  The Casimir-Lifshitz interaction between metamaterials is studied using a
model that takes into account the structural heterogeneity of the dielectric
and magnetic properties of the bodies. A recently developed perturbation theory
for the Casimir-Lifshitz interaction between arbitrary material bodies is
generalized to include non-uniform magnetic permeability profiles, and used to
study the interaction between the magneto-dielectric heterostructures within
the leading order. The metamaterials are modeled as two dimensional arrays of
domains with varying permittivity and permeability. In the case of two
semi-infinite bodies with flat boundaries, the patterned structure of the
material properties is found to cause the normal Casimir-Lifshitz force to
develop an oscillatory behavior when the distance between the two bodies is
comparable to the wavelength of the patterned features in the metamaterials.
The non-uniformity also leads to the emergence of lateral Casimir-Lifshitz
forces, which tend to strengthen as the gap size becomes smaller. Our results
suggest that the recent studies on Casimir-Lifshitz forces between
metamaterials, which have been performed with the aim of examining the
possibility of observing the repulsive force, should be revisited to include
the effect of the patterned structure at the wavelength of several hundred
nanometers that coincides with the relevant gap size in the experiments.
