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

Phil. Trans. R. Soc. A.

2021

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)’.

Section: Introductionmentioning

confidence: 99%

A review on the theory of stable dendritic growth

Phil. Trans. R. Soc. A.

2021

“…These numbers give the product ρV to which an additional expression should be found again in a form of stability criterion. * dmitri.alexandrov@usu.ru † peter.galenko@uni-jena.de Using a solvability condition, different scaling ratios σ * were obtained [7][8][9] for a stable dendrite growth mode at small growth Péclet numbers in one-component (pure), binary, and multicomponent systems. The opposite limit of large Péclet numbers was considered in Refs.…”

mentioning

confidence: 99%

“…The solvability theory predicts the second combination of parameters as σ for the thermal problem in the limit of P g 1, where β is the small anisotropy parameter [4][5][6]). When an external flow and impurities are introduced, a family of the Ivantsov paraboloids can still be used as a solution of the Stefan problem [7][8][9], either in the large Reynolds-number limit (potential flow approximation) [10], or in the small Reynolds-number limit (Oseen approximation) [11,12]. As in the case of the pure thermal problem, the solidification mode is determined only by the growth and flow Péclet numbers, which are related to thermal and solute transport by the molecular and convective mechanisms.…”

mentioning

confidence: 99%

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

2013

Phys. Rev. E

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.

“…takes a special value determined from the microscopic solvability theory [18][19][20][21][22][23][24][25]. Here…”

Section: Relaxation Time To a Steady-state Regimementioning

confidence: 99%

A relaxation time of secondary dendritic branches to their steady-state growth

Титова

IOP Conf. Ser.: Mater. Sci. Eng.

2017

Abstract. In the present article, we use the selection theory to estimate the non-stationarity time of evolving dendrites to their steady-state growth. IntroductionThe microstructure predictions in solidification processes have the deep scientific and practical roots (see [1][2][3][4][5] and references therein). A development of experimental methods in solidification processing made possible to reach a much broader range of measurable crystal growth velocities, temperature gradients and cooling rates [6]. For instance, splats and films can be quenched with the cooling rate of the order of 10 7 K/s, the temperature gradients can have the order of 10 8 K/s in laser annealing of sample surfaces, and containerless methods of droplets processing provide the deep undercoolings having the values of 200-400 K prior to the primary crystallization. A large driving force for transformation, arising in such methods, leads to fast solidification from a liquid metastable state as well as the high-speed solid-state transformations of metastable crystalline phases [7]. For instance, the experimentally measured solidification velocity has the order of 10 −1 − 10 2 m/s in droplets processed by the electromagnetic levitation facility [7,8]. As such, the total duration of primary solidification in small droplets is rather short and estimated as [8]: 10 −5 − 10 −3 s.For the analytical calculations of rapid solidification regimes and theoretical estimations of microstructural parameters of dendritic and eutectic crystals, different models of non-equilibrium crystallization are used, as a rule, formally developed for the steady state scenario [9][10][11]. However, the steady state approximation for conditions of rapid solidification of small samples (films, splats, droplets) is questionable and such approach is often criticized. A very common view is that a steady state scenario may be expected to exist but not in rapidly solidifying small samples [12]: the stationary regime of solidification can be reached at long times that is possible in usual experimental circumstances (directional solidification) or well-known classic technologies (continuous casting and low intensive brazing). In other words, during the short periods of time, the steady state is not achieved and, consequently, the quasi-stationary approximation in modeling of rapid solidification does fail. The detailed calculations of solidification regimes lead, however, to the remarkable behavior of the transient time between the non-stationary and stationary regimes of solidification: the non-stationary time sharply depends on the interface