Effect of Local Topological Changes on Resistance in Tunably-Disordered Networks

TL;DR

This paper investigates how local topological changes in tunably disordered 2D networks affect electrical transport, focusing on Delaunay triangulations derived from Lloyd-evolved point clouds. It develops an edge-flip analysis and an analytic Sherman–Morrison-based approximation showing that resistance changes scale with the square voltage drop across the involved edge, $\frac{(V_a - V_b)^2}{I_0^2}$, and are largest near the source and sink. The work contrasts Delaunay flips with Voronoi tessellations, finds strong correlations with local tortuosity, and demonstrates finite-size effects that damp the impact of a single local change on $R_{ extrm{tot}}$. These results emphasize the importance of localized topology for finite samples and have practical implications for designing disordered network metamaterials with tailored transport properties.

Abstract

Disordered materials occur naturally and also provide a broader design space than ordered or crystalline structures. We investigate a two-dimensional disordered network metamaterial constructed from a Delaunay triangulation of an underlying point cloud. Small perturbations in the point cloud induce discrete topological changes. One such change we identify is a Delaunay flip, in which two neighboring Delaunay triangles that form a convex quadrilateral structure with their common edge being one of the two quadrilateral diagonals exchange this diagonal for the other diagonal. These topological changes can cause substantial jumps in the effective resistance measured diagonally across the network, when the change is located near the source or the sink node. The jumps are explained analytically by showing that the change in effective resistance from edge removal or addition depends on the voltage drop across that edge. However, Delaunay flips have less impact on global resistance measurements and in larger networks. These local topological changes are relevant for finite-sized samples and experimentally-measurable properties such as electrical transport. Global characterizations of the network disorder or topology lack the location-specificity of our observed effects on network transport, and thus may be inadequate for predicting certain experimentally measurable transport properties in disordered network metamaterials, highlighting the importance of localized regions in material design.

Figures (10)

• Figure 1: A: Schematic showing the source and sink nodes used to compute $R_{\textrm{eff}}$ diagonally across the network. Nodes are colored by their voltage, computed with Eq. \ref{['eq:LVI']}. B: Effective resistance $R_{\textrm{eff}}$ as a function of Lloyd's iteration number, showing jumps between successive Lloyd's iterations. Each curve follows the evolution of a different initial point cloud.
• Figure 2: A: The progression of $R_{\textrm{eff}}$ under Lloyd's iterations for one simulation, highlighting two jumps and two topological changes. The insets B and C highlight in black the addition of an edge at the bottom left corner of the network; the full networks are shown in panels B and C. The insets D and E highlight in black the Delaunay flip in the top right corner of the network; the full networks are shown in panels D and E.
• Figure 3: Mechanisms of topological changes. Delaunay flip occurs between panel A and C, with panel B illustrating how the Delaunay criterion is violated as a result of one node moving from the gray location. An edge addition at the boundary occurs between panels D and E, where the Voronoi tesselation is shown with blue nodes and gray edges and the Delaunay triangulation for these three complete cells is shown in black.
• Figure 4: A: Simplified lattice network. B: Heatmap where the color of each edge indicates the magnitude of the relative percent change of the effective resistance, $R_{\textrm{eff}}$, when the edge is flipped to the other diagonal of its quadrilateral. Boundary edges are not flipped, and are colored white (i.e. zero change).
• Figure 5: Heatmap where the color of each edge indicates the magnitude of the relative percent change of the effective resistance, $R_{\textrm{eff}}$, when the edge is flipped to the other diagonal of its quadrilateral. Boundary edges and those not part of a convex quadrilateral are not flipped, and are colored white (i.e. zero change).