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State Equation of a Nanocrystal with Vacancies

  • M. N. Magomedov
Article
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Abstract

An expression for the Helmholtz free energy is established and the equation of state is derived for a nanocrystal containing vacancies in the lattice and delocalized (diffusing) atoms. The calculations are performed for bcc (body-centered cubic) iron under the isothermal compression of a nanocrystal along the 300-K and 1000-K isotherms. The changes in the specific surface energy (σ), the probability of vacancy formation (ϕ v ) and the probability of atom delocalization (x d ) are studied depending on the size (N) and shape of the nanocrystal at different temperatures (T) and pressures (P). The size dependences are characterized along two isobars: at atmospheric pressure (P = 1 bar) and at P = 100 kbar. As is shown, two P-points arise in the σ(P) isotherms at T ≤ 300 K, where the specific surface energy is independent of the nanocrystal size, which is σ(N) = σ(∞). As the temperature increases, the P points approach each other and at T ≥ 1000 K they vanish in the isotherms. At atmospheric pressure and T = 300 K the amount of vacancies per atom in a nanocrystal is much lower than that in a macrocrystal; however, at T = 1000 K the shredding of the latter leads to an increase in the probability of vacancy formation. Moreover, the smaller the nanocrystal size, the higher the probability of atom delocalization (as well as the self-diffusion coefficient) at any pressure and temperature. The ratio ϕ v /x d decreases with decreasing size of the nanocrystal, and less than a certain size, there is an no-vacansies self-diffusion, at which the number of delocalized atoms is greater than the amount of vacant cells in the nanocrystal lattice, i.e., ϕ v < x d . As the nanocrystal shape becomes different from the energetically optimal, the size dependences of the lattice properties of the nanocrystal are enhanced.

Keywords

pressure nanocrystal size shape vacancies self-diffusion iron 

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References

  1. 1.
    Z. H. Li and D. G. Truhlar, Chem. Sci. 5 (7), 2605 (2014).CrossRefGoogle Scholar
  2. 2.
    R. V. Chamberlin, Entropy 17 (1), 73 (2015).Google Scholar
  3. 3.
    L. D. Marks and L. Peng, J. Phys.: Condens. Matter 28 (5), 053001 (2016).Google Scholar
  4. 4.
    L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, Vol. 5: Statistical Physics. Part 1 (Pergamon Press, Oxford, 1969).Google Scholar
  5. 5.
    E. A. Moelwyn-Hughes, Physical Chemistry (Pergamon Press, London, New York, Paris, 1957).Google Scholar
  6. 6.
    M. N. Magomedov, Phys. Solid State 46 (5), 954 (2004).CrossRefGoogle Scholar
  7. 7.
    M. N. Magomedov, Study of Interatomic Interactions, Vacancies Formation, and Self-Diffusion in Crystals (Fizmatlit, Moscow, 2010) [in Russian].Google Scholar
  8. 8.
    M. N. Magomedov, Tech. Phys. Lett. 31 (1), 13 (2005).CrossRefGoogle Scholar
  9. 9.
    M. N. Magomedov, Tech. Phys. 58 (6), 927 (2013).CrossRefGoogle Scholar
  10. 10.
    M. N. Magomedov, Nanotechnol. Russ. 9 (5–6), 293 (2014).CrossRefGoogle Scholar
  11. 11.
    M. N. Magomedov, Semiconductors 42 (10), 1133 (2008).CrossRefGoogle Scholar
  12. 12.
    M. N. Magomedov, Vestn. Mosk. Gos. Tekh. Univ. im. N. E. Baumana, Ser. Estestv. Nauki, No. 2, 28 (2013).Google Scholar
  13. 13.
    M. N. Magomedov, Phys. Met. Metallogr. 114 (3), 207 (2013).CrossRefGoogle Scholar
  14. 14.
    M. N. Magomedov, Semiconductors 44 (3), 271 (2010).CrossRefGoogle Scholar
  15. 15.
    J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (John Wiley and Sons, New York, 1954).Google Scholar
  16. 16.
    L. A. Girifalco, Statistical Physics of Materials (Wiley-Interscience, New York, 1973).Google Scholar
  17. 16a.
    L. A. Girifalco, Statistical Mechanics of Solids (Oxford Univ. Press, Oxford, New York, 2000).Google Scholar
  18. 17.
    M. N. Magomedov, Phys. Solid State 45 (1), 32 (2003).CrossRefGoogle Scholar
  19. 18.
    M. N. Magomedov, Tech. Phys. 60 (11), 1619 (2015).CrossRefGoogle Scholar
  20. 19.
    M. N. Magomedov, Nanotechnol. Russ. 10 (1–2), 89(2015).CrossRefGoogle Scholar
  21. 20.
    G. Sharma and M. Kumar, Indian J. Pure Appl. Phys. 54 (4), 251 (2016).Google Scholar
  22. 21.
    C. C. Yang and S. Li, Phys. Rev. B 75 (16), 165413 (2007).CrossRefGoogle Scholar
  23. 22.
    M. A. Shandiz, J. Phys.: Condens. Matter 20 (32), 325237 (2008).Google Scholar
  24. 23.
    G. Guisbiers, J. Phys. Chem. C 115 (6), 2616 (2011).CrossRefGoogle Scholar
  25. 24.
    X. Yu and Z. Zhan, Nanoscale Res. Lett. 9 (1), 1 (2014).CrossRefGoogle Scholar
  26. 25.
    T. N. Zvonareva, A. A. Sitnikova, G. S. Frolova, and V. I. Ivanov-Omskii, Semiconductors 42 (3), 325 (2008).CrossRefGoogle Scholar
  27. 26.
    Q. Jiang, S. H. Zhang, and J. C. Li, Solid State Commun. 130, 581 (2004).CrossRefGoogle Scholar
  28. 27.
    G. Xiong, J. N. Clark, C. Nicklin, J. Rawle, and I. K. Robinson, Sci. Rep. 4, 6765 (2014).CrossRefGoogle Scholar
  29. 28.
    M. N. Magomedov, Tech. Phys. Lett. 28 (5), 430 (2002).CrossRefGoogle Scholar
  30. 29.
    I. F. Golovnev, E. I. Golovneva, and V. M. Fomin, Fiz. Mezomekh. 15 (1), 69 (2012).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Geothermal Problems, Dagestan Scientific CenterRussian Academy of SciencesMakhachkalaRussia

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