Mixing rules and the Casimir force between composite systems

The Casimir-Lifshitz force is calculated between two inhomogeneous composite slabs, each made of a homogeneous matrix with spherical metallic inclusions. The effective dielectric function of the slabs is calculated using several effective medium approximations and we compare the resulting forces as a function of slab separation and filling fraction. We show that the choice of effective medium approximation is critical in making precise comparisons between theory and experiment. The role that the spectral representation of the effective medium plays in making a Wick rotation to the complex frequency axis is also discussed.

Figure 1

Effective dielectric function plotted as a function of the filling fraction for a composite made of a SiO${}_2$ host with spherical Au inclusions. In the top panel the effective dielectric function is evaluated at the imaginary frequency $\zeta =0.02{\omega }_p$, and at $\zeta =0.5{\omega }_p$ for the bottom graph.

Figure 2

Casimir force (normalized to the ideal case) as a function of plate separation between two composite plates made of a homogeneous matrix of SiO${}_2$ with spherical inclusions of radius $a$. The results in this plot are valid provided that condition $4\pi L>a$ is satisfied. The top figure is for low filling fractions $f=0.015$. The Maxwell-Garnett and Bruggeman models give results similar to the lower Wiener bound. For higher filling fraction ($f=0.25$) the difference in the force between the different models is much bigger, in particular at large separations.

Figure 3

Reduction factor as a function of filling fraction when the two plates are at a fixed separation $L=100$ nm. At low filling fraction the difference between the Bruggeman and Maxwell-Garnet models is negligible but as the number of Au particles increases, the Bruggeman model correctly predicts a more metallic behavior, increasing the force between the slabs.