Analytical bead force model for the 3DFM

The magnetic force experienced by a soft magnetic bead is proportional to the strength of the field at the location of the bead. It is also proportional to the gradient of the field at that location. In general the analytical calculation of this force is intractable. However if the generation of the field can be abstractly modeled by a small number of magnetic monopoles, an analytical solution is straightforward. This model is arguably a reasonable approximation for some magnetic pole tip configurations of a three dimensional force microscope (3DFM) [Cumm]. This research note presents a derivation of the analytical solution. 1 The monopole model of the 3DFM To be sure, there are pole geometries for which the monopole approximation is inappropriate. However the symmetric tetrahedral and hexapolar geometries are of particular interest for this instrument, and the monopole approximation is entirely reasonable in these cases. I will not provide any proof or error estimates here, but will give intuitive reasons why the approximation is “pretty good” for the tetrahedral and hexapole geometries of existing designs. First let us look at the tetrahedral design. Figure 1 Here we have the magnetic bead located in the center of a regular tetrahedron. Four cylindrical magnetic iron cores converge towards it from the centers of the four faces of the tetrahedron (Figure 1). The inner ends of the cores are tapered to a fine point with spherical ends, forming four pole tips. The poles are excited such that the sum of the fluxes leaving their tips vanishes (because magnetic monopoles cannot actually exist). The high permeability of the iron provides that the magnetic potential is essentially constant over the extent of each pole tip. Therefore, in a substantial solid angle in the direction of the bead, a spherical magnetomotive isopotential surface is established identical to that of a monopole located at the center of curvature of the pole tip. In the face centered cubic (FCC) hexapole geometry (Figure 2), the physical correspondence is not as neat because the pole tips are not themselves hemispherical. In this case we have six pole tips located at the centers of the six faces of a cube. The bead is located in the volumetric center of the cube. Magnetic flux is conducted to the tips through cores comprising tapered foil or thin magnetic films arranged in two closely spaced parallel planes. The tapers terminate in circular pole tips which present cylindrical profiles in the direction of the bead. Thus, the isopotential shape is cylindrical rather than spherical at a pole tip surface. However field simulations (Figure 3) show that in a solid angle in the direction of the bead, the isopotential shape quickly becomes spherical as one approaches the bead. The center of the quasi spherical isopotential surface patch touching the bead is approximately at the center of curvature of the pole tip. Therefore, a magnetic monopole at TR03-029 UNC Chapel Hill, Department of Computer Science page 2 Leandra Vicci Analytical bead force model for the 3DFM 02 September 2003 this location provides a reasonable approximation of the field from this pole at the bead location. Again, the pole excitations must provide that the sum of the pole fluxes vanishes. Optical axis (dashed line) is perpendicular to two planes, each containing three facecentered-cubic (FCC) points. These points form a pair of parallel equilateral triangles with a cylindrical working volume between them. Magnetic flux is conducted by thin film cores in two parallel planes to the pole tips having centers at the FCC locations Figure 2: The face centered cubic hexapole geometry Figure 3: Field simulation results of the FCC geometry To the extent that these monopole approximations are sufficiently accurate, the problem remains to calculate the field and its gradient at the bead location from an n-monopole model. 2 Notation We use both 3-space vectors and matrices here, including a 1 × n “vector” of pole excitations (strengths). To minimize confusion, it is useful to employ a notation which clearly distinguishes between spatial and matrix vectors. Accordingly, let us represent spatial vectors in italic bold face, e.g., “B ”, and all matrices in roman type, e.g. “q”, while scalars will be in the conventional italic type face, e.g., “r”. A matrix of spatial vectors will be represented by bold roman typeface, e.g. “f ”. Finally, unit vectors will be represented by the letter “u”. TR03-029 UNC Chapel Hill, Department of Computer Science page 3 Leandra Vicci Analytical bead force model for the 3DFM 02 September 2003 3 Magnetic field at the origin The bead is located at the origin, so we calculate the field there generated by n monopoles of excitation strength qj at locations rj , j = 1 . . . n.