Solid-liquid Interface Velocity Research Articles

Atomistic-informed kinetic phase-field modeling of non-equilibrium crystal growth during rapid solidification

A novel method based on molecular dynamics (MD) is developed to make the kinetic phase-field (PF) model quantitative in predicting non-equilibrium crystal growth during rapid solidification. MD-calculated variations of the diffuse solid-liquid (SL) interface width versus interface velocity are used to parameterize the kinetic PF model. Two approaches are adopted to study temperature independent and temperature dependent interfacial properties on the accuracy of predictions. MD simulations of slow and rapid solidification regimes for an fcc metal (Ni) show that the SL interface width decreases by increasing the solidification velocity. Fitting the dynamic response of the interface width to the traveling wave solution of hyperbolic PF equation determines the target SL interfacial properties, namely propagation velocity and diffusion coefficient. Independently, the MD calculations of nonlinearity in velocity versus undercooling is used to validate the atomistic-informed kinetic PF model. Both parabolic and kinetic PF models parameterized by temperature-independent material properties can accurately simulate the linear portion of near-equilibrium crystal growth during solidification. However, they both fail to predict the crystal growth kinetics during rapid solidification. The kinetic PF model parameterized with the temperature-dependent SL interfacial properties can accurately predict both the equilibrium and non-equilibrium crystal growth during slow and rapid solidification. MD simulation results on Ni along with some analytical analysis on the variation of interface width versus interface velocity show that for fcc metals, in general, {110} interface has a smaller propagation velocity in comparison to {100} interface, resulting in a larger non-linear behavior at smaller undercooling.

Selection of solidification pathway in rapid solidification processes

Rapid solidification processing of alloys enables the formation of exotic nonequilibrium microstructures. However, the interrelationship between the processing parameters and the resulting microstructure is yet to be fully understood. In melt spinning (MS) and additive manufacturing (AM) of rapidly solidified alloys, opposite microstructure development sequences were observed. A fine-to-coarse microstructural transition is typically observed in melt-spun ribbons, whereas melt pools in AM exhibit a coarse-to-fine transition. In this paper, the microstructural evolutions during these two processes are investigated using phase-field modeling. The variation of all key variables of the solid-liquid interface (temperature, composition, and velocity) throughout the entire rapid solidification of AM and MS processes was acquired with high accuracy. It is found that the onset of nucleation determines the selection of the solidification pathway and, consequently, the evolution of temperature and velocity of the interface during the rapid solidification. The switching of control mechanisms of the solid-liquid interface, which happens in both processes but in opposite directions, is found to cause the velocity jump and disrupt the microstructure development.

Kinetic coefficient for ice–water interface from simulated non-equilibrium relaxation at coexistence

In the theory of solidification, the kinetic coefficient multiplies the local supercooling to give the solid-liquid interface velocity. The same coefficient should drive interface migration at the coexistence temperature in proportion to a curvature force. This work computes the ice-water kinetic coefficient from molecular simulations starting from a sinusoidal ice-water interface at the coexistence temperature. We apply this method to the basal and prismatic ice planes and compare results to previous estimates from equilibrium correlation functions and simulations at controlled supercooling.

Anomalous kinetics, patterns formation in recalescence, and final microstructure of rapidly solidified Al-rich Al-Ni alloys

From thermodynamical consideration, rather a monotonically increasing crystal growth velocity with increasing undercooling is expected in the crystallization of liquids, mixtures, and alloys [P.K. Galenko and D. Jou, Physics Reports 818 (2019) 1]. By contrast to this general theoretical statement, Al-rich Al-Ni alloys show an anomalous solidification behavior: the solid-liquid interface velocity slows down as the undercooling increases [R. Lengsdorf, D. Holland-Moritz, D. M. Herlach, Scripta Materialia 62 (2010) 365]. It is also found that besides the anomalous growth behaviour, changes in the shape of the recalescence front as the growth front morphology occur. In the light of recent measurements in microgravity with an Al-25at.% Ni alloy sample onboard the International Space Station (ISS) results confirming this anomalous behavior as an unexpected trend in solidification kinetics are presented. The measurements show multiple nucleation events forming the growth front, a mechanism that has been observed for the first time in Al-Ni alloys [D. Herlach et al., Physical Review Materials 3 (2019) 073402; M. Reinartz et al. JOM 74 (2022) 2420] and summarized with detailed analysis in the present publication over a wider range of concentrations. Particularly, the experimental measurements and obtained data directly demonstrate that the growth front does thus not consist of dendrite tips (as in usual rapid solidifying samples), but of newly forming nuclei propagating along the sample surface in a coordinated manner. Theoretical analysis on intensive nucleation ahead of crystal growth front is made using the previously developed model [D.V. Alexandrov, Journal of Physics A: Mathematical and Theoretical 50 (2017) 345101]. Using equations of this model, quantitative calculations confirm the interpretation of experimentally observed propagation of the recalescence front and obtained data on the microstructure of droplets solidified in electromagnetic levitation facility (EML) on the Ground, under reduced gravity during parabolic flights, and in microgravity conditions onboard the ISS.

Effects of local nonequilibrium in rapid eutectic solidification—Part 1: Statement of the problem and general solution

Numerous experimental data on the rapid solidification of eutectic systems exhibit the formation of metastable solid phases with the initial (nominal) chemical composition. This fact is explained by suppression of eutectic decomposition due to diffusionless (chemically partitionless) solidification beginning at a high but a finite growth velocity of crystals. A model considering the diffusionless growth is developed in the present work to analyze the atomic diffusion ahead of lamellar eutectic couples growing into supercooled liquid. A general solution of the model is presented from which two regimes are followed. The first presents a diffusion‐limited regime with the existence of eutectic decomposition if the solid–liquid interface velocity is smaller than the characteristic diffusion speed in the bulk liquid. The second shows suppression of eutectic decomposition under diffusionless transformation from liquid to one‐phase solid if the solid–liquid interface velocity overcomes characteristic diffusion speed in the bulk liquid.

Formation of columnar lamellar colony grain structure in a high Nb-TiAl alloy by electron beam melting

The grain morphology and texture control in electron beam melted (EBM) Ti-47Al-8Nb γ-TiAl alloy is considered. The EBM process window to obtain a columnar lamellar colony (CLC) grain structure was defined following a critical assessment of thermal gradient and liquid-solid interface velocity by using numerical simulation. Experimentally, an epitaxial grain growth during solidification of Ti-47Al-8Nb has been realised by using the optimum EBM parameter sets. The length of the CLC grain structure reached up to ∼600 μm (compared to the powder layer thickness of 70 μm). The texture analysis and phase identification performed using electron backscatter diffraction (EBSD) provided important insights in understanding the solidification and phase transformation processes during the EBM fabrication. It was found that the solidification path for EBM high Nb-TiAl alloy involves the high-temperature α-phase field (i.e. L+β→α and α→α2+γ phase transformation processes). The epitaxial growth of prior β grains and the anchoring effect of residual B2-phase are very likely to be responsible for the formation of CLC microstructure.

Phase Field Simulations of Microstructure Evolution in IN718 Using a Surrogate Ni–Fe–Nb Alloy during Laser Powder Bed Fusion

The solidification microstructure in IN718 during additive manufacturing was modeled using phase field simulations. The novelty of the research includes the use of a surrogate Ni–Fe–Nb alloy that has the same equilibrium solidification range as IN718 as the model system for phase field simulations, the integration of the model alloy thermodynamics with the phase field simulations, and the use of high-performance computing tools to perform the simulations with a high enough spatial resolution for realistically capturing the dendrite morphology and the level of microsegregation seen under additive manufacturing conditions. Heat transfer and fluid flow models were used to compute the steady state temperature gradient and an average value of the solid-liquid (s-l) interface velocity that were used as input for the phase field simulations. The simulations show that the solidification morphology is sensitive to the spacing between the columnar structures. Spacing narrower than a critical value results in continued growth of a columnar microstructure, while above a critical value the columnar structure evolves into a columnar dendritic structure through the formation of secondary arms. These results are discussed in terms of the existing columnar to dendritic transition (CDT) theories. The measured interdendritic Nb concentration, the primary and secondary arm spacing is in reasonable agreement with experimental measurements performed on the nickel-base superalloy IN718.

Matching time and spatial scales of rapid solidification: dynamic TEM experiments coupled to CALPHAD-informed phase-field simulations

A combination of dynamic transmission electron microscopy (DTEM) experiments and CALPHAD-informed phase-field simulations was used to study rapid solidification in Cu–Ni thin-film alloys. Experiments—conducted in the DTEM—consisted of in situ laser melting and determination of the solidification kinetics by monitoring the solid–liquid interface and the overall microstructure evolution (time-resolved measurements) during the solidification process. Modelling of the Cu–Ni alloy microstructure evolution was based on a phase-field model that included realistic Gibbs energies and diffusion coefficients from the CALPHAD framework (thermodynamic and mobility databases). DTEM and post mortem experiments highlighted the formation of microsegregation-free columnar grains with interface velocities varying from ∼0.1 to ∼0.6 m s−1. After an ‘incubation’ time, the velocity of the planar solid–liquid interface accelerated until solidification was complete. In addition, a decrease of the temperature gradient induced a decrease in the interface velocity. The modelling strategy permitted the simulation (in 1D and 2D) of the solidification process from the initially diffusion-controlled to the nearly partitionless regimes. Finally, results of DTEM experiments and phase-field simulations (grain morphology, solute distribution, and solid–liquid interface velocity) were consistent at similar time (μs) and spatial scales (μm).

Verification and validation of a rapid heat transfer calculation methodology for transient melt pool solidification conditions in powder bed metal additive manufacturing

A fundamental understanding of spatial and temporal thermal distributions is crucial for predicting solidification and solid-state microstructural development in parts made by additive manufacturing. While sophisticated numerical techniques that are based on finite element or finite volume methods are useful for gaining insight into these phenomena at the length scale of the melt pool (100–500μm), they are ill-suited for predicting engineering trends over full part cross-sections (>10×10cm) or many layers over long process times (>many days) due to the necessity of fully resolving the heat source characteristics. On the other hand, it is extremely difficult to resolve the highly dynamic nature of the process using purely in-situ characterization techniques [1]. This paper proposes a pragmatic alternative based on a semi-analytical approach to predicting the transient heat conduction during powder bed metal additive manufacturing processes. The model calculations were theoretically verified for selective laser melting of AlSi10Mg and electron beam melting of IN718 powders for simple cross-sectional geometries and the transient results are compared to steady state predictions from the Rosenthal equation. It is shown that the transient effects of the scan strategy create significant variations in the melt pool geometry and solid-liquid interface velocity, especially as the thermal diffusivity of the material decreases and the pre-heat of the process increases. With positive verification of the strategy, the model was then experimentally validated to simulate two point-melt scan strategies during electron beam melting of IN718, one intended to produce a columnar and one an equiaxed grain structure. Through comparison of the solidification conditions (i.e. transient and spatial variations of thermal gradient and liquid-solid interface velocity) predicted by the model to phenomenological CET theory, the model accurately predicted the experimental grain structures.

Localized melt-scan strategy for site specific control of grain size and primary dendrite arm spacing in electron beam additive manufacturing

In addition to design geometry, surface roughness, and solid-state phase transformation, solidification microstructure plays a crucial role in controlling the performance of additively manufactured components. Crystallographic texture, primary dendrite arm spacing (PDAS), and grain size are directly correlated to local solidification conditions. We have developed a new melt-scan strategy for inducing site specific, on-demand control of solidification microstructure. We were able to induce variations in grain size (30 μm–150 μm) and PDAS (4 μm - 10 μm) in Inconel 718 parts produced by the electron beam additive manufacturing system (Arcam®). A conventional raster melt-scan resulted in a grain size of about 600 μm. The observed variations in grain size with different melt-scan strategies are rationalized using a numerical thermal and solidification model which accounts for the transient curvature of the melt pool and associated thermal gradients and liquid-solid interface velocities. The refinement in grain size at high cooling rates (>104 K/s) is also attributed to the potential heterogeneous nucleation of grains ahead of the epitaxially growing solidification front. The variation in PDAS is rationalized using a coupled numerical-theoretical model as a function of local solidification conditions (thermal gradient and liquid-solid interface velocity) of the melt pool.

Analytics on large microstructure datasets using two-point spatial correlations: Coarsening of dendritic structures

We extend the existing framework [1–4] of two-point spatial correlations to allow for the quantification, analyses, interpretation and visualization of microstructure coarsening measured by time-resolved X-ray computed tomography. Specifically, extensions were made to facilitate (i) the incorporation of nonconventional local attributes such as solid-liquid interface, interface curvature, and interface velocity in the description of the local state, and (ii) the efficient computation of bulk spatial correlations when the local attributes are sparsely defined only at special locations in the three-dimensional volume (e.g., solid-liquid interfaces). We have explored multiple variants of spatial correlations, including Pearson correlation coefficients and two-point joint probabilities, and examined their relative merits in providing useful new insights into the coarsening process. Algorithmic enhancements needed to carry out these computations on the large datasets produced in the experiments are also described. The results demonstrate the remarkable ability of these new protocols in automated (unbiased) capture of the four-fold symmetry of the dendritic microstructure, and in providing quantitative and reliable estimates of the characteristic lengths associated with the dendritic microstructure (including the secondary and tertiary dendrite arm spacings, secondary dendrite arm diameter, and the solute diffusion length). These estimated quantities agree well with the direct measurements from the microstructure. The results also indicate that interfaces with high negative and near-zero mean curvature (H) have long range spatial auto-correlations, whereas all values of the interfacial normal velocity (V) are only auto-correlated in the short range in space. For mid-range (positive) values of H and non-extreme values of V, the spatial distributions are essentially random.

Thermographic Microstructure Monitoring in Electron Beam Additive Manufacturing

To reduce the uncertainty of build performance in metal additive manufacturing, robust process monitoring systems that can detect imperfections and improve repeatability are desired. One of the most promising methods for in situ monitoring is thermographic imaging. However, there is a challenge in using this technology due to the difference in surface emittance between the metal powder and solidified part being observed that affects the accuracy of the temperature data collected. The purpose of the present study was to develop a method for properly calibrating temperature profiles from thermographic data to account for this emittance change and to determine important characteristics of the build through additional processing. The thermographic data was analyzed to identify the transition of material from metal powder to a solid as-printed part. A corrected temperature profile was then assembled for each point using calibrations for these surface conditions. Using this data, the thermal gradient and solid-liquid interface velocity were approximated and correlated to experimentally observed microstructural variation within the part. This work shows that by using a method of process monitoring, repeatability of a build could be monitored specifically in relation to microstructure control.

Thermodynamic and kinetic solid-liquid interface properties from transition path sampling.

We perform transition path sampling simulations to determine two of the key quantities in solidification, the solid-liquid interface energy and velocity, in a Lennard-Jones system. Our approach is applicable to a wide range of temperature and pressure conditions, at the melting temperature and out-of-equilibrium. We show that small system sizes are sufficient for good values of interface energies and velocities. The transition path sampling method thus offers an attractive and robust alternative for the evaluation of solid-liquid interface properties.

Effect of solid–liquid interface velocity on the measured upper and lower limits of the liquidus temperature of cobalt–carbon eutectic fixed points

The melting temperatures of the eutectic phase transitions of cobalt–carbon, platinum–carbon and rhenium–carbon alloys are proposed as reference standards for high temperature radiation thermometry. The behaviour of these is such that rather than a unique readily identified feature of the melting curve being assigned a temperature, upper and lower limits will be given. The lower limit will be the point of inflection in the melting curve, a point previously identified as unchanging with the rate at which melting takes place. The upper limit will be based on an extrapolation of the tangent at the point of inflection. The melting curves of cobalt–carbon fixed points are measured and it is found that the upper and lower limits converge as the melting rate is reduced. The point of inflection appears to have a melting rate dependence while the upper limit appears to be much more stable. This may have implications for planned research into assignment of thermodynamic temperature to these fixed points.

Leading Wave Crystallization Induced by Micro-Thermal-Plasma-Jet Irradiation of Amorphous Silicon Films

The crystalline grain growth of silicon induced by micro-thermal-plasma-jet irradiation has been directly observed using a high-speed camera. An oval-shaped molten region (MR) was formed after the solid phase crystallization (SPC), and it was clearly observed that laterally large grains grew perpendicular to the liquid–solid interface. Leading wave crystallization (LWC), which showed intermittent grain growth with a liquid–solid interface velocity as high as 4500 mm/s, was discovered in between the MR and SPC region. From numerical calculation, it has been clarified that the explosive lateral growth of LWC is triggered by the formation of a thin liquid layer and the explosive propagation of the layer is driven by released latent heat.

A cellular automaton model for dendrite solidification of pure substance

A cellular automaton (CA) model was developed to describe microstructural evolution of pure substance. A self-consistent interface energy conservation was used to determine the growth velocity of solid-liquid interface instead of the KGT model as is commonly used in previous CA models. The mesh and time dependency of the morphological evolution was also investigated, and the stability of the model was validated. The free dentritic growth of pure succinonitrile in the undercooled melt was simulated by the present model. The results were compared with Glicksman's isothermal dendtitic growth experiment and LMK marginal stability theory. The model reproduced qualitatively most of the dendritic features observed experimentally. The results were in good agreement with theoretical and experimental results.

A New Growth Kinetics in Simulation of Dendrite Growth by Cellular Automaton Method

A modified cellular automaton model was proposed to simulate the dendrite growth of alloy. Different from previous models, this model used neither an analytical equation(such as KGT model) nor an interface solute gradient equation to solve the velocity of solid-liquid interface, but used the interface solute and energy conservation and thermodynamic equilibrium condition to describe the solid/liquid interface growth kinetics process. In present model, once the temperature field and solute field were solved by finite different method in the entire domain, the material thermodynamic properties can be substituted into four algebraic equations to easily determine the variation of solid fraction, interface temperature and solute concentration, instead of calculating interface moving velocity. As a result, the complexity of the calculation can be largely reduced. The simulated dendrite growth was in a good agreement with the Lipton–Glicksman–Kurz (LGK) model for free dendritic growth in undercooled melts.

Diffusionless crystal growth in a eutectic system during rapid solidification

Experiments on nonequilibrium rapid eutectic growth are surveyed. The applicability limits of the modern theoretical models describing rapid solidification of binary systems are assessed. A problem of rapid eutectic growth when the local equilibrium is violated in the solute diffusion field (in the bulk liquid and at the solid-liquid interface) is formulated. An analytical solution to the problem of rapid lamellar eutectic growth under local nonequilibrium conditions in the solute diffusion field is found. It is shown that the diffusion-limited growth of a eutectic pattern ceases as soon as a chemically homogeneous crystalline phase begins to grow when the critical point V=VD is achieved (V is the solid-liquid interface velocity and VD is the solute diffusion speed in the bulk liquid). At V ≥ VD, eutectic decomposition is suppressed and the nascent homogeneous crystalline phase has the initial (nominal) chemical composition of the binary system.

Diffusionless Crystal Growth in Rapidly Solidifying Eutectic Systems

Using a local nonequilibrium model of solidification, experiments on rapid eutectic growth are analyzed. An analytical solution of a problem of rapid lamellar eutectic growth under local nonequilibrium conditions in the solute diffusion field is found. It is shown that solute diffusion-limited growth of a eutectic pattern is completely finished, and diffusionless growth of the chemically homogeneous crystalline phase begins to proceed at a critical point V = V(D), where V is the solid-liquid interface velocity and V(D) is the solute diffusion speed in bulk liquid. A suppression of eutectic decomposition occurs in the range V > or = V(D) that results in a growth of homogeneous crystal phase with the initial (nominal) chemical composition of the binary system.

Simulation of casting solidification and its grain structure prediction using FEM

Metal casting is one of the basic manufacturing processes. Numerical simulation of solidification is useful in getting high quality castings and minimising product cost and scrap. Finite element analysis in heat transfer can be used for this purpose. In this study, a program for finite element modelling of casting solidification was developed. The salient features of the program are: facility to incorporate latent heat through enthalpy method, incorporation of air gap by coincident node technique, ability to handle non-linear transient heat conduction through temperature dependent material properties, and object oriented programming. The code was written in C++ language and Gaussian elimination method was used to solve the matrices. This program can handle two-dimensional solidification of pure metals and alloys. Using this program, solidification of infinite slab of water was simulated and the results were compared with the reported literature and ANSYS and were found to be in good agreement. Also, solidification of Al–6wt.% Si alloy in sand and metal mould was simulated and compared with the experimental findings and simulation results reported in the literature and ANSYS. For a given geometric profile, the solidification time obtained from the present program was 1200s as against the literature of 1100s and ANSYS package of 900s. So it was found to be in good agreement with the research and experimental findings. The program was limited to 2D heat transfer analysis using linear quadrilateral elements. The program could be extended for predicting the grain structure. From the simulation results, the solid–liquid interface velocity (R) and solidification rate (dT/dt) could be calculated and using these values thermal gradient (G) could be found and there by the most probable grain structure could be predicted.