Pair Connectedness Function Research Articles

Series expansion of the percolation probability for the directed square lattice

By extrapolation from finite lattices, the authors extend the known series for the percolation probability on the directed square lattice from eight terms to 41. Analysing the series, they obtain the estimates qc=0.355 299+or-0.000 001, beta =0.2764+or-0.0001 for the critical probability and the critical exponent. From this, together with scaling relations and previous results on the moments of the pair-connectedness function, the authors conjecture that beta may be exactly 199/720.

Cluster volume and surface area in dispersions of penetrable particles or pores

The complete description of a homogeneous, multiphase dispersion is contained within the infinite set of n-body density distribution functions g(rn) which have been used to calculate macroscopic properties such as interfacial area and specific volume. Certain quantities of interest, however, must take the connectedness of the individual phases into account. This requires the introduction of a complete set of n-body connectedness functions g+n(rn). Until now, only the pair-connectedness function g+2(r2) has been computed. Here, a formalism for the estimation of higher-order connectedness functions from lower order ones is presented. Results are given for the average volume and interfacial area per cluster for a dispersion of randomly placed spheres.

Monte Carlo calculations of cluster statistics in continuum models of composite morphology

We describe a simple and efficient algorithm for sampling physical cluster statistics in Monte Carlo simulations of continuum morphology models. The algorithm produces a variety of information including the pair connectedness function, cluster size distribution, and mean cluster size. The approach can be applied to any system, given a definition of a physical cluster for that system. Results are presented for two types of models commonly used in studies of percolation phenomena; randomly centered spheres and the concentric shell (extended sphere) model. The simulation results are used to assess the accuracy of the predictions of the Percus–Yevick closure of the Ornstein–Zernike equation for the pair connectedness function.

Exact pair-connectedness function of a one-dimensional hard rod fluid

Assuming that two molecules of a fluid are connected (i.e. belong to the same cluster) when their centres are closer than a given distance d, the authors present, in explicit form, the exact pair-connectedness function for a one-dimensional fluid of hard rods of length sigma such that sigma <d<2 sigma and for densities which are compatible with the existence of a region of translational invariance in the system.

Percolation behavior of the square-well fluid

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.

Bulk properties of two-phase disordered media. II. Effective conductivity of a dilute dispersion of penetrable spheres

The calculation of the effective electrical conductivity σ* of a dilute dispersion of equisized spheres of radius R distributed with arbitrary degree of penetrability is considered. It is demonstrated that σ*, through second order in the inclusion volume fraction φ2, can be written in terms of the zero-density limits of the pair-connectedness and pair-blocking functions, and certain polarizability tensors which involve one and two inclusions. Rigorous upper and lower bounds on σ*, through order φ22, are shown to depend upon, among other quantities, the aforementioned pair distribution functions and are evaluated for two models: an interpenetrable-sphere model and a certain sphere distribution in which the minimum distance between sphere centers is greater than or equal to 2R. An approximate expression obtained for the low-density expansion of σ* for dispersions of penetrable spheres always lies between the derived bounds on σ*. The study demonstrates that the effect of connectivity of the inclusion phase on σ*, through second order in φ2, can be substantial relative to the conductivity of dispersions of spheres characterized by a pair-connectedness function that is zero for all sphere separations.

Clustering and percolation in multicomponent systems of randomly centered and permeable spheres

For a multicomponent system of particles in equilibrium, an exact integral equation is derived for the pair connectedness function (which measures the probability that two particles, with centers a distance r apart, are connected). The pair connectedness function, mean cluster size, and percolation thresholds for mixtures of randomly centered (noninteracting) spheres and permeable spheres are then obtained analytically in the Percus–Yevick (PY) approximation. (The permeable-sphere model provides a one-parameter bridge from randomly centered sphere mixtures to hard-sphere mixtures.) For this family of models, connectedness is defined by particle overlap. It is found that, within the PY approximation, a multicomponent mixture of randomly centered spheres percolates at ξ3=π/6∑iρiR3i =1/2, independent of the concentration and size distributions of the particles. For the permeable-sphere case the percolation threshold depends on the relative densities, size, and interparticle permeability among the species. The loci of the percolation thresholds of a binary mixture of permeable spheres are explicitly determined.

Exact equation for the pair-connectedness function

An exact integral equation for the pair-connectedness function gDouble Dagger (12) is derived on the basis of the author's earlier exact equation for the pair-distribution function g(12). The new equation gives rise to a sequence of approximations for gDouble Dagger (12), starting with the Percus-Yevick approximation. Significant aspects of the first two approximations are noted, as is the relevance of the blocking functions g-gDouble Dagger for ionic and chemically associating particles.

Percolation behaviour of permeable and of adhesive spheres

The pair-connectedness function and the average cluster size are determined for two different three-dimensional fluid systems using the Percus-Yevick (PY) approximation (1980). The permeable-sphere model of Blum and Stell (1980) provides a one-parameter bridge from the ideal gas (perfectly penetrable spheres) to the PY hard-sphere fluid. Two of such particles are considered to be 'bound' when their cores overlap. The percolation transition is located as a function of the interpenetrability of the particles, and is found to correspond to an average coordination number z=4. Baxter's adhesive-sphere model (1968) is also investigated in the PY approximation and it is found that at the percolation transition the average coordination number is 2. The boundary between percolating and non-percolating homogeneous thermodynamic states is determined.

Bond percolation on the square lattice: Anisotropic systems

Bond percolation on the square lattice in which horizontal and vertical bonds can have different occupation probabilities is studied within the context of annealed model solutions for the zero-temperature bond-diluted Ising ferromagnet. The percolation probabilities and pair-connectedness functions for this percolation are obtained in terms of the concentration of horizontal and vertical bonds in the system. Also, the critical behavior of these properties near the critical bond concentrations is determined. The neutron scattering from the annealed isotropic bond-diluted Ising ferromagnet is then studied in bond concentration (percolation) and temperature variables near the critical bond concentration point. Comparison of the crossover from thermal to percolative behavior is made with recent experiments by Cowley et al. 4 ) on Rb 2 Co x Mg 1- x F 4 . Finally, certain exact results for anisotropic plaquette percolation on the square lattice are presented.

Cluster structure near the percolation threshold

Derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc. In particular, the author proves the relations p(dpij/dp)=( lambda ij) where p is the bond density, pij is the pair connectedness function and ( lambda ij) is the average number of cutting bonds between i and j. From this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi , diverges as mod p-pc mod -1. The remaining (multiply connected) bonds in the percolating backbone, which lump together in 'blobs', diverge with a dimensionality-dependent exponent. He also shows that in the cell renormalisation group of Reynolds et al. (1978, 1980) the 'thermal' eigenvalue is simply related to the average number of cutting bonds in the spanning cluster. He discusses a percolation model in which the 'blobs' can be controlled by varying a parameter, and study the influence on the critical exponents.

QUANTITATIVE CHARACTERIZATION OF THE LATERAL DISTRIBUTION OF MEMBRANE PROTEINS WITHIN THE LIPID BILAYER

The dependence of the lateral distribution of membrane proteins on the size, protein/lipoid molar ratio, and the magnitude of the interaction potentials has been investigated by computer modeling protein-lipid distributions with Monte Carlo calculations. These results have allowed us to develop a quantitative characterization of the distribution of membrane proteins and to correlate these distributions with experimental observables. The topological arrangement of protein domains, protein plus annular lipid domains, and free lipid domains is described in terms of radial distribution, pair connectedness, and cluster distribution functions. The radial distribution functions are used to measure the distribution of intermolecular distances between protein molecules, whereas the pair connectedness functions are used to estimate the physical extension of compositional domains. It is shown that, at characteristic protein/lipid molar ratios, previously isolated domains become connected, forming domain networks that extend over the entire membrane surface. These changes in the lateral connectivity of compositional domains are paralleled by changes in the calculated lateral diffusion coefficients and might have important implications for the regulation of diffusion controlled processes within the membrane.

Thermal phase transitions at the percolation threshold

We present a simple argument, valid in all dimensions for a wide range of quenched dilute spin models, for the equality of the pair correlation function and the pair connectedness function at the point p = p c , T = 0 is approached along the T axis.