Miedema Model Research Articles

Modification and verification of Miedema model for predicating thermodynamic properties of binary precipitates in multi-element alloys

At present, there is still a certain difference between the calculated and experimental results of the original Miedema formation enthalpy model. Therefore, it is necessary to improve the calculation accuracy of the Miedema model to facilitate the development of the Miedema model and its application in alloy design. To address the problem, a new atomic size factor is introduced to modify the Miedema model in this paper. Then, the accuracy of the modified Miedema model is verified. Firstly, the formation enthalpies of various magnesium-based binary alloys are recalculated using the modified Miedema model. Secondly, the precipitation temperature of MgZn phase in Mg–6Zn-0.6Zr ternary alloy is predicted using the modified Miedema model combined with Chou model, which is compared with the experimental result. Thirdly, the precipitation of binary phases in Mg–Zn–Zr–Y quaternary alloy is analyzed by calculation based on the modified Miedema model. The final results show that the calculated results obtained from the modified Miedema model are in good agreement with the reported results, experimental results and phase diagram results, indicating that the modification is reasonable.

Calculation of Thermodynamic Properties of Mg-Zn-Al Alloy

The thermodynamic properties of Mg, Zn, Al binary alloys and ternary alloys were calculated by Miedema model and Toop model, and magnesium was selected as the asymmetric component. The results show that the mixing enthalpy, excess entropy and excess Gibbs free energy of Al-Zn binary alloy are all greater than zero. The thermodynamic values of Mg-Al and Mg-Zn binary alloys are relatively similar, and both are less than zero. Within the range of composition variation, the excess Gibbs free energy of Mg-Zn-Al ternary alloy is basically negative. In the Mg-poor and Mg-rich regions, the thermodynamic value of the alloy changes greatly. Excess Gibbs free energy and mixing enthalpy of magnesium-zinc-aluminum alloy can be reduced by adding zinc and aluminum.

Microstructural, magnetic, and geometrical thermodynamic investigation of FeCoNi(MnSi)x (0.0, 0.1, 0.25, 0.5, 0.75, 1.0) high entropy alloys

In the current investigation, FeCoNi(MnSi)x (x=0.0,0.1,0.25,0.5,0.75,1.0) high entropy alloys (HEAs) were effectively processed by mechanical alloying (MA) route. The impact of Mn and Si on the FeCoNi system was further analyzed based on structural, morphological, and magnetic properties utilizing X-ray diffraction (XRD), scanning electron microscopy (SEM), and vibrating sample magnetometer (VSM) at various processing stages. The results showed the formation of the Fe-rich BCC (major) + Ni-rich FCC (minor) phase in FeCoNi(MnSi)x (x=0.0,0.1,0.25) alloys however the Ni-rich FCC phase was significantly more recognizable in the case of FeCoNi(MnSi)x (x=0.5,0.75,1.0) HEAs. Furthermore, different symmetrical and asymmetrical geometrical thermodynamic models such as Muggianu, Lück chou, Kohler, Colinet, Hillert, Toop, and General solution model (GSM) were modeled using the Redlich-Kister formalism and then correlated the enthalpy value with the developed phases during MA. In the case of the GSM model, the enthalpy of mixing (ΔHmixGSM) was found to be negative in a whole concentration range of (MnSi)x at 1237 K and exhibited solid solution criteria confirmed from the enthalpy of mixing (ΔHmixMiedema) according to the Miedema model. Moreover, the associated activity coefficient (γa) revealed a negative deviation as Mn-Si content increased. Additionally, the magnetic properties showed a ferromagnetic nature, and its parameters were calculated from the "law of approach to saturation."

Estimating of Surface Tension and Viscosity of Liquid Ag-Cu Alloys

The present paper compares the results obtained from the experiments for the binary liquid Cu-Ag system at a temperature of 1373K. All viscosity models available in the existing literature on the viscosity dependence of the viscosity of binary liquid alloys were used. The same process was carried out for surface tension at a temperature of 1423 K for a dual liquid Cu-Ag system. In the literature survey, the Miedema model showed little use in determining the thermophysical properties of this alloy. Using Miedema model, the mixture enthalpy and excess Gibbs free energy in the aforementioned models were calculated. In the present study, a correspondence was found between surface tension and viscosity values. Surface tension and viscosity values were inversely proportional to the temperature, at a fixed silver fraction xAg, and directly proportional to copper content at a constant temperature. The obtained results were compared to the data in the existing literature.

The interaction of rare earth cerium, neodymium, yttrium and phosphorus

According to the Miedema model, the formation enthalpies of Nd, Y, Ce and P alloys were calculated; the entropies of NdP, YP and CeP were obtained using single gas ion entropy and crystal ion entropy. The results show that the thermodynamic properties of the compound between rare earth and P can be calculated. The free energy of formation of NdP and YP are both greater than zero. Therefore, [P]+[Nd]=NdP(s) and [P]+[Y]=YP(s) cannot be proceed spontaneously in the direction of the positive reaction. When the temperature is less than 964K, ΔGCeP Θ is less than zero and [P]+[Ce]=CeP(s) reaction can be proceed spontaneously in the direction of generating CeP.

Influence of Cr, Mn, Co and Ni Addition on Crystallization Behavior of Al13Fe4 Phase in Al-5Fe Alloys Based on ThermoDynamic Calculations.

Alloying is an effective method to refine coarse grains of an Al13Fe4 phase and strengthen Al-Fe alloys. However, the grain refinement mechanism remains unclear in terms of the thermodynamics. Herein, the influence of M-element, i.e., Cr, Mn, Co and Ni, addition on the activity of Al and Fe atoms, Gibbs free energy of the Al13Fe4 nucleus in Al-Fe melt and the formation enthalpy of an Al13Fe4 phase in Al-Fe alloys is systematically investigated using the extended Miedema model, Wilson equation, and first-principle calculations, respectively. The results reveal that the addition of different M elements increases the activity of Fe atoms and reduces the Gibbs free energy of the Al13Fe4 nucleus in Al-Fe melt, where the incorporation of Ni renders the most obvious effect, followed by Mn, Co, and Cr. Additionally, the formation enthalpy decreases in the following order: Al78(Fe23Cr) > Al78(Fe23Mn) > Al13Fe4 > Al78(Fe23Ni) > Al78(Fe23Co), where the formation enthalpy of Al78(Fe23Ni) is close to Al78(Fe23Co). Moreover, the presence of Ni promotes the nucleation of the Al13Fe4 phase in Al-Fe alloys, which reveals the mechanism of grain refinement from a thermodynamics viewpoint.

Calorimetric Studies of Magnesium-Rich Mg-Pd Alloys.

Solution calorimetry with liquid aluminum as the bath was conducted to measure the enthalpy of a solution of magnesium and palladium as well as the standard formation enthalpies of selected magnesium-palladium alloys. These alloys were synthesized from pure elements, which were melted in a resistance furnace that was placed in a glove box containing high-purity argon and a very low concentration of impurities, such as oxygen and water vapor. A Setaram MHTC 96 Line evo drop calorimeter was used to determine the energetic effects of the solution. The enthalpies of the Mg and Pd solutions in liquid aluminum were measured at 1033 K, and they equaled −8.6 ± 1.1 and −186.8 ± 1.1 kJ/mol, respectively. The values of the standard formation enthalpy of the investigated alloys with concentrations close to the Mg6Pd, ε, Mg5Pd2, and Mg2Pd intermetallic phases were determined as follows: −28.0 ± 1.2 kJ/mol of atoms, −32.6 ± 1.6 kJ/mol of atoms, −46.8 ± 1.4 kJ/mol of atoms, and −56.0 ± 1.6 kJ/mol of atoms, respectively. The latter data were compared with existing experimental and theoretical data from the literature along with data calculated using the Miedema model.

Thermodynamic analysis of the extension of solid solubility in copper‐lead immiscible system prepared by mechanical alloying

AbstractX‐ray diffraction analysis, scanning electron microscopy and thermodynamic calculations in the Miedema's model were used to study the extension of solid solubility in copper‐lead immiscible system prepared by mechanical alloying. The result shows that the nanocrystalline supersaturated solid solution formed during the alloying process. The Gibbs free energy change in this system during the formation of solid solution is calculated and shows to be positive, which means that a thermodynamic barrier exists for the formation of this alloy system in solid state. It has been found that the additional energy stored in nanocrystalline copper‐lead alloy during the alloying process as a result of crystallite size reduction and increased dislocation density, would be high enough to overcome the thermodynamic barrier in the formation of a solid solution.

Vaporization of alloying elements and explosion behavior during laser powder bed fusion of Cu–10Zn alloy

Vaporization of alloying elements is an important phenomenon during laser powder bed fusion (LPBF), especially in the laser interaction with difficult-to-form materials containing the volatile element as main alloying element, which has a significant effect on the chemical composition, microstructure and mechanical properties. However, the mechanisms of the typical phenomenon in vaporization, such as the explosion, and its influence on the forming quality are not fully understood. Here, based on the in situ high-speed high-resolution imaging, we observe the explosion of the molten pool with temporal scale of ~101 μs and spatial scale of ~101–102 μm during LPBF of Cu–10Zn. This micro-explosion can be attributed to the drastic augment of the saturated vapor pressure of Zn component in the molten pool. Theoretical calculations of vaporization amount in the explosion process by Miedema's model and Langmuir's equation, are agreed well with the experimental results of volume increment for vaporization-induced inflation of the droplet spattering. Moreover, we also reveal a new defect formation mechanism, explosion-induced crater, which is detrimental to the forming quality in LPBF. The explosion-induced crater defect can be mitigated by increasing the lifetime of the molten pool, thus improve the continuity and flatness of the melt track. This study is expected to provide the scientific basis for LPBF of difficult-to-form materials to achieve stable forming with fewer defects.

On the Mg-Pb system. Calorimetric studies and thermodynamic calculations

In this paper, the experimentally determined integral and partial molar enthalpy of mixing of liquid Mg-Pb alloys is presented. Five separate measurements were performed at 1012 and 1014 K with the use of MHTC 96 Line Evo line drop calorimeter. The values of integral molar mixing enthalpy are characterized by negative deviations from the ideal solutions in the entire concentration range. Based on obtained results and other thermodynamic quantities available in the literature the Mg-Pb phase diagram was optimized. The calculated and experimentally obtained values of the integral molar mixing enthalpy are also compared with the Miedema model prediction of the integral molar mixing enthalpy. The formation enthalpy for the Mg2Pb phase is computed by ab initio calculations and equals −5978 J/mol of atoms at 0 K. For the Mg2Pb intermetallic phase, the calculated bulk isothermal and adiabatic modulus, as well as a variation of primitive cell volume and thermal expansion coefficient with the temperature, are presented. Moreover, the heat content of the Mg2Pb intermetallic phase is shown and compared with experimental data for temperatures from 0 K to its melting temperature.

Structural and calorimetric studies of magnesium-rich Mg-Pd alloys

Despite the fact that the affinities of both palladium and magnesium with hydrogen have been studied for many years, information about magnesium-palladium alloys is still incomplete. Although palladium is an extremely expensive metal and magnesium palladium alloys will likely never applied as solid-state hydrogen storage materials, the interaction of these alloys with hydrogen may be useful for broadening knowledge or finding applications as hydrogen splitters, catalysts, and hydrogen sensors. In this work, two types of calorimetric techniques were applied to measure the thermodynamic properties of magnesium-rich alloys from the Mg-Pd system (containing 97.7; 85.4; 80.6; 79.9; 72.3; 70.7; 64.5 at% of Mg respectively). The solution calorimetric method was used to determine the heat of solution of liquid magnesium and palladium in liquid tin. Additionally, the standard enthalpies of formation of six alloys corresponding to the intermetallic phases of the Mg-rich region were measured. These alloys were prepared from pure Mg and Pd, which were melted in a glove box filled with high purity argon, with a very low concentration of impurities. All the obtained alloys were structurally analysed by X-ray diffraction (XRD) and scanning electron microscopy (SEM/EDS) methods. Moreover, DSC studies were used to determine the transformation temperatures in the Mg-rich region. The values of the standard formation enthalpy of the studied alloys were determined to be −27.3 ± 0.8 kJ/mol at., −33.4 ± 0.9 kJ/mol at., −35.2 ± 1.4 kJ/mol at., −44.2 ± 0.9 kJ/mol at., −46.0 ± 0.7 kJ/mol at., and −54.3 ± 2.3 kJ/mol at. These values were compared with the data calculated using the Miedema model, as well as with existing literature data for the Mg6Pd and Mg5Pd2 phases.

Construction of magnetron sputtering non-equilibrium Miedema's model for application in Ag-Mo thin films

The Miedema's model is a phenomenology for studying the thermodynamics of transition binary metals. In this paper, the magnetron sputtering non-equilibrium Miedema's model was constructed by using the single atomic collision theory. The Gibbs free energy diagrams of the crystalline and amorphous phase of the Ag-Mo system were calculated by using this model, the curves were convex and were greater than zero. Combined with the valence rule in the Hume-Rothery's rules, it was found that Ag atoms had a tendency to dissolve into Mo lattice under the non-equilibrium magnetron sputtering condition. The experimental results showed that metastable Mo(Ag) solid solution was formed in the Mo-rich region, which corresponds to the calculated Gibbs free energy curves under the constraints of the Hume-Rothery's rule.

Enthalpy analysis of Ce–Mg–Ni–H formation based on extended miedema theory: Investigation of selected Ce2MgNi2–H2

In this paper, an extended Miedema's model is constructed to illustrate its applicability to estimating the solid-solution enthalpies of Ce–Mg–Ni–H hydrides, adopting the range of an optimized stoichiometry alloy in the contour map of solid-solution state enthalpy. Ce2MgNi2 alloy is designed to investigate its hydrogen storage properties, and its main phase is confirmed with X-ray diffraction characterizations. The alloy shows a good activation ability and the pressure component temperature plateau is extremely flat. The formation enthalpy of Ce2MgNi2–H2 is calculated with the extended Miedema theory, with the least enthalpy value of −59.1 kJ/mol for the corresponding hydrogen content of 1.64 wt %. Both experimental and theoretical data of the hydrogen-containing alloy confirm that the thermodynamic enthalpy of the quaternary Ce2MgNi2–H2 is consistent with that of the experimental results. When calculating the formation enthalpy of hydrogen and metal, the enthalpy of the elastic contribution between metal and hydrogen was considered, generally improving the versatility and accuracy of the calculation. Moreover, the extended Miedema's model is used to predict the hydrogen storage performance.

Calorimetric measurements of the Li[sbnd]Pd system

This work presents the first-ever calorimetric measurements of the partial and integral mixing enthalpies of liquid LiPd alloys. These thermodynamic properties were determined with the use of an MHTC 96 drop calorimeter. The measurements for LiPd system were performed at four different temperatures (620 K, 723 K, 824 K and 926 K), up to a maximum palladium concentration equaled xPd = 0.39. The experimentally obtained values were negative in the entire studied concentration range. The values of the integral mixing enthalpies of LiPd liquid alloys are less exothermic than these predicted by the Miedema model. The highest difference between the data obtained experimentally and those calculated by the Miedema model was about 9 kJ/mol for xPd = 0.39. The experimental data of liquid LiPd solutions were elaborated with the use of the Redlich-Kister formalism and Sommer association model. A good correlation between experimental and calculated by both models results in the entire studied concentration range of Pd was observed.

Thermodynamic properties of liquid Mg[sbnd]Pt alloys determined by the calorimetric method

Pioneering thermodynamic measurements of MgPt system are presented. The integral enthalpies of liquid binary alloys were determined using a drop calorimetric method. The obtained results in three independent series were negative in the entire measured concentration range at 1031 K. The studied integral molar mixing enthalpies were compared with the predicted mixing enthalpies based on Miedema's model. The highest difference between the data obtained experimentally and those predicted by the Miedema's model was found to be 3 kJ/mol for xPt = 0.19. Experimental values of liquid MgPt solutions were also elaborated with the use of Redlich-Kister polynomial. A very good agreement between measurement data and those calculated by the Redlich-Kister was found.

Enthalpies of formation of high entropy and multicomponent alloys using oxide melt solution calorimetry

Understanding the thermodynamics of multi-principal element alloys (MPEA), also known as high entropy alloys (HEA) is crucial to addressing their synthesis, properties, stability, materials compatibility, and technological applications. While the thermodynamics of binary and ternary alloys are relatively well investigated both experimentally and theoretically in the last five decades, this is not the case for multicomponent (>4 elements) alloys. The classical calorimetric methods (e.g., liquid metal bath, direct reaction, and flux calorimeter) used for binary and trinary alloys will often encounter difficulties for alloys with more than three elements. Thus, a more flexible and general calorimetric approach is needed. The present work introduces a new methodology for measuring the enthalpy of formation from elements of metals based on high-temperature oxidative calorimetry. This technique is based on the well-developed oxide melt solution calorimetry (OMSC). Using this method, the heats of formation of four binary (AlNi, AlFe, AlCo, and AlFe3), four quaternary CrFeCoNi, AlCrCoNi, AlFeCoNi, AlCrFeCo and four quinary, AlCrFeCoNi, Al0.3CrFeCoNi, CrMnFeCoNi and AlTiVNbTa alloys were obtained. The values in kJ/(mol atom) are: AlNi, −60.11 ± 3.94; AlFe, −23.46 ± 2.39; AlCo, −51.80 ± 1.89; AlFe, −13.72 ± 2.39; CrFeCoNi, −3.56 ± 1.01; AlCrCoNi, −15.30 ± 2.69; AlFeCoNi, −28.82 ± 1.32; AlCrFeCo, −14.27 ± 2.17; AlCrFeCoNi, −20.97 ± 4.17; Al0.3CrFeCoNi, −8.30 ± 2.56; CrMnFeCoNi, −3.27 ± 2.97; AlTiVNbTa,- 11.93 ± 4.80. The results are compared with predicted values from the Miedema model and with available experimental data. The methodology is thus established to be generally useful for determining the thermodynamic properties of MPEAs.

Microstructural characterization and mechanical behavior of nanocomposite Ti-Ni-Nb surface alloys synthesized on TiNi SMA substrate by additive thin-film electron-beam mixing

An important factor limiting the wide application of magnetron-sputtered amorphous or nanocomposite thin-film coatings for improving surface-sensitive properties of structural alloys is the poor adhesion of coating to substrate. This problem can be overcome through the synthesis of surface alloys (SAs) by additive pulsed electron-beam melting of film/substrate systems of high glass-forming ability (GFA). In this work, this approach is applied to the “film (Ti85/70Nb15/30, at.%), 100 nm/substrate (TiNi SMA)” system using low-energy, high-current electron beam (~2.5 μs, ~15 keV, 1.7 J/cm2) at 10 synthesis cycles and 10 pulses per cycle. The melting/solidification conditions were evaluated using numerical modeling of pulsed heating, taking into account the formation of eutectic in the Ti-Ni-Nb system and initial temperature risen up to 473 K to the end of the synthesis. Using surface SEM/EDS, AES, XRD and cross-sectional HRTEM/EDS/SAED analyses it has been found that both SAs of thickness ~1.5–2 μm have depth-graded structure, in which outer Ni-depleted layer is the multiphase nanocomposite with quasicrystals followed by predominantly Ti-Ni-based amorphous sublayer. Beneath, the ~1 μm thick intermediate sublayer with eutectic columnar structures (B2 + Ti2Ni and B2 + Ti3Ni4 for first and second SA, correspondently), monotonic Nb substitution for Ni and diffusion transition to TiNi substrate is formed. Depth-graded structure of modified surface layers provides monotonic changes with depth in nanohardness, elastic modulus, depth recovery ratio and plasticity to the values of the TiNi substrate. For both SAs the evolution of structure in depth is in a reasonable agreement with glass forming composition range evaluated for Ti-Ni-Nb system using Miedema's model.

Effect of La addition on microstructure evolution of hypoeutectic Al–6Si alloys

Solidification experiments were carried out for Al–6Si alloys with different La addition levels. The results demonstrate that the average sizes of the α-Al grains and the eutectic Si have a tendency of first decreasing and then almost keeping constant with the La addition. About (0.04–0.06) wt% La addition is enough to achieve a satisfactory grain refinement and modification. An analytical model based on the Wilson equation, the Miedema model and the thermodynamics was proposed to investigate the mechanisms of the α-Al grain refinement and the eutectic Si modification in the hypoeutectic Al–6Si alloys with the trace La addition. The La effect on the grain refinement of α-Al and the modification of the eutectic Si are mainly attributed to the increase in the efficiency of the heterogeneous nucleation substrates to nucleate α-Al and the amount of La atoms in the eutectic Si. The grain refinement and the modification results achieve the maximum when the La concentration is near its maximum solid solubility in the primary α-Al phase.

Construction of an n-body Fe–Cu potential and its application in atomistic modeling of Fe–Cu solid solutions

By means of the embedded-atom method, a Fe–Cu potential has been constructed through a newly mathematic form of cross potential. The newly constructed Fe–Cu potential has demonstrated to be more reliable than the five reported Fe–Cu potentials. Based on the Fe–Cu potential, the mechanical and thermodynamic properties and the structural stability of Fe–Cu solid solutions in the whole composition range are derived by molecular dynamics simulation. It is found that the heat of formation curves of the FexCu100 − x solid solutions with body-centered-cubic (BCC) and face-centered-cubic (FCC) structures intersect at the point of x = 65, implying that FexCu100 − x solid solutions with FCC and BCC structures are thermodynamically stable when 0 ≤ x ≤ 65 and 65 < x ≤ 100, respectively. In addition, the derived lattice constants, structural stability, elastic constants, elastic moduli, heat capacity, and coefficients of thermal expansion of Fe–Cu solid solutions from the new Fe–Cu potential agree well with the data of the experiments, first-principles calculation, and the Miedema model.

Estimation of the functions of some iron-based ternary systems within Miedema model and comparison with experiment

The Miedema model is used to estimate the enthalpy of intermetallic compounds and the solid solution of ternary systems. The results showed that the chemical enthalpy values of the binary systems (Fe-Al, Fe-B, Fe-V, Fe-Ga, Al-V, and Al-Mn) are negative. Whereas, the chemical enthalpy values of the binary systems (Fe-Mn and Al-Ga) are positive. The most negative enthalpy is observed for Fe-Al-Ga, Fe-Al-B, and Fe-Al-Sn systems with a mole fraction of 50% of Fe. For ternary systems, the enthalpy varies with the mole fractions of the components. Studies of the products of mechanosynthesis - binary alloy and Fe-Al-based ternary systems: Fe65Al35, Fe65Al35-xMx (Mx = Ga, B, Sn; 5 and 10 at. %) and Fe65-yAl35My (My = Mn, V; 3, 5 and 10 at. %) were confirmed. It concluded that the applicability of the Miedema model for prognostic purposes for estimating the enthalpies and the possibility of obtaining multicomponent systems.