Percolation Exponents Research Articles

Modeliranje električne vodljivosti grafitom modificiranog asfaltbetona na temelju mikromehanike

This paper proposes a novel micromechanics theory to model the electrical conductivity of graphite-modified asphalt concrete. The constraint range of electrical conductivity was derived according to the Hashin–Shtrikman (H–S) method. Based on the self-consistent (SC) method and percolation theory, A modified self-consistent (MSC) method was used to develop an electrical conductivity model based on the SC method and percolation theory. The results predicted by the MSC model were compared with the test data by adjusting the aspect ratio and percolation exponent. We highlight this new theory and its application to graphite-modified asphalt concrete, and demonstrate that the predicted values are in close agreement with the test data.

An insight of heart-like systems with percolation

We study the signal percolation through heart-like biological system. Starting from an initial distribution of waiting and inactive cells with probabilities p and (1−p) respectively, the signal propagation is observed in terms of active cells. As the signal enters the system from one end, the number of arrival of active sites at the other end is studied and analysis of the system behaviour is made by varying a few important parameters of the system like pswitch (switching probability from inactive to waiting) and pact (switching probability from waiting to active). In this connection, the non-regular heart rhythms are discussed. Fraction of paths percolating through the system shows a transition from 0 to 1 near p=pc. Some other important quantities like tortuosity and cluster distribution are discussed. Several critical exponents have been obtained and compared the exponents of standard percolation.

Co-spectral radius for countable equivalence relations

Abstract We define the co-spectral radius of inclusions ${\mathcal S}\leq {\mathcal R}$ of discrete, probability- measure-preserving equivalence relations as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. For the proof, we develop a more general version of the 2–3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.

Modeling the Interface Between Phases in Dense Polymer‐Carbon Black Nanoparticle Composites by Dielectric Spectroscopy: Where Are We Now and What are the Opportunities?

AbstractThe macroscopic properties of polymer nanocomposites (PNC) rely largely on the interphase between the polymer chains and the filler particles. One significant difficulty to solve this issue is to quantitatively model the structure‐property correlations due to the interfacial region in these complex materials. While dielectric spectroscopy (DS) measurements are routinely used to characterize the effective permittivity of filled polymers, fitting standard effective medium models and mixing equations to these data remains notoriously difficult to interpret. This is due to the absence of explicit reference to internal length scales characterizing the interfaces in the PNC. As an illustrative example, a two‐level homogenization framework is proposed which enables the extraction of useful information on the impact of a thin interphase confined on a nanometer length scale based on broadband DS data. This model leads to new ways of tuning the interphase so as to optimize the material's response to electric field, a situation relevant for electromagnetic shielding. This approach provides guidance on how to observe directly and experimentally the actual properties of the interface between the phases (as opposed to model‐based inference). Aside from its secure physical foundation in the theory of effective medium, a significant advantage of this approach is that a genetic algorithm (GA) technique applied to this physics‐based model enables the uniqueness of the fit parameters to be considered, as the GA method is robust in terms of finding globally optimum solutions, therefore placing confidence in non‐universal values of the percolation exponents. Recent work in physics‐informed machine learning indicates that the effective dielectric properties of PNC with many degrees of freedom due to their complex morphology can be described by considering only a few degrees of freedom describing the interface features between the phases in these composites.

Isoperimetric lower bounds for critical exponents for long-range percolation

Isoperimetric lower bounds for critical exponents for long-range percolation

Dielectric Depiction of 1-Butyl-3-methylimidazolium Tetrafluoroborate/1-Butyl-3-methylimidazolium 1,4-Bis(2-ethylhexyl) Sulfosuccinate/Benzene Microemulsions: Percolation, Interface Polarization, and Electrical Parameters.

Two dielectric relaxations located at approximately 10 and 300 MHz were analyzed in the microemulsion composed of the polar ionic liquid (PIL) 1-butyl-3-methylimidazolium tetrafluoroborate ([bmim][BF4]), the surface-active ionic liquid (SAIL) 1-butyl-3-methylimidazolium 1,4-bis(2-ethylhexyl) sulfosuccinate ([bmim][AOT]), and benzene. The curve of the dc conductivity vs PIL weight fraction shows two inflection points, namely, 16.69 and 27.90 wt %, which are used to divide the PIL/O, bicontinuous (B.C.), and O/PIL subregions. The critical exponents of percolation u = 0.75, 0.79, and 0.80 are suggested from the scaling dependence of dc conductivity on the PIL weight fraction, together with frequency dependences of both permittivity and loss angle, which all infer that static percolation occurs in the microemulsion. Only one dielectric relaxation at high frequency was observed in the PIL/O subregion, when the concentration increased to the boundary between the PIL/O and B.C. subregions; the second dielectric relaxation appeared at low frequency. The high-frequency relaxation is caused by interfacial polarization. The low-frequency relaxation is attributed to the dipole-oriented polarization of AOT-. When the oil content of the system was reduced, the interface became softer to allow more AOT- to rotate, and the cation shifted from moving around its long axis to rotating along its short axis. The static dielectric constant of [bmim][AOT] was given. The dielectric constants and conductivity of the dispersed and continuous phases in the PIL/O and O/PIL subregions were calculated from the dielectric parameters of high-frequency relaxation.

Percolation and conductivity in evolving disordered media.

Percolation theory and the associated conductance networks have provided deep insights into the flow and transport properties of a vast number of heterogeneous materials and media. In practically all cases, however, the conductance of the networks' bonds remains constant throughout the entire process. There are, however, many important problems in which the conductance of the bonds evolves over time and does not remain constant. Examples include clogging, dissolution and precipitation, and catalytic processes in porous materials, as well as the deformation of a porous medium by applying an external pressure or stress to it that reduces the size of its pores. We introduce two percolation models to study the evolution of the conductivity of such networks. The two models are related to natural and industrial processes involving clogging, precipitation, and dissolution processes in porous media and materials. The effective conductivity of the models is shown to follow known power laws near the percolation threshold, despite radically different behavior both away from and even close to the percolation threshold. The behavior of the networks close to the percolation threshold is described by critical exponents, yielding bounds for traditional percolation exponents. We show that one of the two models belongs to the traditional universality class of percolation conductivity, while the second model yields nonuniversal scaling exponents.

Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation

Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation

Percolation-Based Metal-Insulator Transition in Black Phosphorus Field Effect Transistors.

The existence of a novel phenomenon, such as the metal-insulator transition (MIT) in two-dimensional (2D) systems, affords emerging functional properties that provide new aspects for future electronics and optoelectronics. Here, we report the observation of the MIT in black phosphorus field effect transistors by tuning the carrier density (n) controlled by back-gate bias. We find that the conductivity follows an n dependence as σ(n) ∝ nα with α ∼ 1, which indicates the presence of screened Coulomb impurity scattering at high carrier densities in the temperature range of 10-300 K. As n decreases, the screened Coulomb impurity scattering breaks down, developing strong charge density inhomogeneity leading to a percolation-based transition at the critical carrier density (nC). At low carrier densities (n < nC), the system is in the insulating regime, which is expressed by Mott variable range hopping that demonstrates the role of disorder in the system. In addition, the extracted average values of critical exponent δ are ∼1.29 ± 0.01 and ∼1.14 ± 0.01 for devices A and B, respectively, consistent with the 2D percolation exponent of 4/3, confirming the 2D percolation-based MIT in BP devices. Our findings strongly suggest that the 2D MIT observed in BP is a classical percolation-based transition caused by charge inhomogeneity induced by screened Coulomb charge impurity scattering around a transition point controlled by n through back-gate bias.

Progressing of a power model for electrical conductivity of graphene-based composites

This work presents a power equation for the conductivity of graphene-based polymer composites by the tunneling length, interphase deepness and filler size. The impressions of these factors on the effective concentration and percolation beginning of graphene nano-sheets in nanocomposites are also expressed. The developed equations for percolation beginning and conductivity are examined by the experimented data of some examples, which can guesstimate the interphase depth, tunneling size and percolation exponent. Besides, the impacts of numerous factors on the percolation beginning and conductivity are designed. The developed equation for percolation beginning shows the formation of thick interphase and large tunnels in the reported samples. So, disregarding of tunneling and interphase spaces in polymer graphene nanocomposites overpredicts the percolation beginning. Additionally, the developed model presents the acceptable calculations for the conductivity of samples. Among the mentioned parameters, the concentration and graphene conductivity in addition to the interphase depth induce the strongest effects on the conductivity of composites.

Gate-Controlled Metal to Insulator Transition in Black Phosphorus Nanosheet-Based Field Effect Transistors

Carrier–carrier interactions or disorders strongly affect the quantum localization–delocalization of carriers which leads to the metal to insulator transition (MIT) in two-dimensional (2D) systems. However, the physical origin of MIT in 2D systems remains controversial. Here, we report the MIT in black phosphorus (BP) nanosheet-based devices with and without the encapsulation of hexagonal boron nitride (hBN). In hBN encapsulated BP devices, we perform critical scaling analysis to elucidate the microscopic origin of 2D MIT by considering the significant role of carrier–carrier interactions over disorder. We find the critical exponents zν = 2.49 ± 0.05 and zν = 2.65 ± 0.06 in metallic and insulting phases, respectively, supporting the quantum percolation (zν = 7/3). In hBN unencapsulated BP devices, the Mott variable range hopping dominates the charge transport in the insulating phase, suggesting that the disorder plays a significant role over the carrier–carrier interactions. The extracted conductivity exponent of 1.31 ± 0.01 at 10 K approaches the 2D percolation exponent value of 4/3, which supports the classical percolation-based 2D MIT. Our findings pave the way toward the utilization of BP with and without hBN encapsulation as a model system with which to study the 2D MIT as well as various classical and quantum transports in 2D nanoelectronic devices.

Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on Zd

Consider long-range Bernoulli percolation on Zd in which we connect each pair of distinct points x and y by an edge with probability 1 − exp(−β‖x − y‖−d−α), where α &amp;gt; 0 is fixed and β ⩾ 0 is a parameter. We prove that if 0 &amp;lt; α &amp;lt; d, then the critical two-point function satisfies 1|Λr|∑x∈ΛrPβc(0↔x)⪯r−d+α for every r ⩾ 1, where Λr=[−r,r]d∩Zd. In other words, the critical two-point function on Zd is always bounded above on average by the critical two-point function on the hierarchical lattice. This upper bound is believed to be sharp for values of α strictly below the crossover value αc(d), where the values of several critical exponents for long-range percolation on Zd and the hierarchical lattice are believed to be equal.

Percolation probability and critical exponents for site percolation on the UIPT

AbstractWe derive three critical exponents for Bernoulli site percolation on the uniform infinite planar triangulation (UIPT). First, we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the off-critical exponent for site percolation on the UIPT is $\beta = 1/2$ . Then we establish an integral formula for the generating function of the number of vertices in the root cluster. We use this formula to prove that, at criticality, the probability that the root cluster has at least n vertices decays like $n^{-1/7}$ . Finally, we also derive an expression for the law of the perimeter of the root cluster and use it to establish that, at criticality, the probability that the perimeter of the root cluster is equal to n decays like $n^{-4/3}$ . Among these three exponents, only the last one was previously known. Our main tools are the so-called gasket decomposition of percolation clusters, generic properties of random Boltzmann maps, and analytic combinatorics.

Effects of lattice dilution on the nonequilibrium phase transition in the stochastic susceptible-infectious-recovered model.

We investigate how site dilution, as would be introduced by immunization, affects the properties of the active-to-absorbing nonequilibrium phase transition in the paradigmatic susceptible-infectious-recovered (SIR) model on regular cubic lattices. According to the Harris criterion, the critical behavior of the SIR model, which is governed by the universal scaling exponents of the dynamic isotropic percolation (DyIP) universality class, should remain unaltered after introducing impurities. However, when the SIR reactions are simulated for immobile agents on two- and three-dimensional lattices subject to quenched disorder, we observe a wide crossover region characterized by varying effective exponents. Only after a sufficient increase of the lattice sizes does it become clear that the SIR system must transition from that crossover regime before the effective critical exponents asymptotically assume the expected DyIP values. We attribute the appearance of this exceedingly long crossover to a time lag in a complete recovery of small disconnected clusters of susceptible sites, which are apt to be generated when the system is prepared with Poisson-distributed quenched disorder. Finally, we demonstrate that this transient region becomes drastically diminished when we significantly increase the value of the recovery rate or enable diffusive agent mobility through short-range hopping.

Advancement of the Power-Law Model and Its Percolation Exponent for the Electrical Conductivity of a Graphene-Containing System as a Component in the Biosensing of Breast Cancer

The power-law model for composite conductivity is expanded for graphene-based samples using the effects of interphase, tunnels and net on the effective filler fraction, percolation start and “b” exponent. In fact, filler dimensions, interphase thickness, tunneling distance and net dimension/density express the effective filler fraction, percolation start and “b” exponent. The developed equations are assessed by experimented values from previous works. Additionally, the effects of all parameters on “b” exponent and conductivity are analyzed. The experimented quantities of percolation start and conductivity confirm the predictability of the expressed equations. Thick interphase, large tunneling distance, high aspect ratio and big nets as well as skinny and large graphene nano-sheets produce a low “b” and a high conductivity, because they improve the conduction efficiency of graphene nets in the system. Graphene-filled nanocomposites can be applied in the biosensing of breast cancer cells and thus the developed model can help optimize the performance of biosensors.

Reliable fabrication of transparent conducting films by cascade centrifugation and Langmuir-Blodgett deposition of electrochemically exfoliated graphene.

Electrochemical exfoliation is an efficient and scalable method to obtain liquid-phase graphene. Graphene in solution, obtained through electrochemical exfoliation or other methods, is typically polydisperse, containing particles of various sizes, which is not optimal for applications. We employed cascade centrifugation to select specific particle sizes in solution and prepared thin films from those graphene particles using the Langmuir–Blodgett assembly. Employing centrifugation speeds of 3, 4, and 5 krpm, further diluting the solutions in different volumes of solvent, we reliably and consistently obtained films of tunable thickness. We show that there is a limit to how thin these films can be, which is imposed by the percolation threshold. The percolation threshold is quantitatively compared to results found in literature that are obtained using other, more complex graphene film fabrication methods, and is found to occur with a percolation exponent and percolative figure of merit that are of the same order as results in literature. A maximum optical transparency of 82.4% at a wavelength of 660 nm is obtained for these films, which is in agreement with earlier works on Langmuir–Blodgett assembled ultrasonic-assisted liquid-phase exfoliated graphene. Our work demonstrates that films that are in all respects on par with films of graphene obtained through other solution-based processes can be produced from inexpensive and widely available centrifugal post-processing of existing commercially available solutions of electrochemically exfoliated graphene. The demonstrated methodology will lower the entry barriers for new research and industrial uses, since it allows researchers with no exfoliation experience to make use of widely available graphene materials.

Chloride diffusivity of pure cement paste

A two-step scheme is presented for estimating the chloride diffusivity of pure cement paste. In the scheme, gel pores are included in the solid skeleton, composed of calcium hydroxide, unhydrated cement and calcium silicate hydrate, while capillary pores and the solid skeleton constitute pure cement paste. In combination with percolation theory, the effective medium method was modified to formulate the chloride diffusivities of the solid skeleton and pure cement paste sequentially. Detailed analysis was conducted to calibrate the percolation exponent for the high-diffusivity phase and the chloride diffusivity of the pore solution. Comparisons with various test results were made to verify the proposed scheme. Unlike existing analytical methods, the physical properties, volume fractions and morphological relationships of various phase constituents and the shapes, percolation behaviour and different roles of gel and capillary pores are all taken into account simultaneously. The calibrated chloride diffusivity of the pore solution was found to be considerably close to that derived by molecular dynamics simulation, suggesting that the scheme represents a more realistic pure cement paste. The scheme can reasonably evaluate the chloride diffusivity of pure cement paste at medium and long curing ages.

Directed percolation in random temporal network models with heterogeneities.

The event graph representation of temporal networks suggests that the connectivity of temporal structures can be mapped to a directed percolation problem. However, similarly to percolation theory on static networks, this mapping is valid under the approximation that the structure and interaction dynamics of the temporal network are determined by its local properties, and, otherwise, it is maximally random. We challenge these conditions and demonstrate the robustness of this mapping in case of more complicated systems. We systematically analyze random and regular network topologies and heterogeneous link-activation processes driven by bursty renewal or self-exciting processes using numerical simulation and finite-size scaling methods. We find that the critical percolation exponents characterizing the temporal network are not sensitive to many structural and dynamical network heterogeneities, while they recover known scaling exponents characterizing directed percolation on low-dimensional lattices. While it is not possible to demonstrate the validity of this mapping for all temporal network models, our results establish the first batch of evidence supporting the robustness of the scaling relationships in the limited-time reachability of temporal networks.

Dielectric spectroscopy of poly(ethylene oxide)–carbon nanotube nanocomposites

The dielectric properties of poly(ethylene oxide)–multiwalled carbon nanotube (MWCNT) nanocomposites have been studied over a wide range of frequency (0.1–106 Hz) and temperature (180–300 K). Nanocomposites were prepared by both melt mixing and twin-screw extrusion, and the concentration of MWCNTs was varied from 0 to 5 wt. %. Both the real and imaginary parts of the complex permittivity increase with the increasing MWCNT concentration. We observe a percolation transition in the DC conductivity of the composites above a critical MWCNT concentration pc. The data from the twin-screw extruded samples give a very well-defined value of pc and a percolation exponent of 1.9 ± 0.2, in good agreement with theoretical predictions. In contrast, both the percolation threshold and the critical exponent were more poorly defined for the melt-mixed nanocomposites. This indicates that the conductive properties of these materials can strongly depend on the details of sample preparation. Our data suggest that the dc conductivity of the nanocomposites is due to the conduction along the nanotubes, coupled with thermally activated transport of electrons across thin polymer bridges, which separate the nanotubes. The frequency dependence of the dielectric spectrum was studied as a function of temperature and composition. The primary dielectric relaxation process is due to the motions of electric dipoles on the polymer backbone. At low MWCNT concentrations, the relaxation involves the entire polymer chains and is slowed substantially when a low concentration of MWCNT is added. At higher MWCNT concentrations, the relaxation becomes much faster. We attribute this to binding of the polymer chains to the nanotubes, which reduces the length of the chain segments contributing to the dielectric relaxation.

Recent development on fragmentation, aggregation and percolation

In this article, I have outlined how an accomplished researcher like Robert Ziff has influenced a new generation of researchers across the globe like gravity as an action-at-a-distance. In the 80s Ziff made significant contributions to the kinetics of fragmentation followed by the kinetics of aggregation. Here, I will discuss fractal and multifractal that emerges in fragmentation and aggregation processes where the dynamics is governed by non-trivial conservation laws. I have then discussed my recent works and results on percolation where I made extensive use of Newman–Ziff fast Monte Carlo algorithm. To this end, I have defined entropy which paved the way to define specific heat and show that the critical exponents of percolation obey Rushbrooke inequality. Besides, we discuss how entropy and order parameter together can help us to check whether the percolation is accompanied by order–disorder transition or not. The idea of entropy also help to explain why encouraging smaller cluster to grow faster than larger clusters makes the transition explosive.