Modeling solid solution strengthening in nickel alloys

The yield stress of multicomponent nickel solid solution alloys has not been modeled in the past with respect to the effects of composition and temperature. There have been investigations of the effect on the yield stress of solutes in binary systems at a fixed temperature, but the effects on the yield stress of multiple solute elements and temperature changes have not been investigated. In this article, two different forms of the trough model are considered for nickel-base alloys to determine the most applicable model for solid solution strengthening in the system. The yield stresses of three binary nickel-chromium and three ternary nickel alloys were determined at a range of temperatures. The yield stress of the alloys was then modeled using the Feltham equation. The constants determined in fitting the Feltham equation to the experimental data were then applied to other experimental solid solution alloys and also to published information on commercial solid solution nickel alloys. It was found that the yield stress of the nickel solid solution alloys could be modeled successfully using the Feltham equation.     

Experiments on the solidification structure of alloy castings

Two related experimental programs on the solidification structure of alloy castings are reported. In the first, the grain structure of catalytically clean Ni?Cu alloys is examined as a function of the degree of supercooling below the equilibrium liquidus. For supercoolings greater than 85°C, the Ni?Cu alloys exhibit a structure which is in accord with previous observations in pure nickel,i. e., the structure is found to be coarse and dendritic in the range 85° to 150°C supercooling, but fine and equiaxed for supercoolings greater than 150°C. However, in the lower range of supercooling (<85°C) the structure is fine and equiaxed. It is concluded that solute elements can promote grain formation in certain castings at supercoolings insufficient to cause heterogeneous nucleation. To study the effect of solute elements on the structure of a solidifying material, castings of pure nickel and aluminum are compared with binary alloys of these base materials. Over the composition ranges studied in both alloy systems, the structure is observed to be related to a parameter which includes the slope of the liquidus, the bulk solute concentration, and the solute distribution coefficient. It is shown or is argued that heterogeneous nucleants are not involved and therefore another mechanism must be operating in determining the structure. In any event, these latter experiments suggest that an effective means of controlling the structure of castings is by appropriate selection of alloying additions on the basis of the alloy variables contained in such a parameter.     

Calculation of phase equilibria in the Ga-Bi and Ga-P-Bi systems based on a theory of regular associated solutions

A new theory of regular associated solutions (RAS) has been formulated to provide a consistent representation of binary and ternary phase equilibria in the Ga-P-Bi system. The key postulate of the theory is that the major species in liquid alloys are Ga, P, and Bi atoms and Bi₂ dimers. At first, the gross component activities in the Ga-Bi binary system are approximated from the dimer dissociation constant,K, and the activity coefficients of the three species (identified here with the coefficients for a ternary regular solution). The accord observed between the form of the isothermal activity data and the theory permits the determination ofK and the Ga-Bi interchange energy. These parameters are then employed in the calculation of the enthalpy and entropy of mixing and the prediction of the liquidus curve and asymmetric miscibility gap for the Ga-Bi system, all of which are in good agreement with the experimental findings. Generalization of the enthalpy and entropy of mixing for a binary RAS facilitates the derivation of the activities in the ternary system. Knowledge of the activities leads to the evaluation of the ternary liquidus isotherms over the entire composition range, since the Bi-P interchange energy can be obtained from the GaP-Bi pseudobinary liquidus data. It is found that along the pseudobinary the standard error between calculated and experimental liquidus points is 7° C. Furthermore, in P-rich liquid solutions, at any temperature below ≈R 1380°C, an open miscibility gap intersects the primary liquidus isotherms of Bi-doped GaP. The predicted miscibility gap in the Bi-P system is consistent with the fragmentary evidence. Finally, the paper discusses extensions of the RAS model to other ternary systems involving compound semiconductors wherein association in the liquid is likely.     

Liquidus of Silicon Binary Systems

Thermodynamic knowledge about liquid silicon is crucial for the production of solar-grade silicon feedstock from molten silicon. In the current study, liquidus for silicon binary alloys is formulated using a previously developed method in which the liquidus curve is calculated using two constants. The liquidus measurements for the silicon portion of the silicon alloys with Al, Ca, Mg, Fe, Ti, Zn, Cu, Ag, Au, Pt, Sn, Pb, Bi, Sb, Ga, In, Ni, Pd, Mn, and Rh are reviewed, and the consistent data were used to determine the liquidus constants. The liquidus curves for silicon binary systems are calculated and plotted. It is indicated that the calculated liquidus curves fit well with the experimental data. A correlation between the determined liquidus constants is also observed, which can be used to gain a better understanding of the thermodynamics of the silicon binary melts.     

Phase diagram of the Cu-Ni-Mn system

Part of the phase diagram of the Cu-Ni-Mn system from 0 to 20% Ni and from 30 to 50% Mn is refined with the help of a theoretical analysis and based on the experimental data. It is shown that the assumption that a line exists in the Cu-Ni-Mn ternary system along which the alloys with a zero crystallization range are arranged is erroneous. Only the alloys in the Cu-Mn and Ni-Mn binary systems have a zero crystallization range. The primary assumption that it is impossible to represent the nonequilibrium crystallization by Petrov-Scheil in the absence of a minimum line in the Cu-Ni-Mn system was not confirmed because previous studies were performed for a model system in which a convex liquidus line was present in the polythermal section connecting the minima in binary systems. In this study it is shown that this line is concaved towards lowering the temperature.     

Prediction of the densities of liquid Ni-based superalloys

The molar volume of nickel and the partial molar volumes of Cr, Co, W, Ta, Al, Mo, and Re are computed from the densities of liquid nickel and liquid Ni-based binary alloys measured by the modified sessile drop method (MSDM) and the modified pycnometric method (MPM). The molar volumes of Hf, Nb, and Ti are calculated from the literature density values of these elements. A prediction model for the densities of liquid Ni-based alloys is developed based on the obtained molar/partial molar volumes of the elements in liquid Ni-based alloys. The validity of this model is verified by the density data of liquid Ni-based ternary (Ni-Co-Al), quaternary (Ni-Co-Al-Cr, Ni-Co-Al-Mo, and Ni-Cr-Al-Mo), and commercial alloys measured by both MSDM and MPM. It is also verified by the density values of liquid Ni-based commercial alloys in the literature. It has been shown that the present model can provide reasonable estimation for the density and its temperature dependency of liquid Ni-based alloys. The difference between the predicted value and the measured/cited value is within ±2.5 pct.     

Differential Calculation Model for Liquidus Temperature of Steel

The liquidus temperature of steel is greatly affected by its chemical composition; therefore, the value of the liquidus temperature can vary among different types of steel. In this study, on the basis of the concept of differentiation, the Fe‐i binary phase diagram is analyzed using Photoshop image processing software. The effects of eleven common alloying elements (C, Si, Mn, P, S, Ca, Ni, Cu, Cr, Nb, and Mo, whose content is between 0%～4%) on the melting point of pure iron are analyzed. This analysis proves the following assumption to be incorrect: the relationship between the change in liquidus temperature and the content of element i is a linear relationship; this assumption is the basis of traditional liquidus calculation models. A new model for calculating the liquidus temperature of steel is established in this study. As compared to data from other literature, the model proposed in this paper has good generality and the calculation error of the liquidus temperature of steel is between ?3 and 4 °C, which is acceptable. A statistical analysis of the experimental data for different steel grades (obtained from Laiwu Special Steel) using the developed model showed that 87.2% of the absolute deviation values of the temperature are within 4 °C. This result shows the model to be both reasonable and credible.     

Vaporization of the components of nickel alloys in a vacuum induction furnace

When nickel alloys are smelted in a vacuum induction furnace, the challenge is to refine the metal and retain the alloying elements and the nickel. In the present work, a model is developed for the behavior of all the components of complex alloys during refining in a vacuum induction furnace. The limiting stage is assumed to be the transfer of the components to the gas phase. In that case, the flux of each alloy component to the gas phase is determined from the Langmuir equation. The model adequately predicts the evaporation of the components of complex nickel alloys during refining in a vacuum induction furnace. The proposed algorithm may be used to calculate the change in mass and composition of nickel alloys over time in the course of refining in a vacuum induction furnace of specific capacity, with specified residual pressure and temperature.     

Alloys and phase equilibria in the Al-Ti-Rh system. II. Melting diagram of the TiRh-Rh-AlRh partial system

Data on the solidus surface of the partial TiRh-Rh-AlRh system and results of metallography, x-ray diffraction, and differential thermal and electron microprobe analyses of its as-cast alloys are used to project the liquidus surface of this system onto the concentration triangle and examine the processes occurring in the crystallization of its alloys for the first time. This makes it possible to plot the melting diagram of the partial TiRh-Rh-AlRh system. Its liquidus surface is completed with four surfaces of primary crystallization of solid solutions based on rhodium and phases based on binary compounds formed in the bounding binary Al-Rh and Ti-Rh systems. In the partial TiRh-Rh-AlRh system, there are two invariant four-phase equilibria involving liquid; one of them is peritectic (at 1714°C) and the other is eutectic (at 1675°C). These four-phase equilibria also include five monovariant three-phase equilibria involving a liquid phase.     

An analytical solution of directional solidification with mushy zone

An analytical heat transfer model is presented that describes the temperature distribution and the positions of solidus and liquidus isotherms in the unidirectional solidification of binary alloys. The proposed technique employs the mathematical expedient of replacing the interfacial thermal resistance by equivalent layers of material. The application of the model is demonstrated by comparison with experimental data and with a finite difference method.     

The liquidus surfaces of ternary systems involving compound semiconductors: II. Calculation of the liquidus isotherms and component partial pressures in the Ga−As−Zn and Ga−P−Zn systems

The previously derived liquidus equation for anAB compound (GaAs or GaP) incorporating a dilute solute (zinc) in equilibrium with a ternary regular melt was applied to the available liquidus data in the Ga?As and Ga?As?Zn and in the Ga?P and Ga?P?Zn systems. The ternary interchange energies for both systems were computed by the method of multidimensional leastsquares. Utilizing these interchange energies, the liquidus isotherms in the Ga?As?Zn and Ga?P?Zn systems were calculated over a wide temperature range from the nonlinear liquidus equation by a numerical method. By a combination of the ternary regular activity coefficients with the vapor pressures of the pure components and the liquidus isotherms, the component partial pressures along the liquidus isotherms were determined. The ternary liquidus and partial pressure isotherms are in good agreement with the experimental data of Panish, Koster and Ulrich, and Shihet al. It is further shown that in the binary limit, the results of these calculations are also in accord with liquidus and partial pressure measurements for the Ga?As and Ga?P systems.     

The Strength of the Spatially Interconnected Eutectic Network in HPDC Mg-La,Mg-Nd,and Mg-La-Nd Alloys

3D numerical images of the intergranular percolating eutectic of two binary alloys, Mg-0.62 at. pctLa and Mg-0.60 at. pctNd, created using dual beam FIB tomography, were incorporated into an FEM code to model their tensile behavior. Due to its high volume fraction (29.9 pct), the behavior of the Mg-La network was akin to that of a stretch-dominated micro-truss structure, whereas the Mg-Nd’s, with a relatively low volume fraction (7.5 pct), mimicked that of a bending-dominated structure. The 3D network contributed some 37 MPa to the strength of the Mg-La alloy casting, whereas it only added about 1.4 MPa to the Mg-Nd’s. The model predictions based on the binary alloys were verified using cast-to-shape specimens of the Mg-La and two ternary Mg-La-Nd alloys, subjected to a flash-annealing aiming at breaking up the continuity of the 3D network, while preserving the rest of the microstructure unchanged. The flash-annealed specimens exhibited a decrease in strength that matched closely the computed values. Implications regarding alloy design involving the eutectic network and solid solution hardening of more complex alloys are discussed.     

The surface tensions of Pb,Sn, and Pb-Sn alloys

The surface tensions of 99.9999 pct Pb and Sn and of alloys made from lead and tin of the same purity have been measured by the sessile-drop method. The surface tension of lead was determined from the melting point, the surface tension of tin from 200°C (32°C of supercooling) and the surface tensions of the Pb−Sn alloys from the respective liquidus temperatures, all, up to a temperature of about 560°C. Within the limits of experimental error, data for both pure metals plot in a rectilinear fashion with negative slopes, thus: γ_lead=472.7−0.085t(±5) dyne/cm γ_tin=569.0−0.080t((±5) dyne/cm The surface tensions of Pb−Sn alloys are not straight-line functions of temperature and the evidence indicates that the temperature coefficients of surface tension are slightly positive at the liquidus temperature.     

Experimental studies on hydrogen solubility in liquid ternary iron-nickel-chromium alloys

The hydrogen solubilities in liquid ternary iron-nickel-chromium alloys have been experimentally determined by Sieverts’ method. The experimental data have been evaluated with the help of regression analysis (Gauß-Jordan) and using own published empirical equations (polynomials). With the help of experimentally determined hydrogen solubilities and the coefficients of the polynomials, the concentration dependencies of hydrogen solubilities, interaction coefficients, enthalpies and entropies of hydrogen solution have been determined. The hydrogen solubility increases with increasing temperature and with an increase in the nickel and chromium concentration. The results have been represented as isothermal planes. The hydrogen solubilities in liquid ternary iron-nickel-chromium alloys (x_Cr ≤ 50%) have been predicted with the help of “Central Atoms” model, assuming the concentration dependencies of the “model parameter λ” and selecting the value of the coordination number (Z’ = 10) for each of the “reference elements” iron or nickel. The comparison between the experimentally determined and the predicted hydrogen solubilities in iron-nickel-chromium alloys confirms that the prediction of hydrogen solubilities in liquid system iron-nickel-chromium can be qualitatively made over a wide range of alloy concentrations (x_Cr = 50%) with the aid of “Central Atoms” model and with the assumption of concentration dependencies of the “model parameter λ”.     

Study of the surface properties of nickel-based melts by the constrained drop method: I. Surface tension

The constrained drop method is used to study the surface tension σ of the following melts at 1773–1923 K and p _Ar = 0.1 MPa: nickel of various grades (with various oxygen contents), binary Ni-Al (Re) alloys, and a complex Ni-Re-(W, Mo, Co) alloy. The value of σ of liquid nickel is shown to decrease with increasing oxygen content in it. The additions of aluminum (6%) and rhenium (3–7%) to nickel in binary alloys weakly change its surface tension. Alloying elements (W, Mo, Co) in Ni-Re-(W, Mo, Co) alloys insignificantly affect σ of their melts.     

Volume effects and associations in liquid alloys

The regular associated solution model for binary systems has been modified by incorporating the size of the complex as an explicit variable. The thermodynamic properties of the liquid alloy and the interactions between theA _μB type of complex and the unassociated atoms in anA-B binary have been evaluated as a function of relative size of the complex using the activity coefficients at infinite dilution and activity data at one other composition in the binary. The computational procedure adopted for determining the concentration of clusters and interaction energies in the associated liquid is similar to that proposed by Lele and Rao. The analysis has been applied to the thermodynamic mixing functions of liquid Al-Ca alloys believed to contain Al₂Ca associates. It is found that the size of the cluster significantly affects the interaction energies between the complex and the unassociated atoms, while the equilibrium constant and enthalpy change for the association reaction exhibit only minor variation, when the equations are fitted to experimental data. The interaction energy between unassociated free atoms remains virtually unaltered as the size of the complex is varied between extreme values. Accurate data on free energy, enthalpy, and volume of mixing at the same temperature on alloy systems with compound forming tendency would permit a rigorous test of the proposed model.     

Volume effects and associations in liquid alloys

The regular associated solution model for binary systems has been modified by incorporating the size of the complex as an explicit variable. The thermodynamic properties of the liquid alloy and the interactions between theA _μB type of complex and the unassociated atoms in anA-B binary have been evaluated as a function of relative size of the complex using the activity coefficients at infinite dilution and activity data at one other composition in the binary. The computational procedure adopted for determining the concentration of clusters and interaction energies in the associated liquid is similar to that proposed by Lele and Rao. The analysis has been applied to the thermodynamic mixing functions of liquid Al-Ca alloys believed to contain Al₂Ca associates. It is found that the size of the cluster significantly affects the interaction energies between the complex and the unassociated atoms, while the equilibrium constant and enthalpy change for the association reaction exhibit only minor variation, when the equations are fitted to experimental data. The interaction energy between unassociated free atoms remains virtually unaltered as the size of the complex is varied between extreme values. Accurate data on free energy, enthalpy, and volume of mixing at the same temperature on alloy systems with compound forming tendency would permit a rigorous test of the proposed model.     

Contraction of aluminum alloys during and after solidification

A technique for measuring the linear contraction during and after solidification of aluminum alloys was improved and used for examination of binary and commercial alloys. The effect of experimental parameters, e.g., the length of the mold and the melt level, on the contraction was studied. The correlation between the compositional dependences of the linear contraction in the solidification range and the hot tearing susceptibility was shown for binary Al-Cu and Al-Mg alloys and used for the estimation of hot tearing susceptibility of 6XXX series alloys with copper. The linear thermal contraction coefficients for binary and commercial alloys showed complex behavior at subsolidus temperatures. The technique allows estimation of the contraction coefficient of commercial alloys in a wide range of temperatures and could be helpful for computer simulations of geometrical distortions during directchill (DC) casting.     

Liquidus Temperature Depression in Cryolitic Melts

The electrolyte in Hall-Héroult cells for the manufacture of primary aluminum nominally contains only cryolite (Na₃AlF₆) with additions of AlF₃, CaF₂, and Al₂O₃. However, impurities are present, entering the process with the feedstock. The effect on the liquidus temperature by the impurities cannot be calculated correctly by the well-known equation for freezing-point depression in binary systems simply because the electrolyte cannot be regarded as a binary system. By extending the equation for freezing-point depression to the ternary system NaF-AlF₃-B, it appeared that the acidity of the impurity B plays a major role. Some calculations were made using an ideal Temkin model, and for most types of impurities, the effect on the liquidus temperature will be larger in an industrial electrolyte than what can be estimated from the equation for freezing-point depression in cryolite. Experimental data on the liquidus temperature in the system Na₃AlF₆-AlF₃-Al₂O₃-CaF₂-MgF₂ show that the effect of MgF₂ on the liquidus temperature increases strongly with decreasing NaF/AlF₃ molar ratio, and it is suggested that MgF₂ forms an anion complex, probably MgF ₄ ^2? .     

Thermography of the phase transformations in nickel-based eutectic alloys at high pressures and temperatures

The phase transformations in STEMET 1301A and STEMET 1311 nickel solder alloys are studied by differential barothermal analysis and metallography at temperatures up to 1100°C and pressures up to 120MPa. When the pressure increases, the temperature of the onset of crystallization of the initial amorphous STEMET 1301A alloy decreases and that of the STEMET 1311 alloy increases. At high pressures, the solidus temperatures of the alloys changes insignificantly and the liquidus temperature of the STEMET 1301A alloy increases by 30°C. The solidification ranges of both alloys at 120 MPa are found to broaden substantially.