Percolation Model Research Articles

Correlation length critical exponent as a function of the percolation radius for one-dimensional chains in bond problems

A one-dimensional lattice percolation model is constructed for the problem of constraints flowing along non-nearest neighbors. In this work, we calculated the critical exponent of the correlation length in the one-dimensional bond problem for a percolation radius of up to 6. In the calculations, we used a method without constructing a covering lattice or an adjacency matrix (to find the percolation threshold). The values of the critical exponent of the correlation length were obtained in the one-dimensional bond problem depending on the size of the system and at different percolation radii. Based on original algorithms that operate on a computer faster than standard ones (associated with the construction of a covering lattice), these results are obtained with corresponding errors.

Thermodynamic and transport properties of matter formed in pp, p-Pb, Xe–Xe and Pb–Pb collisions at the Large Hadron Collider using color string percolation model

To have a better understanding of the matter formed in ultra-relativistic collisions, using the color string percolation model, we have estimated initial energy density, mean free path, squared speed of sound, shear viscosity to entropy density ratio (η/s), bulk viscosity to entropy density ratio and bulk modulus for pp, p-Pb, Xe–Xe and Pb–Pb collisions at the Large Hadron Collider (LHC). These observables are studied as a function of final state charged particle multiplicity density normalized by nuclear overlap area and initial state percolation temperature. The results from this work are compared with the results from well known QCD models and with the transport properties of matter found in day-to-day life. These observables allow us to conclude an array of important findings about the matter formed in high-energy collisions at the LHC. For the high-multiplicity events, the estimated η/s are comparable with that is observed for heavy-ion collisions, which is an indication of a strongly coupled quark gluon plasma. This makes LHC high-multiplicity pp collision events as probable candidates for the search of quark-gluon plasma droplets.

Exponential rate for the contact process extinction time

We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer line, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of Z d or R d in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton-Watson trees grown up to finite generations.

The oriented swap process and last passage percolation

AbstractWe present new probabilistic and combinatorial identities relating three random processes: the oriented swap process (OSP) on n particles, the corner growth process, and the last passage percolation (LPP) model. We prove one of the probabilistic identities, relating a random vector of LPP times to its dual, using the duality between the Robinson–Schensted–Knuth and Burge correspondences. A second probabilistic identity, relating those two vectors to a vector of “last swap times” in the OSP, is conjectural. We give a computer‐assisted proof of this identity for after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman–Greene correspondence. The conjectural identity provides precise finite‐n and asymptotic predictions on the distribution of the absorbing time of the OSP, thus conditionally solving an open problem posed by Angel, Holroyd, and Romik.

Sharpness of the Phase Transition for the Orthant Model

The orthant model is a directed percolation model on mathbb {Z}^{d}, in which all clusters are infinite. We prove a sharp threshold result for this model: if p is larger than the critical value above which the cluster of 0 is contained in a cone, then the shift from 0 that is required to contain the cluster of 0 in that cone is exponentially small. As a consequence, above this critical threshold, a shape theorem holds for the cluster of 0, as well as ballisticity of the random walk on this cluster.

High performance and low power consumption resistive random access memory with Ag/Fe2O3/Pt structure

Due to magnetic field tunability and the abundance of iron in the Earth’s crust, iron oxide-based resistive random access memory (RRAM) is considered to be low cost and potential for multi-level storage. However, the relatively high operation voltage (>1 V) and small storage window (<100) limit its application. In this work, the devices with simple Ag/Fe2O3/Pt structure exhibit typical bipolar resistive switching with ultralow set voltage (V set) of 0.16 V, ultralow reset voltage (V reset) of −0.04 V, high OFF/ON resistance ratio of 103, excellent cycling endurance more than 104 and good retention time longer than 104 s. Each major parameter has about an order of magnitude improvement compared to the previous data. The devices demonstrate outstanding stable low power consumption quality. Based on the analysis of the experimental results, a percolation model of silver ion migration was established and confirmed that low operation voltage is attributed to the amorphous oxide layer with large porosity. During electrical testing, the compliance current (I c) and maximum reset voltage (V max) can also affect the device performance. This discovery suggests Fe2O3 memristor has significant potential for application and provides a new idea for the realization of high-performance low-power RRAM.

Morphological, thermal, and mechanical properties of cellulose nanocrystal reinforced poly(lactic acid) and poly(butylene adipate‐co‐terephthalate): A comparative study on common and novel solvent casting methods

AbstractThe mechanical and thermal properties of semicrystalline (sc) and amorphous (a) poly(lactic acid), PLA, and poly(butylene adipate‐co‐terephthalate), PBAT, and their nanocomposites containing 1 and 3 wt% CNCs, prepared through solvent casting methods using one (N,N‐dimethylformamide [DMF]) or two (dimethyl sulfoxide (DMSO), and tetrahydrofuran (THF)) solvents were analyzed. Differential scanning calorimetry (DSC) showed that the total amount of crystals of the scPLA/CNC nanocomposites increased, whereas it decreased in the PBAT/CNC systems. In both cases, the crystallization temperature increased with CNC content. In tensile experiments, the Young modulus and yield strength of all nanocomposites were found to increase by incorporating CNCs, more significantly for the samples prepared using one solvent. The elongation at break of both PLA nanocomposites increased when prepared via one solvent, while it decreased for the two solvent methods as well as for PBAT nanocomposites prepared by both methods. The impact properties of the samples prepared by the two solvent methods decreased. In contrast, for the one solvent method, incorporating 3 wt% CNCs improved the impact properties by 32% and 9% in scPLA and aPLA, respectively, but decreased by 4% in PBAT nanocomposites. Also, in dynamic mechanical thermal analysis (DMA) the storage modulus of scPLA and PBAT/CNC systems increased significantly, especially in the rubbery region (5–85 MPa and 105–155 MPa, respectively). Using a percolation model, the strength of the percolating CNC was found to be dependent on temperature and affected by traces of solvent mostly in the scPLA nanocomposites.

Fully-connected bond percolation on $${\mathbb {Z}}^d$$

We consider the bond percolation model on the lattice \({\mathbb {Z}}^d\) (\(d\ge 2\)) with the constraint to be fully connected. Each edge is open with probability \(p\in (0,1)\), closed with probability \(1-p\) and then the process is conditioned to have a unique open connected component (bounded or unbounded). The model is defined on \({\mathbb {Z}}^d\) by passing to the limit for a sequence of finite volume models with general boundary conditions. Several questions and problems are investigated: existence, uniqueness, phase transition, DLR equations. Our main result involves the existence of a threshold \(0<p^*(d)<1\) such that any infinite volume model is necessary the vacuum state in subcritical regime (no open edges) and is non trivial in the supercritical regime (existence of a stationary unbounded connected cluster). Bounds for \(p^*(d)\) are given and show that it is drastically smaller than the standard bond percolation threshold in \({\mathbb {Z}}^d\). For instance \(0.128<p^*(2)<0.202\) (rigorous bounds) whereas the 2D bond percolation threshold is equal to 1/2.

Constrained percolation in two dimensions

We prove absence of infinite clusters and contours in a class of critical constrained percolation models on the square lattice. The percolation configuration is assumed to satisfy certain hard local constraints, but only weak symmetry and ergodicity conditions are imposed on its law. The proofs use new combinatorial techniques exploiting planar duality. Applications include absence of infinite clusters of diagonal edges for critical dimer models on the square-octagon lattice, as well as absence of infinite contours and infinite clusters for critical XOR Ising models on the square grid. We also prove that there exists at most one infinite contour for high-temperature XOR Ising models, and no infinite contour for low-temperature XOR Ising model.

Dynamic thermoelectromechanical characterization of carbon nanotube nanocomposite strain sensors

Recent advancements in materials science and bioinspired designs have led to the development of wearable strain sensors with high sensitivity and stretchability. However, the majority of strain sensors are simultaneously responsive to strain and temperature variations. Therefore, it is necessary to invent effective strategies to decouple this unwanted crosstalk for accurate and noise-free strain sensing in complex environmental conditions. Herein, a temperature-dependent 3D percolation model is presented capable of predicting the sensory response of stretchable strain sensors made of carbon nanotubes-elastomer nanocomposites under transient changes in strain and temperature. The nanocomposite strain sensors are fabricated by the vacuum filtration process, and their dynamic thermoelectromechanical properties are systematically investigated. The experimental results are then used to initiate, develop, and validate the proposed model. After the model validation, further studies are conducted to assess the dynamic response of strain sensors containing different amounts of carbon nanotubes at high temperatures, their signal-to-noise ratio, and temperature coefficient of resistance. Finally, to simulate a real-world scenario, virtual displacement and temperature profiles are generated, and the corresponding dynamic response of a strain sensor is predicted utilizing the modified 3D percolation model.

Analysis of flow characteristics of the dual media shale gas reservoir with the elastic outer boundary

In order to more accurately describe the seepage characteristics of shale gas reservoirs, in this paper, an elastic outer boundary condition is introduced, and a new dual-media shale gas seepage model is established to describe the seepage characteristics of shale gas reservoirs more accurately, while considering the adsorption and desorption process. Combining Laplace transformation and Similar Structure Method, the solution of the percolation model is obtained in Laplace space. Furthermore, the solutions are inverted into real space with the Stehfest numerical inversion method. Type curves of dimensionless pressure and dimensionless pressure derivative can be used to evaluate the reservoir characteristics. The results show that the conventional three kinds of outer boundary conditions (infinite, constant pressure and closed) are actually three special cases of elastic outer boundary. The introduction of elastic outer boundary conditions not only expands the scope of data interpretation, but also closer to the actual situation of the outer boundary of the reservoir. The theory of similar structure greatly simplifies the complex process of solving the model and will have a far-reaching impact on the development of well test analysis software in the future.

Possible two-component spin-singlet pairings in Sr2RuO4

Recent experiments suggest a multi-component pairing function in Sr2RuO4, which appears to be inconsistent with the absence of an apparent cusp in the transition temperature (Tc) as a function of the uniaxial strain. We show, however, that the theoretical cusp in Tc for a multi-component pairing can be easily smeared out by the spatial inhomogeneity of strain, and the experimental data can be reproduced qualitatively by a percolation model. This shed new light on multi-component pairings. We then perform a thorough group-theoretical classification of the pairing functions, taking the spin-orbit coupling into account. We list all 13 types of two-component spin-singlet pairing functions, with 8 of them belonging to the Eg representation. In particular, we find two types of intra-orbital pairings in the Eg representation ($k_xk_z$, $k_yk_z$) are favorable in view of most existing experiments.

Study on unsteady seepage characteristics and production decline of tight channel sandstone gas reservoirs

Tight channel sandstone gas reservoirs are mainly characterized by low porosity and permeability to ultra-low porosity and permeability, strong heterogeneity, and slow energy supply rate in the later stage, resulting in small gas well control reserves, and fast decline rate of gas well pressure and production. Therefore, it is urgent to carry out targeted research on production decline. In this paper, from the perspective of percolation mechanics, a mathematical model of low-velocity non-Darcy unstable percolation in tight channel sandstone gas reservoir is established, a diagram of the relationship between non-Darcy effect, flow boundary characteristics, and dynamic bottom-hole flow is established, and the dynamic influencing factors of bottom hole flow are analyzed. From a seepage mechanics point of view, this paper establishes a tight sandstone gas reservoir river unstable low-speed non-darcy seepage flow mathematical model of non-darcy effect and flow boundary characteristics and the relationship between the bottom hole flow dynamic chart and analysis of bottom hole flow dynamic influence factors, research shows that start-up pressure gradient, skin factor, the closed boundary of tight sandstone gas reservoir river bottom hole flow dynamic effect is bigger, The research content of this paper has certain guiding significance for realizing the effective development of tight channel sandstone gas reservoir.

Degassing and gas percolation in basaltic magmas

Due to their generally low eruptive melt viscosities and concomitant high diffusivities of volatiles, basaltic magmas degas relatively efficiently. This relative efficiency, combined with variations in style, extent, timing and length scales of degassing govern the range of eruptive styles observed at basaltic volcanoes. The result is a surprising complexity of degassing regimes and products in basaltic volcanism. In particular, the transition between closed- and open-system degassing at low pressure at the percolation threshold may strongly affect the type of eruption. Here we aim to better understand degassing and gas percolation processes in basaltic magmas and their implications for eruptive style. Combining new and literature data, we present a database of vesicle metrics in basaltic rocks including vesicularity, vesicle number density, vesicle size distribution (and its polydispersivity), vesicle connectivity and permeability. We combine these textural and petrophysical data with a numerical model of percolation for systems having polydisperse vesicle size distributions. Using this model, we also evaluate different definitions of vesicle connectivity inherent to different measurement techniques. Our results show that polydispersivity exerts a strong control on the percolation threshold of basaltic magmas and consequently on eruptive style. Intermediate to highly polydisperse bubble networks are more typical of Hawaiian activity and are characterized by higher values of percolation threshold. This results in delayed coalescence and an increase in magma vesicularity hindering the formation of large decoupled and buoyant bubbles, which in turn can promote magma acceleration, fragmentation by inertia below the percolation threshold and sustained fountaining activity. Bubble populations with lower polydispersivity, typical of Strombolian eruptions, promote early coalescence prior to fragmentation, which in turn may lead to the formation of large decoupled slugs or gas pockets and/or plugs at the surface via outgassing. Further, we discuss the implications of our findings for Plinian, violent Strombolian, Surtseyan, deep submarine and effusive basaltic eruptions.

Numerical study of triple‐phase boundary length in high‐temperature proton exchange membrane fuel cell

Catalyst layer (CL) is the most important component in the high-temperature proton exchange membrane fuel cell (HT-PEMFC), as it offers reaction sites for hydrogen oxidation reaction (HOR) and oxygen reduction reactions (ORR). As the electrochemical reactions take place at gas/electron/proton interface, the length of this triple-phase boundary (TPB) in the CL governs the rate of electrochemical reactions. As the TPB in the HT-PEMFC has not been studied yet, a numerical non-isothermal 3D model and a percolation micro model were developed in this work to investigate the effect of CLs' properties on both the TPB length and cell performance. The volume fractions, particle size, and particle size ratio of ionomer and Pt/C were selected for optimization. The results show that peak current density could be achieved at an ionomer volume fraction of 52%. Further improvement of cell performance requires smaller particle size and smaller particle size ratio of ionomer. Novelty statement A three-dimensional non-isothermal model of HT-PEMFC was developed. By adopting the percolation model, length of triple phase boundary of HT-PEMFC was successfully quantified. The particle parameters including the particle size and ionomer volume fraction can be optimized to increase the length of triple phase boundary, thus achieving higher cell performance.

An upper bound for the bond percolation threshold of the cubic lattice by a growth process approach

AbstractWe reduce the upper bound for the bond percolation threshold of the cubic lattice from 0.447 792 to 0.347 297. The bound is obtained by a growth process approach which views the open cluster of a bond percolation model as a dynamic process. A three-dimensional dynamic process on the cubic lattice is constructed and then projected onto a carefully chosen plane to obtain a two-dimensional dynamic process on a triangular lattice. We compare the bond percolation models on the cubic lattice and their projections, and demonstrate that the bond percolation threshold of the cubic lattice is no greater than that of the triangular lattice. Applying the approach to the body-centered cubic lattice yields an upper bound of 0.292 893 for its bond percolation threshold.

Phase transition in the diffusion and bootstrap percolation models on regular random and Erdős-Rényi networks

The diffusion and bootstrap percolation models were studied in regular random and Erdős-Rényi networks using the modified Newman-Ziff algorithms. We calculated the percolation threshold and the order parameter of the percolation transition (strength of the giant cluster) and its derivatives. The percolation transitions are classified by the results. The diffusion percolation with a small k has a double transition, and the bootstrap percolation with m≥3 has the first-order percolation transition. The diffusion percolation with a large k and the bootstrap percolation with a small m show the second-order percolation transition. Particularly, third-order percolation transitions were discovered in the bootstrap percolation of m=2 in regular random networks.

Significantly enhanced room-temperature TCR and MR of La0.67K0.33-xSrxMnO3 ceramics by adjusting Sr content

Strontium (Sr)-doped La0.67K0.33-xSrxMnO3 (LKSMO) ceramics were obtained by traditional sol-gel method. With increasing Sr content, full width at half maximum (FWHM) for temperature-dependent resistivity curves increased, resistivity decreased, and peak resistivity temperature (TP) shifted toward high temperature. Microstructure of LKSMO ceramics was characterized by X-ray diffraction, scanning electron microscopy, X-ray photoemission spectroscopy, energy dispersive spectrometry, and Fourier transform infrared spectroscopy, etc. Connection between microstructure and properties of ceramics was studied by combining double exchange mechanism, Jahn-Teller effect, and phenomenological percolation model. Electrical transport mechanism of ferromagnetic metal (FMM) and paramagnetic insulator (PMI) regions was employed to explain temperature coefficient of resistivity (TCR) and magnetoresistance (MR) behaviors of as-obtained samples. Various scattering factors, effective mass of itinerant electron, and energy difference between FMM and PMI phases affected FWHM of ρ-T curves in Sr-doped LKSMO ceramics. In sum, these findings provide physical mechanism and experimental guidance for synthesis of high-TCR and MR polycrystalline ceramics at room temperature.

3D Simulation Modeling for the Electrical Conductivity of Carbon Nanotube Networks in Polymer Nanocomposites

In this paper, we developed a three-dimensional percolation model to investigate the effects of the concentration and morphology of CNTs (carbon nanotubes) on the electrical conductivity of the nanocomposites. In the model, we judged the connections between CNTs by range search algorithm based on KD-Tree structure. At the same time, DIJKSTRA-Melissa algorithm was applied to efficiently find all the conductive paths instead of finding conductive network in traditional methods. From the simulation results, CNTs with higher aspect ratio were easier to form the conductive network. In a certain range of CNT’s concentration, the relationship between the conductivity of the conductive network and the carbon nanotubes was basically consistent with the classical percolation theory. To verify our simulation model, the morphological, electrical properties of Carbon nanotubes (CNTs)/poly(dimethyl siloxane) (PDMS) nanocomposites with different aspect ratio (AR) of MWNTs were systematically studied. In conclusion, these unique advantageous properties could be exploited to suggest potential applications of artificial electronic skin.

Exact formula for bond percolation on cliques.

We present exact solutions for the size of the giant connected component of complex networks composed of cliques following bond percolation. We use our theoretical result to find the location of the percolation threshold of the model, providing analytical solutions where possible. We expect the results derived here to be useful to a wide variety of applications including graph theory, epidemiology, percolation, and lattice gas models, as well as fragmentation theory. We also examine the Erdős-Gallai theorem as a necessary condition on the graphicality of configuration model networks comprising clique subgraphs.