Monte Carlo simulations of electrical percolation in multicomponent thin films with nanofillers

Effect of shape anisotropy on percolation of aligned and overlapping objects on lattices

We investigate the percolation transition of aligned, overlapping, anisotropic shapes on lattices. Using the recently proposed lattice version of excluded volume theory, we show that shape-anisotropy leads to some intriguing consequences regarding the percolation behavior of anisotropic shapes. We consider a prototypical anisotropic shape-rectangle-on a square lattice and show that, for rectangles of width unity (sticks), the percolation threshold is a monotonically decreasing function of the stick length, whereas, for rectangles of width greater than two, it is a monotonically increasing function. Interestingly, for rectangles of width two, the percolation threshold is independent of its length. We show that this independence of threshold on the length of a side holds for d-dimensional hypercubiods as well as for specific integer values for the lengths of the remaining sides. The limiting case of the length of the rectangles going to infinity shows that the limiting threshold value is finite and depends upon the width of the rectangle. This "continuum" limit with the lattice spacing tending to zero only along a subset of the possible directions in d dimensions results in a "semicontinuum" percolation system. We show that similar results hold for other anisotropic shapes and lattices in different dimensions. The critical properties of the aligned and overlapping rectangles are evaluated using Monte Carlo simulations. We find that the threshold values given by the lattice-excluded volume theory are in good agreement with the simulation results, especially for larger rectangles. We verify the isotropy of the percolation threshold and also compare our results with models where rectangles of mixed orientation are allowed. Our simulation results show that alignment increases the percolation threshold. The calculation of critical exponents places the model in the standard percolation universality class. Our results show that shape anisotropy of the aligned, overlapping percolating units has a marked influence on the percolation properties, especially when a subset of the dimensions of the percolation units is made to diverge.

A subbands study on the resistivity of field-effect CNT-based piezoresistive nanocomposites

In this paper, an analytical model based on the percolation theory has been developed to predict the subbands effect on the effective electrical resistivity of carbon nanotubes (CNT)-based polymer nanocomposites. The CNTs are considered as randomly distributed or aligned channel material in the polymer transmitting electrons through tunneling. The tunneling effect takes into account the electron transmission between each connected pair of CNTs to evaluate electrical resistivity. The modeling approach contains two steps of primary prediction of resistivity and further calculation of CNTs' displacements and subsequent change of the resistance. A good agreement is found between the analytical model predictions and experimental data when the tunneling behavior was considered in the percolation transition region. The effect of CNT diameter, orientation state, and subbands on the resistivity has been investigated. The results depict that subbands increment is a collateral benefit to the aspect ratio in decreasing the resistivity. The analytical results demonstrate that a random CNT dispersion leads to a decreased piezoresistivity, while an increased strain range depicts a more non-linear behavior.

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.

Engineering microcracks in MWCNT/elastomer bilayers for high-performance stretchable sensor development

Stretchable strain sensors in motion detection, health monitoring, and human-machine interfaces are limited by device sensitivity, linearity, hysteresis, stability, and reproducibility in addition to stretchability. Engineering defect structures in sensing material is an effective approach in modulating the material's physical properties, particularly those associated with mechanical responses. Here, we demonstrate that bilayers of carbon nanotubes deposited on an elastomer substrate are mechanically coupled. The microcrack size, density, and distribution in the nanotube thin film can be engineered through uniaxial tensile training to exhibit highly tunable and stable piezoresistive responses with sensitivity, linearity, range, and reproducibility. These responses far exceeding those in uniform metallic films, patterned structures, and composites. In addition, numerical analyses performed on a two-dimensional network model of the cracked nanotube film provide quantitative explanations of how crack configuration, and evolvement under strain, lead to the significant enhancements in stretchable sensor performance using current bilayer structures.

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.

Improved Stretchable and Sensitive Fe Nanowire-Based Strain Sensor by Optimizing Areal Density of Nanowire Network

Flexible strain sensors, when considering high sensitivity and a large strain range, have become a key requirement for current robotic applications. However, it is still a thorny issue to take both factors into consideration at the same time. Here, we report a sandwich-structured strain sensor based on Fe nanowires (Fe NWs) that has a high GF (37–53) while taking into account a large strain range (15–57.5%), low hysteresis (2.45%), stability, and low cost with an areal density of Fe NWs of 4.4 mg/cm². Additionally, the relationship between the contact point of the conductive network, the output resistance, and the areal density of the sensing unit is analyzed. Microscopically, the contact points of the conductive network directly affect the sensor output resistance distribution, thereby affecting the gauge factor (GF) of the sensor. Macroscopically, the areal density and the output resistivity of the strain sensor have the opposite percolation theory, which affects its linearity performance. At the same time, there is a positive correlation between the areal density and the contact point: when the stretching amount is constant, it theoretically shows that the areal density affects the GF. When the areal density reaches this percolation threshold range, the sensing performance is the best. This will lay the foundation for rapid applications in wearable robots.

Conductance-strain behavior in silver-nanowire composites: network properties of a tunable strain sensor

Highly flexible and conductive nano-composite materials are promising candidates for stretchable and flexible electronics. We report on the strain-resistance relation of a silver-nanowire photopolymer composite during repetitive stretching. Resistance measurements reveal a gradual change of the hysteretic resistance curves towards a linear and non-hysteretic behavior. Furthermore, a decrease in resistance and an increase in electrical sensitivity to strain over the first five stretching cycles can be observed. Sensitivity gauge factors between 10 and 500 at 23% strain were found depending on the nanowire concentration and stretching cycle. We model the electrical behavior of the investigated silver nanowire composites upon repetitive stretching considering the strain induced changes in the local force distribution within the polymer matrix and the tunnel resistance between the nanowires by using a Monte Carlo method.

Simulation tool for predicting and optimizing the performance of nanoparticle based strain sensors

In this work a Monte-Carlo tool simulating platinum nanoparticle (NP) based strain-sensors, on flexible substrates, is presented. The tool begins by randomly placing the NPs on the simulation area, with the ability to tune the NP surface coverage. After the calculation of the conductive paths that were generated in the previous step, the whole system is represented with an equivalent circuit; the NPs and the NP clusters act as nodes and the inter-particle gaps as resistances. The effective resistance is then calculated with the use of a Laplacian Matrix, which has proven extremely effective in significantly reducing the overall computational time. The simulation results are then benchmarked with experimental measurements from actual strain-sensing devices. The software is capable of predicting the strain-sensitivity for different NP sizes as well as surface coverages, emerging as a powerful computational tool for design-optimization of NP based devices in polymeric substrates, while it could well be extended to other nanocomposite materials used in flexible or stretchable electronic applications.

Fabrication, Functionalization, and Application of Carbon Nanotube-Reinforced Polymer Composite: An Overview

A novel class of carbon nanotube (CNT)-based nanomaterials has been surging since 1991 due to their noticeable mechanical and electrical properties, as well as their good electron transport properties. This is evidence that the development of CNT-reinforced polymer composites could contribute in expanding many areas of use, from energy-related devices to structural components. As a promising material with a wide range of applications, their poor solubility in aqueous and organic solvents has hindered the utilizations of CNTs. The current state of research in CNTs-both single-wall carbon nanotubes (SWCNT) and multiwalled carbon nanotube (MWCNT)-reinforced polymer composites-was reviewed in the context of the presently employed covalent and non-covalent functionalization. As such, this overview intends to provide a critical assessment of a surging class of composite materials and unveil the successful development associated with CNT-incorporated polymer composites. The mechanisms related to the mechanical, thermal, and electrical performance of CNT-reinforced polymer composites is also discussed. It is vital to understand how the addition of CNTs in a polymer composite alters the microstructure at the micro- and nano-scale, as well as how these modifications influence overall structural behavior, not only in its as fabricated form but also its functionalization techniques. The technological superiority gained with CNT addition to polymer composites may be advantageous, but scientific values are here to be critically explored for reliable, sustainable, and structural reliability in different industrial needs.

Electrical and network properties of flexible silver-nanowire composite electrodes under mechanical strain

Flexible and conductive silver-nanowire photopolymer composites are fabricated and studied under mechanical strain. The initial resistances of the unstretched flexible composites are between 0.27 Ω mm-1 and 1.2 Ω mm-1 for silver-nanowire concentrations between 120 μg cm-2 and 40 μg cm-2. Stretching of the samples leads to an increased resistance by a factor of between 72 for 120 μg cm-2 and 343 for 40 μg cm-2 at elongations of 23%. In order to correlate network morphology and electrical properties, micrographs are recorded during stretching. The Fiber Image Network Evaluation (FINE) algorithm determines morphological silver-nanowire network properties under stretching. For unstretched and stretched samples, an isotropic nanowire network is found with only small changes in fiber orientation. Monte-Carlo simulations on 2D percolation networks of 1D conductive wires and the corresponding network resistance due to tunneling of electrons at nanowire junctions confirm that the elastic polymer matrix under strain exhibits forces in agreement with Hooke's law. By variation of a critical force distribution the resistance curves are accurately reproduced. This results in a model that is dominated by quantum-mechanical tunneling at nanowire junctions explaining the electrical behavior and the sensitivity of nanowire-composites with different filler concentrations under mechanical strain.

Modelling electrical conduction in nanostructure assemblies through complex networks

Carrier transport processes in assemblies of nanostructures rely on morphology-dependent and hierarchical conduction mechanisms, whose complexity cannot be captured by current modelling approaches. Here we apply the concept of complex networks to modelling carrier conduction in such systems. The approach permits assignment of arbitrary connectivity and connection strength between assembly constituents and is thus ideal for nanostructured films, composites and other geometries. Modelling of simplified rod-like nanostructures is consistent with analytical solutions, whereas results for more realistic nanostructure assemblies agree with experimental data and reveal conduction behaviour not captured by previous models. Fitting of ensemble measurements also allows the conduction properties of individual constituents to be extracted, which are subsequently used to guide the realization of transparent electrodes with improved performance. A global optimization process was employed to identify geometries and properties with high potential for transparent conductors. Our intuitive discretization approach, combined with a simple solver tool, allows researchers with little computational experience to carry out realistic simulations.

Synergy effect in hybrid nanocomposites based on carbon nanotubes and graphene nanoplatelets

Hybrid nanocomposites reinforced with a mixture of graphene nanoplatelets (GNPs) and carbon nanotubes (CNTs) have shown improvement in filler dispersion while providing a cost-effective alternative to CNT monofiller composites. Depending on their composition, hybrid composites can exhibit electrical performance superior to either of the constituent monofiller composites due to synergistic effects. In this work, we develop a three-dimensional tunneling-based continuum percolation model for hybrid nanocomposites filled with hardcore particles of elliptical GNPs and cylindrical CNTs. Using Monte Carlo simulations, parametric studies of the filler content, composition and morphology are carried out to analyze the conditions required for synergy in percolation onset and electrical conductivity. Our results suggest that for hybrid systems with well-dispersed fillers, the electrical performance is linked to the number of tunneling junctions per filler inside the percolated network of the nanocomposites. More importantly, hybrid composites filled with specific morphology of GNP and CNT, exhibit synergy in their electrical performance when the monofiller composites of each of those exact fillers have similar percolation onset values. The simulations results are in agreement with relevant experimental data on hybrid nanocomposites.

3D-Printed Sugar Scaffold for High-Precision and Highly Sensitive Active and Passive Wearable Sensors

In this study, a pairing of a previously unidentified 3D printing technique and soft materials is introduced in order to achieve not only high-resolution printed features and flexibility of the 3D-printed materials, but also its light-weight and electrical conductivity. Using the developed technique and materials, high-precision and highly sensitive patient-specific wearable active or passive devices are fabricated for personalized health monitoring. The fabricated biosensors show low density and substantial flexibility because of 3D microcellular network-type interconnected conductive materials that are readily printed using an inkjet head. Using high-resolution 3D scanned body-shape data, on-demand personalized wearable sensors made of the 3D-printed soft and conductive materials are fabricated. These sensors successfully detect both actively changing body strain signals and passively changing signals such as electromyography (EMG), electrodermal activity (EDA), and electroencephalogram EEG. The accurately tailored subject-specific shape of the developed sensors exhibits higher sensitivity and faster real-time sensing performances in the monitoring of rapidly changing human body signals. The newly developed 3D printing technique and materials can be widely applied to various types of wearable, flexible, and light-weight biosensors for use in a variety of inexpensive on-demand and personalized point-of-care diagnostics.

Modeling the electrical resistivity of polymer composites with segregated structures

Hybrid carbon nanotube composites with two different types of fillers have attracted considerable attention for various advantages. The incorporation of micro-scale secondary fillers creates an excluded volume that leads to the increase in the electrical conductivity. By contrast, nano-scale secondary fillers shows a conflicting behavior of the decreased electrical conductivity with micro-scale secondary fillers. Although several attempts have been made in theoretical modeling of secondary-filler composites, the knowledge about how the electrical conductivity depends on the dimension of secondary fillers was not fully understood. This work aims at comprehensive understanding of the size effect of secondary particulate fillers on the electrical conductivity, via the combination of Voronoi geometry induced from Swiss cheese models and the underlying percolation theory. This indicates a transition in the impact of the excluded volume, i.e., the adjustment of the electrical conductivity was measured in cooperation with loading of second fillers with different sizes.