does change its $h_e$ at a given $R$. For example, one can choose a Tef $\alpha=0.142$ and a Si ($\alpha=0.14$) to obtain the same equilibrium surface-to-center height $L_e$ over a wide range of sphere radii, as shown in upper-right inset of [Fig. \[fig:fig6\]]{}. Alternatively, one can choose a hollow teflon sphere to match the equilibrium surface-surface separations $h_e$ for equal sphere radii, as shown in the lower-right inset of [Fig. \[fig:fig6\]]{}.

![Surface–surface equilibrium height $h_e$ (units of nm) for the hollowed–sphere geometry of [Fig. \[fig:fig5\]]{}, consisting of either a Si (top) or teflon (bottom) hollowed sphere (fill-fraction $\alpha$) suspended in ethanol above a $H = 15$ nm ITO layered gold plate, as a function of sphere radius $R$ (in units of $\mu$m). Solid/dashed lines correspond to stable/unstable equilibria. $h_e$ is plotted for different values of $\alpha$, denoted in the figure. The top inset plots the center–surface separation $L_e$ (in units of nm) as a function of $R$ of a hollowed teflon/Si (red/blue lines) sphere suspended again above a gold plate, for $\alpha = 0.14/0.142$. The lower inset shows $h_e$ for both teflon and Si spheres for $R \in [8.6,10]\mu$m.[]{data-label="fig:fig6"}](Figure6){width="1.0\columnwidth"}

![Average $\langle h \rangle$ (thick lines) and equilibrium $h_e$ (thin lines) 