MaxwellGarnett

Maxwell-Garnett theory, or the Maxwell-Garnett effective medium approximation, is a model used to estimate the macroscopic dielectric response of a composite material containing small inclusions embedded in a host medium. It arises from treating the inclusions as individual dipoles polarized by an external electric field and assumes that the inclusions are small compared to the wavelength of light and that interactions between inclusions are negligible. The theory is most accurate for dilute systems where the volume fraction of inclusions is moderate to low.

For spherical inclusions, the effective permittivity εeff of the composite is given by

εeff = εh [ (εi + 2εh) + 2f(εi − εh) ] / [ (εi + 2εh) − f(εi − εh) ],

where εh is the permittivity of the host, εi the permittivity of the inclusions, and f the volume fraction of inclusions. The formula can be extended to dynamic or dispersive media and, in principle, to anisotropic or non-spherical inclusions by incorporating appropriate depolarization factors, though the simple spherical form is the most commonly cited.

Limitations include its reliance on non-interacting, small, randomly distributed inclusions and the quasi-static approximation. At higher volume fractions, or when particle interactions, non-spherical shapes, aggregation, or percolation effects become significant, other effective medium theories such as Bruggeman’s model may be more appropriate. The Maxwell-Garnett approach remains widely used in optics, nanocomposites, and metamaterials to model the dielectric behavior of composites across a range of frequencies.