1. Non-statistical thermodynamic optimization: an extravagance or a useful tool? Cong (Leo) Dai

2. 2 Outline Thermodynamic modeling CALPHAD method Principles and examples Advantages and deficiencies Kantorovich idea Conclusions and future efforts

3. 3 Thermodynamic modeling PhasSage, www.factsage.com PhasSage; http://honghua66.en.ecplaza.net; Solutionizing Heat Treatment Furnace (27982050)http://honghua66.en.ecplaza.net A B C

4. 4 What is needed to use thermodynamic modeling? Analytical descriptions of the Gibbs energies of stoichiometric compounds and solutions Publications including tables of thermochemical data Databases A program minimizing the Gibbs energy of a system Thermo-Calc FactSage Pandat MALT2 Reliability of the analytical representations of the Gibbs energies?

5. 5 Phase diagrams of the Fe–Nb system TCFE2 databaseTCFE6 database BA

6. 6 Constructing the Gibbs energies

7. 7 CALPHAD technique is not unique Step 1: a model The coin is a disc Step 2: identify unknown model’s parameters Radius r, thickness h, density  ; they all must be positive Step 3: collect all available experimental data r, h, d, l, a, V, m,  Step 4: solve an optimization problem

8. 8 Finding r, h and  Non-linear least squares problem with linear constraints

9. 9 Almost the same happens in CALPHAD Experimental data 1.Enthalpies of mixing at 1150°C 2.(Ag)+L / L liquidus temperatures 3.L / L+(Cu) liquidus temperatures 4.Chemical potential of Cu in (Ag) 5.Chemical potential of Ag in (Cu)

10. 10 Building the Gibbs energies of the phases A traditional CALPHAD method rests on a statistical foundation

11. 11 Statistical description of data Measurement is non-repeated Measurement is insufficient Measurement errors are affected by random errors and systematical errors

12. 12 Kantorovich idea(1962) Instead of minimizing the sum of squared derivations, simultaneous inequalities should be used, which can be solved by linear programming methods. Kantorovich, L.V. Sib. Mat. Zh., 1962, vol.3, No.5, p. 701.

13. 13 Interval data Random errors All information Systematic errors All possible outcomes of an experiment belong to a finite interval! Any value in the same interval data is equally acceptable!

14. Non-statistical approach 14 1.Error analysis 2.Postulate a model 3.Solve inequalities

15. 15 Data from Kawakami Kawakami, M.: Sci. Rep. Res. Inst. Tohoku Univ. 7 (1930) 351. Reach the required temperature Remove the porcelain tube Measure the temperature variation

16. 16 Mixing Enthalpy of liquid Mg-Ag alloys Kawakami, M.: Sci. Rep. Res. Inst. Tohoku Univ. 7 (1930) 351. AB

17. 17 Mixing Enthalpy of liquid Mg-Ag alloys No feasible region.

18. 18 Data from Gran J. Gran et al./ CALPHAD 36 (2012 89-93) Heat the furnace Hold at a predetermined temperature Quenched in an Argon-stream Analyzed by ICP-AES analysis (Inductively Coupled Plasma- Atomic Emission Spectrometry)

19. Activity of Mg in liquid Mg-Ag alloys 19 J. Gran et al./ CALPHAD 36 (2012 89-93)

20. 20 Activity of Mg in liquid Mg-Ag alloys No feasible region. Comparison Future work!

21. Conclusions 21 CALPHAD technique is successful but not always satisfactory. A non-statistical approach is established and applied to thermodynamic optimization.

22. Future efforts 22 Comparisons in liquid Mg-Ag alloys. Rules of assigning interval data. Non-statistical approach applies to other interesting thermodynamic assessments.

23. 23 Acknowledgements Supervisor: - Dr. Dmitri Malakhov Thanks for your attention!