Transport properties of arrays of elliptical cylinders N. A. Nicorovici and R. C. McPhedran Department of Theoretical Physics, School of Physics, University of Sydney, New South Wales 2006, Australia Received 20 March 1996 We apply the Rayleigh method to derive a formulation yielding the effective dielectric tensor of a periodic composite consisting of an array of elliptical cylinders placed in a matrix of unit dielectric constant. We consider three types of composites having this structure: dilute noncritical, dilute critical, and concentrated composites. For dilute noncritical composites we comment on our result in relation to the competing forms of the Maxwell-Garnett formula, which have been proposed previously. We also discuss the case of dilute critical and concentrated composites of solid elliptical inclusions, and comment on geometrical constraints on the validity of the Rayleigh equations. S1063-651X9600908-7 PACS numbers: 03.50.De, 41.20.Cv, 78.20.Bh, 78.20.Ci I. INTRODUCTION Analysis concerning the transport properties of inhomo- geneous systems is of fundamental theoretical interest, but also plays an important role in optimal designs of industrial products. For example, many modern structural materials de- pend on the use of composite materials. It is also possible to produce columnar thin films which are highly conducting. In particular, thin films containing metal and voids in an ob- lique columnar structure exhibit angular dependent optical properties 1,2. By forming the columns at an angle and coating them with metal, highly conducting elliptical cylin- ders can be produced. Such a capacitive grid can manifest strong angular selectivity as a result of the basic asymmetry of the component ellipses. Materials with such characteris- tics may have uses in filters, windows in residential and com- mercial buildings, and car windscreens 1–3. Here, using the terminology for the calculation of the di- electric constant, we analyze the transport properties of a two-dimensional two-phase composite material consisting of a rectangular array of elliptical cylinders placed in a matrix of unit dielectric constant with the principal axes of the ellipses coinciding with the periodicity axes of the array. To obtain the effective dielectric constant of the array we use the method devised by Lord Rayleigh 4, for rectangular arrays of circular cylinders. Rayleigh’s method has been extended to include an arbitrary number of terms and applied in stud- ies of arrays of cylinders 5–7, arrays of coated cylinders 8–10, lattices of spheres 11,12, and lattices of coated spheres 10. This method has also been used in problems of elastostatics 13,14. Note that the same formalism and the results are immediately applicable to many other transport coefficients including those listed by Batchelor 15; e.g., thermal conductivity, electrical conductivity, magnetic per- meability, mobility, permeability of a porous medium, modulus of torsion in a cylindrical geometry, and effective mass in a bubbly flow. The technique used in this paper represents an extension of Rayleigh’s method to noncircular boundaries of the inclu- sions. In this paper we apply the Rayleigh method to such boundaries. Generally, Rayleigh’s method involves the cal- culation of certain static lattice sums, which contain informa- tion about the geometry of the array. We show that, impos- ing certain geometrical constraints on the aspect ratios of the unit cell and inclusions, only the lattice sums in polar coor- dinates are needed. With the exception of the lattice sum  2 , which is conditionally convergent, all the other polar lattice sums are absolutely convergent and can be evaluated with arbitrary high accuracy 4,7,14,16,17. The lattice sum  2 is related with the depolarization field and methods for evaluation have been devised 4,7,14,16,17. Structures involving rectangular arrays are intrinsically of interest since they permit inclusions to come close to touch- ing, even when the area fraction of the inclusions is small e.g., if inclusions are in close proximity along the y axis but well spaced along the x axis. It is thus necessary to distin- guish the concepts of critical composites, where inclusions come close to each other, from dilute composites, where the area fraction of the inclusions is small. We study all three cases: dilute noncritical composites, dilute critical compos- ites, and concentrated critical composites. For dilute noncritical composites, structured as rectangu- lar arrays of elliptical cylinders, we derive a Maxwell- Garnett type formula and comment on our result in relation to other formulas of a Maxwell-Garnett type, which have been proposed previously. In this way, we solve a contro- versy concerning the correct form of this relationship for composites with elliptical inclusions 18–20. For dilute critical composites, we exhibit a correction to the Maxwell- Garnett formula which renders it useful even in the region where the array dielectric constant is large. For concentrated systems, we establish the geometrical constraints on the va- lidity of our method. II. RAYLEIGH’S IDENTITY FOR ELLIPTICAL CYLINDERS We consider a rectangular array of infinitely long ellipti- cal cylinders embedded into a host medium of dielectric con- stant  0 . The dielectric constant of the cylinders (  ) is speci- fied as its ratio to the dielectric constant of the host medium, so that we will use  0 =1. The cross sections of the cylinders are ellipses with the foci distance 2 c . The semiaxes of the cross section of the cylinders are denoted by r 1 and r 2 Fig. 1. Note that the discussion which follows implicitly assumes r 1 r 2 : the modifications required to deal with r 2 r 1 are PHYSICAL REVIEW E AUGUST 1996 VOLUME 54, NUMBER 2 54 1063-651X/96/542/194513/$10.00 1945 © 1996 The American Physical Society