Percolation Model Research Articles

Evaluation of nuclear magnetic resonance spectroscopy for characterisation and quantitation of water-soluble polymers in river water

Water-soluble polymers (WSPs) are commonly used in industrial, commercial, agricultural and pharmaceutical products and their molecular weights and concentrations vary considerably. Methods commonly used in the analysis of WSPs are often for pure products or formulations with only a few other high MW constituents. These methods, like size exclusion chromatography (SEC) or Gel Permeation Chromatography coupled with Mass Spectrometry (MS) can be frustrated by the impact of the necessary separation steps prior to identification and the limitations of MS when identifying and quantifying polymers. To that end, the employment of a Nuclear Magnetic Resonance (NMR) method to identify, characterize and quantify WSPs in the real-world is reported for the first time. Samples were taken from fourteen UK inland river sites, concentrated via air-drying, freeze-drying or vacuum-drying and analyzed using 1D 1H NMR and 2D 1H Diffusion Ordered Spectroscopy (DOSY) NMR analysis. Seven of the river sites showed the presence of polyethylene glycol (PEG) with a range of molecular weights, evidencing the application of these techniques in analysis of WSPs. Soil percolation models evidenced the proof of principle that these techniques can also be used for the detection of polyacrylamide (PAM) and polyacrylic acid (PAA). This work should better enable the evaluation of the biological impact of WSPs on aquatic organisms in future studies.

Speed of sound of a non-equilibrium medium formed at LHC energies

We estimate the squared speed of sound for the hot and dense QCD states formed in ion collisions at very high energies by exploring the implications of small-bounded and geometry effects in the String Percolation Model. The squared sound velocity shows signals of a local minimum (knee point) below the critical temperature consistent with the softest point in the equation of state and the onset of quark deconfinement that characterizes the quark-gluon plasma phase transition.

Extending the Passive Region of CrFeNi‐Based High Entropy Alloys

AbstractThis study provides principles for designing new corrosion resistant high entropy alloys. The theoretical framework is a percolation model developed by Newman and Sieradzki that predicts the ability of an alloy to passivate, i.e., to form a protective surface oxide, based on its composition. Here, their model is applied to more complex materials than previously, namely amorphous CrFeNiTa and CrFeNiW alloys. Furthermore, the model describes a more complex passivation process: reforming the oxide layer above the transpassive potential of Cr. The model is used to predict the lowest concentration of Ta or W required to extend the passive region, yielding 11–14 at% Ta and 14–17 at% W. For CrFeNiTa, experiments reveal a threshold value of 13–15 at% Ta, which agrees with the prediction. For CrFeNiW, the experimentally determined threshold value is 37–45 at% W, far above the predicted value. Further investigations explore why the percolation model fails to describe the CrFeNiW system; key factors are the higher nobility and the pH sensitivity of W. These results demonstrate some limitations of the percolation model and offer complementary passivation criteria, while providing a design route for combining the properties of the 3d transition metal and refractory metal groups.

Vibrational-mechanical properties of the highly-mismatched Cd1−xBexTe semiconductor alloy: experiment and ab initio calculations

The emerging CdTe–BeTe semiconductor alloy that exhibits a dramatic mismatch in bond covalency and bond stiffness clarifying its vibrational-mechanical properties is used as a benchmark to test the limits of the percolation model (PM) worked out to explain the complex Raman spectra of the related but less contrasted Zn1−xBex-chalcogenides. The test is done by way of experiment (xle 0.11), combining Raman scattering with X-ray diffraction at high pressure, and ab initio calculations (x ~ 0–0.5; x~1). The (macroscopic) bulk modulus {B}_{0} drops below the CdTe value on minor Be incorporation, at variance with a linear {B}_{0} versus x increase predicted ab initio, thus hinting at large anharmonic effects in the real crystal. Yet, no anomaly occurs at the (microscopic) bond scale as the regular bimodal PM-type Raman signal predicted ab initio for Be–Te in minority (x~0, 0.5) is barely detected experimentally. At large Be content (x~1), the same bimodal signal relaxes all the way down to inversion, an unprecedented case. However, specific pressure dependencies of the regular (x~0, 0.5) and inverted (x~1) Be–Te Raman doublets are in line with the predictions of the PM. Hence, the PM applies as such to Cd1−xBexTe without further refinement, albeit in a “relaxed” form. This enhances the model’s validity as a generic descriptor of phonons in alloys.

Stepwise percolation behavior induced by nano-interconnection in electrical conductivity of polymer composites

The electrical conductivity of polymer composites incorporating a continuous nanocarbon network is reported to be 1–2 logarithmic orders higher than that of the saturation region in a typical percolation model, indicating the existence of an undiscovered region. In this study, polymer composites with high filler content (∼50 vol%) are fabricated by powder mixing and in situ polymerization including sophisticated humidity control by promoting heat convection using an air fryer. Stepwise percolation behavior is observed, wherein the electrical conductivity of the composite with a high filler content exceeds that of the percolation plateau and further increases (3828% increase at 40 vol% multi-walled carbon nanotube (MWCNT) compared to 35 vol%), and the stepwise behavior is caused by the connected filler network (nano-interconnection). A novel stepwise percolation model that describes the behavior by considering both the tunneling resistance and the nano-interconnection is proposed. In addition, the stepwise percolation behavior induces high levels of electromagnetic interference shielding (236.36% increase at 50 vol% graphene nanoplatelet relative to 40 vol%) and humidity sensing (55.59% improvement at 40 vol% MWCNT compared to 35 vol%) properties. Therefore, derivation of the stepwise percolation behavior based on the nano-interconnection can be a new breakthrough in applications of conductive composites.

Two typical analytic models for reverse bond percolation on real networks

Two typical analytic models for reverse bond percolation on real networks

Percolation on Triangulations: A Bijective Path to Liouville Quantum Gravity

We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by certain 2D lattice paths. Our bijection parallels in the discrete setting the mating-of-trees framework of LQG and Schramm-Loewner evolutions (SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two correspondences allows us to relate uniform site-percolated triangulations to 8 / 3 \sqrt {8/3} -LQG and SLE 6 _6 . In particular, we establish the convergence of several functionals of the percolation model to continuous random objects defined in terms of 8 / 3 \sqrt {8/3} -LQG and SLE 6 _6 . For instance, we show that the exploration tree of the percolation converges to a branching SLE 6 _6 , and that the collection of percolation cycles converges to the conformal loop ensemble CLE 6 _6 . We also prove convergence of counting measure on the pivotal points of the percolation. Our results play an essential role in several other works, including a program for showing convergence of the conformal structure of uniform triangulations and works which study the behavior of random walk on the uniform infinite planar triangulation.

The phase transition for planar Gaussian percolation models without FKG

The phase transition for planar Gaussian percolation models without FKG

Investigation of magnetic properties and colossal magnetoresistance in nanocrystalline doped manganite

Here in, we report structural, magnetic, and magneto-transport properties of nanocrystalline La0.6Ag0.2Bi0.2MnO3 prepared using citrate sol–gel method. By using scanning and transmission electron microscopy measurements, the morphology and particle size of the sample have been confirmed. The Mn2p x-ray photoelectron spectroscopy spectra revealed the nanoparticles contained the coexistence of Mn3+ and Mn4+ ions with Mn3+/Mn4+ ratio of 2:1. Field-cooled and zero field-cooled magnetization protocols with temperature span of 5 K–300 K, confirm the paramagnetic (PM) to ferromagnetic (FM) phase transition at critical temperature, ∼ 146 K. The complete investigation of isothermal magnetization (), Arrott plots, and magnetocaloric effect as well as quantitative analysis of second-order phase transition has been studied. The criticality at the PM-FM transition was examined for the sample, and the obtained critical exponents were verified for their reliability through the utilization of the scaling hypothesis and Kouvel-Fisher plot. We observed a large magnetic entropy change (∼7 J-Kg−1K−1) at upon 5 T magnetic field strength. The renormalized magnetic entropy change plots are found to collapse onto a single curve, thus verifying the universality of the sample. Above the metal–insulator transitions the electrical resistivity shows a small polaron hopping conduction mechanism, however, at low temperatures scattering mechanism dominates and the whole range was explained by the universal percolation model. The colossal value of negative MR is found to be 88% at 168 K under an applied field strength of 2 T. As a result of our experimental data, we can grasp the intuitive understanding of magnetic as well as transport properties in Bi-doped manganite systems potential for magnetic sensors and spintronics applications.

Humidity-Induced electrical conduction in fluropolymer-based photovoltaic backsheets

Understanding the charge transport phenomena in photovoltaic (PV) backsheets is an important step in studying their insulating properties and the related degradation in the field. Temperature and relative humidity (RH) are two environmental factors that have a significant impact on the insulating properties of backsheets. In this work, the charge transport in a commonly used backsheet with Polyethylene terephthalate (PET) as the core insulating layer is studied as a function of temperature and RH. DC conductivity of backsheet samples is considered as the transport parameter for the analysis. The conductivity pre-exponential factor (σ0) and the activation energy (Ea) in the Arrhenius equation follow the Meyer-Neldel relation in the measurement range studied. To the best of our knowledge, this is the first time the compensation effect in the DC conductivity of backsheets subjected to various temperature and RH conditions has been reported. The physical significance of Ea in the context of electronic conduction is explored, and the variation of σ0 and Ea with RH is explained by a percolation model.

A Volume Fracturing Percolation Model for Tight Reservoir Vertical Wells

Based on the non-linear seepage characteristics of tight reservoirs and the reconstruction mode of vertical wells with actual volume fracturing, a seven-area percolation model for volume fracturing vertical wells in tight reservoirs is established. Laplace transform and Pedrosa transform are applied to obtain analytical solutions of bottom hole pressure and vertical well production under a constant production regime. After verifying the correctness of the model, the influence of the fracture network parameters on the pressure and production is studied. The research results indicate that as the permeability modulus increases, the production of volume fracturing vertical wells decreases. The penetration ratio of the main crack and the half-length of the main crack have a small impact on production, while the diversion capacity of the main crack has a significant impact on the initial production, but it is ultimately limited by the effective volume of the transformation. Under constant pressure conditions, the greater the width and permeability of the ESRV region, the higher the vertical well production rate is. The smaller the aspect ratio of the ESRV region, the higher the mid-term yield and the faster the yield decrease. The research results show guiding significance for the design of vertical well volume fracturing in tight reservoirs.

Simulation of the role of agglomerations in the tunneling conductivity of polymer/carbon nanotube piezoresistive strain sensors

A 3D Monte Carlo method paired with a percolation model is carried out to predict the electrical resistivity and piezoresistive sensitivity of carbon nanotube (CNT)-polymer piezoresistive sensors. CNTs are randomly dispersed into the insulating layer of a polymer matrix having agglomerated morphologies in some parts. The electrical resistivity of nanocomposite with a defined agglomeration state is determined by considering the tunneling effect between each connected pair of CNTs. The influence of various parameters such as agglomeration radius, percentage, intrinsic properties and geometries of CNTs are investigated. A comparison was made between the analytical results and experimental data. Considering the tunneling behavior in the vicinity of the percolation transition, a good agreement is obtained between the analytical results and experimental data. Monte Carlo simulations indicated that the tunneling resistance with a random distribution of CNTs depends on the level of agglomeration. Results revealed that piezoresistive sensitivity was diminished by larger agglomerations.

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.

Advancing and retreating fronts in a changing climate: a percolation model of range shifts

Climate change causes considerable shifts in the geographic distribution of species worldwide. Most data on range movements, however, derive from relatively short periods, within which it is difficult to distinguish directional shifts from random fluctuations. For detecting a shift, it is indispensable to delineate the range precisely. We propose a new method for the delineation based on percolation theory. We suggest marking the boundary between the connected and fragmented occurrence of the species (the hull). We demonstrate the advantages of this connectivity‐based method on simulated examples in which a metapopulation is advancing vs retreating along an environmental gradient with different velocities. The simulations show that the hull is a fractal and has the same dimension (7/4) even when the front is advancing or retreating relatively fast, compared to the generation time. It is particularly robust in the retreating (trailing) edge. Accordingly, we propose marking the range edge at the mean position of the hull, the 'connectivity limit' of the species. Theoretical considerations suggest that the position of the connectivity limit is statistically more reliable than those limits that are delineated according to the outermost occurrences, and the connectivity‐based method is broadly applicable to real‐life data.

A grid-based percolation model for the electrode of the solid oxide fuel cell

ABSTRACT For solid oxide fuel cells, the percolation model is a mathematical tool to predict the percolating properties (percolating probability, total and effective three-phase boundaries (TPB), etc.) of an electrode. Here, a grid-based 3D percolation model is proposed. Compared with the traditional analytic percolation models, it is more comprehensive because it additionally accounts for the active TPBs near the electrolyte–electrode interface and the percolating probability of pore. Moreover, compared with the pixel-based 3D reconstruction models, this model consumes much less time and memory, which makes large domain size simulation efficient. To characterize the experimental repeatability and reproducibility of the percolating properties among numerous electrodes, distribution profile is introduced to the simulation where quantities of numerical samples are generated and counted. Our model results match well with the reported ones. The optimal porosity is 30%–35% for our studied cases. Our model suggests that the pore percolating probability could not be neglected in the percolation simulations. Finally, domain size effect is investigated. TPB density becomes converged when the domain. size is at least 12 times the particle diameter. This model provides a practical and flexible access to the large domain simulations of the electrode percolating properties.

New Properties and Sets Derived from the 2-Ball Fractal Dust

Due to their practicality and convenient parametrization, fractals derived from iterated function systems (IFSs) constitute powerful tools widely used to model natural and synthetic shapes. An IFS can generate sets other than fractals, extending its application field. Some of such sets arise from IFS fractals by adding minimal modifications to their defining rule. In this work, we propose two modifications to a fractal recently introduced by the authors: the so-called 2-ball fractal dust, which consists of a set of balls diminishing in size along an iterative process and delimited by an enclosing square. The proposed modifications are (a) adding a resizer parameter to introduce an interaction between the generator and generated ball elements and (b) a new fractal embedded into the 2-ball fractal dust, having the characteristic of filling zones not covered by the previous one. We study some numerical properties of both modified resulting sets to gain insights into their general properties. The resulting sets are geometrical forms with potential applications. Notably, the first modification generates an algorithm capable of producing geometric structures similar to those in mandalas and succulent plants; the second modification produces shapes similar to those found in nature, such as bubbles, sponges, and soil. Then, although a direct application of our findings is beyond the scope of this research, we discuss some clues of possible uses and extensions among which we can remark two connections: the first one between the parametrization we propose and the mandala patterns, and the second one between the embedded fractal and the grain size distribution of rocks, which is useful in percolation modeling.

Robustness of directed higher-order networks.

In complex systems, from human social networks to biological networks, pairwise interactions are insufficient to express the directed interactions in higher-order networks since the internal function is not only contained in directed pairwise interactions but rather in directed higher-order interactions. Therefore, researchers adopted directed higher-order networks to encode multinode interactions explicitly and revealed that higher-order interactions induced rich critical phenomena. However, the robustness of the directed higher-order networks has yet to receive much attention. Here, we propose a theoretical percolation model to analyze the robustness of directed higher-order networks. We study the size of the giant connected components and the percolation threshold of our proposed model by the theory and Monte-Carlo simulations on artificial networks and real-world networks. We find that the percolation threshold is affected by the inherent properties of higher-order networks, including the heterogeneity of the hyperdegree distribution and the hyperedge cardinality, which represents the number of nodes in the hyperedge. Increasing the hyperdegree distribution of heterogeneity or the hyperedge cardinality distribution of heterogeneity in higher-order networks will make the network more vulnerable, weakening the higher-order network's robustness. In other words, adding higher-order directed edges enhances the robustness of the systems. Our proposed theory can reasonably predict the simulations for percolation on artificial and real-world directed higher-order networks.

Theoretical Prediction of Electrical Conductivity Percolation of Poly(lactic acid)-Carbon Nanotube Composites in DC and RF Regime.

Polymer composites based on polylactic acid (PLA) reinforced with 0.25-5 wt.% of carbon nanotubes (CNTs) were synthesized by melt blending. The static (DC) and microwave (RF) electrical conductivity have been investigated on the PLA-CNT composites. The electrical percolation threshold has been theoretically determined using classical models of percolation in order to predict the conductivity of the different nanocomposites. Through the fitting process, it has been found that the percolation threshold is obtained at 1 wt.% of CNTs in the DC regime and reached below 0.25 wt.% of CNTs in the microwave regime. Among the Mamunya, McLachlan, or GEM models, the McCullough model remarkably fits the experimental DC and RF electrical conductivities. The obtained results are correlated to the electrical properties of a range of CNT-based composites, corresponding to the percolation threshold required for a three-dimensional network of CNTs into the polymer matrix.

Porosity and conductivity in ballistic deposition with power-law distributed noise

The ballistic deposition model of rod-like particles with power-law distributed noise is studied by the Monte Carlo simulation. In this modified model instead of particles with fixed unit sizes, vertical rods whose length follows the power-law distribution [Formula: see text] are deposited, where [Formula: see text] denotes the power-law strength exponent. This deposition leads to porous rock structures with varying porosity and conductivity. The time evolution of the surface roughness and the porosity of the resulting structures are studied. The conductivity of the structures is calculated using the parallel resistors in the percolation model using a random walk algorithm. Finally, we discuss the relation between porosity, conductivity and the strength exponent of the power-law noise. The results show that the surface roughness increases as a pseudo-step function versus deposition time for [Formula: see text], which leads to an observable reduction in porosity and conductivity. By increasing the [Formula: see text] exponent, the growth exponent of [Formula: see text] for the Gaussian model appears. The conductivity increases as [Formula: see text] versus porosity, [Formula: see text], and remains constant for [Formula: see text] which the value of [Formula: see text] has been identified for BD.

Nature of the Metal Insulator Transition in High-Mobility 2D_Si-MOSFETs.

Our investigation focuses on the analysis of the conductive properties of high-mobility 2D-Si-MOSFETs as they approach the critical carrier density, nsc (approximately 0.72×1011 cm-2), which marks the metal insulator transition (MIT). In close proximity to the nsc, the conductivity exhibits a linear dependence on the temperature (T). By examining the extrapolated conductivity at the absolute zero temperature (T = 0), denoted as σ0, as a function of the electron density ns, we identify two distinct regimes with varying σ0(ns) patterns, indicating the existence of two different phases. The transition from one of these two regimes to another, coinciding with nsc, is abrupt and serves as the focus of our investigation. Our aim is to establish the possibility of a percolation type transition in the 2D-Si-MOSFETs' sample. In fact, we observed that the model of percolation is applicable only for densities very close to nsc*=n2 (where n2 is the linear extrapolation of σ0), indicating the percolation type transition essentially represents a phase transition at the zero temperature.