Scale-Invariant Shapes

Method

 Casimir force between multiple objects of different shapes and material properties can be related to scattering amplitudes of electromagnetic fields from individual bodies.

 "Scattering theory approach to electrodynamic Casimir forces,"

S.J. Rahi, T. Emig, N. Graham, R.L. Jaffe, and M. Kardar, Phys. Rev. D 80, 085021 (2009)

  Shapes amenable to exact treatment of electromagnetic scattering:

 Plane, sphere, cylinder (circular, parabolic, elliptic), and

 Sharp shapes wedge, and cone,

"Analytical results on Casimir forces for conductors with edges and tips,"

M.F. Maghrebi, S.J. Rahi, N. Graham, T. Emig, R.L. Jaffe, & M. Kardar, PNAS 108, 6867 (2011)

 Wedge:

 Cone:

 For a thin cone (needle), the force vanishes as the inverse logarithm of the opening angle

 There is a strong temperature dependence (above figure is at a distance of 1 micrometer)

 The force is of the order of femto Newtons at this separation, but increases rapidly at shorter separations.

Next (Entropic polymer force) Outline QED Casimir force Thermal Critical Casimir Thermal Casimir in Helium Membrane inclusions Scale-invariant shapes Entropic force in polymers Phantom polymers Self-avoiding polymers Summary