radius, the Casimir–Polder energy the Si sphere would be determined by a Green’s function at the Si location [@Lifshitz80], and if the Si sphere is at comparable distance $d_e \sim h_e$ from both the teflon plate and the sphere, one would in general expect the response to a point-dipole source at the Si location (the Green’s function) to depend non-additively on the teflon sphere and the plate even for an infinitesimal Si sphere. However, in the present case we do not observe any non-additive effect on the Si-sphere $h_e$, because the factor of three(approximately) difference between $h_e$ and $d_e$ is already sufficient to eliminate three-body effects (as in [Fig. \[fig:fig2\]]{}).

[Figure \[fig:fig4\]]{} also exhibits a bifurcation of stable (solid lines) and unstable (dashed lines) equilibria that causes the stable $h$ equilibrium to vanish for Si at large radii. In order to utilize larger spheres to reduce the effects of Brownian motion in the next section, one can consider instead a geometry of hollow air-filled spherical shells with outer radius $R$ and shell thickness $\alpha R$ (so that $\alpha=1$ gives a solid sphere). Such hollow microspheres are readily fabricated with a variety of materials [@Wilcox]. As [Fig. \[fig:fig4\]]{} shows, decreasing the 