Dendritic to globular morphology transition in ternary alloy solidification

The evolution of solidification microstructures in ternary metallic alloys is investigated by adaptive finite element simulations of a general multicomponent phase-field model. A morphological transition from dendritic to globular growth is found by varying the alloy composition at a fixed undercooling. The dependence of the growth velocity and of the impurity segregation in the solid phase on the composition is analyzed and indicates a smooth type of transition between the dendritic and globular growth structures.Comment: 4 pages, 2 figure

Morphological stability diagram for slowly and rapidly solidifying binary systems

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. © 2020, The Author(s)

Travelling-wave amplitudes as solutions of the phase-field crystal equation

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 tanh 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 tanh solution of travelling waves is based on the function of hyperbolic tangent. Together with its set of particular solutions, the general tanh 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’. © 2018 The Author(s) Published by the Royal Society. All rights reserved.Russian Science Foundation, RSF: 16-11-10095Alexander von Humboldt-Stiftung: 116077950WM1541Data accessibility. This article has no additional data. Authors’ contributions. All authors contributed equally to the 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), Alexander von Humboldt Foundation (ID 1160779) and the German Space Center Space Management under contract no. 50WM1541

Diffuse interface models of solidification with convection: The choice of a finite interface thickness

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. © 2020, The Author(s)

Kinetic transition in the order–disorder transformation at a solid/liquid interface

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: 16-11-1009550WM1541Deutsche Forschungsgemeinschaft, DFGData 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)

The boundary integral theory for slow and rapid curved solid/liquid interfaces propagating into binary systems

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. © 2018 The Author(s) Published by the Royal Society. All rights reserved.Russian Science Foundation, RSF: 16-11-1009550WM1541Data accessibility. This article has no additional data. Authors’ contributions. All authors contributed equally to the present review 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) and the German Space Center Space Management under contract no. 50WM1541

Modeling and simulation of heat/mass transport, nucleation and growth kinetics in phase transformations

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

Effects of Local Nonequilibrium in Rapid Eutectic Solidification—Part 1: Statement of the Problem and General Solution

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 suppression of eutectic decomposition due to diffusionless (chemically partitionless) solidification beginning at a high but a finite growth velocity of crystals. A model considering the diffusionless growth is developed in the present work to analyze the atomic diffusion ahead of lamellar eutectic couples growing into supercooled liquid. A general solution of the model is presented from which two regimes are followed. The first presents a diffusion-limited regime with the existence of eutectic decomposition if the solid–liquid interface velocity is smaller than the characteristic diffusion speed in the bulk liquid. The second shows suppression of eutectic decomposition under diffusionless transformation from liquid to one-phase solid if the solid–liquid interface velocity overcomes characteristic diffusion speed in the bulk liquid. © 2020 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons Ltd.This work was also financially supported by the German Space Center Space Management (No. 50WM1941), the Science and Technology Program of Shaanxi Province (No. 2016KJXX‐87), and the Foundation of Shaanxi Provincial Department of Education (No. 18JS050)