Muggianu Model Research Articles

Calorimetric Studies and Thermodynamic Modeling of Ag-Mg-Ti Liquid Alloys.

Due to the absence of thermodynamic data concerning the Ag-Mg-Ti system in the existing literature, this study aims to fill this gap by offering the outcomes of calorimetric investigations conducted on ternary liquid solutions of these alloys. The measurements were performed using the drop calorimetry method at temperatures of 1294 K and 1297 K for the liquid solutions with the following constant mole fraction ratio: xAg/xMg = 9/1, 7/3, 1/1, 3/7 [(Ag0.9Mg0.1)1-xTix, (Ag0.7Mg0.3)1-xTix, (Ag0.5Mg0.5)1-xTix, (Ag0.3Mg0.7)1-xTix)], and xAg/xTi = 19/1 [(Ag0.95Ti0.05)1-xMgx]. The results show that the mixing enthalpy change is characterized by negative deviations from the ideal solutions and the observed minimal value equals -13.444 kJ/mol for the Ag0.95Ti0.05 alloy and xMg = 0.4182. Next, based on the thermodynamic properties of binary systems described by the Redlich-Kister model and the determined experimental data from the calorimetric measurements, the ternary optimized parameters for the Ag-Mg-Ti liquid phase were calculated by the Muggianu model. Homemade software (TerGexHm 1.0) was used to optimize the calorimetric data using the least squares method. Next, the partial and molar thermodynamic functions were calculated and are presented in the tables and figures. Moreover, this work presents, for comparative purposes, the values of the enthalpy of mixing of liquid Ag-Mg-Ti alloys, which were calculated using Toop's model. It was found that the best agreement between the modeled and experimental data was observed for the cross-sections xAg/xTi = 19/1 [(Ag0.95Ti0.05)1-xMgx] and xAg/xMg = 9/1 [(Ag0.9Mg0.1)1-xTix]. The results of the experiments presented in this paper are the first step in the investigation and future evaluation of the thermodynamics of phases and the calculation of the phase diagram of the silver-magnesium-titanium system.

Phase Equilibria of the In–Pd–Sn System at 500 °C and 800 °C: Experimental Study and CALPHAD Modeling

Phase equilibria in the In-Pd-Sn system were investigated by a combination of key experiments and thermodynamic modeling. Partial isothermal sections at 500 °C and 800 °C of the In-Pd-Sn system for Pd contents above 66 at.% have been plotted experimentally using scanning electron microscopy with energy-dispersive X-ray spectroscopy (SEM/EDX) and X-ray diffraction (XRD). The solubility of the third component in binary compounds InPd3 and Pd3Sn was determined. The new ternary compound τ1 was found in Pd contents ranging from 20 to 25 at.% and at Sn contents varying from 5 to approximately 17 at.% Sn. This compound crystallizes in an Al3Ti-type tetragonal structure. Isostructural InPd2 and Pd2Sn phases from the In-Pd and Pd-Sn binary compositions form a continuous phase field in the ternary system at both temperatures. The temperatures of the solidus, liquidus, and phase transitions of the alloys along the Pd-In50Sn50 line were measured using DTA/DSC. Thermodynamic calculation of the In-Pd-Sn ternary system is performed using the CALPHAD method using the Thermo-Calc® software. The thermodynamic properties of the disordered fcc and liquid phases were described by the Redlich-Kister-Muggianu model. To describe intermetallic phases, namely, InPd3, Pd3Sn, τ1 and Pd2(InxSn1-x), a two-sublattice models was used. Thermodynamic description of the In-Pd-Sn system obtained in this study is in good agreement both with our results and the published experimental data.

Calorimetric studies and thermodynamic modelling of liquid Mg-Pb-Pd alloys

The thermodynamic properties of liquid Mg-Pb-Pd alloys had not yet been studied. Therefore, with the use of a MHTC 96 drop calorimeter, the first time, calorimetric measurements of the partial and integral mixing enthalpies of liquid Mg-Pb-Pd alloys were performed. Calorimetric measurements of five series of liquid alloys with compositions (Mg0.75Pb0.25)1-xPdx, (Mg0.50Pb0.50)1-xPdx (two series of this alloy were evaluated), (Mg0.25Pb0.75)1-xPdx, and (Pb0.90Pd0.10)1-xMgx were performed at a temperature of 1012 K. Thereafter, based on the obtained data on the mixing enthalpy and taking into account the thermodynamic properties of binary subsystems described by the Redlich-Kister equation, ternary excess term of the Mg-Pb-Pd liquid phase was evaluated by the Muggianu model. Using home-made software (TerGexHm), thermodynamic optimization resulting in ternary optimized parameters was done. The presented calorimetric measurements represent the first step in the investigation and assessment of ternary Mg-Pb-Pd system. Overall, the intention of the authors is to model the phase equilibria in both binary and ternary systems.

Mixing Enthalpies of Liquid Ag-Mg-Pb Alloys: Experiment vs. Thermodynamic Modeling.

A drop calorimetric method was used to measure liquid Ag–Mg–Pb alloys. The partial and integral mixing enthalpies of the investigated alloys were determined at a temperature of 1116 K. The experiments were performed for four separate series starting from binary alloys with a constant xMg/xPb ratio of 1/3, 1, 3 ((Mg0.25Pb0.75)1−xAgx, (Mg0.50Pb0.50)1−xAgx, (Mg0.75Pb0.25)1−xAgx) and xAg/xMg ratio of 1/3 (Ag0.25Mg0.75)1−xPbx. Next, the ternary interaction parameters were determined using the Muggianu model, the thermodynamic properties of binary systems in the form of the Redlich-Kister equations and the values of the mixing enthalpy changes, which were determined in this study. The partial mixing enthalpies of Ag, Mg, and Pb were calculated based on the binary and elaborated ternary interaction parameters for the same intersections in which the measurements were conducted. It was found that the ternary Ag-Mg-Pb liquid solutions are characterized by negative deviations from the ideal solutions, with a maximal value slightly lower than –13 kJ/mol for alloys with the ratio (Mg0.75Pb0.25) and xAg = 0.4166.

Thermodynamic Properties of Gd–Sn and Gd–Sn–Ni Melt Systems

Enthalpies of mixing of liquid alloys are determined via isoperibol calorimetry for the Gd–Sn system throughout the range of concentrations at 1873, 1640, and 1510 K, and for the Gd–Sn–Ni system along the radial section with a constant ratio of atomic fractions of tin and nickel: xSn/xNi = 0.68/0.32 at 1873 ± 5 K up to xGd ∼ 0.25. It is found that the melts of the Gd–Sn system form with the release of a large amount of heat ( $$\Delta {{{H}}_{{\min }}}$$ = −68.4 ± 0.4 kJ/mol at xSn = 0.45). The calculations are performed using the geometric and analytical Redlich–Kister–Muggianu models of ΔH for melts of the ternary Gd–Sn–Ni system from similar data for binary boundary subsystems at 1873 K. The results show the minimum enthalpy of mixing of these ternary melts was established for the Gd0.55Sn0.45 alloy. It is shown that the experimentally investigated enthalpies of mixing of the Gd–Sn–Ni melts and calculated data using the Redlich–Kister–Muggianu model with ternary interaction parameter L = 500 kJ/mol agree satisfactorily with one another.

Calorimetric study and thermodynamic description of Ga-Ge-Li liquid alloys

This publication contains the thermodynamic results received by the drop calorimetry method. The experiments were conducted for four different cross sections, at the temperature of 1080 K. The investigated alloys were as follows: (Ga0.75Li0.25)1-xGex, (Ge0.50Li0.50)1-xGax, (Ga0.50Li0.50)1-xGex, (Ga0.25Li0.75)1-xGex. The mixing enthalpy changes measured for all four cross sections of the Ga-Ge-Li system are characterized by negative deviations from the ideal solutions. The Muggianu model with the ternary interaction parameters was applied to elaborate the experimental data of the mixing enthalpy change with the use of the optimized thermodynamic parameters of the binary systems available in the literature.

Enthalpy of mixing of liquid Li-Sb-Sn alloys

The partial and integral molar enthalpies of mixing of liquid Li-Sb-Sn alloys were determined by drop calorimetry at 879 K along ten sections xLi:xSb ≈ 2:3, xLi:xSb ≈ 3:7, xLi:xSn ≈ 1:1, xLi:xSn ≈ 3:5, xLi:xSn ≈ 1:3, xSb:xSn ≈ 4:1, xSb:xSn ≈ 13:7, xSb:xSn ≈ 1:1, xSb:xSn ≈ 1:3 and xSb:xSn ≈ 3:17. A check of enthalpy values at the crossing points of the ten sections proved high accuracy of the measurements. The behavior of the partial and integral enthalpy of mixing along these sections can be grouped into three types, depending on the starting binary systems. For the description of the integral enthalpy over the entire concentration range our experimental ternary data were fitted on the basis of an extended Redlich-Kister Muggianu model for substitutional solutions. Additionally, a comparison of these results to the extrapolation models of Muggianu and Toop is given. The entire system shows a strong exothermic behavior with a minimum of ΔmixH ≈ −65,000 J mol−1 in the binary LiSb near to Li3Sb. From experimental ternary data the phase boundary of the liquid at 879 K was derived. It showed that high melting Li3Sb dominates the solidification behavior of the Li-Sb-Sn system.

Enthalpies of mixing of liquid Ga-In and Cu-Ga-In alloys

A literature overview of Cu-Ga-In system shows that the information about the liquid phase of this system does not exist. Therefore, using MHTC 96 Setaram high temperature drop calorimeter, partial and integral enthalpies of mixing of liquid alloys were determined at first in the binary Ga-In system and next in the ternary Cu-Ga-In system. The ternary liquid alloys were investigated along two cross-sections at three temperatures: 1123 K, 1273 K and 1423 K. Experimental data were used to find ternary interaction parameters by applying the Redlich–Kister–Muggianu model for substitutional solutions, and consequently providing a full set of parameters describing the concentration dependence of the enthalpy of mixing. The experimental data indicate that the heat of mixing in this ternary system is temperature dependent, at least in the measured temperature range.

An investigation of influencing of Sb and Bi contents on surface tensions associated with Pb-free Sn-Zn-Sb-Bi quaternary and sub-quaternary solder alloys

ABSTRACTThe surface tensions of ternary and quaternary systems of Sn-based Pb-free solder alloys have been calculated using geometric models, such as Muggianu, Kohler, Chou’s general solution model (GSM), Toop, Guggenheim, ideal Butler and Butler models. It is observed from the calculation carried out in the present work that Sb and Bi contents decrease the surface tension of the solder alloys Sn-Zn-Sb-Bi. It is inferred from the statistical analysis that the best agreement between the experimental results and the corresponding calculated values of the surface tensions is generally observed in GSM and Muggianu models among the geometric models. Whereas Muggianu model is the most appropriate. Relatively good agreements have been observed between models considered in this study and experimental data.

Experimental Investigation and Thermodynamic Extrapolation of the Ga-Ge-Sb Phase Diagram

Phase relations in the Ga-Ge-Sb ternary system were investigated experimentally, using scanning electron microscopy with energy dispersive spectroscopy (SEM–EDS) and differential scanning calorimetry (DSC). Nine ternary samples were prepared by melting of pure elements under an argon atmosphere. The compositions of the prepared alloys were situated along three cross sections with the molar ratios Ga:Ge = 1, Ga:Sb = 1 and Ge:Sb = 1. Phase transition temperatures of the investigated samples were determined using DSC. The microstructure of the samples was analyzed using a scanning electron microscope with EDS and co-existing phases were identified. Binary thermodynamic data from literature, successfully used for phase diagram calculations of binary Ga-Ge, Ga-Sb and Ge-Sb systems, were used for the prediction of phase equilibria in the ternary Ga-Ge-Sb system. The activities of Ga in the liquid phase were calculated using the Redlich–Kister–Muggianu model and compared with experimental data reported in the literature. The liquidus projection, invariant equilibria and three vertical sections of the Ga-Ge-Sb ternary system were calculated and compared with the experimental results and a good agreement was observed.

Mixing enthalpies of liquid Bi–Ni and Ag–Bi–Ni alloys

Partial and integral enthalpies of mixing of liquid Bi-Ni alloys were determined at 1273K, while enthalpies of mixing of ternary Ag–Bi–Ni alloys were determined at 1173K over a broad composition range along six sections: x(Ag)/x(Bi)=3/2, 1, 2/3, 3/7, 1/4; x(Bi)/x(Ni)=4. Measurements were carried out using Calvet type microcalorimeters and drop calorimetric technique. It was found that integral enthalpies of mixing of binary Bi-Ni alloys are small exothermic, whereas integral mixing enthalpies of ternary Ag-Bi-Ni alloys change sign from negative to positive with decreasing Bi content. The ternary data were fitted on the basis of an extended Redlich–Kister–Muggianu model for substitutional solutions, and additional ternary interaction parameters were determined. From experimental ternary data the range of liquid at 1173K was deduced.

Enthalpies of mixing of liquid Ag–Ga, Au–Ga and Ag–Au–Ga alloys

A literature overview of Ag-Au-Ga system shows that the information about the liquid phase of this system does not exist. Therefore, using MHTC 96 Setaram high temperature drop calorimeter, partial and integral enthalpies of mixing of liquid alloys were determined at first in the binary systems Ag-Ga, Au-Ga and finally in ternary Ag-Au-Ga system. Next, the ternary alloys were investigated along two sections at two temperatures: 1223K and 1323K. Experimental results were used to find ternary interaction parameters by applying the Redlich–Kister–Muggianu model for substitutional solutions, and consequently a full set of parameters describing the concentration dependence of the enthalpy of mixing was derived. The experimental results indicate that the heat of mixing in this system is temperature independent, at least in the measured temperature range.

Thermodynamic Properties of Liquid Silver-Antimony-Tin Alloys Determined from Electrochemical and Calorimetric Measurements

The thermodynamic properties of liquid Ag-Sb-Sn alloys were obtained through use of the drop solution calorimetric method and electromotive force (emf) measurements of galvanic cells with a yttria stabilized zirconia (YSZ) solid electrolyte. The experiments were carried out along Ag0.25Sb0.75, Ag0.5Sb0.5 and Ag0.75Sb0.25 sections of the ternary system in the temperature range from 973 K to 1223 K. From the measured emf, the tin activity in liquid solutions of Ag-Sb-Sn was determined for the first time. The partial and integral enthalpy of mixing were determined from calorimetric measurements at two temperatures. These measurements were performed along two cross-sections: Sb0.5Sn0.5 at 912 K and 1075 K, and Ag0.75Sb0.25 at 1075 K. Both experimental data sets were used to find ternary interaction parameters by applying the Redlich–Kister–Muggianu model of the substitutional solution. Consequently, the set of parameters describing the thermodynamic properties of the liquid phase was derived.

Measurement of Activity of Indium in Liquid Bi-In-Sn Alloys by EMF Method

The electrochemical technique based on a molten salt electrolyte galvanic cell has been used to measure the activity of indium in liquid Bi-In-Sn alloys in the temperature range of 723 K to 855 K along three ternary sections. The activity of tin in Bi-Sn binary alloys has also been measured by the same technique in the above temperature range. The activity of indium in Bi-In-Sn alloys shows negative deviation from Raoult’s law for most of the compositions and slight positive deviations for a few indium-rich compositions. The ternary excess molar free energies have been calculated by Darken’s treatment. Isoactivity curves at 813 K in the ternary Bi-In-Sn alloys were derived by combining the activity data of In-Sn and Bi-In alloys. The values of excess molar free energy obtained in this study are compared with those calculated from the Muggianu model at 813 K.

Geometric modelling of viscosity of copper-containing liquid alloys

In this work, viscosities of ternary Au–Ag–Cu and Al–Cu–Si liquid alloys have been calculated as a function of gold, aluminium and copper compositions for the sections Au–Ag–Cu (xAg/xCu = 0.543 at 1373 K), Alx(Cu50–Si50)(1–x) and Cux(Al50–Si50)(1–x) at 1375 K using Chou’s general solution model, Muggianu, Kohler, Toop, Hillert, Budai et al., Kozlov et al., Schick et al. and Kaptay et al. models. The present study finds that a comparison of the predicted values of viscosities associated with the geometric and physical models indicate good mutual agreement. The Muggianu model indicates the best agreement with the results obtained for Au–Ag–Cu and Alx–Cu50–Si50 alloy systems and the Kaptay et al. model, which is a physical model, indicates the best agreement with the results obtained for Al50–Cux–Si50.

O68] Comparison of Muggianu model,Toop model and General solution model for predicting the thermodynamic properties of Mg-Al-Zn system

O68] Comparison of Muggianu model,Toop model and General solution model for predicting the thermodynamic properties of Mg-Al-Zn system

Thermodynamics and characterization of shape memory Cu–Al–Zn alloys

The thermodynamic properties and the microstructure, hardness and electrical conductivity of shape memory alloys (SMAs) belonging to ternary Cu–Al–Zn system were studied by Muggianu model and experiment, respectively. The isothermal section of phase diagram at 293 K was calculated using Thermo-Calc software. Experiments were conducted by X-ray diffraction, light optic microscopy, scanning electron microscopy with energy dispersive X-ray spectrometry, hardness and electrical conductivity measurements. The calculated values of thermodynamic properties indicate that Cu shows good miscibility with Al and Zn in all investigated alloys. The microstructural analysis of samples reveals that the structure consists of large and polygonal grains.

Enthalpy of mixing of liquid Ag–Bi–Cu alloys at 1073 K

The Ag–Bi–Cu system is among those ternary systems which have not been fully studied yet, in particular the thermodynamic description of the liquid phase is missing. Partial and integral enthalpies of mixing of liquid ternary Ag–Bi–Cu alloys were determined over a broad composition range along six sections: x(Ag)/x(Bi)=0.25, 1, 4; x(Ag)/x(Cu)=1.5; x(Bi)/x(Cu)=1.86, 4. Measurements were carried out at 1073K using two Calvet type microcalorimeters and drop calorimetric technique. It was found that integral enthalpies of mixing are small and endothermic, similarly to limiting binary alloys. The ternary data were fitted on the basis of an extended Redlich–Kister–Muggianu model for substitutional solutions. There are no significant additional ternary interactions.

Enthalpies of mixing of liquid Ag–Ga, Cu–Ga and Ag–Cu–Ga alloys

Abstract Using MHTC 96 Setaram high temperature drop calorimeter, partial and integral enthalpies of mixing of liquid alloys were determined in the Ag–Ga, Cu–Ga and in the ternary Ag–Cu–Ga system. The ternary alloys were investigated along two cross-sections at two different temperatures: 1128 K and 1273 K, respectively. Experimental data were used to find ternary interaction parameters by applying the Redlich–Kister–Muggianu model for substitutional solutions, and a full set of parameters describing the concentration dependence of the enthalpy of mixing was derived. The experimental data indicate that the heat of mixing in this system is slightly temperature dependent, at least in the measured temperature range.

Calorimetric studies of Cu–Li, Li–Sn, and Cu–Li–Sn

Integral molar enthalpies of mixing were determined by drop calorimetry for Cu–Li–Sn at 1073K along five sections xCu/xSn≈1:1, xCu/xSn≈2:3, xCu/xSn≈1:4, xLi/xSn≈1:1, and xLi/xSn≈1:4. The integral and partial molar mixing enthalpies of Cu–Li and Li–Sn were measured at the same temperature, for Li–Sn in addition at 773K. All binary data could be described by Redlich–Kister-polynomials. Cu–Li shows an endothermic mixing effect with a maximum in the integral molar mixing enthalpy of ∼5300J·mol−1 at xCu=0.5, Li–Sn an exothermic minimum of ∼ −37,000J·mol−1 at xSn∼0.2. For Li–Sn no significant temperature dependence between 773K and 1073K could be deduced. Our measured ternary data were fitted on the basis of an extended Redlich–Kister–Muggianu model for substitutional solutions. Additionally, a comparison of these results to the extrapolation model of Chou is given.