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 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 fp of the system that belongs to a percolating cluster drops sharply at pc = 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 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 ( φ ) ∝ φ κ ?> with κ ≈ 1 ?> , at p = 0 to a bending-dominated behaviour with κ ≈ 2 ?> at p = 1.


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 ⩽ ⩽ p 0 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 0 p c 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 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, ϕ ϕ ∝ κ K ( ) with κ ≈ 1, at p = 0 to a bending-dominated behaviour with κ ≈ 2 at p = 1.
Percolation is a fundamental model of statistical physics and probability theory [1], with a wealth of scientific and engineering applications [2]. The fundamental question of percolation theory is the existence of connected components whose size is of the order of the system size (percolating clusters), in disordered structures that result from randomly inserting or removing local structural elements. It owes its generality, and hence importance, partially to the strong universality of the percolation transition. In the majority of lattice and continuum models, the transition from non-percolating to percolating structures is a continuous second-order phase transition in the insertion (or deletion) probability p, characterized by the same critical exponents that are independent of lattice type, symmetry, coordination, particle shape, etc [1]. Exceptions are non-equilibrium directed percolation models [3,4] and negative-weight percolation [5], both with different critical exponents, and explosive percolation where a bias for the formation of small clusters leads to a first order transition [6,7] or at least to unusual finite size scaling [8].
We propose a simple statistical model, here referred to as vertex split model or linked loop model, defined for the three-dimensional diamond network. (The diamond network is the crystallographic net with cubic symmetry Fd m 3 consisting of a single type of edge and vertex. Four edges meet at every vertex, forming tetrahedral angles [9].) Rather than deleting spatial elements from the diamond network, such as bonds or vertices, the random operation consists of reducing the vertex coordination by replacing, with probability p, each four-coordinated vertices with pairs of two-coordinated vertices, see figures 1 and 2. This induces a transition from a fully coordinated crystalline network at p = 0 to a network filled densely with self-avoiding random walks. The two names are motivated by two different perspectives; with reference to the ordered fullyconnected crystalline diamond network at p = 0, vertex splitting is the operation that leads to the transition studied here. From the alternative perspective of the state at p = 1, represented by a dense set of self-avoiding walks, the model may be defined as the random insertion of 'links' between adjacent, infinite or finite loops with probability − p (1 ). Each unsevered vertex of the diamond network has four edges connecting the vertex to four distinct neighbour vertices. Each vertex of the diamond network is split (or severed) with probability p, that is, the fourcoordinated node is replaced by two two-coordinated nodes slightly displaced from the position of the original four-coordinated node, see figure 2. When splitting a node, the three possible configurations for neighbour pairs are selected with equal probability. Note that the parameter p is the probability to degrade a four-link, opposite to the conventional use of p in bond/site percolation models as the probability to create a bond or site.
1.07 0.04, significantly different from the site percolation value −0.9318. 3 Note however that the network itself has cubic symmetry and the severing process induces no anisotropy. In fact, in terms of the effective linear-elastic properties, the system becomes more isotropic with increasing p, such that the difference between the two shear moduli (in a system with initially cubic symmetry) vanishes for large p, see insert in figure 3(B) in [10]. This mechanical isotropy relates presumably also to higher structural isotropy, in a statistical sense.
was not investigated here); system size is measured in the number L 3 of unit cells, each comprising eight vertices of the network, see figure 2. Figure 3 shows the fraction of percolating edges f p as function of p, for different system sizes L, suggesting a transition at p c = 1. Over the entire interval ∈ p [0, 1]we find the system to represent percolating configurations. However, p c = 1 represents a critical point where the fraction f p of edges belonging to percolating    The insert of figure 4 shows the number of percolating clusters at = = p p 1 c as a function of system size L. Importantly, in contrast to bond or site percolation, the number of percolating clusters at p c is not 1, but grows (within the limits of our numerical resolution) linearly with system size, ∝ N L p . The three possible types of unbranched self-avoiding paths for systems with lateral periodic boundary conditions are closed loops, percolating clusters (traversing the system in z-direction) and u-turns, i.e. clusters that return to the same end (bottom or top) of the network from where they emanated. The probability that a cluster emanating from one of the 4L 2 sites at z = 0 percolates appears to be inversely proportional to the system height, ∝ L 1 . As any u-turn cluster occupies two sites at z = 0, N p must be even or zero.
Figures 4-8 support the claim that the transition at p c = 1 is a phase transition with scaling behaviour given by power-law decay for the characteristic quantities listed in table 1. The scaling exponents are significantly different from the critical exponents of conventional bond or site percolation, substantiating  [1,11,12], with the percolation threshold for the diamond network reproduced from [13]. For β and ν we have verified that our implementation of the site percolation model reproduces these results. Note that we have not been able to obtain consistent estimates for the exponent ν prescribing the decay of the correlation lengths, for the vertex split model. The two estimates based on finite size scaling of quantities at = p p c yield values around 1.0 whereas the estimate based on finite size scaling of quantities at < p 1 yields ν = 0.54, differing by a factor of very close to 2. We speculate that this discrepancy is likely due to the multiplicity of the percolating cluster at = p p c , or to the possibility that the transition is not of second order. Note the discussion relating to β in item (iii) in the main text. For the scaling exponents of the vertex split model, error bars combine variances of the data (statistical error of fit) with variations when changing fitting ranges and system size. All data is for periodic boundary conditions in the lateral directions; see [14] for data for open boundary conditions.

Site
Vertex split Figure   Threshold  the claim that the transition of the vertex split model is different from the universality class of standard percolation. We comment on some aspects of the numerical extraction of these exponents from the numerical data. First, the observation that for = p p c , = f L ( ) const p shows no statistically significant dependence on L is possible without having to resort to finite size scaling ( figure 4); even for small systems, such as L = 100, we observe only small differences in f p from the value for large systems, at = = p p 1 c . Note however the complication that f p does not drop to 0 but to a finite value > f 0 p c at p c and that a non-percolating phase (characterized by f p = 0) does not exist; this raises the question if the appropriate order parameter is f p (L) or the difference Second, the exponent τ, describing the decay of the cluster size distribution 〈 〉 n s ( ) s at = p p c is substantially harder to determine, requiring the use of finite size scaling, despite large system sizes. The data in figure 6, is determined by using finite size scaling for n s : the system-size independent value n s is extracted from the simulation data n s (L) for a system of size L by = + p p c , this is not the case in the model studied here. Therefore, the question of how to treat percolating clusters is in principle important for the determination of τ. However, we find that for systems with the periodic boundary conditions described here, after the described finite size scaling the value of τ is the same regardless of whether one takes percolating clusters into account for 〈 〉 n L ( ) (vi) In analogy to standard percolation, one may expect the critical behaviour to be independent of the type of underlying network; this expectation should be verified by an analysis of node severing of other fourcoordinated networks, such as the crystalline nbo network [9] or the network of plateau edges in random isotropic or sheared foams [19,20].

Mechanical properties
The remainder of this paper addresses mechanical properties of linear-elastic realizations of the networks with split (or severed) vertices. In porous or cellular structures, the existence of a solid percolating cluster is a prerequisite for mechanical stability, that is, for finite values of the effective linear-elastic moduli. The relationship between percolation critical behaviour and effective elastic properties (those relevant for sample sizes much larger than the micro-structural length scale) is well-known, leading to a power-law decay of the effective elastic moduli near p c [2,12]. We employ a voxel-based finite element method [21,22] to evaluate the effective linear-elastic properties of network solids based on the vertex split model (Some preliminary results, for ≪ p p c far from the percolation critical point, have been published in [10]). Figure 9 shows that near p c , the effective bulk modulus K (the resistance to hydrostatic compression) is commensurate with a power-law decay, ∝ | − | K p p c f c , with an exponent ≈ f 3.0 c , significantly different from the known exponent f c = 3.75 [12] for site percolation (An analysis of site percolation with the FEM scheme used yields = ± f 3.6 0.1 c ). The change in network structure that occurs as p varies from 0 to 1 is reflected in the density dependence of the linear elastic bulk modulus. It is frequently observed that the effective elastic moduli scale as power-laws in the solid volume fraction ϕ of the cellular structure; specifically, for the limit ϕ → 0 of thin beams, the bulk modulus follows ϕ ∝ κ K with κ = 1 and the shear moduli ϕ ∝ γ G with γ = 2 [23]; note that structures with Figure 10. For fixed p, the effective bulk modulus K and the shear moduli G 1 and G 2 obey power-laws as function of solid volume fraction ϕ, see insert. The exponents for the shear moduli, ϕ ∝ γ G i i with i = 1, 2 are found to be close to the literature value 2, and constant as function of p. By contrast, the exponent for the bulk modulus ϕ ∝ κ K changes from the expected value κ ≈ 1 at p = 0 to κ ≈ 2 for ≈ p 1. The insert shows that, near p c = 1, the data follows a power-law with exponent ≈3.0 (determined by straight-line fitting to all data for ∈ p [0.1, 0.5]), different from the site percolation value f c = 3.75 [12].
cubic symmetry, such as the crystallographic diamond network, have three independent elastic moduli, the bulk modulus K and two shear moduli G 1 and G 2 . For the vertex split model, figure 10 shows that the effective exponent κ of the bulk modulus varies from a value near 1 (as expected) at p = 0 to a value close to 2 when all nodes are disconnected at p = 1. The exponents of the shear moduli remain close to the expected value of 2.
This behaviour is somewhat rationalized by the observation that, in ordered cellular structures in the thin beam limit, linear behaviour of elastic moduli is associated with strut compression being the dominant deformation mode, whereas quadratic behaviour is associated with strut bending or torsion [24][25][26]. The network solids corresponding to the vertex split model appear to undergo a transition from being compressiondominated when fully four-coordinated at p = 0 to being bending-dominated in the terminal state (at p = 1) which corresponds to a dense set of self-avoiding polymers.
In conclusion, we have demonstrated that randomly severing the four-coordinated vertices of a diamond network leads to a transition, manifest in the fraction of clusters that are percolating. The transition, which is reminiscent of a percolation transition yet with substantially different behaviour to conventional bond/site percolation, occurs at p c = 1 when all nodes have been split. While the analysis of this paper has clearly demonstrated that the transition does not follow the critical behaviour of standard bond/site percolation, more research is needed to gain a complete understanding of the critical behaviour of this model.