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This article considers the hydrodynamic problem of an oblique flow of a viscous incompressible fluid around the tip of a dendritic crystal. Approximate analytical solutions of Oseen's hydrodynamic equations are obtained in 2D and 3D cases using special curvilinear coordinates. It is shown that the projections of the fluid velocity change significantly with a change in the flow slope and Reynolds number. The theory developed in this work has a limiting transition to the previously known solutions for the rectilinear (without tilt) fluid flow around a dendrite. © 2021 John Wiley & Sons, Ltd. Russian Foundation for Basic Research, РФФИ: 20-08-00199; Ministry of Education and Science of the Russian Federation, Minobrnauka: FEUZ-2020-0057 This work contains two parts, theoretical and numerical. The first of them was supported by the Russian Foundation for Basic Research (project no. 20-08-00199). The second part was supported by the Ministry of Science and Higher Education of the Russian Federation (project no. FEUZ-2020-0057).

This article proposes an analytical model to understand the rod growth of eutectic in the bulk undercooled melt. Based on the previous derivations of the lamellar eutectic growth models, relaxing the assumptions of small Péclet numbers, the model is derived by considering melt kinetic and thermal undercoolings. The intent of this model is to predict the transitions in eutectic pattern for conditions of the low and high growth velocities. In addition to investigation of the transition between lamellar and rod eutectic patterns, mathematical simplifications of solving Bessel function are presented as well, which is the most important priority to model calculation. © 2021 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd. Deutsche Forschungsgemeinschaft, DFG: GA 1142/11-1; Natural Science Foundation of Shaanxi Provincial Department of Education: 18JS050; Key Science and Technology Program of Shaanxi Province: 2016KJXX-87 This work was supported by the Key Science and Technology Program of Shaanxi Province (No. 2016KJXX-87) and the Natural Science Foundation of Shaanxi Provincial Department of Education (No. 18JS050). P. K. G. acknowledges financial support of the German Research Foundation (Deutsche Forschungsgemeinschaft [DFG]) under Project GA 1142/11-1. The authors thank J. T. Cao and A. L. Bao for their help in this work. Open Access funding enabled and organized by Projekt DEAL.

A theory for crystal nucleation and growth with the recalescence front is developed. The theory is based on the saddle-point technique for evaluating a Laplace-type integral as well as the small parameter method for solving the moving boundary heat transfer problem. The theory developed shows the U-shaped behavior of the growth velocity–melt undercooling curve. The ordinary upward branch of this curve is caused by the growth dictated by heat transport and the predominant crystal growth, while the unusual downward branch demonstrates the anomalous behavior caused by the predominant nucleation and attachment kinetics of the growing crystals to the phase interface. Such a U-shaped behavior of the growth velocity–melt undercooling curve is consistent with experimental data carried out on the ground, under reduced gravity during parabolic flights, and in the microgravity conditions onboard the International Space Station [M. Reinartz et al., JOM 74, 2420 (2022); P.K. Galenko et al., Acta Mater. 241, 118384 (2022)].

The classical eutectic growth models are based on the use of eutectic composition. These models neglect the effect of the primary phase formation and their direct using in the rapid solidification process of off-eutectic (hypoeutectic and hypereutectic) alloys is absent. Combing the effect of the primary phase in the eutectic transformation and an off-eutectic composition, the solidification growth model is derived in the present work. The effect of the model and materials parameters on solidification kinetics is discussed in comparison with experimental data. Computational results on the off-eutectic growth model show that the model agrees well with experimental data on solidification kinetics of Ni-B and Ti-Si alloys.

A stationary growth of the dendritic tip having the tip shape of an elliptical paraboloid in a binary liquid is considered. A method for constructing a selection criterion that allows one to obtain the growth rate and two radii of the dendritic tip as functions of undercooling is proposed. Selection constants are found theoretically for the n-fold crystalline symmetry. The theory is in good agreement with experimental data obtained on water and water solution previously described in literature.

The kinetics of dendrite crystal growth in pure Ni and diluted Ni–B melts is analyzed. Using a model for anisotropic dendrites rapidly growing from undercooled melts, numerical solutions for dendrite tip velocity V and tip radius R are presented. The influence of boron content on tip velocity V and tip radius R of the dendrite growing in Ni-B alloys is shown in comparison with the growth kinetics of pure Ni–dendrites. Particularly, we show that there is a well-known regime of growth in which the impurity (boron) decreases the velocity of the alloying Ni–B–dendrite regarding the nickel dendrite velocity. In addition to this well-established regime, we also show quantitatively that there is a tiny amount of boron which enhances the dendrite velocity in comparison with the velocity of pure nickel dendrite. Explanations of this non-linearity are given using the results of calculations for dendrite tip radii and dendrite growth velocity for alloying Ni–B and pure Ni systems.

The phase field model is developed by the effective mobility approach to slow and rapid solidification. The phase field model equations are reduced to the hodograph equation for solid-liquid interface movement which is applied to the problem of solute trapping in a binary alloy. A specific method based on the one-point Cauchy problem is developed for solution of the hodograph equation with the solute diffusion equation. The method is tested in comparison with the rapid solidification of Si–0.1 at.% As alloy previously analyzed experimentally and using phase field modelling.

The paper deals with the analysis of stable thermo-solutal dendritic growth in the presence of intense convection. The n-fold symmetry of crystalline anisotropy as well as the two- and three-dimensional growth geometries are considered. The steady-state analytical solutions are found with allowance for the convective-type heat and mass exchange boundary conditions at the dendritic surface. A linear morphological stability analysis determining the marginal wavenumber is carried out. The new stability criterion is derived from the solvability theory and stability analysis. This selection criterion takes place in the regions of small undercooling. To describe a broader undercooling diapason, the obtained selection criterion, which describes the case of intense convection, is stitched together with the previously known selection criterion for the conductive-type boundary conditions. It is demonstrated that the stitched selection criterion well describes a broad diapason of experimental undercoolings.

The dendritic form is one of the most common forms of crystals growing from supercooled melts and supersaturated solutions. In recent decades, an analytical theory has been developed that describes a stable dendrite growth mode under the conditions of a conductive heat and mass transfer process. However, in experiments, the growth of dendritic crystals is often observed under the conditions of convective fluid flow. In the present work, the theory of the growth of dendritic crystals is developed taking into account the convective mechanism of heat and mass transfer at the crystal-melt interface. A stable mode of dendritic growth in the case of intense convective flows near the steady-state growing dendritic tip is analyzed. The selection theory determining a stable growth mode in the vicinity of parabolic solutions as well as the undercooling balance condition are used to find the dendrite tip velocity and its tip diameter as functions of the melt undercooling. It is shown that the theoretical predictions in the case of convective boundary conditions are in agreement with experimental data for small undercoolings. In addition, the convective and conductive heat and mass transfer mechanisms near the growing dendritic surfaces are compared. Our calculations show that the convective boundary conditions essentially influence the stable mode of dendritic growth.

Using the developed sharp-interface model of solidification we quantitatively estimate influence of tiny amount of impurity and forced convection on kinetics of dendritic growth. As tested systems, we choose Ni and Ni–Zr dendrites which are growing into a stagnant undercooled melt, the melt with incoming forced flow and with/without impurity (that is small amount of Zr diluted in a pure Ni). The model predictions and comparisons allow us to quantitatively estimate predominant influence of impurity (chemical segregation), thermal influence (due to temperature inhomogeneity) and hydrodynamic effects (due to investigated laminar and forced melt flow) on the dendrite growth kinetics.

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 process of rapid solidification of a binary mixture is considered in the framework of local nonequilibrium model (LNM) based on the assumption that there is no local equilibrium in solute diffusion in the bulk liquid and at the solid-liquid interface. According to LNM the transition to complete solute trapping and diffusionless solidification occurs at a finite interface velocity $V=V_D$, where $V_D$ is the diffusion speed in bulk liquid. In the present work, the boundary conditions at the phase interface moving with the velocity $V$ close to $V_D$ ($V \lesssim V_D$) have been derived to find the non-equilibrium solute partition coefficient. In the high-speed region, its comparison with the partition coefficient from the work [Phys. Rev. E 76 (2007) 031606] is given., Comment: 8 pages, 2 figures, sent to the EPJ-ST

The phase-field crystal (PFC-model) is a powerful tool for modelling of the crystallization in colloidal and metallic systems. In the present work, the modified hyperbolic phase-field crystal model for binary systems is presented. This model takes into account slow and fast dynamics of moving interfaces for both concentration and relative atomic number density (which were taken as order parameters). The model also includes specific mobilities for each dynamical field and correlated noise terms. The dynamics of chemical segregation with origination of mixed pseudo-hexagonal binary phase (the so-called ‘triangle phase’) is used as a benchmark in two spatial dimensions for the developing model. Using the free energy functional and specific lattice vectors for hexagonal crystal, the structure diagram of co-existence of liquid and three-dimensional hexagonal phase for the binary PFC-model was carried out. Parameters of the crystal lattice correspond to the hexagonal boron nitride (BN) crystal, the values of which have been taken from the literature. The paper shows the qualitative agreement between the developed structure diagram of the PFC model and the previously known equilibrium diagram for BN constructed using thermodynamic functions. This article is part of the theme issue ‘Transport phenomena in complex systems (part 2)’.

Numerous experimental data on the rapid solidification of eutectic systems exhibit the formation of metastable solid phases with the initial (nominal) chemical composition. This fact is explained by the suppression of eutectic decomposition due to diffusionless (chemically partitionless) solidification beginning at a high but finite growth velocity of crystals. In the present work, a model is suggested for the diffusionless growth to analyse the atomic diffusion in the rod eutectic couples growing into supercooled liquid. A simplified calculating method for the equation related to the Bessel function in the solution of the growth of rod eutectics is obtained. This method can also be used in the calculation of other rod eutectic growth models. This article is part of the theme issue ‘Transport phenomena in complex systems (part 2)’.

This article is devoted to the study of the tip shape of dendritic crystals grown from a supercooled liquid. The recently developed theory (Alexandrov & Galenko 2020Phil. Trans. R. Soc. A378, 20190243. (doi:10.1098/rsta.2019.0243)), which defines the shape function of dendrites, was tested against computational simulations and experimental data. For a detailed comparison, we performed calculations using two computational methods (phase-field and enthalpy-based methods), and also made a comparison with experimental data from various research groups. As a result, it is shown that the recently found shape function describes the tip region of dendritic crystals (at the crystal vertex and some distance from it) well.This article is part of the theme issue ‘Transport phenomena in complex systems (part 1)’.

The results of molecular dynamics (MD) simulations of the crystallization process in one-component materials and solid solution alloys reveal a complex temperature dependence of the velocity of the crystal–liquid interface featuring an increase up to a maximum at 10–30% undercooling below the equilibrium melting temperature followed by a gradual decrease of the velocity at deeper levels of undercooling. At the qualitative level, such non-monotonous behaviour of the crystallization front velocity is consistent with the diffusion-controlled crystallization process described by the Wilson–Frenkel model, where the almost linear increase of the interface velocity in the vicinity of melting temperature is defined by the growth of the thermodynamic driving force for the phase transformation, while the decrease in atomic mobility with further increase of the undercooling drives the velocity through the maximum and into a gradual decrease at lower temperatures. At the quantitative level, however, the diffusional model fails to describe the results of MD simulations in the whole range of temperatures with a single set of parameters for some of the model materials. The limited ability of the existing theoretical models to adequately describe the MD results is illustrated in the present work for two materials, chromium and silicon. It is also demonstrated that the MD results can be well described by the solution following from the hodograph equation, previously found from the kinetic phase-field model (kinetic PFM) in the sharp interface limit. The ability of the hodograph equation to describe the predictions of MD simulation in the whole range of temperatures is related to the introduction of slow (phase field) and fast (gradient flow) variables into the original kinetic PFM from which the hodograph equation is obtained. The slow phase-field variable is responsible for the description of data at small undercoolings and the fast gradient flow variable accounts for local non-equilibrium effects at high undercoolings. The introduction of these two types of variables makes the solution of the hodograph equation sufficiently flexible for a reliable description of all nonlinearities of the kinetic curves predicted in MD simulations of Cr and Si. This article is part of the theme issue ‘Transport phenomena in complex systems (part 1)’.

This review article summarizes the main outcomes following from recently developed theories of stable dendritic growth in undercooled one-component and binary melts. The nonlinear heat and mass transfer mechanisms that control the crystal growth process are connected with hydrodynamic flows (forced and natural convection), as well as with the non-local diffusion transport of dissolved impurities in the undercooled liquid phase. The main conclusions following from stability analysis, solvability and selection theories are presented. The sharp interface model and stability criteria for various crystallization conditions and crystalline symmetries met in actual practice are formulated and discussed. The review is also focused on the determination of the main process parameters—the tip velocity and diameter of dendritic crystals as functions of the melt undercooling, which define the structural states and transitions in materials science (e.g. monocrystalline-polycrystalline structures). Selection criteria of stable dendritic growth mode for conductive and convective heat and mass fluxes at the crystal surface are stitched together into a single criterion valid for an arbitrary undercooling.This article is part of the theme issue ‘Transport phenomena in complex systems (part 1)’.

Motivated by an important application of dendritic crystals in the form of an elliptical paraboloid, which widely spread in nature (ice crystals), we develop here the selection theory of their stable growth mode. This theory enables us to separately define the tip velocity of dendrites and their tip diameter as functions of the melt undercooling. This, in turn, makes it possible to judge the microstructure of the material obtained as a result of the crystallization process. So, in the first instance, the steady-state analytical solution that describes the growth of such dendrites in undercooled one-component liquids is found. Then a system of equations consisting of the selection criterion and the undercooling balance that describes a stable growth mode of elliptical dendrites is formulated and analyzed. Three parametric solutions of this system are deduced in an explicit form. Our calculations based on these solutions demonstrate that the theoretical predictions are in good agreement with experimental data for ice dendrites growing at small undercoolings in pure water. © 2021 IOP Publishing Ltd.

The present theme issue is devoted to recent trends and research directions in the phase transformation phenomena occurring in metastable and heterogeneous materials. All papers are concerned with modern theories, experiments, and computer simulations in the wide area of phase transformations. Particular attention is paid to traditional research domains representing the theoretical background for recent simulations and experiments that are as well specifically highlighted herein. Such rapidly developing research directions as phase-field modeling, laser treatment of surfaces, nanostructures, and influence of external fields on the microstructure formation are specially covered in this issue. © 2020, EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature.

Theoretical predictions by many models of solidification kinetics rest on assumption of the steady-state regime of crystal growth with constant velocity. Such approximation holds if the time for solidification of a region under study is much longer than the non-stationarity time for solidification velocity. Even though a lot of experimental evidences which support the existence of long quasi-stationary periods of crystal growth or solidifying layers with constant velocity exists, direct quantitative estimations of such regimes are given usually for planar fronts or essential simplifications which, for instance, do not take into account the finiteness of solidifying bulk. The present work suggests a method to quantify non-stationary periods of crystal growth in comparison with solidification time of finite bulks. In essence we discuss and quantify limits of applicability of steady state crystal growth theories. With this aim, an acceleration- and velocity-dependent interfacial condition [A. Salhoumi, P.K. Galenko, Physica A 447 (2016) 161] is used for the analysis of various regimes of dendrite growth as particular case of crystalline solidification. The time dependence of the dendrite tip velocity is obtained for the growth from a pure (chemically one component) undercooled liquid. The developed theoretical model shows a drastic reducing the non-stationarity time with the increase of undercooling. The present model can be advanced to the arbitrary undercooling and further compared with data of experimental measurements on crystal growth kinetics.

Rapid solidification is a relevant physical phenomenon in material sciences, whose theoretical analysis requires going beyond the limits of local equilibrium statistical physics and thermodynamics and, in particular, taking account of ergodicity breaking and of generalized formulation of thermodynamics. The ergodicity breaking is related to the time symmetry breaking and to the presence of some kinds of fluxes and gradient flows making that an average of microscopic variables along time is different than an average over some chosen statistical ensemble. In fast processes, this is due, for instance, to the fact that the system has no time enough to explore the whole region of possible microscopic states in the phase space. Similarly to this, systems submitted to strong fluxes may have no time for reaching the whole phase space in local bulks during observable macroscopic time. Rapid solidification, ergodicity breaking and extended thermodynamics actually make a conceptually novel combination in the present overview: ergodicity breaking is expressed in general terms and then extended thermodynamics is formulated as a particular phenomenological expression and applied to describe the dynamics of the phenomenon. Using the formalism of micro- and meso-scopic dynamics we introduce a general view on non-ergodic fast transitions and provide a simplest description of a continuum theory based on the system of hyperbolic equations applicable to rapid solidification. Analysis of non-equilibrium effects, including interface kinetics, solute trapping and solute drag, is presented with their effect on the rapidly moving solid–liquid interface. Special attention is paid to the theory predictions compared with the kinetics obtained in experiments on samples processed by electromagnetic levitation facility and in molecular dynamics simulation.

A nonisothertmal model of spinodal decomposition is proposed for a binary system described by the Ginzburg–Landau energy. The initial stages of spinodal decomposition are investigated for a system of simultaneous equations describing the impurity and temperature redistribution. The dispersion equation and the instability growth rate of spinodal structures are obtained. The temperature dependence of the growth rate and the wave number corresponding to the maximum instability is found. The observed difference in the dispersion relations of isothermal and nonisothermal models shows that a criterion for the effect of temperature fluctuations on the spinodal decomposition is given by a dimensionless parameter Γ; for large values of Γ (Γ ≥ 1000), temperature fluctuations in the form of noise are certainly insufficient for spinodal decomposition due to a change in the dispersion structure of the equations, which suggests the need to take into account the nonisothermal behavior of the system.

Crystals growing in an undercooled or supersaturated system have a shape given by the solution of an integro-differential equation (boundary integral). A new analytical solution of this nonlinear equation is found in the limit of large thermal and chemical Peclet numbers. Based on the saddle-point technique for a Laplace-type integral, our solution describes the tip region of crystals, which is close to a globular and circular shape in the three- and two-dimensional cases. Corresponding to various growth modes and Peclet numbers, the boundary integrals and crystal tips are listed in a table. A special case of chemical dendrite growing in a binary isothermal system is specially analyzed.

A phase field model for small and large driving forces on solidification and melting of a pure metal or binary alloy is formulated. A traveling wave solution of the phase field equation predicts the non-linear behavior in the velocity of the crystal/liquid interface at the large driving force. This non-linearity has the dependence of velocity with saturation or exhibiting the velocity with maximum at a fixed undercooling/superheating. The predicted velocity is compared with the molecular dynamics simulation data for pure Fe that confirms a crucial role of local non-equilibrium in the form of relaxation of gradient flow in the quantitative description of the crystal growth kinetics.

Analysis of phase separation under non-isothermal conditions in undercooled molten Co-Cu droplets has been performed theoretically. The calculated microstructure length scales agree with the Cahn-Hilliard (CH) and Langer-Bar-on-Miller (LBM) models, and experimental data. At moderate cooling rates V c ∼ 10 2 - 10 3 K/s, the wave length λ m ( t ) of the fastest growing mode increases in time ( ∂ λ m / ∂ t > 0 ) in exact correspondence with the isothermal LBM model. At high V c ∼ 10 4 K/s, λ m slowly changes in time ( ∂ λ m / ∂ t ∼ 0 ) in agreement with the non-isothermal CH model. At very high V c ∼ 10 5 - 10 8 K/s, initial decrease of λ m ( ∂ λ m / ∂ t 0 ) is first predicted. Then λ m achieves its maximum value in time ( ∂ λ m / ∂ t = 0 ) and starts to increase ( ∂ λ m / ∂ t > 0 ) at the later stage. The revealed effect is examined and theoretically explained.

The surface tension and viscosity of equilibrium and supercooled liquids of Cu50Zr50 were measured in the containerless electromagnetic levitator ISS-EML in the European space laboratory Columbus on board the International Space Station (ISS) under microgravity using high-speed camera recordings. From 1250 K to 1475 K, the surface tension follows the relation σ(T) = (1.58 ± 0.01) N/m – (3.1 ± 0.6) · 10−4 N/m · K · (T – 1209 K). A frequency shift correction was applied to remove the influence of sample rotation on the measured surface tension. Within the investigated temperature range, the viscosity can be expressed by an Arrhenius temperature dependence η(T) = η0 · exp(EA/kBT), with η0 = (0.08 ± 0.02) mPa·s and EA = (0.58 ± 0.03) eV.

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.

In this paper, we develop a theory of stable dendritic growth in undercooled melts in the presence of conductive and convective heat and mass transfer boundary conditions at the solid/liquid interface of a dendrite. To simplify the matter and construct the analytical theory, conductive and convective mechanisms are considered separately. Namely, the laws for total undercooling and selection criterion defining the stable growth mode (dendrite tip velocity and diameter) are derived for conductive and convective boundary conditions. To describe the case of simultaneous occurrence of these heat and mass transfer mechanisms, we sew together conductive and convective laws using power stitching functions. The generalised selection theory is compared with experimental data for Al24Ge76 and Ti45Al55 undercooled melts.

This article considers the hydrodynamic problem of an oblique flow of a viscous incompressible fluid around the tip of a dendritic crystal. Approximate analytical solutions of Oseen’s hydrodynamic equations are obtained in 2D and 3D cases using special curvilinear coordinates. It is shown that the projections of the fluid velocity change significantly with a change in the flow slope and Reynolds number. The theory developed in this work has a limiting transition to the previously known solutions for the rectilinear (without tilt) fluid flow around a dendrite.

Using the phase field crystal model (PFC model), an analysis of slow and fast dynamics of solid-liquid interfaces in solidification and melting processes is presented. Dynamical regimes for cubic lattices invading metastable liquids (solidification) and liquids propagating into metastable crystals (melting) are described in terms of the evolving amplitudes of the density field. Dynamical equations are obtained for body-centered cubic (bcc) and face-centered cubic (fcc) crystal lattices in one- and two-mode approximations. A universal form of the amplitude equations is obtained for the three-dimensional dynamics for different crystal lattices and crystallographic directions. Dynamics of the amplitude's propagation for different lattices and PFC mode's approximations is qualitatively compared. The traveling-wave velocity is quantitatively compared with data of molecular dynamics simulation previously obtained by Mendelev et al. [Modell. Simul. Mater. Sci. Eng. 18, 074002 (2010)MSMEEU0965-039310.1088/0965-0393/18/7/074002] for solidification and melting of the aluminum fcc lattice.

The growth of a free dendrite having a non-axisymmetric morphology with arbitrary symmetry in a pure substance is considered with the interfacial effect of anisotropy and in the absence of convective flow. A stable mode of thermal dendritic growth with the n-fold crystal symmetry is formulated using the solvability theory. A complete set of nonlinear equations, consisting of the undercooling balance condition and the selection criterion, is derived. Theoretical predictions are compared with phase field modeling data in two and three dimensional geometry obtained for ice dendrites growing in pure water.

The theory of stable dendritic growth within a forced convective flow field is tested against the enthalpy method for a single-component nickel melt. The growth rate of dendritic tips and their tip diameter are plotted as functions of the melt undercooling using the theoretical model (stability criterion and undercooling balance condition) and computer simulations. The theory and computations are in good agreement for a broad range of fluid velocities. In addition, the dendrite tip diameter decreases, and its tip velocity increases with increasing fluid velocity. © 2020, The Author(s). Russian Science Foundation, RSF: 16-11-10095 50WM1941 D.V. Alexandrov and P.K. Galenko acknowledge the support from the Russian Science Foundation (Grant No. 16-11-10095) and support from the German Space Center Space Managment under contract no. 50WM1941.

The thin interface limit of the phase-field model is extended to include transport via melt convection. A double-sided model (equal diffusivity in liquid and solid phases) is considered for the present analysis. For the coupling between phase-field and Navier-Stokes equations, two commonly used schemes are investigated using a matched asymptotic analysis: (i) variable viscosity and (ii) drag force model. While for the variable viscosity model, the existence of a thin interface limit can be shown up to the second order in the expansion parameter, difficulties arise in satisfying no-slip boundary condition at this order for the drag force model. Nevertheless, detailed numerical simulations in two dimensions show practically no difference in dendritic growth profiles in the presence of forced melt flow obtained for the two coupling schemes. This suggests that both approaches can be used for the purpose of numerical simulations. Simulation results are also compared to analytic theory, showing excellent agreement for weak flow. Deviations at higher fluid velocities are discussed in terms of the underlying theoretical assumptions. © 2020 The Author(s) Published by the Royal Society. All rights reserved. European Space Agency, ESA Deutsche Forschungsgemeinschaft, DFG Russian Science Foundation, RSF: 16-11-10095 Deutsche Forschungsgemeinschaft, DFG Data accessibility. This article has no additional data. Authors’ contributions. All the authors have contributed equally to this work. Competing interests. We declare we have no competing interest. Funding. P.K.G. acknowledges the support by the European Space Agency (ESA) under research project MULTIPHAS grant no. (AO-2004) and the German Aerospace Center (DLR) Space Management under contract no. 50WM1541 and also from the Russian Science Foundation under project no. 16-11-10095. A.S. and F.V. acknowledges financial support by the German Research Foundation (DFG) under the project no. Va205/17-1.

The dendritic growth of pure materials in undercooled melts is critical to understanding the fundamentals of solidification. This work investigates two new insights, the first is an advanced definition for the two-dimensional stability criterion of dendritic growth and the second is the viability of the enthalpy method as a numerical model. In both cases, the aim is to accurately predict dendritic growth behavior over a wide range of undercooling. An adaptive cell size method is introduced into the enthalpy method to mitigate against 'narrow-band features' that can introduce significant error. By using this technique an excellent agreement is found between the enthalpy method and the analytic theory for solidification of pure nickel. © 2020 IOP Publishing Ltd. This work was supported by the Russian Science Foundation (Grant No. 16-11-10095).

A phase-field model for small and large driving forces on solidification and melting of a pure substance or alloys is formulated. Derivations of the phase-field model are based on the effective mobility approach and on the kinetic energy approach to analyze fast phase transformation from metastable liquid to solid phase. A hodograph equation (an acceleration-velocity dependent equation of the Gibbs-Thomson type) which predicts the non-linear behavior in the velocity of the crystal-liquid interface is found at the large driving force on transformation and analyzed for different thermodynamic potentials. Traveling wave solutions of this equation are found for double-well and double-obstacle potentials. The velocity-dependent traveling waves as a function of driving force on transformation exhibit non-linearity of the solutions. Namely, in the relationship 'velocity-driving force' exists a maximum at a fixed undercooling which is very well known in the solidification of glass-forming metals and alloys. The predicted solidification velocity is quantitatively compared with the molecular dynamics simulation data obtained by Tang and Harrowell (2013 Nat. Mater. 12 507-11) for the solidification of congruently melting Cu-Zr binary alloy. The comparison confirms a crucial role of local non-equilibrium such as relaxation of gradient flow in the quantitative description of fast phase transformations.

Interface energy and kinetic coefficient of crystal growth strongly depend on the face of the crystalline lattice. To investigate the kinetic anisotropy and velocity of different crystallographic faces we use the hyperbolic (modified) phase field crystal model which takes into account atomic density (as a slow thermodynamic variable) and atomic flux (as a fast thermodynamic variable). Such model covers slow and rapid regimes of interfaces propagation at small and large driving forces during solidification. In example of BCC crystal lattice invading the homogeneous liquid, dynamical regimes of advancing front propagating along the selected crystallographic directions are studied. The obtained velocity and the velocity sequences for different faces are compared with known results., Comment: Accepted manuscript to Journal of Crystal Growth

The dendritic form is one of the most common forms of crystals growing from supercooled melts and supersaturated solutions. In recent decades, an analytical theory has been developed that describes a stable dendrite growth mode under the conditions of a conductive heat and mass transfer process. However, in experiments, the growth of dendritic crystals is often observed under the conditions of convective fluid flow. In the present work, the theory of the growth of dendritic crystals is developed taking into account the convective mechanism of heat and mass transfer at the crystal-melt interface. A stable mode of dendritic growth in the case of intense convective flows near the steady-state growing dendritic tip is analyzed. The selection theory determining a stable growth mode in the vicinity of parabolic solutions as well as the undercooling balance condition are used to find the dendrite tip velocity and its tip diameter as functions of the melt undercooling. It is shown that the theoretical predictions in the case of convective boundary conditions are in agreement with experimental data for small undercoolings. In addition, the convective and conductive heat and mass transfer mechanisms near the growing dendritic surfaces are compared. Our calculations show that the convective boundary conditions essentially influence the stable mode of dendritic growth.

A linear morphological stability of the solid-liquid interface is analyzed for a binary alloy in the limit of low and high crystal growth velocities. Using the result of this analysis, a diagram of morphologies is derived for a whole range of solidification rates with indicating critical growth velocities for the transitions planar front ⇔ cellular/dendritic structure. It is specially noted that the speed of solute diffusion in the bulk liquid limits the absolute chemical stability velocity from the high-rate transition cells/dendrites ⇒ planar front.

Stable growth of a crystal with dendritic morphology with n-fold symmetry is modeled. Using the linear stability analysis and solvability theory, a selection criterion for thermally and solutally controlled growth of the dendrite is derived. A complete set of non-linear equations consisting of the selection criterion and an undercooling balance (which determines the implicit dependencies of the dendrite tip velocity and tip diameter on the total undercooling) is formulated. The growth kinetics of crystals having different lattice symmetry is analyzed. The model predictions are compared with phase field modeling data on ice dendrites grown from pure undercooled water. © 2020, EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature. L.V.T. acknowledges the support from the Ministry of Education and Science of the Russian Federation [grant number 1.12804.2018/12.2]. P.K.G. acknowledges the support by the European Space Agency (ESA) under research project MULTIPHAS (AO-2004) and the German Aerospace Center (DLR) Space Management under contract No. 50WM1541.

The thin interface limit aims at minimizing the effects arising from a numerical interface thickness, inherent in diffuse interface models of solidification and microstructure evolution such as the phase field model. While the original formulation of this problem is restricted to transport by diffusion, we consider here the case of melt convection. Using an analysis of the coupled phase field-fluid dynamic equations, we show here that such a thin interface limit does also exist if transport contains both diffusion and convection. This prediction is tested by comparing simulation studies, which make use of the thin-interface condition, with an analytic sharp-interface theory for dendritic tip growth under convection., 4 pages, 2 figures. Submitted to European Physical Journal E

The free growth of a dendrite crystal in the form of an elliptic paraboloid from a melt is analyzed. The exact solution of the problem is shown to coincide with the well-known Harvey–Cahn solution in the absence of anisotropy and the Gibbs–Thomson effect. A criterion for the stable growth of an elliptic paraboloid dendrite is formulated.