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Graph theory and algebraic topology in the study of phase diagrams of multi-component condensed systems

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Abstract

Based on a constructive generalization of the results of works on the topology of phase diagrams of multi-component condensed systems, the contribution of these studies to the general theory of phase diagrams is presented. General fundamental specific features characteristic of certain classes of diagrams were identified as a result of the systematic study of this topology. The topology of subsolidus isobaric-isothermal phase diagrams, as well as melting diagrams, is considered. This review can be useful for the formulation and solution of new problems in the study of phase diagrams of multi-component systems.

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References

  1. G. V. Belov, G. F. Voronin, V. I. Goryacheva, A. L. Emelina, I. A. Uspenskaya, Matem. Modelirovanie [Math. Modeling], 2006, 18, 67 (in Russian).

    Google Scholar 

  2. E. Yu. Kerimov, Yu. V. Balkova, E. M. Slyusarenko, Chem. Met. Alloys, 2008, 1, 244; DOI: https://doi.org/10.30970/cma1.0037.

    Article  Google Scholar 

  3. E. I. Ardashnikova, Sorosov. Obrazovat. Zhurn. [Soros Educational Journal], 2004, 8, 30 (in Russian).

    Google Scholar 

  4. H. L. Lukas, G. S. Fries, B. Sundman, Computational Thermodynamics: the CALPHAD Method, University Press, Cambridge, 2007, 243 pp.

    Book  Google Scholar 

  5. N. Saunders, P. Miodownik, CALPHAD (Calculation of Phase Diagrams). Pergamon Materials Series, Vol. 1, 1998, 479 pp.

  6. Y. A. Chang, S. Chen, F. Zang, X. Yan, F. Xie, R. Schmid-Fetze, W. A. Oates, Prog. Mater Sci., 2004, 49, 313; DOI: https://doi.org/10.1016/S0079-6425(03)00025-2.

    Article  CAS  Google Scholar 

  7. U. R. Kattner, JOM, 1997, 49, 14.

    Article  CAS  Google Scholar 

  8. N. S. Kurnakov, Vvedenie v fiziko-khimicheskii ana liz [Introduction to Physicochemical Analysis], Izd-vo AN SSSR, Moscow—Leningrad, 1940, 564 pp. (in Russian).

    Google Scholar 

  9. V. Ya. Anosov, S. A. Pogodin, Osnovnye nachala fiziko-khimicheskogo analiza [Basic Principles of Physicochemical Analysis], Izd-vo AN SSSR, Moscow—Leningrad, 1947, 876 pp. (in Russian).

    Google Scholar 

  10. V. P. Radishchev, Mnogokomponentnye sistemy [Multi-component Systems], Izd-vo IONKh AN SSSR, Moscow, 1963, 502 pp. (in Russian).

    Google Scholar 

  11. A. S. Berezhnoy, Mnogokomponentnye sistemy okislov [Multi-component Oxide Systems], Naukova Dumka, Kiev, 1970, 544 pp. (in Russian).

    Google Scholar 

  12. L. A. Serafimov, V. V. Safonov, Zhurn. Neorgan. Khimii [J. Inorg. Chem.], 2000, 45, 279 (in Russian).

    CAS  Google Scholar 

  13. L. A. Serafimov, V. V. Safonov, Zhurn. Neorgan. Khimii [J. Inorg. Chem.], 2000, 45, 682 (in Russian).

    CAS  Google Scholar 

  14. P. P. Fedorov, I. I. Buchinskaya, L. A. Serafimov, Russ. J. Inorg. Chem., 2002, 47, 1249.

    Google Scholar 

  15. P. P. Fedorov, I. I. Buchinskaya, Russ. Chem. Rev., 2012, 81, 1; DOI: https://doi.org/10.1070/RC2012v081n01ABEH004207.

    Article  CAS  Google Scholar 

  16. A. S. Trunin, A. V. Chuvakov, O. E. Morgunova, Dokl. Phys. Chem., 2011, 437, 54; DOI: https://doi.org/10.1134/S0012501611030080.

    Article  CAS  Google Scholar 

  17. A. S. Trunin, O. E. Morgunova, Russ. J. Inorg. Chem., 2012, 57, 1163; DOI: https://doi.org/10.1134/S0036023612080153.

    Article  CAS  Google Scholar 

  18. E. A. Katasonova, A. S. Trunin, O. E. Morgunova, K. D. Surinskii, Vestn. SamGU [Bull. Samara State University], 2015, 146 (in Russian).

  19. V. Lutsyk, V. Vorobjeva, M. Parfenova, Adv. Mater. Res., 2013, 704, 55; DOI: https://doi.org/10.4028/www.scientific.net/AMR.704.55.

    Article  Google Scholar 

  20. V. I. Lutsyk, V. P. Vorob’eva, Russ. J. Inorg. Chem., 2014, 59, 956; DOI: https://doi.org/10.7868/S0044457X14090128.

    Article  CAS  Google Scholar 

  21. O. Ore, Theory of Graphs, American Mathematical Society Collodium Publications, Providence, 1962, 38, 270 pp.; DOI: https://doi.org/10.10090/coll/038.

    Google Scholar 

  22. V. I. Lutsyk, V. P. Vorob’eva, Russ. J. Inorg. Chem., 2014, 59, 1123; DOI: https://doi.org/10.1134/S0036023614100106.

    Article  CAS  Google Scholar 

  23. M. A. Demina, E. M. Egorova, I. K. Garkushin, E. O. Ignat’eva, Russ. J. Inorg. Chem., 2022, 67, 1611; DOI: https://doi.org/10.1134/S0036023622100436.

    Article  CAS  Google Scholar 

  24. V. I. Kosyakov, V. A. Shestakov, E. V. Grachev, Russ. J. Phys. Chem., 2009, 83, 1266.

    Article  CAS  Google Scholar 

  25. V. I. Kosyakov, Russ. J. Inorg. Chem., 2010, 55, 1779; DOI: https://doi.org/10.1134/S003602361011015X.

    Article  CAS  Google Scholar 

  26. V. I. Kosyakov, Russ. J. Inorg. Chem., 2010, 55, 1786; DOI: https://doi.org/10.1134/S0036023610110161.

    Article  CAS  Google Scholar 

  27. V. I. Kosyakov, V. A. Shestakov, E. V. Grachev, Russ. J. Phys. Chem. A, 2019, 93, 2131; DOI: https://doi.org/10.1134/S0036024419110153.

    Article  CAS  Google Scholar 

  28. V. I. Kosyakov, V. A. Shestakov, Russ. J. Inorg. Chem., 2016, 61, 334; DOI: https://doi.org/10.1134/S0036023616030128.

    Article  CAS  Google Scholar 

  29. V. I. Kosyakov, V. A. Shestakov, E. V. Grachev, Dokl. Phys. Chem., 2012, 443, 53; DOI: https://doi.org/10.1134/S0012501612030025.

    Article  CAS  Google Scholar 

  30. V. Kosyakov, V. Shestakov, E. Grachev, MATCH Commun. Math. Comput. Chem., 2013, 69, 795.

    Google Scholar 

  31. V. I. Kosyakov, V. A. Shestakov, E. V. Grachev, V. Yu. Komarov, Russ. J. Inorg. Chem., 2014, 59, 1501; DOI: https://doi.org/10.1134/S0036023614120110.

    Article  CAS  Google Scholar 

  32. V. I. Kosyakov, V. A. Shestakov, E. V. Grachev, V. Yu. Komarov, Russ. J. Inorg. Chem., 2016, 61, 1274; DOI: https://doi.org/10.1134/S0036023616100119.

    Article  CAS  Google Scholar 

  33. V. I. Kosyakov, V. A. Shestakov, E. V. Grachev, Russ. J. Inorg. Chem., 2017, 62, 318; DOI: https://doi.org/10.1134/S003602361703010X.

    Article  CAS  Google Scholar 

  34. V. I. Kosyakov, Russ. J. Phys. Chem. A, 2019, 93, 1635; DOI: https://doi.org/10.1134/S0036024419090085.

    Article  CAS  Google Scholar 

  35. V. A. Shestakov, E. V. Grachev, V. I. Kosyakov, Russ. J. Inorg. Chem., 2021, 66, 1730; DOI: https://doi.org/10.1134/S0036023621110164.

    Article  CAS  Google Scholar 

  36. V. A. Shestakov, E. V. Grachev, V. I. Kosyakov, Russ. J. Phys. Chem. A, 2020, 94, 1083; DOI: https://doi.org/10.1134/S0036024420060205.

    Article  CAS  Google Scholar 

  37. V. A. Shestakov, V. I. Kosyakov, Russ. J. Inorg. Chem., 2021, 66, 401; DOI: https://doi.org/10.1134/S0036023621030165.

    Article  CAS  Google Scholar 

  38. V. I. Kosyakov, V. A. Shestakov, E. V. Grachev, V. Yu. Komarov, Russ. J. Inorg. Chem., 2016, 61, 610; DOI: https://doi.org/10.1134/S0036023616050119.

    Article  CAS  Google Scholar 

  39. V. I. Kosyakov, V. A. Shestakov, E. V. Grachev, Russ. J. Inorg. Chem., 2010, 55, 611; DOI: https://doi.org/10.1134/S0036023610040194.

    Article  CAS  Google Scholar 

  40. V. I. Kosyakov, Dokl. Chem., 2000, 374, 199.

    Google Scholar 

  41. V. I. Kosyakov, V. A. Shestakov, Russ. J. Inorg. Chem., 2017, 62, 795; DOI: https://doi.org/10.1134/S0036023617060122.

    Article  CAS  Google Scholar 

  42. V. I. Kosyakov, Russ. J. Inorg. Chem., 2017, 62, 987; DOI: https://doi.org/10.1134/S0036023617070117.

    Article  CAS  Google Scholar 

  43. V. A. Shestakov, E. V. Grachev, Russ. J. Inorg. Chem., 2022, 67, 488; DOI: https://doi.org/10.1134/S0036023622040179.

    Article  CAS  Google Scholar 

  44. V. A. Shestakov, V. I. Kosyakov, E. V. Grachev, Vestn. Kazan. Tekhnolog. Un-ta [Bull. Kazan Technolog. University], 2010, 67 (in Russian).

  45. L. Soliev, Prognozirovanie stroeniya diagramm fazovykh ravnovesii mnogokomponentnykh vodnosolevykh sistem metodom translyatsii [Prediction of the Structure of Phase Equilibrium Diagrams of Multi-component Water-salt Systems Using the Translation Method], Moscow, 1987, 28 pp.; dep. v VINITI 20.12.87, No. 8990-V87 (in Russian).

  46. L. Soliev, Russ. J. Inorg. Chem., 2020, 65, 212; DOI: https://doi.org/10.1134/S0036023620020187.

    Article  CAS  Google Scholar 

  47. D. Orse, in Computer Modeling of Phase Diagrams, Ed. L. Bennett, ASM/TMS, 1986.

  48. V. I. Kosyakov, Russ. J. Inorg. Chem., 2008, 53, 946; DOI: https://doi.org/10.1134/S0036023608060223.

    Article  Google Scholar 

  49. V. I. Kosyakov, N. A. Pylneva, Z. G. Bazarova, A. M. Yurkin, Mater. Res. Bull., 2001, 36, 573; DOI: https://doi.org/10.1016/S0025-5408(01)00519-0.

    Article  CAS  Google Scholar 

  50. P. A. Akhmedova, B. Yu. Gamataeva, A. M. Gasanaliev, Russ. J. Inorg. Chem., 2009, 54, 779; DOI: https://doi.org/10.1134/S0036023609050179.

    Article  Google Scholar 

  51. A. I. Rasulov, P. A. Akhmedova, B. Y. Gamataeva, A. M. Gasanaliev A. K. Mamedova, Russ. J. Inorg. Chem., 2019, 64, 135; DOI: https://doi.org/10.1134/S0036023619010169.

    Article  CAS  Google Scholar 

  52. P. A. Akhmedova, A. M. Gasanaliev, B. Yu. Gamataeva, P. A. Khizrieva, Russ. J. Inorg. Chem., 2018, 63, 837; DOI: https://doi.org/10.1134/S0036023618060025.

    Article  CAS  Google Scholar 

  53. B. Yu. Gamataeva, R. M. Bagomedova, A. M. Gasanaliev, Russ. J. Inorg. Chem., 2014, 59, 134; DOI: https://doi.org/10.1134/S0036023613120103.

    Article  CAS  Google Scholar 

  54. A. M. Gasanaliev, P. A. Akhmedova, B. Yu. Gamataeva, Fundamental’nye issledovaniya [Fundamental Research], 2014, 1006 (in Russian).

  55. A. M. Gasanaliev, P. A. Akhmedova, B. Yu. Gamataeva, Vestn. Nizhegorod. Un-ta im. N. I. Lobachevskogo [Bull. N. I. Lobachevsky Nizhni Novgorod University], 2014, 130 (in Russian).

  56. I. K. Garkushin, A. V. Burchakov, U. A. Emel’yanova, M. V. Chugunova, Russ. J. Inorg. Chem., 2020, 65, 1040; DOI: https://doi.org/10.1134/S0036023620070086.

    Article  CAS  Google Scholar 

  57. E. M. Egorova, S. S. Likhacheva, I. K. Garkushin, Russ. J. Inorg. Chem., 2019, 64, 247; DOI: https://doi.org/10.1134/S0036023619020049.

    Article  CAS  Google Scholar 

  58. M. A. Sukharenko, I. K. Garkushin, Russ. J. Inorg. Chem., 2016, 61, 355; DOI: https://doi.org/10.1134/S0036023616030220.

    Article  CAS  Google Scholar 

  59. I. K. Garkushin, M. A. Demina, A. A. Chudova, A. V. Nenasheva, Russ. J. Inorg. Chem., 2015, 60, 355; DOI: https://doi.org/10.1134/S0036023615010040.

    Article  Google Scholar 

  60. M. A. Sukharenko, I. K. Garkushin, A. K. Redyushev, I. M. Yaremenko, Russ. J. Inorg. Chem., 2015, 60, 736; DOI: https://doi.org/10.1134/S0036023615060169.

    Article  CAS  Google Scholar 

  61. P. A. Akhmedova, A. M. Gasanaliev, B. Yu. Gamataeva, P. A. Khizrieva, Russ. J. Inorg. Chem., 2017, 62, 1390; DOI: https://doi.org/10.1134/S0036023617100023.

    Article  CAS  Google Scholar 

  62. A. V. Babenko, E. M. Egorova, I. K. Garkushin, Russ. J. Inorg. Chem., 2019, 64, 899; DOI: https://doi.org/10.1134/S0036023619070027.

    Article  CAS  Google Scholar 

  63. N. N. Kiseleva, Komp’yuternoe konstruirovanie neorganicheskikh soedinenii: ispol’zovanie baz dannykh i iskusstvennogo intellekta [Computer-aided Design of Inorganic Compounds: the Use of Databases and Artificial Intelligence], Nauka, Moscow, 2005, 289 pp. (in Russian).

    Google Scholar 

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Funding

This work was financially supported by the Ministry of Science and Higher Education of the Russian Federation (Project No. 121031700314-5).

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  1. Nikolaev Institute of Inorganic Chemistry, Siberian Branch of the Russian Academy of Sciences, 3 prosp. Akad. Lavrent’eva, 630090, Novosibirsk, Russian Federation

    V. A. Shestakov

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Correspondence to V. A. Shestakov.

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Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, Vol. 73, No. 4, pp. 751–760, April, 2024.

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Shestakov, V.A. Graph theory and algebraic topology in the study of phase diagrams of multi-component condensed systems. Russ Chem Bull 73, 751–760 (2024). https://doi.org/10.1007/s11172-024-4190-1

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