**Repulsion/Levitation?**

There have been several proposals for repulsive Casimir forces.

DARPAprogram on Casimir Effect Enhancement (Scientific American)

1. Boundary Conditions & Exotic materials

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

Lifshitz (DLP)theory, unlike boundaries are obtained through an intervening medium with dielectric constant intermediate to the sides, as in the case of liquid helium climbing (wetting) a wall. (dielectric constants: solid > helium > air).

"Verification of Lifshitz theory of the van der Waals Potential using liquid-helium films,"

E.S. Sabisky and C.H. Anderson, Phys. Rev. A

7, 790 (1973)"Measured long-range repulsive Casimir–Lifshitz forces,"

J. N. Munday, F. Capasso & V. A. Parsegian, Nature

457, 170 (2009) (gold, bromobenzene, silica)

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

"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.

2. 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) ]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 = - 0.09235is 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 parallelepiped changes sign with aspect ratio:

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

"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 halfs of a cut sphere attract)

Constrained repulsion?

"Casimir repulsion between metallic objects in vacuum,"

M. Levin, A.P. McCauley, A.W. Rodriguez, M.T.H. Reid, S.G. Johnson, Phys. Rev. Lett.

105,090403 (2010)