Conductivity Fluctuations in Two-Component Finite Systems
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CONDUCTIVITY FLUCTUATIONS IN TWO-COMPONENT FINITE SYSTEMS
JORGEN AXELL AND JOHAN HELSING Department of Theoretical Physics, The Royal Institute of Technology, S-100 44 STOCKHOLM, Sweden.
ABSTRACT In all applications and experimental setups one has to do measurements on finite samples with some average properties. Due to statistical fluctuations in the realization of the specific sample one has, there will be a spread in the measured properties. In this paper we consider electrical conduction in a two-phase material, where the two phases have different conductivities. We study the size dependence of the fluctuations in the effective conductivity and give the dependence of the sample size and the dependence of the resistance ratio, especially when the latter is close to one. A simple model and numerical simulations are presented.
INTRODUCTION Composite materials are of increasing technological interest. Since they give rise to interesting physical problems physicists have for a long time paid them attention. Much work [1] has been done to determine various transport and mechanical properties for inhomogeneous systems. Most work has focussed on bulk properties in infinite systems. If one performs measurements of, e.g., the electrical conductivity in a finite system with a binary distribution of conductivities one will obtain results scattered around some mean. This scattering will depend on the sample size, the resistance ratio and concentrations of the two components. In this paper we distinguish between two different kinds of fluctuations in finite samples. The first is called concentration fluctuations and arises because in a finite sample where the grains take any of two different conductivities with fixed probabilities the concentration Pi of phase one will vary. The second type, configuration fluctuations, arises because different configurations of the constituents give different effective conductivities in a sample where the total concentrations are fixed, i.e. in a sample without concentration fluctuations. We give in this paper models for determining these two kinds of fluctuations and compare the models with numerical simulations on resistor networks. Further details will be given elsewhere [2].
Mat. Res. Soc. Symp. Proc. Vol. 195. 01990 Materials Research Society
142
MODEL SYSTEMS
Definitions We have chosen the random checker-board as a model system. The linear system size, N, is the number of squares along one side. The squares are assigned either a conductivity at with probability p or a conductivity 02 (< at) with probability 1p. Since the total conductance, L, in a system is a homogeneous function of the conductivities [3], L(0l ,02) = 21(01/0o2,1), we may without loss of generality set 02=1. A convenient variable [4] for the resistance ratio used in series expansions is e=(al02)/al. If we use a fixed probability p for assigning a square the conductance 0l, we will in a finite realization of a system get a concentration 0, which will be described by a binomial distribution. The standard deviatio
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