Computer Generation of Disordered Networks with Targeted Structural Properties

TL;DR

This work extends the Wooten-Weaire-Winer (WWW) algorithm to generate disordered networks with arbitrary coordination by replacing the traditional bond-bending term with angle-repulsive constraints in a generalized Keating energy and by tuning disorder through a bond-bending constant $\beta$ and a triangular heating profile. A rich 42-metric framework assesses similarity, homogeneity, isotropy, and topology in both direct and reciprocal space, while a feedforward neural network predicts order metrics from algorithm inputs, enabling targeted network generation. The method is validated by statistically reproducing several disordered biophotonic networks responsible for structural color, demonstrating the approach’s capability to control short-range order and to explore structure–property relations, including potential photonic band-gap emergence in disordered networks. The study lays groundwork for linking network topology to optical properties via simulations (e.g., FDTD, PDOS) and for extending the framework with reciprocal-space terms to directly influence scattering phenomena, with broad applicability across materials, biology, and social systems.

Abstract

Disordered spatial networks are model systems that describe structures and interactions across multiple length scales. Scattering and interference of waves in these networks can give rise to structural phase transitions, localization, diffusion, and band gaps. The study of these complex phenomena requires efficient numerical methods to computer-generate disordered networks with targeted structural properties. In the established Wooten-Weaire-Winer algorithm, a series of bond switch moves introduces disorder into an initial network. Conventional strain energies that govern this evolution are limited to 3D networks with coordination numbers of no more than four. We extend the algorithm to arbitrary coordination number statistics by introducing bond repulsion in the Keating strain energy. We tune the degree and type of disorder introduced into initially crystalline networks by varying the bond-bending force constant in the strain energy and the temperature profile. The effects of these variables are analyzed using a list of order metrics that capture both direct and reciprocal space. A feedforward neural network is trained to predict the structural characteristics from the algorithm inputs, enabling targeted network generation. As a case study, we statistically reproduce four disordered biophotonic networks exhibiting structural color. This work presents a versatile method for generating disordered networks with tailored structural properties. It will enable new insights into structure-property relations, such as photonic band gaps in disordered networks.

Figures (44)

• Figure 1: A Monte Carlo move is shown for a 2D, three-valent network. a A chain of four vertices (black) is randomly selected. b A bond switch reconnects the vertices while maintaining all coordination numbers. c The stress introduced by the bond switch is relaxed by translating the vertices.
• Figure 2: In our extended WWW algorithm, we introduce disorder to networks by successively heating, cooling, and quenching them. The quench continues until it is no longer possible to reduce the strain energy through additional Monte Carlo moves.
• Figure 3: a A section of the skeletonized, disordered photonic network of the Pachyrhynchus congestus mirabilis weevil gives rise to blue structural color (PCM blue). b The periodic ctn network has similar coordination number statistics as PCM blue, as shown in c. d and e show sections of the StV green and StV blue biological networks, respectively. f The periodic $\textbf{bcu}_\mathrm{mod}$ network has coordination number statistics similar to those of StV green and StV blue, as shown in g. There StA orange network in h and the periodic $\textbf{pcu}_\mathrm{mod}$ network in k have similar coordination number statistics, as shown in l.
• Figure 4: We applied the extended WWW algorithm to the periodic diamond dia (a), gyroid dia (b), and lcs (c) networks, which have wide photonic band gaps.
• Figure 5: a The number of accepted Monte Carlo moves in the ctn network increases with increasing maximal heating temperature $T_\mathrm{max}$ and with decreasing heating gradient $\Delta T$. The black markers correspond to networks with ten or fewer accepted moves and show that the melting transition begins at $T_\mathrm{max} \,{\gtrsim}\, T_\mathrm{melt}$. All values of $\beta \,{\in}\, [0,10]$ are considered. b The isotropy metric $h_\mathbf{b}$ effectively measures the melting transition of the initial ctn network.