Geometry-Induced Casimir Suspension of Oblate Bodies in Fluids

We predict that a low-permittivity oblate body (disk-shaped object) above a thin metal substrate (plate with a hole) immersed in a fluid of intermediate permittivity will experience a metastable equilibrium (restoring force) near the center of the hole. Stability is the result of a geometry-induced transition in the sign of the force, from repulsive to attractive, that occurs as the disk approaches the hole—in planar or nearly planar geometries, the same material combination yields a repulsive force at all separations, in accordance with the Dzyaloshinskiĭ-Lifshitz-Pitaevskiĭ condition of fluid-induced repulsion between planar bodies. We explore the stability of the system with respect to rotations and lateral translations of the disks and demonstrate interesting transitions (bifurcations) in the rotational stability of the disks as a function of their size. Finally, we consider the reciprocal situation in which the disk-plate materials are interchanged and find that in this case the system also exhibits metastability. The forces in the system are sufficiently large to be observed in experiments and should enable measurements based on the diffusion dynamics of the suspended bodies.

Figure 1

Room-temperature Casimir energy $U$ of a PTFE disk (thickness $t=10\mathrm{nm}$) suspended above a Au plate (thickness $h=10\mathrm{nm}$ and inner and outer diameters $W=1\mu \mathrm{m}$ and $D=2W$) immersed in water, as a function of vertical separation $d$ (normalized by $W$). $U$ is normalized by the energy in the coplanar configuration $U_0\equiv U(d=0)$ and plotted for multiple aspect ratios $\Lambda =L/t$, where $L$ is the disk diameter. Also shown is the energy of a sphere of diameter $20t$ (dashed black line). Top insets: $U$ as a function of rotation angle $\theta$ (left) and lateral translations $s$ (right) for multiple $\Lambda$. Bottom inset: Unstable equilibrium separation $d_c$ (red circles), along with the energy $U_0$ (green line) and corresponding energy barrier $\Delta =U(d_c)-U_0$ (blue line), normalized by $k_{B}T\approx 25\mathrm{meV}$, as a function of $L$. Both $d_c$ and $L$ are normalized by $W$.

Figure 2

Room-temperature Casimir $U$ (solid lines) and Casimir-Polder $U_{\mathrm{CP}}$ (dashed lines) energies, normalized by $k_{B}T\approx 25\mathrm{meV}$, for the disk-plate geometry of Fig. 1, as a function of $d$ (normalized by $W$), plotted for multiple aspect ratios $\Lambda =L/t$. $U_{\mathrm{CP}}$ is obtained from the polarizability of the disks, as determined by Eq. (2). Top inset: Diagonal components of the photon GF of the plate evaluated along the axis of symmetry, for both finite $D=2W$ (solid lines, evaluated numerically) and semi-infinite $D\to \infty$ PEC (dashed lines, evaluated analytically [33]) plates. Bottom inset: Casimir integrand $U(i\xi )$, in arbitrary units, as a function of imaginary frequency $i\xi$ at three different separations $d=\{0,0.2,0.4\}W$, for $\Lambda =60$.

Figure 3

Room-temperature Casimir energy $U$ as a function of $d$ for the disk-plate geometry of Fig. 1, but with the Au and PTFE materials interchanged. $U$ is normalized by $U_0$ and plotted for multiple values of $\Lambda$. Top insets: $U$ as a function of rotation angle $\theta$ (left) and lateral translations $s$ (right) for multiple $\Lambda$. Bottom inset: Unstable equilibrium separation $d_c$ (red circles), along with $U_0$ (green line) and $\Delta =U(d_c)-U_0$ (blue line), normalized by $k_{B}T\approx 25\mathrm{meV}$, as a function of $L$. Both $d_c$ and $L$ are normalized by $W$.