Two-point matrix probability function for two-phase random media: Computer simulation results for impenetrable spheres

Certain kinds of two-phase random media, such as suspensions, porous media, and composite materials, are characterized by a discrete particle phase that is distributed throughout a continuous matrix phase (fluid, solid, or void). A fundamental understanding of the effective bulk properties of such systems rest upon knowledge of certain statistical quantities that describe how the phases are spatially distributed. In a series of recent papers,I-5 Torquato and Stell have studied a set of n-point matrix probability functions Sn which provide such a description of the microstructure. Once low-order Sn, such as S2 and S3' are known, then effective bulk properties, the diffusion coefficient,6 thermal conductivity, 7-10 and elastic moduli 10, II for example, can be estimated by evaluating integrals over these Sn probability functions. The general n-point function Sn may be interpreted physically as the probability associated with randomly throwing n points into the disordered system such that all n points fall in the matrix phase. The one-point function SI is therefore merely the volume fraction t/J of the matrix. For a homogeneous and isotropic medium the two-point function S2 depends on both the matrix volume fraction and the relative distance r between the two points randomly thrown into the system. Similarly, the three-point function depends upon the size and shape of the triangle whose vertices are the three, randomly thrown points (r ,s,t ). For random media in which completely impenetrable (hard) spheres form the discrete phase, Torquato and Ste1l5 have given formal analytic expressions for the twoand three-point functions. These expressions involve integrals over the twoand three-body distribution functions g 2( r) and g3(r ,s,t), respectively, of the impenetrable spheres. Torquato and Stell were able to evaluate the integral for S2 using the Verlet-Weis modification of the analytic solution13,14 to the Percus-Yevick equation for the hard-sphere two-body distribution function g2(r). These computations for S2 were performed at six values of the matrix volume fraction ¢ = 0.38, 0.5, 0.6, 0.7, 0.8, and 0.9. This paper reports what we believe to be the first computer simulation results for any n-point matrix probability function. Molecular dynamics simulations were performed on the hard-sphere system at the matrix (void) fractions studied by Torquato and Stell and the two-point function was evaluated from the resulting hard-sphere trajectories. The theoretical values for S2(r) are found to be in generally good agreement with the computer simulation results.