Abstract
Nanoscale molecular transport governs mass diffusion and responsiveness in soft porous crystals, where guest adsorption induces host deformation and alters rigidity. Surface-mediated adsorption generates inhomogeneous adsorbate distributions, leading to spatial variations in stiffness—elastic heterogeneity—whose role in adsorption kinetics remains poorly understood. Here, we show that elastic heterogeneity governs adsorption kinetics, giving rise to size-dependent uptake, surface creasing, and anomalous dynamic scaling distinct from established scaling laws. Stress relaxation near corners accelerates adsorption, while on surfaces, creases emerge at flexible unadsorbed regions compressed between rigid adsorbed domains. The resulting lateral correlations of adsorbates exhibit a breakdown of scale invariance between global and local fluctuations. These findings provide a mechanistic foundation for controlling adsorption and deformation kinetics via elastic heterogeneity. Our work opens a route to engineering responsive materials, where mechanical feedback is harnessed to control cooperative molecular transport and drive macroscopic shape changes under external perturbations.
Similar content being viewed by others
Data availability
Input files to generate all of the figures are openly available at GitHub (https://github.com/kmitsumoto51/mof_kinetics). All other raw and processed data generated during this study are available from the corresponding author upon request.
Code availability
The computer codes used in this study are available from the corresponding author upon request.
References
Mehrer, H.Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-controlled Processes (Springer Science & Business Media, 2007).
Lee, D. S., Wingreen, N. S. & Brangwynne, C. P. Chromatin mechanics dictates subdiffusion and coarsening dynamics of embedded condensates. Nat. Phys. 17, 531–538 (2021).
Mukhopadhyay, A. & Sheldon, B. W. Deformation and stress in electrode materials for Li-ion batteries. Prog. Mater. Sci. 63, 58–116 (2014).
Yabuuchi, N., Kubota, K., Dahbi, M. & Komaba, S. Research development on sodium-ion batteries. Chem. Rev. 114, 11636–11682 (2014).
Alefeld, G. & Völkl, J. (eds.) Hydrogen in Metals I (Springer, 1978).
Li, Y. & Tanaka, T. Phase transitions of gels. Annu. Rev. Mater. Res. 22, 243–277 (1992).
Doi, M. Gel dynamics. J. Phys. Soc. Jpn. 78, 052001 (2009).
Coussy, O.Mechanics and Physics of Porous Solids (John Wiley & Sons, 2010).
Mitra, S., Sharma, V. K. & Mukhopadhyay, R. Diffusion of confined fluids in microporous zeolites and clay materials. Rep. Prog. Phys. 84, 066501 (2021).
Horike, S., Shimomura, S. & Kitagawa, S. Soft porous crystals. Nat. Chem. 1, 695–704 (2009).
Horike, S., Umeyama, D. & Kitagawa, S. Ion conductivity and transport by porous coordination polymers and metal–organic frameworks. Acc. Chem. Res. 46, 2376–2384 (2013).
Furukawa, H., Müller, U. & Yaghi, O. M. "Heterogeneity within order” in metal-organic frameworks. Angew. Chem. Int. Ed. 54, 3417–3430 (2015).
Lim, D.-W. & Kitagawa, H. Proton transport in metal–organic frameworks. Chem. Rev. 120, 8416–8467 (2020).
Krishna, R. Diffusion in porous crystalline materials. Chem. Soc. Rev. 41, 3099–3118 (2012).
Kärger, J., Ruthven, D. M. & Theodorou, D. N. Diffusion in Nanoporous Materials (Wiley-VCH, 2012).
Coudert, F.-X. Recent advances in stimuli-responsive framework materials: Understanding their response and searching for materials with targeted behavior. Coord. Chem. Rev. 539, 216760 (2025).
Odoh, S. O., Cramer, C. J., Truhlar, D. G. & Gagliardi, L. Quantum-chemical characterization of the properties and reactivities of metal-organic frameworks. Chem. Rev. 115, 6051–6111 (2015).
Guyer, R. A. & Kim, H. A. Theoretical model for fluid-solid coupling in porous materials. Phys. Rev. E 91, 042406 (2015).
Jawahery, S. et al. Adsorbate-induced lattice deformation in IRMOF-74 series. Nat. Commun. 8, 13945 (2017).
Gor, G. Y., Huber, P. & Bernstein, N. Adsorption-induced deformation of nanoporous materials—a review. Appl. Phys. Rev. 4, 011303 (2017).
Rogge, S. M., Waroquier, M. & Van Speybroeck, V. Unraveling the thermodynamic criteria for size-dependent spontaneous phase separation in soft porous crystals. Nat. Commun. 10, 4842 (2019).
Kolesnikov, A. L., Budkov, Y. A. & Gor, G. Y. Models of adsorption-induced deformation: ordered materials and beyond. J. Phys.: Condens. Matter 34, 063002 (2022).
Schaper, L. & Schmid, R. Simulating the structural phase transitions of metal-organic frameworks with control over the volume of nanocrystallites. Commun. Chem. 6, 233 (2023).
Kolesnikov, A. L., Shkolin, A. V., Men’shchikov, I. E. & Gor, G. Y. Kinetics of adsorption-induced deformation of microporous carbons. Langmuir 40, 23806–23815 (2024).
Sakata, Y. et al. Shape-memory nanopores induced in coordination frameworks by crystal downsizing. Science 339, 193–196 (2013).
Kärger, J. et al. Microimaging of transient guest profiles to monitor mass transfer in nanoporous materials. Nat. Mater. 13, 333–343 (2014).
Cho, H. S. et al. Extra adsorption and adsorbate superlattice formation in metal-organic frameworks. Nature 527, 503–507 (2015).
Sakaida, S. et al. Crystalline coordination framework endowed with dynamic gate-opening behaviour by being downsized to a thin film. Nat. Chem. 8, 377–383 (2016).
Hosono, N., Terashima, A., Kusaka, S., Matsuda, R. & Kitagawa, S. Highly responsive nature of porous coordination polymer surfaces imaged by in situ atomic force microscopy. Nat. Chem. 11, 109–116 (2019).
Delen, G., Monai, M., Meirer, F. & Weckhuysen, B. M. In situ nanoscale infrared spectroscopy of water adsorption on nanoislands of surface-anchored metal-organic frameworks. Angew. Chem. Int. Ed. 60, 1620–1624 (2021).
Yamada, E. et al. Three-dimensional visualization of adsorption distribution in a single crystalline particle of a metal-organic framework. J. Am. Chem. Soc. 146, 9181–9190 (2024).
Watanabe, S. et al. Size-dependent guest-memory switching of the flexible and robust adsorption characteristics of layered metal-organic frameworks. Sci. Adv. 10, eadr1387 (2024).
Hanikel, N. et al. Evolution of water structures in metal-organic frameworks for improved atmospheric water harvesting. Science 374, 454–459 (2021).
Bavykina, A. et al. Metal–organic frameworks in heterogeneous catalysis: Recent progress, new trends, and future perspectives. Chem. Rev. 120, 8468–8535 (2020).
Kreno, L. E. et al. Metal-organic framework materials as chemical sensors. Chem. Rev. 112, 1105–1125 (2012).
Horcajada, P. et al. Metal-organic frameworks in biomedicine. Chem. Rev. 112, 1232–1268 (2012).
Danowski, W. et al. Unidirectional rotary motion in a metal–organic framework. Nat. Nanotech. 14, 488–494 (2019).
Eshelby, J. D. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. A 241, 376–396 (1957).
Onuki, A. Phase Transition Dynamics (Cambridge University Press, 2002).
Mitsumoto, K. & Takae, K. Elastic heterogeneity governs asymmetric adsorption-desorption in a soft porous crystal. Proc. Natl. Acad. Sci. 120, e2302561120 (2023).
Mitsumoto, K. & Takae, K. Adsorption superlattice stabilized by elastic interactions in a soft porous crystal. Phys. Rev. Res. 6, L012029 (2024).
Krug, J. & Spohn, H. Kinetic Roughening of Growing Surfaces. In Godrèche, C. (ed.) Solids Far From Equilibrium, 479–582 (Cambridge University Press, 1991).
Barabási, A.-L. & Stanley, H. E.Fractal Concepts in Surface Growth (Cambridge University Press, 1995).
Takeuchi, K. A. & Sano, M. Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence. J. Stat. Phys. 147, 853–890 (2012).
Tanaka, T. et al. Mechanical instability of gels at the phase transition. Nature 325, 796–798 (1987).
Matsuo, E. S. & Tanaka, T. Patterns in shrinking gels. Nature 358, 482–485 (1992).
Politi, P., Grenet, G., Marty, A., Ponchet, A. & Villain, J. Instabilities in crystal growth by atomic or molecular beams. Phys. Rep. 324, 271–404 (2000).
Aqua, J. N., Berbezier, I., Favre, L., Frisch, T. & Ronda, A. Growth and self-organization of SiGe nanostructures. Phys. Rep. 522, 59–189 (2013).
López, J. M., Rodríguez, M. A. & Cuerno, R. Superroughening versus intrinsic anomalous scaling of surfaces. Phys. Rev. E 56, 3993–3998 (1997).
Ramasco, J. J., López, J. M. & Rodríguez, M. A. Generic dynamic scaling in kinetic roughening. Phys. Rev. Lett. 84, 2199–2202 (2000).
López, J. M., Castro, M. & Gallego, R. Scaling of local slopes, conservation laws, and anomalous roughening in surface growth. Phys. Rev. Lett. 94, 166103 (2005).
Krause, S. et al. The effect of crystallite size on pressure amplification in switchable porous solids. Nat. Commun. 9, 1573 (2018).
Cogswell, D. A. & Bazant, M. Z. Coherency strain and the kinetics of phase separation in LiFePO4 nanoparticles. ACS Nano 6, 2215–2225 (2012).
Coudert, F.-X., Jeffroy, M., Fuchs, A. H., Boutin, A. & Mellot-Draznieks, C. Thermodynamics of guest-induced structural transitions in hybrid organic-inorganic frameworks. J. Am. Chem. Soc. 130, 14294–14302 (2008).
Wang, F. & Landau, D. P. Efficient, multiple-range random walk algorithm to calculate the density of states. Phys. Rev. Lett. 86, 2050–2053 (2001).
Landau, D. P. & Binder, K. A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, 2014).
Amar, J. G., Lam, P.-M. & Family, F. Groove instabilities in surface growth with diffusion. Phys. Rev. E 47, 3242–3245 (1993).
Edwards, S. F. & Wilkinson, D. R. The surface statistics of a granular aggregate. Proc. R. Soc. Lond. A 381, 17–31 (1982).
Kardar, M., Parisi, G. & Zhang, Y.-C. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986).
Soriano, J. et al. Anomalous roughening of viscous fluid fronts in spontaneous imbibition. Phys. Rev. Lett. 95, 104501 (2005).
Herring, C. Surface Tension as a Motivation for Sintering. In Kingston, W. E. (ed.) The Physics of Powder Metallurgy, 143–179 (MacGraw-Hill, New-York, 1951).
Mullins, W. W. Theory of thermal grooving. J. Appl. Phys. 28, 333–339 (1957).
Wolf, D. E. & Villain, J. Growth with surface diffusion. EPL 13, 389–394 (1990).
Das Sarma, S. & Tamborenea, P. A new universality class for kinetic growth: One-dimensional molecular-beam epitaxy. Phys. Rev. Lett. 66, 325–328 (1991).
Xia, H., Tang, G. & Lan, Y. Numerical analysis of long-range spatial correlations in surface growth. Phys. Rev. E 94, 062121 (2016).
Onuki, A. Theory of phase transition in polymer gels. Adv. Polym. Sci. 109, 63–121 (1993).
Acknowledgements
The authors would like to thank Kazumasa A. Takeuchi and Tetsuo Yamaguchi for valuable discussions. This work was supported by Inamori Research Grants and the JSPS KAKENHI Grant No. JP24K00594, JP25H01978, and JP25K17354.
Author information
Authors and Affiliations
Contributions
K.M. and K.T. conceived the project, K.M. performed numerical simulations and analysed the data, and K.M. and K.T. discussed the results and wrote the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Communications Physics thanks thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
About this article
Cite this article
Mitsumoto, K., Takae, K. Elastic heterogeneity governs anomalous dynamic scaling in a soft porous crystal. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02508-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005-026-02508-8
