Mean-field elastic moduli of a three-dimensional, cell-based vertex model
The mechanics of a foam depends on bubble shape, bubble network topology, and the material at hand, be it metallic or polymeric, for example. While the shapes of bubbles are the consequence of minimizing surface area for a given bubble volume in a space-filling packing, if one were to consider biolo...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2024-01-01
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| Series: | New Journal of Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1088/1367-2630/ad3099 |
| _version_ | 1797213079375708160 |
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| author | Kyungeun Kim Tao Zhang J M Schwarz |
| author_facet | Kyungeun Kim Tao Zhang J M Schwarz |
| author_sort | Kyungeun Kim |
| collection | DOAJ |
| description | The mechanics of a foam depends on bubble shape, bubble network topology, and the material at hand, be it metallic or polymeric, for example. While the shapes of bubbles are the consequence of minimizing surface area for a given bubble volume in a space-filling packing, if one were to consider biological tissue as a foam-like material, the zoology of observed shapes of cells perhaps motivates different energetic contributions. Building on earlier two-dimensional results, here, we focus on a mean field approach to obtain the elastic moduli for an ordered, three-dimensional vertex model. We use the space-filling shape of a truncated octahedron and an energy functional containing a restoring surface area spring and a restoring volume spring. The tuning of the three-dimensional shape index exhibits a rigidity transition via a compatible–incompatible transition. Specifically, for smaller shape indices, both the target surface area and volume cannot be achieved, while beyond some critical value of the three-dimensional shape index, they can be, resulting in a zero-energy state. In addition to analytically determining the location of the transition in mean field, we find that the rigidity transition and the elastic moduli depend on the parameterization of the cell shape. This parameterization effect is more pronounced in three dimensions than in two dimensions given the zoology of shapes that a polyhedron can take on (as compared to a polygon). We also uncover nontrivial dependence of the elastic moduli on the deformation protocol in which some deformations result in affine motion of the vertices, while others result in nonaffine motion. Such dependencies on the shape parameterization and deformation protocol give rise to a nontrivial shape landscape and, therefore, nontrivial mechanical response even in the absence of topology changes. |
| first_indexed | 2024-04-24T10:52:34Z |
| format | Article |
| id | doaj.art-d25075fcf3174bc4b6d48f04e82ead2f |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-04-24T10:52:34Z |
| publishDate | 2024-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-d25075fcf3174bc4b6d48f04e82ead2f2024-04-12T10:37:33ZengIOP PublishingNew Journal of Physics1367-26302024-01-0126404300910.1088/1367-2630/ad3099Mean-field elastic moduli of a three-dimensional, cell-based vertex modelKyungeun Kim0https://orcid.org/0000-0001-5211-434XTao Zhang1https://orcid.org/0000-0002-7636-0913J M Schwarz2Department of Polymer Science and Engineering, Shanghai Jiao Tong University , Shanghai 200240, People’s Republic of China; Physics Department, Syracuse University , Syracuse, NY 13244, United States of AmericaDepartment of Polymer Science and Engineering, Shanghai Jiao Tong University , Shanghai 200240, People’s Republic of ChinaPhysics Department, Syracuse University , Syracuse, NY 13244, United States of America; Indian Creek Farm , Ithaca, NY 14850, United States of AmericaThe mechanics of a foam depends on bubble shape, bubble network topology, and the material at hand, be it metallic or polymeric, for example. While the shapes of bubbles are the consequence of minimizing surface area for a given bubble volume in a space-filling packing, if one were to consider biological tissue as a foam-like material, the zoology of observed shapes of cells perhaps motivates different energetic contributions. Building on earlier two-dimensional results, here, we focus on a mean field approach to obtain the elastic moduli for an ordered, three-dimensional vertex model. We use the space-filling shape of a truncated octahedron and an energy functional containing a restoring surface area spring and a restoring volume spring. The tuning of the three-dimensional shape index exhibits a rigidity transition via a compatible–incompatible transition. Specifically, for smaller shape indices, both the target surface area and volume cannot be achieved, while beyond some critical value of the three-dimensional shape index, they can be, resulting in a zero-energy state. In addition to analytically determining the location of the transition in mean field, we find that the rigidity transition and the elastic moduli depend on the parameterization of the cell shape. This parameterization effect is more pronounced in three dimensions than in two dimensions given the zoology of shapes that a polyhedron can take on (as compared to a polygon). We also uncover nontrivial dependence of the elastic moduli on the deformation protocol in which some deformations result in affine motion of the vertices, while others result in nonaffine motion. Such dependencies on the shape parameterization and deformation protocol give rise to a nontrivial shape landscape and, therefore, nontrivial mechanical response even in the absence of topology changes.https://doi.org/10.1088/1367-2630/ad3099vertex modelstissue mechanicsfoam-like materials |
| spellingShingle | Kyungeun Kim Tao Zhang J M Schwarz Mean-field elastic moduli of a three-dimensional, cell-based vertex model New Journal of Physics vertex models tissue mechanics foam-like materials |
| title | Mean-field elastic moduli of a three-dimensional, cell-based vertex model |
| title_full | Mean-field elastic moduli of a three-dimensional, cell-based vertex model |
| title_fullStr | Mean-field elastic moduli of a three-dimensional, cell-based vertex model |
| title_full_unstemmed | Mean-field elastic moduli of a three-dimensional, cell-based vertex model |
| title_short | Mean-field elastic moduli of a three-dimensional, cell-based vertex model |
| title_sort | mean field elastic moduli of a three dimensional cell based vertex model |
| topic | vertex models tissue mechanics foam-like materials |
| url | https://doi.org/10.1088/1367-2630/ad3099 |
| work_keys_str_mv | AT kyungeunkim meanfieldelasticmoduliofathreedimensionalcellbasedvertexmodel AT taozhang meanfieldelasticmoduliofathreedimensionalcellbasedvertexmodel AT jmschwarz meanfieldelasticmoduliofathreedimensionalcellbasedvertexmodel |