Applications of the Correlation Length

Abstract

The correlation length is the system–dependent parameter, which defines the structure of the dominant current–carrying (electric or fluid) paths. Refer back to Fig. 1.3. The typical separation of the nodes is represented in this figure, and this separation is equal to the correlation length, x. The physical reason for this is that x describes the size of the largest holes above the percolation threshold. Furthermore, the tortuosity of the backbone of the largest clusters below the percolation threshold is the same as the tortuosity of the links above the percolation threshold. The influence of the blobs in calculating the conductivity is rather secondary since the most resistive elements that cannot be avoided tend to be found in the portions of links without blobs – by definition there is no alternative to the paths through these (except, in the case of critical path analysis, to go to more resistive elements). In fact, as just suggested, x can be used to describe the structure of such paths in two different contexts: 1) near the percolation transition it gives a characteristic separation of the only possible paths of interconnected medium, which can be used to transport, e.g., air, water, or electrical current, 2) far from the percolation threshold, application of critical path analysis involves an optimization which leads to a calculation of the separation of the paths along which the dominant transport occurs. In either case, x³ is effectively the representative elementary volume, or REV, because in each case x defines the length scale of the heterogeneity relevant for transport. In earlier chapters critical path analysis was used to generate explicit expressions for the correlation length. As long as the numerical coefficient in the proportionality from percolation theory is not available, however, calculations using the correlation length cannot reliably yield precise numerical coefficients for specific systems so the results are given only in terms of system parameters. In this chapter some systems are treated for which there is little or no information regarding “microscopic” variability, and the expressions derived contain further unknown constants. Thus the development here is only diagnostic and not predictive. In chapter (3) an example of this type of argument (originally due to Shklovskii) is given in the problems with an at least semi–quantitative prediction.