Inverse Transport Problems for Composite Media
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INVERSE TRANSPORT PROBLEMS FOR COMPOSITE MEDIA
ROSS C. MCPHEDRAN* AND GRAEME W. MILTON** *School of Physics, University of Sydney Sydney N.S.W. 2006 Australia
"Courant
Institute, New York University
251 Mercer Street, New York, NY 10012, U.S.A.
ABSTRACT For two-component ceramic metal composites we address the inverse problem of estimating the volume fraction occupied by each component from measurements of the transmission and absorption coefficients at various frequencies of the applied radiation. We devise the algorithms, each based on representation formulas and bounds for the complex dielectric constant that were developed by Bergman and Milton. One inversion algorithm works remarkably well when the measurements are exact but fails completely when small errors are present. The second random throw algorithm works well when the measurement errors are large but is computationally intensive and is limited to small data sets. The third method based on single and two-point inversion, while not so accurate, is simple and efficient. The last two algorithms are implemented with experimental data and good agreement is obtained with the experimentally measured true volume fraction.
1. INTRODUCTION Characterizing the microgeometry of a two-phase composite is no easy task and there are many geometric parameters, such as grain size, grain eccentricity, surface area, and so forth, which represent important features of the microstructure. However, one of the most basic parameters, without questions is the volume fraction occupied by each constitutent. Our attention will be focussed on recovering this parameter from measurements of the transmission and absorption
Mat. Res. Soc. Symp. Proc. Vol. 195. 01990 Materials Research Society
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coefficients of ceramic-metal composite films (cermets) at various frequencies of the applied radiation. Such films are used as selective absorbers for solar heating. In figure 1 we show an electron micrograph of a cermet film.
Figure 1. An electronmicrograph of a cermet film. The dark areas represent the metallic phase.
The data is obtained from studies of the optical properties of MgF 2 -Ag and Ag-air composite films. (If silver films are made sufficiently thin, they develop voids.) Reflectance and transmittance measurements on the films are used to deduce the refractive index n, and absorption coefficient ke as a function of wavelength X : these optical constants represent the real and imaginary parts of the complex refractive index h,. Estimates of the measurement errors then lead to error estimates Snh and 8k,. The transport coefficient corresponding to the optical constants is the complex dielectric constant, the square of the complex refractive index:
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.2 (n5 + 1k6)2 •e= n,=
(1)
Typical plots of n, and of k, as a function of wavelength, with error bars, are illustrated in figure 2. In addition to the errors in the effective transport coefficients, there are inevitable errors in the values assumed for the optical constants of the two phases of the cermets. It is difficul
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