Liquidus Slope Research Articles

The fractional latent heat of crystallizing magmas

The fractional latent heat of a crystallizing system is simply the latent heat of fusion divided by the total enthalpy. When plotted against temperature, this function displays robust pulses with the successive saturation of each incoming crystal phase. Each pulse endures for tens of degrees, corresponding to tens to hundreds of thousands of years in large magma bodies. From an original 1963 development by P.J. Wyllie using a synthetic system, I show that the liquidus slopes (degrees per gram of solid produced) decrease discontinuously at the arrival of a new phase, and their inverse, the crystal productivity, undergoes sharp upward pulses at the same time. The overall liquidus slope increases continuously, interrupted by small downward jumps, with the evolution of the multicomponent melt. The crystal productivity pulses feed the fractional latent heat pulses, which dominate the crystallization history of a melt. These elementary relationships govern the near-solidus growth of dihedral angles in cumulates as they relate to the liquidus events. The latent heat ordinarily approximates to about 80% of the total enthalpy budget, but jumps to 100% (by definition) when solidification occurs by isothermal adcumulus growth and approaches 100% when mafic phases such as augite and Fe-Ti oxides are over-produced (relative to their equilibrium saturation) in layered intrusions. The feedback of latent heat to a self-regulating cooling history of large magma bodies is deduced here in principle. The overall results help to clarify the crystallization history of mafic magmas. In particular, they support the solidification of floor cumulates by interchange with parent magma and without the help of compaction.

Uncertainties in the SPRT Subranges of ITS-90: Topics for Further Research

The CCT has completed the guide summarizing the uncertainties in the realization of the SPRT subranges of ITS-90 between the triple point of neon (24.5561 K) and the freezing point of silver (961.78 °C). This article identifies aspects of standard platinum resistance thermometry where either data or models are lacking and further research is required. In the calibration of SPRTs, the two main concerns are the need for data on liquidus slopes for the different impurities in the fixed points and improved understanding of the impact of the thermal environment of the fixed point on the realized temperature. In the use of SPRTs, the two largest sources of uncertainty are Types 1 and 3 non-uniqueness and oxidation. The causes of Type 3 non-uniqueness are not yet understood, especially at low temperatures, and there is a paucity of data for the high-temperature subranges. In respect of oxidation, there is a need for validation of the models developed in the 1980s, especially in light of the reduced partial pressure of oxygen used in modern SPRTs. A range of other effects including vacancy effects in SPRTs, isotopic effects in fixed points, and improved statistical methods are discussed.

Thermodynamic modeling of the BaO-SiO2 and SrO-SiO2 binary melts

The evaluation of the thermodynamic properties and the phase diagrams for the binary BaO-SiO2 and SrO-SiO2 systems is carried out using a structural model for silicate melts and glasses. This thermodynamic model is based on the assumption that an addition of metal oxides to silica results in the depolymerization of the silicon-oxygen network, with a characteristic free energy change. A least squares optimization program permits all available thermodynamic and phase diagram data to be optimized simultaneously. In this manner, data for the above binary systems have been analysed and represented with a small number of parameters. The resulting equations represent the thermodynamic and phase diagram data for the alkaline-earth oxide-silica systems within error limits for most of the experimental data. In particular, the measured limiting liquidus slope, at XSiO2 = 1, is well reproduced.

Effect of Pr doping on the growth and superconducting properties of(Y1−xPrx)Ba2Cu3O7−δ

(Y1−xPrx)Ba2Cu3O7−δ (Pr-YBCO, x<0.2) single crystals and single domain bulks were grown by the top-seededsolution-growth (TSSG) and the top-seeding melt-textured growth(TSMTG) methods, respectively. The superconducting critical temperature(Tc) was determined by magnetic measurements and the critical current density(Jc) was estimated by applying the modified Bean model. For a low-level substitution(x<0.025), an increasein the Jc value as compared with that for unsubstituted YBCO was found while no apparent change in theTc value was noticed. On the other hand, Pr substitution in YBCO leads to: (i) increasing liquid solubilityof the total rare earth element in the Ba–Cu–O solution, (ii) increasing peritectic temperature(Tp), and (iii) a decrease in the liquidus slope near theTp. As a result, the driving force for YBCO growth and the region of singlegrain growth are enhanced. Study of the compositional correlation betweenliquid and solid additionally indicates that the Pr distribution coefficient islarger than 1 in the Pr-YBCO system. Thus the Pr content and then theTp value of Pr-YBCO in the bulk reduce with the growth time during TSMTG. The lowerTp valuein the final crystallized part is beneficial to a slow cooling process. Using the TSMTG method, Pr-YBCO(x = 0.02) monodomains up to 42 mm in diameter were grown with a high growth rate.

Comparison and Extension of Free Dendritic Growth Models through Application to a Ag-15 Mass Pct Cu Alloy

The formulations of existing free dendritic growth models were compared, and an extended model was proposed that employs a subregular solution model to compute the driving force for dendritic growth without Henrian restrictions. These models were also applied to a Ag-15 mass pct Cu alloy to numerically compare their predictions. Models that address only the thermal, solutal, and curvature supercoolings do not properly account for the interface kinetics, even with modifications with the kinetic partition coefficient and liquidus slope. It is only in models that account for the interfacial driving force, −ΔG *, that the kinetic supercooling is properly addressed. All of the models in comparison yield numerically similar predictions for the solutal growth regime, but models that employ the kinetic partition coefficient and liquidus slope, but do not address the interfacial driving force, fail to correctly describe the thermal control regime. The solutal-to-thermal transition is characterized by a rapid increase of interfacial driving force, which causes the tip temperature T * to increase with increasing growth rate V. The criterion for the transition stage is given as \( d\ln( - \Delta G^* )/d\ln\,V >1 \).

Modeling of grain refinement: Part III. Al–7Si–0.3Mg aluminum alloy

Following Part I [X. Yao, et al., J. Mater. Res.23(5), 1282 (2008)] and Part II [X. Yao, et al., J. Mater. Res.23(5), 1292 (2008)] the cellular automation–finite control volume method (CAFVM) model was used to study the grain formation and microstructure morphology resulting from solidification of a commercial Al–Si–Mg alloy with Al–Ti–B grain refiner additions. The model incorporates the effect of the introduced solute Ti and the alloying elements of Si and Mg on the growth restriction factor, constitutional undercooling, and nucleation parameters. With respect to grain refinement, it is found that the alloying elements, Si and Mg, play a role that is similar to Ti qualitatively while different quantitatively. Accordingly, a concept of “equivalent solute” determined by phase diagram parameters such as the solute partitioning coefficient and the liquidus slope is proposed to clarify the effect of each solute in the alloy on grain formation during solidification. Based on the calculations and on comparison to the experimental data, a possible mechanism of grain refinement in this alloy system is proposed.

Molecular dynamics study of equilibrium concentration profiles and the gradient energy coefficient in Cu-Pb nanodroplets

Atomistic simulations were used to equilibrate the concentration profiles in $5.3\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ liquid droplets for an embedded atom method model of Cu-Pb. The strong tendency of Pb to surface segregate establishes a nonuniform profile, which decays over a length comparable to the particle radius. With the free energy vs composition determined from separate Monte Carlo simulations, the composition profile can be analyzed in terms of the Cahn-Hilliard diffuse interface model and the gradient energy coefficient, $\ensuremath{\kappa}$, can be obtained. Results from three different temperatures indicate that $\ensuremath{\kappa}$ lies in the range $1.0--1.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}\phantom{\rule{0.3em}{0ex}}\mathrm{J}∕\mathrm{m}$. In addition to the gradient energy coefficient, various aspects of phase equilibria at the nanoscale have been investigated. The critical point of the liquid-liquid miscibility was found to decrease by $\ensuremath{\approx}50\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ for the $5.3\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ particle size, the melting point of pure Cu is suppressed by $\ensuremath{\approx}110\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, and the liquidus slope is shallower than that of the bulk system.

Galena stability to 26 KBAR

The melting point of galena was measured by differential thermal analysis to 25 kbar. The melting point is 1191 °C at 5.9 kbar and increases to ∼1315 °C by 25 kbar along a concave-down trajectory. The form of the PbS melting curve resembles those of similarly structured compounds that show a range of initial slopes and curvatures. Initial liquidus slopes of several B1-structured compounds including PbS correlate well with melting volume change. However, the roughly 25 J/mol*K entropy of melting required for PbS by the volume of melting, initial liquidus slope, and Clapeyron equation is approximately double literature estimates. Similar discrepancies exist for the other B1-structured compounds. Liquidus curvature correlates well with the initial ratio of liquid to solid compressibility. PbS liquid compressibility of ∼12*10-12 cm2/ dyne is estimated from galena compressibility and initial melting slope. The transformation of galena from cubic to orthorhombic structure with pressure was determined by electrical resistance measurements up to 1000 °C and occurs at about 26 kbar with little temperature variation. Galena9s maximum stability is at the inferred triple point among galena, liquid, and orthorhombic-structured PbS at ∼26 kbar and ∼1315 °C.

Influence of an axial high magnetic field on the liquid–solid transformation in Al–Cu hypoeutectic alloys and on the microstructure of the solid

The influence of an axial high magnetic field (up to 10 T) on the liquid–solid interface morphology and microstructure of the solid has been investigated experimentally during Bridgman growth of Al–Cu hypoeutectic alloys. It is found that the field causes the interface to become destabilized and irregular, and promotes planar–cellular and cellular–dendritic transformation. The field has a great influence on the cellular and dendrite array morphology. Indeed, the field causes severe distortion in the cellular array and enhances cell branching. The field makes the morphology of the dendrite array more complex and, with the increase of the magnetic field intensities and decrease of the growth velocities, the dendrites become broken and orientate with the 〈1 1 1〉-direction along the solidification direction instead of the 〈1 0 0〉-direction. Furthermore, the field also enlarges the primary dendrite spacing and promotes the branching of the dendrites to form higher-order arms. The above phenomena may be attributed to the change of the equilibrium partition coefficient k and the liquidus slope m L caused by the field, the magnetic anisotropy of the α-Al crystal and the flow created by the thermoelectromagnetic convection.

Prediction of hot tearing tendency for multicomponent aluminum alloys

It is of practical importance to be able to predict the hot tearing tendency for multicomponent aluminum alloys. Hot tearing is one of the most common and serious defects that occurs during the casting of commercial aluminum alloys, almost all of which are multicomponent systems. For many years, the main criterion applied to characterize the hot tearing tendency of an alloy system was based on the solidification interval. However, this criterion cannot explain the susceptibility-composition relation between the limits of the pure base metal and the eutectic composition. Clyne and Davies correlated the susceptibility-composition relationship in binary systems based on the concept of the existence of critical time periods during the solidification process when the structure is most vulnerable to cracking. The Scheil equation was used in their model using constant partition coefficient and constant liquidus slope estimated from the phase diagram. In the current study, the authors followed Clyne and Davies’ general idea, and directly coupled the Scheil solidification simulation with phase diagram calculation via PanEngine, a multicomponent phase equilibria calculation interface, and extended the model to higher order systems. The predicted hot tearing tendencies correlated very well with the experimental results of multicomponent aluminum alloys.

Effect of solute on the growth rate and the constitutional undercooling ahead of the advancing interface during solidification of an alloy and the implications for nucleation

A framework is presented for modeling the nucleation in the constitutionally supercooled liquid ahead of the advancing solid/liquid interface. The effects of temperature gradient, imposed velocity, slope of liquidus, and initial concentration have been taken into account in this model by considering the effect of interface retardation, which is caused by solute buildup at the interface. Furthermore, the effect of solute concentration on the chemical driving force for nucleation has been considered in this model. The model is used for describing the nucleation of Al–Si and Al–Cu alloys. It was found that the solute of Si has a significant impact on the chemical driving force for nucleation in Al–Si alloys whereas Cu has almost no effect in Al–Cu alloys.

SIE and OME as tools to estimate the uncertainty in the correction for the effect of impurities applied to metal–carbon eutectic systems

For high-purity metal fixed points the sum of individual estimates (SIE) and the overall maximum estimate (OME) have been introduced as tools to estimate the uncertainty in the correction for the effect of impurities. In this paper we investigate the feasibility of applying these tools to metal–carbon (M–C) eutectics as well on the basis of material collected for the eutectics Fe–C and Co–C. This would be of interest since it would obviate the need for modelling the full freezing or melting curve, which requires detailed knowledge of the material properties involved. For SIE we need only the liquidus slopes pertinent to the detected impurities, whereas for OME under certain conditions detailed knowledge of the impurity parameters would not be needed at all.

The effect of kinetics on the interface stability under the non-equilibrium condition

The kinetic effect, in combination with the effect of relaxation, was considered in the interface stability criterion under the non-equilibrium solidification. Employing a fresh mathematical derivation, a more general marginal stability criterion was obtained by incorporating the non-equilibrium kinetics concluding the kinetic undercooling and the liquidus slope in the kinetic phase diagram. It was shown that the interface stability is correlated with the kinetic effect, i.e. by incorporation in the slope of non-equilibrium liquidus, but is independent of the kinetic undercooling.

On crystallization at the inner core boundary

It is widely believed that a mushy layer develops at the inner core boundary (ICB) while the outer core material freezes onto the inner core. This view is obtained in reference to metallurgical or analog experiments on solidification carried out in the laboratory where morphological instability almost always follows constitutional supercooling. However, morphological instability is not an automatic consequence of supercooling: supercooling and stable solid–liquid interface may coexist under certain conditions such as might perhaps occur at the ICB. If these conditions are realized, there could be both solidification on the ICB and crystallization of small solid particles in the supercooled layer (slurry layer). We made a linear stability analysis of a unidirectionally advancing solid–liquid interface under the ICB conditions. Several values of the liquidus slope as a function of the concentration of light material ( m C ) were used for the stability analysis. It was confirmed that supercooling and stability may coexist if | m C | is low enough, although too low to be realistic. The estimated amount of supercooling in the layer is of order 0.1 K, and this provides an estimate of the size and population of solid particles in the slurry layer. However, even if | m C | were small enough, the existence of a slurry layer at the ICB would be unlikely due to (i) the difficulty of continuously supplying enough nuclei in the layer and (ii) the Rayleigh–Taylor instability due to the compositional buoyancy. It is, therefore, most probable that a mushy layer exists at the ICB.

Effect of Anisotropy of Surface Energy on the Growth Direction of Solid Phase in Constrained Growth Condition

The growth direction of solid phase, which has a great influence on the grain selection, has been investigated. Transparent organic systems, succinonitrile-H 2 O alloy and pivalic acid-H 2 O alloy have been used for unidirectional solidification. Essential parameters of phase diagrams, distribution coefficient of H 2 O (k) and liquidus slope (m) have been determined experimentally. The relationship is analyzed between dimensionless growth velocity (V/V c , where V c is the critical growth velocity for constitutional supercooling) and dimensionless growth direction (π'). The change in π' with V/V c is common and similar for both alloy systems. π' increases rapidly at low growth velocity region. Then, π' increases slowly and approaches unity with increasing V/V c . When the anisotropy of surface energy increases, the growth direction of solid phase becomes close to the preferred growth direction even at low growth velocity region, where cellular interface is observed.

Thermodynamic Modeling of Alkali Metal Oxide‐Silica Binary Melts

The evaluation of the thermodynamic properties as well as the phase diagrams for the binary Na2O–SiO2, K2O–SiO2, and Li2O–SiO2 systems are carried out with a structural model for silicate melts and glasses. This thermodynamic model is based on the assumption that each metallic oxide produces the depolymerization reaction of silica network with a characteristic free‐energy change. A least squares optimization program permits all available thermodynamic and phase diagram data to be optimized simultaneously. In this manner, data for these binary systems have been analyzed and represented with a small number of parameters. The resulting equations represent the thermodynamic and phase diagram data for these alkali metal oxide–silica systems within experimental error limits. In particular, the measured limiting liquidus slope at is well reproduced.

Modelling dendrite evolution under rapid solidification conditions

The evolution of a single dendrite is simulated by the cellular automaton (CA) method, in which the influences of interface velocity on the solute partition coefficient ahead of the interface and liquidus slope are investigated under rapid solidification conditions. Furthermore, the kinetic undercooling ahead of the interface is also taken into account, which is neglected under normal solidification conditions. The simulated dendrite morphology is compared with that of the equilibrium conditions and it is shown that the interface easily loses stability around the dendritic tip causing more and more secondary dendrite arms to appear. In addition, the solute partition coefficient increases when the interface velocity increases, which leads to a decrease in solute microsegregation.

Simulation of constrained dendrite growth of multicomponent alloys using a Calphad method

The conventional theory of constrained dendrite growth for binary alloys has been extended to multicomponent alloys based on the Calphad method by consideration of solute interactions in each phase. The variable solute partition coefficients and liquidus slopes under different tip undercooling were calculated in detail for a series of Al-Si-Mg alloys. Their influence on the kinetics of dendrite growth was demonstrated quantitatively. The most important microstructure scale, the primary dendrite spacing, was predicted in several Al-Si-Mg alloys. By comparing with the experimental results of former researchers, it is found that the present method coupled with Calphad is a satisfactory way for the prediction of primary dendrite arm spacing.

Prediction Technology on Grain Structure Defects in Directional Solidification Casting. Application of Cellular Automaton Method for the Prediction of Solidification Grain Structures.

A method to predict grain structure defects in directional solidification precise casting has been developed based on a solidification process simulation with a three-dimensional CAFE (cellular automaton-finite element) model. In our paper we studied which parameters of the dendrite growth model in the CAFE model are important by sensitive analyses. Then, the dendrite growth model was advanced by evaluating quantitatively liquid diffusion coefficients dependent on temperature and liquidus slope dependent on liquid concentrations which have much effect on the growth model using molecular dynamics simulations and Thermo-Calc calculations. The method to forecast grain structure defects has been applied to directional solidification casting of a simple shape and a gas turbine blade, and has been verified with comparison between the simulated results and experimental results on locations or grain directions of defects, solidification microstructure. It was found that our method could predict a stray grain defect in directional solidification precise casting.

A Parametric Study of Solute Redistribution DuringTransient Liquid Phase Diffusion Bonding Process

A parametric study is performed to investigate the solute redistribution during the transient liquid phase (TLP) diffusion bonding process. The macroscopic solute diffusion in the liquid and the solid phase, as well as the solid transformation to the liquid due to solute macrosegregation, are considered in this study. The effects of the following parameters are considered: ratio of solute diffusivity in liquid and solid state alloy (ξ = Dl/Ds), holding temperatures (θ), a combined parameter related to solidus and liquidus slopes in the phase diagram (ϕ), and the re‐melting and re‐solidification time (τ). The thickness of the pure liquid zone and the mushy zone of the TLP diffusion bonding process are demonstrated with respect to the above‐mentioned parameters. It is shown numerically that the holding time, the holding temperature, and solute diffusivity ratio influence the solute distribution strongly, which in turn influences the liquid zone and mushy zone thickness significantly. It is concluded that for the TLP diffusion bonding process, the optimal technique parameters are high holding temperature, long holding time, and a large liquidus and solidus temperature slope ratio (ml/ms) of the interlayer material.