Stable Suspension and Dispersion-Induced Transitions from Repulsive Casimir Forces Between Fluid-Separated Eccentric Cylinders

We numerically demonstrate a stable mechanical suspension of a silica cylinder within a metallic cylinder separated by ethanol, via a repulsive Casimir force between the silica and the metal. We investigate cylinders with both circular and square cross sections, and show that the latter exhibit a stable orientation as well as a stable position, via a method to compute Casimir torques for finite objects. Furthermore, the stable orientation of the square cylinder undergoes a 45° transition as the separation length scale is varied, which is explained as a consequence of material dispersion.

Figure 1

Casimir force $F_x$ in the $x$ direction, per unit $z$ length, on a ${\mathrm{SiO}}_2$ cylinder suspended within a perfectly metallic cylinder (inset), separated by fluid (ethanol), as a function of the $x$ displacement from equilibrium (eccentricity) $d$ [in units of $a=0.5(D-s)$] for both circular (solid line) and square (dashed line) cylinders. $d=0$ is seen to be a stable equilibrium.

Figure 2

Casimir torque ${\tau }_z$, per unit $z$ length, on the inner square rod of eccentric square cylinders as a function of the angle $\theta$ with respect to the $x$ axis (see right-hand insets) for two material choices as shown in the top insets. Error bars are estimates of the effect of finite grid resolution.

Figure 3

Derivative of the Casimir torque in the $z$ direction with respect to $\theta$, $d{\tau }_z/d\theta$, in units of $\hslash c/a^2$, evaluated at $\theta \approx 0°$, as a function of length scale $a=0.5(D-s)$. The solid line is a fit to the exact Casimir torque, computed by our numerical method (solid squares), and the dashed line is the torque as computed by a PFA approximation. Both curves display a change (dispersion-induced transition) in the stable orientation of the square.

Figure 4

Casimir force $F_x$ in the $x$ direction, per unit area, between a planar ${\mathrm{SiO}}_2$ slab suspended between two perfect-metal plates (solid red line and black line) or gold half-spaces (dashed red line), separated by a fluid (ethanol), as a function of the dimensionless $x$ displacement $d$ from equilibrium. The force is plotted also at two length scales $a\equiv 0.5(D-s)=0.0955\mu \mathrm{m}$ (red lines) and $a=0.6\mu \mathrm{m}$ (black line). The solid blue line shows the same quantity for the case of a lithium-niobate slab and a gold half-space, for $a=0.5\mu \mathrm{m}$. Top inset: $\epsilon (i\xi )$ for ${\mathrm{SiO}}_2$ (black line), ${\mathrm{LiNbO}}_3$ (solid blue line and dashed blue line), ethanol (orange line), and gold (red dashed line) as a function of imaginary frequency $\xi$. Bottom inset: schematic of geometry.