In Chan Kim

Efficient simulation technique to compute effective properties of heterogeneous media

We present a new simulation technique to ‘‘exactly’’ yield effective transport properties of disordered heterogeneous media in which the transport process is governed by a steady‐state diffusion equation. Hence, the algorithm, which is based upon simulating the Brownian motion of a diffusing particle, can be applied to determine the effective electrical and thermal conductivity, dielectric constant, magnetic permeability, diffusion coefficient associated with flow past fixed obstacles, and the trapping rate associated with diffusion‐controlled reactions among sinks. The simulation method is shown to have a very fast execution time. The technique is illustrated by computing the trapping rate associated with diffusion‐controlled reactions; it is demonstrated to have an execution time that is at least an order of magnitude faster than previous simulation methodologies.

Determination of the effective conductivity of heterogeneous media by Brownian motion simulation

A new Brownian motion simulation technique developed by Torquato and Kim [Appl. Phys. Lett. 55, 1847 (1989)] is applied and further developed to compute ‘‘exactly’’ the effective conductivity σe of n‐phase heterogeneous media having phase conductivities σ1, σ2, ..., σn and volume fractions φ1, φ2, ..., φn. The appropriate first passage time equations are derived for the first time to treat d‐dimensional media (d=1, 2, or 3) having arbitrary microgeometries. For purposes of illustration, the simulation procedure is employed to compute the transverse effective conductivity σe of a two‐phase composite composed of a random distribution of infinitely long, oriented, hard cylinders of conductivity σ2 in a matrix of conductivity σ1 for virtually all volume fractions and for several values of the conductivity ratio α=σ2/σ1, including perfectly conducting cylinders (α=∞). The method is shown to yield σe accurately with a comparatively fast execution time.

Effective conductivity of suspensions of hard spheres by Brownian motion simulation

A generalized Brownian motion simulation technique developed by Kim and Torquato [J. Appl. Phys. 68, 3892 (1990)] is applied to compute ‘‘exactly’’ the effective conductivity σe of heterogeneous media composed of regular and random distributions of hard spheres of conductivity σ2 in a matrix of conductivity σ1 for virtually the entire volume fraction range and for several values of the conductivity ratio α=σ2/σ1, including superconducting spheres (α=∞) and perfectly insulating spheres (α=0). A key feature of the procedure is the use of first‐passage‐time equations in the two homogeneous phases and at the two‐phase interface. The method is shown to yield σe accurately with a comparatively fast execution time. The microstructure‐sensitive analytical approximation of σe for dispersions derived by Torquato [J. Appl. Phys. 58, 3790 (1985)] is shown to be in excellent agreement with our data for random suspensions for the wide range of conditions reported here.

Effective conductivity, dielectric constant, and diffusion coefficient of digitized composite media via first-passage-time equations

We generalize the Brownian motion simulation method of Kim and Torquato [J. Appl. Phys. 68, 3892 (1990)] to compute the effective conductivity, dielectric constant and diffusion coefficient of digitized composite media. This is accomplished by first generalizing the first-passage-time equations to treat first-passage regions of arbitrary shape. We then develop the appropriate first-passage-time equations for digitized media: first-passage squares in two dimensions and first-passage cubes in three dimensions. A severe test case to prove the accuracy of the method is the two-phase periodic checkerboard in which conduction, for sufficiently large phase contrasts, is dominated by corners that join two conducting-phase pixels. Conventional numerical techniques (such as finite differences or elements) do not accurately capture the local fields here for reasonable grid resolution and hence lead to inaccurate estimates of the effective conductivity. By contrast, we show that our algorithm yields accurate estimates of the effective conductivity of the periodic checkerboard for widely different phase conductivities. Finally, we illustrate our method by computing the effective conductivity of the random checkerboard for a wide range of volume fractions and several phase contrast ratios. These results always lie within rigorous four-point bounds on the effective conductivity.

Effective conductivity of suspensions of overlapping spheres

An accurate first‐passage simulation technique formulated by the authors [J. Appl. Phys. 68, 3892 (1990)] is employed to compute the effective conductivity σe of distributions of penetrable (or overlapping) spheres of conductivity σ2 in a matrix of conductivity σ1. Clustering of particles in this model results in a generally intricate topology for virtually the entire range of sphere volume fractions φ2 (i.e., 0≤φ2≤1). Results for the effective conductivity σe are presented for several values of the conductivity ratio α=σ2/σ1, including superconducting spheres (α=∞) and perfectly insulating spheres (α=0), and for a wide range of volume fractions. The data are shown to lie between rigorous three‐point bounds on σe for the same model. Consistent with the general observations of Torquato [J. Appl. Phys. 58, 3790 (1985)] regarding the utility of rigorous bounds, one of the bounds provides a good estimate of the effective conductivity, even in the extreme contrast cases (α≫1 or α≂0), depending upon whether the...

Diffusion of finite‐sized Brownian particles in porous media

The effective diffusion coefficient De for porous media composed of identical obstacles of radius R in which the diffusing particles have finite radius βR (β≥0) is determined by an efficient Brownian motion simulation technique. This is accomplished by first computing De for diffusion of ‘‘point’’ Brownian particles in a certain system of interpenetrable spherical obstacles and then employing an isomorphism between De for this interpenetrable sphere system and De for the system of interest, i.e., the one in which the Brownian particles have radius βR. [S. Torquato, J. Chem. Phys. 95, 2838 (1991)]. The diffusion coefficient is computed for the cases β=1/9 and β=1/4 for a wide range of porosities and compared to previous calculations for point Brownian particles (β=0). The effect of increasing the size of the Brownian particle is to hinder the diffusion, especially at low porosities. A simple scaling relation enables one to compute the effective diffusion coefficient De for finite β given the result of De f...

Trapping and flow among random arrays of oriented spheroidal inclusions

The effective trapping rate k associated with diffusion‐controlled reactions among random distributions of spatially correlated and uncorrelated, oriented spheroidal traps of aspect ratio e is determined from Brownian motion simulations. Data for k are obtained for prolate cases (e=2, 5, and 10), oblate cases (e=0.1, 0.2, and 0.5), and spheres (e=1) over a wide range of trap volume fractions (φ2) and satisfy recently obtained rigorous lower bounds on k for this statistically anisotropic model. The results for the trapping rate for correlated traps always bounds from above corresponding results for uncorrelated traps. Generally, the trapping rate k, for fixed φ2, increases with decreasing aspect ratio e, showing a precipitous rise in k as the spheroids become disklike. Using a recent theorem due to Torquato [Phys. Rev. Lett. 64, 2644 (1990)], data for the trapping rate k can be employed to infer information about the fluid permeability tensor K associated with slow viscous flow through porous media compose...

Bounds on the effective properties of polydispersed suspensions of spheres: An evaluation of two relevant morphological parameters

Expressions for the two microstructural parameters that appear in the variational third‐order bounds [G. W. Milton, Phys. Rev. Lett. 46, 542 (1981)] for the effective conductivity and elastic moduli of composite media are derived analytically to first order in the sphere concentration c for random well‐mixed dispersions of impenetrable spheres with an arbitrary size distribution. These relations lead to rigorous bounds on the effective properties which are exactly valid to order c2 for such models. The apparent linear behavior of the microstructural parameters up to moderately high c enables one to apply the bounds beyond second‐order in c, however. Employing these results, the effect of polydispersivity on the effective properties is examined. It is worth noting that, under some conditions, polydispersivity can actually lead to a slight decrease of the shear modulus, whereas, for highly conducting particles, polydispersivity always increases the effective conductivity.

Effective conductivity of composites containing spheroidal inclusions: Comparison of simulations with theory

We determine, by first‐passage‐time simulations, the effective conductivity tensor σe of anisotropic suspensions of aligned spheroidal inclusions with aspect ratio b/a. This is a versatile model of composite media, containing the special limiting cases of aligned disks (b/a=0), spheres (b/a=1), and aligned needles (b/a=∞), and may be employed to model aligned, long‐ and short‐fiber composites, anisotropic sandstones, certain laminates, and cracked media. Data for σe are obtained for prolate cases (b/a=2, 5, and 10) and oblate cases (b/a=0.1, 0.2, and 0.5) over a wide range of inclusion volume fractions and selected phase conductivities (including superconducting inclusions and perfectly insulating ‘‘voids’’). The data always lie within second‐order rigorous bounds on σe due to Willis [J. Mech. Phys. Solids 25, 185 (1977)] for this model. We compare our data for prolate and oblate spheroids to our previously obtained data for spheres [J. Appl. Phys. 69, 2280 (1991)].

Comparison of analytic and numerical results for the mean cluster density in continuum percolation

Recently a number of techniques have been developed for bounding and approximating the important quantities in a description of continuum percolation models, such as 〈nc〉/ρ, the mean number of clusters per particle. These techniques include Kirkwood–Salsburg bounds, and approximations from cluster enumeration series of Mayer–Montroll type, and the scaled‐particle theory of percolation. In this paper, we test all of these bounds and approximations numerically by conducting the first systematic simulations of 〈nc〉/ρ for continuum percolation. The rigorous Kirkwood–Salsburg bounds are confirmed numerically in both two and three dimensions. Although this class of bounds seems not to converge rapidly for higher densities, averaging an upper bound with the corresponding lower bound gives an exceptionally good estimate at all densities. The scaled‐particle theory of percolation is shown to give extremely good estimates for the density of clusters in both two and three dimensions at all densities below the perc...