Repulsion/Levitation?

 There have been several proposals for repulsive Casimir forces. (Scientific American)

1. Repulsion via Geometry & Shape

Abraham-Lorentz (+Casimir) model of electron: A spherical conducting shell of radius R , with charge e

[ "Introductory remarks on quantum electrodynamics," H.B.G. Casimir, Physics 19, 846 (1956) ] (offline)

"Casimir Mouse-trap" for the fine-structure constant: Balancing the repulsive Coulomb energy, with the (presumed attractive) Casimir energy:

"Quantum Electromagnetic Zero-Point Energy of a Conducting Spherical Shell and the Casimir Model for a Charged Particle,"

T.H. Boyer, Phys. Rev. 174, 1764 (1968): 2A = - 0.09235  is negative!  [also obtained by R. Balian and B. Duplantier (1977)]

Not only for spheres, but also rectangular boxes:

["Vacuum means of energy-momentum ...,"  S.G. Mamaev and N.N. Trunov, Sov. Phys. J. (USA) 22, 51 (1979)]

The "Casimir energy" of a parallelopiped changes sign with aspect ratio:

This putative repulsive force has been proposed as a means of levitation:

       ,     but

 "Attractive Casimir forces in a closed geometry,"

M. P. Hertzberg, R. L. Jaffe, M. Kardar, and A. Scardicchio, Phys. Rev. Lett. 95, 250402 (2005):

In the physically accessible geometry of a piston, the partition is always attracted to the closer side.

 ,

"Opposites Attract: A Theorem about the Casimir Force,"

O. Kenneth and I. Klich, Phys. Rev. Lett. 97, 160401 (2006) (two halves of a cut sphere attract)

2. Boundary Conditions & Exotic materials

 Similar materials (boundary conditions) attracts, but sufficiently dissimilar boundaries may repel.

"Van der Waals forces and zero-point energy for dielectric and permeable materials,"

T.H. Boyer, Phys. Rev. A 9, 2078 (1974) (large permittivity and large permeability repel)

A material with large permeability is required for repulsion, but in ordinary materials permeability is close to one.

Metamaterials, incorporating arrays of microengineered circuitry mimic, at certain frequencies, a strong magnetic response and have been proposed as candidates for Casimir repulsion across vacuum.

Metamaterials have also been proposed as invisibility cloaks.

But invisibility cloaks are limited to a band of wavelengths, while Casimir force depends on all frequencies ...

Next (Earnshaw's thm.) Outline Casimir force Repulsion? Earnshaw's theorem Heat Engine Friction Summary

  Unlike boundaries can be obtained through an intervening fluid with dielectric constant intermediate to the sides:

 "Measured long-range repulsive Casimir–Lifshitz forces,"Munday, Capasso & Parsegian, Nature 457, 170 (2009)

(gold, bromobenzene, silica)

  Immersing MEMs in fluids is not practical.  Is repulsion across vacuum possible?