Beyond the simple Proximity Force Approximation: geometrical effects on   the non-retarded Casimir interaction
  We study the geometrical corrections to the simple Proximity Force Approximation for the non-retarded Casimir force. We present analytical results for the force between objects of various shapes and substrates, and between pairs of objects. We compare the results to those from more exact numerical calculations. We treat spheres, spheroids, cylinders, cubes, cones, and wings; the analytical PFA results together with the geometrical correction factors are summarized in a table. 
Bo E. Sernelius and C. E. Rom\'an-Vel\'azquez