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Chemical potential of iron in systems of low dimensionality

  • David Büchner ORCID logo EMAIL logo , Sun Myung Kim ORCID logo , Jan Philipp Hofmann ORCID logo , Vera Krewald ORCID logo , Marc Armbrüster ORCID logo and Rolf Schäfer ORCID logo EMAIL logo
Published/Copyright: November 12, 2025
Zeitschrift für Physikalische Chemie
From the journal Zeitschrift für Physikalische Chemie

Abstract

The influence of particle size on heterogeneous equilibria is investigated using Fe clusters as an example. Considering a thermodynamic cycle, the change in free enthalpy for the transfer of an Fe atom from a dispersed system to the bulk is analyzed with the aid of experimental data from molecular beam experiments. Dissociation energies of mass-selected clusters are used for this purpose. It is shown that predictions within the framework of equilibrium thermodynamics, considering the free surface, are quantitatively correct for nanoscale clusters with less than 100 atoms. The effects on the redox behavior as well as the dependence of the work function on the cluster size are also addressed and discussed. In comparison with quantum chemical studies based on density functional theory, the simple thermodynamic model is surprisingly robust in predicting the chemical behavior of Fe in systems with reduced dimensionality.

Keywords: iron; cluster; chemical potential; Gibbs–Thomson equation; size effects; thermodynamic cycles
Corresponding authors: David Büchner and Rolf Schäfer, Eduard-Zintl-Institute, TU Darmstadt, 64287 Darmstadt, Germany, E-mail: (D. Büchner), (R. Schäfer).

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors state no conflict of interest.

  6. Research funding: This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Grant No. CRC 1487, “Iron, upgraded!” – Project No. 443703006.

  7. Data availability: Data available on request.

References

1. Thomson, W. Proc. R. Soc. Edinburgh, Sect. A 1870, 7, 63–68; https://doi.org/10.1017/s0370164600041729.Search in Google Scholar

2. Ostwald, W. Z. Phys. Chem. 1900, 34, 495–503.10.1515/zpch-1900-3431Search in Google Scholar

3. Mittasch, A. Z. Phys. Chem. 1902, 40, 1–83.10.1515/zpch-1902-4002Search in Google Scholar

4. Pawlow, P. Z. Phys. Chem. 1909, 65, 1–35.10.1515/zpch-1909-6502Search in Google Scholar

5. Buffat, P.; Borel, J. P. Phys. Rev. A 1976, 13, 2287–2298; https://doi.org/10.1103/physreva.13.2287.Search in Google Scholar

6. Henglein, A. Ber. Bunsen-Ges. Phys. Chem. 1990, 94, 600–603; https://doi.org/10.1002/bbpc.19900940513.Search in Google Scholar

7. Hilpert, K.; Gingerich, K. A. Ber. Bunsen-Ges. Phys. Chem. 1980, 84, 739–745; https://doi.org/10.1002/bbpc.19800840810.Search in Google Scholar

8. Plieth, W. J. J. Phys. Chem. 1982, 86, 3166–3170; https://doi.org/10.1021/j100213a020.Search in Google Scholar

9. Gibbs, J. W. On the Equilibrium of Heterogeneous Substances (1878). In The Collected Works of J. Willard Gibbs; Longmans: New York, 1928.Search in Google Scholar

10. Becker, R. Theorie der Wärme; Ludwig, W., Ed., 3rd ed.; Springer-Verlag: Berlin, 1985.10.1007/978-3-662-10440-8Search in Google Scholar

11. Derry, G. N.; Kern, M. E.; Worth, E. H. J. Vac. Sci. Technol., A 2015, 33, 060801; https://doi.org/10.1116/1.4934685.Search in Google Scholar

12. Kawano, H. Prog. Surf. Sci. 2022, 97, 100583; https://doi.org/10.1016/j.progsurf.2020.100583.Search in Google Scholar

13. Wulff, G. Z. Kristallogr. Cryst. Mater. 1901, 34, 449–530; https://doi.org/10.1524/zkri.1901.34.1.449.Search in Google Scholar

14. Defay, R.; Prigogine, I. Surface Tension and Adsorption; Longmans: London, 1966.Search in Google Scholar

15. Wortis, M. Equilibrium Crystal Shapes and Interfacial Phase Transitions. In Chemistry and Physics of Solid Surfaces VII; Vanselow, R., Howe, R., Eds.; Springer-Verlag: Berlin, 1988.10.1007/978-3-642-73902-6_13Search in Google Scholar

16. Pimpinelli, A.; Villain, J. Physics of Crystal Growth; University Press: Cambridge, 1998.10.1017/CBO9780511622526Search in Google Scholar

17. Landau, L. D. Collected Papers of L. D. Landau: The Equlibrium Form of Crystalls; Pergamon Press: Oxford, 1965.Search in Google Scholar

18. Howie, A.; Marks, L. D. Philos. Mag. A 1984, 49, 95–109; https://doi.org/10.1080/01418618408233432.Search in Google Scholar

19. Suryanto, B. H. R.; Wang, Y.; Hocking, R. K.; Adamson, W.; Zhao, C. Nat. Commun. 2019, 10, 5599; https://doi.org/10.1038/s41467-019-13415-8.Search in Google Scholar PubMed PubMed Central

20. Armbrüster, M.; Kovnir, K.; Friedrich, M.; Teschner, D.; Wowsnick, G.; Hahne, M.; Gille, P.; Szentmiklósi, L.; Feuerbacher, M.; Heggen, M.; Girgsdies, F.; Rosenthal, D.; Schlögl, R.; Grin, Y. Nat. Mater. 2012, 11, 690–693.10.1038/nmat3347Search in Google Scholar PubMed

21. Wan, X.; Liu, Q.; Liu, J.; Liu, S.; Liu, X.; Zheng, L.; Shang, J.; Yu, R.; Shui, J. Nat. Commun. 2022, 13, 2963; https://doi.org/10.1038/s41467-022-30702-z.Search in Google Scholar PubMed PubMed Central

22. Kepp, K. P. J. Phys. Chem. A 2019, 123, 6536–6546; https://doi.org/10.1021/acs.jpca.9b05140.Search in Google Scholar PubMed

23. Zhu, S.; Xie, K.; Lin, Q.; Cao, R.; Qiu, F. Adv. Colloid Interface Sci. 2023, 315, 102905; https://doi.org/10.1016/j.cis.2023.102905.Search in Google Scholar PubMed

24. Makov, G.; Nitzan, A.; Brus, L. E. J. Chem. Phys. 1988, 88, 5076–5085; https://doi.org/10.1063/1.454661.Search in Google Scholar

25. Plieth, W. J. Surf. Sci. 1985, 156, 530–535; https://doi.org/10.1016/0039-6028(85)90615-6.Search in Google Scholar

26. Yang, S.; Knickelbein, M. B. J. Chem. Phys. 1990, 93, 1533–1539; https://doi.org/10.1063/1.459131.Search in Google Scholar

27. Rohlfing, E. A.; Cox, D.; Kaldor, A.; Johnson, K. H. J. Chem. Phys. 1984, 81, 3846–3851; https://doi.org/10.1063/1.448168.Search in Google Scholar

28. Rohlfing, E. A.; Cox, D. M.; Kaldor, A. Chem. Phys. Lett. 1983, 99, 161–166; https://doi.org/10.1016/0009-2614(83)80551-x.Search in Google Scholar

29. Błoński, P.; Kiejna, A. Surf. Sci. 2007, 601, 123–133; https://doi.org/10.1016/j.susc.2006.09.013.Search in Google Scholar

30. Jin, H.; Blackwood, D. J.; Wang, Y.; Ng, M.-F.; Tan, T. L. Corros. Sci. 2022, 196, 110029; https://doi.org/10.1016/j.corsci.2021.110029.Search in Google Scholar

31. Ozawa, S.; Suzuki, S.; Hibiya, T.; Fukuyama, H. J. Appl. Phys. 2011, 109, 014902; https://doi.org/10.1063/1.3527917.Search in Google Scholar

32. Morohoshi, K.; Uchikoshi, M.; Isshiki, M.; Fukuyama, H. ISIJ Int. 2011, 51, 1580–1586; https://doi.org/10.2355/isijinternational.51.1580.Search in Google Scholar

33. Ozawa, S.; Takahashi, S.; Suzuki, S.; Sugawara, H.; Fukuyama, H. Jpn. J. Appl. Phys. 2011, 50, 11RD05; https://doi.org/10.7567/jjap.50.11rd05.Search in Google Scholar

34. Miedema, A. R. Z. Metallkd. 1978, 69, 287–292; https://doi.org/10.1515/ijmr-1978-690501.Search in Google Scholar

35. Marlton, S. J. P.; Liu, C.; Watkins, P.; Buntine, J. T.; Bieske, E. J. J. Chem. Phys. 2023, 159, 024302; https://doi.org/10.1063/5.0155548.Search in Google Scholar PubMed

36. Loh, S. K.; Lian, L.; Hales, D. A.; Armentrout, P. B. J. Phys. Chem. 1988, 92, 4009–4012; https://doi.org/10.1021/j100325a001.Search in Google Scholar

37. Loh, S. K.; Hales, D. A.; Lian, L.; Armentrout, P. B. J. Chem. Phys. 1989, 90, 5466–5485; https://doi.org/10.1063/1.456452.Search in Google Scholar

38. Lian, L.; Su, C.-X.; Armentrout, P. B. J. Chem. Phys. 1992, 97, 4072–4083; https://doi.org/10.1063/1.463912.Search in Google Scholar

39. Leitner, J.; Sedmidubskỳ, D. World J. Chem. Educ. 2017, 5, 206–209.10.12691/wjce-5-6-4Search in Google Scholar

40. Miedema, A. R.; Gingerich, K. A. J. Phys. B 1979, 12, 2081–2095; https://doi.org/10.1088/0022-3700/12/13/005.Search in Google Scholar

41. Wagman, D. D. J. Phys. Chem. Ref. Data 1982, 11 (Supplement No. 2).Search in Google Scholar

42. Bachels, T.; Schäfer, R. Chem. Phys. Lett. 2000, 324, 365–372.10.1016/S0009-2614(00)00622-9Search in Google Scholar

43. Bachels, T.; Schäfer, R.; Güntherodt, H.-J. Phys. Rev. Lett. 2000, 84, 4890–4893.10.1103/PhysRevLett.84.4890Search in Google Scholar PubMed

44. Schäfer, R. Z. Phys. Chem. 2003, 217, 989–1002.10.1524/zpch.217.8.989.20420Search in Google Scholar

45. Bobadova-Parvanova, P.; Jackson, K. A.; Srinivas, S.; Horoi, M.; Köhler, C.; Seifert, G. J. Chem. Phys. 2002, 116, 3576–3587; https://doi.org/10.1063/1.1445113.Search in Google Scholar

46. Köhler, C.; Seifert, G.; Frauenheim, T. Chem. Phys. 2005, 309, 23–31.10.1016/j.chemphys.2004.03.034Search in Google Scholar

47. Ma, Q.-M.; Xie, Z.; Wang, J.; Liu, Y.; Li, Y.-C. Solid State Commun. 2007, 142, 114–119; https://doi.org/10.1016/j.ssc.2006.12.023.Search in Google Scholar

48. Aktürk, A.; Sebetci, A. AIP Adv. 2016, 6, 055103.10.1063/1.4948752Search in Google Scholar

49. Gutsev, G. L.; Weatherford, C. A.; Jena, P.; Johnson, E.; Ramachandran, B. R. J. Phys. Chem. A 2012, 116, 10218–10228; https://doi.org/10.1021/jp307284v.Search in Google Scholar PubMed

50. Liu, J.-R.; Die, D.; Kuang, X.-Y. Int. J. Quantum Chem. 2023, 123, e27206; https://doi.org/10.1002/qua.27206.Search in Google Scholar

51. Elliott, J. A.; Shibuta, Y.; Wales, D. J. Philos. Mag. 2009, 89, 3311–3332; https://doi.org/10.1080/14786430903270668.Search in Google Scholar

52. Liu, T.-D.; Fan, T.-E.; Zheng, J.-W.; Shao, G.-F.; Sun, Q.; Wen, Y.-H. J. Nanopart. Res. 2016, 18, 1–16.10.1007/s11051-016-3361-xSearch in Google Scholar

53. Angelié, C.; Soudan, J.-M. J. Chem. Phys. 2017, 146, 174303.10.1063/1.4982252Search in Google Scholar PubMed

54. Jana, R.; Caro, M. A. Phys. Rev. B 2023, 107, 245421; https://doi.org/10.1103/physrevb.107.245421.Search in Google Scholar

55. Yang, W.-H.; Yu, F.-Q.; Huang, R.; Shao, G.-F.; Liu, T.-D.; Wen, Y.-H. J. Chem. Inf. Model. 2023, 63, 6727–6739; https://doi.org/10.1021/acs.jcim.3c01331.Search in Google Scholar PubMed

56. Haynes, W. M.; Lide, D. R.; Bruno, T. J. CRC Handbook of Chemistry and Physics; CRC Press: Boca Raton, 2016.10.1201/9781315380476Search in Google Scholar

57. López-Moreno, S.; Hernández-Vázquez, E. E.; Ponce-Tadeo, A. P.; Ricardo-Chávez, J. L.; Morán-López, J. L. J. Chem. Phys. 2025, 162, 104304.10.1063/5.0234648Search in Google Scholar PubMed

58. Hua, H.; Liu, Y.; Wang, D.; Li, Y. Anal. Chem. 2018, 90, 9677–9681; https://doi.org/10.1021/acs.analchem.8b02644.Search in Google Scholar PubMed

59. Espinoza, R.; Cahua, D. V.; Magro, K.; Nguyen, S. C. J. Phys. Chem. Lett. 2024, 15, 12243–12247; https://doi.org/10.1021/acs.jpclett.4c02998.Search in Google Scholar PubMed PubMed Central

Received: 2025-09-05
Accepted: 2025-10-06
Published Online: 2025-11-12

© 2025 Walter de Gruyter GmbH, Berlin/Boston

https://doi.org/10.1515/zpch-2025-0139
Keywords for this article
iron; cluster; chemical potential; Gibbs–Thomson equation; size effects; thermodynamic cycles
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