Research & Development

Casimir Force Simulation and Nanomachines

Background of the Casimir Effect

The Casimir Effect is a so-called macro quantum effect. Properties at the nano scale that are governed by quantum mechanics lead to a force between macroscopic objects. H.B.G Casimir predicted the effect as early as 1948. He published two prescriptions that led to similar results. The first attempt worked out by Casimir and Polder was based on the classical Van der Waals interaction caused by induced dipoles. The second calculation being based on quantum dynamics predicted the attraction of two uncharged, perfectly conducting plates in vacuum. The proposed reason was the lowering of the local vacuum energy density between the plates. This can easily be understood as shown in Fig. 1. In free vacuum there are fluctuations electromagnetic fields of arbitrary frequency. When bringing together the two plates at a small distance the electrical fields at the surfaces have to vanish due to conductivity. Hence the possible waves are restricted to those having a wavelength that is an integer ratios of the plate distance. The energy density being calculated by summing Heisenbergs quantum mechanical energy relation E=Vw over all possible waves. In our everyday world we find a simple analogue to that in music. The possible modes of a guitar string are restricted to twice the gripped length plus all harmonics that, in fact, have integer ratios of the base wave length.

Fig. 1.: Quantization of vacuum fields between conducting parallel plates. In free vacuum arbitrary wavelengths are permitted while between the plates only integer rationals of the plate distance are possible.

Due to the exremely small distances (kleiner 1µm) at which the Casimir force reaches measureable levels some 50 years had to pass until an experimental verification succeeded in 1996. Since then a wealth of experiments has been conducted all over the globe and the experimental inputs help to understand the details of the underlying physics better.

Numerical Simulation of Casimir Force

Despite 60 years of theoretical development no method has been found to calculate the Casimir effect for geometries other than that of two parallel plates. In pushing forward nanotechnology scientists and engineers enter a world of scales far below the micrometer level. Designing and developing the tools of tomorrows everyday life it gets increasingly important to estimate the forces being caused by the Casimir effect. In order to understand why that is important one should bear in mind that the Casimir force in the micrometer scale is negligible but at distances of 10 nm it accounts for ~100kN/m². This is a pressure of 10 tons per square meter! Reducing distances further increases the forces dramatically. In fact sticking and the resulting high friction is one of the most challenging topics in nano technology and tribology.

To overcome the problem of calculation AIT has developed the numerical tool CasimirSim based on Casimir’s initial approach: the Casimir Polder approximation. This software is the first of its kind, being capable of computing Casimir forces in really arbitrary geometries for MEMS and NEMS (Nanoscaled Machines), and any kind of material (see Fig. 1). It features state-of-the-art mesh technology and a high degree of parametrizability while being easy to use. Designed for a broad range of applications this is the ideal tool for scientists and engineers in both research and industry. The software is available for Linux/UNIX systems as well as Windows (link mit Produkte page).

Fig. 2.: Casimir Force between Grooves and a Nanomotor using Casimir Forces Only

CasimirSim has an integrated mesh generator for rectangular grids that enables, together with parametrizable geometry templates, arbitrary geometries to be set up in a fast and intuitive way. 7 simple shapes are provided for that purpose that can arbitrarily be combined and cut in order to create even the most complicated structures of your application. For a typical example of what is possible see Fig. 2.

In addition, CasimirSim also supports an external open source mesh generator called Gmsh by Christophe Geuzaine and Jean-Franēois Remacle (see geuz.org/gmsh/). Gmsh is a platform-independent tool that provides a nice user interface for quickly constructing simple geometries. It also features a high level scripting language that enables fast and parametrizable creation of arbitrary objects. The 3D mesh output of Gmsh can readily be used as input for CasimirSim.

Fig 3.: Tetragonal meshes: A slice through the grid (left) and a 3D -view onto partial structures.

Regarding the data post processing CasimirSim supports native formats of the industrial standard applications AMTEC Tecplot® and the open source version of the IBM Data Explorer, OpenDX

Recent Publications:

• Sedmik, R., Tajmar, M., "CasimirSim - A Tool to Compute Casimir Polder Forces for Nontrivial 3D Geometries ", Proceedings of the STAIF-2007 Conference, AIP Conference Proceedings, Vol. 880, 2007, pp. 1148
  
  
• Sedmik, R., Vasiljevich, I., and Tajmar, M., "Detailed Parametric Study of Casimir Forces in the Casimir Polder Approximation for Nontrivial 3D Geometries", Journal of Computer-Aided Materials Design, In Press (2006)
  
  
• Tajmar, M., "Finite-Element Simulation of Casimir Forces in Arbitrary Geometries", International Journal of Modern Physics C, Vol. 15, No. 10, 2004, pp. 1387-1396