Specific Examples of Critical Path Analysis

Abstract

At the end of Chap. 1 the general technique of critical path analysis was introduced. This technique will actually form the basis of most of the remainder of this course. However, there are some subtleties to the technique, and its application cannot be so easily generalized as sometimes assumed. To some extent every case or system must be evaluated separately. In typical solid-state applications conductances are connected between sites, which are located randomly in space (why?) meaning that the critical bond fractions from the lattice models of Chap. 1 are not directly applicable. Nevertheless, when transport is locally defined according to the probability per unit time that, say, an electron moves from one clearly defined site to another, then it is clear that the appropriate form of percolation theory to apply is bond percolation, even though one may not know from the results of Chap. 1 what an appropriate bond percolation threshold is. But in porous media it is not initially obvious even which form of percolation theory, bond, site, or continuum, should be used. While network models, which distinguish carefully between pore bodies and pore throats, and for which it is known that the chief limitation to flow comes through the pore throats, clearly require a bond percolation approach, it has been my contention that use of a continuous random fractal model requires application of continuum percolation theory. In each of these cases, for which an important random component of the model is already linked with the topology of the connections, the critical percolation probability becomes a major issue, especially when the distribution of local flow (or transport) rates is very wide, and particularly since one of the basic components of the analysis is supposed to be the determination of the critical path.