Beyond the percolation universality class: the vertex split model for tetravalent lattices. (29th April 2015)

• Record Type: Journal Article
• Authors: Nachtrab, Susan; Hoffmann, Matthias J F; Kapfer, Sebastian C; Schröder-Turk, Gerd E; Mecke, Klaus
• Is Part Of: New journal of physics. Volume 17:Number 4(2015:Apr.)
• Publication Date: 2015-04-29
• Subjects: percolation -- networks in phase transitions -- mechanical properties of solids; Physics -- Periodicals; Physics; Periodicals; 530.05
• DOI: 10.1088/1367-2630/17/4/043061
• Languages: English
• ISSNs: 1367-2630

Abstract

Abstract: 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 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 . This transition is reminiscent of a percolation transition yet with distinct differences to standard percolation behaviour, including a finite mass of the percolating clusters at the critical point. Application of finite size scaling approach for standard percolation yields scaling exponents for 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, with, at p = 0 to a bending-dominated behaviour with at p = 1.