Energy Formulation for Infinite Structures: Order Parameter for Percolation, Critical Bonds, and Power-Law Scaling of Contact-Based Transport

Investigating heterogeneous materials’ microstructure, often simulated using periodic images, is crucial for understanding their physical traits. We propose a generic spring-based representation for periodic two-component structures. The equilibrium energy in this framework serves as an order parameter, offering an analytical expression for wrapping and introducing the concept of critical bonds. We show that these minimum bonds for depercolation can be efficiently detected. The number of critical bonds scales with system size, accurately capturing contact-based transport’s scaling. This approach holds potential to analyze functional robustness of networks.

Figure 1

Energy of spring system as an order parameter. Two different cases, (a) and (c), that were generated using two-dimensional DLCA with the same parameters but different initial conditions, are presented. The structures are juxtaposed against their horizontal and vertical images, and we observe that (a) is connected to its own image, forming an infinite structure and (c) spans multiple simulation boxes, including the neighbouring diagonal one, but is not connected to itself. Considering the bonds between various sites to be Hookean springs with zero equilibrium length, the positions for the respective configurations, at equilibrium in (b),(d), are calculated using Eq. (1), and while (a) deforms to a strained structure as shown in (b), the individual clusters in (b) collapse to point sized structures (d).

Figure 2

Visual representation of the proposed algorithm to estimate $n_c$. We consider an example of a single two-dimensional cluster percolating in both directions as shown in (a), wherein the bonds which pass through the boundaries of the simulation box are highlighted with dotted lines. Given an initial configuration (a), the equilibrium position of the sites is calculated using Eq. (1), and the new bond lengths are calculated based on this new configuration as shown in (b). Following this the longest bond in (b) is deleted (highlighted in red) and the new positions of sites is recalculated using Eq. (1) leading to another configuration shown in (c). The same procedure is repeated, which results in a linelike structure as depicted in (d), which indicates that this structure is not percolating in the transverse direction anymore. Finally, deleting the longest bond for the configuration in (d), leads to complete collapse of the structure to a point, and therefore depercolation of the system in (e). In this particular case, ${\widehat{n}}_c=n_c=3$.

Figure 3

Comparison of scaling exponents. We find that the values of $⟨{\widehat{n}}_c⟩\sigma /L^{D-1}$ versus $\varphi -{\varphi }_c$ on a log-log scale fit with a straight line. The results for two-parametric fits are represented by filled symbols, for three-parametric fits by open symbols (Supplemental Material, Sec. S6 [21]) and the gray shaded regions represent the bounds suggested by previously established experiments and theory. We study two different classes of systems. (a) Disordered insulator-conductor composites, wherein the system is modeled using RSP and electrical conductivity follows a power law above the threshold [25]. For 2D we find $1.4\le \gamma \le 1.57$ while the value reported in literature is $\simeq 1.3$ both from theory and experiments [26], and in 3D we find $1.87\le \gamma \le 1.92$, and while $\gamma \approx 1.8$ [27] from experiments and $\gamma \approx 2$ from theory [15, 25]. (b) Silica aerogels, wherein the microstructure is modeled using three-dimensional DLCA [28]. It has been established experimentally that Young’s modulus scales with a power-law exponent of 3.6 with respect to density [29, 30], and there have been numerous attempts to establish this result numerically [31, 32]. We find that for the theoretical value of ${\varphi }_c=0$, $\gamma \approx 3.6$ for all values of $L$. For the distribution of ${\widehat{n}}_c$ values see Supplemental Material, Fig. S1 [21].

Figure 4

Functional fragility of complex networks. We consider the Watts-Strogatz model [34] with $N=50$ nodes, wherein the connections have been rewired with probability $p=0.5$ and with average degree being for (a) $K=4$ and (b) $K=2$, and with three sources (red) and three sinks (blue), wherein these sets of nodes have been selected randomly with the additional caveat that no source is directly connected to a sink. We observe that for (a) the algorithm correctly determines that the most efficient method compromising the network’s functionality is by isolating all its sources (or sinks), while for (b), it can be achieved by attacking particular edges of the relevant groups, in a combination with some intermediate edges. We found that for an average over 100 ensembles the ratio of targeted attacks on the sources and sinks were, for (a) 95.02% and (b) 57.57%, suggesting that for Watts-Strogatz model the functional robustness is more reliant on the overall structure for lower values of $K$. The experiment was repeated for $N=1000$ nodes and a proportional number of sinks and sources, and found the results to be consistent, indicating that the results are system size invariant. Additionally, when the connections of sources and sinks are safeguarded via a penalty, the functional robustness of the network increases with increasing $p$; results from the same are presented in Supplemental Material, Sec. S7 [21].