bulk phases are taken at the interface and the quantities characterizing interfacial surface such as the surface temperature and the surface heat flux. Introduction of the surface temperature as an independent variable allows us to describe the scattering energy at the interface. For the steady-state motion of the planar interface the expression for the temperature discontinuity across the phase boundary has been obtained. Effect of Kapitza resistance on the interface velocity is considered. It is shown that heat resistance leads to non-linearity in solidification kinetics, namely, in "velocity-undercooling" relationship. The conditions of the steady--state motion of the planar interface has been found., 26 pages, regular article submitted to Physica A

This review article summarizes current theories of the steady-state growth mode of dendrites in the form of elliptical paraboloids. The shape of dendrite tips is analyzed, temperature and solute concentration distributions are described in its vicinity, and a solution of the hydrodynamic problem of a viscous incompressible fluid flowing against a dendrite tip is developed. A significant difference in analytical solutions describing a dendrite tip as an elliptic paraboloid as compared to an axisymmetric morphology is shown. The system of nonlinear equations for determining the stationary velocity of dendrite growth and the radii of curvature of the dendrite tip along the major and minor axis of the ellipse, respectively, is derived. The developed theory is compared with experimental data on the growth of ice crystals consisting of D2O or H2O.

A free dendrite growing in a pure substance is considered with the interfacial effect of anisotropy and convective flow. A stable mode of dendritic growth with the six-fold crystal symmetry is studied using the solvability theory. We demonstrate that the obtained selection criterion for a stable mode of dendritic growth is a function of surface energy stiffness, arbitrary values of Peclet numbers and convective flow intensity. To predict the dendrite tip velocity V and its tip radius R a model of dendrite growth with the six-fold symmetry is formulated. We show that the model equations can be reduced to the growth kinetics with the low Peclet numbers, which exhibit the explicit relationships “tip velocity - undercooling”. The model predictions are compared with experimental data on ice dendrites grown from pure undercooled water on board of the International Space Station (under microgravitational conditions, μ g ) and on the Ground (under terrestrial conditions, 1 g).

The boundary integral method is developed for fast anisotropic interfaces. A general integro-differential equation for curved interfaces controlled by heat and mass transport is derived and applied to the problem of rapid dendritic growth. A selection criterion for the steady-state mode of growing parabolic interfaces is obtained and, in common solution with the undercooling balance, it is compared with experimental data on rapid dendritic solidification of deeply supercooled liquid droplets. In this comparison, transitions from solute diffusion-limited to thermo-solutal regime and, finally, to pure thermally controlled regime of the anisotropic dendrite are discussed and revealed. Limiting cases of known selection criteria for anisotropic dendrites growing at small and high growth Peclet numbers are provided.

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.

Traveling wave solutions of the hyperbolic Cahn–Allen equation are obtained using the first integral method, which follows from well-known Hilbert–Nullstellensatz theorem. The obtained complete class of traveling waves consists of continual and singular solutions. Continual solutions are represented by tanh -profiles and singular solutions exhibit unbounded discontinuity at the origin of coordinate system. With the neglecting inertia of the dynamical system, the obtained traveling waves include the previous solutions for the parabolic Cahn–Allen equation.

Although a resistivity saturation (minimum conductivity) is often observed in disordered metallic solids, such phenomena in the corresponding liquids are not known. Here we report a saturation of the electrical resistivity in ${\mathrm{Zr}}_{64}{\mathrm{Ni}}_{36}$ and ${\mathrm{Cu}}_{50}{\mathrm{Zr}}_{50}$ liquids above a dynamical crossover temperature for the viscosity (${T}_{A}$). The measurements were made for the levitated liquids under the microgravity conditions of the International Space Station. Based on recent molecular dynamics simulations, the saturation is likely due to the ineffectiveness of electron-phonon scattering above ${T}_{A}$ when the phonon lifetime becomes too short compared to the electron relaxation time. This is different from the conventional resistivity saturation mechanisms in solids.

The dendrite growth velocity in Al₇₅Ni₂₅ melts has been measured in a containerless procedure as a function of undercooling using an electromagnetic levitation technique both in the Earth laboratory and in Space on board the International Space Station. The growth shows an anomalous behavior inasmuch as the growth velocity decreases with increasing undercooling, confirming previous experiments on Earth. Within the scatter of experimental data, results obtained on Earth and in Space do not show significant differences. Thus, convection effects as the origin of the anomalous growth characteristics can be excluded. However, high-speed video recording exhibits multiple nucleation events in front of the growing solid-liquid interface. This effect is identified as the origin of the anomalous dendrite growth characteristics in undercooled melts of Al-rich Al-Ni melts.

Thermodynamic driving forces and growth rates in rapid solidification are analysed. Taking into account the relaxation time of the solute diffusion flux in the model equations, the present theory uses, in a first case, the deviation from local chemical equilibrium, and ergodicity breaking. The second case of ergodicity breaking may exist in crystal growth kinetics of rapidly solidifying glass-forming metals and alloys. In this case, a theoretical analysis of dendritic solidification is given for congruently melting alloys in which chemical segregation does not occur. Within this theory, a deviation from thermodynamic equilibrium is introduced for high undercoolings via gradient flow relaxation of the phase field. A comparison of the present derivations with previously verified theoretical predictions and experimental data is given. This article is part of the theme issue 'Heterogeneous materials: Metastable and nonergodic internal structures'. ©2019 The Author(s) Published by the Royal Society.

Results of a study on microstructural evolution of eutectic Sn-57 wt.% Bi processed with cooling rates of 10−2, 1 K s−1and approximately 105 K s−1are presented. In order to distinguish different mechanisms of microstructure formation, a comparison with microstructures of different hypoeutectic alloys with compositions down to below the maximum solubility of Bi in Sn–Bi is undertaken. It is found that at the cooling rates of 10−2and 1 K s−1, coupled eutectic growth occurs, leading to lamellar structures with different length scales. At the rapid quenching rates of approximately 105 K s−1, structure formation in the eutectic alloy is qualitatively different. Partitionless solidification resulting in a supersaturated solid solution with the initial composition is observed in both eutectic and hypoeutectic alloys. It is shown that the observed microstructure of the rapidly solidified alloys forms by the decomposition of the supersaturated solid solution.This article is part of the theme issue ‘Heterogeneous materials: metastable and non-ergodic internal structures’.

A stable mode of anisotropic dendrite growing in a forced convective flow with n-fold crystalline symmetry is studied for low, moderate and rapid tip velocities (for arbitrary Peclet numbers). A generalized selection criterion determining a stable combination for the dendrite tip velocity and dendrite tip diameter is obtained.

We present an analysis of a free dendrite growing in a binary mixture under non-isothermal conditions. The stable growth mode is analyzed through the solvability condition giving the stability criterion for the dendrite tip as a function of the thermal Peclet number, P T , and ratio, W = V / V D , of the dendrite velocity V and solute diffusion speed V D in bulk liquid. We extend previous studies limited to small values of the Peclet numbers, by considering the effect of the anisotropy of surface energy for the needle-like dendrite growing at arbitrary Peclet numbers and under local non-equilibrium solute diffusion described by a hyperbolic type of transport equation. Transitions in growth regimes, namely, from solute diffusion-limited to thermo-solutal regime and, finally, to pure thermally controlled regime of the anisotropic dendrite are derived and revealed. Limiting cases of known criteria for anisotropic dendrite growing at small and high growth Peclet numbers are provided. A comparison with the previously obtained criterion of marginal stability of rapidly growing dendrite is made.

The results of microstructural study of rapidly solidified Sn-Bi alloys obtained at the melt cooling rate of 105 K/s with the compositions of Sn-X wt. % Bi (X = 13, 20, 30, 43) are presented. Microstructural studies are carried out using scanning electron microscopy; a grain structure is analyzed by an electron backscatter diffraction technique. It is found out that the crystallization of all investigated alloys proceeds by a chemically partionless mechanism which results in the formation of a supersaturated solid solution of bismuth in a tin lattice with the original composition. Observations of the solid solution decomposition process at room temperature shows that decomposition proceeds by both continuous and discontinuous mechanisms in alloys with bismuth concentration not higher than the limit of solubility of bismuth in a tin (20 wt. %). Needle-like coherent bismuth inclusions are formed in the volume of a tin grain as a result of continuous decomposition. Discontinuous decomposition rate increases with the increasing concentration of bismuth in the alloy. In hipoeutectic alloys with bismuth concentration higher than the solubility limit, decomposition occurs by discontinuous mechanism. Complete decomposition proceeds by several stages and results in formation of areas with different degrees of microstructure fineness.

A phase-field model is proposed for the solidification and melting of a binary concentrated solution (melt). The method of extended irreversible thermodynamics is used to derive thermodynamically consistent equations and to take into account low and high phase transformation rates. The Hillert parallel construction is applied to determine the driving forces of the solidification or melting of a binary system. When thermodynamic equilibrium states are described, the Gibbs potentials are assumed to be known. They are specified from the existing thermodynamic information databases or from experimental data. The results of a numerical investigation of the proposed model are presented.

Boundary conditions for the solid–liquid interface of the solidifying pure melt have been derived. In the derivation the model of Gibbs interface is used. The boundary conditions include both the state quantities of bulk phases are taken at the interface and the quantities characterizing interfacial surface such as the surface temperature and the surface heat flux. Introduction of the surface temperature as an independent variable allows us to describe the scattering energy at the interface. For the steady-state motion of the planar interface the expression for the temperature discontinuity across the phase boundary has been obtained. Effect of Kapitza resistance on the interface velocity is considered. It is shown that heat resistance leads to non-linearity in solidification kinetics, namely, in “velocity-undercooling” relationship. The conditions of the steady-state motion of the planar interface have been found.

Using a phase-field model for fast phase transformations we derive an interface condition for the rapidly moving solid–liquid interface. The model is described by equations for the hyperbolic transport and fast interface dynamics, which are reduced to a sole equation of the phase field with the driving force given by deviations of temperature and concentration from their equilibrium values within the diffuse interface. It is shown that the obtained interface condition presents the acceleration- and velocity-dependent Gibbs–Thomson interfacial condition. This condition is identical to the advanced Born–Infeld equation for the hyperbolic motion by mean curvature with the driving force. As a limiting case, the interface condition presents “velocity-driving force” relationships found earlier as traveling wave solutions for slow and fast phase field profiles. Predictions of the analytical solutions are qualitatively compared with literature data of atomistic simulations on crystal growth kinetics.

The phase field crystal model provides a continual description of the atomic density over the diffusion time of reactions. We consider a homogeneous structure (liquid) and a perfect periodic crystal, which are constructed from the one-mode approximation of the phase field crystal model. A diagram of 2D structures is constructed from the analytic solutions of the model using atomic density functionals. The diagram predicts equilibrium atomic configurations for transitions from the metastable state and includes the domains of existence of homogeneous, triangular, and striped structures corresponding to a liquid, a body-centered cubic crystal, and a longitudinal cross section of cylindrical tubes. The method developed here is employed for constructing the diagram for the homogeneous liquid phase and the body-centered iron lattice. The expression for the free energy is derived analytically from density functional theory. The specific features of approximating the phase field crystal model are compared with the approximations and conclusions of the weak crystallization and 2D melting theories.

The present article is focused on the shapes of dendritic tips occurring in undercooled binary systems in the absence of convection. A circular/globular shape appears in limiting cases of small and large Péclet numbers. A parabolic/paraboloidal shape describes the tip regions of dendrites whereas a fractional power law defines a shape behind their tips in the case of low/moderate Péclet number. The parabolic/paraboloidal and fractional power law shapes are sewed together in the present work to describe the dendritic shape in a broader region adjacent to the dendritic tip. Such a generalized law is in good agreement with the parabolic/paraboloidal and fractional power laws of dendritic shapes. A special case of the angled dendrite is considered and analysed in addition. The obtained results are compared with previous experimental data and the results of numerical simulations on dendritic growth. This article is part of the theme issue ‘Patterns in soft and biological matters’.

Convection plays an essential role in the growth of dendrites. It may influence the transport of heat and substances as well as mechanical deformation of dendritic crystals. In this paper, the effect of convective heat and mass transport that substantially changes the dendrite tip diameter and its tip velocity is demonstrated. Using currently developed models for dendrite growth with flow and the selection theory in the case of n-fold crystal symmetry, we present the results of commutations and quantitatively compare them with experimental data. Namely, we give the critical analysis of experimental data for solidifying T i 45 A l 55 samples processed in electromagnetic levitators on the Ground and in microgravity conditions. The theory is tested against experimental data for both tip velocity and tip diameter. It is shown that convection plays an essential role in formation of habitus of crystals, morphological stability and growth kinetics of crystal interfaces.

We analyze the crystal growth kinetics of rapidly solidifying glass-forming Cu50Zr50 alloy melts. Formulating a dendrite growth model we predict all features of crystallization kinetics in Cu50Zr50 from thermodynamically controlled growth (governed by the change in Gibbs free energy on solidification) to the kinetically limited regime (governed by atomic attachment at the solid/liquid interface). Comparing critical undercoolings observed in the crystallization kinetics with experimental data on melt viscosity, atomistic simulation's data on liquid microstructure and theoretically predicted dendrite growth velocity allows us to conclude that the dendrite growth kinetics strongly depends on the cluster structure changes of the melt.

The steady-state analytical solutions describing the thermo-solutal growth of a dendritic crystal in the form of an elliptical paraboloid with allowance for a forced convective flow are found. The system of transcendental equations consisting of the undercooling balance and the stability criterion that describes a stable mode of growth of sections of elliptical dendrites is formulated and analyzed. It is shown that the temperature distributions in the melt, the growth and flow Peclet numbers as well as the dendrite tip diameter and tip velocity in the case of elliptical crystals essentially differ from their analogs in the case of symmetric shapes of dendrites in the form of paraboloids of revolution.

The dynamics of the diffuse interface between liquid and solid states is analysed. The diffuse interface is considered as an envelope of atomic density amplitudes as predicted by the phase-field crystal model (Elder et al. 2004 Phys. Rev. E 70 , 051605 ( doi:10.1103/PhysRevE.70.051605 ); Elder et al. 2007 Phys. Rev. B 75 , 064107 ( doi:10.1103/PhysRevB.75.064107 )). The propagation of crystalline amplitudes into metastable liquid is described by the hyperbolic equation of an extended Allen–Cahn type (Galenko & Jou 2005 Phys. Rev. E 71 , 046125 ( doi:10.1103/PhysRevE.71.046125 )) for which the complete set of analytical travelling-wave solutions is obtained by the method (Malfliet & Hereman 1996 Phys. Scr. 15 , 563–568 ( doi:10.1088/0031-8949/54/6/003 ); Wazwaz 2004 Appl. Math. Comput. 154 , 713–723 ( doi:10.1016/S0096-3003(03)00745-8 )). The general solution of travelling waves is based on the function of hyperbolic tangent. Together with its set of particular solutions, the general solution is analysed within an example of specific task about the crystal front invading metastable liquid (Galenko et al. 2015 Phys. D 308 , 1–10 ( doi:10.1016/j.physd.2015.06.002 )). The influence of the driving force on the phase-field profile, amplitude velocity and correlation length is investigated for various relaxation times of the gradient flow. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.

Transport processes around phase interfaces, together with thermodynamic properties and kinetic phenomena, control the formation of dendritic patterns. Using the thermodynamic and kinetic data of phase interfaces obtained on the atomic scale, one can analyse the formation of a single dendrite and the growth of a dendritic ensemble. This is the result of recent progress in theoretical methods and computational algorithms calculated using powerful computer clusters. Great benefits can be attained from the development of micro-, meso- and macro-levels of analysis when investigating the dynamics of interfaces, interpreting experimental data and designing the macrostructure of samples. The review and research articles in this theme issue cover the spectrum of scales (from nano- to macro-length scales) in order to exhibit recently developing trends in the theoretical analysis and computational modelling of dendrite pattern formation. Atomistic modelling, the flow effect on interface dynamics, the transition from diffusion-limited to thermally controlled growth existing at a considerable driving force, two-phase (mushy) layer formation, the growth of eutectic dendrites, the formation of a secondary dendritic network due to coalescence, computational methods, including boundary integral and phase-field methods, and experimental tests for theoretical models—all these themes are highlighted in the present issue. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.

The boundary integral method for propagating solid/liquid interfaces is detailed with allowance for the thermo-solutal Stefan-type models. Two types of mass transfer mechanisms corresponding to the local equilibrium (parabolic-type equation) and local non-equilibrium (hyperbolic-type equation) solidification conditions are considered. A unified integro-differential equation for the curved interface is derived. This equation contains the steady-state conditions of solidification as a special case. The boundary integral analysis demonstrates how to derive the quasi-stationary Ivantsov and Horvay–Cahn solutions that, respectively, define the paraboloidal and elliptical crystal shapes. In the limit of highest Péclet numbers, these quasi-stationary solutions describe the shape of the area around the dendritic tip in the form of a smooth sphere in the isotropic case and a deformed sphere along the directions of anisotropy strength in the anisotropic case. A thermo-solutal selection criterion of the quasi-stationary growth mode of dendrites which includes arbitrary Péclet numbers is obtained. To demonstrate the selection of patterns, computational modelling of the quasi-stationary growth of crystals in a binary mixture is carried out. The modelling makes it possible to obtain selected structures in the form of dendritic, fractal or planar crystals. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.

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.

Using a model of local non-equilibrium diffusion during rapid solidification of a binary system, the isosolutal shapes of growing crystals in steady-state approximation are obtained. It is found that for crystals growing with constant velocity along a selected coordinate direction, two isosolutal growth shapes can occur. These are: the parabolic platelet in two-dimensional case and the paraboloid of revolution in three-dimensional case. In the isothermal case of diffusionless solidification, when the velocity of solidification is equal to or greater than the solute diffusive speed in the bulk system, these interfaces can have an arbitrary configuration. Special attention is given to mathematical transformations from parabolic (paraboloidal) coordinates to usual Cartesian coordinates for Ivantsov solutions extended to the case of rapid dendritic growth in which the solidification velocity V is comparable with the solute diffusion speed V D in bulk liquid.

The properties of a two dimensional crystalline phase invading a metastable or unstable liquid state are examined using the amplitude expansion formulation of the hyperbolic and parabolic phase-field crystal model. When the amplitudes are real and equal to each other, analytic expressions are derived for the profile of a steady state liquid–solid front traveling at constant velocity. Numerical simulations of the full amplitude formulation are conducted and compared with the analytic results. Close to the melting transition the analytic results for the liquid–solid profile, velocity and width are in quantitative agreement with the numerical results and disagree far from the transition.

A thermo-diffusional problem of a free dendrite growing in a binary mixture is considered analytically. Effects of the anisotropy and convective flow on the stable mode of the dendrite with four-fold crystal symmetry are studied. Special analysis is given for the parabolic dendrite growing at arbitrary Péclet numbers and with small anisotropy of surface energy and atomic kinetics. The stable growth mode is analyzed through the solvability condition giving the stability criterion for the dendrite tip velocity V and dendrite tip diameter ρ as a function of growth Péclet number, Pg, flow Péclet number, Pf, and Reynolds number, Re. Using the obtained criterion of stability, a complete sequence of transitions in growth regimes (namely, from solute diffusion-limited to thermally controlled and further to kinetically-limited regimes) of the anisotropic dendrite is derived and revealed. Limiting cases to known criteria for small and high growth Péclet numbers of the solidifying system with and without convective fluid flow are found. Two-dimensional solidification regimes and scalings obtained are discussed for their extension to three-dimensional dendritic growth.

A phase field model of locally nonequilibrium solidification of a supercooled melt of a pure substance has been proposed. To derive thermodynamically consistent equations of the model, the method of extended irre� versible thermodynamics has been used. It has been assumed that Gibbs potentials describing thermodynam� ically equilibrium states in terms of experimental data are known. The limit of a sharp interface of the model has been considered and an analytical dependence of the velocity of a flat solidification front on the temper� ature has been obtained.

We present the first numerical results on three dimensional structures predicted by the modified phase field crystal equation. The computations are performed using parallel algorithms based on isogeometric analysis, a generalization of the finite element method. The evolution of crystal structures to their steady equilibrium state is predicted for various atomic densities and temperatures. These steady structures are consistent with the phase diagram predicted earlier using one-mode approximations of analytical solutions to the classical parabolic phase-field crystal equation.

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 A50B50 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 congruently melting alloy with order–disorder transition (Hartmann et al. 2009 Europhys. Lett. 87, 40007 (doi:10.1209/0295-5075/87/40007)). This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’. © 2018 The Author(s) Published by the Royal Society. All rights reserved. Russian Science Foundation, RSF: 16-11-10095 50WM1541 Deutsche Forschungsgemeinschaft, DFG Data accessibility. This article has no additional data. Authors’ contributions. All the authors contributed equally to the present research paper. Competing interests. The authors declare that they have no competing interests. Funding. This work was supported by the Russian Science Foundation (grant no. 16-11-10095), the German Space Center Space Management (under contract number 50WM1541) and the Deutsche Forschungsgemeinschaft (DFG) (under grant no. Re1261/8-2).

Containerless processing of droplets and drops by atomization and electromagnetic levitation are applied to undercool metallic melts and alloys prior to solidification. Heterogeneous nucleation on crucible walls is completely avoided giving access to large undercoolings. Experiments are performed both under terrestrial (1 g) conditions and in reduced gravity (µg) as well. Microgravity conditions are realized by the free fall of small droplets during atomization of a spray of droplets, individual drops in a drop tube and by electromagnetic levitation of drops during parabolic flights, sounding rocket missions, and using the electro-magnetic levitator multi-user facility on board the International Space Station. The comparison of both sets of experiments in 1 g and µg leads to an estimation of the influence of forced convection on dendrite growth kinetics and microstructure evolution., We thank ESA for financial support of all institutions listed on the cover page. DMH thanks the German Research Foundation DFG for additional support within the Contract HE1601/26. P.K.G. is grateful to the German Space Center Space Management under Contract Number 50WM1541 and the Russian Science Foundation under Contract Number 16-11-10095 for financial support.

The analytical solution describing an inclined viscous flow around a growing dendrite in the Oseen hydrodynamic approximation is constructed in the two- and three-dimensional cases. The velocity components in the vicinity of dendritic tip are found in the parabolic cylinder and paraboloid of revolution coordinates. The obtained solution transforms to the previously obtained solution by Dash and Gill in the case of axisymmetric flow.The analytical solution describing an inclined viscous flow around a growing dendrite in the Oseen hydrodynamic approximation is constructed in the two- and three-dimensional cases. The velocity components in the vicinity of dendritic tip are found in the parabolic cylinder and paraboloid of revolution coordinates. The obtained solution transforms to the previously obtained solution by Dash and Gill in the case of axisymmetric flow.

This paper is based on analytical solutions found in Part 1. The analytical solution describing an inclined viscous flow around a growing dendrite in the Oseen hydrodynamic approximation is illustrated. The velocity components as functions of the parabolic coordinates in the vicinity of dendritic tip are plotted. We show that the inclined flow substantially changes the hydrodynamic solutions around the growing steady-state dendrite in comparison with the previously obtained solution by Dash and Gill in the case of axisymmetric flow.This paper is based on analytical solutions found in Part 1. The analytical solution describing an inclined viscous flow around a growing dendrite in the Oseen hydrodynamic approximation is illustrated. The velocity components as functions of the parabolic coordinates in the vicinity of dendritic tip are plotted. We show that the inclined flow substantially changes the hydrodynamic solutions around the growing steady-state dendrite in comparison with the previously obtained solution by Dash and Gill in the case of axisymmetric flow.

In the present article, we use the selection theory to estimate the non-stationarity time of evolving dendrites to their steady-state growth.

Fronts dynamics of periodic crystalline state, which invades the homogeneous state (liquid phase), are analysed. These fronts are considered as traveling waves of atomic density amplitudes. The propagation of amplitudes is described by the hyperbolic equation of an extended Allen-Cahn type for which the complete set of analytical traveling-wave solutions are obtained by tanh-method. The set of solutions includes previously known traveling waves for the parabolic Allen-Cahn equation of both extended and standard form.

An influence of small amounts of impurity on the dendrite growth is investigated analytically and numerically. For investigations, a model for anisotropic dendrites rapidly growing in pure and binary melts is used. The dependence of the dendrite growth velocity and of the dendrite tip radius from undercooled melt from amounts of impurity is analyzed. Calculations are carried out for the pure nickel and binary nickel-zircon diluted melts.

The Phase Field Crystal model in hyperbolic formulation (modified PFC or MPFC), is investigated as one of the most promising techniques for modeling the formation of crystal patterns. MPFC is a convenient and fundamentally based description linking nano-and meso-scale processes in the evolution of crystal structures. The presented model is a powerful tool for mathematical modeling of the various operations in manufacturing. Among them is the definition of process conditions for the production of metal castings with predetermined properties, the prediction of defects in the crystal structure during casting, the evaluation of quality of special coatings, and others. Our paper presents the structure diagram which was calculated for the one-mode MPFC model and compared to the results of numerical simulation for the fast phase transitions. The diagram is verified by the numerical simulation and also strongly correlates to the previously calculated diagrams. The computations have been performed using software based on the effective parallel computational algorithm.

Melt flow is often quoted as the reason for a discrepancy between experiment and theory on dendritic growth kinetics at low undercoolings. But this flow effect is not justified for glass-fluxed melts where the flow field is weaker. In the present work, we modeled the thermal history, flow pattern and dendritic structure of a glass-fluxed nickel sample by magnetohydrodynamics calculations. First, the temperature distribution and flow structure in the molten and undercooled melt were simulated by reproducing the observed thermal history of the sample prior to solidification. Then the dendritic structure and surface temperature of the recalescing sample were simulated. These simulations revealed a large thermal gradient crossing the sample, which led to an underestimation of the real undercooling for dendritic growth in the bulk volume of the sample. By accounting for this underestimation, we recalculated the dendritic tip velocities in the glass-fluxed nickel melt using a theory of three-dimensional dendritic growth with convection and concluded an improved agreement between experiment and theory. © 2016 Elsevier B.V.

A model for anisotropic growth of a dendritic crystal in a binary mixture under non-isothermal conditions is presented. A criterion for a stable growth mode is given for the dendrite tip as a function of the thermal Peclet number and the ratio between the velocities of dendrite growth and solute diffusion in the liquid bulk. Limiting cases of known criteria for anisotropic dendrite growth at low and high growth Peclet numbers are provided. The inclusion of forced convective flow extends the range of theoretical predictions, especially to low growth velocities, thus eliminating systematic discrepancies between earlier models and observed experimental data, as shown by a comparison of model predictions with measured growth velocities in Ti–55 at% Al alloys solidified under electromagnetic levitation.