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Meeks K, Tencer J, Pantoya ML. Percolation of binary disk systems: Modeling and theory. Phys Rev E 2017; 95:012118. [PMID: 28208494 DOI: 10.1103/physreve.95.012118] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/14/2016] [Indexed: 06/06/2023]
Abstract
The dispersion and connectivity of particles with a high degree of polydispersity is relevant to problems involving composite material properties and reaction decomposition prediction and has been the subject of much study in the literature. This work utilizes Monte Carlo models to predict percolation thresholds for a two-dimensional systems containing disks of two different radii. Monte Carlo simulations and spanning probability are used to extend prior models into regions of higher polydispersity than those previously considered. A correlation to predict the percolation threshold for binary disk systems is proposed based on the extended dataset presented in this work and compared to previously published correlations. A set of boundary conditions necessary for a good fit is presented, and a condition for maximizing percolation threshold for binary disk systems is suggested.
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Affiliation(s)
- Kelsey Meeks
- Sandia National Laboratories, Albuquerque, New Mexico 87123, USA
- Mechanical Engineering Department, Texas Tech University, Lubbock, Texas 79409-1021, USA
| | - John Tencer
- Sandia National Laboratories, Albuquerque, New Mexico 87123, USA
| | - Michelle L Pantoya
- Mechanical Engineering Department, Texas Tech University, Lubbock, Texas 79409-1021, USA
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Gouéré JB, Marchand R. Nonoptimality of constant radii in high dimensional continuum percolation. ANN PROBAB 2016. [DOI: 10.1214/14-aop974] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Sasidevan V. Continuum percolation of overlapping disks with a distribution of radii having a power-law tail. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:022140. [PMID: 24032808 DOI: 10.1103/physreve.88.022140] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/25/2013] [Revised: 06/05/2013] [Indexed: 06/02/2023]
Abstract
We study the continuum percolation problem of overlapping disks with a distribution of radii having a power-law tail; the probability that a given disk has a radius between R and R+dR is proportional to R(-(a+1)), where a>2. We show that in the low-density nonpercolating phase, the two-point function shows a power-law decay with distance, even at arbitrarily low densities of the disks, unlike the exponential decay in the usual percolation problem. As in the problem of fluids with long-range interaction, we argue that in our problem, the critical exponents take their short-range values for a>3-η(sr) whereas they depend on a for a<3-η(sr) where η(sr) is the anomalous dimension for the usual percolation problem. The mean-field regime obtained in the fluid problem corresponds to the fully covered regime, a≤2, in the percolation problem. We propose an approximate renormalization scheme to determine the correlation length exponent ν and the percolation threshold. We carry out Monte Carlo simulations and determine the exponent ν as a function of a. The determined values of ν show that it is independent of the parameter a for a>3-η(sr) and is equal to that for the lattice percolation problem, whereas ν varies with a for 2<a<3-η(sr). We also determine the percolation threshold of the system as a function of the parameter a.
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Affiliation(s)
- V Sasidevan
- Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India
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Russo E, Treves A. Cortical free-association dynamics: distinct phases of a latching network. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:051920. [PMID: 23004800 DOI: 10.1103/physreve.85.051920] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/02/2011] [Indexed: 06/01/2023]
Abstract
A Potts associative memory network has been proposed as a simplified model of macroscopic cortical dynamics, in which each Potts unit stands for a patch of cortex, which can be activated in one of S local attractor states. The internal neuronal dynamics of the patch is not described by the model, rather it is subsumed into an effective description in terms of graded Potts units, with adaptation effects both specific to each attractor state and generic to the patch. If each unit, or patch, receives effective (tensor) connections from C other units, the network has been shown to be able to store a large number p of global patterns, or network attractors, each with a fraction a of the units active, where the critical load p_{c} scales roughly like p_{c}≈CS^{2}/aln(1/a) (if the patterns are randomly correlated). Interestingly, after retrieving an externally cued attractor, the network can continue jumping, or latching, from attractor to attractor, driven by adaptation effects. The occurrence and duration of latching dynamics is found through simulations to depend critically on the strength of local attractor states, expressed in the Potts model by a parameter w. Here we describe with simulations and then analytically the boundaries between distinct phases of no latching, of transient and sustained latching, deriving a phase diagram in the plane w-T, where T parametrizes thermal noise effects. Implications for real cortical dynamics are briefly reviewed in the conclusions.
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Affiliation(s)
- Eleonora Russo
- SISSA, Cognitive Neuroscience, via Bonomea 265, 34136 Trieste, Italy.
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Brunini VE, Schuh CA, Carter WC. Percolation of diffusionally evolved two-phase systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:021119. [PMID: 21405830 DOI: 10.1103/physreve.83.021119] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/10/2010] [Indexed: 05/03/2023]
Abstract
Percolation thresholds and critical exponents for universal scaling laws are computed for microstructures that derive from phase-transformation processes in two dimensions. The computed percolation threshold for nucleation and growth processes, p(c)≈0.6612, is similar to those obtained by random placement of disks and greater than that of spinodal decomposition, p(c)≈0.4987. Three critical exponents for scaling behavior were computed and do not differ significantly from universal values. The time evolution of a characteristic microstructural length was also computed: For spinodal decomposition, this length grows according to a power law after a short incubation period; for nucleation and growth, there are several transitions in the nature of the growth law. We speculate that the transitions in nucleation and growth derive from competing effects of coalescence at short times and then subsequent coarsening. Short-range order is present, but different, for both classes of microstructural evolution.
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Affiliation(s)
- Victor E Brunini
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
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Quintanilla JA, Ziff RM. Asymmetry in the percolation thresholds of fully penetrable disks with two different radii. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:051115. [PMID: 18233631 DOI: 10.1103/physreve.76.051115] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2007] [Indexed: 05/25/2023]
Abstract
We perform simulations of continuum gradient percolation to measure the percolation threshold of systems of fully penetrable disks of two different radii. As a function of volumetric proportion v, the percolation threshold phic(v,lambda) is approximately symmetric about v=0.5 for a fixed ratio lambda of the radii. However, the difference from symmetry is strongly statistically significant. We also improve, by an order of magnitude, the measurement of the percolation threshold for disks of equal radius: phic*=0.676 347 5(6).
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Affiliation(s)
- John A Quintanilla
- Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203, USA.
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Corsi A, Gujrati PD. Percolation of particles on recursive lattices using a nanoscale approach. I. Theoretical foundation at the atomic level. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:061121. [PMID: 17280052 DOI: 10.1103/physreve.74.061121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2006] [Revised: 10/06/2006] [Indexed: 05/13/2023]
Abstract
Powdered materials of sizes ranging from nanometers to micrometers are widely used in materials science and are carefully selected to enhance the performance of a matrix. Fillers have been used in order to improve properties, such as mechanical, rheological, electrical, magnetic, thermal, etc., of the host material. Changes in the shape and size of the filler particles are known to affect and, in some cases, magnify such enhancement. This effect is usually associated with an increased probability of formation of a percolating cluster of filler particles in the matrix. In this series of papers, we will consider lattice models. Previous model calculations of percolation in polymeric systems generally did not take into account the possible difference between the size and shape of monomers and filler particles and usually neglected interactions or accounted for them in a crude fashion. In our approach, the original lattice is replaced by a recursive structure on which calculations are done exactly and interactions as well as size and shape disparities can be easily taken into account. Here, we introduce the recursive approach, describe how to derive the percolation threshold as a function of the various parameters of the problem, and apply the approach to the analysis of the effect of correlations among monodisperse particles on the percolation threshold of a system. In the second paper of the series, we tackle the issue of the effect of size and shape disparities of the particles on their percolation properties. In the last paper, we describe the effects due to the presence of a polymer matrix.
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Affiliation(s)
- Andrea Corsi
- The Department of Physics and The Department of Polymer Science, The University of Akron, Akron, Ohio 44325, USA
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Quintanilla J. Measurement of the percolation threshold for fully penetrable disks of different radii. PHYSICAL REVIEW E 2001; 63:061108. [PMID: 11415069 DOI: 10.1103/physreve.63.061108] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2000] [Indexed: 11/07/2022]
Abstract
We perform simulations of gradient percolation to study the percolation threshold for systems of homogeneous fully penetrable disks of variable radii. We find that, if the radii follow a uniform distribution, the percolation threshold is 0.686 610+/-0.000 007. We also investigate binary dispersions, studying the influence of constitutive parameters on the percolation threshold and suggesting an empirical formula for the threshold. We find that, with the appropriate parameters, a percolation threshold of approximately 0.76 can be achieved. The minimal threshold of 0.676 339+/-0.000 004 is achieved by disks of equal radius.
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Affiliation(s)
- J Quintanilla
- Department of Mathematics, University of North Texas, Denton, Texas 76203, USA.
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Lorenz B, Orgzall I, Heuer HO. Universality and cluster structures in continuum models of percolation with two different radius distributions. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/26/18/032] [Citation(s) in RCA: 64] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Network Approach to Void Percolation in a Pack of Unequal Spheres. PHYSICAL REVIEW LETTERS 1996; 77:1785-1788. [PMID: 10063171 DOI: 10.1103/physrevlett.77.1785] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
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Nolte DD, Pyrak-Nolte LJ. Stratified continuum percolation: Scaling geometry of hierarchical cascades. PHYSICAL REVIEW A 1991; 44:6320-6333. [PMID: 9905762 DOI: 10.1103/physreva.44.6320] [Citation(s) in RCA: 30] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Chiew YC, Stell G, Glandt ED. Clustering and percolation in multicomponent systems of randomly centered and permeable spheres. J Chem Phys 1985. [DOI: 10.1063/1.449490] [Citation(s) in RCA: 73] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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