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Theoretical background of formation enthalpy of amorphous phase

Refs.

[M1] H. Bakker, Enthalpies in Alloys: Miedema’s Semi-Empirical Model, Trans Tech Publications, Zurich, 1998.

[M2] A. R. Miedema and A. K. Niessen, Private communications 1989

Theoretical background of formation enthalpy of statistical solid solution

 Refs.

[M1] H. Bakker, Enthalpies in Alloys: Miedema’s Semi-Empirical Model, Trans Tech Publications, Zurich, 1998.

[M2] D. J. Eshelby, Solid State Physics (F. Seitz and D. Turnbull eds, Acdemic Press, New York, 1956), Vol. 16, 276.

[M3] A. R. Miedema and A. K. Niessen, Private communications 1989

Notices to the applications of Miedema’s model

It is pleased that some people would like to use Miedema’s model to calculate the formation enthalpy. However, notices must be announced to avoid the misunderstanding of Miedema’s theory. For example, some researchers may make the following mistakes when applying the Miedema’s model

1) “the proposed extension to Miedema’s model lacks symmetry for the interchange of metals, whereas Miedema’s model does.” [M1] This is an incorrect understanding to Miedema’s theory because, such the symmetry is also not shown in the original Miedema’s model, and the asymmetry is an important physical feature of Miedema’s model [M2]. Please refer to the following comparison,

2) If one didn’t notice the absolute (“| |”) symbol in the Eq. (2) below, one may get problem which is simply due to the incorrect application of the formulae (1) and (2) [M1] [M2]. The clarification is given below for your attention: 

3) In addition, to be noticed is the composition dependence must be kept in mind, i.e. the solute concerntration must be used to avoid confusing, and accordingly  must be the solute related quantity.

4) Notice that the proposed atomic size difference factor S(c) in [M2][M3] should be corrected as (i.e. using mole concentration cA or cB to replace surface concentration of csA and csB) to get consistent values. Surface concentrations are used for the calculation of f(c) factor. However, atomic size difference factor may be adopted in different forms to improve the statistical agreement with experimental values. In such case the surface concentration may be used for comparison.

More clarification can be found in Ref. [M1][M2] [M3] 

[M1] X.Q. Chen, R. Podloucky, P. Rogl, and W. Wolf, Appl. Phys. Lett. 86 (2005) 216103.

[M2] R.F. Zhang, B.X. Liu, Applied Physics Letters 86/21 (2005) 216104.

[M3] R.F. Zhang, S. H. Sheng, and B.X. Liu, Chem. Phys. Lett. 442 (2007) 511–514

Theoretical background of Standard formation enthalpy

Predicting some fundamental materials properties, e.g., the crystalline structures, formation enthalpies (or energies), bulk modulus, etc. has been well-known to be a vital important issue for the physicists, chemists, and materials scientists [1,2]. In the past century, several empirical rules based on simple physical parameters, such as the electronegativity, atomic size, valence-electron, etc. [3], have been proposed to predict the phase formation. However, these empirical rules do not have semi-/or quantitative formulation. Later in earlier 1970s, Miedema et al. [4–7] developed a semi-quantitative model to predict the phase formation, by combining the intrinsic properties of the constituent elements, i.e., the work functions, molar volumes and electron densities at the boundary of the Wigner–Seitz (WS) cell. Afterwards, Miedema’s model concerning the formation enthalpy was further improved by consideration of the effect of atomic size difference on the contact interface between two dissimilar WS cells [8]. At that time, about 292 self-consistent experimental values are reported by Kleppa’s research group [9,10] and used to test its validity. Recently, the research group continuously reported another 70 experimental values of the formation enthalpies [11], providing an opportunity for further testifying the improved model. Besides, the ab initio calculation is a very powerful means for the prediction of some materials properties [1,12], due to its nature of parameter-free and solid physical insight included. Its predictive ability for the crystalline structure has been comprehensively tested by a highthroughput data-mining (HTDM) technique integrated into the ab intio method (abbreviated as ‘ab initio HTDM method’) proposed by Ceder et al. [2,13,14]. With this method, the crystalline structures of 89 compounds at ground state were correctly predicted, and about 195 values of the formation energies (i.e., the formation enthalpy) at ground state were reported. With these data in hand, it is therefore possible to make a throughout comparison with the predicted formation enthalpy by the improved model. The improved model proposed by Zhang et al. [8] is further analyzed and further developed by a statistical approach of the parameter optimization, followed by some comprehensively statistical comparisons of the improved model with the published experimental data, those deduced from the original Miedema’s model as well as from the ab initio HTDM method [15].

In order to calculate the formation enthalpy of the intermetallic compounds, three important parameters, i.e., the electronegativity difference, the electron-density mismatch, and the atomic size difference [4–8] should be included in the calculation. First, the electronegativity difference  between the two constituent metals corresponds to a negative contribution to the formation enthalpy. It provides a major driving force for the formation of an intermetallic compound by the charge transfer, which decreases the contact potential difference between two dissimilar metals. Second, the discontinuity of the electron density at the boundaries between the dissimilar atoms, must be smoothed and therefore contribute a positive parameter in the formation enthalpy. Third, the atomic size difference would frequently lower the contact area between the two dissimilar WS cells and decrease the binding energy between the two dissimilar atoms, by the way of decreasing the package density of the crystalline lattice, because the electron cloud would become further away from the nuclei. Accordingly, the formation enthalpy with element B as the solute can be expressed by [8],

where f(c) is a function of alloy composition, which has considered the effect of the chemical short-range-order (CSRO) in an ordered intermetallic compound on the formation enthalpy, and is expressed by

Where the CSRO parameter for the intermetallic compound and is taken as a constant, 8. The pre-factor S(c) is defined to describe the effect of the atom size difference on the contact interface and the bonding energy, and is expressed as,

It should be emphasized that  is an absolute value and therefore is always positive. Consequently, the pre-factor S(c) is a value within a range of 0 < S(c) < 1. csA and csB could be mole concentration (cA, cB) or surface concentration as defined in f(c). In the present software, the mole concentration and uncorrected mole volume is used.

The Gamma parameter is a composition-independent constant and defined as

Concerning the physical meaning of the other parameters, the readers are referred to some previously published papers [4–8].

[1] D. de Fontaine, in: F. Seitz, D. Turnbull (Eds.), Solid State Physics, vol. 47, Academic, New York, 1994.

[2] S. Curtarolo, D. Morgan, G. Ceder, CALPHAD 29 (2005) 163, and references therein.

[3] P. Villars, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds: Principles and Applications, vol. 1, John Wiley & Sons ltd, London, 1995, p. 227.

[4] A.R. Miedema, P.F. de Chatel, in: L.D. Bennet (Ed.), Proc. Symposium Theory of Alloy Phase Formation, American Society for Metals, New Orleans, 1979.

[5] A.R. Miedema, P.F. de Chatel, F.R. de Boer, Physica 100 (1980) 1.

[6] A.R. Miedema, A.K. Niessen, F.R. de Boer, R. Boom, W.C.M. Matten, Cohesion in Metals: Transition Metal Alloys, North- Holland, Amsterdam, 1989.

[7] H. Bakker, Enthalpies in Alloys: Miedema’s Semi-Empirical Model, Trans Tech Publications, Zurich, 1998.

[8] R.F. Zhang, B.X. Liu, Appl. Phys. Lett. 81 (2002) 1219; R.F. Zhang, B.X. Liu, Appl. Phys. Lett. 86 (2005) 216104; R.F. Zhang, B.X. Liu, Phil. Mag. Lett. 85 (2005) 283.

[9] J.W. Wang, Q.T. Guo, O.J. Kleppa, J. Alloy. Compd. 313 (2000) 77.

[10] Q.T. Guo, O.J. Kleppa, J. Alloy. Compd. 321 (2001) 169.

[11] S.V. Meschel, O.J. Kleppa, J. Alloy. Compd. 350 (2003) 205;S.V. Meschel, O.J. Kleppa, J. Alloy. Compd. 363 (2004) 237;S.V. Meschel, O.J. Kleppa, J. Alloy. Compd. 388 (2005) 91;S.V. Meschel, O.J. Kleppa, J. Alloy. Compd. 416 (2006) 93;S.V. Meschel, O.J. Kleppa, J. Alloy. Compd. 415 (2006) 143.

[12] J.H. Zhu, C.T. Liu, L.M. Pike, P.K. Liaw, Intermetallics 10 (2002) 579.

[13] S. Curtarolo, D. Morgan, K. Persson, J. Rodgers, G. Ceder, Phys. Rev. Lett. 91 (2003) 135503.

[14] D. Morgan, G. Ceder, S. Curtarolo, Mater. Sci. Technol. 16 (2005) 296.

[15] R.F. Zhang, S. H. Sheng, and B.X. Liu, Chem. Phys. Lett. 442 (2007) 511–514.

Comparisons of ab initio, ZSL and Experiments

Comparison of ab initio and ZSL model