Analytical results on Casimir forces for conductors with edges and tips

Abstract

Casimir forces between conductors at the submicron scale are paramount to the design and operation of microelectromechanical devices. However, these forces depend nontrivially on geometry, and existing analytical formulae and approximations cannot deal with realistic micromachinery components with sharp edges and tips. Here, we employ a novel approach to electromagnetic scattering, appropriate to perfect conductors with sharp edges and tips, specifically wedges and cones. The Casimir interaction of these objects with a metal plate (and among themselves) is then computed systematically by a multiple-scattering series. For the wedge, we obtain analytical expressions for the interaction with a plate, as functions of opening angle and tilt, which should provide a particularly useful tool for the design of microelectromechanical devices. Our result for the Casimir interactions between conducting cones and plates applies directly to the force on the tip of a scanning tunneling probe. We find an unexpectedly large temperature dependence of the force in the cone tip which is of immediate relevance to experiments.fluctuations | quantum electrodynamics T he inherent appeal of the Casimir force as a macroscopic manifestation of quantum "zero-point" fluctuations has inspired many studies over the decades that followed its discovery (1). Casimir's original result (2) for the force between perfectly reflecting mirrors separated by vacuum was quickly extended to include slabs of material with specified (frequency-dependent) dielectric response (3). Quantitative experimental confirmation, however, had to await the advent of high-precision scanning probes in the 1990s (4-7). Recent studies have aimed to reduce or reverse the attractive Casimir force for practical applications in micron-sized mechanical machines, where Casimir forces may cause components to stick and the machine to fail. In the presence of an intervening fluid, experiments have indeed observed repulsion due to quantum (8) or critical thermal (9) fluctuations. Metamaterials, fabricated designs of microcircuitry, have also been proposed as candidates for Casimir repulsion across vacuum (10).Although there have been many studies of the role of materials (dielectrics, conductors, etc.), the treatment of shapes and geometry has remained comparatively less investigated. Interactions between nonplanar shapes are typically calculated via the proximity force approximation (PFA), which sums over infinitesimal segments treated as locally parallel plates (11). This approximation represents a serious limitation because the majority of experiments measure the force between a sphere and a plate with precision that is now sufficient to probe deviations from PFA in this and other geometries (12, 13). Practical applications are likely to explore geometries further removed from parallel plates. Several numerical schemes (14, -16), and even an analog computer (17), have recently been developed for computing Casimir forces in general geometries. However, analytical formulae for quick...

Polymer-mediated entropic forces between scale-free objects

“…[58] can be remedied by exclusion of the entire interior of the cone, as was done by Cardy [31,55] for the wedge geometry. The analogous computations for a cone, described in the following section, are more complicated, and build upon more recent results pertaining to the electrodynamic Casimir interactions between conducting cones and plates [64].…”

Effect of curvature and confinement on the Casimir-Polder interaction

“…Examples include the effect of surface curvature [6], sharp edges and tips [7], orientation dependence [8], and confinement [9,10]. The non-additivity of fluctuation induced forces complicates the study of these effects.…”

Material dependence of the wire-particle Casimir interaction

“…The scattering formalism provides a powerful tool to calculate the Casimir interaction between objects of general shape and material properties [7,8]. There is much recent research activity based on the scattering formalism, e.g., for edges and tips [9,10], anisotropic particles [11], wires and plates [12][13][14][15], spheres and plates [16], and periodic structures [17][18][19]. For some further recent examples, see Ref.…”