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13 pages, 11431 KiB

Open AccessArticle

Pulsed and Static Magnetic Field Influence on Metallic Alloys during Solidification

by Mikus Milgrāvis, Ivars Krastiņš, Imants Kaldre, Matīss Kalvāns, Andris Bojarevičs and Toms Beinerts

Crystals 2023, 13(2), 259; https://doi.org/10.3390/cryst13020259 - 2 Feb 2023

Cited by 5 | Viewed by 2230

Abstract

Electromagnetic methods can be used to affect the solidification of metallic alloys. Combined alternating and static (DC) magnetic fields can induce pressure waves in liquid metals in a contactless manner, refining grain structure, preventing component segregation, and dispersing added particles. Here, the possibility [...] Read more.

Electromagnetic methods can be used to affect the solidification of metallic alloys. Combined alternating and static (DC) magnetic fields can induce pressure waves in liquid metals in a contactless manner, refining grain structure, preventing component segregation, and dispersing added particles. Here, the possibility of using a pulsed magnetic field in combination with a DC magnetic field to improve the solidification structure of metals is investigated. This enables achieving higher pressure amplitude and liquid movement within the whole crucible volume at the same power consumption. This can be a prospective way to achieve a fine-grained and a more homogeneous microstructure of metallic alloys as well as disperse added particles. In this work, analytical description of the phenomena and numerical modeling of the pressure amplitude and melt motion has been performed. A series of experiments have been carried out to demonstrate the effect of such interaction on the solidified metal microstructure. It is found that a combination of pulsed and DC magnetic fields creates strong compression and expansion of the liquid metal. Metals solidified under such interaction exhibit finer grain structure in Sn-Pb alloy and improved TiB₂ particle distribution in 6061 aluminum alloy. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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15 pages, 18172 KiB

Open AccessArticle

The Impact of Two and Three Dimensional Assumptions on Coupled Structural Mechanics and Microstructure Solidification Modelling

by Peter Soar, Andrew Kao and Koulis Pericleous

Crystals 2023, 13(1), 114; https://doi.org/10.3390/cryst13010114 - 8 Jan 2023

Cited by 2 | Viewed by 1451

Abstract

It is usual for computational efficiency to simulate growing alloy dendrites during solidification using a two-dimensional model. However, the fidelity of such simulations is to be questioned, since observations show that three-dimensional models lead to significantly more realistic results in comparison to experiments [...] Read more.

It is usual for computational efficiency to simulate growing alloy dendrites during solidification using a two-dimensional model. However, the fidelity of such simulations is to be questioned, since observations show that three-dimensional models lead to significantly more realistic results in comparison to experiments under many situations. Even in thin sample cases, the properties affecting, for example, mechanical behaviour are intrinsically three-dimensional. However, partly due to the lack of published work on the, topic the impact of 2D assumptions on the evolution and structural mechanical behaviour of dendrites has not been properly explored. In this study, solidification using the Cellular Automata (CA) method was coupled to a Finite Volume Structural Mechanics Solver (FVSMS) capable of both 2D and 3D modelling, applied to a selection of representative problems which clearly demonstrate that structural mechanics is another factor in the modelling of dendrites where two-dimensional assumptions can lead to significantly altered behaviour when compared to three-dimensional reality. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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13 pages, 438 KiB

Open AccessArticle

Non-Fermi Liquid Behavior in the Three-Dimensional Hubbard Model

by Samuel Kellar, Ka-Ming Tam and Juana Moreno

Crystals 2023, 13(1), 106; https://doi.org/10.3390/cryst13010106 - 6 Jan 2023

Cited by 1 | Viewed by 1714

Abstract

We present a numerical study on the non-Fermi liquid behavior of a three-dimensional strongly correlated system. The Hubbard model in a simple cubic lattice is simulated by the dynamical cluster approximation; in particular, the quasi-particle weight is calculated at finite dopings for a [...] Read more.

We present a numerical study on the non-Fermi liquid behavior of a three-dimensional strongly correlated system. The Hubbard model in a simple cubic lattice is simulated by the dynamical cluster approximation; in particular, the quasi-particle weight is calculated at finite dopings for a range of temperatures. By fitting the quasi-particle weight to the marginal Fermi liquid form at finite doping near the putative quantum critical point, we find evidence of a separatrix between Fermi liquid and non-Fermi liquid regions. Our results suggest that a marginal Fermi liquid and possibly a quantum critical point exist in the non-symmetry broken solution of the three-dimensional interacting electron systems. We also calculate the spectral function, close to the half-filling, and we obtain evidence of pseudogap. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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11 pages, 6108 KiB

Open AccessEditor’s ChoiceArticle

Observation of Pattern Formation during Electromagnetic Levitation Using High-Speed Thermography

by Yindong Fang, Dongmei Liu, Yongfu Zhu, Peter K. Galenko and Stephanie Lippmann

Crystals 2022, 12(12), 1691; https://doi.org/10.3390/cryst12121691 - 22 Nov 2022

Cited by 4 | Viewed by 2056

Abstract

Electromagnetic levitation (EML) was employed for studying the velocity and morphology of the solidification front as a function of undercooling of metallic materials. The limitation of the EML technique with respect to low melting alloys that emit outside the visible light spectrum was [...] Read more.

Electromagnetic levitation (EML) was employed for studying the velocity and morphology of the solidification front as a function of undercooling of metallic materials. The limitation of the EML technique with respect to low melting alloys that emit outside the visible light spectrum was overcome by employing state-of-the-art high-speed mid-wavelength infrared cameras (MWIR cameras) with a photon detector. Due to the additional thermography contrast provided by the emission contrast of the solid and liquid phases, conductor, and semi-conductor, the pattern formation of Al-based alloys was studied in detail, revealing information on the nucleation, phase selection during solidification, and the influence of convection. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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11 pages, 367 KiB

Open AccessArticle

Anomalous Dynamics of Recalescence Front in Crystal Growth Processes: Theoretical Background

by Dmitri V. Alexandrov, Peter K. Galenko and Liubov V. Toropova

Crystals 2022, 12(12), 1686; https://doi.org/10.3390/cryst12121686 - 22 Nov 2022

Cited by 3 | Viewed by 1405

Abstract

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 [...] Read more.

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)]. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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12 pages, 498 KiB

Open AccessArticle

Mathematical Modeling of the Solid–Liquid Interface Propagation by the Boundary Integral Method with Nonlinear Liquidus Equation and Atomic Kinetics

by Ekaterina A. Titova, Dmitri V. Alexandrov and Liubov V. Toropova

Crystals 2022, 12(11), 1657; https://doi.org/10.3390/cryst12111657 - 17 Nov 2022

Cited by 2 | Viewed by 1456

Abstract

In this paper, we derive the boundary integral equation (BIE), a single integrodifferential equation governing the evolutionary behavior of the interface function, paying special attention to the nonlinear liquidus equation and atomic kinetics. As a result, the BIE is found for a thermodiffusion [...] Read more.

In this paper, we derive the boundary integral equation (BIE), a single integrodifferential equation governing the evolutionary behavior of the interface function, paying special attention to the nonlinear liquidus equation and atomic kinetics. As a result, the BIE is found for a thermodiffusion problem of binary melt crystallization with convection. Analyzing this equation coupled with the selection criterion for a stationary dendritic growth in the form of a parabolic cylinder, we show that nonlinear effects stemming from the liquidus equation and atomic kinetics play a decisive role. Namely, the dendrite tip velocity and diameter, respectively, become greater and lower with the increasing deviation of the liquidus equation from a linear form. In addition, the dendrite tip velocity can substantially change with variations in the power exponent of the atomic kinetics. In general, the theory under consideration describes the evolution of a curvilinear crystallization front, as well as the growth of solid phase perturbations and patterns in undercooled binary melts at local equilibrium conditions (for low and moderate Péclet numbers). In addition, our theory, combined with the unsteady selection criterion, determines the non-stationary growth rate of dendritic crystals and the diameter of their vertices. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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10 pages, 323 KiB

Open AccessArticle

On the Theory of Unsteady-State Operation of Bulk Continuous Crystallization

by Eugenya V. Makoveeva, Dmitri V. Alexandrov and Alexander A. Ivanov

Crystals 2022, 12(11), 1634; https://doi.org/10.3390/cryst12111634 - 14 Nov 2022

Cited by 12 | Viewed by 1602

Abstract

Motivated by an important application in the chemical and pharmaceutical industries, we consider the non-stationary growth of a polydisperse ensemble of crystals in a continuous crystallizer. The mathematical model includes the effects of crystal nucleation and growth, fines dissolution, mass influx and withdrawal [...] Read more.

Motivated by an important application in the chemical and pharmaceutical industries, we consider the non-stationary growth of a polydisperse ensemble of crystals in a continuous crystallizer. The mathematical model includes the effects of crystal nucleation and growth, fines dissolution, mass influx and withdrawal of product crystals. The steady- and unsteady-state solutions of kinetic and balance equations are analytically derived. The steady-state solution is found in an explicit form and describes the stationary operation mode maintained by the aforementioned effects. An approximate unsteady-state solution is found in a parametric form and describes a time-dependent crystallization scenario, which tends toward the steady-state mode when time increases. It is shown that the particle-size distribution contains kinks at the points of fines dissolution and product crystal withdrawal. Additionally, our calculations demonstrate that the unsteady-state crystal-size distribution has a bell-shaped profile that blurs with time due to the crystal growth and removal mechanisms. The analytical solutions found are the basis for investigating the dynamic stability of a continuous crystallizer. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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9 pages, 5691 KiB

Open AccessArticle

The Tip of Dendritic Crystal in an Inclined Viscous Flow

by Ilya O. Starodumov, Ekaterina A. Titova, Eugeny V. Pavlyuk and Dmitri V. Alexandrov

Crystals 2022, 12(11), 1590; https://doi.org/10.3390/cryst12111590 - 8 Nov 2022

Cited by 6 | Viewed by 1562

Abstract

We study the flow around the tip of a dendritic crystal by an inclined stream of viscous incompressible liquid. The tip shape is chosen accordingly to recent theory [Phil. Trans. R. Soc. A 2020, 378, 20190243] confirmed by a [...] Read more.

We study the flow around the tip of a dendritic crystal by an inclined stream of viscous incompressible liquid. The tip shape is chosen accordingly to recent theory [Phil. Trans. R. Soc. A 2020, 378, 20190243] confirmed by a number of experiments and computations [Phil. Trans. R. Soc. A 2021, 379, 20200326]. Our simulations have been carried out for a 0, 30, 60, and 90-degree flow slope to the dendrite axis. We show that the stream inclination has a significant effect on the hydrodynamic flow and shear stress. In particular, a transition from laminar to turbulent currents on the upstream side of the dendritic crystal may occur in an inclined hydrodynamic flow. This leads to the fact that the heat and mass transfer mechanisms on the upstream and downstream sides of a growing dendritic crystal may be different. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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11 pages, 512 KiB

Open AccessArticle

Variational Theory of Crystal Growth in Multicomponent Alloys

by Maxim V. Dudorov, Alexander D. Drozin and Vasiliy E. Roshchin

Crystals 2022, 12(11), 1522; https://doi.org/10.3390/cryst12111522 - 26 Oct 2022

Cited by 4 | Viewed by 1422

Abstract

The provisions for a new variational theory of crystal growth in multicomponent metal melts were formulated. The developed theory is the generalization of the previously conducted studies of crystal growth under conditions of deviation from local equilibrium at the phase boundary. The description [...] Read more.

The provisions for a new variational theory of crystal growth in multicomponent metal melts were formulated. The developed theory is the generalization of the previously conducted studies of crystal growth under conditions of deviation from local equilibrium at the phase boundary. The description of the methods of non-equilibrium thermodynamics of interrelated physico-chemical processes occurring in the initial phase, on the interface of phases and inside the growing crystal, was compared with the variational description of the crystal growth as a macrobody. The developed approach made it possible to find the general expression for the crystal growth rate, considering the influence of thermal and diffusion processes, as well as taking into account the influence of nonstationary effects associated with deviation from the local equilibrium on the surface of the growing nucleus. The justification of the new method showed that when the condition of the local equilibrium on the surface of the growing crystal is satisfied, the resulting equations take the form of expressions that can be obtained by constructing the equation of a mass and internal energy balance for the system under consideration. As an example, the problem of crystal growth from a melt of eutectic composition was considered. The equation of the growth rate of the two-component nucleus of the stoichiometric composition was obtained, taking into account the influence of the local non-equilibrium effects on growth. The expressions obtained were compared with the known equations of the solute trapping theory. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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12 pages, 575 KiB

Open AccessArticle

Mathematical Modeling of Nucleation and Growth Processes of Ellipsoidal Crystals in Binary Melts

by Margarita A. Nikishina and Dmitri V. Alexandrov

Crystals 2022, 12(10), 1495; https://doi.org/10.3390/cryst12101495 - 21 Oct 2022

Viewed by 1723

Abstract

The transient behavior of an ensemble of ellipsoidal particles in a supercooled binary melt is considered. The model laws, based on the Fokker-Planck type kinetic equation for the particle-volume distribution function, the thermal and mass integral balances for the binary melt temperature and [...] Read more.

The transient behavior of an ensemble of ellipsoidal particles in a supercooled binary melt is considered. The model laws, based on the Fokker-Planck type kinetic equation for the particle-volume distribution function, the thermal and mass integral balances for the binary melt temperature and solute concentration, as well as the corresponding boundary and initial conditions, are formulated and solved analytically. We show that the temperature and concentration increase with time due to the effects of impurity displacement and latent heat emission by the growing ellipsoidal particles. These effects are also responsible for metastability reduction. As this takes place, increasing the initial solute concentration in a metastable binary melt increases the intensity of its desupercooling. The theory is developed for arbitrary nucleation frequency with special consideration of two important nucleation kinetics according to the Meirs and Weber-Volmer-Frenkel Zel’dovich mechanisms. An analytical solution to the integrodifferential model equations is found in a parametric form. The theory contains all limiting transitions to previously developed analytical approaches. Namely, it contains the growth of spherical crystals in binary melts and ellipsoidal crystals in single-component melts. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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11 pages, 3449 KiB

Open AccessArticle

Numerical Studies of Batch and Inline High Shear Melt Conditioning Technology Using Different Rotors

by Gerard Serge Bruno Lebon, Jayesh B. Patel and Zhongyun Fan

Crystals 2022, 12(9), 1299; https://doi.org/10.3390/cryst12091299 - 14 Sep 2022

Viewed by 1711

Abstract

When casting aluminum alloy billets, high shear melt conditioning (HSMC) technology refines the resulting grain size, reduces the number of defects, and improves mechanical properties without the need to add polluting and expensive chemical grain refiners. These resultant improvements spring from the high [...] Read more.

When casting aluminum alloy billets, high shear melt conditioning (HSMC) technology refines the resulting grain size, reduces the number of defects, and improves mechanical properties without the need to add polluting and expensive chemical grain refiners. These resultant improvements spring from the high shear rates that develop in the rotor–stator gap and the stator holes facing the leading edge of the rotor. Despite the growing literature on rotor–stator mixing, it is unclear how the different rotor–stator parameters affect the performance of high shear treatment. To upscale this technology and apply it to processes that involve large melt volumes, an understanding of the performance of the rotor–stator design is crucial. In this paper, we present the results of computational fluid dynamics (CFD) studies of high shear melt conditioning in continuous and batch modes with different rotor designs. These studies build upon our earlier work by studying the effect of rotor variation in a stator design consisting of rows of small apertures at different rotor speeds spanning from 1000 to 10,000 revolutions per minute. While no clear-cut linear pattern emerges for the rotor performance (as a function of the design parameters), the rotor geometry is found to affect the distributive mixing of microparticles, but it is insignificant with regards to their disintegration. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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12 pages, 2443 KiB

Open AccessArticle

Structural Characterization and Dynamics of a Layered 2D Perovskite [NH₃(CH₂)₅NH₃]MnCl₄ Crystal near Phase Transition Temperature

by Ae Ran Lim and Yeji Na

Crystals 2022, 12(9), 1298; https://doi.org/10.3390/cryst12091298 - 14 Sep 2022

Cited by 2 | Viewed by 1988

Abstract

[NH₃(CH₂)₅NH₃]MnCl₄ crystals are grown via slow evaporation, and the crystal undergoes a phase transition at 298 K (T_C) according to differential scanning calorimetry, and the structures determined via X-ray diffraction at 173 [...] Read more.

[NH₃(CH₂)₅NH₃]MnCl₄ crystals are grown via slow evaporation, and the crystal undergoes a phase transition at 298 K (T_C) according to differential scanning calorimetry, and the structures determined via X-ray diffraction at 173 and 333 K are orthorhombic systems in the space group Imma. These results differed slightly from those previously reported, and the reasons for this are analyzed. The thermal stability is relatively high, with a thermal decomposition temperature of approximately 570 K. The ¹H spin-lattice relaxation times t_1ρ exhibited very large variations, as indicated by the large thermal displacement around the ¹H atoms, suggesting energy transfer at ~T_C, even if no structural changes occurred. The influences of the chemical shifts of ¹H of NH₃ and short t_1ρ of C1 adjacent to NH₃ in cation are insignificant, indicating a minor change in the N−H⋯Cl hydrogen bond related to the coordination geometry of the MnCl₆ octahedron. These properties will be make it a potential application for eco-friendly solar cells. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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12 pages, 634 KiB

Open AccessEditor’s ChoiceArticle

Selection Criterion of Stable Dendritic Growth for a Ternary (Multicomponent) Melt with a Forced Convective Flow

by Dmitri V. Alexandrov, Sergei I. Osipov, Peter K. Galenko and Liubov V. Toropova

Crystals 2022, 12(9), 1288; https://doi.org/10.3390/cryst12091288 - 12 Sep 2022

Cited by 1 | Viewed by 1763

Abstract

A stable growth mode of a single dendritic crystal solidifying in an undercooled ternary (multicomponent) melt is studied with allowance for a forced convective flow. The steady-state temperature, solute concentrations and fluid velocity components are found for two- and three-dimensional problems. The stability [...] Read more.

A stable growth mode of a single dendritic crystal solidifying in an undercooled ternary (multicomponent) melt is studied with allowance for a forced convective flow. The steady-state temperature, solute concentrations and fluid velocity components are found for two- and three-dimensional problems. The stability criterion and the total undercooling balance are derived accounting for surface tension anisotropy at the solid-melt interface. The theory under consideration is compared with experimental data and phase-field modeling for Ni₉₈Zr₁Al₁ alloy. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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15 pages, 564 KiB

Open AccessArticle

Analysis of Smoluchowski’s Coagulation Equation with Injection

by Eugenya V. Makoveeva, Dmitri V. Alexandrov and Sergei P. Fedotov

Crystals 2022, 12(8), 1159; https://doi.org/10.3390/cryst12081159 - 17 Aug 2022

Cited by 11 | Viewed by 2633

Abstract

The stationary solution of Smoluchowski’s coagulation equation with injection is found analytically with different exponentially decaying source terms. The latter involve a factor in the form of a power law function that plays a decisive role in forming the steady-state particle distribution shape. [...] Read more.

The stationary solution of Smoluchowski’s coagulation equation with injection is found analytically with different exponentially decaying source terms. The latter involve a factor in the form of a power law function that plays a decisive role in forming the steady-state particle distribution shape. An unsteady analytical solution to the coagulation equation is obtained for the exponentially decaying initial distribution without injection. An approximate unsteady solution is constructed by stitching the initial and final (steady-state) distributions. The obtained solutions are in good agreement with experimental data for the distributions of endocytosed low-density lipoproteins. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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13 pages, 6328 KiB

Open AccessFeature PaperArticle

Compositional Glass: A State with Inherent Chemical Disorder, Exemplified by Ti-rich Ni₃(Al,Ti)₁ D0₂₄ Phase

by Nikolai A. Zarkevich, Timothy M. Smith, Eli N. Baum and John W. Lawson

Crystals 2022, 12(8), 1049; https://doi.org/10.3390/cryst12081049 - 28 Jul 2022

Cited by 4 | Viewed by 2063

Abstract

A compositional glass is a state with an unavoidable disorder in chemical compositions on each site, characterized by frustration and freezing of the compositional degrees of freedom at low temperature. From this state a full atomic long-range order is unachievable by a reasonable [...] Read more.

A compositional glass is a state with an unavoidable disorder in chemical compositions on each site, characterized by frustration and freezing of the compositional degrees of freedom at low temperature. From this state a full atomic long-range order is unachievable by a reasonable thermodynamic treatment. There is a similarity between a spin glass (a magnetic state with disorder in spin orientations) and a compositional glass (with disorder in site occupations by chemical elements): both have frustrated ground states and a frozen disorder at low temperatures T < T_f (here T_f is called the freezing temperature). While it is possible to perform a ground-state search in a compositional glass, the resulting set of the fully ordered structures does not adequately represent the real solid with an inherent atomic disorder. Compositional glasses constitute a class of materials, which is insufficiently understood, but is of high industrial importance. Some of the phases in the precipitated alloys (including steels, high-entropy alloys, and superalloys) might be compositional glasses, and their better understanding would facilitate materials design. Due to their strength at high operating temperatures, superalloys are used in combustion engines and particularly in jet turbine engines. Precipitation strengthening of nickel superalloys is an area of active research. Local phase transformations inside Ni₃Al-based precipitates are of particular interest due to their impact on creep strength. In the Ni₃(Al_1−xTi_x)₁ ternary system, the competing phases are Ni₃Al-type L1₂ (γʹ) and Ni₃Ti-type D0₂₄ (η), while D0₁₉ (χ) is higher in energy. These three phases differ by the stacking of atomic layers: locally, the last two look like the internal and external stacking faults in L1₂. We compute enthalpies of disordered and ordered Ni₃(Al_1−xTi_x)₁ ternary structures, examine phase stability, investigate the ground states and competing structures, and predict that the Ti-rich Ni₃(Al_1−xTi_x)₁ D0₂₄ phase is a compositional glass with the atomic disorder on the Al/Ti sublattice. To resolve apparent contradictions among the previous experiments and to confirm our prediction, we perform X-ray diffraction and scanning electron microscopy analysis of the cast Ni₃(Ti_0.917Al_0.083)₁ sample. Our measurements appear to confirm the ab initio computed results. Our results elucidate properties of compositional glasses and provide a better understanding of precipitation strengthening mechanisms in Ni superalloys. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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Review

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15 pages, 1145 KiB

Open AccessEditor’s ChoiceReview

A Stable Mode of Dendritic Growth in Cases of Conductive and Convective Heat and Mass Transfer

by Liubov V. Toropova, Peter K. Galenko and Dmitri V. Alexandrov

Crystals 2022, 12(7), 965; https://doi.org/10.3390/cryst12070965 - 11 Jul 2022

Cited by 8 | Viewed by 3074

Abstract

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, [...] Read more.

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 Al

_{24}

Ge

_{76}

and Ti

_{45}

Al

_{55}

undercooled melts. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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20 pages, 562 KiB

Open AccessReview

Evolution of a Polydisperse Ensemble of Spherical Particles in a Metastable Medium with Allowance for Heat and Mass Exchange with the Environment

by Dmitri V. Alexandrov, Alexander A. Ivanov, Irina G. Nizovtseva, Stephanie Lippmann, Irina V. Alexandrova and Eugenya V. Makoveeva

Crystals 2022, 12(7), 949; https://doi.org/10.3390/cryst12070949 - 6 Jul 2022

Cited by 22 | Viewed by 1950

Abstract

Motivated by a wide range of applications in various fields of physics and materials science, we consider a generalized approach to the evolution of a polydisperse ensemble of spherical particles in metastable media. An integrodifferential system of governing equations, consisting of a kinetic [...] Read more.

Motivated by a wide range of applications in various fields of physics and materials science, we consider a generalized approach to the evolution of a polydisperse ensemble of spherical particles in metastable media. An integrodifferential system of governing equations, consisting of a kinetic equation for the particle-size distribution function (Fokker–Planck type equation) and a balance equation for the temperature (concentration) of a metastable medium, is formulated. The kinetic equation takes into account fluctuations in the growth/reduction rates of individual particles, the velocity of particles in a spatial direction, the withdrawal of particles of a given size from the metastable medium, and their source/sink term. The heat (mass) balance equation takes into account the growth/reduction of particles in a metastable system as well as heat (mass) exchange with the environment. A generalized system of equations describes various physical and chemical processes of phase transformations, such as the growth and dissolution of crystals, the evaporation of droplets, the boiling of liquids and the combustion of a polydisperse fuel. The ways of analytical solution of the formulated integrodifferential system of equations based on the saddle-point technique and the separation of variables method are considered. The theory can be applied when describing the evolution of an ensemble of particles at the initial and intermediate stages of phase transformation when the distances between the particles are large enough, and interactions between them can be neglected. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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19 pages, 553 KiB

Open AccessReview

Nucleation and Growth of an Ensemble of Crystals during the Intermediate Stage of a Phase Transition in Metastable Liquids

by Liubov V. Toropova, Eugenya V. Makoveeva, Sergei I. Osipov, Alexey P. Malygin, Yang Yang and Dmitri V. Alexandrov

Crystals 2022, 12(7), 895; https://doi.org/10.3390/cryst12070895 - 24 Jun 2022

Cited by 8 | Viewed by 1690

Abstract

In this paper, an analytical method of solving the integro-differential system of kinetic and balance equations describing the evolution of an ensemble of crystals during the intermediate phase of the bulk crystallization process is described. The theory is developed for kinetic equations of [...] Read more.

In this paper, an analytical method of solving the integro-differential system of kinetic and balance equations describing the evolution of an ensemble of crystals during the intermediate phase of the bulk crystallization process is described. The theory is developed for kinetic equations of the first- and second order corresponding to the absence and presence of fluctuations in particle growth rates. The crystal-size distribution function as well as the dynamics of metastability reduction in a supercooled melt (supersaturated solution) are analytically found using the saddle-point and the Laplace transform methods. The theory enables us to obtain the crystal-size distribution function that establishes in a supercooled (supersaturated) liquid at the beginning of the final stage of a phase transformation process when Ostwald ripening, coagulation and fragmentation of crystals are able to occur. Full article

(This article belongs to the Special Issue Phase Transition in External Fields)

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