Structure of supercritical water

This review paper gives the more up to date information on the structure of supercritical water, at different thermodynamic states, for densities ranging from 0.2 to 0.7 g.cm⁻³, studied by neutron scattering. The experimental partial pair correlation functions g_OH(r), G_HH(r) and g_OO(r) are compared with the results of molecular dynamics simulations using the (SPCE) model potential for water. The results confirm that hydrogen bonding is still present in dense supercritical water and are in good agreement with data obtained by NMR and Raman spectroscopies.     

A theory of exciton dynamics with a percolation threshold

Mode-coupling methods are used to obtain an integral equation for the exciton diffusion constant in isotopically mixed crystals. Numerical solutions are obtained for nearest neighbor hopping in one- and two-dimensional square lattices. These solutions exhibit percolation thresholds.Supported by NSF Grant CHE78-09704.     

The tortuosity of occupied crossings of a box in critical percolation

We consider the length of an occupied crossing of a box of size [0,n]×[0, 3n] ^D–1 (in the short direction) in standard (Bernoulli) bond percolation on ^D at criticality. Let ¦s _n¦ be the length of the shortest such crossing. It is believed that ¦s _n¦ ^1+c in some sense for somec>0. Here we show that if the correlation length(p) satisfies (p)p _c}–p) ^– for some <1, then with a probability tending to 1, ¦s _n¦>/C ₁ n ^1/(logn)^–(1–)/. The assumption (p)C ₃(p _c–p)^– with <1 has been rigorously established^(1,2) for largeD, but cannot hold⁽³⁾ forD=2. In the latter case, let ¦l _n¦ be the length of the lowest occupied crossing of the square [0,n]². We outline a proof ofP _pc(¦l_n¦ n ^1+c)n ^– for somec, >0. We also obtain a result about the length of optimal paths in first-passage percolation.     

Structure and dynamics of water: from ambient to supercritical

Our recent works on supercritical water are reviewed. In order to elucidate the hydrogen bonding state of supercritical water, the proton chemical shift of the water proton is measured at temperatures up to 400 °C and densities of 0.19, 0.29, 0.41, 0.49, and 0.60 g/cm³. The magnetic susceptibility correction is made in order to express the chemical shift relative to an isolated water molecule in dilute gas. The chemical shift is then related to the average number of hydrogen bonds in which a water molecule is involved. It is found that the hydrogen bonding persists at supercritical temperatures and that the average number of hydrogen bonds is at least one for a water molecule at the densities larger than the critical. The density dependence of the chemical shift at supercritical temperatures is analyzed on the basis of statistical thermodynamics. It is shown that the hydrogen bonding is spatially more inhomogeneous at lower densities. The dipole moment of water at supercritical states is also estimated from the number of hydrogen bonds. The dynamical counterpart of our structural study of supercritical water has been performed by NMR relaxation measurements. Using D₂O, we measured the spin-lattice relaxation time and determined the reorientational relaxation time as a function of the water density and temperature. It is then found that while the reorientational relaxation time decreases rapidly with the temperature in the subcritical condition, it is a weak function of the density in the supercritical conditions.     

Core percolation in random graphs: a critical phenomena analysis

We study both numerically and analytically what happens to a random graph of average connectivity α when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated vertices plus an induced subgraph we call the core. In the thermodynamic limit of an infinite random graph, we compute analytically the dynamics of leaf removal, the number of isolated vertices and the number of vertices and edges in the core. We show that a second order phase transition occurs at α = e = 2.718 ... : below the transition, the core is small but above the transition, it occupies a finite fraction of the initial graph. The finite size scaling properties are then studied numerically in detail in the critical region, and we propose a consistent set of critical exponents, which does not coincide with the set of standard percolation exponents for this model. We clarify several aspects in combinatorial optimization and spectral properties of the adjacency matrix of random graphs. Received 31 January 2001 and Received in final form 26 June 2001     

Scaling relation for determining the critical threshold for continuum percolation of overlapping discs of two sizes

We study continuum percolation of overlapping circular discs of two sizes. We propose a phenomenological scaling equation for the increase in the effective size of the larger discs due to the presence of the smaller discs. The critical percolation threshold as a function of the ratio of sizes of discs, for different values of the relative areal densities of two discs, can be described in terms of a scaling function of only one variable. The recent accurate Monte Carlo estimates of critical threshold by Quintanilla and Ziff [Phys. Rev. E76, 051115 (2007)] are in very good agreement with the proposed scaling relation.     

Statistics of directed self-avoiding walks on percolation clusters at the percolation threshold

We here study directed self-avoiding walks on site diluted square lattice at the percolation threshold by two parameter real space renormalization group method. We found \(v_\parallel ^{p_c } = 1.00\) and \(v_ \bot ^{p_c } = 0.4348\) from cell-to-cell transformation method. This \(v_ \bot ^{p_c } \) value is then compared with the modified Alexander-Orbach formula that \(v_ \bot ^{p_c } = {{d_S } \mathord{\left/ {\vphantom {{d_S } {2d_L }}} \right. \kern-0em} {2d_L }}\) whered _s is the fracton dimension andd _L is the spreading dimension of the infinite directed percolation cluster.     

随机多孔介质逾渗模型渗透率的临界标度性质

研究了一类非零键渗透率满足均匀分布的随机多孔介质逾渗模型-数值计算了该模型系统渗透率在临界点处的标度指数-结果表明该指数并不能看作是普适常数,而与均匀分布的参数有关-这意味着即使非零键渗透率值的概率密度函数满足负一阶矩存在条件,系统渗透率在逾渗临界点处的标度指数仍然依赖于分布函数的具体参数,并不是常数-这一数值结果与Sahimi对此问题的结论不同- 关键词： 逾渗 随机多孔介质 标度指数 渗透率     

A criterion for percolation threshold in a random array of plates

We give a dimensionless criterion for percolation of a random assembly of 3-dimensional flat disks of radius r with a density N per unit volume as (Nr³)_c ～0.15 to 0.3. The invariant at threshold is obtained by analogy with excluded volume results for arrays of needles or with the active filling factor approach used for spheres. As the result can be applied to the permeability of fractured rocks, we discuss the results for polydisperse assemblies and we show how the invariant can be measured from a random cut of the array of disks.     

The Curie temperature of dilute ferromagnetic alloys near the percolation threshold

Upper limit for the Curie temperature T_c(x) of the dilute Heisenberg ferromagnet near percolation threshold is derived. Qualitative arguments are put forward that this limit gives a correct dependence of T_c on the concentration of magnetic atoms x.     

The Hall effect in granular metal films near the percolation threshold

We have measured the Hall effect in the granular metals Au-SiO₂ and W-Al₂O₃ near their percolation thresholds. Our results indicate that when conduction is by electronic tunneling between isolated grains, the Hall coefficient is negative for both cermets. In the bulk, Au has a negative coefficient and W a positive one.     

Random percolation as a gauge theory

Three-dimensional bond or site percolation theory on a lattice can be interpreted as a gauge theory in which the Wilson loops are viewed as counters of topological linking with random clusters. Beyond the percolation threshold large Wilson loops decay with an area law and show the universal shape effects due to flux tube quantum fluctuations like in ordinary confining gauge theories. Wilson loop correlators define a non-trivial spectrum of physical states of increasing mass and spin, like the glueballs of ordinary gauge theory. The crumbling of the percolating cluster when the length of one periodic direction decreases below a critical threshold accounts for the finite temperature deconfinement, which belongs to 2D percolation universality class.