On applicability of differential mixing rules for statistically homogeneous and isotropic dispersions

The classical differentialmixing rules are assumed to be independent effective-medium approaches, applicable to certain classes of systems. In the present work, the inconsistency of differentialmodels formacroscopically homogeneous and isotropic systems is illustratedwith amodel for the effective permittivity of simple dielectric systems of impenetrable balls. The analysis is carried out in terms of the compact group approach reformulated in away that allows one to analyze the role of different contributions to the permittivity distribution in the system. It is shown that the asymmetrical Bruggemanmodel (ABM) is physically inconsistent since the electromagnetic interaction between previously added constituents and those being added is replaced by the interaction of the latter with recursively formed effectivemedium. The overall changes in the effective permittivity due to addition of one constituent include the contributions fromboth constituents and depend on the system structure before the addition. Ignoring the contribution fromone of the constituents, we obtain generalized versions of the original ABMmixing rules. They still remain applicable only in a certain concentration ranges, as is shownwith theHashin-Shtrikman bounds. The results obtained can be generalized tomacroscopically homogeneous and isotropic systemswith complex permittivities of constituents.