Many-body electrostatic interactions in electrorheological fluids

We present a general method based on a multipole-expansion theory that allows us to calculate efficiently and accurately the electrostatic forces and the dielectric constant of an assembly of spheres. This method is applied to the study of two aspects which play an important role in the behavior of electrorheological (ER) fluids. The first one concerns the calculation of the principal values ${\mathrm{\epsilon }}_{\mathrm{\parallel }}$ and ${\mathrm{\epsilon }}_{\mathrm{\perp }}$ of the dielectric tensor of the body-centered-tetragonal (bct) lattice, and the calculation of the induced dipole on the particles in this lattice. These are rigorous calculations on physical properties of interest in the study of ER fluids. These results support the idea that the columnlike aggregates which have been found in ER fluids should have a bct structure. Although calculations based on the dipolar approach were previously presented, no results are available that confirm this idea rigorously. The second point concerns an exact analytical derivation of an expression describing the many-body electrostatic forces among spherical polarizable particles in terms of the multipole moments. We have compared this force expression, in the case of two-particle interactions, to some results from the literature. It agrees very well with some analytical two-particle expressions for perfectly conducting spheres and also with some recent results concerning the interactions between two polarizable spheres. Furthermore, we present results for three-particle contributions to the electrostatic force and show that these contributions are unexpectedly large. In particular, the rate of divergence of the force between two conducting spheres can be considerably changed by the presence of a third one.