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Topological Properties of a Self-Assembled Electrical Network via Ab Initio Calculation

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Topological Properties of a Self-Assembled Electrical Network via Ab Initio Calculation

C Stephenson et al. Sci Rep.

Abstract

Interacting electrical conductors self-assemble to form tree like networks in the presence of applied voltages or currents. Experiments have shown that the degree distribution of the steady state networks are identical over a wide range of network sizes. In this work we develop a new model of the self-assembly process starting from the underlying physical interaction between conductors. In agreement with experimental results we find that for steady state networks, our model predicts that the fraction of endpoints is a constant of 0.252, and the fraction of branch points is 0.237. We find that our model predicts that these scaling properties also hold for the network during the approach to the steady state as well. In addition, we also reproduce the experimental distribution of nodes with a given Strahler number for all steady state networks studied.

Conflict of interest statement

The authors declare no competing financial interests.

Figures

Figure 1
Figure 1
Left: Experimental steady state for N = 512. Right: Numerically calculated state after t = 120 s for N = 289.
Figure 2
Figure 2. Top: Number of termini Δ1 as a function of total number of conductors N after t = 120 s.
Bottom: Number of termini Δ1 as a function of number of conductors connected to the ground electrode M during the formation process out of total N = 324. In both plots the slope of the black line is the experimentally measured value 0.252.
Figure 3
Figure 3
Top: Number of branch points B as a function of total number of conductors N after t = 120 s. Bottom: Number of branch nodes B as a function of number of conductors connected to the ground electrode M during the formation process out of total N = 324. In both plots the slope of the black line the experimentally measured value 0.237.
Figure 4
Figure 4. Number of nodes σj with Strahler number j as a function of total number of nodes.
Black lines are the experimentally observed relations.

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