Mathematical modelling of electrical conductivity of films based on metallic nanowires

The talk presents the results of dissertation research aimed at improving methods for predictive modelling of the DC electrical conductivity of transparent conducting films based on metallic nanowires. The study proposes a mathematical model of a random nanowire network represented as an undirected, irregular, weighted graph. The graph’s edges are categorised into groups corresponding to different sources of resistance: the resistance of the wires themselves and the contact resistance between them. For each group, the edge weights obey their own distribution law. A numerical method was developed to determine these edge-weight distributions, enabling the transition from the geometric and physical parameters of nanowires to an equivalent random resistor network. Special attention is paid to a model of a mixture of conducting nanorings and nanorods. When calculating electrical conductivity using computer simulation, both wire resistance and contact resistance were taken into account. It is shown that the electrical conductivity of such a mixture is practically insensitive to the proportions of rings and rods when the wire resistance and contact resistance are of the same order of magnitude. Systems based on arcs (half-rings), which correspond to realistic methods for producing nanorings by bending and closing nanowires, were also investigated. A modified wall-following algorithm is proposed for identifying the current-carrying backbone of a percolation cluster on planar graphs. Its key advantage is that it does not require traversing every edge of the graph, thereby significantly reducing computational costs, especially near the percolation threshold. The algorithm enables the extraction of the backbone, the identification of bridges, and the analysis of the current-carrying structure of the network. Seamless networks based on crack patterns are also examined. A phenomenological model for the arrangement of cell centroids and a recursive algorithm that reproduces the hierarchical crack structure are proposed. Numerical experiments were conducted, including the solution of sparse systems of linear equations with approximately 10⁶ equations, and a statistical analysis of the results was performed. Dependences of electrical conductivity on concentration, filler shape, and contact resistance were identified.