Percolation Probability Research Articles

Effect of CNT additives on the electrical properties of derived nanocomposites (experimentally and numerical investigation)

In this work, two simulations models have been developed to study the electrical percolation and the electrical conductivity of epoxy-based nanocomposite containing Multi-walled Carbon Nanotubes. The models are based on resistor-model and finite element analysis. The former was evaluated using MATLAB code and the finite element analysis using DIGIMAT software. The maximum tunnelling distance and its influence on the percolation probability and final electrical conductivity were studied. Electrical measurements on the samples were conducted for numerical validation. The experimental data showed a percolation achievement around 2 wt%, which was confirmed in the numerical simulations. This study provides evidence of the effectiveness of the resistor model and finite element method approach to predict the electrical conductivity of nanocomposites.

Exact results for average cluster numbers in bond percolation on infinite-length lattice strips.

We calculate exact analytic expressions for the average cluster numbers 〈k〉_{Λ_{s}} on infinite-length strips Λ_{s}, with various widths, of several different lattices, as functions of the bond occupation probability p. It is proved that these expressions are rational functions of p. As special cases of our results, we obtain exact values of 〈k〉_{Λ_{s}} and derivatives of 〈k〉_{Λ_{s}} with respect to p, evaluated at the critical percolation probabilities p_{c,Λ} for the corresponding infinite two-dimensional lattices Λ. We compare these exact results with an analytic finite-size correction formula and find excellent agreement. We also analyze how unphysical poles in 〈k〉_{Λ_{s}} determine the radii of convergence of series expansions for small p and for p near to unity. Our calculations are performed for infinite-length strips of the square, triangular, and honeycomb lattices with several types of transverse boundary conditions.

Aligning silver nanowire films with cellulose nanocrystal nematics

Anisotropic plasmonic films are a desirable material for many optoelectronic applications. Here, we propose a method to align silver nanowires (AgNWs) with the help of uniaxial nematic alignment of cellulose nanocrystal (CNC) liquid crystals (LCs) that can preserve their LC orientation in solid film. AgNWs are doped into uniaxial nematic CNC LCs, where AgNWs are oriented parallel to the director of shear- aligned CNCs without aggregation. The AgNWs orientations are determined by polarized optical and dark field microscopic images. The alignment of AgNWs is characterized by the scalar order parameter S, and the measured S around 0.59 for aligned AgNWs presents an improved anisotropy with the assistance of uniaxial nematic CNC LCs. The electrical property of aligned AgNWs is examined by the four-probe method and exhibits the maximum ratio of anisotropic sheet resistance around 5, consistent with the simulated percolation probability performance using the Monte Carlo computation. Our results show that the uniaxial nematic CNC LCs can act as an effective template in aligning AgNWs, which is compatible with 3D printing and microfluidics, and allows for the preparation of low-cost, innovative optical materials and devices.

Percolation networks inside 3D model of the mineralized collagen fibril

Bone is a hierarchical biological material, characterized at the nanoscale by a recurring structure mainly composed of apatite mineral and collagen, i.e. the mineralized collagen fibril (MCF). Although the architecture of the MCF was extensively investigated by experimental and computational studies, it still represents a topic of debate. In this work, we developed a 3D continuum model of the mineral phase in the framework of percolation theory, that describes the transition from isolated to spanning cluster of connected platelets. Using Monte Carlo technique, we computed overall 120 × 106 iterations and investigated the formation of spanning networks of apatite minerals. We computed the percolation probability for different mineral volume fractions characteristic of human bone tissue. The findings highlight that the percolation threshold occurs at lower volume fractions for spanning clusters in the width direction with respect to the critical mineral volume fractions that characterize the percolation transition in the thickness and length directions. The formation of spanning clusters of minerals represents a condition of instability for the MCF, as it could be the onset of a high susceptibility to fracture. The 3D computational model developed in this study provides new, complementary insights to the experimental investigations concerning human MCF.

Computational Study of the Curviness Percolation Threshold in Nanotube/Nanowire Networks for Flexible and Transparent Conductors

Two-dimensional (2D) or quasi-2D networks of one-dimensional (1D) nanoelements, such as carbon nanotubes, metal nanowires, and graphene nanoribbons, have attracted significant research interest recently for next-generation flexible and transparent conductors, as a replacement for indium tin oxide (ITO), which suffers from brittleness, scarcity, high cost, and slow deposition. These nanotube/nanowire networks can be used in many device applications, such as touch screens, flat panel displays, solar cells, light-emitting diodes (LEDs), and wearable flexible electronics. They also have applications in thin film transistors and resistive switching memory.At the high optical transmittance values required for flexible and transparent conductors, the electrical properties of nanotube/nanowire networks are governed by percolation transport, which deals with the formation of long-range connectivity in random networks. As a result, Monte Carlo simulations need to be employed to theoretically calculate, predict, and optimize the electrical properties of 1D nanoelement networks [1,2]. In most computational work, nanotubes/nanowires in such networks have been modeled as straight sticks. However, experimentally deposited nanotubes/nanowires exhibit some degree of curviness.There has been some computational work on the density percolation threshold of carbon nanotube/nanofiber networks and nanocomposites at fixed curviness, where the network undergoes an insulator-to-conductor phase transition [3-7]. These studies have shown that curviness increases the density percolation threshold of random networks. However, the curviness percolation threshold at fixed density has not been investigated.In this work, we employ Monte Carlo simulations to compute the curviness percolation threshold in 2D networks consisting of curvy nanotubes or nanowires. We generate curvy nanotubes/nanowires using third order Bezier curves, which are parametric functions specified by the values of four control points. We introduce the concept of the curviness angle, which puts a geometrical constraint on the locations of the two intermediate control points of the Bezier curve, and hence on the statistical distribution of nanotube/nanowire curviness in the network. For a more universal definition of the degree of curviness, we define the curl ratio as the ratio of the actual length of the nanotube/nanowire to the effective straight length between its two ends. Using simulation data, we establish an empirical relationship between curviness angle and curl ratio.We first study the percolation probability as a function of curviness angle at fixed density for randomly distributed nanotube/nanowire networks. We extract the curviness percolation threshold at fixed density characterized by the critical curviness angle and the critical curl ratio using finite size scaling analysis. We find that the critical curviness angle and curl ratio increase with increasing density. We also find that there is a minimum (maximum) density below (above) which the percolation probability is 0 (1) regardless of the curviness angle. Second, we find that nanotube/nanowire alignment significantly changes the curviness percolation threshold, resulting in a reversal of the percolation probability versus curviness angle curve.These results show that nanotube/nanowire curviness plays a significant role in determining the percolation threshold in 2D or quasi-2D networks, films, or nanocomposites consisting of 1D nanoelements. These results further show that computational studies are an essential tool for providing insights into the insulator-to-conductor transition in nanotube/nanowire networks, which are promising candidates for a wide range of applications such as flexible transparent conductors, thin film transistors, and resistive switching memory.[1] N. Fata, S. Mishra, Y. Xue, Y. Wang, J. Hicks, and A. Ural, J. Appl. Phys. 128, 124301 (2020).[2] J. Hicks, J. Li, C. Ying, and A. Ural, J. Appl. Phys. 123, 204309 (2018).[3] Y. B. Yi, L. Berhan, and A. M. Sastry, J. Appl. Phys. 96, 1318 (2004).[4] L. Berhan and A. M. Sastry, Phys. Rev. E 75, 041121 (2007).[5] C. Li and T.-W. Chou, Appl. Phys. Lett. 90, 174108 (2007).[6] Y. Yu, S. Song, Z. Bu, X. Gu, G. Song, and L. Sun, J. Mater. Sci. 48, 5727 (2013).[7] H. M. Ma, X.-L. Gao, and T. B. Tolle, Appl. Phys. Lett. 96, 061910 (2010). Figure 1

The critical probability for Voronoi percolation in the hyperbolic plane tends to 1/2

AbstractWe consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster tends to 1/2 as the intensity of the Poisson process tends to infinity. This confirms a conjecture of Benjamini and Schramm [5].

A Two-Dimensional Partitioning of Fracture–Matrix Flow in Fractured Reservoir Rock Using a Dual-Porosity Percolation Model

Fractures and micropores have varying contributions to the gas permeability of fractured reservoirs. The quantification of the contribution of fractures and micropores that form a dual-porosity system for gas permeability is critical when attempting to accurately evaluate gas production. However, due to insufficient knowledge of fracture–matrix flow partitioning in such dual-porosity systems, it is challenging for previous models to quantitatively characterize the fracture heterogeneity and accurately evaluate the gas flow and permeability in fractured rocks. In this study, we propose a dual-porosity percolation model to quantitatively investigate the contributions of fractures and matrix micropores towards the gas permeability of fractured rocks. Using percolation theory, we establish fracture networks with complex heterogeneity, which are characterized by various fracture densities and percolation probabilities within a porous matrix with various fracture/matrix permeability ratios. The compressible Navier–Stokes and Brinkman equations were adopted to describe the gas flow in the fractures and porous matrix, respectively. The simulation results indicate that the gas permeability of the dual-porosity system has an exponential relationship with the fracture density and matrix permeability. The contribution of fractures and matrix micropores toward gas permeability can be classified by establishing a two-dimensional partitioning of the fracture–matrix flow related to the fracture heterogeneity and fracture/matrix permeability ratio. The contribution of matrix micropores cannot be neglected if the fracture density is lower than a critical value.

Coastlines and percolation in a model for hierarchical random deposition

We revisit a known model in which (conducting) blocks are hierarchically and randomly deposited on a D-dimensional substrate according to a hyperbolic size law with the block size decreasing by a factor λ>1 in each subsequent generation. In the first part of the paper the number of coastal points (in D=1) or coastline segments (in D=2) is calculated, which are points or lines that separate a region at “sea level” and an elevated region. We find that this number possesses a non-universal character, implying a Euclidean geometry below a threshold value Pc of the deposition probability P, and a fractal geometry above this value. Exactly at the threshold, the geometry is logarithmic fractal. The number of coastline segments in D=2 turns out to be exactly twice the number of coastal points in D=1. We comment briefly on the surface morphology and derive a roughness exponent α. In the second part, we study the percolation probability for a current in this model and two extensions of it, in which both the scale factor and the deposition probability can take on different values between generations. We find that the percolation threshold Pc is located at exactly the same value for the deposition probability as the threshold probability of the number of coastal points. This coincidence suggests that exactly at the onset of percolation for a conducting path, the number of coastal points exhibits logarithmic fractal behaviour.

Percolation perspective on sites not visited by a random walk in two dimensions

We consider the percolation problem of sites on an L×L square lattice with periodic boundary conditions which were unvisited by a random walk of N=uL^{2} steps, i.e., are vacant. Most of the results are obtained from numerical simulations. Unlike its higher-dimensional counterparts, this problem has no sharp percolation threshold and the spanning (percolation) probability is a smooth function monotonically decreasing with u. The clusters of vacant sites are not fractal but have fractal boundaries of dimension 4/3. The lattice size L is the only large length scale in this problem. The typical mass (number of sites s) in the largest cluster is proportional to L^{2}, and the mean mass of the remaining (smaller) clusters is also proportional to L^{2}. The normalized (per site) density n_{s} of clusters of size (mass) s is proportional to s^{-τ}, while the volume fraction P_{k} occupied by the kth largest cluster scales as k^{-q}. We put forward a heuristic argument that τ=2 and q=1. However, the numerically measured values are τ≈1.83 and q≈1.20. We suggest that these are effective exponents that drift towards their asymptotic values with increasing L as slowly as 1/lnL approaches zero.

Performance degradation prediction of direct internal reforming solid oxide fuel cell due to Ni-particle coarsening in composite anode

Long-term stability and durability of solid oxide fuel cell (SOFC) are the major challenges for fuel flexibility and commercialization. In this study, a fully coupled multi-field model is developed by coupling a modified Ni-particle coarsening model and a microstructure model for predicting the performance degradation of methane direct reforming (DIR) SOFC due to Ni-particle coarsening. The effect of Ni-particle coarsening on percolation probability of Ni-particle, effective TPB area, effective electron-conductivity, maximum power density, and the degradation rate of DIR-SOFC are quantitatively investigated under different operating conditions. The results indicate that to enhance the electrical performance and to reduce the degradation rate, the optima of the operating conditions and anode microstructure for the long-term DIR-SOFC are as follows: 750 °C for the cell operating temperature, 1.0 for anode inlet steam to carbon ratio (S/C), 0.3 A/cm2 for the operating current density, and corresponding to Ni-particle with an initial diameter of 0.6 μm, the optimal YSZ-particle diameter is 1.0 μm.

Critical Percolation on Scale-Free Random Graphs: New Universality Class for the Configuration Model

In this paper, we study the critical behavior of percolation on a configuration model with degree distribution satisfying an infinite second-moment condition, which includes power-law degrees with exponent $$\tau \in (2,3)$$ . It is well known that, in this regime, many canonical random graph models, such as the configuration model, are robust in the sense that the giant component is not destroyed when the percolation probability stays bounded away from zero. Thus, the critical behavior is observed when the percolation probability tends to zero with the network size, despite of the fact that the average degree remains bounded. In this paper, we initiate the study of critical random graphs in the infinite second-moment regime by identifying the critical window for the configuration model. We prove scaling limits for component sizes and surplus edges, and show that the maximum diameter the critical components is of order $$\log n$$ , which contrasts with the previous universality classes arising in the literature. This introduces a third and novel universality class for the critical behavior of percolation on random networks, that is not covered by the multiplicative coalescent framework due to Aldous and Limic (Electron J Probab 3(3):1–59, 1998). We also prove concentration of the component sizes outside the critical window, and that a unique, complex giant component emerges after the critical window. This completes the picture for the percolation phase transition on the configuration model.

Evolution of Shape and Volume Fraction of Superconducting Domains with Temperature and Anion Disorder in (TMTSF)2ClO4

In highly anisotropic organic superconductor (TMTSF)2ClO4, superconducting (SC) phase coexists with metallic and spin-density wave phases in the form of domains. Using the Maxwell-Garnett approximation (MGA), we calculate the volume ratio and estimate the shape of these embedded SC domains from resistivity data at various temperature and anion disorder, controlled by the cooling rate or annealing time of (TMTSF)2ClO4 samples. We found that the variation of cooling rate and of annealing time affect differently the shape of SC domains. In all cases the SC domains have oblate shape, being the shortest along the interlayer z-axis. This contradicts the widely assumed filamentary superconductivity along the z-axis, used to explain the anisotropic superconductivity onset. We show that anisotropic resistivity drop at the SC onset can be described by the analytical MGA theory with anisotropic background resistance, while the anisotropic Tc can be explained by considering a finite size and flat shape of the samples. Due to a flat/needle sample shape, the probability of percolation via SC domains is the highest along the shortest sample dimension (z-axis), and the lowest along the sample length (x-axis). Our theory can be applied to other heterogeneous superconductors, where the size d of SC domains is much larger than the SC coherence length ξ, e.g., cuprates, iron-based or organic superconductors. It is also applicable when the spin/charge-density wave domains are embedded inside a metallic background, or vice versa.

Scaling limits of crossing probabilities in metric graph GFF

We consider metric graph Gaussian free field (GFF) defined on polygons of $\delta\mathbb{Z}^2$ with alternating boundary data. The crossing probabilities for level-set percolation of metric graph GFF have scaling limits. When the boundary data is well-chosen, the scaling limits of crossing probabilities can be explicitly constructed as "fusion" of multiple SLE$_4$ pure partition functions.

A note on the local weak limit of a sequence of expander graphs

We show that the local weak limit of a sequence of finite expander graphs with uniformly bounded degree is an ergodic (or extremal) unimodular random graph. In particular, the critical probability of percolation of the limiting random graph is constant almost surely. As a corollary, we obtain an improvement to a theorem by Benjamini-Nachmias-Peres (2011) in [4] on locality of percolation probability for finite expander graphs with uniformly bounded degree where we can drop the assumption that the limit is a single rooted graph.

Mean conservation of nodal volume and connectivity measures for Gaussian ensembles

We study in depth the nesting graph and volume distribution of the nodal domains of a Gaussian field, which have been shown in previous works to exhibit asymptotic laws. A striking link is established between the asymptotic mean connectivity of a nodal domain (i.e. the vertex degree in its nesting graph) and the positivity of the percolation probability of the field, along with a direct dependence of the average nodal volume on the percolation probability. Our results support the prevailing ansatz that the mean connectivity and volume of a nodal domain is conserved for generic random fields in dimension d=2 but not in d≥3, and are applied to a number of concrete motivating examples.

Análise do potencial de lixiviação do inseticida sulfoxaflor em solos

A utilização de agrotóxicos na agricultura é justificada pela necessidade de eliminação de diversas pragas, todavia, pode promover contaminação ambiental. Dentre esses produtos, está o princípio ativo sulfoxaflor, desenvolvido pela empresa Dow Agrosciences, atingindo grande sucesso no combate aos parasitas que se alimentam de seiva. Porém, tal composto é capaz de atingir negativamente espécies não-alvo, como formigas, minhocas e abelhas. Poucos estudos avaliaram a relação entre o sulfoxaflor e os recursos hídricos, especificamente os subterrâneos. Assim, a presente pesquisa visou analisar o potencial de tal agrotóxico em lixiviar no perfil do solo, atingindo os aquíferos. Para tanto, valeu-se de quatro modelos analíticos que, com parâmetros físico-químicos, fornecem output válido, que servem de base para estudos futuros mais aprofundados, são eles: GSI, LIX, RLPI e LEACH. Os quatro índices indicaram, cada um em seu espectro de valores, que o sulfoxaflor tem capacidade de lixiviar. GSI se destacou, pois alocou-o em classe de maior probabilidade de percolação. Ressalta-se, porém, que este estudo é preliminar, sugerindo-se análises mais robustas para que as conclusões a respeito dos impactos do uso e lixiviação do inseticida para os recursos hídricos subterrâneos sejam confirmadas.

Experimental Study and Percolation Analysis on Seepage Characteristics of Fractured Coal and Sandstone Based on Real-Time Micro-CT

Sandstone and coal are the two most common types of reservoirs in nature. The permeability of sandstone in oil-bearing formations controls its oil and gas production; the permeability of the coal seam containing gas has a crucial influence on the gas drainage efficiency. One of the main factors affecting rock permeability is the spatial distribution and connectivity of pores and fissures in the rock. In this paper, a small-sized sample with a diameter of 5 mm and a height of 10 mm was used for the test. The rock samples under different stress states were scanned in real-time during the seepage testing. Based on 2D images, a 3D digital sample was reconstructed. We extracted the pores and fissures from the 3D digital sample, studied the size and distribution of the largest cluster in the sample, and revealed the influence of confining pressure and seepage pressure on the percolation probability and permeability of the sample. The research results show that brittle sandstone and plastic coal, two types of rocks with completely different properties of mechanics, have obvious differences in the spatial distribution of the largest clusters. Under the same stress state, in brittle sandstone-like rocks, the connectivity of the fissures is the primary factor affecting permeability, and the pores are the auxiliary factor; for plastic rocks such as coal, the situation is just the opposite, pores are the primary factor affecting permeability, and fissures are the auxiliary factor. The research results answer the question: Hydraulic fracturing technology can increase the oil and gas production of sandstone reservoirs but cannot increase the drainage efficiency of coalbed methane.

Research on risk propagation method of multimodal transport network under uncertainty

Taking multimodal transport network as the research object, a quantitative method of risk propagation based on improved percolation theory is proposed. This paper analyzes the applicability and limitations of the percolation theory for this problem, and improves such four aspects of nodes and edges as the state, the initial load, the percolation probability and the evaluation index of percolation effect, so that they are universal to the risk propagation of the transport network. Taking Sichuan–Tibet​ region as an example for empirical analysis, this paper simulates and analyzes the risk propagation law of the multimodal transport network under different attack types and load preferences. The results show that the risk propagation effect in the multimodal transport network will become stronger with the increase of attack scale and attack intensity, and it will show an exponential growth trend with the increase of attack scale. When other conditions are the same, the order of strength of the three kinds of attack law according to the risk propagation effect is: the intentional attack based on the loads > the random attack > the intentional attack based on the failure probability; the risk propagation effect of the load with preferences is stronger than that without preferences. Therefore, managers can control the risk of multimodal transport network in Sichuan and Tibet from the perspective of controlling attack types and balancing load preference. The research improves the method of risk propagation simulation, makes up for the shortcomings of existing research which neglects the subjectivity of load and the dynamics of risk propagation, and makes it closer to the reality, which is of strong practical value and theoretical significance.

Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution

Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. We prove that if G is nonamenable and p > p_c(G) then there exists a positive constant c_p such thatPp(n≤|K|<∞)≤e-cpn\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathbf {P}_p(n \\le |K| < \\infty ) \\le e^{-c_p n} \\end{aligned}$$\\end{document}for every nge 1, where K is the cluster of the origin. We deduce the following two corollaries:Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini et al. (in: Random walks and discrete potential theory (Cortona, 1997), symposium on mathematics, XXXIX, Cambridge University Press, Cambridge, pp 56–84, 1999).For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of p throughout the supercritical phase.

Irreversible bilayer adsorption of straight semirigid rods on two-dimensional square lattices: Jamming and percolation properties.

Numerical simulations and finite-size scaling analysis have been performed to study the jamming and percolation behavior of straight semirigid rods adsorbed on two-dimensional square lattices. The depositing objects can be adsorbed on the surface forming two layers. The filling of the lattice is carried out following a generalized random sequential adsorption (RSA) mechanism. In each elementary step, (i) a set of k consecutive nearest-neighbor sites (aligned along one of two lattice axes) is randomly chosen and (ii) if each selected site is either empty or occupied by a k-mer unit in the first layer, then a new k-mer is then deposited onto the lattice. Otherwise, the attempt is rejected. The process starts with an initially empty lattice and continues until the jamming state is reached and no more objects can be deposited due to the absence of empty site clusters of appropriate size and shape. A wide range of values of k (2≤k≤64) is investigated. The study of the kinetic properties of the system shows that (1) the jamming coverage θ_{j,k} is a decreasing function with increasing k, with θ_{j,k→∞}=0.7299(21) the limit value for infinitely long k-mers and (2) the jamming exponent ν_{j} remains close to 1, regardless of the size k considered. These findings are discussed in terms of the lattice dimensionality and number of sites available for adsorption. The dependence of the percolation threshold θ_{c,k} as a function of k is also determined, with θ_{c,k}=A+Bexp(-k/C), where A=θ_{c,k→∞}=0.0457(68) is the value of the percolation threshold by infinitely long k-mers, B=0.276(25), and C=14(2). This monotonic decreasing behavior is completely different from that observed for the standard problem of straight rods on square lattices, where the percolation threshold shows a nonmonotonic k-mer size dependence. The differences between the results obtained from bilayer and monolayer phases are explained on the basis of the transversal overlaps between rods occurring in the bilayer problem. This effect (which we call a "cross-linking effect"), its consequences on the filling kinetics, and its implications in the field of conductivity of composites filled with elongated particles (or fibers) are discussed in detail. Finally, the precise determination of the critical exponents ν,β, and γ indicates that, although the increasing in the width of the deposited layer drastically affects the behavior of the percolation threshold with k and other critical properties (such as the crossing points of the percolation probability functions), it does not alter the nature of the percolation transition occurring in the system. Accordingly, the bilayer model belongs to the same universality class as two-dimensional standard percolation model.