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.
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Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Use of Large Language Models, AI and Machine Learning Tools: None declared.
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Conflict of interest: The authors state no conflict of interest.
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Research funding: This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Grant No. CRC 1487, “Iron, upgraded!” – Project No. 443703006.
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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.Suche in Google Scholar
2. Ostwald, W. Z. Phys. Chem. 1900, 34, 495–503.10.1515/zpch-1900-3431Suche in Google Scholar
3. Mittasch, A. Z. Phys. Chem. 1902, 40, 1–83.10.1515/zpch-1902-4002Suche in Google Scholar
4. Pawlow, P. Z. Phys. Chem. 1909, 65, 1–35.10.1515/zpch-1909-6502Suche in Google Scholar
5. Buffat, P.; Borel, J. P. Phys. Rev. A 1976, 13, 2287–2298; https://doi.org/10.1103/physreva.13.2287.Suche in Google Scholar
6. Henglein, A. Ber. Bunsen-Ges. Phys. Chem. 1990, 94, 600–603; https://doi.org/10.1002/bbpc.19900940513.Suche 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.Suche in Google Scholar
8. Plieth, W. J. J. Phys. Chem. 1982, 86, 3166–3170; https://doi.org/10.1021/j100213a020.Suche 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.Suche in Google Scholar
10. Becker, R. Theorie der Wärme; Ludwig, W., Ed., 3rd ed.; Springer-Verlag: Berlin, 1985.10.1007/978-3-662-10440-8Suche 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.Suche in Google Scholar
12. Kawano, H. Prog. Surf. Sci. 2022, 97, 100583; https://doi.org/10.1016/j.progsurf.2020.100583.Suche in Google Scholar
13. Wulff, G. Z. Kristallogr. Cryst. Mater. 1901, 34, 449–530; https://doi.org/10.1524/zkri.1901.34.1.449.Suche in Google Scholar
14. Defay, R.; Prigogine, I. Surface Tension and Adsorption; Longmans: London, 1966.Suche 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_13Suche in Google Scholar
16. Pimpinelli, A.; Villain, J. Physics of Crystal Growth; University Press: Cambridge, 1998.10.1017/CBO9780511622526Suche in Google Scholar
17. Landau, L. D. Collected Papers of L. D. Landau: The Equlibrium Form of Crystalls; Pergamon Press: Oxford, 1965.Suche in Google Scholar
18. Howie, A.; Marks, L. D. Philos. Mag. A 1984, 49, 95–109; https://doi.org/10.1080/01418618408233432.Suche 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.Suche 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/nmat3347Suche 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.Suche 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.Suche 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.Suche 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.Suche in Google Scholar
25. Plieth, W. J. Surf. Sci. 1985, 156, 530–535; https://doi.org/10.1016/0039-6028(85)90615-6.Suche in Google Scholar
26. Yang, S.; Knickelbein, M. B. J. Chem. Phys. 1990, 93, 1533–1539; https://doi.org/10.1063/1.459131.Suche 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.Suche 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.Suche in Google Scholar
29. Błoński, P.; Kiejna, A. Surf. Sci. 2007, 601, 123–133; https://doi.org/10.1016/j.susc.2006.09.013.Suche 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.Suche 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.Suche 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.Suche 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.Suche in Google Scholar
34. Miedema, A. R. Z. Metallkd. 1978, 69, 287–292; https://doi.org/10.1515/ijmr-1978-690501.Suche 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.Suche 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.Suche 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.Suche 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.Suche in Google Scholar
39. Leitner, J.; Sedmidubskỳ, D. World J. Chem. Educ. 2017, 5, 206–209.10.12691/wjce-5-6-4Suche 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.Suche in Google Scholar
41. Wagman, D. D. J. Phys. Chem. Ref. Data 1982, 11 (Supplement No. 2).Suche in Google Scholar
42. Bachels, T.; Schäfer, R. Chem. Phys. Lett. 2000, 324, 365–372.10.1016/S0009-2614(00)00622-9Suche in Google Scholar
43. Bachels, T.; Schäfer, R.; Güntherodt, H.-J. Phys. Rev. Lett. 2000, 84, 4890–4893.10.1103/PhysRevLett.84.4890Suche in Google Scholar PubMed
44. Schäfer, R. Z. Phys. Chem. 2003, 217, 989–1002.10.1524/zpch.217.8.989.20420Suche 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.Suche in Google Scholar
46. Köhler, C.; Seifert, G.; Frauenheim, T. Chem. Phys. 2005, 309, 23–31.10.1016/j.chemphys.2004.03.034Suche 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.Suche in Google Scholar
48. Aktürk, A.; Sebetci, A. AIP Adv. 2016, 6, 055103.10.1063/1.4948752Suche 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.Suche 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.Suche in Google Scholar
51. Elliott, J. A.; Shibuta, Y.; Wales, D. J. Philos. Mag. 2009, 89, 3311–3332; https://doi.org/10.1080/14786430903270668.Suche 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-xSuche in Google Scholar
53. Angelié, C.; Soudan, J.-M. J. Chem. Phys. 2017, 146, 174303.10.1063/1.4982252Suche in Google Scholar PubMed
54. Jana, R.; Caro, M. A. Phys. Rev. B 2023, 107, 245421; https://doi.org/10.1103/physrevb.107.245421.Suche 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.Suche in Google Scholar PubMed
56. Haynes, W. M.; Lide, D. R.; Bruno, T. J. CRC Handbook of Chemistry and Physics; CRC Press: Boca Raton, 2016.10.1201/9781315380476Suche 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, 104304.10.1063/5.0234648Suche 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.Suche 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.Suche in Google Scholar PubMed PubMed Central
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