Theory of dendrite dynamics

Pelcé, Pierre; Bensimon, David

doi:10.1016/0920-5632(87)90022-3

Cited by 54 publications

(86 citation statements)

References 14 publications

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“…It defines the criterion for the stable mode of the dendrite tip growth in a three component melt taking into account the surface energy anisotropy (parameter β) and the nonisothermal nature of the liq uid. Criterion (21) is a generalization of the results obtained earlier for one component [15,35,36] and binary [24] melts disregarding convection. Figures 2 and 3 show the dependences of both parts of expression (21) on the Peclet growth number.…”

Section: Discussionmentioning

confidence: 99%

“…This allows us to select a stable regime of dendrite growth in terms of velocity V of its tip with radius of curvature ρ (i.e., with superimposed symmetry of the crystal lattice taking into account anisotropy of the preferred direction of crystal growth) [19,20]. Further, we will use the microscopic solvabil ity condition obtained in [35], which can be written in the form (i is the imaginary unity)…”

Section: Microscopic Solvability Conditionmentioning

confidence: 99%

“…(see, for example, [35]) and following [23], we can write solvability condition (8) in the form (19) where constant C is normalized to dimensionless fac tor VPρ 2 /(2d 0 D T ). Further, we estimate integral (19) in the weak anisotropy limit using the method developed in [23].…”

Section: Criterion Of Stable Growth Of the Dendrite Tipmentioning

confidence: 99%

“…The solution to this equation will be sought in the vicinity of the Mullins-Sekerka solution [15,35,36] with constant amplitude g(y c ) = = const. Substi tuting T l0 into the right hand side of the first expression in (14), we obtain the first approximation for g(y c ): (15) In this expression, strong inequality V/D T Ӷ k is taken into account (here, k is estimated in accordance with the Mullins-Sekerka theory as 10 7 m -1 [15,35,36] Cl and V/D T is estimated as 10 2 m -1 for binary metallic systems).…”

Section: Linear Analysis Of Stabilitymentioning

confidence: 99%

“…It should be noted that wavenumber (18) for C i = 0 (D C ∞) and B i = 0 (D B ∞) is transformed to the corresponding Mullins-Sekerka expression for the one component system [15,35,36]. Expression (18) also contains a correct limiting transition to the wave number for the binary system [24] for…”

Section: Linear Analysis Of Stabilitymentioning

confidence: 99%

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Selection of stable growth conditions for the parabolic dendrite tip in crystallization of multicomponent melts

Alexandrov

Pinigin

2013

Tech. Phys.

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Abstract-Free growth of a dendrite crystal in a stationary multicomponent melt is investigated. A mathe matical model of the process is developed and its analytic solution is constructed. A stability criterion is obtained for a 2D parabolic dendrite and the refined relation is derived for the growth rate of the dendrite tip taking into account the anisotropy of surface tension at the solid phase-melt interface. It is shown that the accumulation of impurity in the front of a dendrite makes it tip thinner and reduces its velocity. The theory contains limiting transitions to the familiar dendrite growth criteria for one component and binary systems.

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Section: Discussionmentioning

confidence: 99%

Section: Microscopic Solvability Conditionmentioning

confidence: 99%

Section: Criterion Of Stable Growth Of the Dendrite Tipmentioning

confidence: 99%

Section: Linear Analysis Of Stabilitymentioning

confidence: 99%

Section: Linear Analysis Of Stabilitymentioning

confidence: 99%

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Selection of stable growth conditions for the parabolic dendrite tip in crystallization of multicomponent melts

Alexandrov

Pinigin

2013

Tech. Phys.

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

Toropova

Alexandrov

Galenko

2020

Math Methods in App Sciences

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The paper deals with the analysis of stable thermo-solutal dendritic growth in the presence of intense convection. The n-fold symmetry of crystalline anisotropy as well as the two-and three-dimensional growth geometries are considered. The steady-state analytical solutions are found with allowance for the convective-type heat and mass exchange boundary conditions at the dendritic surface. A linear morphological stability analysis determining the marginal wavenumber is carried out. The new stability criterion is derived from the solvability theory and stability analysis. This selection criterion takes place in the regions of small undercooling. To describe a broader undercooling diapason, the obtained selection criterion, which describes the case of intense convection, is stitched together with the previously known selection criterion for the conductive-type boundary conditions. It is demonstrated that the stitched selection criterion well describes a broad diapason of experimental undercoolings.

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Mathematical modeling of dendrite growth in an Al–Ge alloy with convective flow

Toropova

Rettenmayr

Galenko

et al. 2021

Math Methods in App Sciences

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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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Theory of dendrite dynamics

Cited by 54 publications

References 14 publications

Selection of stable growth conditions for the parabolic dendrite tip in crystallization of multicomponent melts

Selection of stable growth conditions for the parabolic dendrite tip in crystallization of multicomponent melts

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

Mathematical modeling of dendrite growth in an Al–Ge alloy with convective flow

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