Miedema's model explained

Miedema's model is a semi-empirical approach for estimating the heat of formation of solid or liquid metal alloys and compounds in the framework of thermodynamic calculations for metals and minerals.[1] It was developed by the Dutch scientist Andries Rinse Miedema (15 November 1933 – 28 May 1992)[2] while working at Philips Natuurkundig Laboratorium. It may provide or confirm basic enthalpy data needed for the calculation of phase diagrams of metals, via CALPHAD or ab initio quantum chemistry methods. For a binary system composed by elements A and B, a generic Miedema Formula could be cast as

\DeltaH=f(ElementA,PhiA,nWSA,VA,ElementB.PhiB,nwSB,VB)

where terms Phi and nwS are explained and reported below. For a binary system the physical picture could be simplified by considering a relatively simpler function of the difference of these three physical parameters

\DeltaH=f(PhiA-PhiB,nWSA-nWSB,VA-VB)

resulting in a more complex form

DeltaH=x[A]*(2*f[AB]*V[A,alloy](2/3)*(-P*DeltaPhi2+Q*DeltanWS2-R)/(1/3*nWS[A]+1/3*nWS[B])+deltaH[trans])

[3]

History

Miedema introduced his approach in several papers, beginning in 1973 in Philips Technical Review Magazine with "A simple model for alloys".[4] [5]

Miedema described his motivation with "Reliable rules for the alloying behaviour of metals have long been sought. There is the qualitative rule that states that the greater the difference in the electronegativity of two metals, the greater the heat of formation - and hence the stability. Then there is the Hume-Rothery rule, which states that two metals that differ by more than 15% in their atomic radius will not form substitutional solid solutions. This rule can only be used reliably (90 % success) to predict poor solubility; it cannot predict good solubility. The author has proposed a simple atomic model, which is empirical like the other two rules, but nevertheless has a clear physical basis and predicts the alloying behaviour of transition metals accurately in 98 % of cases. The model is very suitable for graphical presentation of the data and is therefore easy to use in practice."

Free web based applications include Entall [6] and Miedema Calculator.[7] The latter was reviewed and improved in 2016, with an extension of the method.[8] [9] The original Algol program[10] was ported to Fortran.[11]

Informatics-guided classification of miscible and immiscible binary alloy systems

Miedema's approach has been applied to the classification of miscible and immiscible systems of binary alloys. These are relevant in the design of multicomponent alloys. A comprehensive classification of alloying behavior for 813 binary alloy systems consisting of transition and lanthanide metals.[12] "Impressively, the classification by the miscibility map yields a robust validation on the capability of the well-known Miedema’s theory (95% agreement) and shows good agreement with the HTFP method (90% agreement)." These 2017 results demonstrate that "a state-of-the art physics-guided data mining can provide an efficient pathway for knowledge discovery in the next generation of materials design".

Appendix: Basic Miedema Model Parameters

This Table, reports the three main Miedema parameters for the elements of the Periodic table for whom the model is applicable.

These are original parameters [13] which are after page 24 of the book after F.R. De Boer, R. Boom, W.C.M. Mattens, A.R. Miedema and A.K. Niessen Cohesion in Metals. Transition Metal Alloys (1988),[14]

Element
PhiVolt
nWS(density units)^1/3
V(2/3)cm
Group123456789101112131415
Period
1
1HElement
5,2PhiVolt
1,5nWS(density units)^1/3
1,42V(2/3)cm
34567
2LiBeBCN
2,855,055,36,246,86
0,981,6701,7501,7701,650
5,532,882,82,24,15
1112131415
3NaMgAlSiP
2,73,454,24,75,55
0,8201,1701,3901,5001,650
8,275,814,644,24,15
192021222324252627282930313233
4KCaScTiVCrMnFeCoNiCuZnGaGeAs
2,252,553,253,84,254,654,454,935,15,24,454,14,14,554,8
0,6500,9101,2701,5201,6401,7301,6101,7701,7501,7501,4701,3201,3101,3701,440
12,778,826,094,124,123,743,783,693,553,523,74,385,194,65,2
373839404142434445464748495051
5RbSrYZrNbMoTcRuRhPdAgCdInSnSb
2,12,43,23,454,054,655,35,45,45,454,354,053,94,154,4
0,6000,8401,2101,4101,6401,7701,8101,8301,7601,6701,3601,2401,1701,2401,260
14,6510,487,345,814,894,454,214,64,14,294,725,536,286,436,6
555671727374757677787980818283
6CsBaLuHfTaWReOsIrPtAuHgTlPbBi
1,952,323,64,054,85,25,45,555,655,154,23,94,14,15
0,5500,8101,4501,6301,8101,8501,8501,8301,7801,5701,2401,1201,1501,160
16,8611,325,654,894,54,284,154,174,364,75,836,676,947,2
The above list of parameters should be considered as a starting point, which could yield such data (results after Fortran program made available by Emre Sururi Tasci improved data may be found in more recent publications;[15] possibly, in the near future, improvement or insisight of these data could be provided by the extended Calphad databases open collections available at NIMS[16] For instance for Fe-X binary phase diagrams, a list of available databases is as presented in this link https://cpddb.nims.go.jp/cpddb/fe-elem/fe_index.htm and more specifically in this table:
colspan="3"
Binary Iron Systems
Fe-AgFe-GdFe-PFe-Tm
Fe-AlFe-HFe-PrFe-V
Fe-AuFe-HoFe-PtFe-Yb
Fe-BFe-IrFe-SbFe-Zn
Fe-CFe-LaFe-ScFe-Zr
Fe-CdFe-LuFe-Si
Fe-CeFe-MnFe-Sm
Fe-CoFe-MoFe-Sn
Fe-CrFe-NFe-Ta
Fe-CuFe-NdFe-Tb
Fe-DyFe-NiFe-Th

See also

Notes and References

  1. Web site: Thermodynamic Data for Mineral Technology. 1984. 27 November 2017. 1 March 2017. https://web.archive.org/web/20170301092803/http://digicoll.manoa.hawaii.edu/techreports/PDF/USBM-677.pdf. dead.
  2. Web site: . Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW) : Levensbericht A.R. Miedema, in: Levensberichten en herdenkingen, 1993, Amsterdam . 61–66 . Dwc.knaw.nl . 2017-02-28.
  3. Boom R., de Boer F.R.; (2020) Enthalpy of formation of binary solid and liquid Mg alloys – Comparison of Miedema-model calculations with data reported in literature https://doi.org/10.1016/j.calphad.2019.101647
  4. Miedema. A.R.. 1973. A simple model for alloys. I. Rules for the alloying behaviour of transition metals. Philips Technical Review. 33. 149–160.
  5. Miedema. A.R.. 1973. A simple model for alloys. Il, The influence of ionicity on the stability and other physical properties of alloys. Philips Technical Review. 33. 196–202.
  6. Web site: Miedema calculator of standard formation enthalpy . Entall.imim.pl . 2017-02-28.
  7. Web site: Welcome to >>> Miedema Calculator | Homepage organized by Dr. Zhang . Zrftum.wordpress.com . 2017-02-28.
  8. 10.1016/j.cpc.2016.08.013 . Miedema Calculator: A thermodynamic platform for predicting formation enthalpies of alloys within framework of Miedema's Theory . Computer Physics Communications . 209 . 58–69 . 2016 . Zhang . R.F. . Zhang . S.H. . He . Z.J. . Jing . J. . Sheng . S.H. . 2016CoPhC.209...58Z .
  9. Book: Gokcen, N. A.. Statistical Thermodynamics of Alloys (simple presentation). 1986. 978-1-4684-5053-8. 255–76. Appendix B. Springer . https://link.springer.com/content/pdf/bbm%3A978-1-4684-5053-8%2F1.pdf.
  10. A.K. Niessen, F.R. de Boer, R. Boom, P.F. de Châtel, W.C.M. Mattens, A.R. Miedema. 1983. Model predictions for the enthalpy of formation of transition metal alloys II. Calphad. 7. 1, January–March. 51–70. 10.1016/0364-5916(83)90030-5.
  11. Web site: Hex, Bugs and More Physics Emre S. Tasci » Blog Archive » Miedema et al.'s Enthalpy code — 25 years after... hexbugsmorephysics.wordpress.com. 28 July 2008 . 2020-09-06.
  12. Zhang. R. F.. Kong. X. F.. Wang. H. T.. Zhang. S. H.. Legut. D.. Sheng. S. H.. Srinivasan. S.. Rajan. K.. Germann. T. C.. 2017-08-29. An informatics guided classification of miscible and immiscible binary alloy systems. Scientific Reports. en. 7. 1. 9577. 10.1038/s41598-017-09704-1. 28851941. 5575349. 2017NatSR...7.9577Z . 2045-2322. free.
  13. Li . Cai . Yuan . Ye . Li . Fei . Wei . Qiang . Huang . Yuan . 2022-02-15 . Modification and verification of Miedema model for predicating thermodynamic properties of binary precipitates in multi-element alloys . Physica B: Condensed Matter . en . 627 . 413540 . 10.1016/j.physb.2021.413540 . 2022PhyB..62713540L . 244027262 . 0921-4526.
  14. Book: Cohesion in metals : transition metal alloys . 1988 . North-Holland . F. R. de Boer . 0-444-87098-9 . Amsterdam . 17650206.
  15. Boom . R. . de Boer . F. R. . 2020-03-01 . Enthalpy of formation of binary solid and liquid Mg alloys – Comparison of Miedema-model calculations with data reported in literature . Calphad . en . 68 . 101647 . 10.1016/j.calphad.2019.101647 . 213361596 . 0364-5916.
  16. ABE . Taichi; HASHIMOTO . CPDDB . 2022-06-20 . mdr.nims.go.jp . 2007 . Japanese . 10.48505/nims.3060.

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