presence of the plate typically acts to either increase or decrease the sphere–sphere interaction, depending on the sign of the dominant sphere–plate interaction, as explained above.

In particular, [Fig. \[fig:fig3\]]{} shows the isolated Si–PS and Tef–Si sphere–sphere equilibrium separation $d_e$ as a function of the radius $R$ of the spheres. As a consequence of its material dispersion (similar to phenomena observed in [@RodriguezMc10:PRL]), the Si–PS combination exhibits a bifurcation at $R \approx 55$ nm where the stable and unstable equilibria, such that there is no equilibrium for larger $R$ (the interaction is purely attractive). The Tef–Si combination exhibits no such bifurcation (even if we extend the plot to $R=300$ nm), because it has no unstable equilibrium: the interaction is purely repulsive for small separations and attractive for large separations. Therefore, if the Si–PS radius is above or below the 55 nm bifurcation, the presence of the plate can shift this bifurcation and lead to a bifurcation as a function of $h$ as in [Fig. \[fig:fig2\]]{}, whereas no such bifurcation with $h$ appears for Tef–Si.

In the Si–PS–Au case of a gold plate with Si–PS spheres, the sphere–plate interactions turns out to be primarily repulsive, which should push the bifurcation in 