Free Dendrite Research Articles

PARAMETERS OF THE PERTURBED STATE OF THE DENDRITE TIP IN DEEPLY SUPERCOOLED NICKEL AND COPPER MELTS

The object of study is the morphological stability of the tip of a free dendrite in a supercooled melt of a pure substance. Areas of deep supercooling are considered: for nickel DT > 166 K, for copper – DT > 180 K. A distinctive feature of the processes being studied is the presence of two propagation velocities of small disturbances (velocities of “sound”). The dependences of these rates on the supercooling of the melt were determined. The periodic and coordinate-aperiodic modes of perturbation of the growth line have been studied in detail. For these stable regimes, the possibility of observing the same speed of the disturbance wave in two processes, differing from each other in the size of the spatial inhomogeneity of the background in front of the wave and the characteristic wave attenuation times, was discovered. It is shown that aperiodic instability appears if, after the passage of the wave front, the width of the growth line inhomogeneity zone decreases. The resonant excitation mode of the dendrite tip demonstrates important differences between the properties of the nickel and copper melts. First of all, this relates to the temperature dependences of the resonant frequency and the speed of a standing wave formed in the vicinity of the vertex. Numerical calculations are presented and graphic information is presented illustrating the patterns of growth of nickel and copper dendrites.

Modeling for free dendrite growth based on physically-informed machine learning method

In order to overcome limitations of traditional theoretical and numerical models, based on machine learning (ML) method, three models were developed for dendrite growth in undercooled alloy melts to predict solid-liquid interfacial velocity, including purely data-driven model-1, model-2 based on Galenko-Danilov (GD) model and model-3 coupled with an extended GD model newly proposed by introducing thermo-kinetic correlation. A thoroughly comparative analysis was carried out by applying the three ML models to existing data of five alloys. Results verify that model-3 has the relatively better fitting ability to experimental data in case of interpolation, due to the varied effective kinetic coefficient introduced and the thermo-kinetic correlation considered. Both cases of complete and partial extrapolations were discussed. It is concluded that on the whole the two physics-based ML models, especially model-3, are superior to the purely data-driven ML model for extrapolation ability. Thus, ML model-3 is finally proposed in modeling dendrite growth.

VS4/MoS2 cathodes with long cycle life and high rate performance for hybrid magnesium-lithium batteries

Hybrid magnesium-lithium batteries (MLIBs) have become the focus of research due to the advantages of combining a magnesium metal anode (free dendrite) and Li-driven reaction cathode (fast ion diffusion rate). VS4 with large interchain space can be used as a potential cathode for MLIBs, but its low conductivity and weak structural stability limit the development of MLIBs. In this work, MoS2 was introduced into VS4 to induce the generation of abundant sulfur vacancies, which improved the structural stability and Li+ ion transport properties of VS4. The VS4/MoS2 cathode retains stability and an impressively high reversible capacity of 581.7 mAh·g−1 after 100 cycles, corresponding to a capacity retention of 98.99 % (compared with the second cycle). Meanwhile VS4/MoS2 still has 472.64 mAh·g−1 at high current density of 1500 mA·g−1, which is much higher than that of NF-VS4 (11.9 mAh·g−1) and VF-VS4 (29.8 mAh·g−1). This work points the way to ameliorate Mg-based rechargeable batteries.

SPATIAL-TEMPORAL INHOMOGENEITIES AT THE PHASE BOUNDARY OF HIGH-SPEED
 CRYSTALLIZATION OF A UNDERCOOLED MELT

The object of study is the growth line of a free dendrite in a undercooled melt of a pure substance. The perturbed
 state of the growth line at a finite distance from the top of the dendrite has been studied. It is in this part
 of the crystallization front that the appearance of side branches is observed. Two main variants are considered:
 aperiodic and coordinate-periodic background, along which the perturbation wave propagates. An important role
 of the characteristic size of the zone of spatial inhomogeneity of the background is revealed, and quantitative estimates
 of the threshold values of this size are given. The dependences of the wave velocity on the angle of sharpening
 of the growth line and on the width of the inhomogeneity zone are studied. Examples are given demonstrating
 that the direction of wave movement (from the top to the periphery or from the periphery to the top) affects the morphological
 stability/instability of the growth line. The properties of the perturbation damping parameter are studied
 in detail. Numerical modeling of the properties of the growth process was performed for nickel and copper melts
 at undercoolings equal to 160 K and 180 K respectively. The result of the calculations is the correlation "wave
 speed – taper angle – size of the inhomogeneity zone", "attenuation parameter – taper angle". The presented graphical
 information demonstrates quantitative differences in the growth properties of nickel and copper dendrites.

Grid anisotropy reduction method for cellular automata based solidification models

The reliability of a cellular automata (CA) simulation for a free dendritic growth problem relies heavily on its ability to reduce the artificial grid anisotropy. Hence, a computationally efficient, accurate and elegant cell capturing methodology is essential to achieve reliable results. Therefore, a novel cell capturing method termed limited circular neighbourhood (LCN) is proposed in the present study for solidification models. The LCN method is applied to the canonical test cases with an isotropic growth rate and is compared with other grid anisotropy reducing methods. It is observed that the LCN method is able to capture the growth orientation accurately. Moreover, the mass loss and shape error in the proposed method is significantly reduced as compared with the other methods. In addition, its performance is also evaluated for a free dendrite growth problem in a pure material in which the growth captured by the LCN method is found to be accurate. Finally, its efficacy is also demonstrated in the results presented for a constrained dendritic growth problem in a binary alloy with multiple growth sites.

Induced side-branching in smooth and faceted dendrites: theory and phase-field simulations.

The present work is devoted to the phenomenon of induced side branching stemming from the disruption of free dendrite growth. We postulate that the secondary branching instability can be triggered by the departure of the morphology of the dendrite from its steady state shape. Thence, the instability results from the thermodynamic trade-off between non monotonic variations of interface temperature, surface energy, kinetic anisotropy and interface velocity within the Gibbs-Thomson equation. For the purposes of illustration, the toy model of capillary anisotropy modulation is prospected both analytically and numerically by means of phase-field simulations. It is evidenced that side branching can befall both smooth and faceted dendrites, at a normal angle from the front tip which is specific to the nature of the capillary anisotropy shift applied. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

Non-axisymmetric growth of dendrite with arbitrary symmetry in two and three dimensions: sharp interface model vs phase-field model

The growth of a free dendrite having a non-axisymmetric morphology with arbitrary symmetry in a pure substance is considered with the interfacial effect of anisotropy and in the absence of convective flow. A stable mode of thermal dendritic growth with the n-fold crystal symmetry is formulated using the solvability theory. A complete set of nonlinear equations, consisting of the undercooling balance condition and the selection criterion, is derived. Theoretical predictions are compared with phase field modeling data in two and three dimensional geometry obtained for ice dendrites growing in pure water.

A comparative in situ X-radiography and DNN model study of solidification characteristics of an equiaxed dendritic Al-Ge alloy sample

In situ X-radiography imaging during the isothermal solidification of a 200 µm thin Al-24 at.% Ge sample was performed to determine dendritic growth rates, liquid concentrations and projected solid fractions. The experimental data allowed to follow dendrite tip growth velocities with high accuracy all the way from the initial transient growth related decrease of velocity via the expected increase of velocity during free dendrite growth to its final decrease during neighbor interacted growth. Concentration profiles in front of individual dendrite tips were measured to analyze the increase in far-field solute concentration due to solute accumulation. By time-resolved imaging of almost the entire 12 mm diameter sample, the global melt concentration could be derived. This enabled to estimate the initial nucleation undercooling of this un-refined alloy, which was so far unresolved. The experimental results are compared to a three-dimensional dendrite needle network model that simulates the experimental conditions, such as hexagonal dendrite structures and confined sample geometry. Although the simulation results are very sensitive to variations of the parameters undercooling and dendrite selection constant, one model parameter set is found that reproduces all experimentally measured solidification characteristics. The thorough experiment-model comparison indicates that the dendritic growth rates in this experimental configuration can be lower by a factor of 7 depending on undercooling than in the non-confined case.

PF-LBM Modelling of Dendritic Growth and Motion in an Undercooled Melt of Fe-C Binary Alloy

In the present study, a two-dimensional phase field model coupled with Lattice Boltzmann method (PF-LBM) is proposed to predict the dendritic growth and motion in the melt of Fe-C binary alloy, where the phase field method (PF) is used to calculate the dendritic growth, including the phase field and the concentration field, and the lattice Boltzmann method (LBM) is used to calculate the flow field. The dendrite motion is determined by Newton’s Second Law and tracked by Lagrangian point in a Cartesian coordinate system. Later, the model validations were performed with the benchmark of a solid particle settlement in a stagnant fluid and particle motion in a shear flow, and the results show that the present model is capable of predicting the solid particle motion in the fluid flow. Finally, the model is adopted to investigate the dendritic growth and motion in a forced fluid flow (laminar flow or rotational flow), and the dendrite settlement in a terrestrial environment. The results show that when the forced fluid flow is a laminar flow, the free dendrite would be driven to translate, and the relative velocity between the dendrite and flow fluid decreases, resulting in weak influence of fluid flow on the dendritic growth. When the forced fluid flow is a rotational fluid flow, the dendrite would centrifugally rotate on the domain center with a counterclockwise self-spinning, and the rotation radius becomes larger and larger. For the case of dendrite settlement in a terrestrial environment, the relative movement between the dendrite and melt promotes the downward branch growth, but inhibits the upward branch growth, and two vortices form at the wake region of dendrite. Therefore, the settling dendrite shows a significant asymmetrical morphology.

LiCrTiO4 Nanowires as High-Performance Cathodes for Magnesium–Lithium Hybrid Batteries

Magnesium–lithium hybrid batteries attract considerable attention due to combined advantages of metallic magnesium anode (free dendrite) and Li-driven reaction cathode (fast ion diffusion rate). In this work, nanowired LiCrTiO4 with moderate redox potential is chosen as a potential cathode material for magnesium–lithium hybrid batteries. LiCrTiO4 nanowires are made up of small dispersed nanoparticles. In such a structure, the diffusion/migration barrier of Li+ insertion is reduced and the infiltration of electrolyte is enhanced, leading to faster ion diffusion rates and smaller potential polarization. The Mg2+/Li+ hybrid battery exhibits a high reversible capacity (128.7 mA h g–1 after 100 cycles at 50 mA g–1) and excellent rate performance (even at a high current density of 500 mA g–1, a specific capacity of 110.1 mA h g–1 can be still achieved, corresponding to 67.8% of that at 30 mA g–1). Notably, the ion storage mechanism of LiCrTiO4 nanowires in Mg2+/Li+ hybrid battery is systematically investigated by ex situ X-ray diffraction and ex situ X-ray photoelectron spectroscopy. This research demonstrates that the nanowired LiCrTiO4 cathode is a high-performance candidate for magnesium–lithium hybrid batteries.

Simulation method based on phase-field lattice Boltzmann model for long-distance sedimentation of single equiaxed dendrite

In solidification of metals and alloys under earth’s gravity, the growing free equiaxed dendrites either settle or float because of the difference between the solid and liquid densities. In this study, we developed a two-dimensional computational method that can efficiently simulate the growth of free dendrite settling over a long distance. In the developed method, the growth of the dendrite is expressed by a phase-field method, the liquid flow is computed by a lattice Boltzmann method, and the settling of dendrite is expressed by equations of motion. A moving-frame algorithm is employed to track the dendrite settling over a long distance. The computation is accelerated using a multi-level mesh method and GPU computing. The validity of the developed method was evaluated through simulation of the settling of a single circular particle. Specifically, using the developed method, we simulated the growth of a single dendrite in an isothermal undercooled melt of a binary alloy under gravity, and evaluated the time evolutions of the growth velocity of the primary arms, settling velocity, and crystal orientation of the dendrite. The results confirmed that the developed computational method can efficiently express the dendrite growth and settling over a long distance at a reasonable computational cost.

Dendritic growth with the six-fold symmetry: Theoretical predictions and experimental verification

A free dendrite growing in a pure substance is considered with the interfacial effect of anisotropy and convective flow. A stable mode of dendritic growth with the six-fold crystal symmetry is studied using the solvability theory. We demonstrate that the obtained selection criterion for a stable mode of dendritic growth is a function of surface energy stiffness, arbitrary values of Péclet numbers and convective flow intensity. To predict the dendrite tip velocity V and its tip radius R a model of dendrite growth with the six-fold symmetry is formulated. We show that the model equations can be reduced to the growth kinetics with the low Péclet numbers, which exhibit the explicit relationships “tip velocity - undercooling”. The model predictions are compared with experimental data on ice dendrites grown from pure undercooled water on board of the International Space Station (under microgravitational conditions, μg) and on the Ground (under terrestrial conditions, 1g).

Selection criterion of a stable dendrite growth in rapid solidification

We present an analysis of a free dendrite growing in a binary mixture under non-isothermal conditions. The stable growth mode is analyzed through the solvability condition giving the stability criterion for the dendrite tip as a function of the thermal Péclet number, PT, and ratio, W=V/VD, of the dendrite velocity V and solute diffusion speed VD in bulk liquid. We extend previous studies limited to small values of the Péclet numbers, by considering the effect of the anisotropy of surface energy for the needle-like dendrite growing at arbitrary Péclet numbers and under local non-equilibrium solute diffusion described by a hyperbolic type of transport equation. Transitions in growth regimes, namely, from solute diffusion-limited to thermo-solutal regime and, finally, to pure thermally controlled regime of the anisotropic dendrite are derived and revealed. Limiting cases of known criteria for anisotropic dendrite growing at small and high growth Péclet numbers are provided. A comparison with the previously obtained criterion of marginal stability of rapidly growing dendrite is made.

On the Origin of Grid Anisotropy in the Simulation of Dendrite Growth by a VFT Model

A virtual front tracking model, based on solute and heat diffusion in two dimensions, is chosen to capture the full microstructural behavior of dendritic solidification in a binary alloy. We use a simple method of calculation, easy to perform, with relatively high stable time step, to simulate the dendrite growth in an Al-8 wt pct Mg alloy for which no numerical simulation has been carried out in the past. Local equilibrium at the liquid solid interface and the buildup of solute ahead of the interface are solved, and the dendrite growth process is simulated in isothermal solidification conditions. We show that the artificial grid anisotropy originates from the four cell neighborhood method adopted for capturing the moving front. By a correct neighborhood configuration, a grid independent set of results and expected phenomena are reproduced for a free dendrite growing either aligned or inclined with the grid. The dendrite morphology and orientation, and the growth velocity are explored via physical simulation parameters such as undercooling and surface tension anisotropy.

Investigation on Grain Refinement of AZ31 Alloy by Self-Inoculation Method

The grain refining technology of AZ31 alloy by using self-inoculation method (SIM) has been developed. Results indicated that SIM significantly decreased the grain size of as-cast AZ31 billet from 287 to 58 μm, changed the morphologies and size of the β-Mg17Al12 phase and decreased the solute elements segregation along grain boundaries. The average grain size is sensitive to the diameter of a cast rod; the microstructures of the large diameter rod are relatively inhomogeneous. The grain refinement was mainly attributed to the grain multiplication which derives from heterogeneous nucleation induced by self-inoculants and free crystals and dendrite fragments enhanced by fluid director, combined with a high solidification rate of alloy melt in the metal mold.

Phase field simulation of dendrite growth with boundary heat flux

AbstractBoundary heat flux has a significant effect on solidification behavior and microstructure formation, for it can directly affect the interfacial heat flux and cooling rate during phase transition. In this study, a phase field model for non-isothermal solidification in binary alloys is employed to simulate the free dendrite growth in undercooled melts with induced boundary heat flux, and an anti-trapping current is introduced to suppress the solute trapping due to the larger interface width used in simulations than a real solidifying material. The effect of heat flux input/extraction from different boundaries was studied first. With heat input from boundaries, the temperature can be raised and the dendritic morphology changed with gradient temperature distribution caused by the heat flux input coupling with latent heat release during the liquid-solid phase transition. Also, the concentration distribution can be also influenced by this irregular temperature distribution. Heat flux extraction from the boundaries can decrease the temperature, which results in rapid solidification with small solute segregation and concentration changes in the dendrite structures. Also, dendrite growth manner changes caused by undercooling variation, the result of competition between heat flux and latent heat release from phase transition, are also studied. Results indicate that heat flux in the simulation zone significantly reduces the undercooling, thus slowing down the dendrite formation and enhancing the solute segregation, while large heat extraction can enlarge the undercooling and lead to rapid solidification with large dendrite tip speed and small secondary dendrite arm spacing, while solute segregation tends to be steady. Therefore, the boundary heat flux coupling with the latent heat release from the solidification has an effective influence on the temperature gradient distribution within the simulation zone, which leads to the morphology and concentration changes in the dendritic structure formation.

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 criterion for the growing dendritic tip at the inner core boundary

A free dendrite growth at the Earth’s inner core boundary is analyzed using a sharp interface model. A new selection criterion of the dendrite tip’s stable growth into non-isothermal binary melt with convection and pressure effects for the 2D and 3D axisymmetric models is derived. The criterion obtained combines known analytic results for the dendrite growth under forced convection in a pure system and dendrite growth in a stagnant binary system. The generalized selection criterion represents a condition connecting the main physical parameters of the Earth’s inner core.

Preparation of non-dendritic microstructure of AM60 alloy for rheoforming using self-inoculation method

A rheocasting process, i.e. self-inoculation method (SIM), has been developed for preparing sound semisolid slurry of AM60 alloy. This process involves mixing between alloy melt and particles of solid alloy (self-inoculants in SIM) and subsequent pouring of the mixed melt with a multistream fluid director. Results show that the combined action of self-inoculants and fluid director can create high nucleation and survive rate of the primary α-Mg phase even with high superheat (85°C) in the AM60 alloy. The melt treatment temperature and addition amounts of self-inoculants are the main factors influencing the slurry microstructure. In order to prepare high quality semisolid slurries, proper melt treatment temperature range of 680–700°C and self-inoculants addition range of 5–7% are suggested. The grain multiplication derives from the heterogeneous nucleation induced by self-inoculants and free crystals and dendrite fragments enhanced by the fluid director; globular grain growth and coarsening of particles should be contributed to the microstructure formation.

Growth of a free dendrite in pure substances under modulated flow conditions

We study the free dendrite growth of a pure substance under the influence of a parallel flow, modulated in time using two different approaches, by means of two-dimensional phase-field simulations. In a previous work amplified side-branching was shown to occur for velocity pulses of appropriate duration (H. Neumann-Heyme et al., IOP Conference Series: Materials Science and Engineering 2012 [1]) which is extended for the case of modulated temperature. To efficiently detect the resonant pulse duration corresponding to the naturally preferred sidebranch frequency we make use of a technique in which the pulse duration is ramped during the growth of the dendrite. Geometrical features of the dendrites grown under the different modulation conditions are discussed.