Extended Data Fig. 6: Trapping and release of a flake, prediction of the theoretical model.
From: Tunable critical Casimir forces counteract Casimir–Lifshitz attraction
a–h, Two dimensional potential Utot(ΔT; h, x), i–p, effective lateral potential Ueff(ΔT; x), and q–x, effective lateral probability distribution function Peff(ΔT; x) for a disk-like flake (radius a = 1450 nm, thickness b = 40 nm) hovering on a patterned substrate with hydrophobic, 3-μm-wide, gold-coated stripes alternating periodically with hydrophilic, 3-μm-wide, uncoated silica stripes. The thickness of each gold-coated stripe is a1 = 30nm. As shown in panels a–h, the height of the flake h on the surface is measured with respect to the upper surface of the gold stripes; hence, when considering the height of the flake with respect to the hydrophilic uncoated surface, one must take into account the additional term a1. The profile of the substrate is represented in panels a–h. The value of the two-dimensional total potential is represented with a contour plot. For ΔT = − 1 K, the minimum of the total potential is located over the gold-coated stripe at its centre (for symmetry reasons) about 100nm above the surface. Upon increasing the temperature towards Tc, the minimum of the potential becomes less and less deep, and for ΔT > − 0.1 K, the minimum is located above the uncoated silica stripe. In i–p, the effective lateral potential Ueff(ΔT; x) defined in Eq. (S8) is represented with a black continuous line. The zero of the potential is set at x = 0. For ΔT≤ − 0.12K (i–m), the effective potential has a single minimum at x = 1.5μm; for ΔT≥ − 0.1K (o,p), the effective potential has a single minimum at x = 4.5μm, that is, localized at the centre of the uncoated silica stripe; for ΔT = − 0.1K (n), the effective potential has two local minima. In i–p the effective probability distribution Peff(ΔT; x) defined in Eq. (S9) is represented with a black continuous line. In i–m, Peff(ΔT; x) has a single peak at x = 1.5μm, indicating that the flake is localized on the gold stripe. The peak is initially very sharp, but becomes broader and less high upon increasing the temperature towards Tc. In n, Peff(ΔT; x) has two peaks. In n,o, the peak is again one, this time localized at the centre of the uncoated silica stripe. In this model, the parameters for the electrostatic interaction are λD = 17 nm and P0 = 1.5 kPa for the gold-coated stripes, whereas P0 = 28.8 kPa for the gold-coated stripes.
