of the sphere-plate separation $h$ for two equal-radius spheres, as plotted in [Fig. \[fig:fig2\]]{}. To start with, we consider very small spheres, with radius $R=25$ nm, for which the three-body effects are substantial. The separation $d_h$ at a given $h$ is normalized by $d_\infty$ ($d$ as $h\to\infty$, i.e. in the absence of the plate). Several different material combinations are shown (where $X$–$Y$–$Z$ denotes spheres of materials $X$ and $Y$ and a plate of material $Z$): polystyrene (PS), teflon (Tef), and silicon (Si) spheres with gold (Au), teflon (Tef), and vacuum (air) plates (the latter corresponding to a fluid-gas interface). Depending on the material combinations, we find that $d_h$ can either increase or decrease by as much as $15\%$ as the plate is brought into proximity with the spheres from $h=\infty$ to $h \approx R$. (We expect even larger deviations when $h < R$, but small separations are challenging for this computational method [@Rahi09:PRD] and our results for $h \geq R$ suffice here to characterize the general influence of three-body effects.)

Interestingly, depending on the material combination, the $d_h$ can either increase or decrease as a function of $h$: that is, the proximity of the plate can either increase or decrease 