Extended Data Fig. 7: Effective stiffness predicted by the theoretical model.
From: Tunable critical Casimir forces counteract Casimir–Lifshitz attraction
Stiffness of the lateral trapping (kx) calculated for the effective potential Ueff as a function of ΔT = T − Tc. Here we calculate kx using the variance of the peak of the probability distribution Peff(ΔT; x), that is, \({k}_{x}=\frac{{k}_{{{{\rm{B}}}}}T}{{\sigma }^{2}}\) where \({\sigma }^{2}=\int\nolimits_{{x}_{1}}^{{x}_{2}}{P}_{{{{\rm{eff}}}}}(\Delta T;x)\,{(x-{x}_{{{{\rm{peak}}}}})}^{2}\,{{{\rm{d}}}}x\). The extrema of the integral are defined as x1 = 0, x2 = 3 μm for the ΔTs where the absolute peak of Peff(ΔT; x) falls in the centre of the gold-coated stripe, and as x1 = 3 μm, x2 = 6 μm for the ΔTs where the absolute peak of Peff(ΔT; x) falls in the centre of the uncoated silica stripe. We remark that, for our model, σ is determined not only by the parameters of the interaction, but also by the geometrical characteristics of the system. In particular, the main factor affecting σ in our model is the difference between the diameter of the flake and the width of the stripes. In fact, Ueff(ΔT; x) is necessarily flat over an interval of length w − 2 R, where w is the thickness of the stripe and R is the radius of the disk-like flake. Hence, for the same parameters describing the interaction per unit area, different flake dimensions have a σ that is at least w/2 − R, which increases upon decreasing R.
