Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Collective buckling in metal-organic framework materials

Nico Hahn1,*, Lars Öhrström2, and R. Matthias Geilhufe1

  • *Contact author: nico.hahn@chalmers.se

Phys. Rev. Research 8, 013325 – Published 25 March, 2026

DOI: https://doi.org/10.1103/bpjt-6qr2

Abstract

We develop a framework to describe collective buckling in metal-organic frameworks (MOFs). Starting from the microscopic structure of a single organic linker, we define a buckling coordinate governed by an effective double-well potential. Coupling between linkers is introduced within a dipole-dipole approximation, resulting in an effective lattice Hamiltonian. We analyze the transition between ordered and disordered phases within a mean-field approximation and estimate the critical temperature. As an illustrative example for our theory, we discuss the collective buckling instability for the prototypical cubic framework MOF-5 under different values of uniaxial strain. Our approach provides a quantitative description of collective buckling in framework materials.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (35)

  1. Z. P. Bažant and L. Cedolin, Stability of Structures (World Scientific, Mineola, New York, 2010).
  2. A. Guerra, A. C. Slim, D. P. Holmes, and O. Kodio, Self-ordering of buckling, bending, and bumping beams, Phys. Rev. Lett. 130, 148201 (2023).
  3. A. Schneemann, V. Bon, I. Schwedler, I. Senkovska, S. Kaskel, and R. A. Fischer, Flexible metal–organic frameworks, Chem. Soc. Rev. 43, 6062 (2014).
  4. F.-X. Coudert, Responsive metal–organic frameworks and framework materials: Under pressure, taking the heat, in the spotlight, with friends, Chem. Mater. 27, 1905 (2015).
  5. P. Ying, J. Zhang, and Z. Zhong, Pressure-induced phase transition of isoreticular MOFs: Mechanical instability due to ligand buckling, Microporous Mesoporous Mater. 312, 110765 (2021).
  6. R. Pallach, J. Keupp, K. Terlinden, L. Frentzel-Beyme, M. Kloß, A. Machalica, J. Kotschy, S. K. Vasa, P. A. Chater, C. Sternemann, M. T. Wharmby, R. Linser, R. Schmid, and S. Henke, Frustrated flexibility in metal-organic frameworks, Nat. Commun. 12, 4097 (2021).
  7. S. R. Batten, N. R. Champness, X.-M. Chen, J. Garcia-Martinez, S. Kitagawa, L. Öhrström, M. O’Keeffe, M. P. Suh, and J. Reedijk, Terminology of metal–organic frameworks and coordination polymers (IUPAC recommendations 2013), Pure Appl. Chem. 85, 1715 (2013).
  8. L. Öhrström and F. M. Amombo Noa, Metal-Organic Frameworks (American Chemical Society, Washington, DC, USA, 2021).
  9. H. Furukawa, K. E. Cordova, M. O’Keeffe, and O. M. Yaghi, The chemistry and applications of metal-organic frameworks, Science 341, 1230444 (2013).
  10. Z. Huang and R. M. Geilhufe, Quantum metal-organic frameworks, Small Sci. 4, 2400161 (2024).
  11. W. Jiang, X. Ni, and F. Liu, Exotic topological bands and quantum states in metal–organic and covalent–organic frameworks, Acc. Chem. Res. 54, 416 (2021).
  12. D. Kumar, J. Hellerstedt, B. Field, B. Lowe, Y. Yin, N. V. Medhekar, and A. Schiffrin, Manifestation of strongly correlated electrons in a 2D kagome metal–organic framework, Adv. Funct. Mater. 31, 2106474 (2021).
  13. B. Field, A. Schiffrin, and N. V. Medhekar, Correlation-induced magnetism in substrate-supported 2D metal-organic frameworks, npj Comput. Mater. 8, 227 (2022).
  14. B. Lowe, B. Field, J. Hellerstedt, J. Ceddia, H. L. Nourse, B. J. Powell, N. V. Medhekar, and A. Schiffrin, Local gate control of Mott metal-insulator transition in a 2D metal-organic framework, Nat. Commun. 15, 3559 (2024).
  15. X. Huang, S. Zhang, L. Liu, L. Yu, G. Chen, W. Xu, and D. Zhu, Superconductivity in a copper(II)-based coordination polymer with perfect kagome structure, Angew. Chem. Int. Ed. 57, 146 (2018).
  16. T. Takenaka, K. Ishihara, M. Roppongi, Y. Miao, Y. Mizukami, T. Makita, J. Tsurumi, S. Watanabe, J. Takeya, M. Yamashita, K. Torizuka, Y. Uwatoko, T. Sasaki, X. Huang, W. Xu, D. Zhu, N. Su, J.-G. Cheng, T. Shibauchi, and K. Hashimoto, Strongly correlated superconductivity in a copper-based metal-organic framework with a perfect kagome lattice, Sci. Adv. 7, eabf3996 (2021).
  17. M. F. Ohlrich, E. M. Makaresz, H. L. Nourse, and B. J. Powell, Flat bands and unconventional superconductivity in a simple model of metal-organic frameworks, Phys. Rev. B 111, L100503 (2025).
  18. R. M. Geilhufe, Quantum buckling in metal–organic framework materials, Nano Lett. 21, 10341 (2021).
  19. S. E. Rowley, L. J. Spalek, R. P. Smith, M. P. M. Dean, M. Itoh, J. F. Scott, G. G. Lonzarich, and S. S. Saxena, Ferroelectric quantum criticality, Nat. Phys. 10, 367 (2014).
  20. T. Esswein and N. A. Spaldin, Ferroelectric, quantum paraelectric, or paraelectric? Calculating the evolution from BaTiO3 to SrTiO3 to KTaO3 using a single-particle quantum mechanical description of the ions, Phys. Rev. Res. 4, 033020 (2022).
  21. H. Li, M. Eddaoudi, M. O’Keeffe, and O. M. Yaghi, Design and synthesis of an exceptionally stable and highly porous metal-organic framework, Nature (London) 402, 276 (1999).
  22. C. R. Groom, I. J. Bruno, M. P. Lightfoot, and S. C. Ward, The cambridge structural database, Acta Crystallogr. B 72, 171 (2016).
  23. L. H. N. Rimmer, M. T. Dove, A. L. Goodwin, and D. C. Palmer, Acoustic phonons and negative thermal expansion in MOF-5, Phys. Chem. Chem. Phys. 16, 21144 (2014).
  24. S. Savel'ev, X. Hu, and F. Nori, Quantum electromechanics: Qubits from buckling nanobars, New J. Phys. 8, 105 (2006).
  25. P. Ghosez, J.-P. Michenaud, and X. Gonze, Dynamical atomic charges: The case of ABO3 compounds, Phys. Rev. B 58, 6224 (1998).
  26. X. Gonze and C. Lee, Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory, Phys. Rev. B 55, 10355 (1997).
  27. L.-M. Yang, P. Vajeeston, P. Ravindran, H. Fjellvåg, and M. Tilset, Theoretical investigations on the chemical bonding, electronic structure, and optical properties of the metal-organic framework MOF-5, Inorg. Chem. 49, 10283 (2010).
  28. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  29. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  30. N. Lock, Y. Wu, M. Christensen, L. J. Cameron, V. K. Peterson, A. J. Bridgeman, C. J. Kepert, and B. B. Iversen, Elucidating negative thermal expansion in MOF-5, J. Phys. Chem. C 114, 16181 (2010).
  31. G. Teschl, Ordinary Differential Equations and Dynamical Systems (American Mathematical Society, Providence, Rhode Island, 2012).
  32. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, Course of Theoretical Physics, 3rd ed. (Pergamon Press, Oxford, 1977), Vol. 3.
  33. A. Altland and B. Simons, Condensed Matter Field Theory, 3rd ed. (Cambridge University Press, Cambridge, UK, 2023).
  34. R. Kubo, Generalized cumulant expansion method, J. Phys. Soc. Jpn. 17, 1100 (1962).
  35. NIST Digital Library of Mathematical Functions, edited by F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, Release 1.2.6 of 2026-03-15, https://dlmf.nist.gov/.

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation