The Use of Computer Simulations to Interpret and Understand Electrical Measurements
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Mat. Res. Soc. Symp. Proc. Vol. 500 © 1998 Materials Research Societv
processing power and memory constraints, handle arbitrary shapes, topologies, and electrical properties. It is important to emphasize that the electrical response of a microstructure is indeed a mixture of shape, topology, and electrical properties. Two different topology microstructures can give a similar electrical response depending on the electrical properties of the different phases. In this paper, "electrical properties" means DC conductivity, with the amplitudes of applied fields small enough so that the material response is always linear. The term "computer simulations" means direct construction ofa microstructure in a computer, and then solution of the appropriate electrical equations on this microstructure under some set of boundary conditions. The main kind of simulation that will be discussed in this paper is typified by programs like dc3d.f, whose code and operating manual can be found at http://ciks.cbt.nist.gov/garboczi [1]. This is a finite-difference program for DC linear 3-D steady-state conduction problems, where the microstructure is described by a digital image. This program is especially adapted for working on 3-D digital images. The problem being solved is given by the equation
(1)
V. (oE) = 0 where the current flux density is given by
j = oE
(2)
so that eq. (1) is the steady state charge conservation equation. Besides studying random microstructures, one can also be concerned about particle shapes when a second phase is particulate. In the case of very small particles, it may be difficult to directly image them and so their shape must be inferred from property measurements. One can also be concerned about sample and electrode shape. Simple sample shapes are usually preferred, like cylinders, cubes, etc. because when the electrodes are applied uniformly across the ends, the applied field is uniform in the sample. However, it may be experimentally useful to have an "odd" sample and/or electrode shape. The electrical equations will usually not be analytically solvable for a general shape. In this case as well, computer simulations make it possible to understand the response of these kind of shapes. The remainder of this paper will show how computer simulations can be used at the individual particle level, the microstructural level, and the sample level, to compute the overall electrical signal. INDIVIDUAL PARTICLE LEVEL When a second phase is particulate, and is dilute, so that its volume concentration is small, say at most 5% or so, then much can be inferred about particle shape and electrical properties from the electrical signal. To be more specific, the case is that a second phase, of some conductivity 02 and volume fraction cq,embedded as isolated particles in a matrix of conductivity a, and volume fraction cl. We wish.to infer the shape and properties of the particles from the overall conductivity, where
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a1 is known. In this case, much can be done analytically, for certain shapes. The quantity to be d
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