ECN 2010 31 May Barcelona EFDA programme on

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ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Modelling of phase diagrams

ECN 2010, 31 May, Barcelona 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 OX 14 3 DB, United Kingdom CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials What is necessary to

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials What is necessary to build a phase diagram? q q 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) CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Magnetic Cluster Expansion Hamiltonian

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Monte Carlo simulations q

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Monte Carlo simulations q q 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 CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona Bcc Fe In pure iron, the Curie EFDA programme

ECN 2010, 31 May, Barcelona Bcc Fe In pure iron, the Curie EFDA programme on fusion materials 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona Bcc Cr EFDA programme on fusion materials Disappearance of

ECN 2010, 31 May, Barcelona Bcc Cr EFDA programme on fusion materials 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Magnetic phase diagram M.

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Cr-Cr NN Correlations Antiferromagnetic

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Magnetic phase diagram The

ECN 2010, 31 May, Barcelona 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%). CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials α-γ-δ transitions in Fe

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials α-γ-δ transitions in Fe

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials α-γ-δ transitions in Fe

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials α-γ-δ transitions in Fe

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials α-γ-δ transitions in Fe

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials α-γ-δ transitions in Fe

ECN 2010, 31 May, Barcelona 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 α-γ phase transitions: γ-δ 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Gamma-loop The free energy

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Gamma-loop The free energy difference for small nonzero Cr concentrations is negative, leading to the occurrence of γloop, in good agreement with experiment. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Fe-Ni phase diagram Motivation:

ECN 2010, 31 May, Barcelona 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). CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials The Magnetic Cluster Expansion

ECN 2010, 31 May, Barcelona 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 4 th degree. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Pure Fe Existence of

ECN 2010, 31 May, Barcelona 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. 8 th degree expansion is necessary to take into account both low- and high-spin magnetic states. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Pure Fe (II) At

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Pure Fe (II) At lowest temperatures, pure fcc Fe is in antiferromagnetic highspin configuration. Energetically the low-spin configuration becomes more favourable at about 400 K. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Pure Ni Magnetic order

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Pure Ni Magnetic order disappears at about 550 K (experimental data – 634 K). CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials FCC Fe-Ni ab initio

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials FCC Fe-Ni ab initio

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials FCC Fe-Ni - comparison

ECN 2010, 31 May, Barcelona 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 me. V. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials MCE interactions (me. V)

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials MCE interactions (me. V) Fe-Fe Fe-Ni Ni-Ni 1 st NN -0. 793 1. 516 -13. 153 2 nd NN -10. 827 -2. 710 7. 228 3 rd NN 0. 547 -2. 500 -5. 605 4 th NN 2. 306 1. 649 -6. 744 Up to the 4 th nearest neighbour interactions taken into account. Ferromagnetic interaction for all pairs of species, although for Fe-Ni it is rather weak. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Enthalpy of mixing R.

ECN 2010, 31 May, Barcelona 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). CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Magnetic moment of random

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Building the phase diagram

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Building the phase diagram On the Fe-rich part of the phase diagram, hightemperature calculations can be performed in the limit of very small solubility of Ni in Fe. Common tangent construction is used. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Building the phase diagram

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Building the phase diagram

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Building the phase diagram Intermetallic compounds Fe. Ni and Fe. Ni 3 are stable compared to the random mixtures up to temperatures of about 500600 K. This leads to a maximum in the Curie temperature dependence of the Ni concentration, in agreement with experimental data. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Building the phase diagram

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Fe-Ni phase diagram Maximum

ECN 2010, 31 May, Barcelona 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Summary q An MCE

ECN 2010, 31 May, Barcelona EFDA programme on fusion materials Summary q An MCE fit For Fe-Cr and Fe-Ni systems. q Good description of magnetic properties of pure bcc Fe and Cr, fcc Ni and Fe. q Properties of Fe-Cr: γ-loop, increase of Curie temperature for small Cr concentrations in good agreement with experiment. q Terms up to the 8 th degree used for fcc Fe and Ni to take into account high- and low-spin configurations. q Phase diagram of binary fcc Fe-Ni system obtained. q 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. CCFE is the fusion research arm of the United Kingdom Atomic Energy Authority