1. The Application of MTDATA to the Melting/Freezing Points of ITS-90 Metal Fixed-Points
H Davies, D I Head, J Gray, P Quested
Engineering and Process Control Division
23 November 2006

2. Contents Thermodynamic modelling introduction Sn-X systems
Data availability for Sn-X
Sn-X binary diagrams
Influence of X on Tm(Sn)
Real Sn compositions and simulations
Equilibrium or not ?
Al-X systems
Data availability for Al-X
Non-metals (C, N, O)
Al-X binary diagrams
Influence of X on Tm(Al)
Simulations on freezing of real “pure” Al compositions and doping experiments
A Virtual Measurement System for fixed points ?
Conclusions

3. How a thermodynamic model works
The equilibrium state of a chemical system at a fixed temperature (T), pressure (P) and overall composition can be calculated by minimising its Gibbs energy (G) with respect to the amounts of individual species (unaries) that could possibly form, either as distinct phases or within solutions
Models for the variation of G with T and P for unaries and for the influence of interactions between unaries on G for solutions are needed in order to make such calculations possible
G(T) = A + BT + CTlnT + DT2 + ET3 + F/T
More complex models describe the pressure dependence of G for unaries and the contribution made by magnetism, if appropriate
Other effects such as surface energy (small particles/droplets) and strain can in principle be taken into account

4. Unary data for Sn
Sets of coefficients describing G(T,P) must be derived for chemical species in ALL phases (gas, liquid, different crystalline structures) in which they appear. The most stable phase at chemical equilibrium can then be predicted for a chosen T and P as the one with the lowest Gibbs energy

5. Unary data for Sn
Just showing stable phases
Note diamond phase (grey tin) moving towards stability below room temperature

6. Unary phase diagrams The phase rule: F = C + 2 - P
relates the number of degrees of freedom in a system (F) to the number of components (C) and phases (P). For a one component system in which 3 phases (such as gas, liquid and solid) co-exist the number of degrees of freedom is zero - neither temperature nor pressure can be fixed arbitrarily.

7. Gm(T,x) = Sj xjGj(T) + Gmag + RT Sj xjlnxj + EGm(T,x)
Binary data
The molar Gibbs energy of a solution phase (Gm) can be written:
Gm(T,x) = Sj xjGj(T) + Gmag + RT Sj xjlnxj + EGm(T,x)
where R is the gas constant and Gj is the molar Gibbs energy of component j, present in solution with mole fraction xj. The four terms represent unary contributions, magnetic contributions, ideal entropy contributions and finally additional or excess contributions to the Gibbs energy of the phase resulting from interactions between components during mixing. The Redlich-Kister equation is widely used to model the excess Gibbs energy of mixing:
EGm(T,x) = Sj Sk>j xj xk (0Ljk + 1Ljk(xj-xk) + 2Ljk(xj-xk)2 + 3Ljk(xj-xk)3 + …)
nLjk are coefficients determined to model the measured mixing properties of the phase in question as closely as possible. nLjk may be temperature dependent but anything more complex than a linear temperature dependence is unusual. Different powers of (xj-xk) in the Redlich-Kister equation allow asymmetry in the Gibbs energy of mixing to be modelled.

8. Binary Gibbs energies
Phase equilibria are determined by the relationship between Gibbs energies of phases
Temperature is K therefore BCT and LIQUID phase Gibbs energies are identical at pure Sn
As Pb concentration increases the Gibbs energy of the FCC phase becomes lower (green line) until eventually the Pb rich FCC solid solution phase precipitates. This is reflected in the phase diagram on the next slide

9. Sn-Pb
Full binary phase diagram

10. Where do the solute distribution coefficients come from?
Explicit distribution coefficients are not used in the calculations
They can however be deduced from the underlying thermodynamic data
Sections of phase diagrams up to 3 wt% solute and associated calculated distribution coefficients are shown to right

11. Sn-X systems

12. C, N, O, Na, Al, Si, S, Cl, Tl, Cu, Ag, Pb
Data availability for Sn-X (impurity elements having thermodynamic data for interaction with Sn are underlined – MTSOL, MTSOLDERS, COST531)
Analysed and found above detection limits by NRC using glow discharge mass spectrometry ( )
C, N, O, Na, Al, Si, S, Cl, Tl, Cu, Ag, Pb
Metallic elements not found but with high (> 50 ppb) detection limits
Co, In, Sb
Set of elements indicated from analysis and available for thermodynamic modelling
Ag, Al, Cu, In, Ni (as analogue for Co), Pb, Sb, Si
Full set of elements available for thermodynamic modelling
Ag, Al, Au, Bi, Cu, Ge, In, Ni, Pb, Pd, Sb, Si, Zn

14. Sn-Ag (20, 50, 100, 500 and 1000 ppb)
1000 ppb
20 ppb

15. Sn-Al (20, 50, 100, 500 and 1000 ppb)

16. Sn-Au (20, 50, 100, 500 and 1000 ppb)

17. Sn-Bi (20, 50, 100, 500 and 1000 ppb)

18. Sn-Cu (20, 50, 100, 500 and 1000 ppb)

19. Sn-Ge (20, 50, 100, 500 and 1000 ppb)

20. Sn-In (20, 50, 100, 500 and 1000 ppb)

21. Sn-Ni (20, 50, 100, 500 and 1000 ppb)

22. Sn-Pb (20, 50, 100, 500 and 1000 ppb)

23. Sn-Pd (20, 50, 100, 500 and 1000 ppb)

24. Sn-Sb (20, 50, 100, 500 and 1000 ppb)
20 ppb
1000 ppb

25. Sn-Si (20, 50, 100, 500 and 1000 ppb)
NB: No diamond phase considered

26. Sn-Zn (20, 50, 100, 500 and 1000 ppb)

27. NRC Sn composition simulation
Pb impurity only considered
Tm – Tliq = 15 K
Tm – T50% liq = 25 K

28. NRC Sn composition simulation
Only analysed levels for Ag, Cu, Pb and Si considered
Tm – Tliq = 24 K
Tm – T50% liq = 44 K
If Si is excluded these values become 15 and 25 K

29. NRC Sn composition simulation
Analysed levels for Ag, Cu, Pb and Si + 50% of limits of detection
for others
Tm – Tliq = 84 K
Tm – T50% liq = 147 K

30. NRC Sn composition simulation
Analysed levels for Ag, Cu, Pb and Si + 100% of limits of detection
for others
Tm – Tliq = 142 K
Tm – T50% liq = 250 K

31. NRC Sn composition simulation
Ideal liquid and BCT phase models
Tm – Tliq = 120 K
Tm – T50% liq = 214 K

32. NRC Sn composition simulation
Ideal liquid and pure Sn BCT phase models – Raoults Law assumption
Tm – Tliq = 180 K
Tm – T50% liq = 352 K

33. Summary of NRC analysed Sn composition simulation
(Tm – Tliq) / K
(Tm – T50% liq) / K
Pb only 15 25
Analysed Cu, Ag, Si and Pb 24 44
Analysed excluding Si
Analysed + 50% detection limits 84
147
Analysed + 100% detection limits
142
250
Ideal liquid and BCT solid solution
120
214
Ideal liquid and pure BCT solid
180
352

34. Equilibrium v Non-equilibrium (Scheil)
Sn with 10 ppm Pb
MTDATA can do limiting case non-equilibrium solidification simulation by assuming rapid diffusion in liquid and none in solid
Scheil solidification shows significant lowering of temperature at higher solid fractions
Equilibrium and Scheil are bounds to “true” behaviour ???

35. Pressure = 101.325 kPa + 10 kPa + 20 kPa
Pressure effects
Sn with 10 ppm Pb
Pressure = kPa + 10 kPa + 20 kPa
Pressure dependence of melting is handled naturally by MTDATA
Pressures relate to hydrostatic heads of approximately 0 to 29 cm

36. A quick look at look at Al
(1990 JM analysis, including Ti)

37. A quick look at look at Al
(1990 JM analysis, excluding Ti)

38. Al-X systems

39. C, N, O, Mg, Si, P, S, Cl, Ti, V, Cr, Mn, Fe, Ni, Cu
Data availability for Al-X (impurity elements having thermodynamic data for interaction with Al are underlined – MTAL, MTSOL)
Analysed and found above detection limits by NRC using glow discharge mass spectrometry ( )
C, N, O, Mg, Si, P, S, Cl, Ti, V, Cr, Mn, Fe, Ni, Cu
Metallic elements with high (> 50 ppb) detection limits Au
Set of elements indicated from analysis and available for thermodynamic modelling
C, N, Mg, Si, P, Ti, V, Cr, Mn, Fe, Ni, Cu
Full set of elements available for thermodynamic modelling
Ag, C, Ca, Ce, Cr, Cu, Fe, Ga, Ge, Hg, In, Li, Mg, Mn, Mo, N, Nb, Nd, Ni, P, Pb, Sb, Si, Sn, Ta, Ti, V, W, Y, Zn, Zr

40. Al-Si
Full binary phase diagram

41. Partial binary phase diagram
Al-N
Partial binary phase diagram
Predicted nitrogen solubility in liquid Al near Tm is greater than the 1800 ppb impurity found in NRC analysis

42. Partial binary phase diagram
Al-C
Partial binary phase diagram
Predicted carbon solubility in liquid Al near Tm is approx. 0.3 ppb (w/w)

45. Al-N (20, 50, 100, 500 and 1000 ppb) NRC analysis: 1800 ppb

46. Al-Si (20, 50, 100, 500 and 1000 ppb) NRC analysis: 420 ppb 1000 ppb

47. Al-Cu (20, 50, 100, 500 and 1000 ppb) NRC analysis: 230 ppb 1000 ppb

48. Al-Fe (20, 50, 100, 500 and 1000 ppb) NRC analysis: 220 ppb 1000 ppb

49. NRC Al composition simulation
Mg, Si, Ti, V, Cr, Mn, Fe, Ni, Cu, P and N impurities considered
Tm – Tliq = 2.8 mK
Tm – T50% liq = 5.5 mK

50.  Scheil simulation results
NRC Al composition simulation (no N)
Mg, Si, Ti, V, Cr, Mn, Fe, Ni, Cu, P impurities considered
 Scheil simulation results
Tm – Tliq = 275 K
Tm – T50% liq = 810 K

51. NRC Al composition simulation
Only N impurity considered
Tm – Tliq = 2.3 mK
Tm – T50% liq = 4.7 mK

52. Summary of NRC analysed Al composition simulations
(Tm – Tliq) / K
(Tm – T50% liq) / K
Full analysis considered
2800
5500
N excluded
275
810
Only Al-N
(1800 ppb N)
2300
4700

53. Impurity dependence of the aluminium point J Ancsin, Metrologia 40 (2003) 36–41
Al-X systems in NRC study based on wt% Al + precise impurity additions
Adiabatic calorimeter used to allow fraction melted to be quantified
X = Ag, Zn, Cu, Fe, In, Si, Ti, Mn, Cd, Sb, Ca, and Ni
MTDATA equilibrium calculations carried out for Ag, Si and Ti

54.  J Ancsin, Metrologia 40 (2003) 36–41
Impurity dependence of the aluminium point MTDATA equilibrium simulation
Ag impurity
 J Ancsin, Metrologia 40 (2003) 36–41
36 ppm of Ag impurity
76 ppm of Ag impurity

55.  J Ancsin, Metrologia 40 (2003) 36–41
Impurity dependence of the aluminium point MTDATA equilibrium simulation
Si impurity
 J Ancsin, Metrologia 40 (2003) 36–41
18.4 ppm of Si impurity
44.1 ppm of Si impurity

56.  J Ancsin, Metrologia 40 (2003) 36–41
Impurity dependence of the aluminium point MTDATA equilibrium simulation
Ti impurity
 J Ancsin, Metrologia 40 (2003) 36–41
2.8 ppm of Ti impurity
7.0 ppm of Ti impurity
Experimental data show “wrong” curvature

57. Elevation or reduction of melting point Bi-Sb as a model system
Bi side of diagram acts and as analogue of Al + Ti impurity
Sb side acts as an analogue of Al + Si impurity

58. Elevation or reduction of melting point Bi rich system (Al + Ti impurity analogue)

59. Elevation or reduction of melting point Sb rich system (Al + Si impurity analogue)

60. NPL Virtual Measurement system for Al alloys v1.0
(2005 NRC analysis for Mg-Si-Fe-Cu)
Analysis / ppb (w/w)
Mg 160
Si 420
Fe 220
Cu 230
45 mK abscissa range
Note: The current release of this software was designed for commercial Al-alloys and not modified to handle ultra pure metals over very small temperature ranges
11 mK abscissa range

62. MTDATA calculations for pure metals SGTE UNARY Database v4.4
ITS-90
MTDATA
Isotech*
Tm / K
A / K-1 Ga - In Sn Zn Al Ag Au Cu
*A is the 1st cryoscopic constant (-fusH/RTm2).
Table 2, Isotech Journal of Thermometry, Vol 4, No 2 (1993)

63. Conclusions General Thermodynamic data availability
Equilibrium thermodynamic and limiting case non-equilibrium (Scheil) simulations can help in (a) extrapolating non-constant freezing “plateaux” to true liquidus temperatures and (b) estimating deviations of observed liquidus temperature from true pure element melting point
Thermodynamic data availability
Much more data available for Al-X than for Sn-X (commercial Al-alloys v solders)
Chemical analysis issues
The effect of non-metals such as nitrogen is important for Al
Uncertainty in sample analysis in terms of the measured concentrations or the chemical state of the elements is a significant problem possibly introducing more uncertainty than thermodynamic modelling assumptions (eg real solutions v ideal)
Simulations
Results of Scheil simulations only start to deviate from equilibrium above 70% solid
Non-equilibrium modelling should be better at determining the liquidus temperature from sub-liquidus experimental data near the end of solidification
With the high carbon levels, indicated by the analysis and use of graphite crucibles, the phase Al4C3 should always be present
AlN and Al2O3 should also be present in solids with AlN dissolving significantly in liquid Al
Equilibrium simulations are in close agreements with NRC doping experiments for Ag and Si. Odd curvature of NRC Ti experimental results cannot be explained