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Magnetic guidance is understood as a remote, untethered and contact-free control of the movements of an object via magnetic interactions. The movements should happen on arbitrary trajectories inside a container caused by an external device. The concept of remote magnetic guiding is developed from the underlying physics for bijective force generation over the inner volume of magnet systems. This concept can equally be implemented by electro- or permanent magnets.
So what happens to a small paramagnetic object in an inhomogeneous magnetic field? It is hard to imagine that an object that should be guided through space is not freely movable (at least in two dimensions). If the object has an intrinsic fixed direction of m (e.g., remanent magnetization), it is rotated by the magnetic torque,
Now the question arises why guiding charged particles is so straightforward, while it is so difficult to control the collective spin of electrons in materials magnetically? The reason is the bijective direction (v) of the electron beam, which is just slightly deflected by steering fields. This suggests that a preferred direction would also be beneficial for steering paramagnetic objects. This is tantamount to a magnetic field that just orients (polarizes) the particles without exerting a force on them. For static magnetic fields, this request can be fulfilled by applying a strong but homogeneous magnetic flux density, Bhom, which magnetizes the object along its direction. An additional, small, and spatially-dependent steering or deflecting field can then act as a perturbation but with full directional control (cf. Figure 1c,d). Ideally, this deflecting field will have a linear spatial dependence, i.e., a constant gradient (The fact that G is a tensor is ignored for the moment), ∇ B = G , and the total field in such an experiment is then
With the reasonable assumption that there is no strong spatial variation of the magnetic moment over the sample, one could conclude that Fm = mG (because ∇Bhom = 0). Under certain limits this is correct, but unfortunately magnetism is not quite that simple. Things become a bit more complicated due to Maxwell’s (or Gauss’) law
Hence, there cannot be a single gradient field at any point. Either the field has to be homogeneous or the sum of all its spatial derivatives have to cancel. For the simple case of a perfect quadrupolar field , this could be for instance ∂Bx /∂x = +G and ∂By /∂y = -G consequently the last equation then dictates ∂Bz /∂z = 0. Then a more detailed representation of B(r) will be
The described concept is easy to realize by so-called Halbach cylinders, the homogeneous field will be generated by a Halbach dipole (see Fig. 2a) while the constant gradients can readily be provided by a Halbach quadrupole (see Fig. 2b). The first advantage of such Halbach cylinders is that they provide ideal homogeneous and graded fields (as are assumed for eq. (8)) with simple geometric relations to calculate their field
where Ri is the inner and Ro the outer radius of the hollow cylinders and BR [T] is the remanence of the used permanent magnet material.
Figure 2. Sketch of ideal Halbach cylinders: (a) inner dipole with a homogeneous field of strength, Bhom along the x-axis. (b) inner quadrupole with a circular modulus field which can be decomposed into a two linear field components Bx = Gx in (c) and By = -Gy in (d). The hollow cylinders consist of permanent magnet material with continuously changing magnetization direction (arrows). The poles are encircled. In (a) and (b) the magnetic field is represented by field lines, while in (c) and (d) the arrows are field vectors (the different colors are only for better contrast). Note that the magnetic fields are only inside the hollow cylinders and that there are no stray fields.
The second great advantage is the absence of stray fields, so that they are “no magnets” when approached from the outside. Therefore, the cylinders can be concentrically arranged or nested and mutually rotated without much torque. If two Halbach cylinders of the same type are nested and the geometries are chosen such that they both produce the same field or gradient strength, their combined field can then be varied between zero and twice the value of a single cylinder. This allows to scale the field or force or eventually even switch it off by simple mechanical rotation. This principle is illustrated in Fig. 3.
Figure 3. Coaxial arrangement and rotation of Halbach dipoles (green, upper row) and quadrupoles (red, lower row): (a) and (b) two Halbach dipoles which produce the same field strength, B (central green arrow); are coaxially nested in (c-e) and the outer one is rotated by an angle α. The resulting field is also illustrated by B-arrows. (c) For α = 0° the fields are parallel and the two dipole fields add to 2B . (d) For α = 90° the fields are orthogonal and the two dipole field vectors add to √2|B| at an angle of 45°. (e) For α = 180° the fields are antiparallel and cancel each other. An analog presentation is shown in (f-g) for two nested quadrupoles which produce the same field gradient (i.e. the derivative of the field! The red arrow shows the horizontal component only). (h-j) Same representation as above. Note that the gradient rotates at twice the angle of the quadrupole. (k) The angular dependence of the combined field of both dipoles, BΣ = 2|B||cos(α/2)|, (l) Angular dependence of the gradient strength of the two quadrupoles GΣ = 2|G||cosα|.
If the previous example is realized by a Halbach dipole and a Halbach quadrupole and the quadrupole is rotated by an angle α relative to the dipole, the magnetic field in such a structure is (cf.Fig. 3l)
Using the same arguments as before (i.e. that the field component which is not along BD [here. By] can be ignored for BD >> Gr) the magnetic force is then given by
This means that the magnetic force has a constant strength of |m|GQ and rotates with 2α over the entire volume where the prerequisite BD >> Gr is fulfilled.
In order to completely control the movements of such a guided object, not only the direction but also the amplitude of the force must be controlled. This can easily be done by using a second quadrupole, ideally of a size that produces the same gradient strength in the internal volume as already provided by the first quadrupole. If the direction of the force shall be determined by α and shall not be altered by scaling the force, one quadrupole must be rotated by an angle (α+β/2) and the other by (α-β/2). Hence, generating a force
Such a system allows complete control in two dimensions as demonstrated in this video .
Typical examples of such magnetically guided objects are endoscopic capsules for inspection of the gastrointestinal tract or superparamagnetic nanoparticles suggested for local therapy, which therefore have to be moved through blood vessels.