(units of nm) of a teflon (red) or Si hollowed sphere (shown on the inset) suspended in ethanol above a gold plate, as a function of radius $R$ (units of nm). The equilibria are plotted for different values of the fill-fraction $\alpha$, defined as the ratio of the spherical–shell thickness over the radius of the sphere. Solid/dashed lines correspond to stable/unstable equilibria.[]{data-label="fig:fig4"}](larger_alph_eqbms){width="1.0\columnwidth"}

To begin with, let us consider sphere radii on the order of $10^2$ nm, as in our previous work [@RodriguezMc10:PRL]. We wish to make a bound dicluster, at some separation $d$, of two spheres (Si and teflon) that are suspended above a gold substrate by Casimir repulsion in balance with gravity. Furthermore, suppose that we wish to suspend both spheres at the same equilibrium height $h_e$, and therefore choose the radii of the two spheres to equate their $h_e$ values. In [Fig. \[fig:fig4\]]{}, we plot $h_e$ as a function of radius $R$ for the isolated sphere–plate geometries ($d\to\infty$). For example, with an Si sphere of radius $R=100$ nm, the (stable) equilibrium height is $h_e = 298.17$ nm, whereas to obtain the same $h_e$ value for teflon one needs a much larger teflon sphere of radius $R=217.2$ nm, primarily 