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A percolation theory for designing corrosion-resistant alloys


Iron–chromium and nickel–chromium binary alloys containing sufficient quantities of chromium serve as the prototypical corrosion-resistant metals owing to the presence of a nanometre-thick protective passive oxide film1,2,3,4,5,6,7,8. Should this film be compromised by a scratch or abrasive wear, it reforms with little accompanying metal dissolution, a key criterion for good passive behaviour. This is a principal reason that stainless steels and other chromium-containing alloys are used in critical applications ranging from biomedical implants to nuclear reactor components9,10. Unravelling the compositional dependence of this electrochemical behaviour is a long-standing unanswered question in corrosion science. Herein, we develop a percolation theory of alloy passivation based on two-dimensional to three-dimensional crossover effects that accounts for selective dissolution and the quantity of metal dissolved during the initial stage of passive film formation. We validate this theory both experimentally and by kinetic Monte Carlo simulation. Our results reveal a path forward for the design of corrosion-resistant metallic alloys.

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Fig. 1: Passivation on a 2D topological surface.
Fig. 2: LSV and potential step integrated chronoamperometry results with numerical fits to the theoretical equation, \({\it{h}} = {\it{c}}\left( {{\it{p}}_{{\mathrm{c}}}\left( {\it{h}} \right) - {\it{p}}_{{\mathrm{c}}}^{\mathrm{3D}}} \right)^{ - {\it{\nu }}^{\mathrm{3D}}}\).
Fig. 3: Online ICPMS results for passivation of Fe–Cr and Ni–Cr alloys in 0.1 M H2SO4.
Fig. 4: Representative results of first-principles calculations for the dissociative adsorption of dioxygen on Fe(100), Fe(110), Cr(100), Cr(110) and Cr-doped Fe surfaces.

Data availability

The data used in this study are available from the corresponding author upon request.

Code availability

The KMC and data analysis computer codes used in this study are available from the corresponding author upon request.


  1. 1.

    McCafferty, E. Introduction to Corrosion Science (Springer, 2010).

  2. 2.

    Uhlig, H. H. & Woodside, G. E. Anodic polarization of passive and non-passive chromium–iron alloys. J. Phys. Chem. 57, 280–283 (1953).

    CAS  Article  Google Scholar 

  3. 3.

    Kirchheim, R. et al. The passivity of iron-chromium alloys. Corros. Sci. 29, 899–917 (1989).

    CAS  Article  Google Scholar 

  4. 4.

    King, P. F. & Uhlig, H. H. Passivity in the iron-chromium binary alloys. J. Phys. Chem. 63, 2026–2032 (1959).

    CAS  Article  Google Scholar 

  5. 5.

    Williams, D. E., Newman, R. C., Song, Q. & Kelly, R. G. Passivity breakdown and pitting corrosion of binary alloys. Nature 350, 216–219 (1991).

    CAS  Article  Google Scholar 

  6. 6.

    Diawara, B., Beh, Y.-A. & Marcus, P. Nucleation and growth of oxide layers on stainless steels (FeCr) using a virtual oxide layer model. J. Phys. Chem. C 114, 19299–19307 (2010).

    CAS  Article  Google Scholar 

  7. 7.

    Hamm, D., Ogle, K., Olsson, C.-O., Weber, S. & Landolt, S. Passivation of Fe–Cr alloys studied with ICP-AES and EQCM. Corros. Sci. 44, 1443–1456 (2002).

    CAS  Article  Google Scholar 

  8. 8.

    Davenport, A. J. et al. In situ synchrotron X-ray microprobe studies of passivation thresholds in Fe-Cr alloys. J. Electrochem. Soc. 148, B217–B221 (2001).

    CAS  Article  Google Scholar 

  9. 9.

    Odette, R. & Zinkle, S. Structural Alloys for Nuclear Energy Applications (Elsevier, 2019).

  10. 10.

    Niinomi, M., Nakai, M. & Hieda, J. Development of new metallic alloys for biomedical applications. Acta Biomater. 11, 3888–3903 (2012).

    Article  Google Scholar 

  11. 11.

    Miracle, D. B. & Senkov, O. N. A critical review of high entropy alloys and related concepts. Acta Mater. 122, 448–511 (2017).

    CAS  Article  Google Scholar 

  12. 12.

    George, E. P., Raabe, D. & Ritchie, R. O. High-entropy alloys. Nat. Rev. Mater. 4, 515–534 (2019).

    CAS  Article  Google Scholar 

  13. 13.

    Troparevsky, M. C., Morris, J. R., Kent, P. R. C., Lupini, A. R. & Stocks, G. M. Criteria for predicting the formation of single-phase high entropy alloys. Phys. Rev. X 5, 011041 (2015).

    Google Scholar 

  14. 14.

    Huang, L.-F., Scully, J. R. & Rondenelli, J. M. Modeling corrosion with first principles electrochemical phase diagrams. Annu. Rev. Mater. Res. 49, 53–77 (2019).

    CAS  Article  Google Scholar 

  15. 15.

    Huang, L.-F. & Rondenelli, J. M. Electrochemical phase diagrams of Ni from ab initio simulations: role of exchange interactions on accuracy. J. Phys. Condens. Mat. 29, 475501 (2017).

    Article  Google Scholar 

  16. 16.

    Yu, X. et al. Nonequilibrium solute capture in passivating oxide films. Phys. Rev. Lett. 121, 145701 (2018).

    CAS  Article  Google Scholar 

  17. 17.

    Sherman, Q., Voorhees, P. W. & Marks, L. D. Thermodynamics of solute capture during the oxidation of multicomponent metals. Acta Mater. 181, 584–594 (2019).

    CAS  Article  Google Scholar 

  18. 18.

    Frankenthal, R. P. On the passivity of iron‐chromium alloys I. Reversible primary passivation and secondary film formation. J. Electrochem. Soc. 114, 542–547 (1967).

    CAS  Article  Google Scholar 

  19. 19.

    Bastek, P. D., Newman, R. C. & Kelly, R. G. Measurement of passive film effects on scratched electrode behavior. J. Electrochem. Soc. 140, 1884–1889 (1993).

    CAS  Article  Google Scholar 

  20. 20.

    Sieradzki, K. & Newman, R. C. A percolation model for passivation in stainless steels. J. Electrochem. Soc. 133, 1979–1980 (1985).

    Article  Google Scholar 

  21. 21.

    Shannon, R. D. Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Cryst. A32, 751–767 (1976) .

    CAS  Article  Google Scholar 

  22. 22.

    Liu, M., Aiello, A., Xie, Y. & Sieradzki, K. The effect of short range order on the passivation of Fe-Cr alloys. J. Electrochem. Soc. 165, C830–C834 (2018).

    CAS  Article  Google Scholar 

  23. 23.

    Shante, V. K. S. & Kirkpatrick, S. An introduction to percolation theory. Adv. Phys. 20, 325–357 (1971).

    Article  Google Scholar 

  24. 24.

    Sykes, M. F. & Glen, M. Percolation processes in two dimensions. I. Low-density series expansions. J. Phys. A Math. Gen. 9, 87–95 (1976).

    Article  Google Scholar 

  25. 25.

    Sykes, M. F. & Glen, M. Percolation processes in three dimensions. J. Phys. A Math. Gen. 9, 1705–1712 (1976).

    Article  Google Scholar 

  26. 26.

    Stauffer, D. & Aharony, A. Introduction to Percolation Theory 2nd edn (Taylor & Francis, 1992).

  27. 27.

    Sotta, P. & Long, D. The crossover from 2D to 3D percolation: theory and numerical simulations. Eur. Phys. J. E 11, 375–388 (2003).

    CAS  Article  Google Scholar 

  28. 28.

    Clerc, J. P., Giraud, G., Alexander, S. & Guyon, E. Conductivity of a mixture of conducting and insulating grains: dimensionality effects. Phys. Rev. B 22, 2489–2494 (1980).

    CAS  Article  Google Scholar 

  29. 29.

    Lopes, P. P. et al. Relationships between atomic level surface structure and stability. ACS Catal. 6, 2536–2544 (2016).

    CAS  Article  Google Scholar 

  30. 30.

    Liu, Y. et al. Stability limits and defect dynamics in Ag nanoparticles probed by Bragg coherent diffractive imaging. Nano Lett. 17, 1595–1601 (2017).

    CAS  Article  Google Scholar 

  31. 31.

    Gaskell, G. R. Introduction to the Thermodynamics of Materials 4th edn (Taylor & Francis, 2003).

  32. 32.

    Reynolds, P. J., Stanley, H. E. & Klein, W. Large-cell Monte Carlo renormalization group for percolation. Phys. Rev. B 21, 1223–1245 (1980).

    CAS  Article  Google Scholar 

  33. 33.

    May, T. W. & Wiedmeyer, R. H. A table of polyatomic interferences in ICP-MS. At. Spectrosc. 19, 150–155 (1998).

    CAS  Google Scholar 

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K.S., A.A. and Y.X. acknowledge funding from the National Science Foundation under award DMR-1708459. M.L.T. and K.S. acknowledge funding in part from the Office of Naval Research, Multidisciplinary University Research Initiative programme under award N00014-20-1-2368. D.M.A. and R.C.N. were funded by NSERC (Canada) and UNENE, the University Network of Excellence in Nuclear Engineering. P.P.L. acknowledges support by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division.

Author information




Y.X. and A.A. performed the electrochemical experiments. P.P.L. performed the in situ ICPMS measurement of the dissolution profile for the alloys. Y.X., D.W. and H.Z. performed the first-principles-based calculations, and J.L.H., E.A. and M.L.T. performed the supplementary scanning transmission electron microscopy (STEM) analysis. D.M.A. performed the KMC simulations and the MC-RNG analysis with input from R.C.N. and K.S.; Y.X., R.C.N., A.A. and K.S. analysed and interpreted all the results. K.S. conceived and supervised the study.

Corresponding author

Correspondence to Karl Sieradzki.

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The authors declare no competing interests.

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Peer review information Nature Materials thanks Sannakaisa Virtanen, David Williams and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Information

Supplementary Discussion, Figs. 1–10, Tables 1–3 and refs. 1–20.

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Xie, Y., Artymowicz, D.M., Lopes, P.P. et al. A percolation theory for designing corrosion-resistant alloys. Nat. Mater. (2021).

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