The problem of calculating the electric polarizability tensor ${\alpha }_e$ of objects of arbitrary shape has been reformulated in terms of path integration and implemented computationally. The method simultaneously yields the electrostatic capacity C and the equilibrium charge density. These functionals of particle shape are important in many materials science applications, including the conductivity and viscosity of filled materials and suspensions. The method has been validated through comparison with exact results (for the sphere, the circular disk, touching spheres, and tori), it has been found that ${10}^6$ trajectories yield an accuracy of about four and three significant figures for C and ${\mathbf{\alpha }}_{\mathbf{e}},$ respectively. The method is fast: For simple objects, ${10}^6$ trajectories require about 1 min on a PC. It is also versatile: Switching from one object to another is easy. Predictions have also been made for regular polygons, polyhedra, and right circular cylinders, since these shapes are important in applications and since numerical calculations of high stated accuracy are available. Finally, the path-integration method has been applied to estimate transport properties of both linear flexible polymers (random walk chains of spheres) and lattice model dendrimer molecules. This requires probing of an ensemble of objects. For linear chains, the distribution function of C and of the trace $\left({\mathbf{\alpha }}_{\mathbf{e}}\right),$ are found to be universal in a size coordinate reduced by the chain radius of gyration. For dendrimers, these distribution functions become increasingly sharp with generation number. It has been found that C and ${\alpha }_e$ provide important information about the distribution of molecular size and shape and that they are important for estimating the Stokes friction and intrinsic viscosity of macromolecules.

• Received 26 February 2001

DOI:https://doi.org/10.1103/PhysRevE.64.061401