approximation (PFA), in which the force can be approximated by additive surface–surface “parallel-plate” forces [@bordag01; @Gies; @Derjaguin]. In order to damp the Brownian fluctuations as described in the next section, we actually propose to use much larger ($R>5\,\mu$m) spheres, and we quantify the accuracy of PFA in this regime below. Second, as $h$ becomes large compared to $d$, the effect of the plate becomes negligible and three-body effects disappear; this is apparent in [Fig. \[fig:fig2\]]{}, where $d_e \to
d_e(\infty)$ when $h \gg d_e$. Third, in the limit where one of the spheres is much smaller than the other sphere, then the smaller sphere has a negligible effect on the sphere–plate interaction of the larger sphere, and at least some of the three-body effect disappear as described below. In fact, we find that even for a situation in which one sphere is only a few times smaller than the other, the three-body effects tend to be negligible. For the sphere-radius regime considered in our previous work, we argue below that equal-height suspension of the two spheres leads to a strong asymmetry in sphere radii that tends to eliminate three-body effects.

![Center–surface $L_e$ (solid lines) and surface–surface $h_e$ (dotted lines) equilibrium separation 