Mark A. Knackstedt

Structure-property correlations in model composite materials

We investigate the effective properties (conductivity, diffusivity and elastic moduli) of model random composite media derived from Gaussian random fields and overlapping hollow spheres. The morphologies generated in the models exhibit low percolation thresholds and give a realistic representation of the complex microstructure observed in many classes of composites. The statistical correlation functions of the models are derived and used to evaluate rigorous bounds on each property. Simulation of the effective conductivity is used to demonstrate the applicability of the bounds. The key morphological features which effect composite properties are discussed.

Patterns of fluid flow in naturally heterogeneous rocks

Abstract Patterns of fluid displacement in porous media have previously been studied extensively on networks with uncorrelated random properties. Percolation network models have received particular attention because of their relevance to simulating capillary dominated flow. Much fluid flow in porous media of conomic significance occurs in natural rocks that exhibit long-range correlations in properties. In this paper we present results of invasion percolation in two dimensions with heterogeneity that corresponds to naturally occurring sedimentary rocks. Long-range correlations in properties based on multifractal, fractional Brownian motion and fractional Levy motion models are compared to results on uncorrelated networks. We conclude that results in invasion percolation vary for different long-range correlation in properties and breakthrough of the invasion phase is less sensitive to lattice size on the correlated networks.

Fluid flow across mass fractals and self-affine surfaces

Abstract We use a lattice-gas method to simulate the slow flow of a fluid in systems with fractal surfaces and volumes. Two systems are studied. One is flow in a single three-dimensional fracture with self-affine surfaces. The other is flow across a three-dimensional diffusion-limited aggregate. In both cases, significant deviations from classical results are observed.

Nonuniversal field dependence of the critical behavior of the interface between fluid phases in two dimensions

The equilibrium structure of the interface between two‐dimensional fluid phases in the presence of an external field is investigated. Unlike in three or more dimensions of space, it is found that in two dimensions the interfacial thickness is very sensitive to the external field. In particular, the critical exponent that describes the divergence of the interfacial thickness as the critical point is approached is found to be strongly nonuniversal, i.e., field dependent, in two dimensions, whereas it is universal, i.e., field independent, in three or more dimensions.

External field effect on the critical behavior of the interface between fluid phases

The equilibrium structure of the interface between fluid phases ind dimensions in the presence of an external field is investigated. The equilibrium interface is assumed to consist of an intrinsic interface which undergoes capillary-wave fluctuations. It is found that in two dimensions the interfacial thickness is very sensitive to the choice of external field and intrinsic interface. For an intrinsic interface of a thickness proportional toξ, the bulk correlation length, the exponent co, which describes the divergence of the interfacial thickness as the critical point is approached, depends on the scale of the external field relative toξ and ranges fromω=9/32 toω = 17/32, in contrast to the predictionω=1 of scaling theory. When an intrinsic interface of vanishing thickness is chosen,ω=9/32 for any external field. This is in strong contrast to the results in three or more dimensions, whereω is found to be independent of both the external field and the intrinsic interface and satisfiesμ = (d-1)ω, withμ the critical exponent of the surface tension, in accord with scaling theory.

On the universality of geometrical and transport exponents of rigidity percolation

We develop a three-parameter position-space renormalization group method and investigate the universality of geometrical and transport exponents of rigidity (vector) percolation in two dimensions. To do this, we study site-bond percolation in which sites and bonds are randomly and independently occupied with probabilitiess andb, respectively. The global flow diagram of the renormalization transformation is obtained which shows that thegeometrical exponents of the rigid clusters in both site and bond percolation belong to the same universality class, and possibly that of random (scalar) percolation. However, if we use the same renormalization transformation to calculate the critical exponents of the elastic moduli of the system in bond and site percolation, we find them to be very different (although the corresponding values of the correlation length exponent are the same). This indicates that the critical exponent of the elastic moduli of rigidity percolation may not be universal, which is consistent with some of the recent numerical simulations.

Simple permeability model for natural granular media

A previously published pore scale model provides a simple analytic expression relating the permeability of a natural granular media to its porosity. This model is quantitatively accurate at high to intermediate porosities. It is based on two independent parameters, porosity and average grain size. At low porosities, however, the model is unable to capture the experimentally observed trends. Percolation provides a natural description of the pore closure evident in sedimentary media at low porosities. The use of percolation concepts provides the necessary input to predict the permeability in the porosity range where pore blocking occurs. Thus a simple analytic model, utilizing an experimental value for the percolation threshhold, is derived for the permeability of sandstones and calcite aggregates over the full range of porosity.

Block cluster theory of site percolation on the four- and five-dimensional ordinary hypercubic lattices

Site percolation on the ordinary hypercubic lattice in four and five dimensions is studied using block cluster theory. The exact analytic renormalisation group equations of block cluster theory are obtained by a computer algorithm. They are solved numerically and yield, for the susceptibility exponent, gamma =1.277 (d=4), gamma =1.031 (d=5), and for the correlation length exponent nu =0.566 (d=5).

Crossover from capillary wave to van der Waals regime for fluid interfaces in two dimensions very close to the critical point

An analytic expression is derived for the interfacial profile of a liquid–vapor or a liquid–liquid phase equilibrium by assuming, in accord with current ideas, that the equilibrium interface consists of an intrinsic interface of the nonclassical van der Waals type broadened by capillary wave fluctuations. It is shown that in two dimensions of space, the interfacial thickness exhibits a crossover with change of critical exponent from capillary wave behavior at low and near‐critical temperatures, to van der Waals behavior at temperatures very close to the critical temperature. The location of the crossover temperature is determined by the ratio of the critical amplitudes of the root‐mean squared interfacial thickness and the bulk correlation length. From available experimental data and theory, an estimate of the order of magnitude of the critical amplitude ratio is made, from which the crossover is found to occur in the temperature range 3×10−4 K<(Tc−T)<0.3 K. In the currently accessible temperature range, ...