M odelling of phase diagrams of iron alloys

1. EFDA programme on fusion materials Modelling of phase diagrams of iron alloys M.Yu. Lavrentiev, D. Nguyen-Manh, J. Wrobel, S.L. Dudarev EURATOM/CCFE Fusion Association, Abingdon, Oxfordshire OX14 3DB, United Kingdom

2. EFDA programme on fusion materials What is necessary to build a phase diagram? • Comparison of free energies – energy (enthalpy) and the entropy contribution • An interaction model. Role of magnetism in the interatomic interaction. Long enough range of interaction • Entropy – magnetic and configurational. For high temperatures, configurational entropy may be considered ideal • Vibrational contribution – might be necessary and even dominating (miscibility gap in Fe-Cr)

3. EFDA programme on fusion materials Magnetic Cluster Expansion Hamiltonian The Hamiltonian includes both occupational {σ} and magnetic (vector) {M} variables. For any given configuration (set of occupational variables), the energy of the system is minimized with respect to magnetic moments.

4. EFDA programme on fusion materials Monte Carlo simulations • Allow fast calculation of the enthalpy of magnetic system • Large simulation boxes available • Magnetic entropy difference can be calculated via integration of the specific heat • For high temperatures, configurational entropy may be considered ideal

5. EFDA programme on fusion materials Bcc Fe In pure iron, the Curie temperature can be identified from the maximum of specific heat at about 1100 K, close to experimental value of 1043 K. Magnetic correlations, unlike the magnetic moment, persist at temperatures above the Curie point.

6. EFDA programme on fusion materials Bcc Cr Disappearance of antiferromagnetic order in pure Cr can be seen in the diagram of distribution of magnetic moments at 50 K and 500 K, as well as in nearest neighbour correlations. Order disappears above 300 K, in agreement with Neel temperature of 310 K.

7. EFDA programme on fusion materials Magnetic phase diagram M.Y. Lavrentiev et al. J Phys: Cond Matter 24 (2012) 326001 Maximum of the Curie temperature as a function of Cr content, in agreement with numerous experimental data.

8. EFDA programme on fusion materials Cr-Cr NN Correlations Antiferromagnetic correlations in pure Cr and Fe-Cr alloys rapidly weakens with increasing iron content. For Cr-12.5%Fe system, the nearest neighbour Cr-Cr correlations are already positive.

9. EFDA programme on fusion materials Magnetic phase diagram The resulting magnetic phase diagram shows maximum of the Curie temperature, and a small concentration range where the system is antiferromagnetic (X_Cr > 87.5%).

10. EFDA programme on fusion materials α-γ-δ transitions in Fe and Fe-Cr The Magnetic Cluster Expansion (MCE) M.Y. Lavrentiev et al. PRB 81 (2010) 184202 γ-α magnetic energy difference α−γ transition determines the processing routes, high-T mechanical properties, and radiation damage structures in iron and steels. MCE is a way of modelling this phase transformation starting from first principles.

11. EFDA programme on fusion materials α-γ-δ transitions in Fe and Fe-Cr The Magnetic Cluster Expansion (MCE) M.Y. Lavrentiev et al. PRB 81 (2010) 184202 γ-α magnetic energy difference Entropy term -T(Sfcc-Sbcc) α−γ transition determines the processing routes, high-T mechanical properties, and radiation damage structures in iron and steels. MCE is a way of modelling this phase transformation starting from first principles.

12. EFDA programme on fusion materials α-γ-δ transitions in Fe and Fe-Cr The Magnetic Cluster Expansion (MCE) M.Y. Lavrentiev et al. PRB 81 (2010) 184202 γ-α magnetic free energy difference α−γ transition determines the processing routes, high-T mechanical properties, and radiation damage structures in iron and steels. MCE is a way of modelling this phase transformation starting from first principles.

13. EFDA programme on fusion materials α-γ-δ transitions in Fe and Fe-Cr The Magnetic Cluster Expansion (MCE) M.Y. Lavrentiev et al. PRB 81 (2010) 184202 γ-α magnetic free energy difference γ-α phonon free energy difference α−γ transition determines the processing routes, high-T mechanical properties, and radiation damage structures in iron and steels. MCE is a way of modelling this phase transformation starting from first principles.

14. EFDA programme on fusion materials α-γ-δ transitions in Fe and Fe-Cr The Magnetic Cluster Expansion (MCE) M.Y. Lavrentiev et al. PRB 81 (2010) 184202 phase transitions: γ-δ α-γ α−γ transition determines the processing routes, high-T mechanical properties, and radiation damage structures in iron and steels. MCE is a way of modelling this phase transformation starting from first principles.

15. EFDA programme on fusion materials α-γ-δ transitions in Fe and Fe-Cr The Magnetic Cluster Expansion (MCE) M.Y. Lavrentiev et al. PRB 81 (2010) 184202 phase transitions: γ-δ α-γ α−γ transition determines the processing routes, high-T mechanical properties, and radiation damage structures in iron and steels. MCE is a way of modelling this phase transformation starting from first principles.

16. EFDA programme on fusion materials Gamma-loop The free energy difference for small nonzero Cr concentrations is negative, leading to the occurrence of γ-loop, in good agreement with experiment.

17. EFDA programme on fusion materials Fe-Ni phase diagram Motivation: importance of fcc Fe-Cr-Ni steels. Fe-Ni: good mutual solvability, but presence of intermetallic compound(s). Calculation of phase diagram requires comparison of fcc and bcc free energies. First step – fcc Fe-Ni system only (small solvability of Ni in bcc Fe).

18. EFDA programme on fusion materials The Magnetic Cluster Expansion for Fe-Ni Note that for the fcc system we do not stop the on-site expansion at the 4th degree.

19. EFDA programme on fusion materials Pure Fe Existence of high-spin and low-spin fcc Fe makes it difficult to describe full range of magnetic moments within single magnetic cluster expansion. 8th degree expansion is necessary to take into account both low- and high-spin magnetic states.

20. EFDA programme on fusion materials Pure Fe (II) At lowest temperatures, pure fcc Fe is in antiferromagnetic high-spin configuration. Energetically the low-spin configuration becomes more favourable at about 400 K.

21. EFDA programme on fusion materials Pure Ni Magnetic order disappears at about 550 K (experimental data – 634 K).

22. EFDA programme on fusion materials FCC Fe-Ni ab initio results Ab initio calculations show that mostly high-spin Fe configurations are important for low-energy states of Fe-Ni.

23. EFDA programme on fusion materials FCC Fe-Ni ab initio results Magnetic moment for high-spin configurations decreases almost linearly as a function of Ni content.

24. EFDA programme on fusion materials FCC Fe-Ni - comparison 29 configurations were used in fitting the parameters of MCE Hamiltonian, together with two antiferromagnetic configurations of pure fcc Fe. Mean square deviation of resulting MCE fit is about 12 meV.

25. EFDA programme on fusion materials MCE interactions (meV) Up to the 4th nearest neighbour interactions taken into account. Ferromagnetic interaction for all pairs of species, although for Fe-Ni it is rather weak.

26. EFDA programme on fusion materials Enthalpy of mixing R. Idczak et al. / Physica B 407 (2012) 235–239 Enthalpy of mixing is negative, and higher than the experimental data (but closer to the ab initio results).

27. EFDA programme on fusion materials Magnetic moment of random mixture (collinear) If non-collinear magnetic configurations are forbidden, the dependence of the magnetic moment of the concentration is almost linear, like in the ab initio calculations.

28. EFDA programme on fusion materials Building the phase diagram On the Fe-rich part of the phase diagram, high-temperature calculations can be performed in the limit of very small solubility of Ni in Fe. Common tangent construction is used.

29. EFDA programme on fusion materials Building the phase diagram Agreement with the experimental phase diagram in the low-Ni concentration range is good and can be further improved when taking into account nonzero solubility of Ni in the bcc Fe.

30. EFDA programme on fusion materials Building the phase diagram Intermetallic compounds FeNi and FeNi3 are stable compared to the random mixtures up to temperatures of about 500-600 K. This leads to a maximum in the Curie temperature dependence of the Ni concentration, in agreement with experimental data.

31. EFDA programme on fusion materials Building the phase diagram Ordered structures keep magnetic order until much higher temperatures than the disordered, and than the pure Ni.

32. EFDA programme on fusion materials Fe-Ni phase diagram Maximum in the Curie temperature dependence of the Ni concentration is in a good agreement with experimental data. The resulting phase diagram is very close to the experiment in both low-Ni and high-Ni parts of concentration range.

33. EFDA programme on fusion materials Summary • An MCE fit For Fe-Cr and Fe-Ni systems. • Good description of magnetic properties of pure bcc Fe and Cr, fcc Ni and Fe. • Properties of Fe-Cr: γ-loop, increase of Curie temperature for small Cr concentrations in good agreement with experiment. • Terms up to the 8th degree used for fcc Fe and Ni to take into account high- and low-spin configurations. • Phase diagram of binary fcc Fe-Ni system obtained. • Importance of magnetic interactions in the phase diagrams: pure Fe; gamma-loop in Fe-Cr; opening of the gamma-loop due to attractive Fe-Ni interaction; intermetallic compounds keeping magnetic order up to higher temperatures.