“…Note that expression (3.14) does not work for a three-dimensional crystal in the form of an elliptical paraboloid. In this case, a more general formula should be used [47].…”
Section: Thermo-solutal Dendritic Growth In An Undercooled Melt/solutionmentioning
confidence: 99%
“…The slight discrepancy between theory and modelling for a larger radius is explained by the fact that instead of the Ivantsov solutions for axisymmetric crystals, in this case, it is necessary to use the Horvey–Cahn solutions for non-axisymmetric ice crystals in the form of elliptical paraboloids [35,38]. Figure 5The tip radii in the basal (ρ6, solid line) and perpendicular (ρ2, dashed line) planes ( a ) and growth velocity ( b ) of a dendritic crystal according to the theory (expressions (3.11) and (4.18), lines) and phase-field simulations (symbols) for three-dimensional ice dendrites growing in pure water [47]. (Online version in colour.…”
Section: A Behaviour Of the Main Functionsmentioning
confidence: 99%
“…The tip radii in the basal (ρ6, solid line) and perpendicular (ρ2, dashed line) planes ( a ) and growth velocity ( b ) of a dendritic crystal according to the theory (expressions (3.11) and (4.18), lines) and phase-field simulations (symbols) for three-dimensional ice dendrites growing in pure water [47]. (Online version in colour.…”
Section: A Behaviour Of the Main Functionsmentioning
confidence: 99%
“…Note that expression (3.14) does not work for a three-dimensional crystal in the form of an elliptical paraboloid. In this case, a more general formula should be used [47]. The kinetic contribution T K is of primary importance at moderate and large values of solidification velocities V or melt undercoolings T. This contribution, which grows with increasing velocity, has a power-law dependence…”
Section: (Iii) the Gibbs-thomson And Kinetic Contributionsmentioning
confidence: 99%
“…Here, we have two tip radii: ρ 6 in the basal plane and ρ 2 < ρ 6 in the perpendicular plane. To calculate these radii, we apply the selection criterion (4.18) twice, in each of these planes (using the constant aspect ratio for ice crystals in the form of elliptical paraboloids [47]). As is easily seen, the tip radii and growth rate have the same behaviour: ρ 6 and ρ 2 decrease and V increases with increasing T. The slight discrepancy between theory and modelling for a larger radius is explained by the fact that instead of the Ivantsov solutions for axisymmetric crystals, in this case, it is necessary to use the Horvey-Cahn solutions for non-axisymmetric ice crystals in the form of elliptical paraboloids [35,38].…”
Section: A Behaviour Of the Main Functionsmentioning
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)’.
“…Note that expression (3.14) does not work for a three-dimensional crystal in the form of an elliptical paraboloid. In this case, a more general formula should be used [47].…”
Section: Thermo-solutal Dendritic Growth In An Undercooled Melt/solutionmentioning
confidence: 99%
“…The slight discrepancy between theory and modelling for a larger radius is explained by the fact that instead of the Ivantsov solutions for axisymmetric crystals, in this case, it is necessary to use the Horvey–Cahn solutions for non-axisymmetric ice crystals in the form of elliptical paraboloids [35,38]. Figure 5The tip radii in the basal (ρ6, solid line) and perpendicular (ρ2, dashed line) planes ( a ) and growth velocity ( b ) of a dendritic crystal according to the theory (expressions (3.11) and (4.18), lines) and phase-field simulations (symbols) for three-dimensional ice dendrites growing in pure water [47]. (Online version in colour.…”
Section: A Behaviour Of the Main Functionsmentioning
confidence: 99%
“…The tip radii in the basal (ρ6, solid line) and perpendicular (ρ2, dashed line) planes ( a ) and growth velocity ( b ) of a dendritic crystal according to the theory (expressions (3.11) and (4.18), lines) and phase-field simulations (symbols) for three-dimensional ice dendrites growing in pure water [47]. (Online version in colour.…”
Section: A Behaviour Of the Main Functionsmentioning
confidence: 99%
“…Note that expression (3.14) does not work for a three-dimensional crystal in the form of an elliptical paraboloid. In this case, a more general formula should be used [47]. The kinetic contribution T K is of primary importance at moderate and large values of solidification velocities V or melt undercoolings T. This contribution, which grows with increasing velocity, has a power-law dependence…”
Section: (Iii) the Gibbs-thomson And Kinetic Contributionsmentioning
confidence: 99%
“…Here, we have two tip radii: ρ 6 in the basal plane and ρ 2 < ρ 6 in the perpendicular plane. To calculate these radii, we apply the selection criterion (4.18) twice, in each of these planes (using the constant aspect ratio for ice crystals in the form of elliptical paraboloids [47]). As is easily seen, the tip radii and growth rate have the same behaviour: ρ 6 and ρ 2 decrease and V increases with increasing T. The slight discrepancy between theory and modelling for a larger radius is explained by the fact that instead of the Ivantsov solutions for axisymmetric crystals, in this case, it is necessary to use the Horvey-Cahn solutions for non-axisymmetric ice crystals in the form of elliptical paraboloids [35,38].…”
Section: A Behaviour Of the Main Functionsmentioning
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)’.
A theory of stable dendrite growth in an undercooled binary melt is developed for the case of intense convection. Conductive heat and mass transfer boundary conditions are replaced by convective conditions, where the flux of heat (or solute) is proportional to the temperature or concentration difference between the surface of the dendrite and far from it. The marginal mode of perturbation wavelengths is calculated using the linear morphological stability analysis. Combining this analysis with the solvability theory, we have derived a selection criterion that represents the first condition to define a combination of dendrite tip velocity and tip diameter. The second condition—the undercooling balance—is derived for intense convection. The theory under consideration determines the dendrite tip velocity and tip diameter for low undercooling. This convective theory is combined with the classical theory of dendritic growth (conductive boundary conditions), which is valid for moderate and high undercooling. Thus, the entire range of melt undercooling is covered. Our results are in good agreement with experiments on Al–Ge crystallization.
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