Clinton DeW. Van Siclen
Physicist. Runner. Negroni connoisseur.
This site gives access to many of my papers, representing a variety of physics research areas.
Walker diffusion method for calculation of transport properties of composite materials
The morphology of a multiphase microstructure greatly influences the macroscopic transport properties of the composite material. These properties are shown to be related to the diffusion coefficient of a random nonbiased walker. The proper diffusion rules are found by considering an isomorphic image of the micro-structure in which distinct populations of walkers correspond to the phase domains, with the walker density of a population proportional to the transport coefficient of the corresponding domain. To demonstrate the method, it is applied to disordered two-phase percolating composites.
Anomalous walker diffusion through composite systems
Diffusion of a non-biased walker through a composite (multi-phase) system is shown to be anomalous for length scales less than the correlation length ξ (i.e., when the path length is measured with a ruler of length less than ξ) and Gaussian for length scales greater than ξ. The values of ξ and the fractal dimension dw of the walker path in the anomalous regime reflect the phase properties and phase domain morphology of the composite. They are related to the diffusion coefficient Dw for walker diffusion in the Gaussian regime by Dw ∝ξ2−dw, and to the macroscopic transport coefficient σ through the relation σ ∝Dw. The correlation length ξ thus gives the size above which the composite is effectively homogeneous with respect to the transport property of interest. Walker behaviour is compared for disordered (random), particulate–matrix, and labyrinthine two-phase microstructures.
Walker diffusion method for calculation of transport properties of finite composite systems
A heterogeneous medium may be represented by a scalar field of local transport coefficients e.g., conductivity or by a ‘‘resistor network’’ derived from that scalar field. In either case the effective macroscopic and local microscopic transport properties may be calculated by the walker diffusion method. Some sample calculations for disordered systems are presented to demonstrate the method.
Conductivity exponents at the percolation threshold
Connections are found between the two-component percolation problem and the conductor/insulator percolation problem. These produce relations between critical exponents, and suggest formulae connecting the conductivity exponents in different dimensions. Values for the critical exponents are obtained from calculations on the incipient infinite cluster in two and three dimensions.
Random walker derivation of Archie’s law
Theoretical justification is provided for Archie’s law. This phenomenological equation, having the form of a power law, relates the measured electrical resistivity of electrolyte-saturated rock samples to their connected porosity. Historically it has been important for oil and gas exploration.
Effective conductivity of the multidimensional chessboard
The effective conductivity σ of the two-component checkerboard (chessboard) and analogous higher-dimensional checkered structures, are considered. An algebraic expression for σd (the subscript d indicates the Euclidean dimension of the structure) is derived by accounting for the symmetries of the structures.
Transport properties of the Menger sponge
The Menger sponge is a three-dimensional cube that comprises a self-similar, fractal domain and a non-fractal domain, both of which are continuous. Thus it is a useful heuristic model for natural and engineered fractal systems. For this purpose the effective transport coefficient associated with the transport properties (e.g., electrical conductivity, thermal conductivity) of the sponge is derived. Comparison is made to the Sierpinski sponge.
Fluid infiltration of a heterogeneous medium: A stochastic model
Fluid infiltration of a permeable brick in contact with a pressurized reservoir of fluid is considered. A stochastic model, informed by Darcy’s law and the incompressibility of the fluid, shows how the heterogeneity of the permeability field affects the time evolution of the fluid infiltration. In particular, the cause of “anomalous” (non-Darcian) advance of a plume is determined. The model is applied to bricks that are linear arrays of Sierpinski carpets. These calculated results are compared to experimental results available in the literature, to verify the model and method.
Walker diffusion method for solution of ohmic circuit problems
A probabilistic method is derived for solution of ohmic circuit problems. It is compared to the standard approach, which is construction and solution of a set of coupled, linear equations manifesting Kirchhoff’s laws. An example is made of an electrical circuit that has the complicated connectivity of a bond-and-node Sierpinski triangle, which would be tedious to solve by matrix methods.
Conductivity properties of the Sierpinski triangle
The classic Sierpinski triangle comprised of conducting bonds is multifractal. Thus the critical exponents and dimensions related to the conductivity are obtained asymptotically—that is, in the limit that the correlation length ξ of the recursive triangle goes to infinity.