Fig. 3: Critical Casimir forces overcome Casimir–Lifshitz forces.
From: Tunable critical Casimir forces counteract Casimir–Lifshitz attraction
a,b, The diffusion constant D of a hydrophilic hexagonal flake with a = 700 nm above a hydrophilic surface (symmetric (−, −) boundary conditions) decreases as the temperature T approaches the critical temperature Tc (a, blue squares) while its height h decreases (b, blue squares). The solid blue line is the best fit of the theoretical model (Eq. (8)), from which we extracted heq in Eq. (9). c, The flake is under the action of the Casimir–Lifshitz (CL) force (orange dots) and the critical Casimir force (CCF, blue dots), both of which are attractive and whose strengths increase as the flake approaches the surface, while electrostatic forces (ES, green dots) are repulsive. The orange and blue solid lines are the theoretical (sim) fits of the critical Casimir (Eq. (2)) and Casimir–Lifshitz (Eq. (1)) forces, respectively. exp, experiment; sim, simulation. Note how the critical Casimir force becomes stronger than the Casimir–Lifshitz force at ΔT ≈ −0.3 K. d,e, The diffusion constant D (red circles) for a hydrophilic hexagonal flake with a = 840 nm above a hydrophobic surface (antisymmetric (−, +) boundary conditions) increases upon approaching Tc (d) as its height h (red circles) increases (e). f, The solid red line is the best fit of the theoretical model (Eq. (8)) of the average height heq ≈ 〈h〉 (Eq. (9)) under the action of the Casimir–Lifshitz force (orange dots, attractive), the critical Casimir force (red dots, repulsive) and the electrostatic force (green dots, repulsive). The red, orange and green solid lines are the theoretical values from the fit of the critical Casimir, Casimir–Lifshitz and electrostatic forces, respectively. Data presented as mean values. The error bars on the experimental points in b and e are the s.d. from three measurements. The error range for the theoretical lines in b and e is indicated as shaded areas for the confidence levels of 68%, 86% and 95%. In c and f, only the 68% confidence level error range is shown. The dashed lines in b and e refer to the equilibrium height of particle far away from criticality.
