1. Sources of phase-equilibrium data:
CT-7
Sources of phase-equilibrium data:
Thermal analysis, quantitative metallo-graphy, microprobe measurements, two-phase tie-lines, three-phase tie-lines, X-ray, electron and neutron diffraction

2. Binary phase-diagram data
Measured quantities in binary phase diagram:
Temperatures of invariant (three-phase) equilibria
Points on the boundaries of two-phase fields
measured for samples of known composition by determining
T of phase change
Determining phase composition (1 phase or 2 phases)
for series of samples of different composition annealed at T

3. Phase Diagrams T Liquid TmA TmB a b A xB B P = const. solvus liquidus
a + L
a + AxBy
AxBy + b
AxBy + L
b + L
eutectic
peritectic
AxBy A xB B
Schmetterer C.: 13. Austrian Chemistry Days 2009

4. Phase Diagrams
Technologically important example: Fe-Fe3C

5. Phase Diagrams
Technologically important example: Fe-Fe3C

6. Binary phase-diagram data-cont.
LFS - CT

7. Binary phase-diagram: compositions of phases.
• Rule 1 and 2: If we know T and Co, then we know:
--the composition of each phase.
--the amount of each phase (given in wt%) – lever rule
Cu-Ni system
• Examples:
Adapted from Fig. 9.2(b), Callister 6e.
(Fig. 9.2(b) is adapted from Phase Diagrams of Binary Nickel Alloys, P. Nash (Ed.), ASM International, Materials Park, OH, 1991.) 7
Callister W.D. Materials Science and Engineering. John Wiley 1999.

8. Example: COOLING IN A Cu-Ni BINARY
• Phase diagram:
Cu-Ni system.
Cu-Ni
system
• System is:
--binary
i.e., 2 components:
Cu and Ni.
--isomorphous
i.e., complete
solubility of one
component in
another; a phase
field extends from
0 to 100wt% Ni.
• Consider
Co = 35wt%Ni.
Adapted from Fig. 9.3, Callister 6e. 10
Compare Scheil-Gulliver solidification model
Callister W.D. Materials Science and Engineering. John Wiley 1999.

9. CORED VS. EQUILIBRIUM PHASES
• Ca changes as we solidify.
• Cu-Ni case:
First a to solidify has Ca = 46wt%Ni.
Last a to solidify has Ca = 35wt%Ni.
• Fast rate of cooling:
Cored structure
• Slow rate of cooling:
Equilibrium structure
Influence of diffusion: Homological temperature Th = T / Tm 11
Callister W.D. Materials Science and Engineering. John Wiley 1999.

10. HYPOEUTECTIC & HYPEREUTECTIC
Adapted from Fig. 9.7,
Callister 6e. (Fig. 9.7 adapted from Binary Phase Diagrams, 2nd ed., Vol. 3, T.B. Massalski (Editor-in-Chief), ASM International, Materials Park, OH, 1990.)
(Figs and 9.15 from Metals Handbook, 9th ed.,
Vol. 9, Metallography and Microstructures, American Society for Metals, Materials Park, OH, 1985.)
Adapted from
Fig. 9.15, Callister 6e.
Adapted from Fig. 9.15, Callister 6e. (Illustration only)
Adapted from Fig. 9.12, Callister 6e. 20

11. Thermal analysis
The sample is heated or cooled and its temperature is recorded with time.
When sample is going from a single-phase equilibrium state into a two-phase field, some heat of precipitation is released, which can be sensitively detected by the use of differential thermal analysis (DTA)

12. DTA curves and phase diagram in Ba-Cu system
LFS - CT

13. Tammann triangle Three-phase equilibrium (T=const.):
In thermal analysis - horizontal part of T(t) curve (eutectic dwell time)
Length of this part is proportional to matter reacting at T=const.
Plotting this length vs.mole fraction for eutectic reaction:
Tammann triangle – vertex indicates mole fraction of the eutectic liquid (Sauveur diagram) – determination of xE
In scanning calorimeter – quantitatively correctly measured.
For peritectic reaction – often segregation – use it with caution!

14. Lazerges, M. ; Rietveld, I. ; Corvis, Y. ; et al
Lazerges, M.; Rietveld, I.; Corvis, Y.; et al. THERMOCHIMICA ACTA   497  (1-2)     (2010)

15. Thermal analysis-cont.
Sensitivity of thermal analysis is larger for flat two-phase field boundary than for steep two-phase field boundaries

16. Properties vs. temperature
Thermal analysis:
singularity on H(T) curve is used to identify boundaries between different fields of phase diagram
Other property can be used for this purpose:
Length of sample (T) – dilatometry
Electric conductivity (T) – resistometry
Magnetic susceptibility (T) - magnetometry

17. Properties vs. composition
Isothermal annealing of many samples:
Lattice parameter (xi) – X-ray diffraction
(it is constant in two-phase field and varies in single-phase field)
Electrical conductivity (xi) - resistometry

18. Metallography Optical microscope Objective Objective
Different orientation of grains
Transmission mode
Reflection mode

19. Metallography-cont.
Much of information from metallography is qualitative (single-phase field / two-phase field). Boundaries between single-phase and two-phase fields are mapped by hand. (In binary- as well as in ternary- systems.)
LFS - CT

20. Quantitative metallography:
Metallography-cont.
Quantitative metallography:
In the micrograph of two-phase sample the ratio of areas covered by images of two phases can be measured representing the volume ratio of two phases. If molar molumes of two phases are known, molar ratio can be calculated from volume ratio.
Grinding and polishing errors should be taken into account.

21. Quantitative metallography
Image analysis
Black particles – sigma-phase a b c
Figures were taken after annealing at 700 oC for
a) 500 hrs, b) 3000 hrs, c) 6000 hrs.
Kraus M., Kroupa A., Miodownik P., Svoboda M., Vrestal J.: Int.J.for Mat. Res., accepted

22. Microprobe measurements
Electron gun
Anode
Lenses
Condensors
Screen
Screen

23. Microprobe measurements-cont.
In the electron microprobe, areas of the order of 1 m2 can be chemically analyzed by X-ray spectroscopy (in spite of that the electron beam can be focused to d=50 nm).
(ZAF – corrections, standards or standard-less methods)
(EDX or WDX analysis)
In two-phase samples annealed long enough for they to have large grains enough and being in equilibrium – equilibrium composition of both phases can directly be analyzed (tie-lines).

24. Ternary miscibility gap
Overall composition of two-phase sample analyzed have to be on the determined tie-line.
(K.J.Laidler et.al.: Physical Chemistry, 4.ed., Houghton Mifflin Co., Boston, p.253 )

25. Ternary phase-diagram data
Methods used to localize boundaries in ternary system – in principle the same as for binary ones.
Two independent variables for composition description – two different types of measurements of mole fraction for the same phase

26. The Ternary Phase DiagramB C
Binary system BC T
P=const.
Binary system AC
vertical cut:
Isoplethal Section
horizontal cut:
Isothermal Section
Binary system AB A B C
Schmetterer C.: 13. Austrian Chemistry Days 2009

27. Thermal analysis in ternary system
The temperature of primary crystallisation and the temperature of an invariant equilibrium are found in the same manner as in a binary system.
Tammann triangle – pyramid with two independent composition variables (e.g. mole fractions). Between primary crystallization and invariant equilibrium, the secondary crystallisation may begin with an additional kink in DTA-line.

28. Two-phase tie-lines Similarly as in the binaries,
either the temperature may be measured, where this boundary is at the given composition
or, the composition may be measured where this boundary is found at a given temperature.

29. Two-phase tie-lines-cont.
T = const.
Error of experiment:
LFS - CT

30. Directions of two-phase tie-lines
Tie-lines connects composition points where the chemical potentials i are the same in both phases.
Calculated values of these chemical potentials, however, are given by the description of the previously optimized binary systems. Such a measurements are, therefore, a check of the compatibility of the two binary descriptions.

31. Directions of two-phase tie-lines-cont.
LFS - CT

32. Ternary three-phase equilibria
The experimental methods are the same as for isothermal two-phase equilibria.
Lattice parameters are constant within a three-phase field.
Amounts of phases in equilibrium can be measured by quantitative metallography.

33. Ternary three-phase equilibria
Example: In-Sb-Sn ternary system – 100 oC and 300 oC
Calculation:
binary prediction (only binary data used)
Experiment:
Solubility of In in SbSn phase determined and used for improvement of calculations
Ternary three-phase tie-lines check the binary prediction (note saving of experimental work)
Manasijevic D. et.al.: Journal of Alloys and Compounds 450 (2008)

34. Multicomponent experimental data
Usually not directly used in assessment –
Used as check of extrapolation from the lower-order system (e.g. binary prediction of ternary system) – modification of parameters describing the lower-order systems.

35. Example : Phase diagram Cr-Mo
Cr-Mo system:
Fcc phase is not stable in a binary system –
Parameters for CALPHAD method for it are necessary – have to be assessed from the higher-order system.
Possible influence of performed modifications on lower-order systems must be checked carrefully.

36. X-ray and neutron diffraction
Determination of crystal structures in single-crystalline samples.
Lattice parameters and site occupancies as functions of compositions and temperature can be obtained. (Different scattering factors of X-rays and neutrons: complementary information.)
Ordering can be observed (see example)

37. X-ray and neutron diffraction-example
LFS - CT

38. Rietveld refinement
Determination of site-occupancy parameters by analyzing intensity ratios of X-ray- or neutron-diffraction spectra of polycrystalline samples.

39. Mössbauer spectroscopy
Recoil-free resonance absorbtion of -rays
Measure of local state of ordering and site occupancy (namely Fe and Sn systems - „ Mössbauer“ nuclei Fe57 and Sn119)
(Magnetic moment on the nucleus is sensitive to the local surroundings of it.)

40. Other experimental data related to phase equilibrium
Theoretical and experimental data not directly used in assessments but important:
Phonon spectra – for excess term of Gibbs energy (vibrational contribution to the entropy)
Elastic constants, bulk modulus, thermal expansion –
for derivatives of Gibbs energy

41. Questions for learning
1. What information concerning of phase diagram we can receive using optical metallography?
2. What information concerning of phase diagram we can receive using differential thermal analysis or differential scanning calorimetry?
3. What information concerning of phase diagram we can receive using dilatometry, resistometry, magnetometry?
4. What information concerning of phase diagram we can receive using scannning electron microscopy (with EDX analyser), X-ray diffractometry?
5. What information concerning of phase diagram we can receive using Rietveld refinement and Mössbauer spectroscopy?