Effective conductivity for densely packed highly conducting cylinders

We study the effective heat conductivity 2e of a periodic square array of nearly touching cylinders of conductivity h, embedded in a matrix material of conductivity 1. We construct a sequence of two-point Pad6 approximants for the effective conductivity. As the basis for the construction we use the coefficients of the expansions of 2e at h = 1 and h = oe. The two-point Pad6 approximants form a sequence of rapidly converging upper and lower bounds on the effective conductivity. PACS: 02.00, 65.40.f, 66.70.+f One of the main tasks of the theory of inhomogeneous media is a theoretical prediction of the effective transport properties such as the effective heat or electric conductivity, the effective dielectric constant, and the effective elastic constants. In the present paper we study the effective conductivity 2e of a two-phase material which consists of an infinite regular array of nearly touching cylinders. The conductivity of the cylinders is h and they are embedded in a matrix material of conductivity 1. The effective conductivity of periodic systems has been theoretically investigated in several ways. The usual starting point is the infinite set of Rayleigh's equations [1]. The Rayleigh's equations have been solved either by a truncation method [2, 3] or by a power expansion in the variable ( h 1)/(h+ 1) [4]. The power-series expansion has been used to construct Pad6-approximants and continued fraction representations of the effective conductivity [4, 6]. It has been shown that certain Pad6 approximants form a converging sequence of lower and upper bounds on 2°. Neither of the above mentioned methods yields accurate results for a system of highly conducting, nearly touching cylinders. In order to describe such a system McPhedran and his colleagues [7] have derived an asymptotic formula. However, the validity range of this formula is not known. Moreover, there still remains a certain parameter range which is covered neither by the asymptotic formula nor by the solutions based on Rayleigh's equations. The main purpose of our paper is to fill in this gap. To this end we develop a new approach based on an application of two-point Pad6 approximants. We construct the approximants by using the coefficients of the expansions of 2o at h = 1 and h = oe. As with one-point approximants, we show that certain two-point Pad~ approximants form a sequence of converging lower and upper bounds on the effective conductivity [5]. The convergence in the two-point case is much faster than in the one-point case. This paper is organized as follows: In Sect. 1 we define our model system. In Sect. 2 we construct the power expansion of 2o around h = 1 in a new, concise method. In Sect. 3 we discuss two point Pad6 approximants for 2e. In Sect. 4 we present numerical results for a system of nearly touching cylinders. We conclude with some final remarks in Sect. 5. 1 Array of conductive cylinders In this paper we consider an infinite array of identical, parallel cylinders arranged in a square lattice. Very similar methods to those applied in this paper can be also applied for other periodic arrangements of parallel cylinders. Without loss of generality we may assume that the nearest-neighbor distance of the cylinder axis is equal to 1. We denote the cylinder radius by 0.The difference of the conductivities of the cylinders and the matrix medium is denoted by z = h 1. The continuous temperature distribution in the system considered obeys the conductivity equation of the form: V" (1 + z O 2 ) V T = 0, (1) where 02 is the characteristic function of the volume occupied by the cylinders. The conductivity equation (1) is supplemented by the continuity condition for the nor602 S. Tokarzewski et al. mal component of the heat current J = (1 + z O2)V T at the surface of the cylinders. In order to evaluate the effective heat conductivity we consider the system under the influence of a constant temperature gradient along one of the main square lattice axis, say in the x-direction. The temperature field can be then decomposed into the systematic part T (°) and the periodic part 6 T: T = T (°) + c~ T. (2) Since the problem is linear the amplitude of the temperature field is irrelevant. Therefore we may set