Paper detail

Percolation transition by random vertex splitting of diamond networks?

We propose a statistical model defined on the three-dimensional diamond network 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 of 0 <= p <= 1 the network percolates, yet the fraction fp of the system that belongs to a percolating cluster drops sharply at pc=1 to a finite value. This transition has the signature of a phase transition with scaling exponents for p close to pc that are different from the critical exponents of the second-order phase transition of standard percolation models. As is the case for percolation transitions, this transition significantly affects the mechanical properties of linear-elastic realisations, 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 (with K proportional to phi) at p=0 to a bending-dominated behaviour (with quadratic dependence of K on phi) at p=1.