Percolation threshold of curved linear objects

Bridge percolation: electrical connectivity of discontinued conducting slabs by metallic nanowires

The properties of nanostructured networks of conductive materials have been extensively studied under the lens of percolation theory. In this work, we introduce a novel type of local percolation phenomenon used to investigate the conduction properties of a new hybrid material that combines sparse metallic nanowire networks and fractured conducting thin films on flexible substrates. This original concept could potentially lead to the design of a novel composite transparent conducting material. Using a complementary approach including formal analytical derivations, Monte Carlo simulations and electrical circuit representation for the modelling of bridged-percolating nanowire networks, we unveil the key relations between linear crack density, nanowire length and network areal mass density that ensure electrical percolation through the hybrid. The proposed theoretical model provides key insights into the conduction mechanism associated with the original concept of bridge percolation in random nanowire networks.

Numerical modeling of the effects of the shape and aspect ratio of 3D curved fiber on the percolation threshold and electrical conductivity of conductive polymer composites

For designing conductive polymer composites (CPCs), understanding how the fiber curvature affects the percolation behavior of curved conductive fibers is essential for determining the effective electrical conductivity σeff of the CPCs. In this work, CPCs were considered as a polymer matrix filled with the random packing of overlapped curved spherocylinders. The geometries of the curved spherocylinders were defined, and inter-curved spherocylinder contact-detecting and system-spanning fiber cluster searching algorithms were developed. The finite-size-scaling method was used to explore how the aspect ratio α and bending central angle θ of a curved spherocylinder affect the percolation threshold ϕc of an overlapped curved spherocylinder system in 3D space. The findings suggest that ϕc decreases as α increases and increases initially before declining as θ increases. An empirical approximation formula was proposed to quantify the effect of the curved spherocylinder's morphology, characterized by the dimensionless excluded volume Vdex of the curved spherocylinder, on ϕc. The new rigorous bound for ϕc of the soft-curved spherocylinder system was further proposed. A random resistor network model was constructed, and the reliability of this model was validated by comparing the simulations and published data. Finally, a fitting formula was developed to assess the impacts of the normalized reduced density (η - ηc)/ηc and Vdex on the σeff of CPCs. A distinct linear correlation between σeff and (η - ηc)/ηc was constructed, denoted as σeff ∼ [(η - ηc)/ηc]t(α,θ). An empirical approximation model was proposed to establish the relationship between the fiber shape and conductivity exponent t. Our study may provide a theoretical hint for the design of CPCs.

Optimizing silver nanowire dimensions by the modification of polyol synthesis for the fabrication of transparent conducting films

Transparent conducting films (TCFs) made by the assembly/deposition of silver nanowires (Ag NWs) are widely used to manufacture flexible electronics such as touch screens, heaters, displays, and organic light-emitting diodes. Controlling the dimensions (length and diameter) of the nanowires is key in obtaining TCFs with the desired optoelectronic properties, namely sheet resistance and optical transparency. This work describes a combined experimental and theoretical investigation on the optimization of the NW dimensions to fabricate high-quality TCFs. Ag NWs of different dimensions are synthesized by the modified polyol method and the average diameter and length of the wires are tailored over a wide range, 35-150 nm and 12-130μm respectively, by controlling the synthesis parameters such as reaction conditions, stabilizing agents, and growth promoters. The synthesized NWs are spin coated on glass substrates to form TCFs. Comparing the films with different lengths, but identical diameters, enabled the quantification of the effect of length on the optoelectronic properties of the TCFs. Similarly, the effect of NW diameter is also studied. A non-uniformity factor is defined to evaluate the uniformity of the TCF and the transmittance of the NW network is shown to be inversely proportional to its area coverage. The sheet conductance versus the normalized number density is plotted for the different concentrations of NWs to extract a conductivity exponent that agrees well with the theoretical predictions. For thin film networks, the relation between the transmittance and sheet resistance provides the percolative figure of merit (FoM) as a fitting parameter. A large FoM is desirable for a good-performing TCF and the synthesis conditions to achieve this are optimized.

Percolation threshold and electrical conductivity of conductive polymer composites filled with curved fibers in two-dimensional space

Quantifying the influence of fiber curvature on the percolation behavior of flexible conductive fiber and further on the electrical conductivity of conductive polymer composites (CPCs) is crucial for the design of CPCs. This study considers CPCs as a random packing of soft curved discorectangles (CDCRs) in a polymer matrix. The geometry of CDCR is developed, and an inter-CDCR contact detection algorithm is used to generate a random packing structure of CDCRs. The effects of aspect ratio α and bending central angles θ of CDCR on the percolation threshold ϕc of the overlapped CDCR system in a two-dimensional plane are then investigated using the finite-size scaling method. The result reveals that ϕc decreases monotonically as α grows and increases monotonically as θ rises. A shape-independent power law formula, denoted as ϕc = 2.2015 A-0.8172dex is developed to quantify the relationship between the Adex and ϕc. A comparison of our numerical simulations, published data, and predictions verifies the reliability and universality of the fitting model. Subsequently, a resistor network searching algorithm (RNSA) is proposed to construct the random resistor network model (RRNM). A power law model, denoted as is developed to evaluate the effects of the normalized reduced density (η - ηc)/ηc on the effective conductivity σeff of CPC. Comparing our predictions with data from the literature and our simulation verifies the reliability of our RNSA and the fitting model. This paper's methodology and findings may provide a theoretical hint for the CPC's design.

Electrical conductance of two-dimensional random percolating networks based on mixtures of nanowires and nanorings: A mean-field approach along with computer simulation

We have studied the electrical conductance of two-dimensional (2D) random percolating networks of zero-width metallic nanowires (a mixture of rings and sticks). We took into account the nanowire resistance per unit length and the junction (nanowire-nanowire contact) resistance. Using a mean-field approximation (MFA) approach, we derived the total electrical conductance of these nanowire-based networks as a function of their geometrical and physical parameters. The MFA predictions have been confirmed by our Monte Carlo (MC) numerical simulations. The MC simulations were focused on the case when the circumferences of the rings and the lengths of the wires were equal. In this case, the electrical conductance of the network was found to be almost insensitive to the relative proportions of the rings and sticks, provided that the wire resistance and the junction resistance were equal. When the junction resistance dominated over the wire resistance, a linear dependency of the electrical conductance of the network on the proportions of the rings and sticks was observed.

Tuning the electro-optical properties of nanowire networks

Conductive and transparent metallic nanowire networks are regarded as promising alternatives to Indium-Tin-Oxides (ITOs) in emerging flexible next-generation technologies due to their prominent optoelectronic properties and low-cost fabrication. The performance of such systems closely relies on many geometrical, physical, and intrinsic properties of the nanowire materials as well as the device-layout. A comprehensive computational study is essential to model and quantify the device's optical and electrical responses prior to fabrication. Here, we present a computational toolkit that exploits the electro-optical specifications of distinct device-layouts, namely standard random nanowire network and transparent mesh pattern structures. The target materials for transparent conducting electrodes of this study are aluminium, gold, copper, and silver nanowires. We have examined a variety of tunable parameters including network area fraction, length to diameter aspect ratio, and nanowires angular orientations under different device designs. Moreover, the optical extinction efficiency factors of each material are estimated by two approaches: Mie light scattering theory and finite element method (FEM) algorithm implemented in COMSOL®Multiphysics software. We studied various nanowire network structures and calculated their respective figures of merit (optical transmittance versus sheet resistance) from which insights on the design of next-generation transparent conductor devices can be inferred.

Random nanowire networks: Identification of a current-carrying subset of wires using a modified wall follower algorithm

We mimic random nanowire networks by the homogeneous, isotropic, and random deposition of conductive zero-width sticks onto an insulating substrate. The number density (the number of objects per unit area of the surface) of these sticks is supposed to exceed the percolation threshold, i.e., the system under consideration is a conductor. To identify any current-carrying part (the backbone) of the percolation cluster, we have proposed and implemented a modification of the well-known wall follower algorithm-one type of maze solving algorithm. The advantage of the modified algorithm is its identification of the whole backbone without visiting all the edges. The complexity of the algorithm depends significantly on the structure of the graph and varies from O(sqrt[N_{V}]) to Θ(N_{V}). The algorithm has been applied to backbone identification in networks with different number densities of conducting sticks. We have found that (i) for number densities of sticks above the percolation threshold, the strength of the percolation cluster quickly approaches unity as the number density of the sticks increases; (ii) simultaneously, the percolation cluster becomes identical to its backbone plus simplest dead ends, i.e., edges that are incident to vertices of degree 1. This behavior is consistent with the presented analytical evaluations.