A Century of General Lichtenecker Equation: Between Stringency and Empiricism, Accuracy and Approximability

The general Lichtenecker equation, with exponent values in the range -1≤k≤1, is widely used to describe and predict the effective macroscopic electrical, magnetic, thermal and optical properties of various heterogeneous media. Although it sometimes fits experimental data well, in most cases, however, it is physically incorrect and its predictive ability is often overestimated. Using a rigorous spectral representation, we show that for isotropic composites, except for the trivial one-dimensional case, only two specific forms of this equation, the logarithmic (k=0) and Landau-Lifshitz-Looyenga (k=1/3) mixing laws, are dimensionally consistent, corresponding to spatial dimensions d=2 and d=3, respectively. Furthermore, the requirement of phase-interchange symmetry severely restricts the applicability of the Lichtenecker equation, rendering it unsuitable for most real composite systems.