Scaling theory of percolation clusters

Handbook of SiC properties for fuel performance modeling

A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential $\ensuremath{\psi}(\mathrm{x})$. For rapidly decaying correlations of $\ensuremath{\psi}$, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.

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Percolation, statistical topography, and transport in random media

Diffusion in disordered systems does not follow the classical laws which describe transport in ordered crystalline media, and this leads to many anomalous physical properties. Since the application of percolation theory, the main advances in the understanding of these processes have come from fractal theory. Scaling theories and numerical simulations are important tools to describe diffusion processes (random walks: the 'ant in the labyrinth') on percolation systems and fractals. Different types of disordered systems exhibiting anomalous diffusion are presented (the incipient infinite percolation cluster, diffusion-limited aggregation clusters, lattice animals, and random combs), and scaling theories as well as numerical simulations of greater sophistication are described. Also, diffusion in the presence of singular distributions of transition rates is discussed and related to anomalous diffusion on disordered structures.

Diffusion in disordered media

REVIEW ARTICLE: Percolation theory

The Vickers indentation fracture toughness test, or VIF, is addressed by considering its origins and the numerous equations that have been applied along with the technique to estimate the fracture resistance, or the KIc of ceramics. Initiation and propagation of cracks during the VIF test are described and contrasted with the pre-cracking and crack growth for internationally standardized fracture toughness tests. It is concluded that the VIF test technique is fundamentally different than standard fracture toughness tests. The VIF test has a complex three-dimensional crack system with substantial deformation residual stresses and damage around the cracks. The VIF test relates to an ill-defined crack arrest condition as opposed to the rapid crack propagation of the standardized fracture toughness tests.

Previously published fracture toughness results employing the VIF technique are reviewed. These reveal serious discrepancies in reported VIF fracture toughness values. Finally, recent fracture resistance measurements by the VIF technique for the Standard Reference Material SRM 2100 are presented. These are compared with standardized test results for the same material. It is concluded that the VIF technique is not reliable as a fracture toughness test for ceramics or for other brittle materials. What the VIF actually measures in terms of fracture resistance cannot be readily defined. It is recommended that the VIF technique no longer be acceptable for the fracture toughness testing of ceramic materials.

On the Vickers Indentation Fracture Toughness Test

The hardness and brittleness of ceramic materials are interrelated. Hard materials are more apt to fracture in the vicinity of an indentation during a hardness test, while softer materials tend to plastically deform to the indenter shape without fracturing. Measured hardness, in turn, is affected by both specimen deformation and fracture processes. This interrelationship is examined by means of extensive Vickers hardness testing. A new index of brittleness is proposed.

Indentation brittleness of ceramics: a fresh approach

The uniaxial strength of engineering ceramics is often measured by the well-known flexure strength test method there is a risk that flexure data are not representative of the properties of fabricated components. Reliability estimates for components based upon statistical extrapolation techniques from flexure data may not be valid. This paper reviews the problem and judges the usefulness of flexure data for design purposes. It is shown that some of the limitations of flexure data apply; to other modes of testing, including direct tension testing

Design Data for Engineering Ceramics: A Review of the Flexure Test

The strength of 1500 sintered alumina and reaction-bonded silicon nitride flexure specimens was measured by seven laboratories in four countries. Consistent results were obtained only with a well-defined standard test method. Older procedures were less consistent or gave erroneous results. A simple statistical procedure allowed a quick estimate of consistency, and outlier samples could be appraised for the likelihood of their occurrence.

Advanced Structural Ceramics: A Round Robin

Cluster statistics in two- and three-dimensional site percolation problems are derived here by Monte Carlo methods. The average number n, of percolation clusters with s occupied sites each is calculated by up to 19 runs on a 4000 X 4000 triangular lattice near pc. Our data support the two-exponent scaling assumption n, as-'f(z'), where z'= ( p/pc - 1)s". At the percolation threshold p = pc we find for s up to lo6 a rough agreement with the expected power law n, as-' over 12 decades in n, ; we can approximate the leading correction term near 5-10' by n,cXs-'(l-1.2 s-~'~). If the ratio U, = n,(p)/n,(p,) is plotted against z', then all data follow the same curve U, = f(z') for different p. This scaling function f(z') has a finite slope at z' = 0, has a maximum f(zk, = -0.8) = 5 for p below pc, and decays rapidly for z'+*m. For 5-+m at fixed p this rapid decay corresponds to In n, Cc -s"' above pc and In n, a --s below pc. Apart from finite-size corrections we find the second moment x =Zs2n, diverges as 1 p -pC(-', with y = 2.4, on both sides of the phase transition; the amplitude ratio x(p p,) is about 200. The fraction of occupied sites belonging to the infinite cluster vanishes as (p -p,)'. with p -0.13. In three dimen- sions using system sizes up to 400 x 400 x 400 the two-exponent scaling function is also supported, with the same universal function f(z') valid for both the simple cubic and BCC lattices. f(z') has a maximum f(z;, = -0.8) = 1.6. The amplitude ratio is approximately 11. Our conclusions are in general consistent with but more complete than other recent Monte Carlo work by Stoll and Domb, Leath and Reich, and Nakanishi and Stanley.

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George D. Quinn

Papers

On the Vickers Indentation Fracture Toughness Test

Indentation brittleness of ceramics: a fresh approach

Design Data for Engineering Ceramics: A Review of the Flexure Test

Advanced Structural Ceramics: A Round Robin

Monte Carlo experiments on cluster size distribution in percolation