Bifurcations in a sidebranch surface of a free-growing dendrite (original) (raw)
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Phase-field simulations of dendritic crystal growth in a forced flow
2001
Convective effects on free dendritic crystal growth into a supercooled melt in two dimensions are investigated using the phase-field method. The phase-field model incorporates both melt convection and thermal noise. A multigrid method is used to solve the conservation equations for flow. To fully resolve the diffuse interface region and the interactions of dendritic growth with flow, both the phase-field and flow equations are solved on a highly refined grid where up to 2.1 million control volumes are employed. A multiple time-step algorithm is developed that uses a large time step for the flow-field calculations while reserving a fine time step for the phase-field evolution. The operating state ͑velocity and shape͒ of a dendrite tip in a uniform axial flow is found to be in quantitative agreement with the prediction of the Oseen-Ivantsov transport theory if a tip radius based on a parabolic fit is used. Furthermore, using this parabolic tip radius, the ratio of the selection parameters without and with flow is shown to be close to unity, which is in agreement with linearized solvability theory for the ranges of the parameters considered. Dendritic sidebranching in a forced flow is also quantitatively studied. Compared to a dendrite growing at the same supercooling in a diffusive environment, convection is found to increase the amplitude and frequency of the sidebranches. The phase-field results for the scaled sidebranch amplitude and wavelength variations with distance from the tip are compared to linear Wentzel-Kramers-Brillouin theory. It is also shown that the asymmetric sidebranch growth on the upstream and downstream sides of a dendrite arm growing at an angle with respect to the flow can be explained by the differences in the mean shapes of the two sides of the arm.
Numerical Simulation of Dendritic Solidification with Convection: Two-Dimensional Geometry
Journal of Computational Physics, 2002
A front tracking method is presented for simulations of dendritic growth of pure substances in the presence of flow. The liquid-solid interface is explicitly tracked and the latent heat released during solidification is calculated using the normal temperature gradient near the interface. A projection method is used to solve the Navier-Stokes equations. The no-slip condition on the interface is enforced by setting the velocities in the solid phase to zero. The method is validated through a comparison with an exact solution for a Stefan problem, a grid refinement test, and a comparison with a solution obtained by a boundary integral method. Three sets of two-dimensional simulations are presented: a comparison with the simulations of Beckermann et al. (J. Comput. Phys. 154, 468, 1999); a study of the effect of different flow velocities; and a study of the effect of the Prandtl number on the growth of a group of dendrites growing together. The simulations show that on the upstream side the dendrite tip velocity is increased due to the increase in the temperature gradient and the formation of side branches is promoted. The flow has the opposite effect on the downstream side. The results are in good qualitative agreement with published experimental results,even though only the two-dimensional aspects are examined here.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2018
Motivated by important applications in materials science and geophysics, we consider the steady-state growth of anisotropic needle-like dendrites in undercooled binary mixtures with a forced convective flow. We analyse the stable mode of dendritic evolution in the case of small anisotropies of growth kinetics and surface energy for arbitrary Péclet numbers and n -fold symmetry of dendritic crystals. On the basis of solvability and stability theories, we formulate a selection criterion giving a stable combination between dendrite tip diameter and tip velocity. A set of nonlinear equations consisting of the solvability criterion and undercooling balance is solved analytically for the tip velocity V and tip diameter ρ of dendrites with n -fold symmetry in the absence of convective flow. The case of convective heat and mass transfer mechanisms in a binary mixture occurring as a result of intensive flows in the liquid phase is detailed. A selection criterion that describes such solidific...
Sharp-interface simulation of dendritic solidification of solutions
International Journal of Heat and Mass Transfer, 2002
A numerical method is developed for the simulation of solidification of solutions/alloys. The heat and species transport equations are solved with appropriate interface conditions. The interface shape and thermal and solutal fields are calculated in a fully coupled manner. The effects of capillarity are included in the interfacial dynamics. The present mixed Eulerian–Lagrangian framework treats the immersed phase boundary as a sharp solid–fluid interface and a conservative finite-volume formulation allows boundary conditions at the moving surface to be exactly applied. We first compare the planar growth results with published one-dimensional numerical results. We then show that the method can compute the breakdown of the solid–liquid interface due to the Mullins–Sekerka instability. The dendritic growth of the crystals under various growth parameters is computed.
Metallurgical and Materials Transactions B, 2014
A three-dimensional (3-D) cellular automata (CA) model coupled with the finite-element (FE) method has been proposed to simulate dendrite growth with various crystallographic orientations during solidification. The model introduces a new tracking neighborhood method to resolve the mesh dependency caused by the cubic lattice in the CA model for simulating 3-D dendrite growth. The migration of the solid-liquid (SL) interface is associated with the dendritic preferential orientation and the driving force for the phase transition. The latter is obtained from a thermodynamics database. The local curvature and anisotropy of the surface energy are also incorporated to describe the growth kinetics of the SL interface. The solute transfer is calculated using the FE method. A numerical simulation has been performed on a Fe-1.5 wt pct C alloy. The grain morphologies with various crystallographic orientations and the solute distribution during isothermal solidification are studied and discussed. The simulation results are compared with analytical solutions and experimental results, which are in good agreement.
Modeling of a transition to diffusionless dendritic growth in rapid solidification of a binary alloy
Computational Materials Science, 2009
Diffusionless growth of dendritic crystals results in microsegregation-free microstructures with an initial (nominal) chemical composition of solidifying systems. Normally, a transition from chemically partitioned growth to diffusionless solidification is accompanied by the morphological transition in crystal shape with the appearance of nonlinearity in the kinetic behavior of growing crystals. This phenomenon is discussed using a model of local non-equilibrium rapid solidification. Considering the transition from the solute diffusion-limited growth to purely thermally controlled growth of dendritic crystals, the model predicts the abrupt change of growth kinetics with the break points in the ''dendrite tip velocity-undercooling" and ''dendrite tip radius-undercooling" relationships. It is shown that the abrupt change of growth kinetics occurs with the ending of the transition to purely thermally controlled growth and the onset of diffusionless solidification. To predict the dendrite growth kinetics in a whole region of undercooling, numeric analysis shows that the model has to take into account both anisotropies of solid-liquid interfacial properties. These are anisotropy of surface energy and anisotropy of atomic kinetics of solidification.
Selected mode for rapidly growing needle-like dendrite controlled by heat and mass transport
Acta Materialia, 2017
The boundary integral method is developed for fast anisotropic interfaces. A general integro-differential equation for curved interfaces controlled by heat and mass transport is derived and applied to the problem of rapid dendritic growth. A selection criterion for the steady-state mode of growing parabolic interfaces is obtained and, in common solution with the undercooling balance, it is compared with experimental data on rapid dendritic solidification of deeply supercooled liquid droplets. In this comparison, transitions from solute diffusion-limited to thermo-solutal regime and, finally, to pure thermally controlled regime of the anisotropic dendrite are discussed and revealed. Limiting cases of known selection criteria for anisotropic dendrites growing at small and high growth P eclet numbers are provided.
Model of Dendritic Solidification
Dendritic structures are one of the most frequent patterns in nature, that appear in crystalline systems (such as metals), advanced ceramics and neural systems of the living organisms. Embedded in the core of many industrially important processes such as casting, there is a strong relationship between the material properties and the solidified dendritic microstructures. In our proposed study microstructural simulations were performed using MICRESS® code, which implements a multi-phase-field approach and essentially provides a mathematical solution to the coupled partial differential equations of phase-field and diffusion. In the context of the study, MICRESS® was used to simulate the morphological transitions and the solute segregations during directional solidification under a wide spectrum of temperature gradients, solidification rates and chemical compositions. The proposed model is integrated with an in-situ experimental study of solidification in a high-temperature laser-scanning confocal microscope. Such in-situ studies are able to provide an experimental mean to measure some of the crucial physical properties of the studied alloys, resulting in more refined microstructural simulations of such phenomena. For instance we demonstrated a real-time method for the calculation of Gibbs-Thomson coefficient and the solid-liquid interfacial energy of the studied alloys by analysing the shape of the grain-boundary grooves at the solid-liquid interface. The interface mobility parameter was calibrated so to achieve a realistic interface velocity aligned with the regime experimentally observed in real-time in the confocal microscopy.
Journal of Crystal Growth, 1989
The dynamics of spontaneous pattern formation are studied experimentally by observing and recording the evolution of ice crystal patterns which grow freely in a supercooled melt. The sequence of evolution to dendrites is recorded in real time using cine-micrography. In the range of subcoolings from 0.06 to 0.290 C, all the patterns evolved as follows: Smooth disk -. Perturbed disk -. Disk dendrite -. Partially developed dendrite -. Fully developed dendrite. The initial smooth disk, the main branch and the side branches all developed perturbations beyond a critical size which depends on the subcooling. The combined effect of the destabilizing thermal gradients ahead of the growing crystal and the stabilizing Gibbs-Thompson capillarity effect dictates the critical size of the unstable structures in terms of the mean curvature of the interface. Detailed analysis of the evolving patterns was done using digital image analysis on the PRIME computer to determine both the manner in which the dendritic growth process replicates itself and the role which the shape and the movement of the interface play in the pattern formation process. Total arc length ST, total area A and the complexity ratio~= ST/V~of evolving patterns were computed as a function of time and undercooling for each crystal image. These results permitted us to make some comparisons with theoretical models on pattern evolution. Three distinct phases of evolution were identified: the initial phase when the crystal structure is smooth and free of any perturbations and the complexity ratio is almost a constant, an intermediate phase when the crystal structure develops perturbations which grow quickly in number and in size and the complexity ratio increases rapidly and a final phase when the pattern approaches that of a fully developed dendrite which, on a global scale grows in a shape-perserving manner and has a slowly increasing complexity ratio which seems to approach an asymptote. Two factors were found to be responsible for the symmetric dendritic patterns. These are: first, hexagonal symmetry due to the hexagonal closed packed structure, leads to strong anisotropy in molecular attachment kinetics and in surface free energy; second, the competition among side branches causes smaller side branches to melt when they are trapped between larger ones which generate latent heat and prevent the small branches from gaining access to the fresh cold fluid ahead of them. These two factors lead to a channelling effect which prevents the growth of perturbations from occurring randomly and thus directs the evolving crystal structure into patterns which are regular and reproducible. Theoretical models which are local in nature fail to take into account side branch competition, and this is one of their major weaknesses.