Showing papers on "Percolation published in 1986"
TL;DR: In this article, the percolation models of immiscible displacement in porous media are discussed, with emphasis on the critical behavior, including fractal nature of the nonwetting fluid configuration at breakthrough in drainage, and the size distribution of the residual non-wetting clusters in imbibition.
Abstract: The observable consequences of percolation models of immiscible displacement in porous media are discussed, with emphasis on the critical behavior. At the microscopic level, these include the fractal nature of the nonwetting fluid configuration at breakthrough in drainage, and the size distribution of the residual nonwetting clusters in imbibition. At the macroscopic level, it is suggested that percolation ideas are consistent with the usual multiphase Darcy equations, and critical behaviors of the relative permeability and capillary pressure curves are obtained. By using these results, predictions are made for the shape of the fluid saturation profiles near the percolation thresholds in the presence of buoyancy or viscous pressure gradients. Finally, it is pointed out that very close to the percolation thresholds, the diverging correlation length requires these macroscopic ideas to be modified. A simple way of doing this is suggested.
208 citations
TL;DR: In this article, it was shown that no percolation is possible if for short bonds Kx,y≦p = 1, and the occupation probability of a bond −Kx,Y, has a slow power decay as a function of the bond's length.
Abstract: We consider one dimensional percolation models for which the occupation probability of a bond −Kx,y, has a slow power decay as a function of the bond's length. For independent models — and with suitable reformulations also for more general classes of models, it is shown that: i) no percolation is possible if for short bondsKx,y≦p =1. This dichotomy resembles one for the magnetization in 1/|x−y|2 Ising models which was first proposed by Thouless and further supported by the renormalization group flow equations of Anderson, Yuval, and Hamann. The proofs of the above percolation phenomena involve (rigorous) renormalization type arguments of a different sort.
175 citations
160 citations
TL;DR: In this article, the moments of the cluster size distribution for single events in a finite-size percolation model and in multifragmentation of atomic nuclei are studied for both systems.
Abstract: The moments of the cluster size distribution are studied for single events in a finite-size percolation model and in multifragmentation of atomic nuclei. It is shown that both systems break up in roughly the same way.
143 citations
TL;DR: In this paper, the existence of spontaneous magnetization in one-dimensional independent bond percolation models is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size.
Abstract: Consider a one-dimensional independent bond percolation model withpj denoting the probability of an occupied bond between integer sitesi andi±j,j≧1. Ifpj is fixed forj≧2 and\(\mathop {\lim }\limits_{j \to \infty }\)j2pj>1, then (unoriented) percolation occurs forp1 sufficiently close to 1. This result, analogous to the existence of spontaneous magnetization in long range one-dimensional Ising models, is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size. Oriented percolation is shown to occur forp1 close to 1 if\(\mathop {\lim }\limits_{j \to \infty }\)jspj>0 for somes<2. Analogous results are valid for one-dimensional site-bond percolation models.
128 citations
TL;DR: In this article, the authors considered a new percolation process, "percolation with trapping", in which one species (the displaced fluid) starts out at occupation fraction p = 1, but as p decreases only the infinite (connected) cluster is depleted; the finite (disconnected) clusters remain the same as when they are first detached from the infinite cluster.
Abstract: A novel form of percolation is considered which is motivated by models of the displacement of one fluid by another from a porous medium. The physical idea is that if the displaced phase is incompressible, then regions of it which are surrounded by the displacing fluid become 'trapped' and cannot subsequently be invaded. The authors thus consider a new percolation process, 'percolation with trapping', in which one species (the displaced fluid) starts out at occupation fraction p=1, but as p decreases only the infinite (connected) cluster is depleted; the finite (disconnected) clusters remain the same as when they are first detached from the infinite cluster. It is argued that the critical behaviour of percolation with trapping can be understood in terms of ordinary percolation exponents. In particular, the size distribution of the finite clusters at the end of the process has the same power law behaviour as in ordinary percolation. Relations with the process of invasion percolation are discussed.
111 citations
TL;DR: The protonic conduction process described here for low hydration and previously for high hydration is viewed as percolative proton transfer along threads of hydrogen-bonded water molecules, which explains the invariance of h(c) to change in pH and solvent.
Abstract: The framework of percolation theory is used to analyze the hydration dependence of the capacitance measured for protein samples of pH 3-10, at frequencies from 10 kHz to 4 MHz For all samples there is a critical value of the hydration at which the capacitance sharply increases with increase in hydration level The threshold hc = 015 g of water per g of protein is independent of pH below pH 9 and shows no solvent deuterium isotope effect The fractional coverage of the surface at hc is in close agreement with the prediction of theory for surface percolation We view the protonic conduction process described here for low hydration and previously for high hydration as percolative proton transfer along threads of hydrogen-bonded water molecules A principal element of the percolation picture, which explains the invariance of hc to change in pH and solvent, is the sudden appearance of long-range connectivity and infinite clusters at the threshold hc The relationship of the protonic conduction threshold to other features of protein hydration is described The importance of percolative processes for enzyme catalysis and membrane transport is discussed
103 citations
TL;DR: In this article, the authors propose a model for percolation in liquids, the model of extended spheres, which permits connectivity to be studied in the context of, but independently from, liquid structure.
Abstract: Problems involving percolation in liquids (i.e., involving connectivity of some sort) range from the metal–insulator transition in liquid metals to the properties of supercooled water. A common theme, however, is that connectivity can be distinguished from interaction and that one should not be slighted in order to describe the other. In this paper we suggest a model for percolation in liquids—the model of extended spheres—which permits connectivity to be studied in the context of, but independently from, liquid structure. This model is solved exactly in the Percus–Yevick approximation, revealing the existence of an optimum liquid structure for percolation. We analyze this behavior by first deriving an explicit diagrammatic representation of the Percus–Yevick theory for connectivity and then studying how the various diagrams contribute. The predictions are in excellent qualitative agreement with recent Monte Carlo calculations.
97 citations
96 citations
TL;DR: In this paper, the fractal dimensionality of the perimeter of the square lattice is found to depend on the size of adsorbent particles used to measure it: if the vacant perimeter sites have only nearest-neighbour connectivity then the perimeter has dimension De = 1.37+or 0.03, instead of that of the hull (Dh=1.75), and the mass of internal dangling sites, singly and doubly connected sites, the number of 'blobs' and the average linear distance between entry into the exit from a 'blob
Abstract: Site percolation clusters are simulated at the percolation threshold on the square lattice. An algorithm for walks around each cluster is used to obtain information on its fractal geometry. The fractal dimensionality of the external perimeter is found to depend on the size of adsorbent particles used to measure it: if the vacant perimeter sites have only nearest-neighbour connectivity then the perimeter has dimension De=1.37+or-0.03, instead of that of the hull (Dh=1.75). The authors also measured the mass of internal dangling sites (surrounded by the backbone's 'blobs'), singly and doubly connected sites, the number of 'blobs' and the average linear distance between entry into the exit from a 'blob'.
92 citations
TL;DR: In this article, the percolation transition in AOT/water/isooctane microemulsions has been studied by means of dielectric spectroscopy.
Abstract: The percolation transition in AOT/water/isooctane microemulsions has been studied by means of dielectric spectroscopy. At the percolation threshold the conductivity of the system increases sharply and the static dielectric permittivity attains a large maximum. It is shown that the frequency dependence of the permittivity can be well described by simple scaling relations of percolation theory. The critical exponent u of the frequency dependence has been found to be equal to u=0.62±0.02 and independent of temperature and microemulsion composition. The temperature dependence of the critical volume fraction for percolation has been related to the temperature dependence of the permittivity of the system far away from the percolation threshold.
TL;DR: In this paper, a review of various mechanisms which occur during immiscible displacements in 2-dimensional networks of interconnected capillaries is given, and the physical laws governing meniscus equilibrium and flow conditions lead to different statistical models (mean field, percolation, DLA).
Abstract: A review is given of various mechanisms which occur during immiscible displacements in 2-dimensional networks of interconnected capillaries. We show how the physical laws governing meniscus equilibrium and flow conditions lead to different statistical models (mean field, percolation, DLA…).
TL;DR: In this article, the authors apply the model to a two-dimensional surface stretched in one direction, and show that the cracks developing in the material exhibit properties similar to each other, at least at the molecular level.
Abstract: Interest is growing in the wide variety of fractal objects1 found in nature; these include electric discharge patterns2, infinite percolation clusters3,4, branched polymers5,6 and surface irregularities7. The question of whether surface cracks, such as those that occur in protective coatings, also lead to fractal-like structures is another problem of great interest. Here we study this question theoretically, making use of a molecular model for material failure, introduced previously8, which is based on the kinetic theory of fracture9,10. The model is applied to a two-dimensional surface stretched in one direction, and our results show that the cracks developing in the material exhibit properties similar to each other, at least at the molecular level. For a wide range of values of the elastic constants of the material, the fractal dimensionality is found to have a universal value, D = 1.27 ± 0.02.
TL;DR: In this article, a review is given of the studies of the dimensionality characteristics of percolation clusters, and scaling relationships between different dimensionalities, as well as relationships between dimensionalities and conventional critical exponents are discussed.
Abstract: A review is given of the studies of the dimensionality characteristics of percolation clusters. The purely geometric nature of a percolation phase transition and the great variety of the quantities exhibiting critical behavior make this geometric approach both informative and useful. In addition to the fractal dimensionality of a cluster and its subsets (such as the backbone, hull, and other dimensionalities), it is necessary to introduce additional characteristics. For example, the maximum velocity of propagation of excitations is determined by the chemical dimensionality of a cluster, and the critical behavior of the conductivity, diffusion coefficient, etc., is determined by spectral (or other related to it) dimensionalities. Scaling relationships between different dimensionalities, as well as relationships between dimensionalities and conventional critical exponents are discussed.
TL;DR: In this paper, a random walk model for describing mixed alkali effects in ionic conductors of the β-alumina type is proposed. But the model is restricted to the case of a tracer ion.
Abstract: We investigate a random walk model for describing mixed alkali effects in ionic conductors of the β‐alumina type. In our model, the observed drastic variation in the transport properties of the mixed crystals is related to critical properties near a percolation threshold, which originates from the blocking of conduction paths by complexes containing the substituted ions. The motion of a tracer ion and the conductivity are obtained by using the Monte Carlo technique and are discussed in relation to experiments. It is pointed out that anomalous diffusion, which occurs near percolation, leads to a characteristic frequency dependence of the dynamic conductivity, which should be experimentally accessible.
TL;DR: Application de la methode, permettant de resoudre le probleme du ralentissement critique dans les simulations numeriques des systemes physiques, au cas du reseau de resistances aleatoire 2D au seuil de percolation.
Abstract: Critical slowing down poses a major obstacle to reaching the steady-state distribution in large-scale numerical simulations. We demonstrate how to alleviate this problem by means of Fourier acceleration, a method consisting of updating in $k$ space with a $k$-dependent time step. The method is general and applicable to a wide range of problems. We demonstrate its use by numerical experiments on random resistor networks at the percolation threshold.
TL;DR: In this paper, the dielectric properties of carbon black filled crosslinked polyethylene composites are investigated and the authors show that the dissipation factor-concentration curves are bell-shaped with maximum values at approximately the percolation concentration.
Abstract: This paper reports results on the dielectric properties of carbon black filled crosslinked polyethylene composites. These systems are shown to follow percolative type models. The dielectric constant increases slowly, with carbon black concentration, up to roughly the percolation concentration and then increases rapidly over the whole concentration ranges studied. The dissipation factor-concentration curves are bell-shaped with maximum values at approximately the percolation concentration. The dielectric properties of these systems are discussed in terms of interfacial Maxwell-Wagner polarization effects.
TL;DR: A percolation model combined with Monte Carlo method has been applied to two-dimensional misoriented short fiber composites with the aim of predicting the effective electrical conductivity of the composite.
Abstract: A percolation model combined with Monte Carlo method has been applied to two‐dimensionally misoriented short fiber composites with the aim of predicting the effective electrical conductivity of the composite. Then the threshold volume fraction of fiber (VFth) at and above which the composite becomes conductive was computed for several cases of misoriented short fiber composites, i.e., from completely random to reasonably well oriented short fiber composites. It was found in this study that the larger the fiber aspect ratio and the more randomly fibers are oriented, the smaller the value of VFth becomes.
TL;DR: In this paper, the properties of random resistor and flow networks are studied as a function of the density, p, of bonds which permit transport, and it is shown that percolation is sufficient for bulk transport, in the sense that the conductivity and flow capacity are bounded away from zero whenever p exceeds an appropriately defined threshold.
Abstract: The properties of random resistor and flow networks are studied as a function of the density,p, of bonds which permit transport. It is shown that percolation is sufficient for bulk transport, in the sense that the conductivity and flow capacity are bounded away from zero wheneverp exceeds an appropriately defined percolation threshold. Relations between the transport coefficients and quantities in ordinary percolation are also derived. Assuming critical scaling, these relations imply upper and lower bounds on the conductivity and flow exponents in terms of percolation exponents. The conductivity exponent upper bound so derived saturates in mean field theory.
TL;DR: In this article, a numerical procedure is used to solve the connectedness Ornstein-Zernike equation in the Percus-Yevick approximation for the square-well fluid system.
Abstract: A numerical procedure is used to solve the connectedness Ornstein–Zernike equation in the Percus–Yevick approximation for the square‐well fluid system. The pair‐connectedness function, direct connectedness function, and average cluster size are calculated in order to locate the percolation transition from scaling law behavior.
TL;DR: In this article, a recently proven equivalence between certain random walks with memory and the hull of percolation clusters in two dimensions is used to estimate the fractal dimension of the latter by means of Monte Carlo simulations.
Abstract: A recently proven equivalence between certain random walks with memory and the hull of percolation clusters in two dimensions is used to estimate the fractal dimension of the latter by means of Monte Carlo simulations. The authors calculate the value DH=1.750+or-0.002, in agreement with a recent conjecture that DH=7/4 exactly.
TL;DR: In this article, the authors discussed the relation between the critical behavior of elastic percolation networks in which the bond-bending forces are present, and the percolations conductivity.
Abstract: The author discusses the relation between the critical behaviour of elastic percolation networks in which the bond-bending forces are present, and that of percolation conductivity. They propose that if the elastic and geometrical thresholds of the network are equal, the critical exponent f of the elastic moduli, is given by, f=t+2 nu , where t is the conductivity exponent and nu the correlation length exponent. This predicts that f(d=2) approximately=3.96, in complete agreement with the most recent and accurate estimate of f. They also discuss the applicability of the present elastic percolation models to real systems such as gels.
TL;DR: Network representations and percolation theory are used to develop expressions for pore closure time and the evolution of accessible volume, effective diffusivity, and conversion rates of noncatalytic gas-solid reactions.
Abstract: Previous models of noncatalytic gas-solid reactions are based primarily on parallelpore representations, thus excluding important topological effects of the porous medium. This paper utilizes network representations and percolation theory to develop expressions for pore closure time and the evolution of accessible volume, effective diffusivity, and conversion rates.
TL;DR: In this paper, the authors present a method for analyzing currentvoltage relationships to obtain information on the dominant conduction mechanisms of CoAl2O3 and AuAl 2O3 composite films, based on the analysis of the derivative of the logarithmic conductivity with respect to inverse applied electrical field.
Abstract: We present a method for analyzing current‐voltage relationships to obtain information on the dominant conduction mechanisms. The method is based on the analysis of the derivative of the logarithmic conductivity with respect to inverse applied electrical field. To illustrate the method we apply it to the study of Co‐Al2O3 and Au‐Al2O3 composite films. The former material displays space‐charge‐limited conduction, which is due to a high density of trap states in the oxide matrix. On the other hand, Au‐Al2O3 shows evidence of percolation and tunneling between metal particles.
TL;DR: In this article, the authors analyzed the mass and radius of gyration Rg of the burning tree clusters as a function of time, with alpha and beta being the fractal exponents.
Abstract: Forest fires under directional constraints, such as wind or local topography, generalise the bond percolation problem of symmetrical fires. The authors analyse the mass M and the radius of gyration Rg of the burning tree clusters as a function of time. M approximately talpha and Rg approximately tbeta , with alpha and beta being the fractal exponents.
TL;DR: In this paper, the authors investigated the influence of the structure of carbon-black particles when transferred over into the polymer matrix on the conductivity level and attempted to explain the change observed in the percolation threshold.
Abstract: Several reports have appeared recently on conductive polymeric composites based on mixtures of nonconductive polymers with conductive solid microadditives [1-4] Among the various combinations, polymer-carbon-black composites are of special interest because they combine the good application properties of commerical polymers (processability, low cost) with relatively high conductivity, thus offering an interesting type of light material The low concentration of the additive required to form a conductive network is specially attractive In a previous letter we have reported [5] on the electrical conductivity (a) variation ofpolycarbonate (PC)/carbon composites as a function of the concentration of the microadditive The influence of film thickness and temperature dependence on the conductivity level reached was also examined [5] In these systems the supermolecular structure of the polymer matrix has only a minor influence on the carrier mobility [6, 7] The conductivity mainly depends on the percolation threshold for the conducting particles In this system, carriers are transported above the percolation threshold through the carbon network predominantly involving a tunnelling conduction mechanism [6, 8] The specific type of initial chain-like carbon-black structure, defined through the aggregation of primary particles (few nanometres in size) in strongly bonded aggregates of several hundreds of nanometres and weakly bonded agglomerates of these aggregates of several micrometres, presumably plays an important part in defining the onset of the conductivity threshold [9] The purpose of the present letter is twofold: first, to report on the influence of the structure of carbon-black particles when transferred over into the polymer matrix on the conductivity level; secondly, to attempt to explain the change observed in the percolation threshold We have used two types of carbon-black materials: XE2 from Philips Petroleum (A) having a welldeveloped chain-like structure (dibutyl phthalate absorption (DBP) - 400 cc/100 g) and acetylene carbon black from SEA Tudor (B) with a lessdeveloped structure (DBP-~ 140 cc/100 g) Several mixtures of polycarbonate (bisphenol A) from Bayer with different carbon-black concentrations were prepared using a plastograph Polycarbonate was melted at the working temperature of 230 ° C Thereafter the filler was added and finally the mixture was mechanically stirred at 50 rpm, for 10 m until homogeneous materials were obtained Compressionmoulded films (T = 230°C, p = 50bar), 500#m thick and 2 x 2cm of lateral dimensions, were