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The classical differential mixing rules are assumed to be independenteffective-medium approaches, applicable to certain classes of systems. In thepresent work, the inconsistency of differential models for macroscopicallyhomogeneous and isotropic systems is illustrated with a model for the effectivepermittivity of simple dielectric systems of impenetrable balls. The analysisis carried out in terms of the compact group approach reformulated in a waythat allows one to analyze the role of different contributions to thepermittivity distribution in the system. It is shown that the asymmetricalBruggeman model (ABM) is physically inconsistent since the electromagneticinteraction between previously added constituents and those being added isreplaced by the interaction of the latter with recursively formed effectivemedium. The overall changes in the effective permittivity due to addition ofone constituent include the contributions from both constituents and depend onthe system structure before the addition. Ignoring the contribution from one ofthe constituents, we obtain generalized versions of the original ABM mixingrules. They still remain applicable only in a certain concentration ranges, asis shown with the Hashin-Shtrikman bounds. The results obtained can begeneralized to macroscopically homogeneous and isotropic systems with complexpermittivities of constituents.

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