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Excess Thermodynamic Properties of Solutions in Ln-Ru (Ln = Nd, Gd, Dy) Binary Systems Based on Quadratic, Exponential and Combined Models Supported by Ab-Initio Calculations

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Abstract

The systematic thermodynamic optimization of the Ln-Ru (where Ln = Nd, Gd, and Dy) binary systems is performed using the CALPHAD approach. Using this approach, experimental information published in literature, including both phase equilibria and thermodynamic data, are critically evaluated. The thirteen intermediate phases with stoichiometries, such as Ln3Ru, Nd5Ru2, Ln7Ru3, Nd5Ru3, Ln73Ru27, Ln2Ru, and LnRu2, which have been reported for the three binary Ln-Ru systems are considered as stoichiometric compounds. An exponential and a quadratic model are used to describe the temperature dependence of the excess quantities for the Liquid, (αNd), (βNd), (αDy), (βDy), (αGd), (βGd), and (Ru) solution phases. The results were compared with those from a combined quadratic-exponential temperature dependence description of the excess energies. Using the Thermo-Calc software, self-consistent sets of thermodynamic parameters were obtained to describe the Gibbs energies of the numerous phases in the Ln-Ru binary systems. The calculated results are in good agreement with the available phase equilibria and thermodynamic data.

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References

  1. J.H. Rademaker, R. Kleijn, and Y. Yang, Recycling as a Strategy Against Rare Earth Element Criticality: A Systemic Evaluation of the Potential Yield of NdFeB Magnet Recycling, Environ. Sci. Technol., 2013, 47, p 10129–10136. https://doi.org/10.1021/es305007w

    Article  ADS  Google Scholar 

  2. K.M. Goodenough, J. Schilling, E. Jonsson, P. Kalvig, N. Charles, J. Tuduri, E.A. Deady, M. Sadeghi, H. Schiellerup, A. Müller, G. Bertrand, N. Arvanitidis, D.G. Eliopoulos, R.A. Shaw, K. Thrane, and N. Keulen, Europe’s Rare Earth Element Resource Potential: An Overview of REE Metallogenetic Provinces and Their Geodynamic Setting, Ore Geol. Rev., 2016, 72, p 838–856. https://doi.org/10.1016/j.oregeorev.2015.09.019

    Article  Google Scholar 

  3. Ad hoc Working Group, Report on critical raw materials for the EU, European Commission, 2015.

  4. X. Du, and T.E. Graedel, Science of the Total Environment Uncovering the End Uses of the Rare Earth Elements, Sci. Total Environ., 2013, 462, p 781–784. https://doi.org/10.1016/j.scitotenv.2013.02.099

    Article  ADS  Google Scholar 

  5. L. Schlapbach, and A. Züttel, Hydrogen-Storage Materials for Mobile Applications, Nature, 2001, 414, p 353–358. https://doi.org/10.1038/35104634

    Article  ADS  Google Scholar 

  6. K. Buschow, S. Jaakkola, S. Parviainen, S. Penttila, and K.H.J. Buschow, Reports on Progress in Physics Intermetallic Compounds of Rare-Earth and 3D Transition Metals Intermetallic Compounds of Rare-Earth and 3D Transition Metals, Rep. Prog. Phys., 1977, 93, p 1179. https://doi.org/10.1088/0034-4885/40/10/002

    Article  Google Scholar 

  7. K.H.J. Buschow, Permanent Magnet Materials Based on Tetragonal Rare Earth Compounds of the Type RFe12-xMx, J. Magn. Magn. Mater., 1991, 100, p 79–89. https://doi.org/10.1016/0304-8853(91)90813-P

    Article  ADS  Google Scholar 

  8. K. Binnemans, P.T. Jones, B. Blanpain, T. Van Gerven, Y. Yang, A. Walton, and M. Buchert, Recycling of Rare Earths: A Critical Review, J. Clean. Prod., 2013, 51, p 1–22. https://doi.org/10.1016/j.jclepro.2012.12.037

    Article  Google Scholar 

  9. N. Saunders, and A.P. Miodownik, CALPHAD (Calculation of Phase Diagrams): A Comprehensive Guide. Pergamon/Elsevier, Oxford, 1998.

    Google Scholar 

  10. L. Kaufman, and H. Bernstein, Computer Calculation of Phase Diagrams (With Special Reference to Refractory Metals). Academic Press Inc, New York, 1970.

    Google Scholar 

  11. H. Lukas, S. Fries, and B. Sundman, Computational Thermodynamics: The Calphad Method. Cambridge University Press, Cambridge, UK, 2007.

    Book  MATH  Google Scholar 

  12. G. Kaptay, Materials Equilibria in Macro-, Micro- and Nano-systems, Raszter Ny. (2011).

  13. R. Schmid-Fetzer, D. Andersson, P.Y. Chevalier, L. Eleno, O. Fabrichnaya, U.R. Kattner, B. Sundman, C. Wang, A. Watson, L. Zabdyr, and M. Zinkevich, Assessment Techniques, Database Design and Software Facilities for Thermodynamics and Diffusion, Calphad, 2007, 31, p 38–52. https://doi.org/10.1016/j.calphad.2006.02.007

    Article  Google Scholar 

  14. G. Kaptay, Nano-Calphad: Extension of the Calphad Method to Systems with Nano-phases and Complexions, J. Mater. Sci., 2012, 47, p 8320–8335. https://doi.org/10.1007/s10853-012-6772-9

    Article  ADS  Google Scholar 

  15. B. Jansson, Computer Operated Methods for Equilibrium Calculations and Evaluation of Thermochemical Model Parameters. Report, Royal Institute of Technology, Stockholm Sweden, 1984.

  16. B. Sundman, B. Jansson, A. J-O, THE THERMO-CALC DATABANK SYSTEM, Calphad 9 (1985) 153–190. https://doi.org/10.1016/0364-5916(85)90021-5.

  17. S.L. Chen, S. Daniel, F. Zhang, Y.A. Chang, W.A. Oates, and R. Schmid-Fetzer, On the Calculation of Multicomponent Stable Phase Diagrams, J. Phase Equilibria, 2001, 22, p 373–378. https://doi.org/10.1361/105497101770332910

    Article  Google Scholar 

  18. G. Kaptay, A New Equation for the Temperature Dependence of the Excess Gibbs Energy of Solution Phases, Calphad, 2004, 28, p 115–124. https://doi.org/10.1016/j.calphad.2004.08.005

    Article  Google Scholar 

  19. T. Abe, K. Ogawa, and K. Hashimoto, Analysis of Miscibility Gaps Based on the Redlich–Kister Polynomial for Binary Solutions, Calphad, 2012, 38, p 161–167. https://doi.org/10.1016/j.calphad.2012.06.006

    Article  Google Scholar 

  20. G. Kaptay, On the Tendency of Solutions to Tend Toward Ideal Solutions at High Temperatures, Metall. Mater. Trans. A Phys. Metall. Mater. Sci., 2012, 43, p 531–543. https://doi.org/10.1007/s11661-011-0902-x

    Article  ADS  Google Scholar 

  21. G. Kaptay, On the Abilities and Limitations of the Linear, Exponential and Combined Models to Describe the Temperature Dependence of the Excess Gibbs Energy of Solutions, Calphad, 2014, 44, p 81–94. https://doi.org/10.1016/j.calphad.2013.08.007

    Article  Google Scholar 

  22. J.F. Elliott, and C.H.P. Lupis, Correlation Between Excess Entropy and Enthalpy Functions, Trans. Met. Soc. AIME, 1966, 236, p 130.

    Google Scholar 

  23. A. Palenzona, and F. Canepa, The Phase Diagrams of the La-Ru and Nd-Ru Systems, J. Less-Common Met., 1990, 157, p 307–313. https://doi.org/10.1016/0022-5088(90)90186-N

    Article  Google Scholar 

  24. P. Sharifrazi, R.C. Mohanty, and A. Raman, Intermediate Phases in Some Rare Earth Metal-Iridium Systems, Z. Metallkd., 1984, 75, p 801–805. https://doi.org/10.1515/ijmr-1989-800309

    Article  Google Scholar 

  25. H. Okamoto, Nd-Ru (Neodymium-Ruthenium), J. Phase Equilibria, 1991, 12, p 250–252. https://doi.org/10.1007/BF02645729

    Article  Google Scholar 

  26. O. Loebich Jr., and E. Raub, Die Legierungen des Rutheniums mit Gadolinium und Dysprosium und ihre Magnetischen Eigenschaften, J. Less-Common Met., 1976, 46, p 7–15. https://doi.org/10.1016/0022-5088(76)90172-7

    Article  Google Scholar 

  27. K.H. Hellwege, Strukturdaten der Elemente und intermetallischen Phasen Landolt-Börnstein, New Series, Group 3, Vol. 6. Springer, Berlin, 1971.

    Google Scholar 

  28. R.M. Bozorth, B.T. Matthias, H. Suhl, E. Corenzwit, and D.D. Davis, Magnetization of Compounds of Rare Earths with Platinum Metals, Phys. Rev., 1959, 115, p 1595–1596. https://doi.org/10.1103/PhysRev.115.1595

    Article  ADS  Google Scholar 

  29. O. Loebich Jr., and E. Raub, Magnetic Properties of Alloys of Rh with Lanthanides (in German), Magn. Prop. Alloy Rh Lanthanides., 1975, 10, p 1017–1022.

    Google Scholar 

  30. O. Loebich Jr., and E. Raub, Das magnetische Verhalten der Legierungen des Palladiums Mit Gadolinium, Dysprosium und Holmium, J. Less-Common Met., 1973, 31, p 111–118. https://doi.org/10.1016/0022-5088(73)90134-3

    Article  Google Scholar 

  31. A.E. Berkowitz, F. Holtzberg, and S. Methfessel, New Ferromagnetic 5:2 Compounds in the Rare-Earth—Palladium Systems, J. Appl. Phys., 1964, 35, p 1030–1031. https://doi.org/10.1063/1.1713364

    Article  ADS  Google Scholar 

  32. R. Ferro et al., Bericht der Eur. Atomgemeinschaft, EUR-1894i. (1964).

  33. R.D. Hutchens, V.U.S. Rao, J.E. Greedan, W.E. Wallace, and R.S. Craig, Magnetic and Electrical Characteristics of REPd3 Intermetallic Compounds, J. Appl. Phys., 1971, 42, p 1293–1294. https://doi.org/10.1063/1.1660221

    Article  ADS  Google Scholar 

  34. W.E. Gardner, J. Penfold, T.F. Smith, and I.R. Harris, The Magnetic Properties of Rare Earth-Pd3 Phases, J. Phys. F Met. Phys., 1972, 2, p 133–150. https://doi.org/10.1088/0305-4608/2/1/019

    Article  ADS  Google Scholar 

  35. J. Pierre, and E. Siaud, Structure Cristalline et Propriétés Magnétiques du Composé GdPd, Compt. Rend. Acad. Sci., 1968, 266, p 1483–1485.

    Google Scholar 

  36. Q. Guo, and O.J. Kleppa, Standard Enthalpies of Formation of Neodymium Alloys, Nd + Me (Me-Ni, Ru, Rh, Pd, Ir, Pt), by High-Temperature Direct Synthesis Calorimetry, Metall. Mater. Trans. B., 1995, 26B, p 275–279. https://doi.org/10.1007/BF02660969

    Article  ADS  Google Scholar 

  37. A.L. Shilov, L.N. Padurets, and M.E. Kost, Formation Enthalpy Determination for Intermetallic Compounds and Their Hydrides from Differential Thermal Analysis Data, Russ. J. Phys. Chem., 1983, 57, p 555–560.

    Google Scholar 

  38. N. Selhaoui, J. Charles, L. Bouirden, and J.C. Gachon, Optimization of La-Ru System, Ann. Chim. Sci. Des Mater., 1999, 24, p 97–104.

    Article  Google Scholar 

  39. A. Debski, R. Debski, and W. Gasior, New Features of Entall Database: Comparison of Experimental and Model Formation Enthalpies, Arch. Metall. Mater., 2014, 59, p 1337–1343. https://doi.org/10.2478/amm-2014-0228

    Article  Google Scholar 

  40. F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema, and A.K. Niessen, Cohesion in metals. Transition metal alloys. North-Holland, Amsterdam, 1988.

    Google Scholar 

  41. A. Jain, S.P. Ong, G. Hautier, W. Chen, W.D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K.A. Persson, Commentary: The Materials Project: A Materials Genome Approach to Accelerating Materials Innovation, APL Mater., 2013, 1, p 011002. https://doi.org/10.1063/1.4812323

    Article  ADS  Google Scholar 

  42. S. Kirklin, J.E. Saal, B. Meredig, A. Thompson, J.W. Doak, M. Aykol, S. Rühl, and C. Wolverton, The Open Quantum Materials Database (OQMD): Assessing the Accuracy of DFT Formation Energies, npj Comput. Mater., 2015. https://doi.org/10.1038/npjcompumats.2015.10

    Article  Google Scholar 

  43. R.F. Zhang, S.H. Sheng, and B.X. Liu, Predicting the Formation Enthalpies of Binary Intermetallic Compounds, Chem. Phys. Lett., 2007, 442, p 511–514. https://doi.org/10.1016/j.cplett.2007.06.031

    Article  ADS  Google Scholar 

  44. N. Selhaoui, and O.J. Kleppa, Standard Enthalpies of Formation of Lanthanum Alloys, La + Me (Me = Ru, Rh, Pd, Os, Ir, Pt), by High-Temperature Calorimetry, J. Alloys Compd., 1993, 191, p 155–158. https://doi.org/10.1016/0925-8388(93)90289-Y

    Article  Google Scholar 

  45. C. Oses, E. Gossett, D. Hicks, F. Rose, M.J. Mehl, E. Perim, I. Takeuchi, S. Sanvito, M. Scheffler, Y. Lederer, and O. Levy, Supporting Information: AFLOW-CHULL Manual AFLOW-CHULL: Cloud-Oriented Platform for Autonomous Phase Stability Analysis, J. Chem. Inf. Model., 2018, 58, p 2477–2490. https://doi.org/10.1021/acs.jcim.8b00393

    Article  Google Scholar 

  46. Q. Guo, and O.J. Kleppa, Standard Enthalpies of Formation of Dysprosium Alloys, Dy + Me (Me ≡ Ni, Ru, Rh, Pd, Ir, and Pt), by High-Temperature Direct Synthesis Calorimetry, Metall. Mater. Trans., 1996, 27B, p 417–422. https://doi.org/10.1007/BF02914906

    Article  Google Scholar 

  47. K. Choudhary, K.F. Garrity, A.C.E. Reid, B. Decost, A.J. Biacchi, A.R.H. Walker, Z. Trautt, J. Hattrick-simpers, A.G. Kusne, A. Centrone, A. Davydov, J. Jiang, R. Pachter, G. Cheon, E. Reed, A. Agrawal, X. Qian, V. Sharma, H. Zhuang, and S.V. Kalinin, The Joint Automated Repository for Various Integrated Simulations (JARVIS ) for Data-Driven Materials Design, Npj Comput. Mater., 2020, 6, p 1–13. https://doi.org/10.1038/s41524-020-00440-1

    Article  ADS  Google Scholar 

  48. A. Takeuchi, and A. Inoue, Mixing Enthalpy of Liquid Phase Calculated by Miedema’s Scheme and Approximated with Sub-regular Solution Model for Assessing Forming Ability of Amorphous and Glassy Alloys, Intermetallics, 2010, 18, p 1779–1789. https://doi.org/10.1016/j.intermet.2010.06.003

    Article  Google Scholar 

  49. W. Kohn, and L.J. Sham, Self-Consistent Equations Including Exchange and Correlation Effects, Phys. Rev., 1965, 140, p A1133–A1138. https://doi.org/10.1103/PhysRev.140.A1133

    Article  MathSciNet  ADS  Google Scholar 

  50. P. Hohenberg, and W. Kohn, Inhomogeneous Electron Gas, Phys. Rev., 1964, 136, p B864. https://doi.org/10.1103/PhysRev.136.B864

    Article  MathSciNet  ADS  Google Scholar 

  51. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G.L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. De Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A.P. Seitsonen, A. Smogunov, P. Umari, and R.M. Wentzcovitch, QUANTUM ESPRESSO: A Modular and Open-Source Software Project for Quantum Simulations of Materials, J. Phys. Condens. Matter., 2009. https://doi.org/10.1088/0953-8984/21/39/395502

    Article  Google Scholar 

  52. J.P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation made Simple, Phys. Rev. Lett., 1996, 77, p 3865–3868. https://doi.org/10.1103/PhysRevLett.77.3865

    Article  ADS  Google Scholar 

  53. C. Colinet, Ab-Initio Calculation of Enthalpies of Formation of Intermetallic Compounds and Enthalpies of Mixing of Solid Solutions, Intermetallics, 2003, 11, p 1095–1102. https://doi.org/10.1016/S0966-9795(03)00147-X

    Article  Google Scholar 

  54. A.T. Dinsdale, SGTE Data for Pure Elements, Calphad, 1991, 15, p 317–425. https://doi.org/10.1016/0364-5916(91)90030-N

    Article  Google Scholar 

  55. M. Hillert, and M. Jarl, A Model for Alloying Effects in Ferromagnetic Metals, Calphad, 1978, 2, p 227–238. https://doi.org/10.1016/0364-5916(78)90011-1

    Article  Google Scholar 

  56. O. Kister, and A.T. Redlich, Algebraic Representation of Thermodynamic Properties and the Classification of Solutions, Ind. Eng. Chem., 1948, 40, p 345–348. https://doi.org/10.1021/ie50458a036

    Article  Google Scholar 

  57. G. Kaptay, The Exponential Excess Gibbs Energy Model Revisited, Calphad, 2017, 56, p 169–184. https://doi.org/10.1016/j.calphad.2017.01.002

    Article  Google Scholar 

  58. N. Selhaoui, and O.J. Kleppa, Standard Enthalpies of Formation of Cerium Alloys, Ce + Me (Me= Ru, Rh, Pd, Ir, Pt) and of Lutetium Alloys, Lu+ Me (Me= Rh, Pd, Ir, Pt) by Hightemperature Calorimetry, Z. Metallkd., 1993, 84, p 744–747. https://doi.org/10.1515/ijmr-1993-841103

    Article  Google Scholar 

  59. A. Palenzona, and F. Canepa, The Phase Diagram of the Sm-Ru System, J. Less-Common Met., 1989, 155, p L31–L33. https://doi.org/10.1016/0022-5088(89)90246-4

    Article  Google Scholar 

  60. N. Selhaoui, J. Charles, O.J. Kleppa, L. Bouirden, and J.C. Gachon, The Ruthenium-Yttrium System: An Experimental Calorimetric Study with a Phase Diagram Optimization, J. Solid State Chem., 1998, 138, p 302–306. https://doi.org/10.1006/jssc.1998.7795

    Article  ADS  Google Scholar 

  61. G.F. Voronin, Thermodynamic Properties of Intermediate Phases with Narrow Regions of Homogeneity, J. Russ. J. Phys. Chem., 1976, 50, p 607–611.

    Google Scholar 

  62. B. Predel, Phase Equilibria, Crystallographic and Thermodynamic Data of Binary Alloys, Landolt-Börnstein, New Series, Vol. 5, 1st edn. Springer, Berlin, 1993.

    Google Scholar 

  63. T.B. Massalski, H. Okamoto, P. Subramanian, L. Kacprzak, and W.W. Scott, Binary Alloy Phase Diagrams. ASM International, Materials Park, 1986. https://doi.org/10.31399/asm.hb.v03.a0006247

    Book  Google Scholar 

  64. H. Okamoto, Ru-Sm (Ruthenium-Samarium), J. Phase Equilibria Diffus., 1991, 12, p 253–254. https://doi.org/10.1007/BF02645731

    Article  Google Scholar 

  65. V.P. Vassiliev, A. Benaissa, and A.F. Taldrik, Thermodynamics Analysis of the Rare Earth Metals and Their Alloys with Indium in Solid State, J. Alloys Compd., 2013, 572, p 118–123. https://doi.org/10.1016/j.jallcom.2013.03.063

    Article  Google Scholar 

  66. V.P. Vassiliev, and V.A. Lysenko, A New Approach for the Study of Thermodynamic Properties of Lanthanide Compounds, Electrochim. Acta., 2016, 222, p 1770–1777. https://doi.org/10.1016/j.electacta.2016.11.075

    Article  Google Scholar 

  67. V.P. Vassiliev, V.A. Lysenko, and M. Gaune-Escard, Relationship of Thermodynamic Data with Periodic Law, Pure Appl. Chem., 2019, 91, p 879–893. https://doi.org/10.1515/pac-2018-0717

    Article  Google Scholar 

  68. V.N. Eremenko, V.G. Khorujaya, P.S. Martsenyuk, and K.Y. Korniyenko, The Scandium-Ruthenium Phase Diagram, J. Alloys Compd., 1995, 217, p 213–217. https://doi.org/10.1016/0925-8388(94)01321-7

    Article  Google Scholar 

  69. H. Okamoto, Ru-Sc (Ruthenium-Scandium), J. Phase Equilibria Diffus., 2008, 29, p 387–388. https://doi.org/10.1007/s11669-008-9343-7

    Article  Google Scholar 

  70. A. Palenzona, The Phase Diagram of the Ce-Ru System, J. Alloys Compd., 1991, 176, p 241–246. https://doi.org/10.1016/0925-8388(91)90031-P

    Article  Google Scholar 

  71. N. Selhaoui, J. Charles, L. Bouirden, and J.C. Gachon, Optimization of the Binary Ce–Ru System, J. Alloys Compd., 1998, 269, p 166–172. https://doi.org/10.1016/S0925-8388(98)00125-X

    Article  Google Scholar 

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We acknowledge Professor Lorie Wood from the University of Colorado (USA) for the language help.

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Authors and Affiliations

  1. Thermal and Energy Research Team (ERTE), National School of Arts and Crafts, Mohammed V University, Rabat, Morocco

    S. Kardellass

  2. Laboratory of Thermodynamics and Energy (LTE), Faculty of Sciences, Ibn-Zohr University, Agadir, Morocco

    S. Kardellass, K. Mahdouk & N. Selhaoui

  3. Chemistry Department, Lomonosov Moscow State University, Moscow, Russia, 119991

    V. P. Vassiliev

  4. Materials, Energy and Acoustics Team (MEAT), EST Salé, Mohammed V University in Rabat, Avenue Prince Héritier, BP 227, 11000, Medina, Morocco

    N. Laaroussi

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Kardellass, S., Vassiliev, V.P., Mahdouk, K. et al. Excess Thermodynamic Properties of Solutions in Ln-Ru (Ln = Nd, Gd, Dy) Binary Systems Based on Quadratic, Exponential and Combined Models Supported by Ab-Initio Calculations. J. Phase Equilib. Diffus. 44, 43–75 (2023). https://doi.org/10.1007/s11669-022-01023-x

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