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Alexandrov DV, Galenko PK. A review on the theory of stable dendritic growth. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20200325. [PMID: 34275358 DOI: 10.1098/rsta.2020.0325] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 03/13/2021] [Indexed: 06/13/2023]
Abstract
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)'.
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Affiliation(s)
- Dmitri V Alexandrov
- Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, Ural Federal University, Ekaterinburg, 620000, Russian Federation
| | - Peter K Galenko
- Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, Ural Federal University, Ekaterinburg, 620000, Russian Federation
- Physikalisch-Astronomische Fakultät, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany
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Kao A, Toropova LV, Alexandrov DV, Demange G, Galenko PK. Modeling of dendrite growth from undercooled nickel melt: sharp interface model versus enthalpy method. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:194002. [PMID: 31931496 DOI: 10.1088/1361-648x/ab6aea] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
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.
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Affiliation(s)
- A Kao
- Centre for Numerical Modelling and Process Analysis, University of Greenwich, Old Royal Naval College, Park Row, London SE10 9LS, United Kingdom
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Selection Criterion of Stable Mode of Dendritic Growth with n-Fold Symmetry at Arbitrary Péclet Numbers with a Forced Convection. ACTA ACUST UNITED AC 2019. [DOI: 10.1007/978-3-030-13720-5_17] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register]
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Microstructures Evolution and Micromechanics Features of Ni-Cr-Si Coatings Deposited on Copper by Laser Cladding. MATERIALS 2018; 11:ma11060875. [PMID: 29882889 PMCID: PMC6025212 DOI: 10.3390/ma11060875] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/30/2018] [Revised: 04/28/2018] [Accepted: 04/29/2018] [Indexed: 12/03/2022]
Abstract
Three Ni-Cr-Si coatings were synthesized on the surface of copper by laser cladding. The microstructures of the coatings were characterized by optical microscopy (OM), X-ray diffraction (XRD), and scanning electron microscopy (SEM) with an energy dispersive spectrometer (EDS). According to the analysis results of phase compositions, Gibbs free energy change and microstructures, the phases of three coatings appeared were Cr3Si+γ-Ni+Cuss (Coating 1, Ni-26Cr-29Si), Cr6Ni16Si7+Ni2Si+Cuss (Coating 2, Ni-10Cr-30Si) and Cr3Ni5Si2+Cr2Ni3+Cuss (Coating 3, Ni-29Cr-16Si). The crystal growth in the solidification process was analyzed with a modified model, which is a combination of Kurz-Giovanola-Trivedi (KGT) and Lipton-Kurz-Trivedi (LKT) models. The dendrite tip undercooling in Coating 2 was higher than those of Coating 1 and Coating 3. Well-developed dendrites were found in Coating 2. A modification of Hunt’s model was adopted to describe the morphological differences in the three coatings. The results show that Coating 1 was in the equiaxed dendrite region, while Coatings 2 and 3 were in the columnar dendrite region. The average friction coefficients of the three coatings were 0.45, 0.5 and 0.4, respectively. Obvious plastic deformation could be found in the subsurface zone of Coatings 2 and 3.
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Galenko PK, Nizovtseva IG, Reuther K, Rettenmayr M. Kinetic transition in the order-disorder transformation at a solid/liquid interface. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2018; 376:rsta.2017.0207. [PMID: 29311206 PMCID: PMC5784098 DOI: 10.1098/rsta.2017.0207] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 09/04/2017] [Indexed: 05/25/2023]
Abstract
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. JETP26, 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'.
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Affiliation(s)
- P K Galenko
- Otto-Schott-Institut für Materialforschung, Physikalisch-Astronomische Fakultät, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany
| | - I G Nizovtseva
- Otto-Schott-Institut für Materialforschung, Physikalisch-Astronomische Fakultät, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany
- Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modelling, Ural Federal University, Ekaterinburg 620000, Russian Federation
| | - K Reuther
- Otto-Schott-Institut für Materialforschung, Physikalisch-Astronomische Fakultät, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany
| | - M Rettenmayr
- Otto-Schott-Institut für Materialforschung, Physikalisch-Astronomische Fakultät, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany
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Jou D, Galenko PK. Coarse-graining for fast dynamics of order parameters in the phase-field model. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2018; 376:rsta.2017.0203. [PMID: 29311202 PMCID: PMC5784094 DOI: 10.1098/rsta.2017.0203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 09/07/2017] [Indexed: 05/25/2023]
Abstract
In standard descriptions, the master equation can be obtained by coarse-graining with the application of the hypothesis of full local thermalization that is equivalent to the local thermodynamic equilibrium. By contrast, fast transformations proceed in the absence of local equilibrium and the master equation must be obtained with the absence of thermalization. In the present work, a non-Markovian master equation leading, in specific cases of relaxation to local thermodynamic equilibrium, to hyperbolic evolution equations for a binary alloy, is derived for a system with two order parameters. One of them is a conserved order parameter related to the atomistic composition, and the other one is a non-conserved order parameter, which is related to phase field. A microscopic basis for phenomenological phase-field models of fast phase transitions, when the transition is so fast that there is not sufficient time to achieve local thermalization between two successive elementary processes in the system, is provided. In a particular case, when the relaxation to local thermalization proceeds by the exponential law, the obtained coarse-grained equations are related to the hyperbolic phase-field model. The solution of the model equations is obtained to demonstrate non-equilibrium phenomenon of solute trapping which appears in rapid growth of dendritic crystals.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.
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Affiliation(s)
- D Jou
- Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain
| | - P K Galenko
- Physikalisch-Astronomische Fakultät, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany
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Herlach DM, Palberg T, Klassen I, Klein S, Kobold R. Overview: Experimental studies of crystal nucleation: Metals and colloids. J Chem Phys 2016; 145:211703. [DOI: 10.1063/1.4963684] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Dieter M. Herlach
- Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany
| | - Thomas Palberg
- Institut für Physik, Johannes Gutenberg Universität Mainz, 55099 Mainz, Germany
| | - Ina Klassen
- Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany
- Projektträger Jülich, Forschungszentrum Jülich GmbH, 52425 Jülich, Germany
| | - Stefan Klein
- Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany
| | - Raphael Kobold
- Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany
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Galenko P. Solute trapping and diffusionless solidification in a binary system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:031606. [PMID: 17930255 DOI: 10.1103/physreve.76.031606] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2007] [Indexed: 05/11/2023]
Abstract
Numerous experimental data on the rapid solidification of binary systems exhibit the formation of metastable solid phases with initial (nominal) chemical composition. This fact is explained by complete solute trapping leading to diffusionless (chemically partitionless) solidification at a finite growth velocity of crystals. Special attention is paid to developing a model of rapid solidification which describes a transition from chemically partitioned to diffusionless growth of crystals. Analytical treatments lead to the condition for complete solute trapping which directly follows from the analysis of the solute diffusion around the solid-liquid interface and atomic attachment and detachment at the interface. The resulting equations for the flux balance at the interface take into account two kinetic parameters: diffusion speed VDI on the interface and diffusion speed VD in bulk phases. The model describes experimental data on nonequilibrium solute partitioning in solidification of Si-As alloys for the whole range of solidification velocity investigated.
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Affiliation(s)
- Peter Galenko
- German Aerospace Center, Institute of Materials Physics in Space, Cologne 51170, Germany.
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Fan J, Greenwood M, Haataja M, Provatas N. Phase-field simulations of velocity selection in rapidly solidified binary alloys. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:031602. [PMID: 17025638 DOI: 10.1103/physreve.74.031602] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/11/2006] [Revised: 06/13/2006] [Indexed: 05/12/2023]
Abstract
Time-dependent simulations of two-dimensional isothermal Ni-Cu dendrites are simulated using a phase-field model solved with a finite-difference adaptive mesh refinement technique. Dendrite tip velocity selection is examined and found to exhibit a transition between two markedly different regimes as undercooling is increased. At low undercooling, the dendrite tip growth rate is consistent with the kinetics of the classical Stefan problem, where the interface is assume to be in local equilibrium. At high undercooling, the growth velocity selected approaches a linear dependence on melt undercooling, consistent with the continuous growth kinetics of Aziz and with a one-dimensional steady-state phase-field asymptotic analysis of Ahmad [Phys. Rev. E 58, 3436 (1998)]. Our simulations are also consistent with other previously observed behaviors of dendritic growth as undercooling is increased. These include the transition of dendritic morphology to absolute stability and nonequilibrium solute partitioning. Our results show that phase-field models of solidification, which inherently contain a nonzero interface width, can be used to study the dynamics of complex solidification phenomena involving both equilibrium and nonequilibrium interface growth kinetics.
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Affiliation(s)
- Jun Fan
- Department of Materials Science and Engineering and Brockhouse Institute for Materials Research, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L7
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Galenko PK, Danilov DA. Linear morphological stability analysis of the solid-liquid interface in rapid solidification of a binary system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:051608. [PMID: 15244830 DOI: 10.1103/physreve.69.051608] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2003] [Indexed: 05/24/2023]
Abstract
The interface stability against small perturbations of the planar solid-liquid interface is considered analytically in linear approximation. Following the analytical procedure of Trivedi and Kurz [Acta Metall. 34, 1663 (1986)]], which is advancing the original treatment of morphological stability by Mullins and Sekerka [J. Appl. Phys. 35, 444 (1964)]] to the case of rapid solidification, we extend the model by introducing the local nonequilibrium in the solute diffusion field around the interface. A solution to the heat- and mass-transport problem around the perturbed interface is given in the presence of the local nonequilibrium solute diffusion. Using the developing local nonequilibrium model of solidification, the self-consistent analysis of linear morphological stability is presented with the attribution to the marginal (neutral) and absolute morphological stability of a rapidly moving interface. Special consideration of the interface stability for the cases of solidification in negative and positive thermal gradients is given. A quantitative comparison of the model predictions for the absolute morphological stability is presented with regard to experimental results of Hoglund and Aziz [D. E. Hoglund and M. J. Aziz, in Kinetics of Phase Transformations, edited by M.O. Thompson, M. J. Aziz, and G. B. Stephenson, MRS Symposia Proceedings No. 205 (Materials Research Society, Pittsburgh, 1991), p. 325] on critical solute concentration for the interface breakdown during rapid solidification of Si-Sn alloys.
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Affiliation(s)
- P K Galenko
- Institute of Space Simulation, German Aerospace Center, Cologne 51170, Germany.
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Cao C, Lu X, Wei B. Peritectic solidification under high undercooling conditions. CHINESE SCIENCE BULLETIN 1999. [DOI: 10.1007/bf02885858] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Li D, Herlach DM. Direct Measurements of Free Crystal Growth in Deeply Undercooled Melts of Semiconducting Materials. PHYSICAL REVIEW LETTERS 1996; 77:1801-1804. [PMID: 10063175 DOI: 10.1103/physrevlett.77.1801] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
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Barth M, Wei B, Herlach DM. Crystal growth in undercooled melts of the intermetallic compounds FeSi and CoSi. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:3422-3428. [PMID: 9979149 DOI: 10.1103/physrevb.51.3422] [Citation(s) in RCA: 86] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Schwarz M, Karma A, Eckler K, Herlach DM. Physical mechanism of grain refinement in solidification of undercooled melts. PHYSICAL REVIEW LETTERS 1994; 73:1380-1383. [PMID: 10056778 DOI: 10.1103/physrevlett.73.1380] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
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Wei B, Herlach D, Feuerbacher B, Sommer F. Dendritic and eutectic solidification of undercooled CoSb alloys. ACTA ACUST UNITED AC 1993. [DOI: 10.1016/0956-7151(93)90200-c] [Citation(s) in RCA: 61] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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