Miedemas Model

We will briefly discuss here the two principal semi-empirical models Miedema s model and the semi-empirical band structure model. 

Miedema and coworkers have developed a model which gives reasonably good agreement between calculated and experimental heat of formation [67-70]. Consider a binary alloy AB where element A easily forms a hydride and element B does not form a hydride. Element A forms a hydride following the reaction 

At a given temperature T, this reaction has a hydrogen equilibrium pressure H. Consider now a solid solution of this metal with a non-hydride forming metal (B). The relative partial molar free energy of A in the alloy is AGa- If the resulting alloy AB reacts with hydrogen as 

the hydrogen pressure equilibrium of reaction (4.16) is given by 

Since AG is always negative, P is always higher than P. Therefore, the formation of a solid solution with a non-forming hydride has the effect of destabilizing the AH, hydride. The above relation holds when the alloy decomposes upon hydrogenation. However, most of the time, the AB alloy will not decompose upon hydrogenation but instead react as a ternary hydride. 

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Miedema model calculated heats vs. experimental heats of formation,... 

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. 

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... 

The Miedema Model and Other Semi-Empirical Methods 170... 

The Miedema mode and other semi-empiricai methods. The Miedema model was originally devised as a tool for merely predicting the sign of the heat of solution at the equiatomic composition. Therefore Eq. (6.36) does not contain any concentration-dependent terms (Miedema 1973, Miedema et al. 1973). However, the treatment was extended in subsequent publications, and modifications were made to include ordering contributions and asymmetric effects (Miedema et al. 1975, de Boer et al. 1988). Additional fimctions f c A,c and (c, cb, VmA, Vme) were kept very simple and did not include additional parameters other than (Ki), which had already been used to determine (n s). 

More general is the rule of reversed stability (Miedema model) the more stable an intermetallic compound, the less stable is the corresponding hydride, and the other tvay around [36]. This model is based on the fact that hydrogen can only participate on a bond with a neighboring metal atom if the bonds between the metal atoms are at least partially broken (Figures 5.25 and 5.26). 

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]. 

The formation of the magnetically interesting Heusler and related compounds has been discussed recently in terms of the Miedema model (Miedema et al. 1980) which describes the heat of formation of a given compound in terms of energy effects occurring at atomic cell boundaries. The heat of solution of one element in another is given by... 

Williams et al. (1980), while not detracting from the general utility of the Miedema model, have shown that its success results from a dominant chemical trend which is well described by a model of d-bonding due to Pettifor (1970, 1978, 1979). Both the crystal structure and dependence of the heats of formation on atomic number can be attributed to the dependence of the total energy on a geometry-related variation in the d-band density of states. 

Stability (Miedema model) The more stable an intermetallic compound the less stable is the corresponding hydride and vice versa [67]. 

Table 2.2. Values of the Atomic Parameters in the Miedema Model...

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 two-parameter models of Miedema (Section 2.6) and Pearson (this section) are both based on the idea of electronegativity and its equalization on bond formation but each interprets the positive contribution to AH in a different way in the Miedema model it means equalizing the electron density at the surface of the atom, while in the Pearson model it is called chemical hardness and also expresses resistance against charge transfer. 

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. 

The phlogiston parameters and rj of all the elements could be derived from the ionization energy and the electron affinity as in the Pearson model, but they also can be determined by fitting the theoretical function of the bond energy with experimentally observed values as in the Miedema model. Given the bond length Rq the parameter x and rj for every element can be derived with a least-squares fit. The existence of any compound, its bond energy, and polarity can be simply estimated with these parameters. 

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