non-monotonic effect [@RahiRo07; @Rodriguez07:PRL].)

[Figure \[fig:fig2\]]{} also exhibits the interesting phenomenon of bifurcations, in which stable equilibria (solid lines) and unstable equilibria (dashed lines) appear/disappear at some critical $h$ for certain materials and geometries, which is discussed in more detail in [Sec. \[sec:bifurcations\]]{}. As the sphere radius $R$ increases, all of these three-body effects rapidly decrease, eventually entering an additive regime in which three-body effects are negligible and in which a parallel-plate/PFA approximation eventually becomes valid, as described in [Sec. \[sec:additive\]]{}.

Bifurcations {#sec:bifurcations}
------------

![Equilibrium separation $d_e(\infty)$ (units of nm) between a Si sphere and either a teflon(Tef) or polystyrene(PS) sphere immersed in ethanol as a function of their equivalent radii $R$. Solid/dashed lines denote stable/unstable equilibria.[]{data-label="fig:fig3"}](new_Silicon-plsty_dvsR){width="1.0\columnwidth"}

In the case of PS and Si spheres suspended above either a gold or teflon plate, one can observe the emergence or disappearance of a stable (solid) and unstable (dashed) pair of equilibria as $h$ decreases from $h=\infty$, respectively, as evidenced by the blue curves in [Fig. \[fig:fig2\]]{} (teflon plate) and [Fig. \[fig:fig2\]]{} (inset) (Au plate). This can be qualitatively explained by the fact that the isolated sphere–sphere interactions exhibit a natural bifurcation for sufficiently-large spheres, in conjunction with the fact that the 