Radial distribution function for hard disks from the BGY2 theorya)

Measuring many-body distribution functions in fluids using test-particle insertion

We derive a hierarchy of equations, which allow a general n-body distribution function to be measured by test-particle insertion of between 1 and n particles. We apply it to measure the pair and three-body distribution functions in a simple fluid using snapshots from Monte Carlo simulations in the grand canonical ensemble. The resulting distribution functions obtained from insertion methods are compared with the conventional distance-histogram method: the insertion approach is shown to overcome the drawbacks of the histogram method, offering enhanced structural resolution and a more straightforward normalization. At high particle densities, the insertion method starts breaking down, which can be delayed by utilizing the underlying hierarchical structure of the insertion method. Our method will be especially useful in characterizing the structure of inhomogeneous fluids and investigating closure approximations in liquid state theory.

Finite-element method for calculation of the effective permittivity of random inhomogeneous media

The challenge of designing new solid-state materials from calculations performed with the help of computers applied to models of spatial randomness has attracted an increasing amount of interest in recent years. In particular, dispersions of particles in a host matrix are scientifically and technologically important for a variety of reasons. Herein, we report our development of an efficient computer code to calculate the effective (bulk) permittivity of two-phase disordered composite media consisting of hard circular disks made of a lossless dielectric (permittivity epsilon2) randomly placed in a plane made of a lossless homogeneous dielectric (permittivity epsilon1) at different surface fractions. Specifically, the method is based on (i) a finite-element description of composites in which both the host and the randomly distributed inclusions are isotropic phases, and (ii) an ordinary Monte Carlo sampling. Periodic boundary conditions are employed throughout the simulation and various numbers of disks have been considered in the calculations. From this systematic study, we show how the number of Monte Carlo steps needed to achieve equilibrated distributions of disks increases monotonically with the surface fraction. Furthermore, a detailed study is made of the dependence of the results on a minimum separation distance between disks. Numerical examples are presented to connect the macroscopic property such as the effective permittivity to microstructural characteristics such as the mean coordination number and radial distribution function. In addition, several approximate effective medium theories, exact bounds, exact results for two-dimensional regular arrays, and the exact dilute limit are used to test and validate the finite-element algorithm. Numerical results indicate that the fourth-order bounds provide an excellent estimate of the effective permittivity for a wide range of surface fractions, in accordance with the fact that the bounds become progressively narrower as more microstructural information is incorporated. Future directions of the active field of computational studies of the structure-property relations for composite systems are briefly discussed.

Correlated light scattering by a dense distribution of condensation droplets on a window pane

An analytical model of the scattering structure factor for an assembly of noninteracting hard disks has recently appeared in the literature [Phys. Rev. A 42, 5978-5989 (1990)]. We employ this model to calculate correlated light scattering by monodispersions and binary mixtures of condensation droplets on a window pane. We find that an area fraction of ƒ ≥ 0.6 is required for producing the near-forward direction scattering suppression and that a moderately wide polydispersion of droplet sizes is capable of producing the experimentally observed bright ring of colored light.