Enthalpy Miedema model

The Miedema s parameters and the Miedema model and formula proved to be useful in an approximate evaluation of the formation enthalpy of alloys, in the estimate of the formation capability of intermetallic compounds, etc. 

Miedema s model and parameters and prediction of compound formation capability. A semi-empirical approach to the evaluation of the compound formation capability and of the heats of formation of alloys was proposed by Miedema and co-workers. This resulted in a model which became very popular and, especially because there was scarcity of experimental data, was frequently used in the evaluation, even if approximate and several times incorrect, of the formation enthalpies. The model suggested for energy effects in alloys is well known essentially it is based on the definition of two parameters. By assigning two coordinates... 

As a conclusive comment to the Miedema model, especially with reference to the enthalpy evaluation, we have to underline that while it may be useful in order to define a reference behaviour, however, its approximate (in a way qualitative) character cannot be forgotten. A critical discussion on the application and limits of this model has been published for instance by Chen et al. (2004) see the comments on the thermochemistry of the Laves phases in 3.9.3, see also a few more remarks on this subject in 4.4.7.1. 

Miedema s theory and structural information. The Miedema model for energy effects in alloys, presented in 2.2.1.3 has been very useful in an evaluation, albeit approximate, of the formation enthalpies and in the prediction of compound formation capability. For an example of the application and limits of this model, see the comments on the thermochemistry of the Laves phases reported in 3.9.3. However notice that the general usefulness of the Miedema approaches has diminished with time, both for its inherent approximation and for... 

For the calculation of the net adsorption enthalpies of transactinides on metal surfaces the partial molar enthalpies of solution and the enthalpy of displacement are required. These values can be obtained using the semi-empirical Miedema model [66-70] and the Volume-Vacancy or Surface-Vacancy model [32,70,71]. Data for these calculations are given in [34,72,73]. 

Figure 2.17. Several elements arranged according to their parameter values in the Miedema model. The sign of the reaction enthalpy is indicated in the quadrant bounded by the P/Q lines of the group of metals. The stable binary (metallic) compounds can be identified at a glance. The metallic form of hydrogen is also placed with the metals. Hydrogen has a negative charge in lithium hydride and a positive charge in the stable compound with palladium.

The Linnett model is not parameterized as the Pearson and Miedema models are and bond enthalpies cannot be calculated from atomic data. However, it is strong on molecular structures and provides a simple (but qualitative) explanation for them without the need for much resonance and configuration interaction. 

Some aspects of the mentioned relationships have been presented in previous chapters while discussing special characteristics of the alloying behaviour. The reader is especially directed to Chapter 2 for the role played by some factors in the definition of phase equilibria aspects, such as compound formation capability, solid solution formation and their relationships with the Mendeleev Number and Pettifor and Villars maps. Stability and enthalpy of formation of alloys and Miedema s model and parameters have also been briefly commented on. In Chapter 3, mainly dedicated to the structural characteristics of the intermetallic phases, a number of comments have been reported about the effects of different factors, such as geometrical factor, atomic dimension factor, etc. on these characteristics. 

By contrast, the macroscopic atom model of Miedema (Miedema et al. 1975) starts with a descriptions of the solid state which is then modified to describe the liquid state (Boom et al. 1976a, 1976b). In their model the enthalpy of formation at 0.5 mole fraction, Hc=o.s is given to a first approximation by ... 

Table G. 1 reproduces values calculated by Miedema s model (Niessen et al. 1983) for the partial enthalpy of solution at infinite dilution of a liquid metal solute i in a liquid metal solvent i, AH, (in kJ/mole). For a i-j alloy, the regular solution parameter k can be approximated by [AHj(j( + AHJ(l)]/2.

The value of the maximum dissociation energy in a homonuclear diatomic transition metal molecule has been predicted (2) to be (600 40) kJ mol-1 for ditantalum. The dissociation energies of homonuclear transition metal molecules have been predicted by various methods of which the most recent is the cell model of Miedema (7). This proposes a relation between the enthalpy of vaporisation of the solid metal, AH ap, the dissociation energy, D°, of the diatomic molecule and the surface energy of the metal, Y° as follows,... 

Fig. 3.26. Maximum negative enthalpies of mixing Affm in Co5STM45 alloys calculated by Miedema s model [3.59] and observed glass-forming ranges AGFR [3.57]...

The general trend of the enthalpies of formation of these compounds appears similar to that observed for compounds of uranium [1968KUB] and a number of lanthanides [1980BOR/BOR] with arsenic, antimony, and bismuth more generally, these trends are also in satisfactory agreement with those deduced from the model of Miedema et a/. [1977MIE/BOE]. 

Miedema, A. R., de Boer, F. R., Boom, R., Model predictions for the enthalpy of formation of transition metal alloys, CALPHAD Comput. Coupling Phase Diagrams Thermochem., 1, (1977), 341-349. Cited on page 334. 

In addition to the experimental thermodynamic values, Niessen and de Boer (1981) calculated the enthalpies of formation at 0 K according to a semiempirical model proposed by Miedema et al. (1980) for the carbides R2C, RC and RC2, where R = La, Sc, Y,AHf, room temperature = 99,132 and 111 for La2C, Sc2CandY2C 100,132 and 112 for LaC, ScC and YC 180, 210 and 192(kJmol ) for LaC2, ScC2 and YC2, respectively. Comparison of the predicted heats of formation for LaC2 and YC2 with the experimental values shows these predicted values to be inaccurate by about 100%. 

One of the most useful models in evaluating both enthalpy of formation and volume effects in intermetalllc compounds is Miedema s approach (Miedema Niessen, 1982 de Boer et al., 1988). It uses an atomic cellular model of the alloy, based on the concept of Wigner-Seitz (W.S.) cell. The intermetalllc compound AB is considered as being formed by the atomic W.S. cells of the pure metals A and B (seeRgureS). 

Crystallization temperature (T ) for several amorphous alloys and formation enthalpies (AH) calculated by means of Miedema s model for the corresponding (hypothetical) intermetallic compounds. The AH values are given in units of kj per mol of alloy (Miedema et al., 1980). 

Miedema s model is an empirical theory for calculating heat of mixing in various binary systems both for the solid state (Miedema et al., 1975) and liquid (Boom et al., 1976). This model involves the calculations of the formation enthalpy of metallic glasses (amorphous... 

Kleppa and his coworkers measured the enthalpies of formation of a series of intermetallic compounds between rare earth metals and transition metals by accurate high temperature calorimetric method [62]. Based on these data, we have used SVR-based feature selection method (this method is described in Chapter 4 of this book) to search the optimal set of atomic parameters for support vector regression. It has been found that an effective feature set suitable for the computerized prediction of the enthalpy of formation of these compounds includes only two parameters Miedema s A< ) and the ratio of atomic radii R1/R2 (here the values of the atomic radius proposed by Teatum are used). By SVR and leave-one-out (LOO) cross-validation, the results of prediction are rather good. Table 7.1 shows the comparison between the experimental values and the predicted values of the enthalpy of formation of these compounds. The results of calculation by Miedema s model [104] are also listed for comparison. 

Formation enthalpies of intermetallic compounds of the rare earth metals La, Y and Sc with 3d, 4d and 5d elements (in kj/mol). The values listed represent model calculations according to the results publi ed by Miedema et al. (1980). Light rare earth elements interpolate between La and Y, heavy rare earth... 

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