1. Dynamic methods to construct phase diagrams
Lection 9
Dynamic methods to construct phase diagrams
Dynamic methods of phase equilibrium studies – DTA, HF-DSC
a. Unary system;
b. Binary and ternary systems
c. TGA;
d. Heat capacity measurements using DSC

2. Thermal Analysis: Dynamic Methods
Measurement of physical property of a substance as a function of temperature using controlled temperature program
Method
Measured property
Application
Differential Thermal Analysis (DTA)
Temperature difference
Phase reactions, phase transformations
Differential Scanning Calorimetry (DSC)
Heat flow
Specific heat, Heat of transition
Thermal Gravimetric Analysis (TGA)
Mass Change
Reactions with the gas phase, decomposition reactions
Dilatometry
Size Change
Phase transformations, Thermal expansion
Thermomechanical Analysis (TMA)
Mechanical properties
Materials testing

3. DTA vs. DSC DTA DSC Heat-flux Power compensation
DT between sample and reference
DT between sample and reference
Power compensation to keep the same temperature in both furnaces
T and DH of transformation
More robust, measurements can be done in wider T range, in more aggressive environment (oxidation atmosphere), possible combination with TGA to measure mass change
T and DH of transformation, Cp measurements
T and DH of transformation, Cp measurements
HF-DSC is more sensitive than DTA, possible to measure heat capacity
More effective, since response time is shorter than in HF-DSC
DTA/HF-DSC signal is difference between sample and reference thermocouples. It is usually given in mV. For some devices it is given as temperature difference; this means that reference table or equation was used for recalculation.
The temperature is not measured in the sample, but at the bottom of crucible. Temperature calibration is necessary.

4. The DTA Signal
Heat is transferred between furnace, crucible, sample (reference) and thermocouple
Steady –state condition ∆Φ 𝑆𝑅 =− 𝐴𝜆 Δ𝑙 Δ 𝑇 𝑆𝑅 =−𝐾Δ𝑇
DFSR – difference in heat flow rate, l – thermal conductivity
TW is furnace temperature, TC is temperature of crucible
TT is temperature of thermocouple, TS is sample temperature
Idealized curve
Real curve

5. DTA signal for unary system
DTA responses to melting and freezing of pure substance: a- onset temperature, b- peak at temperature c. Due to dynamic character of experiment, the temperature distribution is never completely homogeneous. The temperature is not measured in the sample, but at the bottom of crucible. That is why temperature correction is necessary.
Temperature calibration: establishment of relation between measured temperature Tmeas indicated by the instrument and the true temperature Ttr.
At least three substances (usually pure metals) with melting temperature covering temperature range of interest should be selected. Mass should be corresponded to recommended mass for measurement in this instrument. Measurement should be done with different heating rate b and extrapolated to b=0.

6. Temperature calibration
Measured temperature of thermal event depends on mass and cooling rate b.
Extrapolated T melting of Sn to b=0 for different sample mass
Substance
T, °C Sn Al Ag
961.78 Au Pd
1554.8
Temperature correction for Ga, In and Sn

7. Enthalpy calibration KH is instrument sensitivity for Ga, In, Sn
𝑚 𝑠𝑎𝑚𝑝𝑙𝑒 ∆ 𝐻 𝑠𝑎𝑚𝑝𝑙𝑒 = 𝐾 𝐻 𝑡1 𝑡2 ∆𝑇 𝑡 𝑑𝑡

8. Recommended values of temperatures and enthalpies of melting of metals
Enthalpy calibration is establishment of relationship between the enthalpy change DHmeas (peak area A) measured by the instrument and true enthalpy change DtrsH absorbed or released by the sample as a result of transition at transition temperature Ttrs: DtrsH=KH(Ttrs)DHmeas.
Enthalpy calibration factors for each calibration substance are represented as a function of transition temperature. Provided the dependence on heating rate b and sample mass are negligible (within scatter of individual experiments) the enthalpy calibration factor KH(Ttr, m, b) give the enthalpy calibration function KH(T).
Element
Tmelt (°C)
DHm(J/g) Ga
29.764
80.07 In
28.62 Sn
60.38 Zn
108.09 Al
399.87 Ag
961.78
104.61 Au
64.58

9. Heat capacity measurement using DSC: classical three-step method
Empty crucible measurement F0(T)
Calibration substance measurement (e.g. synthetic sapphire, Pt)
𝐶 𝑃,𝑐 𝑚 𝑐 𝛽= 𝐾 Φ (𝑇)( Φ 𝑐 − Φ 0 )
KF(T) – calibration function
Sample measurements
𝐶 𝑃,𝑆 𝑚 𝑆 𝛽= 𝐾 Φ (𝑇)( Φ 𝑆 − Φ 0 )
All three measurements have to be performed under exactly the same conditions
Good thermal contact between sample and crucible is required
𝐶 𝑃,𝑆 = 𝑚 𝑐 𝑚 𝑠 ∆Φ 𝑆 ∆Φ 𝑐 𝐶 𝑃,𝑐
DFS=FS-F0, DFc=Fc-F0, - heat flow rate difference between substance S / calibration substance c and zero-line F0(T), mc-mass of calibration substance, ms – mass of sample, CP,c – heat capacity of calibration substance

10. Binary systems
Equilibrium: example Sn-Bi system

11. Binary system
Equilibrium: Example Sn-Bi system

12. Scheil Solidification
Fast diffusion in liquid
Slow diffusion in solid
Local equilibrium

13. Latent heat: Equilibrium vs. Scheil solidification
Equilibrium solidification
Scheil solidification

14. Example: Ag-Cu system. Comparison of calculated dHS/dTS with DTA results
Phase diagram and calculated dHS/dTS for 1 and 5 mass.% Cu.
dHS/dTS (J/kgK)
DTA results at different heating rates: black -15 K/min, red – 10 K/min, blue – 5 K/min.

15. Example: binary system Ag-Cu
Comparison of experimental DTA with calculated dHS/dTS
for alloys with 9, 23 and 28 mass.% Cu:
Black line – heating rate 15 K/min, red – 10 K/min, blue 5 K/min.
dHS/dTS (J/kgK)

16. General DTA curve analysis for binary system
Alloy Ag-15%Cu: dHS/dTS vs. TS using equilibrium enthalpy. Delta function is eutectic, vertical jump is liquidus.
DTA scan for melting and freezing at 5 K/min for Ag-15%Cu alloy:
Important points are labeled by i, not important by n.

17. Eutectic reactions (L a+b) vs. Peritectic reactions (L+b a)
The kinetics of peritectic reaction is different from eutectic because the diffusion rate is very different in liquid and substitutional solids. If only interstitial diffusion is required the peritectic reaction occurs more easily. If one assume that no diffusion occurs in the solid upon cooling, solidification merely switch from freezing of high temperature phase L b to freezing of low temperature phase L a. Then b phase usually surrounds a phase resulting in coarser two phase microstructure than eutectic one.

18. Phase diagram with eutectic and peritectic reactions
Example Au-Sn system
Sn-rich part of Au-Sn phase diagram (a). Hs and dHs/dTs curves for Sn-25%Au calculated for equilibrium (black) and Scheil (red) conditions (b). DTA scan for Sn-25%Au (c)
(c)

19. Types of reactions in ternary systemSn
E2: LAg3Sn+(Sn)+Cu6Sn5
U5:L+Cu3Sn  Ag3Sn+Cu6Sn5
P2: L+(Ag)+Cu10Sn3  Cu3Sn

20. Example: system Ag-Cu-Sn
Part of liquidus surface
U5 L+Cu3Sn=Ag3Sn+Cu6Sn5, E2 L=Ag3Sn+(Sn)+Cu6Sn5
Phase fraction diagram Sn-25%Ag-10%Cu
Vertical section at 25 mass% Ag
Latent heat release in equilibrium
Latent heat release in Scheil conditions
dH/dT in equilibrium

21. Example Ag-Cu-Sn Phase fraction Sn-25%Ag-27%Cu
Latent heat release in equilibrium
dHS/dT in equilibrium
Latent heat release in Scheil conditions

22. Vertical section construction using DTA/DSC

23. TGA: Thermo Gravimetric Analysis
Determination of the mass as function of temperature. It is applicable for all reaction involving gas phase
Decomposition reactions
Vaporization and sublimation
Desorption
Oxidation and reduction
Corrosion
Small sample
Gas flow essential
Different temperature modes
Linear heating, isothermal, stepwise isothermal

24. Solid lines- TG, dashed lines - DTG

25. TGA: thermogravimetric analysis
TGA measure mass loss or gain with temperature and/or oxygen partial pressure. TGA can be used to investigate red/ox reactions in M/Oxide system and decomposition reactions accompanied by gas formation (e.g. CaCO3CaO+CO2 (gas))
778°C 3Mn2O3  2aMn3O4+0.5O2
1167°C aMn3O4  bMn3O4
1458°C bMn3O4  3MnO+0.5O2
Calculated phase diagram of the Mn-O system
pO2=10-2 bar

26. Good praxis for DTA experiments
Problems
Influence of mass and heating rate: Larger mass and larger heating rate produce larger peak, but make detection of closely spaced thermal events more difficult.
Sample shape is typically not conform to the shape of the sample cup. The thermal contact area between the sample and crucible will change during melting process.
Undercooling: Many metals and alloys are prone to undercooling before the nucleation of solid phase start from melt. Nucleation temperature can differ from liquidus up to 100 or more degree depending on nature of alloy system and other factors. Determination of melting on heating is more reliable.
Good praxis for DTA experiments
Calibration:
Based on the melting point of pure substances. Crucible, standard material, heating rate, sample mass, atmosphere are kept constant
Characterisation:
The composition of samples and crystal structure have to be investigated before and after the measurement
Combination:
DTA experiments tell us that something is happening at a specific temperature. They do not tell us, what is happening. Combination with other methods like X-ray diffraction, spectroscopy, microscopic investigation and composition analysis (e.g. Electron probe microanalysis) are required to interpret the results