Beyond the percolation universality class: the vertex split model for tetravalent lattices - Nachtrab, Susan; Hoffmann, Matthias; Kapfer, Sebastian; Schröder-Turk, Gerd; Mecke, Klaus (New Journal of Physics, 2015)
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 <= p <= 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) > 0 of the percolating clusters at the critical point. Application of finite size scaling approach for standard percolation yields scaling exponents for p -> 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 phi. 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) alpha phi(k) with kappa approximate to 1, at p = 0 to a bending-dominated behaviour with kappa approximate to 2 at p = 1.
Keywords: percolation; networks in phase transitions; mechanical properties of solids