force is determined by the ordering of the $\varepsilon_k(i\kappa)$ values at imaginary frequencies $\omega=i\kappa$ (where $\varepsilon_k$ is purely real and positive for any causal passive material [@Dzyaloshinskii61]). If the ordering changes for different values of $\kappa$, then there are competing repulsive and attractive contributions to the force. At larger or smaller separations, smaller or larger values of $\kappa$, respectively, dominate the contributions to the total force, and so the force can change sign with separation. For example, if $\varepsilon_1 <
\varepsilon_2 < \varepsilon_3$ for large $\kappa$ and $\varepsilon_1 <
\varepsilon_3 < \varepsilon_2$ for small $\kappa$, then the force may be repulsive for small separations and attractive for large separations, leading to a stable equilibrium at an intermediate nonzero separation. Alternatively, for a sphere–plate geometry in which the sphere is pulled downwards by gravity, a purely repulsive Casimir force (which dominates at small separations) will also lead to a stable suspension. These basic ideas were exploited in our previous work [@RodriguezMc10:PRL] to design sphere–sphere and sphere–plate geometries exhibiting a stable non-touching configuration. The effects of material dispersion are further modified by an interplay with geometric effects (which set additional length-scales beyond that of the separation), as well as by nonzero-temperature effects 