Parabolic Dendrite Research Articles

The thermal field around a parabolic dendrite growing in inclined melt flow

The thermal field around a parabolic dendrite growing in inclined melt flow

A semi-implicit conservative sharp-interface method for liquid-solid phase transition

The solidification of an undercooled liquid is physically unstable. The dominating instability modes are affected by both the evolving temperature field in the solid and liquid phases, and characteristics of the phase interface such as the curvature and the propagation velocity. To capture the instability mode, therefore both the temperature field and the interface have to be represented accurately in a numerical model of the phase-change process. In this work, we develop conservative interface exchange terms for a sharp-interface formulation of liquid-solid phase transition. Conservation at the interface is maintained by explicit formulation of interface fluxes into both solid and liquid phases. We propose a semi-implicit level-set formulation to evolve the phase interface. A new formulation for the interface surface in a cut cell is derived, which includes the Stefan condition. We achieve low numerical dissipation by an explicit third-order Runge-Kutta scheme for time discretization, and a novel WENO-like (Weighted Essentially Non-Oscillatory) interface-gradient reconstruction. This distinguishes our level-set based sharp-interface model from previous level-set based approaches, which rely on finite-difference based interface treatment, and thus do not ensure discrete conservation at the interface. The flux terms in our approach take into account surface-tension and kinetic effects on the interface temperature (Gibbs-Thomson relation). The Stefan condition provides a relation between interface fluxes of mass and energy, and the interface-propagation velocity. Computational efficiency is maintained by a multiresolution approach for local mesh adaptation, and an adaptive local time-stepping scheme.We present one- and two-dimensional simulation results for the growth of a planar solidification front and a single parabolic dendrite affected by surface tension. The results agree well with experimental and analytical reference data, showing that the model is capable to capture both stable (planar) and unstable (dendritic-like) growth processes in the heat-diffusion dominated regime. The convergence order for successively finer meshes in the one-dimensional case is one for the interface location and the temperature field, outperforming previously reported level-set based approaches. We present numerical data of a growing crystal with four-fold symmetry. Our results indicate that the artificial dissipation of the underlying numerical scheme affects its capability to reproduce consistently physical tip-splitting instabilities. The proposed low-dissipation scheme is able to resolve such instabilities. Finally, we demonstrate the capability of the method to simulate multiple growing crystals with anisotropic surface-tension and kinetic effects.

Аналитическое решение задачи об обтекании параболического дендрита наклонным потоком вязкой жидкости в приближении Осеена

Аналитическое решение задачи об обтекании параболического дендрита наклонным потоком вязкой жидкости в приближении Осеена

Selection criterion of stable dendritic growth at arbitrary Péclet numbers with convection

A free dendrite growth under forced fluid flow is analyzed for solidification of a nonisothermal binary system. Using an approach to dendrite growth developed by Bouissou and Pelcé [Phys. Rev. A 40, 6673 (1989)], the analysis is presented for the parabolic dendrite interface with small anisotropy of surface energy growing at arbitrary Péclet numbers. The stable growth mode is obtained from the solvability condition giving the stability criterion for the dendrite tip velocity V and dendrite tip radius ρ as a function of the growth Péclet number, flow Péclet number, and Reynolds number. In limiting cases, the obtained stability criterion presents known criteria for small and high growth Péclet numbers of the solidifying system with and without convective fluid flow.

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

Free growth of a dendrite crystal in a stationary multicomponent melt is investigated. A mathematical 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.

Numerical simulation of Zn coating solidification

A numerical model, which simulates nucleation and growth of Zn grains, has been developed in order to describe quantitatively the solidification of Zn coatings during the hot-dipping process. The inputs of the model are the nucleation distribution, which has been measured by electron backscattered diffraction (EBSD), and the dendritic growth kinetics, calculated with an analytical model of a parabolic dendrite tip modified to account for the interactions with the coating interfaces. The model predicts the shapes of the grain envelopes as a function of the grain orientation and the texture induced by growth. Three types of grain envelopes have been evidenced, depending on the angle between the c-axis and the normal to the coating plane. Moreover, it has been shown that growth reinforces the already existing {00.1} nucleation texture, in good agreement with experimental data. The model also predicts the cooling curve, including recalescence, and the grain size. Thus, it is used to describe the effects of Pb additions on solidification. In particular, it has been shown that Pb increases the nucleation undercooling and strongly decreases the density of active nuclei, thus resulting in a much larger grain size.

Variational algorithms and pattern formation in dendritic solidification

We present a completely new variational algorithm for computing dendritic solidification. This algorithm reproduces the Gibbs-Thomson relation as a balance between bulk and surface energy and is able to operate in the infinite-mobility limit with no unphysical time-step restriction. It may be used with arbitrary non-smooth surface energy functions and may include finite kinetic mobility. We perform computations with isotropic and anisotropic surface energy; from a small irregular initial seed we generate radial tip-splitting structures for isotropic energy and parabolic dendrites with side-branching for anisotropic energy. For anisotropic energy, the final structure is determined by the material and environmental properties; the initial shape is forgotten. For the parabolic dendrite tips, we obtain agreement with the Ivantsov solutions within a few percent and proper dimensional scaling of lengths and velocities with surface energy.

Buoyancy effects of a growing, isolated dendrite

An axisymmetric dendrite of pure material solidifies downward into an undercooled melt. Surface energy and kinetic undercooling are negligible. The Ivantsov [Dokl. Akad. Nauk SSSR 58 (1947) 567] parabolic dendrite is modified by buoyant convection. We construct an approximate solution to the growth/convection problem in powers of a buoyancy parameter G. The solution depends on Prandtl number P and Stefan number S (undercooling). When P and/or S are large enough, buoyancy enhances growth and distorts the dendrite by sharpening the tip and widening the base. These results compare well with the experiments on succinonitrile ( P = 23) of Huang and Glicksman [Acta Met. 29 (1981) 701] and the local theory of Ananth and Gill [J. Crystal Growth 91 (1988) 587] up to G ≈ 1000, but overpredict convective effects for larger G. When P and S are small enough, buoyancy slows growth and flattens the tip. Physical explanations are given for the differences in buoyant effects at different P. The results suggest that near-tip effects of buoyancy should be different in metallics than in organics.

Pattern selection in two-dimensional dendritic growth

Abstract We present an analytical treatment of the problem of pattern selection in a fully non-local symmetrical model of dendritic crystal growth. Simplifications of mathematical equations are based on the assumption that anisotropies of surface energy and kinetic effects are small. Selection rules for growth velocity and instability increments are derived at arbitrary Peclet numbers. For a dendrite growing in a channel, a double-valued velocity versus undercooling dependence is obtained. The upper branch of the solution is stable and changes into a free dendrite with increased channel width. Interplay between surface energy and kinetic effects results in morphological transition from surface energy dendrite to dense branching morphology and then to kinetic dendrite. In the framework of the boundary-layer model it is shown that at deep undercooling parabolic dendrite turns into angular dendrite and then into planar front.

Self-consistent theory of dendritic growth with convection

A theory is described which treats dentritic growth with forced convection in the melt as a free boundary problem. This approach yields self-consistent solutions for the rate of propagation of an isothermal interface and the temperature and velocity fields surrounding it. The solutions to the free boundary problem reduce to Ivantsov's pure conduction solution when the Reynolds number is zero. It is shown that the Oseen, Stokes and potential flow approximation of the Navier-Stokes equations for convection in a subcooled melt yield exact solutions for the shape preserving growth of a parabolic dendrite. The results given here, and our previous results, yield an infinite family of solutions. Therefore microscopic solvability theory is used to separate the growth velocity, V G, and tip radius, R, for a fixed subcooling, Δ T. The theoretical results which include convection and the solvability relationship agree well with the experimental data of Huang and Glicksman and with the results of Ananth and Gill who used the experimental data directly with the convection theory to separate V G and R. In contrast, Ivantsov's theory, which neglects convection, yields large errors when compared with these experiments, which are correlated well by 2αd 0 V GR 2.1 = 0.03 . This implies that the solvability relationship, 2αd 0 V GR 2 = 60 ϵ 7 4 , in contrast to the heat transfer calculation, is not affected profoundly by convection in the melt in the range of the subcooling used in the experiments. However, the value of V G R 2 decreases slightly, for a fixed degree of anisotropy, ϵ, as the subcooling gets smaller and convection increases in intensity. Forced flow experiments in which flow velocities are orders of magnitude larger than those in natural convection, would help to clarify the effect of convection on interfacial stability.

A Model for Silicon Dendrite Growth During Laser/Plasma Deposition from a Silane Discharge

AbstractA model for dendrite growths in polycrystalline Si films formed during laser/plasma deposition with a silane discharge and a pulsed KrF laser has been developed. The model assumes a thin (less than 10 nm) amorphous silicon (a-Si) film is deposited on a substrate prior to phase transformation due to laser heating. The observed dendritic structure of the overall polycrystalline Si films is attributed to Si crystals shooting from an excessively supercooled Si liquid bath. Supercooled liquid forms since the melting point for a-Si can be reached at relatively low KrF laser fluences. Latent heat evolved at the solid-liquid interface induces an interface temperature higher than that of the melt and the requisite negative temperature gradient for absolute bath supercooling. Since the formation of an undercooled liquid by fast melting a-Si is also an important first step in explosive crystal regrowth studies, these results may have important implications for crystal growth and transient annealing. A conical approximation model is used in this study to characterize the stability of the dendrite tip in terms of local temperature gradients, i.e., the degree of undercooling at the tip of the parabolic dendrite. The degree of undercooling and hence the tip radius appears to be significantly affected by small changes in the laser fluence. Stability criteria lead to a relationship between regrowth velocity, V, and the tip radius, R, of the form VR2= constant. The size and stability of the dendrite tip is determined from a balance between the destabilizing force due to thermal diffusion and the stabilizing capillary force. Based on the observed tip radii formed at laser fluences from 0.13 to 0.25 J/cm2, the model predicts regrowth velocities in a range between 2.0 and 20 m/s – values consistent with transient annealing studies of a-Si

Interface morphology during crystallization—I. Single filament, unconstrained growth from a pure melt

Steady-state dendrite growth theories are compared with the experimental data in various metallic and non-metallic systems. The influence of sidebranches and the effect of interface kinetics on the dendrite growth rates are discussed. A dimensionless kinetic parameter, q, is defined and it is found that when q < 1, the interface kinetic effect becomes significant. Furthermore, the theory of parabolic dendrite is extended to study the effect of anisotropy in the interface kinetic coefficient on the growth rate of needle and plate dendrites. The results show that the anisotropy effect is quite significant under growth conditions of small Péclet numbers.