A Quantitative Analysis of Cermet Resistivity Data Using the General Effective Media (GEM) Equation
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A QUANTITATIVE ANALYSIS OF CERMET RESISTIVITY DATA USING THE GENERAL EFFECTIVE MEDIA (GEM) EQUATION
DAVID S. McLACHLAN Physics Department, University of the Witwatersrand, P 0 Wits 2050, Johannesburg, South Africa. ABSTRACT A General Effective Media (GEM) equation is used to quantitatively describe the resistivity of the W - A1 203 and Ni - Si0 2 cermet systems as a function of the relative volume fractions. The two percolation morphology parameters (Oc and t) characterise the microstructure. A constant resistivity p, is assumed for the metal (W or Ni) while, in some cases, it is necessary that the resistivity of the insulator be modeled as the tunneling of electrons, through an oxide barrier, from a thermally excited charged grain to a neutral grain. Theory Neither effective media equations [1] or the percolation equations [1,2] cani quantitative describe the resistivity of the W - A'203 and Ni - Si0 2 cermet systems as a function of volume fraction over the complete range at both 4.2 and 300 K. The objective of this paper is to show that a new general effective media (GEM) equation can do just this, provided the correct expressions are used for the resistivities of the two components. The Generalized Effective Media (GEM) equation, which contains the two percolation theory parameters, Oc = 1 - fc and t, was derived as an interpolation between the Bruggeman symmetric and asymmetric effective media theories [3] Here, 0c is critical (percolation) volume fraction at which the high conductivity (ah) component first forms a continuous path and t is an exponent. In this equation f(¢) will be used to denote the volume fraction of the low (high) conductivity component (0 = 1-f). The GEM equation written in terms of electrical conductivity is:
f(atl / t-oam1/t) Crtl/t+(fc
(1-fc))am1/t
( 1-f)(O'hl/t-mI /t) ahl/t+(fc/(1-fc)) ml/t
Mat. Res. Soc. Symp. Proc. Vol. 195. ©1990 Materials Research Society
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200
Here, ae is the conductivity of the low conductivity component, ah is the conductivity of high conductivity component and am is the conductivity of the medium itself. Equation 1 reduces to the symmetric and asymmetric theories and has the mathematical form of the percolation equation in the appropriate limits [3]. The GEM equation can also be viewed as a matched asymptotic expression as it interpolates between the two percolation equations. am=ah(( 1 -f/fc)t[ar=O] and Pm=Ph( 1-O/c) tah=®,pt=0]
where
Ph = 1a, p, = 1/ah and pm = l/am.
The GEM equation has been used to accurately fit conductivity data [3-6]. It has also been used to conductivities and permeability experimental data from sample (0.14 - 0.95 volume fraction of nickel), using parameters fc and t [4].
a large amount of electrical fit electrical and thermal a series of sintered nickel the same two morphology
In some instances, the resistivity of the low field high resistivity component must be modeled as the tunneling of electrons, through the oxide barrier, from a thermally excited charged metallic grain to a neutral one [Reference 7 and the reference
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