Abstract

Fundamental to the development of electrorheological nanofluids is a precise knowledge of the electrostatic force of attraction between uncharged particles subject to an applied external electric field. In order to design more effective materials, more complicated geometries have been proposed containing both conducting and dielectric materials. Accordingly, solution methods are needed which address the issues of accurately calculating the force of attraction for closely spaced, polarised, dielectric particles; while at the same time are amenable to the material boundaries of various particle geometries. In this paper the method of multipole re-expansion is employed for the problem of two spherical particles consisting of a core, coated with a conducting layer which in turn is surrounded by an outer dielectric coating of arbitrary size. These particles are imbedded in a dielectric medium and are subject to an externally applied electric field. An exact solution is given which has not appeared in the literature previously, and which can be exploited to efficiently determine accurate numerical values to be used for the design of new electrorheological fluids. Also the solution in various limits is analysed to recover firstly the solution for homogeneous dielectric particles, secondly to produce an exact solution for conducting media and thirdly to determine an accurate approximation for conducting spheres with very thin coatings, and which provides the opportunity to make use of existing conducting media theory. Quite remarkably, the solution for thin coatings is shown to differ from the conducting solution only by a factor of 1-1/k, where k is the dielectric constant. As might be expected, the force would approach zero for coatings with very low dielectric constants, or approach the conducting solution when the dielectric constant is large. It is conjectured that this result may be applicable to bodies of any shape.

Keywords: Laplace's equation; Electrostatic potential; Spherical harmonics; Electrostatic force; Re-expansion method; ER (electrorheological) fluids

Article Outline

1. Introduction

2. Problem description

3. Problem solution

3.1. Governing equation, Legendre expansions

3.2. Converting between the coordinate systems

3.3. Boundary conditions at the core/coating interfaces

3.4. Surface charge on the conducting layer

3.5. Boundary conditions at the particle surface

3.6. Solution for the potential

3.7. Solution for the electrostatic force

4. Limiting cases of the solution

5. Numerical results

6. Conclusions

Acknowledgements

References