Skip to main content
SpringerLink
Log in

Kinetic Monte Carlo approach to Schottky defects in noble metal nanoclusters

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

The vacancy concentration dependence on temperature and diameter of noble metal (gold, silver, and copper) nanoclusters is investigated using a Kinetic Monte Carlo method. Icosahedral and decahedral nanoclusters are studied, with diameters up to 3.73 nm for icosahedral clusters and up to 6.65 nm for decahedral clusters. The cohesive energy is calculated using a coordination number approach, resulting in a linear relation with cluster size. Random Schottky defects are frozen into the clusters at low temperatures (100–600 K) and we find that the vacancy concentration increases with smaller diameters and higher temperatures. We develop a model for this behavior, which explains the temperature and size dependence. This model predicts silver icosahedra to have the highest concentration of vacancies in the clusters studied. Vacancy concentrations are related to the ratio of surface/interior sites based on nearest neighbor calculations. The modified enthalpy and entropy of constant diameter clusters are derived from a logarithmic model for the Gibbs energy. Melting entropy and enthalpy are calculated in this coordination type model and compare well with previously published molecular dynamics results.

Graphical Abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. J. Frenkel, Z. Phys. 35(8/9), 652–669 (1926)

    Article  CAS  Google Scholar 

  2. C. Wagner, W. Schottky, Z. Phys. Chem. 11(2/3), 163–210 (1930)

    CAS  Google Scholar 

  3. Y. Kraftmakher, Phys. Rep. 299, 79–188 (1998)

    Article  CAS  Google Scholar 

  4. R.O. Simmons, R.W. Balluffi, Phys. Rev. 125(3), 862–872 (1962)

    Article  Google Scholar 

  5. R.O. Simmons, R.W. Balluffi, Phys. Rev. 119(2), 600–605 (1960)

    Article  CAS  Google Scholar 

  6. R.O. Simmons, R.W. Balluffi, Phys. Rev. 129(4), 1533–1544 (1963)

    Article  CAS  Google Scholar 

  7. T. Mori, M. Meshii, Acta Metall. 12, 104–106 (1964)

    Article  CAS  Google Scholar 

  8. T. Uefuji, Y. Shimomura, T. Kimo, Jpn. J. Appl. Phys. 16(6), 909–918 (1977)

    Article  CAS  Google Scholar 

  9. C.C. Battaile, Comput. Methods Appl. Mech. Eng. 197, 3386–3398 (2008)

    Article  Google Scholar 

  10. M. Müller, K. Albe, Acta Mater. 55, 3237–3244 (2007)

    Article  Google Scholar 

  11. A.S. Barnard, N.P. Young, A.I. Kirkland, M.A. van Huis, H. Xu, ACS Nano 3(6), 1431–1436 (2009)

    Article  CAS  Google Scholar 

  12. A.L. Gonzalez, C. Noguez, J. Beranek, A.S. Barnard, J. Phys. Chem. C 118, 9128–9136 (2014)

    Article  CAS  Google Scholar 

  13. G. Guisbiers, S. Mejia-Rosales, S. Khanal, F. Ruiz-Zepeda, R.L. Whetton, M.J. Yacaman, Nano Lett. 14, 6718–6726 (2014)

    Article  CAS  Google Scholar 

  14. Ph Buffat, J.-P. Borel, Phys. Rev. A 13(6), 2287–2298 (1976)

    Article  CAS  Google Scholar 

  15. Y. Xia, Y. Xiong, B. Lim, S.E. Skrabalak, Angew. Chem. Int. Ed. 48, 60–103 (2009)

    Article  CAS  Google Scholar 

  16. S.R. Plant, L. Cao, R.E. Palmer, J. Am. Chem. Soc. 136, 7559–7562 (2014)

    Article  CAS  Google Scholar 

  17. Q. Zhang, J. Xie, J. Yang, J.Y. Lee, ACS Nano 3(1), 139–148 (2009)

    Article  CAS  Google Scholar 

  18. M. Tsuji, M. Ogino, R. Matsuo, H. Kumagae, S. Hikino, T. Kim, S.H. Yoon, Cryst. Growth Des. 10, 296–301 (2010)

    Article  CAS  Google Scholar 

  19. F. Silly, M.R. Castell, ACS Nano 3(4), 901–906 (2009)

    Article  CAS  Google Scholar 

  20. I. Shyjumon, M. Gopinadhan, O. Ivanova, M. Quaas, H. Wulff, C.A. Helm, R. Hippler, Eur. Phys. J. D 37, 409–415 (2006)

    Article  CAS  Google Scholar 

  21. H.R. Alarifi, M. Atis, C. Ozdogan, A. Hu, M. Yavuz, Y. Zhou, J. Phys. Chem. C 117(23), 12289–12298 (2013)

    Article  CAS  Google Scholar 

  22. F. Delogu, Phys. Rev. B 72, 205418 (2005)

    Article  Google Scholar 

  23. H.H. Kart, H. Yildirim, S. Ozdemir Kart, T. Cagin, Mater. Chem. Phys. 147, 204–212 (2014)

    Article  CAS  Google Scholar 

  24. T. Imaoka, H. Kitazawa, W.-J. Chun, K. Yamamoto, Angew. Chem. Int. Ed. 54, 9810–9815 (2015)

    Article  CAS  Google Scholar 

  25. H. Delavari, H.H. Madaah Hosseini, A. Simchi, Chem. Phys. 383, 1–5 (2011)

    Article  Google Scholar 

  26. H. Delavari, H.R.M. Hosseini, A. Simchi, Physica B 406, 3777–3780 (2011)

    Article  Google Scholar 

  27. G. Guisbiers, G. Abudukelimu, J. Nanoparticle Res. 15, 1431 (2013)

    Article  Google Scholar 

  28. A. Cervellino, C. Giannini, A. Guagliardi, J. Appl. Crystallogr. 36, 1148–1158 (2003)

    Article  CAS  Google Scholar 

  29. C. Kittel, Introduction to Solid State Physics, 8th edn. (Wiley, Hoboken, 2005)

    Google Scholar 

  30. W. Martienssen, H. Warlimont, Springer Handbook of Condensed Matter and Materials Data (Springer, Berlin, 2005)

    Book  Google Scholar 

  31. Q. Jiang, H.M. Lu, Surf. Sci. Rep. 63, 427–464 (2008)

    Article  CAS  Google Scholar 

  32. J.M. Zhang, X.L. Song, X.J. Zhang, K.W. Xu, V. Ji, Surf. Sci. 600, 1277–1282 (2006)

    Article  CAS  Google Scholar 

  33. Y.Y. Andreev, Russ. J. Phys. Chem. 79(2), 179–185 (2005)

    CAS  Google Scholar 

  34. F.H. Kaatz, A. Bultheel, Comput. Mater. Sci. 99, 73–80 (2015)

    Article  CAS  Google Scholar 

  35. C. Mottet, G. Treglia, B. Legrand, Surf. Sci. 383, L719–L727 (1997)

    Article  CAS  Google Scholar 

  36. X. Shao, Y. Xiang, W. Cai, Chem. Phys. 305, 69–75 (2004)

    Article  CAS  Google Scholar 

  37. K. Laasonen, E. Panizon, D. Bochicchio, R. Ferrando, J. Phys. Chem. C 117, 26405–26413 (2013)

    Article  CAS  Google Scholar 

  38. G. Guisbiers, J. Phys. Chem. C 115, 2616–2621 (2011)

    Article  CAS  Google Scholar 

  39. G.A. de Wijs, G. Kresse, M.J. Gillan, Phys. Rev. B 57(14), 8223–8234 (1998)

    Article  Google Scholar 

  40. D. Fuks, S. Dorfman, Phys. Rev. B 50(22), 16340–16345 (1994)

    Article  CAS  Google Scholar 

  41. A. Glensk, B. Grabowski, T. Hickel, J. Nuegebauer, Phys. Rev. X 4, 011018 (2014)

    Google Scholar 

  42. H. Omid, H. Delavari, H.R.M. Hosseini, J. Phys. Chem. C 115, 17310–17313 (2011)

    Article  CAS  Google Scholar 

  43. W. Luo, W. Hu, S. Xiao, J. Phys. Chem. C 112, 2359–2369 (2008)

    Article  CAS  Google Scholar 

  44. M.A. Sandiz, A. Safaei, Mater. Lett. 62, 3954–3956 (2008)

    Article  Google Scholar 

  45. Z.W. Wang, R.E. Palmer, Nano Lett. 12, 91–95 (2012)

    Article  CAS  Google Scholar 

  46. M. Azubel, J. Koivisto, S. Malola, D. Bushnell, G.L. Hura, A.L. Koh, H. Tsunoyama, T. Tsukuda, M. Pettersson, H. Hakkinen, R.D. Kornberg, Science 345, 909–912 (2014)

    Article  CAS  Google Scholar 

  47. D. Bahena, N. Bhattarai, U. Santiago, A. Tlahuice, A. Ponce, S.B.H. Bach, B. Yoon, R.L. Whetton, U. Landman, M.J. Yacaman, J. Phys. Chem. Lett. 4, 975–981 (2013)

    Article  CAS  Google Scholar 

  48. Z.Y. Li, N.P. Young, M. Di Vece, S. Palomba, R.E. Palmer, A.L. Bleloch, B.C. Curley, R.L. Johnston, J. Jiang, J. Yuan, Nature 451, 46–49 (2008)

    Article  CAS  Google Scholar 

Download references

Acknowledgments

The MATLAB file Cluster Generator available from MATLAB Central was invaluable in assisting the simulation of Schottky defects in these clusters. We thank the reviewers and editor for suggestions that improved the manuscript.

Author information

Authors and Affiliations

  1. Mesalands Community College, 911 South 10th Street, Tucumcari, NM, 88401, USA

    Forrest H. Kaatz

  2. Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001, Heverlee, Belgium

    Adhemar Bultheel

  3. Actinium Chemical Research, Via Casilina, 1626/A, 00133, Rome, Italy

    Ottorino Ori

  4. Laboratory of Computational and Structural Physical-Chemistry for Nanosciences and QSAR, West University of Timişoara, Timişoara, Romania

    Ottorino Ori

Authors
  1. Forrest H. Kaatz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adhemar Bultheel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ottorino Ori
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Forrest H. Kaatz.

Ethics declarations

Conflicts of interest

The authors declare no competing financial interests.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kaatz, F.H., Bultheel, A. & Ori, O. Kinetic Monte Carlo approach to Schottky defects in noble metal nanoclusters. J Math Chem 55, 34–49 (2017). https://doi.org/10.1007/s10910-016-0667-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-016-0667-y

Keywords

Search

Navigation