Convective and conductive selection criteria of a stable dendritic growth and their stitching

Toropova, Liubov V.; Alexandrov, Dmitri V.; Galenko, Peter

doi:10.1002/mma.6830

Cited by 8 publications

(7 citation statements)

References 48 publications

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“…To find this velocity independently, we need to use the selection/stability theory. The criterion of stable dendritic growth in the case of convective-type boundary conditions takes the form of [37,41] right left right left right left right left right left right left3pt0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278emσ∗=σconv∗=2d0DTρ2V=σ0nαd5/nAn5/n(1+b1DTβ1)()1+μnτfalse~1nn+52(n−1)21+ν1nαd3/nAn3/nρb1+3αd1/4An2/nPgβ1DT21/4d02+2…”

Section: Intense Fluid Flows: Different Interfacial Boundary Conditionsmentioning

confidence: 99%

“…Note that the linear analysis of stability in the case of convective boundary conditions is carried out similarly to the stability analysis with the conductive boundary conditions (see §4). To save space, we will not dwell on this theory in detail here but refer the interested reader to the articles [37,41]. In the next subsection, we present only the basic laws governing the stable mode of dendritic growth under convective boundary conditions (7.1) and (7.2).…”

Section: Intense Fluid Flows: Different Interfacial Boundary Conditionsmentioning

confidence: 99%

“…This combination for a purely thermal problem with the fourfold crystalline symmetry was derived by Langer & Hong [23] in the form of σ∗=σ0αd7/4, where αd and σ0 represent the surface energy stiffness and a selection constant. Then this criterion was generalized to binary and multicomponent systems [24,25], forced and potential convective flows [26,27], arbitrary Péclet numbers [28], high pressure at Earth’s inner core boundary [29], attachment kinetics of atoms at the solid/liquid interface [30], sixfold crystalline symmetry [31,32], rapid crystallization [33–35], arbitrary crystalline symmetry [36,37], non-axisymmetric tip shape in the form of elliptical paraboloid [38,39], and natural convection around dendrites [37,40,41]. This review summarizes these studies and pays the most attention to the stable mode of dendritic growth, which is selected by the selection/stability parameter and the melt undercooling condition.…”

Section: Introductionmentioning

confidence: 99%

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A review on the theory of stable dendritic growth

Alexandrov

Galenko

2021

Phil. Trans. R. Soc. A.

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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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Section: Intense Fluid Flows: Different Interfacial Boundary Conditionsmentioning

confidence: 99%

Section: Intense Fluid Flows: Different Interfacial Boundary Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A review on the theory of stable dendritic growth

Alexandrov

Galenko

2021

Phil. Trans. R. Soc. A.

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“…Following references [47,48], let us write down the twodimensional selection criterion that describes the stable crystal growth mode with conductive heat and mass transfer boundary conditions at the dendrite surface. Considering the case of low and moderate velocities (Péclet numbers [32]), we have…”

Section: Stability Criterion For An Elliptical Paraboloid: Low and Mo...mentioning

confidence: 99%

Dendrite tips as elliptical paraboloids

Alexandrov¹,

Титова²,

Galenko³

et al. 2021

J. Phys.: Condens. Matter

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

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“…Using these algebraic equations, it is possible to independently determine the velocity and radius of curvature of the dendritic tip depending on the system undercooling (the driving force of dendritic growth). Then this theory was extended to binary melts [10,11], dendritic growth under conditions of convective fluid flow [12][13][14][15][16][17], and also to processes of local nonequilibrium (rapid) crystallization [18][19][20]. 3D phase-field simulation of non-axisymmetric ice dendrite growth [21,22].…”

Section: Introductionmentioning

confidence: 99%

Dendritic growth of ice crystals: a test of theory with experiments

Toropova

Титова

Alexandrov

et al. 2021

J. Phys.: Condens. Matter

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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.

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Convective and conductive selection criteria of a stable dendritic growth and their stitching

Cited by 8 publications

References 48 publications

A review on the theory of stable dendritic growth

A review on the theory of stable dendritic growth

Dendrite tips as elliptical paraboloids

Dendritic growth of ice crystals: a test of theory with experiments

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