Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Hierarchical amorphous ordering in colloidal gelation

Nature Physics volume 19, pages 1171–1177 (2023)Cite this article

Subjects

Abstract

Amorphous gels are formed in various soft matter and biomatter when phase separation is dynamically arrested without crystallization. The dynamic arrest in gelation has been attributed to glass transition, but a microscopic foundation is lacking. To address this issue, we experimentally study the gelation of a sticky colloid model using the single-particle-level dynamic analysis of in situ confocal microscopy observations. We show that, during gelation, individual colloids first aggregate into tetrahedra, which then grow to form poly-tetrahedral clusters. Subsequently, pentagonal bipyramids are formed as sets of five tetrahedra, and finally, these pentagonal bipyramids self-catalyse secondary growth to form medium-range amorphous order and arrest dynamics. This hierarchical ordering is primarily driven by local potential energy, not free energy. Thus, the amorphous ordering and arrest mechanism fundamentally differ between gels formed by phase separation and glasses formed homogeneously. These findings will deepen our understanding of two types of amorphous solid, namely, gel and glass.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Our experimental setup.
Fig. 2: Pentagonal bipyramids in a dilute gel.
Fig. 3: Hierarchical ordering of pentagonal bipyramids in gelation.
Fig. 4: Elementary process forming a pentagonal bipyramid cluster and its impact on dynamical arrest.

Similar content being viewed by others

Data availability

The data supporting the findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The computer codes used in this study are available from the corresponding authors upon reasonable request.

References

  1. Verhaegh, N. A. M., Asnaghi, D., Lekkerkerker, H. N. W., Giglio, M. & Cipelletti, L. Transient gelation by spinodal decomposition in colloid-polymer mixtures. Physica A 242, 104–118 (1997).

    ADS  Google Scholar 

  2. Anderson, V. J. & Lekkerkerker, H. N. Insights into phase transition kinetics from colloid science. Nature 416, 811–815 (2002).

    ADS  Google Scholar 

  3. Poon, W. The physics of a model colloid–polymer mixture. J. Phys.: Condens. Matter 14, R859 (2002).

    ADS  Google Scholar 

  4. Campbell, A. I., Anderson, V. J., van Duijneveldt, J. S. & Bartlett, P. Dynamical arrest in attractive colloids: the effect of long-range repulsion. Phys. Rev. Lett. 94, 208301 (2005).

    ADS  Google Scholar 

  5. Manley, S. et al. Glasslike arrest in spinodal decomposition as a route to colloidal gelation. Phys. Rev. Lett. 95, 238302 (2005).

    ADS  Google Scholar 

  6. Zaccarelli, E. Colloidal gels: equilibrium and non-equilibrium routes. J. Phys.: Condens. Matter 19, 323101 (2007).

    Google Scholar 

  7. Lu, P. J. et al. Gelation of particles with short-range attraction. Nature 453, 499–503 (2008).

    ADS  Google Scholar 

  8. Royall, C. P., Williams, S. R., Ohtsuka, T. & Tanaka, H. Direct observation of a local structural mechanism for dynamic arrest. Nat. Mater. 7, 556–561 (2008).

    ADS  Google Scholar 

  9. Lekkerkerker, H. N. & Tuinier, R. Depletion interaction. in Colloids and the Depletion Interaction 57–108 (Springer, 2011).

  10. Lu, P. J. & Weitz, D. A. Colloidal particles: crystals, glasses, and gels. Annu. Rev. Condens. Matter Phys. 4, 217–233 (2013).

    ADS  Google Scholar 

  11. Whitaker, K. A. et al. Colloidal gel elasticity arises from the packing of locally glassy clusters. Nat. Commun. 10, 2237 (2019).

    ADS  Google Scholar 

  12. Matter, F., Luna, A. L. & Niederberger, M. From colloidal dispersions to aerogels: how to master nanoparticle gelation. Nano Today 30, 100827 (2020).

    Google Scholar 

  13. Mezzenga, R., Schurtenberger, P., Burbidge, A. & Michel, M. Understanding foods as soft materials. Nat. Mater. 4, 729–740 (2005).

    ADS  Google Scholar 

  14. Tanaka, H. & Nishikawa, Y. Viscoelastic phase separation of protein solutions. Phys. Rev. Lett. 95, 078103 (2005).

    ADS  Google Scholar 

  15. Vekilov, P. G. Nucleation. Cryst. Growth Des 10, 5007–5019 (2010).

    Google Scholar 

  16. Banani, S. F., Lee, H. O., Hyman, A. A. & Rosen, M. K. Biomolecular condensates: organizers of cellular biochemistry. Nat. Rev. Mol. Cell Biol. 18, 285–298 (2017).

    Google Scholar 

  17. Shin, Y. & Brangwynne, C. P. Liquid phase condensation in cell physiology and disease. Science 357, eaaf4382 (2017).

    Google Scholar 

  18. Berry, J., Brangwynne, C. P. & Haataja, M. Physical principles of intracellular organization via active and passive phase transitions. Rep. Prog. Phys. 81, 046601 (2018).

    ADS  Google Scholar 

  19. Tanaka, H. Viscoelastic phase separation in biological cells. Commun. Phys. 5, 167 (2022).

    Google Scholar 

  20. Lu, P. J., Conrad, J. C., Wyss, H. M., Schofield, A. B. & Weitz, D. A. Fluids of clusters in attractive colloids. Phys. Rev. Lett. 96, 028306 (2006).

    ADS  Google Scholar 

  21. Furukawa, A. & Tanaka, H. Key role of hydrodynamic interactions in colloidal gelation. Phys. Rev. Lett. 104, 245702 (2010).

    ADS  Google Scholar 

  22. Varga, Z., Wang, G. & Swan, J. The hydrodynamics of colloidal gelation. Soft Matter 11, 9009–9019 (2015).

    ADS  Google Scholar 

  23. Tsurusawa, H., Leocmach, M., Russo, J. & Tanaka, H. Direct link between mechanical stability in gels and percolation of isostatic particles. Sci. Adv. 5, eaav6090 (2019).

    ADS  Google Scholar 

  24. Tateno, M. & Tanaka, H. Numerical prediction of colloidal phase separation by direct computation of Navier–Stokes equation. npj Comput. Mater. 5, 40 (2019).

    ADS  Google Scholar 

  25. Tsurusawa, H., Arai, S. & Tanaka, H. A unique route of colloidal phase separation yields stress-free gels. Sci. Adv. 6, eabb8107 (2020).

    ADS  Google Scholar 

  26. Tsurusawa, H., Russo, J., Leocmach, M. & Tanaka, H. Formation of porous crystals via viscoelastic phase separation. Nat. Mater. 16, 1022–1028 (2017).

    ADS  Google Scholar 

  27. Rouwhorst, J., Ness, C., Stoyanov, S., Zaccone, A. & Schall, P. Nonequilibrium continuous phase transition in colloidal gelation with short-range attraction. Nat. Commun. 11, 3558 (2020).

    ADS  Google Scholar 

  28. Tateno, M., Yanagishima, T. & Tanaka, H. Microscopic structural origin behind slowing down of colloidal phase separation approaching gelation. J. Chem. Phys. 156, 084904 (2022).

    ADS  Google Scholar 

  29. Hsiao, L. C., Newman, R. S., Glotzer, S. C. & Solomon, M. J. Role of isostaticity and load-bearing microstructure in the elasticity of yielded colloidal gels. Proc. Natl Acad. Sci. USA 109, 16029–16034 (2012).

    ADS  Google Scholar 

  30. Zhang, S. et al. Correlated rigidity percolation and colloidal gels. Phys. Rev. Lett. 123, 058001 (2019).

    ADS  Google Scholar 

  31. Nabizadeh, M. & Jamali, S. Life and death of colloidal bonds control the rate-dependent rheology of gels. Nat. Commun. 12, 4274 (2021).

    ADS  Google Scholar 

  32. Sciortino, F., Bansil, R., Stanley, H. E. & Alstrøm, P. Interference of phase separation and gelation: a zeroth-order kinetic model. Phys. Rev. E 47, 4615 (1993).

    ADS  Google Scholar 

  33. Tanaka, H. Viscoelastic phase separation. J. Phys.: Condens. Matter 12, R207 (2000).

    ADS  Google Scholar 

  34. Prasad, V., Semwogerere, D. & Weeks, E. R. Confocal microscopy of colloids. J. Phys.: Condens. Matter 19, 113102 (2007).

    ADS  Google Scholar 

  35. Royall, C. P., Faers, M. A., Fussell, S. L. & Hallett, J. E. Real space analysis of colloidal gels: triumphs, challenges and future directions. J. Phys.: Condens. Matter 33, 453002 (2021).

    ADS  Google Scholar 

  36. Zhang, T. H., Klok, J., Tromp, R. H., Groenewold, J. & Kegel, W. K. Non-equilibrium cluster states in colloids with competing interactions. Soft Matter 8, 667–672 (2012).

    ADS  Google Scholar 

  37. Richard, D., Hallett, J., Speck, T. & Royall, C. P. Coupling between criticality and gelation in ‘sticky’ spheres: a structural analysis. Soft Matter 14, 5554–5564 (2018).

    ADS  Google Scholar 

  38. Doye, J. P., Wales, D. J. & Berry, R. S. The effect of the range of the potential on the structures of clusters. J. Chem. Phys. 103, 4234–4249 (1995).

    ADS  Google Scholar 

  39. Sedgwick, H., Egelhaaf, S. & Poon, W. Clusters and gels in systems of sticky particles. J. Phys.: Condens. Matter 16, S4913 (2004).

    ADS  Google Scholar 

  40. Kohl, M., Capellmann, R., Laurati, M., Egelhaaf, S. & Schmiedeberg, M. Directed percolation identified as equilibrium pre-transition towards non-equilibrium arrested gel states. Nat. Commun. 7, 11817 (2016).

    ADS  Google Scholar 

  41. Helgeson, M. E. et al. Homogeneous percolation versus arrested phase separation in attractively-driven nanoemulsion colloidal gels. Soft Matter 10, 3122–3133 (2014).

    ADS  Google Scholar 

  42. Frank, F. C. Supercooling of liquids. Proc. R. Soc. Lond. A 215, 43–46 (1952).

    ADS  Google Scholar 

  43. Reichert, H. et al. Observation of five-fold local symmetry in liquid lead. Nature 408, 839–841 (2000).

    ADS  Google Scholar 

  44. Sheng, H., Luo, W., Alamgir, F., Bai, J. & Ma, E. Atomic packing and short-to-medium-range order in metallic glasses. Nature 439, 419–425 (2006).

    ADS  Google Scholar 

  45. Hu, Y., Li, F., Li, M., Bai, H. & Wang, W. Five-fold symmetry as indicator of dynamic arrest in metallic glass-forming liquids. Nat. Commun. 6, 8310 (2015).

    ADS  Google Scholar 

  46. Tanaka, H., Tong, H., Shi, R. & Russo, J. Revealing key structural features hidden in liquids and glasses. Nat. Rev. Phys. 1, 333–348 (2019).

    Google Scholar 

  47. Yang, Y. et al. Determining the three-dimensional atomic structure of an amorphous solid. Nature 592, 60–64 (2021).

    ADS  Google Scholar 

  48. Yuan, Y. et al. Three-dimensional atomic packing in amorphous solids with liquid-like structure. Nat. Mater. 21, 95–102 (2022).

    MathSciNet  ADS  Google Scholar 

  49. Meng, G., Arkus, N., Brenner, M. P. & Manoharan, V. N. The free-energy landscape of clusters of attractive hard spheres. Science 327, 560–563 (2010).

    ADS  Google Scholar 

  50. Weitz, D. & Oliveria, M. Fractal structures formed by kinetic aggregation of aqueous gold colloids. Phys. Rev. Lett. 52, 1433 (1984).

    ADS  Google Scholar 

  51. Lin, M. et al. Universality in colloid aggregation. Nature 339, 360–362 (1989).

    ADS  Google Scholar 

  52. Bonacci, F. et al. Contact and macroscopic ageing in colloidal suspensions. Nat. Mater. 19, 775–780 (2020).

    ADS  Google Scholar 

  53. Bonacci, F., Chateau, X., Furst, E. M., Goyon, J. & Lemaître, A. Yield stress aging in attractive colloidal suspensions. Phys. Rev. Lett. 128, 018003 (2022).

    ADS  Google Scholar 

  54. Koeze, D. J., Hong, L., Kumar, A. & Tighe, B. P. Elasticity of jammed packings of sticky disks. Phys. Rev. Research 2, 032047 (2020).

    ADS  Google Scholar 

  55. Schwen, E. M., Ramaswamy, M., Cheng, C.-M., Jan, L. & Cohen, I. Embedding orthogonal memories in a colloidal gel through oscillatory shear. Soft Matter 16, 3746–3752 (2020).

    ADS  Google Scholar 

  56. Tanaka, H. Bond orientational order in liquids: towards a unified description of water-like anomalies, liquid-liquid transition, glass transition, and crystallization. Eur. Phys. J. E 35, 113 (2012).

    Google Scholar 

  57. Tong, H. & Tanaka, H. Structural order as a genuine control parameter of dynamics in simple glass formers. Nat. Commun. 10, 5596 (2019).

    ADS  Google Scholar 

  58. Tong, H. & Tanaka, H. Role of attractive interactions in structure ordering and dynamics of glass-forming liquids. Phys. Rev. Lett. 124, 225501 (2020).

    ADS  Google Scholar 

  59. Leocmach, M. & Tanaka, H. Roles of icosahedral and crystal-like order in the hard spheres glass transition. Nat. Commun. 3, 974 (2012).

    ADS  Google Scholar 

  60. Malins, A., Eggers, J., Royall, C. P., Williams, S. R. & Tanaka, H. Identification of long-lived clusters and their link to slow dynamics in a model glass former. J. Chem. Phys. 138, 12A535 (2013).

    Google Scholar 

  61. Hallett, J. E., Turci, F. & Royall, C. P. Local structure in deeply supercooled liquids exhibits growing lengthscales and dynamical correlations. Nat. Commun. 9, 3272 (2018).

    ADS  Google Scholar 

  62. Auer, S. & Frenkel, D. Prediction of absolute crystal-nucleation rate in hard-sphere colloids. Nature 409, 1020–1023 (2001).

    ADS  Google Scholar 

  63. Nguyen, V., Schoemaker, F., Blokhuis, E. & Schall, P. Measurement of the curvature-dependent surface tension in nucleating colloidal liquids. Phys. Rev. Lett. 121, 246102 (2018).

    ADS  Google Scholar 

  64. Bosma, G. et al. Preparation of monodisperse, fluorescent PMMA–latex colloids by dispersion polymerization. J. Colloid Interface Sci. 245, 292–300 (2002).

    ADS  Google Scholar 

  65. Klein, S. M., Manoharan, V. N., Pine, D. J. & Lange, F. F. Preparation of monodisperse PMMA microspheres in nonpolar solvents by dispersion polymerization with a macromonomeric stabilizer. Colloid Polym. Sci. 282, 7–13 (2003).

    Google Scholar 

  66. Crocker, J. C. & Grier, D. G. Methods of digital video microscopy for colloidal studies. J. Colloid Interface Sci. 179, 298–310 (1996).

    ADS  Google Scholar 

  67. Leocmach, M. Colloids: stable 2015. Zenodo https://doi.org/10.5281/zenodo.31286 (2015).

Download references

Acknowledgements

H. Tanaka acknowledges receipt of Grants-in-Aid for Specially Promoted Research (JSPS KAKENHI grant no. JP20H05619) and Scientific Research (A) (JSPS KAKENHI grant no. JP18H03675) from the Japan Society for the Promotion of Science (JSPS).

Author information

Authors and Affiliations

  1. Institute of Industrial Science, University of Tokyo, Tokyo, Japan

    Hideyo Tsurusawa & Hajime Tanaka

  2. Research Center for Advanced Science and Technology, University of Tokyo, Tokyo, Japan

    Hajime Tanaka

Authors
  1. Hideyo Tsurusawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hajime Tanaka
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

H. Tsurusawa and H. Tanaka conceived the project. H. Tanaka supervised the project. H. Tsurusawa performed the experiments and analysed the data. Both authors discussed the results and wrote the manuscript.

Corresponding authors

Correspondence to Hideyo Tsurusawa or Hajime Tanaka.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Confirmation of ‘stress-free’ gelation.

In-situ data of a dilute gel sample with ϕc = 0.053 and cp = 2.2 mg/g is analysed. a, The averaged contact number, 〈Nc〉, is plotted as a function of time. The dotted line indicates the isostatic condition of 〈Nc〉 = 6. This sample became isostatic at t = 2070 s. The region of t ≥ 2070 s is coloured in yellow. b, The radius gyration of the largest cluster (\({L}_{\max }\)) is plotted as a function of time. Lsys is half the system’s size. If we set the percolation threshold as \({L}_{\max }/{L}_{{{{\rm{sys}}}}}=0.8\), this sample percolated at t = 5990 s (magenta arrow). If we employ a more severe threshold of \({L}_{\max }/{L}_{{{{\rm{sys}}}}}=0.95\), the percolation time shifts to 12000 s. c, The edge-to-edge distances of the biggest cluster along X, Y, or Z directions are plotted as a function of time. These distances are normalised by the full length of the system (2Lsys). The biggest cluster spanned the system along the X, Y, and Z direction at t = 5000 s, 4640 s, and 7250 s, respectively. Due to the finite size effect, the percolation time has arbitrariness in an approximate range from 5000 s to 12000 s. Regardless of the percolation threshold, percolation occurs ‘after’ 〈Nc〉 overwhelms 6. Thus, this gel sample is classified as a stress-free gel25.

Extended Data Fig. 2 Three-dimensional visualisations of the early stage of gelation.

Typical 3D images reconstructed from a gel sample with ϕc = 0.053 and cp = 2.2 mg/g. In the visualisations by cluster length, small dots show the particles in small clusters with their mass below 15. For each cluster, cluster length is calculated as its radius gyration. P is the fraction of the symmetric order in all the particles (see Methods). At t = 0 s, colloids start phase separation. At t = 180 s, the first pentagonal bipyramid is formed. At t = 420 s, the fraction of tetrahedra overwhelms 50%. The late stage is shown in the following figure.

Extended Data Fig. 3 Three-dimensional visualisations of the late stage of gelation.

At t = 1380 s, the fraction of pentagonal bipyramid clusters overwhelms the fraction of isolated pentagonal bipyramids. At t = 5990 s, the radius gyration of the largest cluster reaches the percolation threshold of \({L}_{\max }/{L}_{{{{\rm{sys}}}}}=0.8\).

Extended Data Fig. 4 Optimisation of the threshold parameters to identify pentagonal bipyramids.

a, Radial distribution function, g(r), calculated from a gel sample. The plot is obtained by averaging over 10 timesteps (from t = 9230 s to t = 10040 s). We experimentally determine the diameter of our colloids (σ) as 2150 nm such that g(r) becomes maximum at r/σ = 1. b, g(r) zoomed near the first minimum. We set the bond threshold as r/σ = 1.295. c, The histogram of the angles between two connecting bonds (θbond). d, The number of regular pentagons detected by our algorithm. e, The number of pentagonal bipyramids detected by our algorithm. In d and e, we fix as δθplane = 10∘ and δθrot = 10∘. The bond threshold, dbond, and the angular threshold, δθedge, are varied. See Supplementary Methods about the parameters’ optimisation. A gel sample with ϕc = 0.053 and cp = 2.2 mg/g at t = 10040 s is analysed.

Extended Data Fig. 5 Impact of the planar and angular criteria on detecting five-fold symmetry.

To evaluate the net contribution of the planar and angular criteria, we directly compare our method with the topological cluster classification (TCC) method. For the ‘TCC-like’ analysis, the planar and angular criteria are disabled by the following settings: δθplane = δθrot = δθedge = 180∘. For our method, the optimal thresholds are set as follows: δθplane = 10∘, δθrot = 10∘, and δθedge = 12∘. a, The number of the regular pentagons detected by the two methods. dbond is the upper threshold to define a bond. b, The number of pentagonal bipyramids detected by the two methods. c, The distributions of the planar, rotation, and edge angles detected by the two methods. In c, the data is analysed in the pentagonal bipyramids that are detected by the threshold of dbond/σ = 1.295 (see the arrows in b). A gel sample with ϕc = 0.053 and cp = 2.2 mg/g at t = 10040 s is analysed.

Extended Data Fig. 6 Formation of pentagonal mono-pyramids.

a, 3D models of pentagonal pyramids. b, The number of each pyramid type is plotted as a function of time. More than 98% of pentagonal planes form bipyramids during the gelation process. An alternative model against Fig. 3b is the structural evolution between (i) a pentagon (not-pyramid), (ii) a pentagonal mono-pyramid, and (iii) a pentagonal bipyramid. However, the experimental data rejects this alternative model. A gel sample with ϕc = 0.053 and cp = 2.2 mg/g at t = 10040 s is analysed.

Extended Data Fig. 7 The role of poly-tetrahedra in increasing the contact number of a gel.

For each time step, the fractions of the tetrahedron, 2-tetrahedra, and 3-tetrahedra are plotted as functions of 〈Nc〉. The fractions of the poly-tetrahedral orders monotonically increase as 〈Nc〉. These fractions are saturated before 〈Nc〉 approaches 6. A gel sample with ϕc = 0.053 and cp = 2.2 mg/g at t = 10040 s is analysed.

Extended Data Fig. 8 High-speed microscopy to access elementary process of the local ordering.

High-speed microscopy acquired time-lapse 3D images every 1.5 s during gelation. The temporal resolution of the high-speed microscopy is even faster than the Brownian motion time of our colloids (τB ~ 2 s). We applied the high-speed microscopy to a gel sample with ϕc = 0.045 and cp = 0.96 mg/g. Since high-speed microscopy limits the scanning volume, the number of particles in the limited 3D volume is typically 400. a, Temporal evolution of the fractions of the tetrahedron, 2-tetrahedra, 3-tetrahedra, a pentagonal bipyramid, and an octahedron in the gel. b, 3D visualisations of the high-speed microscopy at t = − 135 s, 0 s, 157.5 s, 252 s, 390 s, 750 s, and 990 s. Blue particles form pentagonal bipyramids. A small cluster is marked in magenta at t = 157.5 s and 252 s. Figure 4a-f analyse the transformation of this marked cluster. Figure 4g analyses the trajectory data of 400 time steps from t = 390 s to 990 s.

Extended Data Fig. 9 Universality of the hierarchical ordering in the arrested cluster phase separation.

a and b, Temporal changes of the fractions of tetrahedra, 2-tetrahedra, 3-tetrahedra, octahedra, and pentagonal in the formation of non-percolating clusters (a) and dilute gels (b). c, Averaged contact number 〈Nc〉 and the fraction of pentagonal bipyramids are plotted for all time steps and all the phase separation samples in our study.

Supplementary information

Supplementary Information

Supplementary Video 1

Elementary process forming a pentagonal bipyramid cluster. This is the full video for Fig. 4a–f. The video starts at t = 157.5 s and ends at t = 252.0 s. One frame corresponds to 1.5 s. To improve visibility, we correct both drift and rotation of the cluster so that the centre of mass and direction of the rotational axis (the red one) are fixed. A pentagonal ring is coloured blue only when it satisfies all the topological, planar and rotational criteria.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tsurusawa, H., Tanaka, H. Hierarchical amorphous ordering in colloidal gelation. Nat. Phys. 19, 1171–1177 (2023). https://doi.org/10.1038/s41567-023-02063-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-023-02063-x

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing