Local volume fraction fluctuations in heterogeneous media

The volume fractions of multiphase heterogeneous media fluctuate on a spatially local level even for statistically homogeneous materials. A general formulation is given to represent the standard deviation associated with the local volume fraction of statistically homogeneous but anisotropic D‐dimensional two‐phase media for arbitrary‐shaped observation regions. The standard deviation divided by the macroscopic volume fraction, termed the coarseness, is computed for D‐dimensional distributions of penetrable as well as impenetrable spheres, for a wide range of densities and observation‐region sizes. The effect of impenetrability of the particles, for fixed observation‐region size, is to reduce the coarseness relative to that of the penetrable‐sphere model, especially at high densities. For either sphere model, increasing the dimensionality D decreases the coarseness.

[1]  H. R. Anderson,et al.  Scattering by an Inhomogeneous Solid. II. The Correlation Function and Its Application , 1957 .

[2]  I. R. Mcdonald,et al.  Theory of simple liquids , 1998 .

[3]  Salvatore Torquato,et al.  Microstructure of two-phase random media.III: The n-point matrix probability functions for fully penetrable spheres , 1983 .

[4]  B. E. Bayer.  Relation Between Granularity and Density for a Random-Dot Model , 1964 .

[5]  Salvatore Torquato,et al.  Diffusion‐controlled reactions: Mathematical formulation, variational principles, and rigorous bounds , 1988 .

[6]  S. Torquato,et al.  Effective properties of fiber-reinforced materials: I—Bounds on the effective thermal conductivity of dispersions of fully penetrable cylinders , 1986 .

[7]  S. Torquato,et al.  Computer simulation results for the two-point probability function of composite media , 1988 .

[8]  Salvatore Torquato,et al.  Conductivity tensor of anisotropic composite media from the microstructure , 1990 .

[9]  S. Torquato,et al.  Characterisation of the microstructure of distributions of rigid rods and discs in a matrix , 1985 .

[10]  Salvatore Torquato,et al.  Bulk properties of two‐phase disordered media. I. Cluster expansion for the effective dielectric constant of dispersions of penetrable spheres , 1984 .

[11]  S. Torquato,et al.  Microstructure of two‐phase random media. II. The Mayer–Montroll and Kirkwood–Salsburg hierarchies , 1983 .

[12]  Graeme W. Milton,et al.  Multicomponent composites, electrical networks and new types of continued fraction I , 1987 .

[13]  Edward L. O'Neill,et al.  Introduction to Statistical Optics , 1963 .

[14]  S. Torquato,et al.  Photographic granularity : mathematical formulation and effect of impenetrability of grains , 1990 .