Percolation Model Research Articles

Precision sub-monolayer manipulation of diamond surface chemistry using laser direct write oxidation in air

Manipulation and patterning of diamond surface chemistry is of interest for a wide range of diamond-based technologies. We report the patterned oxidation of hydrogen-terminated diamond surfaces with sub-monolayer (ML) precision by a deep-UV two-photon process performed in air. Using focused laser pulses of photon energy 4.66 eV (266 nm; below the diamond bandgap of 5.47 eV), hydrogen-terminated (001) surfaces were exposed with calibrated doses to remove carbon with a precision of 0.02 ML. The measurement of the electrical properties of the laser-exposed zone between ohmic electrodes enabled monitoring of the transition from a conducting H-terminated surface to insulating O-terminated. The surface resistance increases by more than 7 orders of magnitude for doses corresponding to 0.5 ML, and the I–V characteristics show a transition from linear to nonlinear for doses above 0.30 ML. We show that this behavior agrees well with a surface percolation model for carrier diffusion in which the laser etch rate for the H-terminated top layer is the same as for O-terminated. Hence, this work reveals an ultra-precise method for modifying the sub-monolayer surface chemistry with the practical advantages of a laser-induced mechanism compared to conventional plasma or chemical processing methods.

Message-passing approach to higher-order percolation

Hypergraph describes real-world networks widely because it captures pairwise and multiple nodes’ interactions. Various kinds of damages, such as network attacks, hardware malfunctions, and communication disruptions, may impair the function of those real-world systems. We propose a generalized higher-order percolation model to investigate the robustness of the hypergraph, in which the nodes and hyperedges were randomly removed with probabilities. An accurate approach to studying the higher-order percolation model should overcome non-local tree-like structures and higher-order interactions, which makes the classical mean-field approach invalid. To this end, we develop a message-passing approach in which we first transform the hypergraph into a factor graph then develop a message-passing approach on the factor graph. Through extensive experimental studies on both artificial and real-world hypergraphs, our theory can accurately predict numerical results. The experimental data demonstrate that our theory achieves average accuracy rates in calculating giant connected component (GCC) size of 99.87% for artificial loopless hypergraphs, 99.24% for artificial hypergraphs with loops, and 99.65% for real-world hypergraphs. Our findings provide another way to understand the robustness of hypergraphs, and also provide certain ideas for studying complex systems in various fields.

Evolution of global value-added trade networks and response to risk transmission

To enhance the international division of labor of developing countries and scientifically respond to the risks and conflicts in the network system, it is important to examine the evolutionary characteristics of the global value-added trade network, simulate the impact of risk shocks, and propose corresponding measures. Based on UNCTAD-Eora value-added trade data, this paper measured and evaluated the evolution characteristics of the global value-added trade network from 2003 to 2018 using the social network analysis and value-added decomposition methods. Then we analyze the impact of risk shocks on the evolution of the trade network using the bootstrap percolation model, building global trade networks and proposing countermeasures. The results show that the global value-added trade network has formed a complex structure and structurally stable distribution pattern, with Germany, China, and the U.S. as the core and the most crucial supports. Among which, China's core position is mainly due to the rapid rise in its export center status. The trade benefits of the three core countries are both competitive and complementary along the “One Belt and One Road”. Furthermore, simulations of bootstrap percolation model reveals that the adoption of trade protection policies (caused by poor institutional quality) by different countries will spread and diffuse non-linearly in the network, and the impacts triggered by low-centered countries are comparatively more widespread. By improving defense capabilities and changing the network structure, the “cascading” impacts of trade policy uncertainty can be reduced.

Electrical conductivity of geopolymer-graphite composites: Percolation, mesostructure and analytical modeling

The incorporation of carbon fillers into geopolymers enhances conductivity, forming intricate networks for integrated energy conversion in construction materials. This study employs a novel computational modeling approach to investigate the electrical conductivity of geopolymer-graphite composites. Utilizing a particle-based Monte Carlo method, percolation thresholds in virtual composite mesostructures are simulated, and Bruggeman's effective medium model is applied to predict conductivity. The research explores how mesostructural characteristics, including pore and particle size distribution, influence percolation and conductivity in multi-phase mixtures. The findings indicate that graphite polydispersity increases the percolation threshold from 10 vol% to 24.4 vol%, while the introduction of two additional (segregated) phases, metakaolin and pores, decreases the percolation threshold to 11.9 vol%. The concept of effective volume fraction is introduced to assess the influence of metakaolin and porosity on the percolation threshold. This enabled to establish a linear relationship between the electrical conductivity and effective volume fraction of graphite with a slope of 2 S/m/vol%. In a three-phase mixture, the percolation threshold is 12.18 vol%, with a 2.35% increase due to capillary water drying, enhancing graphite particle aggregation in the dried pores. Finally, Bruggeman's analytical effective medium model is used to simulate conductivity variation with increasing graphite content, aligning well with conductivity measurements. These findings provide valuable insights into percolation behavior and offer a relatively simple quantitative modeling approach to understand the factors influencing the effective conductivity of geopolymer-graphite composites.

Finite-size effect on quantum percolation in topological insulators

We study the finite-size effect on quantum percolation in two-dimensional topological insulators. We demonstrate that the percolation threshold in topological insulators strongly depends on the localization length of the edge states in small clusters due to the finite-size effect. Also, we explain why the percolation threshold in the corresponding classical model determines the lower bound of the quantum percolation threshold in topological insulators. In addition, we extend the percolation model to a more general scenario, where the system is composed of both topological and trivial clusters. We find that the quantum percolation threshold can be less than the classical percolation threshold due to quantum tunneling of the edge states.

Investigation of mesoporous silicon thermal conductivity: Effect of nanographene insertion

In this study, the impact of nanographene incorporation on the thermal properties of mesoporous silicon (PSi) was evaluated using two complementary experimental methods: the temperature gradient (TG) and the photothermal radiometry (MPTR) methods. It is shown that the measured thermal conductivity of the mesoporous silicon (PSi) ranges from 0.10 to 0.68 W/m.K in the case of TG and from 0.37 to 3.02 W/m.K in MPTR and is strongly correlated to the electrochemical etching parameters. These values are much lower than that of crystalline silicon, estimated to be from 100 to 140 W/m.K, depending on the doping rate. They appear to be, however, in the order of magnitude range for the percolation models that also include the in-depth porosity and the crystallite mean radius. This set of experiments on the thermal conductivity was extended to investigate the effect of graphene incorporation in the PSi matrix (G-PSi) as it has seldom been reported in the literature. The results from both methods exhibit significantly higher values (1.7 ± 0.3 W/m.K for TG, and from 0.7 to 2.13 W/m.K for MPTR). This spread of the thermal conductivity values is attributed to the intrinsic working principle of the TG versus the MPTR method as highlighted in the last part of the present paper. Targeting the thermoelectric application of both matrices (PSi, G-PSi), the thermal conductivity remains sufficiently low for them to be considered as very promising materials, keeping in mind the enhancement of the power-factor attributed to the incorporation of graphene.

Crosslinked polymer-in-salt solid electrolyte with multiple ion transport paths for solid-state lithium metal batteries

Low ionic conductivity and unsatisfactory mechanical properties of the solid polymer electrolytes hinder their applications of in solid-state lithium metal batteries. Herein, we design a polymer-in-salt solid electrolyte (PISSE) with multiple Li+ transport paths and crosslinked polymer chain networks, endowing the PISSE with both high mechanical strength and high ionic conductivity. The optimized PISSE can therefore reach ionic conductivity of 3.03×10−4 S cm−1 (25 °C) and the mechanical strength of 0.811MPa. The percolation model explains that the salt-rich clusters account for the extra fast ion transfer paths; whereas, the crosslinking structure compensates the loss of mechanical strength at high salt concentration. Finally, the in situ polymerization of PISSE promotes the electrode-electrolyte interface compatibility, resulting in the Li//PISSE60%@LiFePO4 cells with 71% (initial capacity: 111.8 mAh g−1) of capacity retention after 800 cycles at 0.5 C. Moreover, the PISSE also exhibits compatibility with high voltage cathode LiFe0.2Mn0.8PO4 cells up to 4.3V. Therefore, this work provides a practical strategy to fabricate stable solid electrolytes for next-generation lithium metal batteries.

An improved study of near-well zone flooding for polymer solution through the mechanical shearing and forchheimer flow simulation

Polymer flooding is a vital tertiary oil recovery technology that maintains stable oilfield production during high water cut periods. This technique involves improving the viscosity of injection fluids by using polymer solutions, which help reduce the mobility ratio of oil and water, expand sweep efficiency, and increase swept area. However, highly viscous polymer solutions form complex spatial structures between and within molecules, which leads to significant viscosity loss rates in the near-wellbore region at the well injection end. To address this issue, a near-wellbore shear device with nozzle completion was built to study the shear effects of two types of polymers. The study revealed that the polymer solutions destroyed intermolecular and intramolecular structures after shearing, causing the water cut curves to shift up and left. Based on the Brinkman equation of COMSOL Multiphysics's Carreau model, the percolation model of polymer solution in the near-well zone was constructed. COMSOL simulation results also demonstrated that high-speed shearing at the nozzle orifice was the primary cause of shear degradation. Therefore, it is essential to adopt reasonable injection intensities and proper polymer solution concentrations in oilfields to maximize oil recovery and minimize production costs. This is of crucial engineering significance in guiding oilfield production and optimizing injection parameters and processes.

Higher-order interdependent percolation on hypergraphs

A fundamental concern on the robustness of hypergraphs lies in comprehending how the failure of individual nodes affects the hyperedges they are associated with. To address the issue, we propose a simple but novel percolation model that takes into account the dependency of hyperedges on their internal nodes, where the failure of a single node can lead to the dissolution of its associated hyperedge with a probability β. Based on a newly proposed analytical method of percolation theory on hypergraphs, our research reveals that the impact of mean cardinality on the system robustness varies with β. For a large value of β, a larger mean cardinality increases the fragility of hypergraphs, while for a small β, a larger mean cardinality enhances the robustness of hypergraphs. Additionally, our research uncovers divergent effects of hyperdegree distribution on system robustness between monolayer and double-layer hypergraphs. Specifically, monolayer hypergraphs with scale-free hyperdegree distribution exhibit higher robustness, while Poisson hyperdegree distributions lead to stronger robustness in double-layer hypergraphs. These findings provide valuable insights into the robustness of hypergraphs and its dependency on hyperdegree distributions and mean cardinality, contributing to a more comprehensive understanding of the complexities of robustness in complex systems. Furthermore, the development of the percolation model enriches our understanding of node-hyperedge interactions within complex systems.

3D Percolation Modeling for Connectivity and Permeability of Sandstone with Different Pore Distribution Characteristics

3D Percolation Modeling for Connectivity and Permeability of Sandstone with Different Pore Distribution Characteristics

Cascading failures in interdependent networks with reinforced crucial nodes and dependency groups

Previous studies of group percolation models in interdependent networks with reinforced nodes have rarely addressed the effects of the degree of reinforced nodes and the heterogeneity of group size distribution. In this paper, a cascading failure model in interdependent networks with reinforced crucial nodes and dependency groups is investigated numerically and analytically. For each group, we assume that if all the nodes in a group fail on one network, a node on another network that depends on that group will fail. We find that rich percolation transitions can be classified into three types: discontinuous, continuous, and hybrid phase transitions, which depend on the density of reinforced crucial nodes, the group size, and the heterogeneity of group size distribution. Importantly, our proposed crucial reinforced method has higher reinforcement efficiency than the random reinforced method. More significantly, we develop a general theoretical framework to calculate the percolation transition points and the shift point of percolation types. Simulation results show that the robustness of interdependent networks can be improved by increasing the density of reinforced crucial nodes, the group size, and the heterogeneity of group size distribution. Our theoretical results can well agree with numerical simulations. These findings might develop a new perspective for designing more resilient interdependent infrastructure networks.

Minimal center-of-mass energy required for QGP formation in pp and AA collisions

We discuss the conditions for QGP formation under the Color String Percolation Model. Since the observables in the percolation theory are sensitive to the system size, we expect the finite size effects take a relevant contribution to the estimation of the color string observables, such as the transition temperature or the center of mass energy needed for the QGP formation. We observe that pp collisions (small systems) require around 20 times bigger center of mass energy than heavy ion collisions. Our results are consistent with the experiments claiming that the QGP has been observed.

Soft and hard scales of the transverse momentum distribution in the color string percolation model

In color string models, the transverse momentum distribution (TMD) is obtained through the convolution of the Schwinger mechanism with the string tension fluctuations distribution. Considering a q-Gaussian distribution for these fluctuations, the TMD becomes a hypergeometric confluent function that adequately reproduces the characteristic scales at low and high p T values. In this approach, the hard scale of the TMD is a consequence of considering a heavy-tailed distribution for the string tension fluctuations whose width rises as , multiplicity or centrality increases. In this paper, we introduce broader information of the TMD in the color string percolation model by determining the color suppression factor, which now also depends on the parameters of the q-Gaussian. To this end, we analyze the reported data on pp and AA collisions at different center of mass energies, multiplicities, and centralities. In particular, for minimum bias pp collisions, we found that the q-Gaussian parameters and the effective temperature are monotonically increasing functions of the center of mass energy. Similar results are found for AA collisions as a function of the centrality at fixed . We summarize these results in a phase diagram that indicates the q-Gaussian parameters region allowing the quark–gluon plasma formation.

Agent-based modeling of high-rise building fires reveals self-rescue behaviors and better fire protection designs

It is always challenging to seek external rescue assistance in high-rise building fires. Therefore, it is critical for individuals to master survival skills. For crowd dynamics modeling, previous research focused on numerical simulations and building designs with little attention to the self-rescue mechanism. It is critical to understanding crowd evacuations and better response strategies. We modeled the Grenfell Tower (a high-rise building with a complicated structure) case in 2017. Based on the percolation and social force models, we build an agent-based model to simulate individual behaviors inside. We obtain the optimal solution and robust paralleled outcomes under all counterfactual situations based on precisely matching tangible case outcomes (fire duration, deaths, and injuries). For individuals, mastering self-rescue skills is better at reducing social losses (deaths & injuries). In terms of high-rise buildings design, the central alarm system is also useful to reduce them. Besides, the crowd evacuation guided by the social force model also reduces deaths & injuries. This work provides insight into better high-rise building design and practical response strategies for societies. The central alarm system and fire-proof materials should be used in high-rise buildings. The residents should have routine training in social force-based evacuations and survival (self-rescue) skills to better the evacuation process and outcome under natural disasters and social emergencies.

The breakdown characteristic of porous dielectric discharge based on percolation structure

Porous dielectrics have received increasing attention in plasma sterilization, all-solid-state battery technology, and surface functionalization of biological tissue materials. Due to their complex structure and discharge characteristics, the current researches are hard to quantify the stochastic properties of porous dielectrics. In this paper, we used a percolation structure to simulate the discharge process in a 2D porous dielectric. The simulation results of the 2D percolation model are similar to that of 2D real porous slices, which can characterize the physical properties of the porous dielectric well while greatly reducing the time required for simulation. In addition, simulations on percolation models with different porosity and lattice size are performed. When the porosity and lattice size remain constant, tortuosity and Debye radius are the main factors affecting the breakdown of the percolation model. With the decrease in porosity, the Pashcen curve shifts to the upper right. With the decrease in lattice size, the Pashcen curve moves higher. The results show correlations between random parameters and Paschen curves. This study presents a novel simulation approach for the theoretical analysis of porous dielectric and improves the simulation efficiency at the same time. In addition, this new model is also applied to quantify the impact mechanism of random parameters such as porosity and lattice size on porous dielectric discharge.

The Blume–Emery–Griffiths Model on the FAD Point and on the AD Line

We analyse the Blume–Emery–Griffiths (BEG) model on the lattice Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^d$$\\end{document} on the ferromagnetic-antiquadrupolar-disordered (FAD) point and on the antiquadrupolar-disordered (AD) line. In our analysis on the FAD point, we introduce a Gibbs sampler of the ground states at zero temperature, and we exploit it in two different ways: first, we perform via perfect sampling an empirical evaluation of the spontaneous magnetization at zero temperature, finding a non-zero value in d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document} and a vanishing value in d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}. Second, using a careful coupling with the Bernoulli site percolation model in d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}, we prove rigorously that under imposing +\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$+$$\\end{document} boundary conditions, the magnetization in the center of a square box tends to zero in the thermodynamical limit and the two-point correlations decay exponentially. Also, using again a coupling argument, we show that there exists a unique zero-temperature infinite-volume Gibbs measure for the BEG. In our analysis of the AD line we restrict ourselves to d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} and, by comparing the BEG model with a Bernoulli site percolation in a matching graph of Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^2$$\\end{document}, we get a condition for the vanishing of the infinite-volume limit magnetization improving, for low temperatures, earlier results obtained via expansion techniques.

Distances in 1‖x-y‖2d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{\\Vert x-y\\Vert ^{2d}}$$\\end{document} Percolation Models for all Dimensions

We study independent long-range percolation on Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}^d$$\\end{document} for all dimensions d, where the vertices u and v are connected with probability 1 for ‖u-v‖∞=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert u-v\\Vert _\\infty =1$$\\end{document} and with probability p(β,{u,v})=1-e-β∫u+0,1d∫v+0,1d1‖x-y‖22ddxdy≈β‖u-v‖22d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(\\beta ,\\{u,v\\})=1-e^{-\\beta \\int _{u+\\left[ 0,1\\right) ^d} \\int _{v+\\left[ 0,1\\right) ^d} \\frac{1}{\\Vert x-y\\Vert _2^{2d}}d x d y } \\approx \\frac{\\beta }{\\Vert u-v\\Vert _2^{2d}}$$\\end{document} for ‖u-v‖∞≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert u-v\\Vert _\\infty \\ge 2$$\\end{document}. Let u∈Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u \\in \\mathbb {Z}^d$$\\end{document} be a point with ‖u‖∞=n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert u\\Vert _\\infty =n$$\\end{document}. We show that both the graph distance D(0,u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D(\ extbf{0},u)$$\\end{document} between the origin 0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{0}$$\\end{document} and u and the diameter of the box {0,…,n}d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{0,\\ldots , n\\}^d$$\\end{document} grow like nθ(β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{\ heta (\\beta )}$$\\end{document}, where 0<θ(β)<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\ heta (\\beta ) < 1$$\\end{document}. We also show that the graph distance and the diameter of boxes have the same asymptotic growth when two vertices u, v with ‖u-v‖2>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert u-v\\Vert _2 > 1$$\\end{document} are connected with a probability that is close enough to p(β,{u,v})\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(\\beta ,\\{u,v\\})$$\\end{document}. Furthermore, we determine the asymptotic behavior of θ(β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta (\\beta )$$\\end{document} for large β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}, and we discuss the tail behavior of D(0,u)‖u‖2θ(β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{D(\ extbf{0},u)}{\\Vert u\\Vert _2^{\ heta (\\beta )}}$$\\end{document}.

Infrared dielectric function of GaAs1−xPx semiconductor alloys near the reststrahlen bands

The infrared dielectric function of thick GaAs1−xPx alloy layers grown on (001) GaAs substrates by hydride vapor phase epitaxy was investigated in the reststrahlen region using Fourier-transform infrared ellipsometry. The spectra are influenced by the Berreman artifact at the longitudinal optical phonon frequency of the GaAs substrate and by interference fringes due to the finite layer thickness. The ellipsometric angles were analyzed to determine the dielectric function of the alloy layer. Two-mode behavior, including strong GaAs-like and GaP-like optical phonons, was observed, confirming the results of Verleur and Barker [Phys. Rev. 149, 715 (1966)]. Due to the increased sensitivity of ellipsometry in the reststrahlen region, several weak phonon features could also be seen. The lattice absorption peaks are asymmetric and show side bands at the lower and higher frequencies. A single additional peak, as suggested by the percolation model, does not describe the spectra. The cluster model proposed by Verleur and Barker is a better fit to the data. Due to the broadening of the phonon absorption peaks, the authors were unable to find a unique decomposition into multiple components.

Percolation mechanism of the graphene nanoplatelets/elastomeric flexible sensing nanocomposite under an applied compressive strain

The pressure/resistance sensitivity of the GNP elastomeric nanocomposite is investigated using a finite element percolation model. The simulation model creates an elastomeric RVE, including solid GNP disks, and updates the position and direction of GNPs after the applied pressure. The percolation model predicts the tunneling paths between GNP intersections between the two electrodes. The percolation network between electrodes is promoted through the prevalent contact mechanism between GNPs. The developed numerical model simulates the electrical conductivity and pressure/resistance response in a good correlation with experimental data. Results explain the dominant factors of alignment direction and volume fraction on the resistance response. The resistivity of the nanocomposite reduces significantly with critical distance with a reduced variation rate for lower volume fractions. Results indicate linear pressure/resistance sensitivity under a compressed strain up to 0.16. The results reveal a controllable sensitivity through the appropriate choice of GNP alignment direction, aspect ratio, and volume fraction.

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.