Percolation Point Research Articles

Migration of Triplet Excitations in Chrysene Glass Films

ABSTRACT Research results of delayed fluorescence decay kinetics in solid chrysene films deposited on quartz substrates in vacuum are presented. It is found that at the temperatures which are close to the temperature of liquid nitrogen, phosphorescence and annihilation delayed fluorescence decay kinetics are described by non-exponential function. The data obtained were discussed from the point of formal kinetics and percolation T–T annihilation models.

Prediction of crack deflection in porous/dense ceramic laminates

The arrangement of ceramic layers in laminated structures is an interesting way to enhance the flaw tolerance of brittle ceramic materials. The interfaces are expected to deflect cracks, increasing the fracture energy of the laminate compared to a monolithic material and thus raising the toughness. Laminates have been fabricated with alternating dense and porous layers of the same material, i.e. SiC or B 4C, in order to obtain a good chemical compatibility between the laminas and almost no thermal residual stresses. Porosity, in the porous layers, is achieved by incorporating organic particles which are removed during the debinding step. In this context, the target of this study is to predict the volume fraction of pores, in the porous layer, required to cause crack deflection. The proposed criterion derives from an energy balance. It relies on a two-scale analysis taking into account the laminated structure of the material. It can be written in terms of two relevant material parameters: the ratio of Young's moduli of the dense and porous materials and toughness ratios. A unique function depending on the volume fraction of pores can be used to express the two above-mentioned ratios. Assuming a cubic lattice of spherical voids, the parameters of the porous ceramic depend linearly on the porosity and vanish at the point of percolation of pores. As a consequence, the criterion can be rewritten in term of a single parameter: the porosity. Crack deflection is permitted only for very high values of the porosity. Predicted values agree satisfactorily with experiments on SiC and B 4C. The comparison with the He and Hutchinson criterion shows that this latter underestimates the correct value.

Nanoscale occurrence of Pb in an Archean zircon

We report, for the first time, a direct, atomic-scale characterization of Pb in zircon (4.4–3.1 Ga) from the early Archean Yilgarn craton in Australia using high-resolution HAADF-STEM. Two forms of Pb have been identified: Pb concentrated at ∼3 atom% as a nanoscale patch in zircon structure, and Pb concentrated within the amorphous domain created by fission fragment damage. The first result suggests that the Pb atoms directly substitute for Zr 4+ in the zircon structure, and the latter observation demonstrates that Pb diffusion can occur through amorphous regions created by radiation damage, although volume diffusion is typically considered to be the dominant mechanism for Pb diffusion. Beyond the first percolation point, i.e., when the amorphous domains overlap and form a fully interconnected network of amorphous domains, there is a new pathway for the diffusion of Pb that is faster than volume diffusion through crystalline zircon.

Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution

Analytic approximations for percolation points in two–dimensional and three–dimensional particulate arrays have been reported for only a very few, simple particle geometries. Here, an analytical approach is presented to determine the percolative properties (i.e. statistical cluster properties) of permeable ellipsoids of revolution. We generalize a series expansion technique, previously used by other authors to study arrays of spheres and cubes. Our analytic solutions are compared with Monte Carlo simulation results, and show good agreement at low particle aspect ratio. At higher aspect ratios, the analytic approximation becomes even more computationally intensive than direct simulation of a number of realizations. Additional simulation results, and simplified, closed–form bounding expressions for percolation thresholds are also presented. Limitations and applications of the asymptotic expressions are discussed in the context of materials design and design of sensor arrays.

Two-Dimensional vs. Three-Dimensional Clustering and Percolation in Fields of Overlapping Ellipsoids

Maximum depth-to-particle-dimension ratios in which systems can be treated as two-dimensional (2D) rather than three-dimensional (3D) systems in determining percolative properties have not been reported. This problem is of great technological significance. 3D solutions for percolation in even low-density systems pose much more intensive computational problems than their 2D analogs and also result in significantly different predictions for percolation onset. Moreover, many materials and sensing applications require analysis of domains of finite thickness. Adequate loading of particles is required, e.g., in electrodes in advanced batteries and fuel cells to ensure good conductivity. Adequate deployment of sensors into fields of finite thickness such as oblate, neuronal cells is required, e.g., to detect specific ions. A systematic determination of the effect of these arrangements on percolation properties is needed for both applications. Here, we provide comparisons of cluster sizes, densities, and percolation points among monodisperse, 2D and 3D systems of overlapping ellipsoids by systematically increasing the depth of the 3D system relative to particle dimensions. We investigate the effect of several boundary condition assumptions on the resulting particle orientations, emphasizing the probability of formation of large clusters. A method of experimental determination of percolation onset is also suggested, using the maximum change in cluster size. © 2004 The Electrochemical Society. All rights reserved.

Spreading with immunization in high dimensions

We investigate a model of epidemic spreading with partial immunizationwhich is controlled by two probabilities, namely, for first infections,p0, and forreinfections, p. When the two probabilities are equal, the model reduces to directed percolation, while forperfect immunization one obtains the general epidemic process belonging to theuniversality class of dynamical percolation. We focus on the critical behaviour in thevicinity of the directed percolation point, especially for high number of dimensionsd>2. It is argued that the clusters of immune sites are compact for . This observation implies that a recently introduced scaling argument,suggesting a stretched exponential decay of the survival probability forp = pc, in one spatial dimension, wherepc denotes the critical threshold for directed percolation, should apply for any number ofdimensions and perhaps for d = 4 as well. Moreover, we show that the phase transition line, connectingthe critical points of directed percolation and of dynamical percolation,terminates at the critical point of directed percolation with vanishing slope ford<4 and with finite slope for . Furthermore, an exponent is identified for the temporal correlation length for the case ofp = pc and p0 = pc−ϵ, , which is different from the exponent of directed percolation. We also improve numerical estimates of severalcritical parameters and exponents, especially for dynamical percolation ford = 4,5.

Percolation in deposits for competitive models in (1+1)-dimensions

The percolation behaviour during the deposit formation, when the spanning cluster was formed in the substrate plane, was studied. Two competitive or mixed models of surface layer formation were considered in (1+1)-dimensional geometry. These models are based on the combination of ballistic deposition (BD) and random deposition (RD) models or BD and family deposition (FD) models. Numerically we find, that for pure RD, FD or BD models the mean height of the percolation deposit h ̄ grows with the substrate length L according to the generalized logarithmic law h ̄ ∝( ln(L)) γ , where γ=1 (RD), γ=0.88±0.02 (FD) and γ=1.52±0.02 (BD). For BD model, the scaling law between deposit density p and its mean height h ̄ at the point of percolation of type p−p ∞∝ h ̄ −1/ν h are observed, where ν h =1.74±0.02 is a scaling coefficient. For competitive models the crossover, corresponding to the RD or FD-like behaviour at small L and the BD-like behaviour at large L are observed.

Nonlinear screening and percolative transition in a two-dimensional electron liquid

A novel variational method is proposed for calculating the percolation threshold, the real-space structure, and the thermodynamical compressibility of a disordered two-dimensional electron liquid. Its high accuracy is verified against prior numerical results and newly derived exact asymptotics. The inverse compressibility is shown to have a strongly asymmetric minimum at a density that is approximately the triple of the percolation threshold. This implies that the experimentally observed metal-insulator transition takes place well before the percolation point is reached.

Stiffness exponents for lattice spin glasses in dimensions $\mathsf{d = 3,\ldots,6}$

The stiffness exponents in the glass phase for lattice spin glasses in dimensions $d=3,...,6$ are determined. To this end, we consider bond-diluted lattices near the T=0 glass transition point $p^*$. This transition for discrete bond distributions occurs just above the bond percolation point $p_c$ in each dimension. Numerics suggests that both points, $p_c$ and $p^*$, seem to share the same $1/d$-expansion, at least for several leading orders, each starting with $1/(2d)$. Hence, these lattice graphs have average connectivities of $\alpha=2dp\gtrsim1$ near $p^*$ and exact graph-reduction methods become very effective in eliminating recursively all spins of connectivity $\leq3$, allowing the treatment of lattices of lengths up to L=30 and with up to $10^5-10^6$ spins. Using finite-size scaling, data for the defect energy width $\sigma(\Delta E)$ over a range of $p>p^*$ in each dimension can be combined to reach scaling regimes of about one decade in the scaling variable $L(p-p^*)^{\nu^*}$. Accordingly, unprecedented accuracy is obtained for the stiffness exponents compared to undiluted lattices ($p=1$), where scaling is far more limited. Surprisingly, scaling corrections typically are more benign for diluted lattices. We find in $d=3,...,6$ for the stiffness exponents $y_3=0.24(1)$, $y_4=0.61(2), y_5=0.88(5)$, and $y_6=1.1(1)$. The result for the upper critical dimension, $d_u=6$, suggest a mean-field value of $y_\infty=1$.

Biopolymer mimicry with polymeric wormlike micelles: Molecular weight scaled flexibility, locked‐in curvature, and coexisting microphases

AbstractGiant and stable wormlike micelles formed in water from a series of poly(ethylene oxide) (PEO)‐based diblock copolymer amphiphiles mimicked the flexibility of various cytoskeletal filaments. The worm diameter (d) was found by cryo‐transmission electron microscopy to scale with the length of the hydrophobic chain (Nh) of the copolymer as d ∼ Nh0.61. By fluorescence video imaging of worm dynamics, we also showed that the persistence length (lP) of wormlike micelles scaled as lP ∼ d2.8, consistent with a fluid aggregate (∼d3) rather than a solid rod (∼d4). By polymerizing the unsaturated bonds of assembled copolymers, fluid worms were converted to solid‐core worms, extending the bending rigidity from that of intermediate filament biopolymers to actin filaments and, in principle, microtubules. Through partial crosslinking, polymerized worms further locked in spontaneous curvature at a novel fluid‐to‐solid percolation point. The dynamics of distinct, branched conformations were also imaged for recently discovered Y‐junctioned wormlike micelles composed of diblocks of high molecular weight (&gt;10–15 kg/mol). Finally, block copolymers of hydrophilic weight fraction close to the transition between a vesicle‐ and worm‐former assembled into both structures, allowing encapsulation of wormlike micelles in giant vesicles reminiscent of cytoskeletal filaments enclosed within cells. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 168–176, 2004

Absorption of light by exchange-coupled ions in a 2D antiferromagnet

The optical absorption spectra of Rb2MnxCd1−xCl4 crystals are experimentally studied in the vicinity of a magnon sideband of the exciton band at a manganese content x ranging from 1.0 to 0.4. Additional absorption bands are observed with an increase in the magnetic structural disorder upon replacement of manganese ions by cadmium ions. An analysis of the evolution of the additional absorption bands in a magnetic field during the spin-flop phase transition and the change in the intensity with variations in the manganese content x demonstrates that these bands are associated with the excitation of the exchange-coupled pairs of manganese ions located in different environments in a plane square lattice. The phase boundary between the antiferromagnetic and spin-flop phases is constructed using the results of optical measurements. The manganese content corresponding to the magnetic percolation point is evaluated.

Scaling behavior of diffusion and reaction processes in percolating porous media

We investigate the diffusion-reaction behavior of two-dimensional pore networks at the critical percolation point. Our results indicate the existence of three distinct regimes of reactivity, determined by parameter xi[triple bond]D/(Kl2), where D is the molecular diffusivity of the reagent, K is its chemical reaction coefficient, and l is the length scale of the pore. First, when the diffusion transport is strongly limited by chemical reaction (i.e., D<<K), we recover the classical scaling behavior Phi approximately Lxi(1/2), where Phi is the mass flux of reagent penetrating the pore space and L is the system size. Second, at an intermediate range of xi values, when the process is influenced by the fractal morphology of the percolation cluster, we observe an anomalous diffusion scaling, Phi approximately L(alpha/2)xiB, with an exponent beta approximately 0.34. Third, in the absence of diffusional limitation (D>>K), the flux of reagent reaches a saturation limit Phi(sat) that scales with the system size as Phi(sat) approximately L(alpha), with an exponent alpha approximately 1.89, corresponding to the fractal dimension of the sample-spanning cluster. We then show that the variation of flux Phi calculated for different network sizes at the second and third regimes can be adequately described in terms of the scaling relation, Phi approximately L(alpha)f(xi/L(z)), where the crossover exponent z approximately 2.69 is consistent with the predicted scaling law alpha=2betaz.

Low-temperature hydrothermal alteration of natural metamict zircons from the Eastern Desert, Egypt

AbstractThe chemical and structural alteration of metamict zircon crystals from a 619 ±17 (2σ) Ma old, posttectonic granite in the southern part of the Eastern Desert, Egypt was studied. The crystals show simple oscillatory growth zones with metamictization–induced fractures, which provided pathways for fluid infiltration. Electron and ion microprobe analyses reveal that metamict, i.e. U and Th–rich, areas are heavily enriched in Ca, Al, Fe, Mn, LREE, and a water species, and have lost Zr and Si as well as radiogenic Pb. These chemical changes are the result of an intensive reaction with a low–temperature (120—200°C) aqueous solution. The chemical reactions probably occurred within the amorphous regions of the metamict network. During the zircon–fluid interactions the metamict structure was partially recovered, as demonstrated by micro-Raman and -infrared measurements. A threshold degree of metamictization, as defined empirically by an α–decay dose, Dc, was necessary for zircons to undergo hydrothermal alteration. It is proposed that Dc marks the first percolation point, where the amorphous domains start to form percolating clusters in the metamict network and where bulk chemical diffusion is believed to increase dramatically. The time of the hydrothermal alteration is determined by a lower intercept age of a U-Pb SHRIMP discordia of 17.9 (2σ) Ma, which is in good agreement with an apatite fission track age of 22.2 (2σ) Ma. The hydrothermal alteration event occurred contemporaneously with the main rifting phase of the Red Sea and widespread low- temperature mineralizations along the Red Sea coast.

Critical droplets and phase transitions in two dimensions

In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point T_c of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability p_B<1, the corresponding site-bond clusters keep on percolating at T_c and the exponents do not change, until p_B=p_CK=1-exp(-2J/kT): for this special expression of the bond weight the critical percolation exponents switch to the 2D Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability p_c such that, for any p_B>=p_c, the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for p_c<p_B<=1 but, for p_B=p_c, they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.

Analytical approximation of the two-dimensional percolation threshold for fields of overlapping ellipses

Percolation of particle arrays is of high interest in microstructural design of materials. While there have been numerous contributions to theoretical modeling of percolation in particulate systems, no analytical approximation for the generalized problem of variable aspect-ratio ellipses has been reported. In the present work, we (1) derive, and verify through simulation, an analytical percolation approach capable of identifying the percolation point in two-phase materials containing generalized ellipses of uniform shape and size; and (2) explore the dependence of percolation on the particle aspect ratio. We validate our technique with simulations tracking both cluster sizes and percolation status, in networks of elliptical and circular particles. We also outline the steps needed to extend our approach to three-dimensional particles (ellipsoids). For biological materials, we ultimately aim to provide direct insight into the contribution of each single phase in multiphase tissues to mechanical or conductive properties. For engineered materials, we aim to provide insight into the minimum amount of a particular phase needed to strongly influence properties.

Porosity effect on the dielectric constant and thermomechanical properties of organosilicate films

This letter reports a study of the porosity effect on material properties of methylsilsesquioxane films, including the dielectric constant, thermal conductivity, and thermal stress behavior. In a porosity range from 0% to 50%, both the dielectric constant and thermal conductivity decreased with increasing porosity and no significant change was observed at the percolation point where pores became interconnected. In comparison, the stress–temperature slope also decreased with porosity, but as the porosity approached the percolation point, the slope showed a large drop of 40%, indicating a significant degradation of the thermomechanical properties due to percolation of pores. Assuming the coefficients of thermal expansion remain at 17 ppm/°C within the porosity range, the change in the stress–temperature slope corresponds to a decrease of the biaxial modulus from 7 to 5 GPa around the percolation point.

Magnetic coupling and magnetoresistance in Fe/Si1−xAgx multilayers

In this letter, we present a study on the magnetic coupling and magnetoresistance (MR) properties in Fe/Si1−xAgx multilayers (MLs) with a granular Si1−xAgx spacer layer. We found that with increasing silver content (x) in a silicon matrix, the magnetic state of MLs varies from a nonmagnetic-coupling state to a weak-antiferromagnetic state around the percolation point of the ∼24-Å-thick granular spacer Si1−xAgx. The MR measurements also reveal an abrupt increase of MR near the same percolation point. These variations are ascribed to the formation of the percolation path in the granular spacer.

Formation of interfacial phase and its effects on the magnetic and transport properties of the La0.82Ca0.18MnO3/La0.18Ca0.82MnO3 composite

Formation of interfacial phase in the La0.82Ca0.18MnO3/La0.18Ca0.82MnO3 composite and associated effects have been experimentally studied. Interfacial phase appears when the composite is sintered at temperatures above 950°C, and its composition experiences a considerable change during the sintering according to the x-ray diffraction analysis. Compared with magnetization, the transport property is relatively insensitive to the phase constitution of the composite. The appearance of the third phase and the resulted extra grain boundaries produce only a weak enhancement of magnetoresistance at the percolation point (~26% in volume fraction) of the new phase.

Percolation-related magnetic coupling and magnetoresistance properties in Fe/Si 1- x Ag x multilayers

In this paper, we present a study of the magnetic coupling and magnetoresistance (MR) properties in Fe/Si1-xAgx multilayers with a granular Si1-xAgx spacer layer. We have found that, with increasing silver content (x) in a silicon matrix, the magnetic state of multilayers changes from a nonmagnetic coupling state to weak antiferromagnetic around the percolation point of the ~2.4 nm thick spacer Si1-xAgx. The MR measurements also reveal an abrupt increase of MR near the same percolation point. These changes are ascribed to the formation of the percolation path in the granular spacer.

Cluster percolation and first order phase transitions in the Potts model

The q-state Potts model can be formulated in geometric terms, with Fortuin–Kasteleyn (FK) clusters as fundamental objects. For vanishing external field, the phase transition of the model can be equivalently described as a percolation transition of FK clusters. In this work, we investigate numerically the percolation behaviour along the line of first-order phase transitions of the 3d 3-state Potts model in a non-vanishing external field and find that the percolation strength exhibits a discontinuity along the entire line. The endpoint is also a percolation point for the FK clusters, but the corresponding critical exponents are neither in the Ising nor in the random percolation universality class.