Three-dimensional dendrite-tip morphology at low undercooling

A Novel Approach to Visualize Liquid Aluminum Flow to Advance Casting Science

Turbulent filling of molten metal in sand-casting leads to bi-films, porosity and oxide inclusions which results in poor mechanical properties and high scrap rate of sand castings. Hence, it is critical to understand the metal flow in sand-molds, i.e., casting hydrodynamics to eliminate casting defects. While multiple numerical methods have been applied to simulate this phenomenon for decades, harsh foundry environments and expensive x-ray equipment have limited experimental approaches to accurately visualize metal flow in sand molds. In this paper, a novel approach to solve this challenge is proposed using Succinonitrile (SCN) as a more accurate metal analog in place of water. SCN has a long history in solidification research due to its BCC (Body-Centered-Cubic) crystal structure and dendrite-like solidification (melting temperature ~60 °C) like molten aluminum. However, this is the first reported study on applying SCN through novel casting hydrodynamics to accurately visualize melt flow for casting studies. This paper used numerical simulations and experiments using both water and SCN to identify the critical dimensionless numbers to perform accurate metal flow analog testing. Froude's number and wall roughness were identified as critical variables. Experimental results show that SCN flow testing was more accurate in recreating the flow profile of molten aluminum, thus validating its utility as a metal analog for metal flow research. Findings from this study can be used in future metal flow analysis such as: runner, in-gate and integrated filling-feeding-solidification studies.

Phase-field-crystal investigation of the morphology of a steady-state dendrite tip on the atomic scale

Through phase-field-crystal (PFC) simulations, we investigated, on the atomic scale, the crucial role played by interface energy anisotropy and growth driving force during the morphological evolution of a dendrite tip at low growth driving force. In the layer-by-layer growth manner, the interface energy anisotropy drives the forefront of the dendrite tip to evolve to be highly similar to the corner of the corresponding equilibrium crystal from the aspects of atom configuration and morphology, and thus affects greatly the formation and growth of a steady-state dendrite tip. Meanwhile, the driving force substantially influences the part behind the forefront of the dendrite tip, rather than the forefront itself. However, as the driving force increases enough to change the layer-by-layer growth to the multilayer growth, the morphology of the dendrite tip's forefront is completely altered. Parabolic fitting of the dendrite tip reveals that an increase in the influence of interface energy anisotropy makes dendrite tips deviate increasingly from a parabolic shape. By quantifying the deviations under various interface energy anisotropies and growth driving forces, it is suggested that a perfect parabola is an asymptotic limit for the shape of the dendrite tips. Furthermore, the atomic scale description of the dendrite tip obtained in the PFC simulation is compatible with the mesoscopic results obtained in the phase-field simulation in terms of the dendrite tip's morphology and the stability criterion constant.

Ternary eutectic dendrites: Pattern formation and scaling properties

Extending previous work [Pusztai et al., Phys. Rev. E 87, 032401 (2013)], we have studied the formation of eutectic dendrites in a model ternary system within the framework of the phase-field theory. We have mapped out the domain in which two-phase dendritic structures grow. With increasing pulling velocity, the following sequence of growth morphologies is observed: flat front lamellae → eutectic colonies → eutectic dendrites → dendrites with target pattern → partitionless dendrites → partitionless flat front. We confirm that the two-phase and one-phase dendrites have similar forms and display a similar scaling of the dendrite tip radius with the interface free energy. It is also found that the possible eutectic patterns include the target pattern, and single- and multiarm spirals, of which the thermal fluctuations choose. The most probable number of spiral arms increases with increasing tip radius and with decreasing kinetic anisotropy. Our numerical simulations confirm that in agreement with the assumptions of a recent analysis of two-phase dendrites [Akamatsu et al., Phys. Rev. Lett. 112, 105502 (2014)], the Jackson-Hunt scaling of the eutectic wavelength with pulling velocity is obeyed in the parameter domain explored, and that the natural eutectic wavelength is proportional to the tip radius of the two-phase dendrites. Finally, we find that it is very difficult/virtually impossible to form spiraling two-phase dendrites without anisotropy, an observation that seems to contradict the expectations of Akamatsu et al. Yet, it cannot be excluded that in isotropic systems, two-phase dendrites are rare events difficult to observe in simulations.

Unified derivation of phase-field models for alloy solidification from a grand-potential functional

In the literature, two quite different phase-field formulations for the problem of alloy solidification can be found. In the first, the material in the diffuse interfaces is assumed to be in an intermediate state between solid and liquid, with a unique local composition. In the second, the interface is seen as a mixture of two phases that each retain their macroscopic properties, and a separate concentration field for each phase is introduced. It is shown here that both types of models can be obtained by the standard variational procedure if a grand-potential functional is used as a starting point instead of a free energy functional. The dynamical variable is then the chemical potential instead of the composition. In this framework, a complete analogy with phase-field models for the solidification of a pure substance can be established. This analogy is then exploited to formulate quantitative phase-field models for alloys with arbitrary phase diagrams. The precision of the method is illustrated by numerical simulations with varying interface thickness.

Phase-field study of three-dimensional steady-state growth shapes in directional solidification

We use a quantitative phase-field approach to study directional solidification in various three-dimensional geometries for realistic parameters of a transparent binary alloy. The geometries are designed to study the steady-state growth of spatially extended hexagonal arrays, linear arrays in thin samples, and axisymmetric shapes constrained in a tube. As a basis to address issues of dynamical pattern selection, the phase-field simulations are specifically geared to identify ranges of primary spacings for the formation of the classically observed "fingers" (deep cells) with blunt tips and "needles" with parabolic tips. Three distinct growth regimes are identified that include a low-velocity regime with only fingers forming, a second intermediate-velocity regime characterized by coexistence of fingers and needles that exist on separate branches of steady-state growth solutions for small and large spacings, respectively, and a third high-velocity regime where those two branches merge into a single one. Along the latter, the growth shape changes continuously from fingerlike to needlelike with increasing spacing. These regimes are strongly influenced by crystalline anisotropy with the third regime extending to lower velocity for larger anisotropy. Remarkably, however, steady-state shapes and tip undercoolings are only weakly dependent on the growth geometry. Those results are used to test existing theories of directional finger growth as well as to interpret the hysteretic nature of the cell-to-dendrite transition.

Dendrite engineering on xenon crystals

The experimental work presented focuses on transient growth, morphological transitions, and control of xenon dendrites. Dendritic free growth is perturbed by two different mechanisms: Shaking and heating up to the melting temperature. Spontaneous and metastable multitip configurations are stabilized, coarsening is reduced, leading to a denser sidebranch growth, and a periodic tip splitting is found during perturbation by shaking. On the other hand, heating leads to controlled sidebranching and characteristic transitions of the tip shape. A deterministic behavior is found besides the random-noise-driven growth. The existence of a limit cycle is supported by the findings. Together the two perturbation mechanisms allow a "dendrite engineering"--i.e., a reproducible controlling of the crystal shape during its growth. The tip splitting for dendritic free growth is found not to be a splitting of the tip in two; rather, the respective growth velocities of the main tip and the fins change. The latter then surpass the main tip and develop into new tips. The occurrence of three- and four-tip configurations is explained with this mechanism. Finite-element calculations of the heat flow and the convective flow in the growth vessel show that the idea of a single axisymmetric toroidal convection roll across the whole growth vessel has to be dropped. The main effect of convection under Earth's gravity is the compression of the diffusive temperature field around the downward-growing tip. A model to explain the symmetry of dendritic crystals--e.g., snow crystals--is developed, based on the interaction of crystal shape and heat flow in the crystal.

Three-Dimensional Single Crystal Morphologies of Diffusion Limited Growth in Experiments and Phase Field Simulations

In growth experiments for 3D xenon crystals we have observed three different morphologies: dendrites, doublons and seaweed. The 3D shape of dendrites and doublons is reconstructed by means of refractive reconstruction. Our measurements on the fin thickness of dendrites support the validity of analytical predictions by Brener. We have found that the fins of the doublon morphology can be fitted with the fin predictions for dendrites. Measurements of the doublon gap as a function of the supercooling show that the gap decreases hyperbolically with increasing supercooling. Phase field simulations of 3D doublons reveal that the channel shows an inner structure in the presence of anisotropy of surface tension. A combination of reconstructions and phase field simulations leads to a geometrical description of doublon cross sections.

Quantitative phase-field modeling of two-phase growth

A phase-field model that allows for quantitative simulations of low-speed eutectic and peritectic solidification under typical experimental conditions is developed. Its cornerstone is a smooth free-energy functional, specifically designed so that the stable solutions that connect any two phases are completely free of the third phase. For the simplest choice for this functional, the equations of motion for each of the two solid-liquid interfaces can be mapped to the standard phase-field model of single-phase solidification with its quartic double-well potential. By applying the thin-interface asymptotics and by extending the antitrapping current previously developed for this model, all spurious corrections to the dynamics of the solid-liquid interfaces linear in the interface thickness W can be eliminated. This means that, for small enough values of W, simulation results become independent of it. As a consequence, accurate results can be obtained using values of W much larger than the physical interface thickness, which yields a tremendous gain in computational power and makes simulations for realistic experimental parameters feasible. Convergence of the simulation outcome with decreasing W is explicitly demonstrated. Furthermore, the results are compared to a boundary-integral formulation of the corresponding free-boundary problem. Excellent agreement is found, except in the immediate vicinity of bifurcation points, a very sensitive situation where noticeable differences arise. These differences reveal that, in contrast to the standard assumptions of the free-boundary problem, out of equilibrium the diffuse trijunction region of the phase-field model can (i) slightly deviate from Young's law for the contact angles, and (ii) advance in a direction that forms a finite angle with the solid-solid interface at each instant. While the deviation (i) extrapolates to zero in the limit of vanishing interface thickness, the small angle in (ii) remains roughly constant, which indicates that it might be a genuine physical effect, present even for an atomic-scale interface thickness.

Quantitative phase-field model of alloy solidification

We present a detailed derivation and thin interface analysis of a phase-field model that can accurately simulate microstructural pattern formation for low-speed directional solidification of a dilute binary alloy. This advance with respect to previous phase-field models is achieved by the addition of a phenomenological "antitrapping" solute current in the mass conservation relation [Phys. Rev. Lett. 87, 115701 (2001)]]. This antitrapping current counterbalances the physical, albeit artificially large, solute trapping effect generated when a mesoscopic interface thickness is used to simulate the interface evolution on experimental length and time scales. Furthermore, it provides additional freedom in the model to suppress other spurious effects that scale with this thickness when the diffusivity is unequal in solid and liquid [SIAM J. Appl. Math. 59, 2086 (1999)]], which include surface diffusion and a curvature correction to the Stefan condition. This freedom can also be exploited to make the kinetic undercooling of the interface arbitrarily small even for mesoscopic values of both the interface thickness and the phase-field relaxation time, as for the solidification of pure melts [Phys. Rev. E 53, R3017 (1996)]]. The performance of the model is demonstrated by calculating accurately within a phase-field approach the Mullins-Sekerka stability spectrum of a planar interface and nonlinear cellular shapes for realistic alloy parameters and growth conditions.

Quantitative description of morphological transitions in diffusion-limited growth of xenon crystals

Changes of growth morphologies are induced by a perturbation of the temperature distribution in the surrounding of a growing xenon crystal. Apart from the dendritic morphology seaweed and doublon morphologies are found. We present a method which quantitatively describes growth morphologies by means of rotational, scale, and translational invariant transformations. Evolutions of growth morphologies are represented as paths in the morphology space. The presented method could be of some use for other fields of research where qualitative and quantitative information of different classes of images has to be identified.

Generalized length scales for three-dimensional dendritic growth

The crucial length scale of dendritic growth is the tip radius. Usually it is determined by fitting the data to a theoretical function. We present a method of a generalized tip radius which is entirely based on geometric considerations and is not dependent on an underlying assumption of the shape of the tip. Furthermore the results are stable and the average change of the tip radius between successive images is less than 6%.

Effect of the ratio of solid to liquid conductivity on the stability parameter of dendrites within a phase-field model of solidification

We use a phase-field model of dendritic growth in a pure undercooled melt to examine the effect of the ratio mu=kappa(s)/kappa(l) on the operating point of the needle crystal and hence the stability parameter sigma(*), where kappa(s) and kappa(l) are the thermal conductivities of the solid and liquid phases, respectively. These results are compared with the microscopic solvability calculations of Barbieri and Langer [Phys. Rev. A 39, 5314 (1989)]. We find that in our phase-field model sigma(*) varies much more rapidly with mu than is predicted from solvability theory.

Phase-field formulation for quantitative modeling of alloy solidification

A phase-field formulation is introduced to simulate quantitatively microstructural pattern formation in alloys. The thin-interface limit of this formulation yields a much less stringent restriction on the choice of interface thickness than previous formulations and permits one to eliminate nonequilibrium effects at the interface. Dendrite growth simulations with vanishing solid diffusivity show that both the interface evolution and the solute profile in the solid are accurately modeled by this approach.

Phase-field simulations of dendritic crystal growth in a forced flow

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 shape) 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.