Undercooled Melt Research Articles

Mathematical Modeling of the Solid–Liquid Interface Propagation by the Boundary Integral Method with Nonlinear Liquidus Equation and Atomic Kinetics

In this paper, we derive the boundary integral equation (BIE), a single integrodifferential equation governing the evolutionary behavior of the interface function, paying special attention to the nonlinear liquidus equation and atomic kinetics. As a result, the BIE is found for a thermodiffusion problem of binary melt crystallization with convection. Analyzing this equation coupled with the selection criterion for a stationary dendritic growth in the form of a parabolic cylinder, we show that nonlinear effects stemming from the liquidus equation and atomic kinetics play a decisive role. Namely, the dendrite tip velocity and diameter, respectively, become greater and lower with the increasing deviation of the liquidus equation from a linear form. In addition, the dendrite tip velocity can substantially change with variations in the power exponent of the atomic kinetics. In general, the theory under consideration describes the evolution of a curvilinear crystallization front, as well as the growth of solid phase perturbations and patterns in undercooled binary melts at local equilibrium conditions (for low and moderate Péclet numbers). In addition, our theory, combined with the unsteady selection criterion, determines the non-stationary growth rate of dendritic crystals and the diameter of their vertices.

Crystal nucleation in Au49Ag5.5Pd2.3Cu26.9Si16.3 glass and undercooled melt

Crystal nucleation is an indispensable step in the crystallization of metallic glasses. With the increase of undercooling, different nucleation kinetics from that predicted by the classical nucleation theory (CNT) could be revealed. A prerequisite to addressing the nucleation kinetics is building a well-identified amorphous structure, which is yet arduous. Moreover, due to the limited temporal and spatial resolution of the conventional experimental methods, nucleation during the crystallization of metallic glass cannot be well recognized. In this study, nanocalorimetry with ultrahigh sensitivity and ultrafast scanning rate is used to address the nucleation kinetics in glass and undercooled melt. With the increase in cooling rate, crystallization, homogeneous nucleation, and heterogeneous nucleation are successively distinguished. Accordingly, the critical cooling rates suppressing crystallization and nucleation are estimated, based on which a well-identified amorphous phase for nucleation studies is produced. By applying Tammann’s two-stage crystal nuclei development method, nucleation in glass and undercooled melt is investigated. According to the evolution of crystallization heat and overall heat on the reheating after the annealing, the underlying kinetic mechanism is revealed with the Johnson-Mehl-Avrami (JMA) method. The nucleation half-time indicates that homogeneous nucleation is dominant for the crystallization below Tg. For the annealing at and above Tg, crystallization completes by the inevitable heterogeneous nucleation followed by crystal growth.

Dynamic nucleation in sub-critically undercooled melts during electromagnetic levitation

By classical nucleation theory, sub-critically undercooled melts in electromagnetic levitation are expected to be very stable for extended periods of time. While electromagnetic levitation experiments are typically consistent with classical nucleation theory, there have been several historical instances including experiments on zirconium during the MSL-1 campaign and during the IML-2 experiments where the molten sample solidified at sub-critical undercoolings. While different electromagnetic levitation conditions were present in these anomalous nucleation events, both sets of anomalous nucleation events were attributed to dynamic nucleation. The work presented here investigates more recent experiments in the International Space Station Electromagnetic Levitation facility in which both sets of anomalous nucleation events were replicated and further investigated. The conditions of the MSL-1 solidification events were replicated in the ISS-EML using a pure zirconium sample held in isothermal conditions between 45 °C and 290 °C below the melting temperature. The sample successfully solidified in 18 of these experiments in less than 600 s where classical nucleation theory predicts the melt to remain liquid for very long periods of time. The IML-2 result, was replicated in the ISS-EML in which a Zr64Ni36 sample was used to demonstrate pulse-triggered nucleation events. During these experiments, the excitation pulse triggered solidification in the sample which was held between 59.5 °C and 64.5 °C below the melting temperature. Again, classical nucleation predicts that under these conditions the sample will remain molten. The results of both sets of experiments are consistent with dynamic nucleation theory in which nucleation events are affected by the flow conditions within the drop.

Petrogenesis of zoned and unzoned mafic pegmatites: An insight from the Palaeoproterozoic mafic-ultramafic Hamn intrusion, Northern Norway

Mafic pegmatites have been reported from various geological environments, including ophiolites, layered magmatic intrusions, and volcanic arcs, but their petrogenesis stayed poorly constrained. This study brings new mineralogical and geochemical data obtained from unzoned and zoned gabbroic pegmatites hosted by the 1.8 Ga mafic-ultramafic Hamn intrusion, Northern Norway, with an aim to improve our understanding of the primary magmatic physicochemical factors that control the development of pegmatitic textures in mafic rocks. The unzoned mafic pegmatites in the Hamn intrusion are hosted in gabbronorite and gabbro. The pegmatites differ macroscopically from the gabbronorite only by grain size and the color of feldspars. The bulk chemical compositions, including REE concentrations and rock-forming mineral assemblages of both rocks, are identical. The mineral composition of the pegmatite and host gabbro overlap. However, the pegmatite experienced epidotization and scapolitization, suggesting that H 2 O- and chlorine were essential in the formation. Furthermore, the pyroxene of the pegmatite shows a more distinct negative Eu anomaly than the pyroxene of the host gabbronorite. Amphibole dyklets associated with the unzoned pegmatite pockets consist of Cl-rich pargasite and Cl-scapolite, indicating that the fluid pressure during crystallization of the unzoned pegmatites exceeded the confining pressure and resulted in the fracturing of host rocks and expulsion of a Cl- and H 2 O-rich fluid and residual melt that subsequently formed amphibole veins. The zoned pegmatites of the mafic-ultramafic Hamn intrusion are internally differentiated in terms of grain size, texture, and mineral composition. A characteristic comb-like diopside layering shows a change from numerous small to fewer larger grains from the rim towards the core of the pegmatite pocket. This textural change suggests that the pegmatite-forming melt experienced a transition from a high nucleation rate (N) vs. growth rate (G) ratio to a low N/G ratio. A decreasing degree of undercooling with continuous crystallization from the rim to the core of the pockets can explain this transition. Apart from the comb-like layering, the bifurcating texture of some of the diopside grains is another evidence that the pegmatite-forming melt experienced undercooling. The pegmatites could have formed from a remobilized intercumulus or fractionated melt that was emplaced as residual melt into the colder host rock. Fast heat diffusion towards the host rock could have caused undercooling of the pegmatite-forming melt, which led to the formation of the comb-like and partly extremely coarse-grained texture. In contrast to the unzoned pegmatites, the zoned pegmatites lack evidence of significant involvement of chlorine- and H 2 O. Instead, the fluid inclusion study revealed that the pegmatite-forming melt was enriched in CO 2 . Microthermometry of the CO 2 -bearing inclusions indicates a minimum formation pressure of 647 to 734 MPa, and the titanium-in-quartz geothermometer yields a minimum formation temperature of 753 ± 34 °C for the quartz segregation. • The mafic-ultramafic Hamn intrusion hosts unzoned and zoned gabbroic pegmatites. • The unzoned pegmatites crystallized from a H 2 O and Cl rich mafic melt. • Change in degree of undercooling led to a comb-like texture of the zoned pegmatites. • The zoned pegmatites crystallized from a dry but CO2-enriched mafic melt.

Substrate stimulating technique for Ni-based single crystal superalloy preparation during direction solidification

The grain selector method is a prevailing technique in the manufacturing process of Ni-based single crystal superalloy, however, it cannot control the orientation of single crystal within an appropriate scope. In this study, a novel technique named substrate stimulating method was proposed and DD32 Ni-based single crystal superalloy was prepared. The microstructure evolution was characterized by EBSD and OM, and the temperature field and temperature gradient distribution were predicted by the finite volume method. With a substrate height of 18 mm, the proportion of grains with deviation angles less than 12° and 8° account for 100 % and 99 % at the entrance of spiral selector, compared with 80 % and 51 % by grain selector method, which makes significance sense for optimizing the orientation of single crystal casting and improving the yield of single crystal casting. The morphology and the orientation of the substrate stimulated grains, which are mainly determined by temperature gradient and melt undercooling, could be optimized by adjusting the substrate height and melt pouring temperature.

Effect of addition of La and Ce on solidification behavior of Al-Cu alloys

• The first time applying in situ synchrotron imaging to Al-Cu alloy with La-Ce. • The real time solidification of Al-Cu alloy with La-Ce has been captured. • La-Ce decreases nucleation density of α-Al at a lower cooling rate. • La-Ce leads to grain refinement of α-Al at a higher cooling rate. The effect of La-Ce addition on the solidification behavior of hypoeutectic Al-Cu alloy was investigated using in situ synchrotron X-ray imaging with conventional casting experiments. The results show that La-Ce addition leads to a decrease in nucleation density and coarser grains at a lower cooling rate since La-Ce addition can suppress α-Al nucleation to a certain extent. At a higher cooling rate, melt undercooling can be increased by inhibiting early nucleation of α-Al on heterogeneous particles with large size, resulting in grain refinement.

The Effect of the Shear Flow on Columnar Crystal Growth in an Undercooled Melt

Herein, the effect of the shear flow on the growth of columnar crystals in an undercooled melt is studied. The asymptotic method is used to solve the dynamic model for the growth of a columnar crystal. The resulting asymptotic solution shows that the shear flow significantly changes the interface morphology of the columnar crystal. With the shear effect of the forced flow, the growth rate of the columnar interface increases in the shear direction of the shear flow. As the shear rate of the shear flow further increases, the interface of the columnar crystal is seriously deformed and distorted. The shear flow causes the columnar crystal in the undercooled melt to tend to evolve into smaller crystals in the initial stage of crystal growth. The analytical result provides a prediction of the formation of interface microstructures during solidification through the change of processing parameters.

The Effect of the Shear Flow on the Morphological Pattern of Particles in an Undercooled Melt

The effect of shear flow on the morphological pattern of particles in an undercooled melt is studied by using the asymptotic method. The mathematical model of the particle includes the anisotropic interface kinetic undercooling. The asymptotic solution for the mathematical model of the particle shows that shear flow in an undercooled melt intensifies the deformation and distortion of the particle in the initial stage of crystal growth. Due to the shear flow, the growth rate of the interface increases in the shear direction of the flow and strengthens the inward decay of the part of the interface induced by the anisotropic interface kinetics in the initial stage of crystal growth. As the shear rate of the flow increases, the interface of the particle is seriously deformed and distorted until it breaks into smaller particles. The analytical result provides the prediction of the formation of interface microstructures during solidification through the change of processing parameters.

First evidence of dmisteinbergite (CaAl2Si2O8 polymorph) in high-grade metamorphic rocks

Abstract We identified dmisteinbergite, the rare trigonal polymorph of CaAl2Si2O8, for the first time in high-grade metamorphic rocks. Dmisteinbergite occurs as a crystallization product of silicate melt inclusions (nanogranitoids) in garnet from three host rocks with different protoliths and re-equilibration conditions, i.e., from 1.0 to 4.5 GPa. Raman spectra and compositions of the dmisteinbergite here investigated are overall identical to those of previously characterized artificial and natural dmisteinbergite. In nanogranitoids, this phase coexists with other metastable polymorphs of feldspar (kumdykolite, kokchetavite) and SiO2 (quartz, cristobalite), recently interpreted as the result of undercooling, supersaturation and rapid crystallization of a silicate melt confined in a micrometric pore. Dmisteinbergite formation likely results from a similar process, and thus it should be regarded as a kinetically controlled phase. Moreover, the investigation of dmisteinbergite as well as of other metastable feldspar polymorphs offers new insights into the behavior of natural materials under non-equilibrium conditions.

A Stable Mode of Dendritic Growth in Cases of Conductive and Convective Heat and Mass Transfer

In this paper, we develop a theory of stable dendritic growth in undercooled melts in the presence of conductive and convective heat and mass transfer boundary conditions at the solid/liquid interface of a dendrite. To simplify the matter and construct the analytical theory, conductive and convective mechanisms are considered separately. Namely, the laws for total undercooling and selection criterion defining the stable growth mode (dendrite tip velocity and diameter) are derived for conductive and convective boundary conditions. To describe the case of simultaneous occurrence of these heat and mass transfer mechanisms, we sew together conductive and convective laws using power stitching functions. The generalised selection theory is compared with experimental data for Al24Ge76 and Ti45Al55 undercooled melts.

Rapid Eutectic Growth Kinetics of Undercooled Nb-Si Alloys at Electrostatic Levitation State

The rapid eutectic growth kinetics of undercooled Nb-Si alloys was investigated by using electrostatic levitation (ESL) technique combined with triggering method. For eutectic Nb-17.3 at.% Si alloy, the rapidly solidified microstructure was characterized by (Nb) dendrites and (Nb)+Nb3Si eutectics, which was different from the master alloy composed of complete eutectic structures. The transition of primary phase from the faceted Nb3Si phase to non-faceted (Nb) phase and the formation of complete eutectic microstructure were observed within hypereutectic Nb-18.7 at.% Si alloy with the increase of undercooling. Three distinct solidification paths leading to different microstructures and constituent phases were revealed by controlling the melt undercooling. Furthermore, the phase selection mechanism was clarified from the perspective of nucleation kinetics. Based on the experimental and analytical results, the coupled zone of (Nb)+Nb3Si eutectic was qualitatively predicted. The final microstructures solidified at different undercoolings were all composed of the (Nb) phase and Nb3Si phase, whose crystaline structures were both confirmed by transmission electron microscopy analysis. Besides, it was found that the volume fractions of (Nb) and Nb3Si phases could be undercooling control adjusted by. According to the nano-indentation and Vickers micro-indentation tests, the mechanical properties varied significantly with solidification paths. Compared with the master alloy, the hardness of hypereutectic Nb-18.7 at.% Si alloy at 157 and 473 K undercoolings was decreased by 42.7 % and 9.9 %, while the fracture toughness was enhanced remarkably by 593.1 % and 144.8 %, respectively. This may provide a desirable approach for designing Nb-Si alloys with improved performances.

Viscosity and structural relaxation of silver-doped (GeS2)50(Sb2S3)50

The viscosities of Agx[(GeS2)50(Sb2S3)50]100-x glasses and undercooled melts for x = 5, 7.5, and 15 are studied in the range of 106–1013 Pa·s. The measurements are recorded using a thermomechanical analyzer and two experimental setups, the parallel-plate and penetration methods. The structural relaxation of the composition Ag10[(GeS2)50(Sb2S3)50]90 is studied using differential scanning calorimetry and thermomechanical analysis. The enthalpy and volume relaxation are characterized and described using the Tool-Narayanaswamy-Moynihan (TNM) model. It is verified that both the enthalpy and volume relaxation can be described using similar values of the TNM model parameters. The parameters characterizing the viscous flow and structural relaxation, and their relationship, are discussed.

Dynamical law of the phase interface motion in the presence of crystals nucleation

In this paper, we develop a theory of solid/liquid phase interface motion into an undercooled melt in the presence of nucleation and growth of crystals. A set of integrodifferential kinetic, heat and mass transfer equations is analytically solved in the two-phase and liquid layers divided by the moving phase transition interface. To do this, we have used the saddle-point method to evaluate a Laplace-type integral and the small parameter method to find the law of phase interface motion. The main result is that the phase interface Z propagates into an undercooled melt with time t as Z(t)=sigma sqrt{t}+varepsilon chi t^{7/2} with allowance for crystal nucleation. The effect of nucleation is in the second contribution, which is proportional to t^{7/2} whereas the first term sim sqrt{t} represents the well-known self-similar solution. The nucleation and crystal growth processes are responsible for the emission of latent crystallization heat, which reduces the melt undercooling and constricts the two-phase layer thickness (parameter chi <0).

Effect of anisotropic interface kinetics on morphological pattern of a particle in an undercooled melt

We study the morphological pattern of a particle in an undercooled melt with anisotropic interfacial atoms attachment kinetics by using the asymptotic method. The resulting asymptotic solution reveals the locally inward concave growth of the particle in the initial stage of crystal growth after nucleation and the variation of orientation selection. In the initial stage of crystal growth after nucleation, due to the effect of anisotropic interfacial atoms attachment kinetics, some parts of the interface first grow locally inward. Until the locally inward growth attains a certain distance, these parts start to grow outward with other parts. The locally inward concave growth and subsequent outward growth form the time window of inward growth of the particle, within which the particle has an inner radius less than the critical radius of nucleation. With the anisotropy parameters are systematically changed, the particle changes continuously its preferred growth direction from one preferred growth direction to another preferred growth direction. This orientation selection induced by anisotropic interfacial atoms attachment kinetics displays the rich morphological patterns of the particle with various deformed petal-shaped patterns.

Parallel-GPU AMR implementation for phase-field lattice Boltzmann simulation of a settling dendrite

The motion of the equiaxed grains plays an important role in macrosegregation. To simulate the motion of three-dimensional (3D) equiaxed dendrites growing in an undercooled melt, a large computational domain with small grids is required. In this study, we investigated the morphology and velocity of the settling dendrite and the growth velocity of its primary arms by implementing parallel computing with multiple graphics processing units (GPUs) for the adaptive mesh refinement (AMR) method (parallel-GPU AMR). A multiple-grid (MG) method that assigns different grid sizes to each field was also applied to the parallel-GPU AMR. A large-scale phase-field lattice Boltzmann (PF-LB) parallel-GPU AMR simulation of the settling dendrite was performed. The parallel GPU AMR method developed for the 3D PF-LB solidification model can evaluate the growth and motion of 3D equiaxed dendrite settling in an undercooled melt. The results are achieved with a significant acceleration of the 3D PF-LB simulations of settling dendrite and using a significantly reduced number of GPUs required for the simulation.

Molecular dynamics determination of Two-dimensional nucleation kinetic coefficient for modeling the faceted growth of Si (1 1 1) from an undercooled melt

The discrepancies between kinetic model predictions and experimental observations of two-dimensional (2D) nucleation-mediated growth of silicon limits modeling reliability for existing and new crystal growth processes. Molecular dynamics (MD) simulations were performed to identify the mechanism of evolution of crystallites on a Si (111) facet and semi-quantitatively describe 2D nucleation kinetics using the forced-velocity solidification (FVS) and free-solidification (FS) MD simulations techniques. Both MD models predicted similar nucleation expressions but gave lesser nucleation energy barriers than predicted from Monte Carlo (MC) nucleation model. The estimated nucleation rate from MD was fitted to a polynuclear growth model to estimate a 2D kinetic model and compared to available experimentally reported growth rates. The Si (111) facet velocity model derived from the kinetic coefficient given in this work generally provided more conservative estimates of undercooling and the minimum undercooling that may result in kinetic roughening transition. In addition, the FVS model implemented in this work provided a unique opportunity for qualitatively describing the behavior of a crystal-melt interface and gave a molecular-level perspective on the interface stability criterion for the growth of single-crystal silicon during the horizontal ribbon growth (HRG) process.

Analysis of the boundary integral equation for the growth of a parabolic/paraboloidal dendrite with convection

The growth of a parabolic/paraboloidal dendrite streamlined by viscous and potential flows in an undercooled one-component melt is analyzed using the boundary integral equation. The total melt undercooling is found as a function of the Péclet, Reynolds, and Prandtl numbers in two- and three-dimensional cases. The solution obtained coincides with the modified Ivantsov solution known from previous theories of crystal growth. Varying Péclet and Reynolds numbers we show that the melt undercooling practically coincides in cases of viscous and potential flows for a small Prandtl number, which is typical for metals. In cases of water solutions and non-metallic alloys, the Prandtl number is not small enough and the melt undercooling is substantially different for viscous and potential flows. In other words, a simpler potential flow hydrodynamic model can be used instead of a more complicated viscous flow model when studying the solidification of undercooled metals with convection.

Accelerated crystal growth in a lithia aluminosilicate glass

No viscosity change in the vicinity of the crystal/melt interface is to be expected for crystal growth in silicate systems at constant temperature in a deeply undercooled isochemical melt. In most experimental studies, however, absolute chemical conformity is not realized and the formation of an interfacial zone with presumably strongly altered viscosity will occur. In the present investigation isolated Li2O⋅Al2O3⋅4SiO2 (LAS4) single crystals grew in a slightly alumina deficient base glass at temperatures above glass transition. Due to the depletion of alumina and the enrichment of excess components and impurities of the continuously growing interfacial zone (“shell”), its viscosity will change. Dedicated experiments showed that the viscosity of the interfacial zone is up to 7 orders of magnitude lower than the viscosity of the base glass. The decreased viscosity at the crystal growth front is assumed to cause the experimentally observed increase of the crystallisation velocity with the crystal radius. This behaviour can be quantitatively modelled introducing a linear decrease of the activation enthalpy with the crystal radius. Obviously, the proposed model can be likewise used in the opposite case, i.e. for an increasing viscosity of the interfacial zone during crystal growth. According to the standard concept of a fractional viscosity relation, a coupling factor can formally be extracted from the comparison between the experimentally determined activation enthalpies of the crystal growth velocity and of the base glass viscosity. The resulting unrealistically low numerical value indicates that it is not the high viscosity of the base glass, but the lower viscosity of the interfacial zone that must be considered for the determination of a physically meaningful coupling factor.

Diffusion in metallic glasses and undercooled metallic melts

Diffusion in metallic glasses and undercooled metallic melts

Exploration of crystal growth behavior in Au-based metallic glass by nanocalorimetry

The Au49Ag5.5Pd2.3Cu26.9Si16.3 (at.%) bulk metallic glass is among the promising Au-based metallic glasses with the enhanced glass-forming ability and thermal stability. However, crystal growth kinetics during the crystallization of this metallic glass remains ambiguous. In this study, the rapid crystallization of a Au49Ag5.5Pd2.3Cu26.9Si16.3 bulk metallic glass in a wide temperature range was realized by nanocalorimetry. The underlying thermodynamic and kinetic features of crystal growth were ascertained based on the Cohen and Grest (C&G), Mauro-Yue-Ellison-Gupta-Allan (MYEGA) and Schmelzer models, and the adaptability of these three models was demonstrated. The maximum crystal growth rate (0.037 m/s) and corresponding temperature (595 K) estimated by these three models were similar but different crystal growth behaviors in the deeply undercooled melt were distinguished. With an increase in undercooling, the breakdown of the Stokes-Einstein relation was discovered, and a temperature-dependent decoupling exponent was demonstrated by the Schmelzer model, causing a divergence of crystal growth rate from that estimated by C&G and MYEGA models. Moreover, screw-dislocation-mediated growth was stated as the dominant crystal growth mechanism during the crystallization of the Au49Ag5.5Pd2.3Cu26.9Si16.3 bulk metallic glass.