The Bain and Orthorhombic Paths of the Structural and Phase Transformations and Theoretical Strength of Molybdenum

A. R. Kuznetsov; S. A. Starikov

doi:10.17804/2410-9908.2025.5.053-065

A. R. Kuznetsov, S. A. Starikov

THE BAIN AND ORTHORHOMBIC PATHS OF THE STRUCTURAL AND PHASE TRANSFORMATIONS AND THEORETICAL STRENGTH OF MOLYBDENUM

DOI: 10.17804/2410-9908.2025.5.053-065

An ab initio method is used to study the energy of the Bain and orthorhombic paths of the structural and phase transformations in molybdenum and phonon instability during uniaxial deformation along <001>. The calculation of the phonon spectrum in the whole irreducible Brillouin zone as dependent on strain for the Bain path makes it possible to find the softest phonon spectral regions, responsible for the loss of structural stability. The nature of the stability loss is revealed, and the strain at which stability is lost under both tension and compression is evaluated. The most probable mechanisms governing the structural stability and theoretical strength of molybdenum are discussed. The results can concern experiment situations when small defect-free regions are strained, e. g., in nanostructured materials, or when surface layers are modified by state-of-the-art plastic deformation techniques, or during nanoindentation, or for super-high ductility of Mo-based alloys.

Acknowledgement: The work was performed under the state assignment for the IMP UB RAS from the Russian Ministry of Science and Higher Education. The Uran supercomputer of the IMM UB RAS was used for the calculations.

Keywords: ab initio calculation, Bain and orthorhombic paths, phonon spectrum, structural stability

References:

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А. Р. Кузнецов, С. А. Стариков

БЕЙНОВСКИЙ И ОРТОРОМБИЧЕСКИЙ ПУТИ СТРУКТУРНО-ФАЗОВЫХ ПРЕВРАЩЕНИЙ И ТЕОРЕТИЧЕСКАЯ ПРОЧНОСТЬ МОЛИБДЕНА

В работе методом ab initio изучена энергетика бейновского и орторомбического путей структурно-фазовых превращений в Mo и нестабильности фононов в ходе одноосной деформации вдоль <001>. Расчет фононного спектра во всей неприводимой зоне Бриллюэна в зависимости от деформации для бейновского пути позволил найти наиболее мягкие ветви фононного спектра, ответственные за потерю устойчивости структуры. Выявлен характер потери устойчивости, а также оценена величина деформации, при которой теряется устойчивость как при растяжении, так и при сжатии. Обсуждаются наиболее вероятные механизмы, определяющие устойчивость структуры и теоретическую прочность Mo. Полученные результаты могут относиться к ситуациям в эксперименте, когда деформируются малые свободные от дефектов области, например, в наноструктурированных материалах, при модифицировании поверхностных слоев современными методами пластического деформирования, при наноиндентировании, при сверхвысокой пластичности сплавов на основе Mo.

Благодарность: Работа выполнена в рамках государственного задания Минобрнауки России для ИФМ УрО РАН. Расчеты проведены с использованием суперкомпьютера «Уран» ИММ УрО РАН.

Ключевые слова: первопринципный расчет, бейновский и орторомбический пути, фононный спектр, стабильность структуры

Библиография:

Bain E. C. The Nature of Martensite // Trans. AIME. – 1924. – Vol. 70. – P. 25–35.
Effect of magnetic state on the α-γ transition in iron: first-principles calculations of the Bain transformation path / S. V. Okatov, A. R. Kuznetsov, Yu. N. Gornostyrev, V. N. Urtsev, M. I. Katsnelson // Physical Review B. – 2009. – Vol. 79 (9). – P. 094111–094115. – DOI: 10.1103/RevModPhys.84.945.
Lattice instabilities in metallic elements / G. Grimvall, B. Magyari- Köpe, V. Ozoliņš, K. A. Persson // Review of Modern Physics. – 2012. – Vol. 84 (3). – P. 945–986. – DOI: 10.1103/PhysRevB.79.094111.
Phonon instabilities and the ideal strength of aluminum / D. M. Clatterbuck, C. R. Krenn, Marvin L. Cohen, J. W. Morris, Jr. // Physical Review Letters. – 2003. – Vol. 91 (13). – P. 135501–135504. – DOI: 10.1103/PhyaRevLett.91.135501.
Calculations of theoretical strength: state of the art and history / / J. Pokluda, M. Cern, P. Sandera, M. Sob // Journal of Computer-Aided Materials Design. – 2004. – Vol. 11. – P. 1–28. –DOI: 10.1007/s10820-004-4567-2.
Ab initio calculations of mechanical properties: methods and applications / J. Pokluda, M. Cerny, M. Sob, Y. Umeno // Progress in Materials Science. – 2015. – Vol. 73. – P. 127–158. – DOI: 10.1016/j.pmatsci.2015.04001.
Kuznetsov A. R., Starikov S. A., Sagaradze V. V. Phonon instabilities in a metal on the bain fcc–bcc transformation path // Diagnostics, Resource and Mechanics of materials and structures. – 2022. – Iss. 6. – P. 86–94. – DOI: 10.17804/2410-9908.2022.6.086-094. – URL: http://dream-journal.org/issues/2022-6/2022-6_385.html
Kuznetsov A. R., Starikov S. A. The Bain and orthorhombic paths of the bcc–fcc transformation in a bcc metal // Diagnostics, Resource and Mechanics of materials and structures. – 2023. – Iss. 6. – P. 35–44. – DOI: 10.17804/2410-9908.2023.6.035-044. – URL: http://dream-journal.org/issues/2023-6/2023-6_423.html
Kuznetsov A. R., Starikov S. A. The Bain and orthorhombic paths of structural-phase transformations in a transition metal (V) // Diagnostics, Resource and Mechanics of materials and structures. – 2024. – Iss. 6. – P. 119–130. – DOI: 10.17804/2410-9908.2024.6.119-130. – URL: http://dream-journal.org/issues/2024-6/2024-6_490.html
Ideal strength of bcc molybdenum and niobium / W. Luo, D. Roundy, M. L. Cohen, J. W. Morris, Jr. // Physical Review B. – 2002.– Vol. 66. – 094110. – DOI: 10.1103/PhysRevB.66.094110.
Liu Z., Shang J. First principles calculations of electronic properties and mechanical properties of bcc molybdenum and niobium // Rare Metals. – 2011. – Vol. 30 (suppl. 1). – P. 354–358. – DOI: 10.1007/s12598-011-0302-9.
URL: arXiv:2406.16634 [cond-mat.mtrl-sci]
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Compression curves of transition metals in the Mbar range: experiments and projector augmented-wave calculations / A. Dewaele, M. Torrent, P. Loubeyre, M. Mezouar // Physical Review B: Condensed Matter. – 2008. – Vol. 78 (10). – 104102. – DOI: 10.1103/PhysRevB.78.104102.
Haas P., Tran F., Blaha P. Calculation of the lattice constant of solids with semilocal functionals // Physical Review B. – 2009. – Vol. 79 (8). – 085104. – DOI: 10.1103/physrevb.79.085104.
Ming L., Manghnani M. H. Isothermal compression of BCC transition metals to 100 kbar // Journal of Applied Physics. – 1978. – Vol. 49. – P. 208–212. – DOI: 10.1063/1.324325.
Anomalies in the response of V, Nb, and Ta to tensile and shear loading: ab initio density functional theory calculations / N. Nagasako, M. Jahnátek, R. Asahi, J. Hafner // Physical Review B. – 2010. – Vol. 81. – P. 094108–094121. – DOI: 10.1103/PhysRevB.81.094108.
Katahara K. W., Manghnani M. H., Fisher E. Elastic moduli of paramagnetic chromium and Ti–V–Cr alloys // Journal of Physics F: Metal. Physics. – 1979. – Vol. 9. – P. 773. – DOI: 10.1088/0305-4608/9/11/008.
Dickinson J. M., Armstrong P. E. Temperature dependence of the elastic constants of molybdenum // Journal of Applied Physics. – 1967. – Vol. 38. – P. 602–606. – DOI: 10.1063/1.1709381.
Bolef D. I., De Klerk J. Elastic constants of single crystal Mo and W between 77° and 500°K // Journal of Applied Physics. – 1962. – Vol. 33. – P. 2311–2314. – DOI: 10.1063/1.1728952.
Lattice dynamics and thermodynamics of molybdenum from first-principles calculations / Z.-Y. Zeng, C.-E. Hu, L.-C. Cai, X.-R. Chen, and F.-Q. Jing // The Journal of Physical Chemistry B. – 2010. – Vol. 114. – P. 298–310. – DOI: 10.1021/jp9073637.
Kittel C. Introduction to Solid State Physics. – New York : Wiley, 1996. – 674 p.
Молекулярно-лучевая эпитаксия и гетероструктуры / под ред. Л. Ченга, К. Плога; пер. с англ. под ред. Ж. И. Алферова, Ю. В. Шмарцева. – Москва : Мир, 1989. – 582 с.
Kohn anomaly and phase stability in group VB transition metals / A. Landa, P. Söderlind, I. I. Naumov, J. E. Klepeis, L. Vitos // Computation. – 2018. – Vol. 6. – P. 29. – DOI: 10.3390/computation6020029.
Elasticity of the superconducting metals V, Nb, Ta, Mo, and W at high pressure / L. Koči, Y. Ma, A. R. Oganov, P. Souvatzis, R. Ahuja // Physical Review B. – 2008. – Vol. 77. – 214101. DOI: 10.1103/PhysRevB.77.214101.
Johannes M. D., Mazin I. I. Fermi surface nesting and the origin of Charge Density Waves in metals // Physical Review. B: Condensed Matter. – 2007. – Vol. 77 (16). – 165135. – DOI: 10.1103/PhysRevB.77.165135.

Библиографическая ссылка на статью

Kuznetsov A. R., Starikov S. A. The Bain and Orthorhombic Paths of the Structural and Phase Transformations and Theoretical Strength of Molybdenum // Diagnostics, Resource and Mechanics of materials and structures. - 2025. - Iss. 5. - P. 53-65. -
DOI: 10.17804/2410-9908.2025.5.053-065. -
URL: http://dream-journal.org/issues/content/article_531.html
(accessed: 18.04.2026).