1. Cohn, H. A conceptual breakthrough in sphere packing. Not. Am. Math. Soc. 64, 102–115 (2017).

   Article  MathSciNet  MATH  Google Scholar 

2. Torquato, S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties 2nd edn (Springer, 2013).

3. Corwin, E. I., Clusel, M., Siemens, A. O. N. & Brujić, J. Model for random packing of polydisperse frictionless spheres. Soft Matter 6, 2949–2959 (2010).

4. Desmond, K. W. & Weeks, E. R. Influence of particle size distribution on random close packing of spheres. Phys. Rev. E 90, 022204 (2014).

   Article  Google Scholar 

5. Torquato, S. Basic understanding of condensed phases of matter via packing models. J. Chem. Phys. 149, 020901 (2018).

   Article  Google Scholar 

6. Seckendorff, J. & Hinrichsen, O. Review on the structure of random packed-beds. Can. J. Chem. Eng. 99, S703–S733 (2021).

7. Culligan, K. A., Wildenschild, D., Christensen, B. S. B., Gray, W. G. & Rivers, M. L. Pore-scale characteristics of multiphase flow in porous media: a comparison of air–water and oil–water experiments. Adv. Water Res. 29, 227–238 (2006).

   Article  Google Scholar 

8. Landry, C. J., Karpyn, Z. T. & Piri, M. Pore-scale analysis of trapped immiscible fluid structures and fluid interfacial areas in oil-wet and water-wet bead packs. Geofluids 11, 209–227 (2011).

   Article  Google Scholar 

9. Roozbahani, M. M., Huat, B. B. K. & Asadi, A. The effect of different random number distributions on the porosity of spherical particles. Adv. Powder Technol. 24, 26–35 (2013).

   Article  Google Scholar 

10. Rauter, M., Viroulet, S., Gylfadottir, S. S., Fellin, W. & Lovholt, F. Granular porous landslide tsunami modelling—the 2014 Lake Askja flank collapse. Nat. Commun. 13, 678 (2022).

    Article  Google Scholar 

11. Doerr, F. J. S. & Florence, A. J. A micro-XRT image analysis and machine learning methodology for the characterisation of multi-particulate capsule formulations. Int. J. Pharm. X 2, 100041 (2020).

    Google Scholar 

12. Averardi, A., Cola, C., Zeltmann, S. E. & Gupta, N. Effect of particle size distribution on the packing of powder beds: a critical discussion relevant to additive manufacturing. Mater. Today Commun. 24, 100964 (2020).

13. Walker, D. M. et al. Self-assembly in a near-frictionless granular material: conformational structures and transitions in uniaxial cyclic compression of hydrogel spheres. Soft Matter 11, 2157–2173 (2015).

    Article  Google Scholar 

14. Zhao, S., Evans, T. M. & Zhou, X. Three-dimensional Voronoi analysis of monodisperse ellipsoids during triaxial shear. Powder Technol. 323, 323–336 (2018).

    Article  Google Scholar 

15. Yi, L. Y., Zou, R. P., Pinson, D., Dong, K. J. & Yu, A. B. An assessment of the mathematical models for estimating the coordination number of the packing of multisized particles. Powder Technol. 379, 58–68 (2021).

    Article  Google Scholar 

16. Zhang, C., Zhao, S., Zhao, J. & Zhou, X. Three-dimensional Voronoi analysis of realistic grain packing: an XCT assisted set Voronoi tessellation framework. Powder Technol. 379, 251–264 (2021).

    Article  Google Scholar 

17. Zhao, S., Zhao, J. & Guo, N. Universality of internal structure characteristics in granular media under shear. Phys. Rev. E 101, 012906 (2020).

    Article  Google Scholar 

18. Wilson-Whitford, S. R., Gao, J., Chiara Roffin, M., Buckley, W. E. & Gilchrist, J. F. Microrollers flow uphill as granular media. Nat. Commun. 14, 5829 (2023).

19. Ketcham, R. A., Meth, C., Hirsch, D. M. & Carlson, W. D. Improved methods for quantitative analysis of three-dimensional porphyroblastic textures. Geosphere 1, 42–59 (2005).

20. Videla, A., Lin, C.-L. & Miller, J. D. Watershed functions applied to a 3D image segmentation problem for the analysis of packed particle beds. Part. Part. Syst. Charact. 23, 237–245 (2006).

    Article  Google Scholar 

21. Manoharan, V. N. Colloidal matter: packing, geometry, and entropy. Science 349, 1253751 (2015).

    Article  MathSciNet  MATH  Google Scholar 

22. Lu, F. et al. Unusual packing of soft-shelled nanocubes. Sci. Adv. 5, eaaw2399 (2019).

    Article  Google Scholar 

23. Al-Raoush, R. Microstructure characterization of granular materials. Physica A 377, 545–558 (2007).

    Article  Google Scholar 

24. Batys, P. & Weroński, P. Porosity and tortuosity of layer-by-layer assemblies of spherical particles. Model. Simul. Mater. Sci. Eng. 22, 065017 (2014).

25. Miyabe, K. New moment equations for chromatography using various stationary phases of different structural characteristics. Anal. Chem. 79, 7457–7472 (2007).

    Article  Google Scholar 

26. Larachi, F. et al. X-ray micro-tomography and pore network modeling of single-phase fixed-bed reactors. Chem. Eng. J. 240, 290–306 (2014).

    Article  Google Scholar 

27. Saadatfar, M., Takeuchi, H., Robins, V., Francois, N. & Hiraoka, Y. Pore configuration landscape of granular crystallization. Nat. Commun. 8, 15082 (2017).

    Article  Google Scholar 

28. Steinhaus, H. Mathematical Snapshots 3rd edn (Dover, 2011).

29. Houdoux, D., Amon, A., Marsan, D., Weiss, J. & Crassous, J. Micro-slips in an experimental granular shear band replicate the spatiotemporal characteristics of natural earthquakes. Commun. Earth Environ. 2, 90 (2021).

30. Darling, N. J. et al. Click by click microporous annealed particle (MAP) scaffolds. Adv. Healthc. Mater. 9, e1901391 (2020).

    Article  Google Scholar 

31. Fang, J. et al. Injectable drug-releasing microporous annealed particle scaffolds for treating myocardial infarction. Adv. Funct. Mater. 30, 2004307 (2020).

    Article  Google Scholar 

32. Griffin, D. R. et al. Activating an adaptive immune response from a hydrogel scaffold imparts regenerative wound healing. Nat. Mater. 20, 560–569 (2021).

    Article  Google Scholar 

33. Dumont, C. M. et al. Aligned hydrogel tubes guide regeneration following spinal cord injury. Acta Biomater. 86, 312–322 (2019).

    Article  Google Scholar 

34. Truong, N. F. et al. Microporous annealed particle hydrogel stiffness, void space size, and adhesion properties impact cell proliferation, cell spreading, and gene transfer. Acta Biomater. 94, 160–172 (2019).

    Article  Google Scholar 

35. Matsiko, A., Gleeson, J. P. & O’Brien, F. J. Scaffold mean pore size influences mesenchymal stem cell chondrogenic differentiation and matrix deposition. Tissue Eng. Part A 21, 486–497 (2015).

    Article  Google Scholar 

36. McWhorter, F. Y., Wang, T., Nguyen, P., Chung, T. & Liu, W. F. Modulation of macrophage phenotype by cell shape. Proc. Natl Acad. Sci. USA 110, 17253–17258 (2013).

    Article  Google Scholar 

37. Zadpoor, A. A. Bone tissue regeneration: the role of scaffold geometry. Biomater. Sci. 3, 231–245 (2015).

    Article  Google Scholar 

38. Denais, C. M. et al. Nuclear envelope rupture and repair during cancer cell migration. Science 352, 353–358 (2016).

    Article  Google Scholar 

39. Werner, M. et al. Surface curvature differentially regulates stem cell migration and differentiation via altered attachment morphology and nuclear deformation. Adv. Sci. 4, 1600347 (2017).

    Article  Google Scholar 

40. Natsui, S., Sawada, A., Nogami, H., Kikuchi, T. & Suzuki, R. O. Topological consideration of 3-D local void structure for static holdup site in packed bed. ISIJ Int. 60, 1453–1460 (2020).

    Article  Google Scholar 

41. Shelepova, E. A., Paschek, D., Ludwig, R. & Medvedev, N. N. Comparing the void space and long-range structure of an ionic liquid with a neutral mixture of similar sized molecules. J. Mol. Liq. 299, 112121 (2020).

42. Li, Z., Wang, Y. H., Chow, J. K., Su, Z. & Li, X. 3D pore network extraction in granular media by unifying the Delaunay tessellation and maximal ball methods. J. Pet. Sci. Eng. 167, 692–701 (2018).

    Article  Google Scholar 

43. van der Linden, J. H., Sufian, A., Narsilio, G. A., Russell, A. R. & Tordesillas, A. A computational geometry approach to pore network construction for granular packings. Comput. Geosci. 112, 133–143 (2018).

    Article  Google Scholar 

44. Schaller, F. M. et al. Non-universal Voronoi cell shapes in amorphous ellipsoid packs. Europhys. Lett. 111, 24002 (2015).

45. Weis, S., Schönhöfer, P. W. A., Schaller, F. M., Schröter, M. & Schröder-Turk, G. E. Pomelo, a tool for computing generic set Voronoi diagrams of aspherical particles of arbitrary shape. EPJ Web Conf. 140, 5–8 (2017).

    Article  Google Scholar 

46. Willems, T. F., Rycroft, C. H., Kazi, M., Meza, J. C. & Haranczyk, M. Algorithms and tools for high-throughput geometry-based analysis of crystalline porous materials. Microporous Mesoporous Mater. 149, 134–141 (2012).

    Article  Google Scholar 

47. Li, X. & Li, X. S. Micro–macro quantification of the internal structure of granular materials. J. Eng. Mech. 135, 641–656 (2009).

    Article  Google Scholar 

48. Fu, P. & Dafalias, Y. F. Relationship between void- and contact normal-based fabric tensors for 2D idealized granular materials. Int. J. Solids Struct. 63, 68–81 (2015).

    Article  Google Scholar 

49. Roozbahani, M. M., Borela, R. & Frost, J. D. Pore size distribution in granular material microstructure. Materials 10, 1237 (2017).

    Article  Google Scholar 

50. Caldwell, A. S., Campbell, G. T., Shekiro, K. M. T. & Anseth, K. S. Clickable microgel scaffolds as platforms for 3D cell encapsulation. Adv. Healthc. Mater. 6, 1700254 (2017).

51. Sideris, E. et al. Particle hydrogels based on hyaluronic acid building blocks. ACS Biomater. Sci. Eng. 2, 2034–2041 (2016).

    Article  Google Scholar 

52. Sheikhi, A. et al. Microfluidic-enabled bottom-up hydrogels from annealable naturally-derived protein microbeads. Biomaterials 192, 560–568 (2019).

    Article  Google Scholar 

53. Khorasani, H. et al. A quantitative approach to scar analysis. Am. J. Pathol. 178, 621–628 (2011).

    Article  Google Scholar 

54. Wershof, E. et al. A FIJI macro for quantifying pattern in extracellular matrix. Life Sci. Alliance 4, e202000880 (2021).

55. Jiang, Z. et al. Machine-learning-revealed statistics of the particle-carbon/binder detachment in lithium-ion battery cathodes. Nat. Commun. 11, 2310 (2020).

    Article  Google Scholar 

56. Blum, H. A. in Models for the Perception of Speech and Visual Form (ed. Wathen-Dunn, W.) 362–380 (MIT Press, 1967).

57. Sherbrooke, E. C., Patrikalakis, N. M. & Brisson, E. An algorithm for the medial axis transform of 3D polyhedral solids. IEEE Trans. Vis. Comput. Graph. 2, 44–61 (1996).

    Article  Google Scholar 

58. Pizer, S. M., Siddiqi, K., Székeley, G., Damon, J. N. & Zucker, S. W. Multiscale medial loci and their properties. Int. J. Comput. Vis. 55, 155–179 (2003).

    Article  MATH  Google Scholar 

59. Hesselink, W. H. & Roerdink, J. B. T. M. Euclidean skeletons of digital image and volume data in linear time by the integer medial axis transform. IEEE Trans. Pattern Anal. Mach. Intell. 30, 2204–2217 (2008).

    Article  Google Scholar 

60. Xiong, Q., Baychev, T. G. & Jivkov, A. P. Review of pore network modelling of porous media: experimental characterisations, network constructions and applications to reactive transport. J. Contam. Hydrol. 192, 101–117 (2016).

    Article  Google Scholar 

61. Shaked, D. & Bruckstein, A. M. Pruning medial axes. Comput. Vis. Image Underst. 69, 156–169 (1998).

    Article  Google Scholar 

62. Lindquist, W. B. & Venkatarangan, A. Investigating 3D geometry of porous media from high resolution images. Phys. Chem. Earth A 25, 593–599 (1999).

    Article  Google Scholar 

63. Silin, D. & Patzek, T. Pore space morphology analysis using maximal inscribed spheres. Physica A 371, 336–360 (2006).

    Article  Google Scholar 

64. Jones, A. C. et al. The correlation of pore morphology, interconnectivity and physical properties of 3D ceramic scaffolds with bone ingrowth. Biomaterials 30, 1440–1451 (2009).

    Article  Google Scholar 

65. Chiang, S.-C. The Euclidean Distance Transform. PhD thesis, Purdue Univ. (1992).

66. Liang, Z., Ioannidis, A. & Chatzis, I. Geometric and topological analysis of three-dimensional porous media: pore space partitioning based on morphological skeletonization. J. Colloid Interface Sci. 221, 13–24 (2000).

    Article  Google Scholar 

67. Youssef, S. et al. High resolution CT and pore-network models to assess petrophysical properties of homogeneous and heterogeneous carbonates. In SPE/EAGE Reservoir Characterization and Simulation Conference SPE-111427-MS (SPE, 2007).

68. Thomson, P.-R., Hazel, A. & Hier-Majumder, S. The influence of microporous cements on the pore network geometry of natural sedimentary rocks. Front. Earth Sci. https://doi.org/10.3389/feart.2019.00048 (2019).

69. Hormann, K., Baranau, V., Hlushkou, D., Höltzel, A. & Tallarek, U. Topological analysis of non-granular, disordered porous media: determination of pore connectivity, pore coordination, and geometric tortuosity in physically reconstructed silica monoliths. New J. Chem. 40, 4187–4199 (2016).

    Article  Google Scholar 

70. Jiang, Z. et al. Efficient extraction of networks from three-dimensional porous media. Water Resour. Res. 43, W12S03 (2007).

71. Rabbani, A., Ayatollahi, S., Kharrat, R. & Dashti, N. Estimation of 3-D pore network coordination number of rocks from watershed segmentation of a single 2-D image. Adv. Water Res. 94, 264–277 (2016).

    Article  Google Scholar 

72. Huaimin, D., Jianmeng, S., Likai, C., Naser, G. & Weichao, Y. Characteristics of the pore structure of natural gas hydrate reservoir in the Qilian Mountain Permafrost, Northwest China. J. Appl. Geophys. 164, 153–159 (2019).

    Article  Google Scholar 

73. Dong, H. & Blunt, M. J. Pore-network extraction from micro-computerized-tomography images. Phys. Rev. E 80, 036307 (2009).

    Article  Google Scholar 

74. Medvedev, N. N., Voloshin, V. P., Luchnikov, V. A. & Gavrilova, M. L. An algorithm for three-dimensional Voronoi S-network. J. Comput. Chem. 27, 1676–1692 (2006).

    Article  Google Scholar 

75. Joon Lee, C., Kang, Y.-M., Cho, K.-H. & No, K. T. A robust method for searching the smallest set of smallest rings with a path-included distance matrix. Proc. Natl Acad. Sci. USA 106, 17355–17358 (2009).

    Article  Google Scholar 

76. Jones, J. R., Atwood, R. C., Poologasundarampillai, G., Yue, S. & Lee, P. D. Quantifying the 3D macrostructure of tissue scaffolds. J. Mater. Sci. Mater. Med. 20, 463–471 (2009).

    Article  Google Scholar 

77. Bashoor-Zadeh, M., Baroud, G. & Bohner, M. Geometric analysis of porous bone substitutes using micro-computed tomography and fuzzy distance transform. Acta Biomater. 6, 864–875 (2010).

    Article  Google Scholar 

78. Gostick, J. T. Versatile and efficient pore network extraction method using marker-based watershed segmentation. Phys. Rev. E 96, 023307 (2017).

    Article  Google Scholar 

79. Rong, L. W., Dong, K. J. & Yu, A. B. Lattice-Boltzmann computation of hydraulic pore-to-pore conductance in packed beds of uniform spheres. Chem. Eng. Sci. 224, 115798 (2020).

80. Sweijen, T., Hassanizadeh, S. M., Chareyre, B. & Zhuang, L. Dynamic pore-scale model of drainage in granular porous media: the pore-unit assembly method. Water Resour. Res. 54, 4193–4213 (2018).

    Article  Google Scholar 

81. Riley, L., Cheng, P. & Segura, T. Identification and analysis of 3D-pores in packed particulate materials. Code Ocean https://doi.org/10.24433/CO.4876664.v1 (2023).

Download references