Beyond the percolation universality class: the vertex split model for tetravalent lattices
We propose a statistical model defined on tetravalent three-dimensional lattices in general and the three-dimensional diamond network in particular where the splitting of randomly selected nodes leads to a spatially disordered network, with decreasing degree of connectivity. The terminal state, that...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2015-01-01
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| Series: | New Journal of Physics |
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| Online Access: | https://doi.org/10.1088/1367-2630/17/4/043061 |
| _version_ | 1797751262068867072 |
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| author | Susan Nachtrab Matthias J F Hoffmann Sebastian C Kapfer Gerd E Schröder-Turk Klaus Mecke |
| author_facet | Susan Nachtrab Matthias J F Hoffmann Sebastian C Kapfer Gerd E Schröder-Turk Klaus Mecke |
| author_sort | Susan Nachtrab |
| collection | DOAJ |
| description | We propose a statistical model defined on tetravalent three-dimensional lattices in general and the three-dimensional diamond network in particular where the splitting of randomly selected nodes leads to a spatially disordered network, with decreasing degree of connectivity. The terminal state, that is reached when all nodes have been split, is a dense configuration of self-avoiding walks on the diamond network. Starting from the crystallographic diamond network, each of the four-coordinated nodes is replaced with probability p by a pair of two edges, each connecting a pair of the adjacent vertices. For all values $0\leqslant p\leqslant 1$ the network percolates, yet the fraction f _p of the system that belongs to a percolating cluster drops sharply at p _c = 1 to a finite value $f_{p}^{c}$ . This transition is reminiscent of a percolation transition yet with distinct differences to standard percolation behaviour, including a finite mass $f_{p}^{c}\gt 0$ of the percolating clusters at the critical point. Application of finite size scaling approach for standard percolation yields scaling exponents for $p\to {{p}_{c}}$ that are different from the critical exponents of the second-order phase transition of standard percolation models. This transition significantly affects the mechanical properties of linear-elastic realizations (e.g. as custom-fabricated models for artificial bone scaffolds), obtained by replacing edges with solid circular struts to give an effective density ϕ . Finite element methods demonstrate that, as a low-density cellular structure, the bulk modulus K shows a cross-over from a compression-dominated behaviour, $K(\phi )\propto {{\phi }^{\kappa }}$ with $\kappa \approx 1$ , at p = 0 to a bending-dominated behaviour with $\kappa \approx 2$ at p = 1. |
| first_indexed | 2024-03-12T16:45:56Z |
| format | Article |
| id | doaj.art-cd650987ec5344788603c7d8c5d14208 |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:45:56Z |
| publishDate | 2015-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-cd650987ec5344788603c7d8c5d142082023-08-08T14:16:02ZengIOP PublishingNew Journal of Physics1367-26302015-01-0117404306110.1088/1367-2630/17/4/043061Beyond the percolation universality class: the vertex split model for tetravalent latticesSusan Nachtrab0Matthias J F Hoffmann1Sebastian C Kapfer2Gerd E Schröder-Turk3Klaus Mecke4Institut für Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7, D-91058 Erlangen, GermanyInstitut für Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7, D-91058 Erlangen, GermanyInstitut für Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7, D-91058 Erlangen, GermanyInstitut für Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7, D-91058 Erlangen, Germany; School of Engineering & Information Technology, Mathematics & Statistics, Murdoch University, 90 South St, Murdoch, WA 6150, AustraliaInstitut für Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7, D-91058 Erlangen, GermanyWe propose a statistical model defined on tetravalent three-dimensional lattices in general and the three-dimensional diamond network in particular where the splitting of randomly selected nodes leads to a spatially disordered network, with decreasing degree of connectivity. The terminal state, that is reached when all nodes have been split, is a dense configuration of self-avoiding walks on the diamond network. Starting from the crystallographic diamond network, each of the four-coordinated nodes is replaced with probability p by a pair of two edges, each connecting a pair of the adjacent vertices. For all values $0\leqslant p\leqslant 1$ the network percolates, yet the fraction f _p of the system that belongs to a percolating cluster drops sharply at p _c = 1 to a finite value $f_{p}^{c}$ . This transition is reminiscent of a percolation transition yet with distinct differences to standard percolation behaviour, including a finite mass $f_{p}^{c}\gt 0$ of the percolating clusters at the critical point. Application of finite size scaling approach for standard percolation yields scaling exponents for $p\to {{p}_{c}}$ that are different from the critical exponents of the second-order phase transition of standard percolation models. This transition significantly affects the mechanical properties of linear-elastic realizations (e.g. as custom-fabricated models for artificial bone scaffolds), obtained by replacing edges with solid circular struts to give an effective density ϕ . Finite element methods demonstrate that, as a low-density cellular structure, the bulk modulus K shows a cross-over from a compression-dominated behaviour, $K(\phi )\propto {{\phi }^{\kappa }}$ with $\kappa \approx 1$ , at p = 0 to a bending-dominated behaviour with $\kappa \approx 2$ at p = 1.https://doi.org/10.1088/1367-2630/17/4/043061percolationnetworks in phase transitionsmechanical properties of solids |
| spellingShingle | Susan Nachtrab Matthias J F Hoffmann Sebastian C Kapfer Gerd E Schröder-Turk Klaus Mecke Beyond the percolation universality class: the vertex split model for tetravalent lattices New Journal of Physics percolation networks in phase transitions mechanical properties of solids |
| title | Beyond the percolation universality class: the vertex split model for tetravalent lattices |
| title_full | Beyond the percolation universality class: the vertex split model for tetravalent lattices |
| title_fullStr | Beyond the percolation universality class: the vertex split model for tetravalent lattices |
| title_full_unstemmed | Beyond the percolation universality class: the vertex split model for tetravalent lattices |
| title_short | Beyond the percolation universality class: the vertex split model for tetravalent lattices |
| title_sort | beyond the percolation universality class the vertex split model for tetravalent lattices |
| topic | percolation networks in phase transitions mechanical properties of solids |
| url | https://doi.org/10.1088/1367-2630/17/4/043061 |
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