cells
Review
Magnetic Guiding with Permanent Magnets: Concept,
Realization and Applications to Nanoparticles and Cells
Peter Blümler
Institute of Physics, University of Mainz, Staudingerweg 7, 55099 Mainz, Germany; bluemler@uni-mainz.de;
Tel.: +49-6131-392-4240
Abstract: The idea of remote magnetic guiding is developed from the underlying physics of a concept
that allows for bijective force generation over the inner volume of magnet systems. This concept
can equally be implemented by electro- or permanent magnets. Here, permanent magnets are in
the focus because they offer many advantages. The equations of magnetic fields and forces as well
as velocities are derived in detail and physical limits are discussed. The special hydrodynamics of
nanoparticle dispersions under these circumstances is reviewed and related to technical constraints.
The possibility of 3D guiding and magnetic imaging techniques are discussed. Finally, the first
results in guiding macroscopic objects, superparamagnetic nanoparticles, and cells with incorporated
nanoparticles are presented. The constructed magnet systems allow for orientation, movement, and
acceleration of magnetic objects and, in principle, can be scaled up to human size.
Keywords: steering; magnetic force; magnetic drug targeting (MDT); nanoparticle; SPIO; ferrofluid;
superparamagnetic; ferromagnetic; Halbach magnets; dipole; quadrupole; cells; micro-robots;


 endoscopic capsules; magnetic resonance imaging; MRI; magnetic particle imaging; MPI



Citation: Blümler, P. Magnetic
Guiding with Permanent Magnets:
Concept, Realization and 1. Introduction
Applications to Nanoparticles and In this review, the meaning of magnetic guidance is understood as a remote, unteth-
Cells. Cells 2021, 10, 2708. https:// ered and contact-free control of the movements of an object via magnetic interactions.
doi.org/10.3390/cells10102708 The movements should happen on arbitrary trajectories inside a container caused by an
external device.
Academic Editors: Vitalii Zablotskii Typical examples of such magnetically guided objects are endoscopic capsules for
and Xin Zhang
inspection of the gastrointestinal tract or superparamagnetic nanoparticles suggested for
local therapy, which therefore have to be moved through blood vessels. There are numerous
Received: 6 August 2021
reviews on the subject because this research area is very diverse and the problem has been
Accepted: 2 September 2021
Published: 9 October 2021 tackled from different directions. The following reviews on magnetically guided medical
devices [1–3]; miniature robots [4]; nanoparticles in microfluidics and nanomechanics [5] for
Publisher’s Note: MDPI stays neutral drug delivery [6–8], hyperthermia, and alternative local magnetic therapeutic effects [9,10];
with regard to jurisdictional claims in tissue engineering [11–13]; as well as magnet systems for this purpose [14] are some
published maps and institutional affil- of the most recent (or a book [15] treating most of these topics). There are many more
iations. applications of magnets in biomedicine, e.g., permanent magnets are also used for the
separation of superparamagnetic nanoparticles from solution. If the nanoparticles are
functionalized, specific substances can be removed from the solution. However, this is not
guiding in the sense of the initial definition because the direction of motion of the particles
is not controllable.
Copyright: © 2021 by the author.
It is not the intention of this review to summarize this very active, vast, and diverse
Licensee MDPI, Basel, Switzerland.
This article is an open access article field of research, but rather to discuss a simple and very general concept of magnetic guid-
distributed under the terms and ing that borrows ideas from the treatment of magnetic fields in magnetic resonance imaging
conditions of the Creative Commons (MRI). To avoid confusion, this comparison does not imply that the magnetic fields in MRI
Attribution (CC BY) license (https:// are particularly useful for guiding (although possible [16]), because the typical gradient
creativecommons.org/licenses/by/ fields are too weak. It is rather the physical principle that is similar. Both techniques use a
4.0/). projection of a small deflection or encoding magnetic field tensor on a much stronger and
Cells 2021, 10, 2708. https://doi.org/10.3390/cells10102708 https://www.mdpi.com/journal/cells
Cells 2021, 10, 2708 2 of 31
homogeneous field (component), which generates spatial bijection. In the case of MRI, the
unavoidable concomitant gradient components of the coil systems can be ignored if the
strength of the homogeneous field is much higher (a concept which fails at low magnetic
fields). This physical statement and how magnetic guiding of paramagnetic objects can
be implemented analogously will be explained in the next section. From this perspective,
it does not matter if the magnetic fields for guiding are generated by electro-, permanent,
or hybrid magnets. However, since permanent magnets have so many advantages over
electromagnets, only realizations using exclusively permanent magnets are discussed in
Section 3. The described bijective guiding concept relies on the experimental condition that
the homogeneous field is much stronger than the deflection field, and Section 4 describes
what happens if this is violated. While guiding macroscopic objects does not require
particularly strong magnetic fields, doing alike with superparamagnetic nanoparticles
is a challenge. It is almost unavoidable that their mutual magnetic interactions induce
cluster formation. This changes not only the magnetic force but also the hydrodynamics
as discussed in Section 5. Section 6 then covers possible 3D realizations of such guiding
instruments and magnetic imaging schemes. Finally, some first applications of the tech-
nique are shown before this article concludes. These first applications demonstrate that the
proposed concepts work nicely in two dimensions by guiding macroscopic objects as well
as suspended superparamagnetic nanoparticles or cells with incorporated nanoparticles.
At this point, a few general remarks need to be made. The following line of arguments
will consider paramagnetic materials only, i.e., materials with a relative permeability bigger
than one (µr > 1) or positive magnetic susceptibilities (χ = µr − 1), hence including ferri-,
ferro-, and superparamagnetic materials. In principle, diamagnetic (0 ≤ µr < 1) substances
can be guided with the same instrument as well. Essentially, it would just reverse the sign
in the force equations. However, since the only substances with a diamagnetic permeability
significantly different from 1 are superconductors (with µr = 0) and most applications of
magnetic guiding are typically in biological systems, they will be ignored in the following.
Since most of this article deals with concepts, the problem is that most of the time these
concepts are idealized to the task of guiding a magnetic dipole with a magnetic moment
⇀ ⇀
m [Am2] by an applied magnetic flux density B [T]. Deviations when concerned with real
bulk materials are discussed in Appendix A. At this point, the author wants to apologize
for not always using the completely correct terminology concerning magnetic flux density,
B [T], which in the following is often termed as magnetic field or induction field to improve
readability. Wherever a physical magnetic field, H [A/m], is meant, it is labelled as such.
For most of this conceptual presentation, B = µ0 H is valid with µ ≈ 4π × 10−70 N/A2 is
the vacuum permeability.
To further improve readability, highly technical or mathematical details were sep-
arated in seven appendices, which are found at the end of this article. They might be
particularly useful for readers who want to design their own systems or follow the deriva-
tion of equations.
The author wants to conclude this introduction with a statement about names. The first
system of this kind was nicknamed “MagGuider” (for Magnetic Guiding and Scanner) [17],
but this name should not be used here, because this article rather deals with a concept than
with a particular instrument.
2. Concept of Magnetic Guiding
Magnetic guiding has been an established technique since 1897 [18], when Ferdinand
Braun invented magnetic guidance of charged particles (electrons or ions) by cathode ray
tubes where the electrons are emitted from a cathode into an evacuated tube, accelerated by
an anode, and deflected by magnetic fields (used en masse in analogue oscilloscopes and
⇀ ⇀ ⇀
television screens). The magnetic deflection is based on the Lorentz force F L = q v × B ,
⇀
which is perpendicular to the direction of the magnetic flux density B and the flight
⇀
direction of the particles with charge q and velocity v . However, the situation is very
Cells 2021, 10, 2708 3 of 31
different if an electrically n(eutral paramagnetic mater)ial is exposed to magnetic fields. The
⇀
force is then the gradient ∇ = [∂/∂x, ∂/∂y, ∂/∂z ] of the magnetic field acting on the
⇀
object with a magnetic moment m( ) ( )
⇀ ⇀ ⇀ ⇀ Appendix A ⇀ ⇀
F m = ∇ m · B '
⇀
m · ∇ B . (1)
The right simplified term is usually correct for the applications discussed here, how-
ever, it is not generally the case. Particularly, it assumes that m is not dependent on B,
which depends on the material and the range of B (see Appendix A for a discussion).
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
⇀ ⇀ ⇀
τm = m× B , (2)
⇀ ⇀
until the cross-product becomes zero or m is parallel to B . If the object has initially (at B = 0)
⇀
no preferred direction of m, the actual field will magnetize it (orient the electron spins)
⇀ ⇀ ⇀
along B . Either way, as a result, m points along B , which is very unfortunate with respect
to guiding, because the dot-product in Equation (1) will lose its sign for two parallel vectors
and the material will always move towards higher magnetic fields (cf. Figure 1a,b). This is
an everyday observation, as e.g., paper clips are attracted equally by the north and south
pole of a permanent magnet. For steering this is like using a clipper without a sail. Almost
independently of what one tries with the rudder, the boat will go to where the winds or
currents move it. In electrodynamics, this is also known as Earnshaw’s theorem [19], and
it is the reason why permanent magnets were originally not considered as being useful
for magnetic guidance, because as their name suggests they are permanent and cannot be
switched on or off.
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
⇀ ⇀ ⇀
B(r) = Bhom + G r . (3)
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
⇀ ⇀ ∂B
∇ · ∂Bx y ∂BB z= + + = 0. (4)
∂x ∂y ∂z
Cells 2021, 10, 2708 4 of 31
Figure 1. Illustration of the suggested guiding principle. A small magnetizable sphere serves as the
object to be guided by a large deflecting bar magnet. The colors indicate the magnitude of the local
magnetic flux density (see color bars on the left, Bmax in (a,b) is roughly a quarter of that in (c,d)).
The black lines are field lines. A zoom of the region around the object is shown on the inserts. The
top rows (a,b) just show the field generated by the deflection magnet, while in (c,d) a strong and
homogeneous field is superimposed to the scenario above. The difference between the columns is the
orientation (south- and north-pole) of the deflection magnet. (a,b) Changing the magnet’s orientation
has no effect on the movement of the object (white arrow), because the object is magnetized in
opposite directions as well and just moves to the highest flux density. The additional homogeneous
field in (c,d) essentially keeps the magnetization direction of the object along its horizontal direction.
The field of the deflecting magnet now causes the opposite magnetic “landscape” around the object
and hence it moves in opposite directions. The data were generated using FEMM (www.femm.info)
but should serve for illustration purposes only.
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 (see Appendix B), this could be for instance ∂Bx/∂x = +G
and ∂By/∂y = −G, consequently Equation (4) dictates ∂Bz/∂z = 0. Then a more detailed
representation of Equation (3) wi 
ll be
⇀ Bx(x, y, z)⇀ ⇀ ⇀
B( r ) = By(x, y, z)  = Bhom + G r
Bz(x, y, z)       (5)1 1 0 0 x= Bhom 0  + G 0 −1 0  y   Bhom + Gx= −Gy .
0 0 0 0 z 0
Cells 2021, 10, 2708 5 of 31
The deflecting field is written here in the most general form as a gradient tensor (G,
see Appendix F), which will be needed later. As discussed above, the magnetic moment of
⇀ ⇀ ⇀
an object at r = [x, y, z]T will be oriented parallel to B( r ) (with unit vector êB)
∣∣∣ ∣∣∣ ∣∣∣ ∣⇀ ⇀ ⇀m(x, y, z) = m êB = m∣∣ ∣∣⇀∣B∣ ∣⇀∣B ∣∣
∣
√ ∣∣
∣  
⇀
m∣∣  Bhom + Gx  B ∣ ∣− homGrGy ≈ ∣∣⇀m∣∣
 
(6)
1
= 0  .
(B +Gx)2hom +G2y2 0 0
The last approximation was already motivated in the discussion of Figure 1 and is
the origin of bijection, namely that the homogeneous field must be much stronger than
the local deflection field, so that its tensorial properties can be reduced to a vector via
projection. The condition for this ∣prerequisite is then∣∣∣⇀ ∣∣∣∣ ∣∣∣∣ ∣⇀⇀∣∣∣∣∣∣ ∣Bhom  ∇ ⇀B r ∣∣. (7)
A full treatment will follow but to clarify the concept, it is instructive to continue with
the approximation from Equation (6). Then the magneticforce in Equation (1) simplifies to
⇀ ⇀ ⇀ ∣ ∣ Bhom + Gx⇀ ⇀
F m(x, y, z
∣ ∣
) = ∣(m
∂
∣· ∇ )B ≈ ∣m∣ ∂x −Gy ∣∣ ∣∣ 0
(8)
⇀
= m G êx ,
or more generally only the field component of the deflection field, which is parallel to
⇀
Bhom, determines the direction and amplitude of the magnetic force. It is a very beneficial
feature of this concept that there is no spatial dependence of the force vector in Equation (8),
hence the guiding force is homogeneous or constant over that region where Equation (7) is
fulfilled (cf. also Figure 5). This is an important issue because other systems which guide
an object by moving permanent magnets around the outside of the container (e.g., [20]) or
use electromagnets on opposing ends of the container, also have to consider the non-linear
drop of the magnetic field with distance (depending on their dimensions, the far-field of
permanent magnets drops with an exponent between −2 and −3, and hence the force with
−3 to −4). This can extremely complicate the control because the position of the object has
to be known precisely to estimate speed and direction of motion. This is a problem that
does not exist in the presented concept. Additionally, the movements are also not limited
⇀
to the direction of B . In the following section and Appendix C it is explained how any
⇀
direction can be addressed by rotating the gradient field relative to Bhom.
3. Permanent Magnets with Adjustable Fields
As already said in the introduction, this conceptual idea of magnetic guiding can of
course be implemented with any magnet system consisting of either electro-, permanent
magnets, or hybrids. While Equation (8) motivates this concept, real guiding of objects
along arbitrary paths needs deflection fields, which must become time-dependent in
orientation and amplitude [21]. Since permanent magnets are not typically associated with
these properties, they are often excluded from considerations, although they offer some
important advantages over electromagnets. Particularly, if the devices need to be scaled up,
the enormous power consumption of resistive electromagnets becomes a real problem. For
example, to generate roughly the magnetic field produced by 1 cm3 of modern rare-earth
magnets, several kW of electrical power are already needed and additionally the generated
Cells 2021, 10, 2708 6 of 31
heat must be removed by cooling. This can be estimated from the Amperian loop model,
where a loop of current, I, produces a magnetic moment m = IA, with A as the surface of the
loop. Using Equation (A3) (see Appendix A), this can be rearranged for a cylinder of height
h to I = BR h/µ0. Hence, a cylinder of rare earth material with a remanence BR = 1.3 T and
h = 1 cm height is equivalent to a current of ca. 10 kA. Additionally, the time-dependence
of large coils is very nonlinear, as the inductance scales with the square of the number
of windings and the coil cross-section. Large inductances then result in long delays for
discharging and charging the coil. Together with the temperature and hence resistance
changes associated with time-dependent currents, the resulting magnetic fields are difficult
to calculate and control [22,23]. Superconducting magnets are not an acceptable option
either because they cannot be switched fast. So what about permanent magnets? As the
name suggests, the orientation and strength of a permanent magnet are time-independent,
however, many of them can be arranged to systems such that the direction and strength of
their resulting magnetic fields can be changed by simple mechanical rotation [24].
The most suitable arrangements for this purpose are so-called Halbach cylinders [25].
Their concept, construction, and field calculation is discussed in Appendix B. In order to
implement the concept of magnetic guidance from Section 2, the homogeneous field will
be generated by a Halbach (inner) dipole (see Figure 2a), while the constant gradients can
readily be provided by a Halbach (inner) quadrupole (see Figure 2b). In the following,
the discussion will be limited to ideal systems. Hence, it is sufficient to treat this as a
two-dimensional problem (infinite length in the third dimension, see Appendix B).
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 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,b), the magnetic field is represented by field lines, while in (c,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 first advantage of such Halbach cylinders is that they provide ideal homogeneous
and graded fields (as are assumed for Equation (8)) with simple geometric relations to
calculate their field [ ] [ ]
⇀ Ro 1 1dipole : B(x, y) = BR ln R = B ,i 0 D 0
( ) [ ] [ ] [ ][ ] (9)⇀ 1 0 x 1 0 x
quadrupole : B(x, y) = 2B 1 − 1R R = Gi Ro 0 −1 y Q 0 − ,1 y
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. The strength of the homogeneous field,
BD = Bhom, of the dipole and the strength of the gradient, GQ, produced by the quadrupole,
are now indexed by the type of Halbach magnet they originate from. This will help to
retain an overview when nesting multiple rings.
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 [24]. If two Halbach cylinders of the
Cells 2021, 10, 2708 7 of 31
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
Cells 2021, 10, x FOR PEER REVIEWth e value of a single cylinder. This allows to scale the field or force or eventual8l yof e3v2 en
 switch it off. This principle is illustrated in Figure 3.
 
FigFuirgeu3re.  3C. oCaoxaixailaal rarrarnanggeemmeenntt aanndd rroottaattiioonn ooff HHaalblbacahch didpioploesle (sgr(egerne,e unp, pueprp reorwr)o awn)da qnudadqruuapdorleusp (orleeds, (lorewde,rl roowwe)r: (rao)w ):
and (b) two Halbach dipoles that produce the same field stren⇀gth, B  (central green arrow); are coaxially nested in (c–e) (a,ba)ntdw toheH oaulbtearc hondei piso lreostattheadt bpyro adnu acnegtlhee αs.a T
mhee fireesludltsitnregn fgietlhd, iBs a(clseon itlrlaulsgtrrae⇀t
eend abryr oBw-)a;rarroewcso.a (xci)a Flloyr nαe =s t0e°d, tihne( cfi–eeld) as nadret he
outpearroanlleeli sanrdot tahtee dtwboy dainpoalneg fileeldαs. aTdhde troe s2uBlt.i n(dg) fiFeolrd αi s= a9l0s°o, tihlleu fsiterladtse adreb yortBh-oagroronwals a.n(dc) tFhoe rtwαo= d0ip◦,otlhe efiefiledl dvsecatroersp aadrda∣∣l lel √ ∣⇀ ⇀∣
andtot he2twBo adti pano leanfigelled osfa 4d5d°.t o(e2) BFo. r( dα) =F o1r80α°,= th90e ◦f,iethldesfi aerled asnatriepaorathlloelg aondal caandcetlh eatcwh oodthiepro. lAe nfi ealndavloegct oprsesaedndtattoion 2is∣∣ B ∣∣
at asnhoanwgnl eino f(f4,g5)◦ .fo(er )twFoor nαes=te1d8 0q◦u,atdhreufipeolldess athreata nptriopdaurcaell ethl ea nsdamcaen fcieellde agcrhadoitehnet r(.i.Ae.n, tahnea dloegrivparetisveen otaft tiohne fiisesldh.o Twhne irned(f ,g)
foratwrroown esshtoewdsq tuhaed hrourpizoolenstatlh caotmpproodnuencet othnelys)a. m(he–jfi) eSladmger aredpierenste(ni.tea.t,iothne ads earbiovvaeti.v Neootfe tthheatfi tehled .gTrahdeiernetd raortarotews asth towwicset he
the angle of the quadrupole (cf. Appendix C). (k) The angular dependence of the combined field of both dipoles, horizontal component only). (h–j) Same representation as above. Note that the gradient rotates at twice the angle of the
BΣ = 2 B cos(α / 2) ; (l) angular dependence of the gradient strength of the two quadrupoles GΣ = 2 G cosα . ∣∣
quadrupole (cf. Appendix C). (k) The angular dependence of the combined fie∣ld ∣of both dipoles, BΣ = 2∣∣
∣
⇀∣
B ∣∣|cos(α/2)|;
(l) angular dependence of the gradIife nthtes terxenagmthploef ftrhoemtw Eoququataidornu (p5o)l iess pGut i=nto
∣
2∣ e⇀Gff∣∣e|ccot sby| .a Halbach dipole and a Halbach α
quadrupole, and the quadrupole is rotatedΣ by a∣n a∣ngle α relative to the dipole, the mag-
netic field in such a structure is (cf. Appendix C and Figure 3l) 
If the example from Equation (5) is put into effect by a Halbach dipole and a Halbach
quadrBupo
1
(x, yle), a=ndB the q+uGadru
cpoosl2eαis rostiante2dαbyxan a=ng
lBeDα+rGeQla(txivceosto2αth+e ydsipino2leα,)th.e magnetic
field in such a strDuc0ture iQs   (csfin. A2αppend
 
−coisx2Cα   (10)[ ] a[nyd FigureG3l)Q (x sin 2α − y cos 2α) By using the ⇀arguments of Eq1uations (5) ancdo s(62)α, the fsieinld2 cαom]p[onxen]t that is not along 
B  (i.e., B ) can be B(x, y) = BD y ignored an[dD the0 mag+neGtQic forscien i2sα give−n cboys 2α y      ] (10)Fm = ∇ (m ⋅B ) ≈B∇+mGB= D Qx(x cos 2α+ y sin 2α) .
 ∂ ∂x GQ(x sin 2α− y cos 2α)(  = m   BD +GQ ( cos 2α (11)x cos 2α + y sin 2α)) = m GQ  .
∂ ∂y sin 2α


 
Cells 2021, 10, x. https://doi.org/10.3390/xxxxx www.mdpi.com/journal/cells 
 
Cells 2021, 10, 2708 8 of 31
By using the arguments of Equations (5) and (6), the field component that is not along
BD (i.e., By) can be ignored an∣d ∣the magnetic force is given by⇀ ⇀ ⇀ ⇀
F m = ∇
⇀ · ≈ ∇∣∣⇀(m B) m∣∣Bx
∣∣∣ ∣∣∣[ ] ∣ ∣ [ ] (11)⇀ ∂/∂x ⇀ cos 2α= m (B + G (x cos 2 y sin 2 ∣∣m∣α+ α)) = ∣G .
∂/∂y D Q Q∣ ∣ sin 2α∣∣⇀This means that the force has a constant strength of m∣∣GQ and rotates with 2α over
the entire volume where the prerequisite of Equation (7) is fulfilled. Although it is a bit
counterintuitive that the object moves at twice the angle of which its actuator is rotated,
this concept gives complete control over the direction of a magnetically guided object in
such a magnet system [17].
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 (cf. Appendix B). 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)
 ( ( ) ( )) ( ( ) ( )) ⇀ BD + G(Q x c(os 2(α+)β2 ) + y s(in 2(α+ )β2)) + G(Q x c(os 2(α−)β2 ) + y s(in 2(α−B )β2= )) 
G x sin 2(α+ β β β β
[ Q 2
) − y cos 2(α+ 2 ) + GQ x sin 2(α− 2 ) − y cos 2(α− )] 2
BD + GQ{x(cos(2α+ β) + cos(2α− β)) + y(sin(2α+ β) + sin(2α− β))}=
[ GQ{x(sin(2α+ β) + sin(2α− β))− y(cos(2α+]β) + cos(2α− β))} (12)
BD + GQ{x(2 cosβ cos(2α)) + y(2 cosβ sin(2α))}=
[ GQ{x(2 cosβ sin(2α))− y(2 cosβ cos(]2α))}
BD + 2 cosβ GQ{x cos(2α) + y sin(2α)}= .
2 cosβ GQ{x sin(2α)− y cos(2α)}
This generates a force (∣aga∣in only taking the direction of BD into account)⇀
F m ≈ ∣∣⇀m∣∣ ⇀∇Bx
∣∣ ∣ [ ]⇀∣ ∂/∂x
= ∣m∣ (B
∂/∂y D
+ 2 cosβ GQ(x cos 2α+ y sin 2α)) (13)
∣ ∣ [ ]
⇀ cos 2α
= 2 cosβ ∣∣m∣∣ GQ .sin 2α
From Figure 3l it can clearly be seen that the angle β between the quadrupoles only
scales the force by 2|cosβ|, i.e., from twice the gradient generated by one quadrupole at β
= 0◦ and 180◦ and zero at β = 90◦ and 270◦. At the same time, the direction of the force is
kept constant at 2α. In this way, it is possible to accelerate, decelerate, stop, or move the
object at very sharp angles. This is best illustrated by a video in which a small steel ball (in
very a highly viscous medium to slow down its speed) was used in such a system to write
letters (Figure 9a).
A complete guiding system that allows not only changing the direction and strength
of the magnetic force but also the direction and strength of Bhom will then consist of two
dipoles and two quadrupoles. With such an instrument, all magnetic fields can then be
cancelled in the inner volume, so that also aggregates of nanoparticles might disintegrate
Cells 2021, 10, 2708 9 of 31
(see Section 5). Such a system is treated in a very general way in Appendix D. The
resulting equations may look complicated at first glance, however, they contain only simple
trigonometric functions that can easily be calculated. This is a fact that should not be
underestimated, because there are neither time-dependent nor interdependent terms, nor
non-linearities in these equations. Even if the magnetic fields cannot be made as ideal
as assumed here, they can be measured and tabulated for each ring and used for the
calculation of the local force with high precision.
There is no torque if nested ideal Halbach systems are rotated with respect to each
other [26]. However, the necessary segmentation and truncation to a certain length (see
⇀
Appendix B) also introduces a torque τm∼sin(kα) [27] whose amplitude very much
depends on the geometry and has its main contributions due to cogging at the edges of the
segments. Especially for quadrupoles constructed from polygonal magnets, the torque can
become significant.
It may be useful to evaluate the range or dimensions of magnetic fields, gradients
and instruments sizes. One limit is the demagnetization field, which limits the local
magnetic fields inside the magnet structure. If exceeded, the magnetic material will alter
its magnetization like in the initial polarization process (to some extent, this is discussed in
Appendix B and in [28,29]). However, this process is ignored and a simple guiding system
made from one Halbach quadrupole (Ri = r1, Ro = r2) surrounded by a Halbach dipole
(Ri = r2, Ro = r3) is straightforwardly calculated from Equation (9). Then the field strength,
BD, of an ideal Halbach-dipole and the gradient strength, GQ, of a quadrupole are given by
Equation (9) and combined with Equation (7)( )
GQ = 2B 1 1R r −1 r ,2
B r3D = BR ln r ,2 (14)
BD ≥ GQ r1 .
From this set of(equations,)the following relatio(ns can b)e equated:
r − r
r ≥ r exp 2 2 1 r B3 2 and BD = 2B 1− 1 and G ≤ D (15)r R r Q2 2 r1
The dependency of this equation and the resulting homogeneous and graded fields are
shown in Figure 4 assuming r1 and BR are given and the relation in Equation (15) is treated
as an equation. Although using the equations for ideal Halbach-magnets in Equation (14)
is somewhat naïve (see Appendix B), Figure 4 gives a good estimate for the magnitudes of
the geometric dimensions, magnetic fields, and gradients, which are important to estimate
the achievable magnetic forces on nanoparticles in Section 5.
Cells 2021, 10, 2708 10 of 31
Figure 4. Estimation of the achievable magnetic fields and gradient strengths for a simple guiding system consisting of an
inner Halbach-quadrupole (red) and an outer Halbach-dipole (green) (see sketch of the geometry inserted in (a)). Shown
is the dependence of the homogeneous field, BD (green line), produced by the dipole and the gradient, GQ (dashed red
line), of the quadrupole versus the inner radius of the dipole r2. Both have the same functional dependence but scale with
r1. Therefore, there are two axes on the left in the respective colors. In (a), the internal radius is set to r1 = 5 cm, which
could allow for guiding objects inside a rodent. The innermost radius in (b) of r1 = 40 cm was chosen to host a human. A
remanence BR = 1.3 T was used in both cases. The outermost radius, r3, BD, and GQ are then given by Equation (15).
4. Deviation from Constant Forces
The key requirement for obtaining a constant force over the internal volume of radius
R is given by Equation (7). This defines a prerequisite for ignoring the unavoidable
additional components of the deflection field. In magnetic resonance imaging (MRI), these
components are also named concomitant gradients [30]. Figure 5 gives a visual explanation
of what happens to the force field if the gradients become too strong. The deviations
are of course strongest at large distances from the center and that central line, which is
perpendicular to the anticipated force direction (vertical central line in Figure 5). This is
because the other concomitant gradient field has a zero here as well (cf. Figure 2c,d).
Figure 5. Schematic representation of the effect of concomitant gradients or violating Equation (7). Shown is the force
over the internal area with a radius R for different ratios B/(GR), cf. Equation (7). The magenta-colored arrows ignore the
⇀ ⇀
local field and m always points along B (as in Equation (6), while the blue arrows account for the full field. This shows
the increasing deviation from a homogeneous force direction with increasing gradient and/or distance from the center.
(a) B/(GR) = 10 or δ ◦max = 5.7 , (b) B/(GR) = 5 or δ ◦max = 11.5 , and (c) B/(GR) = 1 or δ = 90◦max . See text for more details.
Nevertheless, the guiding equation can still easily be solved by using Equation (6)
without the approximation of Equation (7) (see also Appendix D for a full treatment). For
Cells 2021, 10, 2708 11 of 31
⇀
instance, the magnetic field, B , in Equation (10) for a dipole with rotated quadrupole gives
then the following orientation [for the local magnetic moment
⇀ ∣∣∣ ∣∣∣ ∣∣∣⇀∣ ⇀⇀ ∣B ∣ ∣∣∣m∣∣∣
∣∣ ]
∣ ∣ ∣ ∣ BD + GQ(x cos 2α+ y sin 2αm x y m )( , ) = ∣⇀∣ = ∣⇀B B ∣ GQ(x sin 2α− y cos 2α)∣∣ ∣∣ √ (16)⇀
with ∣∣B ∣∣ = B2 2D + GQ(x2 + y2) + 2BDGQ(x cos 2α+ y sin 2α) ,
and the force becomes ∣∣∣ ∣⇀m∣ ∣∣G∣ [ ]⇀ ∣ ∣Q∣ ∣ GQx + BD cos 2αF m(x, y) = ∣ ∣ . (17)⇀ GQy + BD sin 2αB
The deviation between the ideal and real case can be described by a deviation angle
(i.e., the angle between the magenta and blue arrows in Figure 5). Its maximum, δmax, is
derived in [17] as ( )
−1 GR Bδmax = sin or G = sin δmax. (18)B R
From this equation, it is obvious that absolute angular precision (δmax = 0) implies
G = 0 and hence no force or deflection, except for the line where the concomitant gradient
is zero. For an angular error of δ ◦max ≤ 1 , a ratio B/(GR) ≈ 60 must be achieved. However,
the situation is not as bad as it seems, because there are simple analytical expressions of
this deviation for every spatial coordinate so that the error can easily be accounted for (see
Appendix D) and usually the real challenge is to guide deep inside (center, or small R) the
body where the precision is naturally higher.
5. Magnetic Force and Velocity
⇀
The magnetic force, F m, on a single particle in such magnet systems with a mag-
⇀ ⇀ ⇀ ⇀ ⇀
netic field B( r ), which has a gradient G( r ) at the spot, r , of the particle, is given by
Equation (1) or (8).
⇀ ∣ ∣⇀ ⇀
F m( r
∣
) = ∣m∣∣ ∣∣ ∣∣ ∣ ∣⇀ ⇀ ⇀ ⇀ ⇀ ⇀ ⇀ ∣⇀ ∣ ⇀ ⇀G( r ) ≈ ∣∣M(B( r ))∣∣ V G( r ) ≈ ∣∣Ms∣∣VG( r ), (19)
where the magnetic moment is expressed by the more commonly used magnetization,
M [A/m], and the particle’s volume, V. However, the magnetization is the volume integral
of all magnetic moments in the object, which do not necessarily all align with the external
⇀ ⇀ ⇀
magnetic flux density, and consequently, M is a function of B( r ). If all magnetic moments
are oriented, the material is saturated with a magnetization, Ms. This is assumed to happen
for the last term in Equation (19) and is usually a valid assumption for most magnetic ma-
terials in nanoparticle synthesis (e.g., for magnetite, Fe3O4: Mms := Ms/ρ ≈ 4–80 Am2/kg
with a density ρ = 1000–5200 kg/m3, depending on if the particle composition has a
saturation induction field of B < 10 mT).
In the following it is further assumed that the particle has the shape of a sphere and
is embedded in a medium of dynamic viscosity, η [Pa s]. Then the medium will exert a
Stokes friction or drag of
⇀ ⇀
F S = 6πηRh v , (20)
Cells 2021, 10, 2708 12 of 31
⇀
on the particle, so that in∣ eq∣uilibrium it w∣ill m∣ove with constant∣velocity v .∣∣∣⇀ ∣∣∣ ∣∣∣⇀Ms V 2 M⇀ ⇀ s∣∣∣ ∣∣∣
∣
⇀ ∣
R3 2 M ∣R2
⇀ sif R=R ∣ ⇀
v = G = G = h G. (21)
6πηRh 9ηRh 9η
The hydrodynamic radius of the particle, Rh, does not have to be identical to the
geometric radius (as assumed for the last approximation) of the magnetic part(icle) of
the spherical object, because many nanoparticles consist of a magnetite core and some
biocompatible shell [31]. However, in most applications of magnetic guiding of nanopar-
ticles, more than just one of such particles is administered. Therefore, it is very likely
that neighboring particles will form chains, because they are all magnetized in the same
direction and will experience dipolar interaction (cf. Figure 6a–d, this clustering can only
be avoided with low concentrations or large Rh together with low Ms so that the interaction
energy is lower than the thermal energy).
From the perspective of transportation, such clustering is advantageous, because a
cluster of n particles experiences an up to n-fold increase in force [32]. Associated with the
clustering is obviously a change of the shape of the guided object, and hence Equation (21)
is no longer valid. If a cloud of superparamagnetic nanoparticles is injected inside a
magnet system designed as suggested above, the strong and homogeneous magnetic field
magnetizes all particles along its direction. If no gradient (=neither force nor velocity) is
present, cluster-formation will only happen on the time-scale of Brownian motion (i.e., self-
diffusion). Gradients will assist a quick cluster-formation as the particles are all moved in
the same direction [33], and hence their inter-distances will become smaller in this direction.
As soon as some larger clusters have formed, they will attract neighboring particles with
increased force. A self-accelerating process starts in which the particles are mainly attracted
to the ends of the forming beaded structure (cf. Figure 6b). This is exactly the same process
that forms “field lines” from iron-filings in the stray field of magnets (cf. Figure 6b,c).
To account for this behavior, the shape of such a beaded chain is approximated
by a long slender body of length Lh = 2nRh (cf. Figure 6a) and its velocity is given by
(Equations (7.10) and (7.21) in [34])
[ ] ( )
⇀ v‖ cosα ⇀ ⇀v = with α ] m, F
v m⊥ sinα
⇀ ∣∣ ∣ ∣ ∣ (22)⇀ ∣ ⇀ ∣⇀ ∣ ⇀
and F m ≈ n ∣∣Ms∣∣VG = n ∣∣Ms∣∣ 43πR3G,
∣∣∣ ( )≈ ∣⇀ ∣∣∣∣ ∣ ∣ln LhR +C ∣⇀ ∣h ‖ ≈ ∣∣ ∣∣ ln(2n C if R R [ ])+v F F ‖ = h‖ m 2πηL m 4πη nR = 2κ ln(2n) + C‖ ,h h
∣∣ ∣∣ ( )L ∣ ∣⇀ ln h +C⊥ ∣⇀ ∣
with v⊥ ≈ ∣∣F Rm∣∣ h4πηL ≈ ∣∣F ∣∣ ln(2n)+C⊥ if R = Rm = h κ[ln(2n) + C⊥] (23)h 8πη nRh
∣∣∣∣ ∣∣∣∣∣⇀ ∣∣∣ ∣⇀∣with κ = R2 Ms G∣∣/6η.
The last approximation assumes that the radius of the magnetic sphere is identical to
its hydrodynamic radius. It is important to realize that both velocity components grow
logarithmically with twice the number of particles in the cluster. The geometric details of
Cells 2021, 10, 2708 13 of 31
the slender body are then provided via the constants C‖ and C⊥ = C‖ + 1 in Equation (23),
which are given by (Equations (7.12)–(7.14) in and (7.23) in [34])
cylinder : C‖ = − 32 + ln 2 ≈ −0.8069 and C⊥ = +0.1931 ,
spheroid : C 1‖ = − 2 = −0.5 and C⊥ = +0.5 , (24)
double cone : C‖ = − 12 + ln 2 ≈ +0.1931 and C⊥ = +1.1931 .
Not surprisingly, the velocity parallel to the long axis v‖ = 2(v⊥ − κ) is roughly
twice as big as perpendicular to it [17]. The geometric features in Equation (24) namely
cylinder and double cone (spindle) are good matches for the shapes that one finds among
the particle aggregates (cf. Figure 6b–d). Such spindle-shaped clusters form at higher local
concentrations after a certain length (n ≥ 15) of the initial line-growth of the cluster. This is
schematically explained in Figure 6e. If there are sufficiently many particles in the vicinity,
such clusters will form in any magnetic field and this behavior is not a particular feature of
the suggested instrument. However, they can easily be studied in them macroscopically [17]
and microscopically [33].
If some typical numbers are plugged in Equation (21) (Mms = 50 Am2/kg, ρ = 2500 kg/m3,
G = 1 T/m, η = 1 mPas) and the velocity is calculated for various R = Rh, however, here the
time, T, to travel 1 mm is given as: R = 1 nm→ T = 1.1 a, R = 100 nm→ T = 1 h, R = 10 µm
→ T = 0.36 min, the latter is roughly the size of an erythrocyte, which has to fit through
all blood vessels. Hence, to travel a biologically relevant distance in decent time, large
particles with strong magnetization should be used. However, the size of the particles used
inside biological systems is often considered of paramount importance, because larger
particles are more likely to be recognized as foreign bodies and will be attacked by the
immune system and might cause clotting. In a way, guiding makes this point somewhat
less important, because the big particles are injected somewhere and subsequently guided
to their target position (like an endoscope). The only processes to worry about are those
which could hamper this transport. In this picture, it also does not matter if the big particle
is solid or an aggregate from a large number of smaller particles. Although the latter sounds
more promising because such agglomerates would be somewhat flexible in shape and can
disintegrate at the target site by removing the homogeneous field (using two dipoles of
equal strength). In this way, a release process may be triggered. Technically, magnetic field
gradients inside humans are limited to 1–10 T/m (cf. Figure 4b), and if all other parameters
are optimized, a speed of 1 mm/s still requires particles with sizes from 100 nm to 1 µm.
Cells 2021, 10, 2708 14 of 31
Figure 6. Behavior of superparamagnetic particles under magnetic guiding conditions: (a) Magne-
⇀
tized by the homogeneous magnetic field, Bhom, the particles form long chains in the same direction.
Ideally they are mono-disperse spherical particles with a hydrodynamic radius, Rh, and then n of
them form a chain of length Lh = n 2Rh. However, if they are moved, the velocity of the chain has
two extremes of being moved either with v‖ along the long axis or direction of the external field or
perpendicular to this direction with v⊥. (b–d) Shows microscopic photographs of typical clusters
of magnetite particles with an average diameter of 30 µm. The yellow arrow indicates the direction
⇀
of Bhom. (b) At low local concentration linear beaded chains (right) are typically formed. After a
certain length particles can also attach from the sides, forming spindle-shaped structures (left). (c) At
higher concentrations such spindles dominate and separate from each other. (d) Such “carpet-like”
structures are also observed at very high concentrations and at fluid surfaces. (e) FEM simulation
(COMSOL 5.5) showing the magnetic force (Equation (1)) in cylindrical coordinates (r, z) around
chains of n particles, which are all magnetized in the vertical (+z) direction. For a small number of
particles; the “repulsive“ part (red) of the force dominates in radial direction. The particles must first
overcome this force, to enter the attractive force minima (blue) closer to the chain. This repulsive
force vanishes at the center for n ≥ 15. Therefore, particles aggregate at the sides of longer chains in a
hexagonal pattern, causing the typical spindle-shapes.
All the above is again valid for ideal Newtonian liquids only. Guiding through a
living organism, one will encounter many more problems, as body liquids—foremost
blood—show complex rheology because they contain living deformable cells. Additionally,
for guiding through blood vessels, the strong, pulsatile flow needs to be overcome. When
dealing with 3D systems (see next section), gravitational and buoyancy effects need to be
considered as well.
6. Possible 3D Designs and Imaging
So far, only instrumentation to guide paramagnetic objects across a plane was pre-
sented. However, almost all applications need to have 3D-control over the object’s trajectory.
The most straightforward solution to this is to use Halbach-spheres [35] instead of cylinders
(see Figure 7a). These are constructed by rotating the cross-section of a Halbach-cylinder
Cells 2021, 10, 2708 15 of 31
around an axis through opposite poles. One may wonder how a sample can be inserted into
a closed sphere, but there are certain angles at which such spheres, likewise cylinders [36],
can be opened without a magnetic force [37]. Nonetheless, other problems will arise when
using rotating nested spheres. Maybe most challenging will be the fact that the stray
field of such Halbach spheres is no longer zero, so that significant torque can be expected.
Alternatively, one could think about a system of rotatable cylinders on a spherical gantry
to get the additional degree of freedom (see Figure 7b). This might be interesting because
it also allows to decouple the force field of a quadrupole from the orientation of a dipole
by simple rotation in a plane where both cylinders are orthogonal. Such a device could
be a reasonable option to move macroscopic and ferromagnetic objects (e.g., catheters,
capsules, or micro-robots) that possess very strong magnetization, so that the guiding fields
do not have to be particularly strong. Applications with nanoparticles cannot afford the
extra space required for full 3D rotation around an elongated object like a human or most
animals, and the magnetic forces between such gimbals will also become significant.
In [17], it was suggested to maximize the field in the center of the machine to keep
the guided object just there while moving the container in the third dimension. However,
this would generate a rather metastable situation, because in the front and back of these
homogeneous regions, the field drops with a strong gradient or force components out of
the central plane.
Therefore, a new design is suggested here. The system with all the features described
and analyzed in Appendix D is sketched in Figure 7c. Different to Appendix D, all magnets
have finite length. To homogenize the dipolar field in the center, the Halbach dipoles are
arranged with a gap (calculated in Appendix E) in which the quadrupole pair is inserted
to compact the apparatus. If the dipole pairs point in the same direction (Figure 7c), the
situation of the ideal system is maintained in a central xy-slice. Here the object can be
moved and accelerated via rotation of the quadrupoles (as demonstrated in [17,33]). Now,
if the quadrupoles are put in a cancelling position (cf. Figure 3j) and one of the dipole pairs
is inverted (see Figure 7d), a gradient ∂Bxy/∂z is generated, which is constant in the central
plane. This will then move the object in the z-direction with velocity vz, which can be scaled
by controlling the strength of the dipole-pairs (cf. Appendix E). In a real system it would be
recommendable that the container in which the object is guided is moved by approximately
the same speed (e.g., on a stage movable along the axis like in CT- or MRI-scanners), but in
opposite direction, to keep the object in the center of the machine.
The instrument in Figure 7c,d then combines 3 degrees of freedom in guiding with
velocities vx, vy (via rotating the quadrupoles, cf. Figures 3h–j,l and 7c), and vz (by bringing
the dipole pairs in opposite direction, cf. Figure 7d) with 2 degrees of freedom in orientation
in x and y, e.g., of nanoparticle chains by co-rotating the dipole pairs (cf. Figure 3c–e,k).
Additionally, all these degrees of freedom can be scaled by the apparatus sketched in
Figure 7c,d. Orientation along z is not possible because ideally there is no such field
component in the center of the machine.
A crucial aspect of guiding is the control of the actual position of the steered object. If
the container in which the object is moved is not transparent, like most biological systems,
a blindfolded tour will probably not end up at the designated target position and maybe
even cause severe inner injuries. Maybe with an exception for some endoscopic or capsules
devices with an onboard camera, the position must be continuously monitored by a non-
invasive imaging technique. Since the proposed magnet system already possesses strong
homogeneous and gradient fields, there are two obvious candidates for this: magnetic
resonance imaging (MRI) and magnetic particle imaging (MPI).
Cells 2021, 10, 2708 16 of 31
Figure 7. Three-dimensional guiding ideas: (a) A Halbach sphere is made by rotating a Halbach cylinder around the axis
through the poles. The magnetic field inside is 4/3 higher than in an ideal Halbach cylinder, but it also produces stray fields
(small vertical arrows). (b) Sketch of 3D gimbal gantry. (c) Axial cross-section through a magnet system of two pairs of
dipoles (green) and two quadrupoles (red). The dipoles and quadrupoles could for instance be concentric Halbach cylinders,
here displayed as axial slices. The dipoles are split in a pair to homogenize the field in the center. If the strength of D1 = D3
and D2 = D4 the system is described in Section 3 and analyzed in Appendix D. In the shown configuration, objects are
oriented in the x-direction and moved in the central xy-plane (lilac arrows). (d) Adding one additional degree of freedom to
the system in (c) by independently rotating the left and right dipoles. If the quadrupoles are put in a cancelling position
and the dipoles are in opposite directions, a gradient along the z-direction (axis) is generated. This system is discussed in
Appendix E. If the guided object moves, then with vz the container (e.g., patient) should be moved via a stage with −vz so
that the object stays in the center.
Magnetic Resonance Imaging (for excellent introductions see [38,39]) uses the magnetic
moments of atomic nuclei (foremost 1H), which are associated with the spin of the nucleons.
To generate an energetic difference between the different orientations of the spin, a very
strong and homogeneous (polarizing) magnetic field is needed. Typically, the energy
differences between these energy levels correspond to a radiofrequency (e.g., ca. 42 MHz/T
for 1H). To excite transitions between the energy levels, a magnetic field must be irradiated
with a frequency that matches the strength of B0 and is oriented perpendicular to it. This is
usually done in the form of short bursts of an AC current of matched frequency in a coil
that surrounds the sample. The nuclear spins then induce a much weaker signal in the
same coil. The signal strength scales with B20 and therefore magnets must be strong for
Cells 2021, 10, 2708 17 of 31
this technique. Spatial resolution is then introduced by constant gradient fields, G, i.e., a
linear variation of the magnetic field component, which is parallel to B0 along the spatial
direction of interest. As said before, the same requirement (B0 > Gr) as in Equation (7) is
necessary to create bijective spatial encoding. Because the frequency of the MRI-signal is
directly proportional to the magnetic field, such a gradient generates an integral projection
of the sample along the selected gradient direction. Therefore, the instruments presented
here possess all requirements for MRI, albeit the gradients discussed so far would be way
too strong for MRI (G < 0.01 T/m) and two quadrupoles would be needed to allow for
guiding at high G and imaging at low G, also ensuring the objects are no longer moving
while being imaged. Since the gradients in the discussed designs can be rotated, projections
at various angles can be easily acquired and reconstructed to Cartesian coordinates using
an inverse Radon transform, aka “back-projection” as used in CT-scanners [40]. The only
extra-equipment would be an rf-coil with an amplifier and the MRI-spectrometer. Because
the contrast of MRI in biological systems mainly stems from the magnetic interactions of the
nuclear spins in water with the direct environment, superparamagnetic nanoparticles cause
a strong disturbing effect and are already standard MRI contrast agents [38,41]. However,
this leads the reader to believe that MRI comes for free with the suggested guiding systems.
The difficulty lies in the field homogeneity required for MRI. While a guiding system
could sufficiently function with a precision of 1%, MRI requires fields that are 2–3 orders
of magnitude more homogeneous. Nevertheless, recently great progress was made in
building functioning MRI systems using Halbach-magnets [42,43].
Magnetic Particle Imaging [44,45] relies on the presence of superparamagnetic particles
to generate a signal. Their magnetization follows a magnetic drive field oscillating at a
suitable frequency (typically in the kHz–MHz range depending on the particle relaxation
behavior) and its time derivative induces a signal in a receive coil. It can be distinguished
from the drive field because the magnetization in such materials does not depend linearly
on the applied field (typically M(H) is described by a Langevin-function) and can be
identified (e.g., via overtones in the spectrum) or filtered from the induced voltage. This
signal is then proportional to the concentration of the particles. Spatial resolution is
obtained by applying strong gradient fields, so that the particles magnetization outside
the region of zero-crossing is saturated, hence it cannot follow the drive signal and no
longer gives a signal. By moving such zero-lines or points through the region of interest,
images of the particle concentration can be reconstructed. The latter are typically those
magnetic fields that consume most of the energy in electromagnetic MPI-devices, but can
be generated with the described apparatus by combining quadrupole and dipole fields as
suggested in [46].
Apparently as the title of this section suggests, none of the discussed devices have
been built yet. Of course, other tracking possibilities exist as well, especially for larger
objects as reviewed in [47].
7. Applications
So far two types of guiding systems were designed, larger ones with Ri ≈ 5 cm
(the original MagGuider from [17] and a system with one dipole and two compensated
quadrupoles, i.e., system M1 in [33], cf. Figure 8a,b) and several smaller ones with
Ri ≈ 1.5 cm made to fit light-microscopes (for instance M2, M3 in [33], cf. Figure 8c,d).
Appendix G contains some practical suggestions for building such magnet systems.
Several objects were guided in 2D with such systems. Being equipped with two
quadrupoles of almost equal strength allows to accelerate the object or even to bring it to a
stop. To illustrate this option, a 1-mm steel ball was used to write the initials of the author’s
employer (cf. Figure 9a or [48]). The steel ball was chosen because it can easily be tracked
visually or by a regular camera, however, in order to slow down its movements in the very
strong gradients of the device shown in Figure 8b it had to be immersed in an extremely
viscous silicon oil. On the other hand, the cobalt ferrite nanoparticles shown in Figure
9b,c had almost 10 times the velocity of that steel ball; however they were immersed in
Cells 2021, 10, 2708 18 of 31
water (with a 50,000 times lower viscosity than the silicon oil used in Figure 9a). They are
only visible (average diameter of 75 nm) because they form very long clusters (n = 105–106).
Several other commercial superparamagnetic nanoparticles were compared under identical
conditions in [17].
In [33], it was shown that not only bare nanoparticles can be moved but also human
and murine cells that have taken up commercial nanoparticles (cf. Figure 9d,e). This
might be important for ex-vivo 3D-printing of tissues or entire organs. In-vivo regenerative
medicine might make use of this (e.g., lesion of nerves, retina, inner ear, etc.) by transporting
specific cells to lesions, building cellular scaffolds, and repairing or engineering tissues
with certain textures.
Figure 8. Examples of two constructed guiding systems: (a) Cut-away view of a schematic drawing
of the larger device. The innermost opening has a diameter of 10 cm, outer diameter is 36 cm, and the
total height is 27.5 cm. It consists of a static Halbach dipole (magnets in green) and two quadrupoles
(darker and lighter red) that can be rotated. In gray are aluminum and in blue brass supports; ball
bearings are in yellow. (b) Constructed system with a homogeneous field of 325 mT and gradients
from 0 to 2.1 T/m. The complete system has a weight of ca. 100 kg. (c) Cut-away view of a schematic
drawing of the smaller device; the larger squares on the light-blue pad have a size of 1 cm. The
magnets in both dipole (green with two magnet layers) and quadrupole (red) are shown in gray.
Supports are in yellow and glass balls, which serve as bearings, are in blue. (d) 3D printed version of
(c) used in a light microscope. It has a homogeneous field of ca. 100 mT and a gradient of ca. 1.3 T/m,
which both can be rotated independently. It weighs 273 g. More information about both systems can
be found in the supporting material of [33].
The last example is like a micro-robot including actuation, however, on a much smaller
scale. In [49], liquid crystalline elastomers, which included iron-oxide particles, were
functionalized with thermo- and photoresponsive groups. The long distance transport
could then be provided by the suggested magnetic guiding system. At the target site,
material properties could then be changed either by heat or light and used for actuation.
Cells 2021, 10, 2708 19 of 31
For instance, the stickiness of the particles was modified, so they could adhere to some
other non-magnetic object, and both could then be moved together magnetically. Bringing
the liquid crystalline elastomer to the isotropic phase by heating weakened the adhesion
and released the object (cf. Figure 9f).
Figure 9. Various applications of the presented guiding systems. (a) A 1-mm steel ball (white arrow) was used to write
the letters JGU using the magnet system presented in Figure 8a,b. The yellow arrows indicate the initial motion. (see
also Supplementary Video S1 in [33,48]). To slow down the speed of the ball it was immersed in very viscous silicon oil
(η = 50 Pas). Nevertheless, it had an average speed of 1.45 mm/s. (b,c) A small cloud (yellow ellipse) of cobalt ferrite
nanoparticles with an average diameter of 75 nm was moved in water with a magnet system consisting of one dipole
and one quadrupole. The direction of the static Bhom = 0.1 T is indicated by a green arrow while the quadrupole with
G = 0.2 T/m can be rotated and is marked by a red arrow. The resulting force or velocity (ca. 14 mm/s) is marked by a
lilac arrow. The underlying squares have a side length of 5 mm. (c) Image taken 3.5 s after the quadrupole (red arrow)
was moved by ca. 45◦ (Supplementary Video S2 in [17]). (d) Murine macrophage cells incubated with 100 nm iron oxide
particles were guided in a system like in Figure 8c,d while being studied with a light microscope. Due to the strong and
homogeneous dipole field, they form long clusters that are reversible when the magnetic field is removed. (e) Close up of
(d) the cells arranged around the connected nanoparticles like meat pieces on a shish kebab (for details see [33]). (f) A liquid
crystalline elastomer with incorporated iron oxide particles (white arrow) is guided over long distances (not shown) with a
system like in Figure 8c,d. When the particles arrive at the target site, actuation can be initialized by a temperature change.
A small piece of copper sticks to it at temperatures up to 110 ◦C (left & middle image) but leaves on the right when 125 ◦C
are exceeded (Supplementary Video S7 of [49]).
8. Conclusions and Outlook
Magnetic guidance of non-charged objects by combining a strong homogeneous
magnetic field with smaller deflection fields is a very general concept that can be put into
practice in various ways using resistive or superconducting electromagnets, by permanent
magnets or hybrid systems. This article focusses on permanent magnets and in particular
Cells 2021, 10, 2708 20 of 31
in Halbach configuration, because they are inexpensive, stronger, and can directly be scaled
up. Of course, there are alternative technical routes to fulfill this task than using nested
dipolar and quadrupolar Halbach cylinders in the presented combinations. Alternative
arrangements might be more useful and compact for certain applications. For instance,
in [50], six permanent magnets are individually rotated to generate the desired fields. This
article limited the discussion to Halbach magnets because the author believes that they
produce magnetic fields closest to the demanded ideals and are easy to calculate and
understand.
If orientation and force velocity do not have to be controlled independently, dipole
and quadrupole can also be combined in a single ring and so forth. In the end, the number
of degrees of freedom needed for a particular experiment decides upon the complexity of
the movable rings of the magnetic device.
The techniques proposed here cannot only be used to guide superparamagnetic
nanoparticles. This particular application is probably the most demanding in terms of
required forces and hence gradient and field strengths. Much weaker fields would be
sufficient to guide objects that contain larger ferromagnetic parts (e.g., endoscopes, mini-
robots). The necessary magnet systems for this application could therefore be much more
spacious than the devices shown in Section 7. It also would not be a problem to power
actuation of such robots by additional rf fields. Nevertheless, guiding superparamagnetic
nanoparticles is still the grand challenge in this field, and a few routes were shown how this
could be accomplished, e.g., by designing larger clusters of nanoparticles. For instance, a
“pill” made from differently functionalized particles could be synthesized. It could contain
some drug-carriers placed in the core and covered with layers of biocompatible material,
all superparamagnetic and weakly cross-linked, such that it decomposes after a certain
time at the target site and deploys its therapeutic payload for local treatment. In order to
get a step ahead in this direction, Section 5 and Figure 4 provide some clues on the physical
limits with today’s technology, and what sizes and shapes are needed to make real tools.
Taken together, the author thinks that the proposed guiding system may serve as a
general solution to the problem of steering by magnetic fields. It is hoped that this mainly
conceptual overview convinces researchers that this methodology has the potential to
simplify magnetic guiding, make it more versatile while reducing its costs and allowing
for human scale applications. To achieve this, much more efforts are needed in designing
proper computer-controlled systems with sufficiently homogeneous fields to implement all
the different modalities that were suggested in this article.
9. Patent
The principle of the presented guiding with permanent magnets is patented in O.
Baun, and P. Blümler “Vorrichtung zur Bewegung von magnetischen Partikeln in einem
Raum mittels magnetischer Kräfte” DE102016014192A1, 2016.
Funding: This work was funded by the German Research Foundation (DFG) via SFB 1066 “Nanodi-
mensional polymer therapeutics for tumor therapy” and by Inneruniversitäre Forschunsgforderung Stufe
I by the Johannes Gutenberg University Mainz.
Acknowledgments: First of all I want to thank Matthias Barz (University of Leiden, The Netherlands)
who (unintentionally?) inspired and catalyzed this whole idea. This happened in the framework
of SFB 1066, and I am indebted to Rudolf Zentel (University of Mainz, Germany) who put this
together and always is a source of inspiration. Olga Baun spent the main part of her Ph.D. thesis
in getting the first prototypes to work and collected many of the data shown in this article. Olga
Vinogradova (Moscow State University, Russia) started the discussion on altered hydrodynamics. I
am also very grateful to Vitalii Zablotskii (FZU Prague, Czech Repubic) for many helpful comments
and suggestions to this article. I want to acknowledge Dieter Sekels and Stefan Hiebel (Sekels GmbH,
Germany) for their motivation, interest, and getting the idea patented. I also want to show my
gratitude to Helmut Soltner (Forschungszentrum Jülich, Germany) who taught me many things
about magnets, magnetic fields, and their calculation, as well as for his careful checking of this
manuscript. Finally and infinitely I want to thank my wonderful wife, Friederike Schmid (Univer-
Cells 2021, 10, 2708 21 of 31
sity of Mainz, Germany)—in this context for physical and mathematical discussions, inspirations,
checking, polishing, and cleaning up my sloppy maths.
Conflicts of Interest: The author declares no conflict of interest.
Appendix A. Magnetic Force
For a detailed thermodynamic treatment, see Sections 2.5.4 and 2.5.5 in [51]. For the
sake of simplicity, the discussion is kept at a “microscopic level” of a magnetic dipole and
⇀ ⇀
non-linear bulk properties (i.e., magnetic field, H, and magnetization, M) are neglected. At
⇀
constant temperature, the force exerted by an external magnetic field with flux density B
⇀
on an object wit(h magn)etic moment m is given by
⇀ ⇀ ⇀
F m = ∇( ⇀m ·)B ( ) ( ) ( )
⇀ ⇀ (A1)
· ∇ ⇀ ⇀
⇀ ⇀ ⇀ ⇀ ⇀ ⇀ ⇀ ⇀
= B m + m · ∇ B + B × ∇×m + m× ∇× B .
Now, th(e advan)tage of limiting the discussion to a magnetic dipole manifests itself in
⇀
∇×⇀
⇀ ⇀
the fact that m = 0 because m (very much in difference to M) is no function of B( r ).
Furthermore, the discussion is restricted to permanent magnets, hence Ampère’s circuital
⇀ ⇀
law ∇× B = 0 be(cause n)o currents are present. Finally, m has no spatial dependence,
⇀ ⇀ ⇀
which also cancels B · ∇ m = 0 and just leaves an expression known as the Kelvin force
( )
⇀ ⇀ ⇀ ⇀
F m = m · ∇ B . (A2)
This approximation is a good choice for building a magnetic apparatus to manipulate
small paramagnetic objects (e.g., superparamagnetic nanoparticles). However, its approx-
imated character has to be kept in mind when strong interactions of the particles need
to be considered or bulk ferromagnetic objects with non-linear M(H)-curves have to be
analyzed (see [51]). If any anisotropy and non-linear behavior of the used bulk material
can be neglected, the magnetic (dipole) moment relates to macroscopic properties as
y ⇀
⇀ ⇀ ⇀
m = M dV ≈ MV ≈ B R V. (A3)
µ0
Appendix B. Field Calculation for Cylindrical Halbach Multipole Magnets
In 1973, John C. Mallinson published the idea of flat permanent magnet arrangements
which would generate a magnetic field only on one of their sides while it cancels on
the other [52]. Many household magnets and magnetic foils have such magnetization
patterns nowadays. Later this idea was generalized to cylindrical [25] and spherical [35]
arrangements with multipolar magnetic fields. Due to his pioneering work in this area,
they are now all named after Klaus Halbach.
Appendix B.1. Ideal Cylindrical Halbach Multipoles
For cylinders, the basic design idea is illustrated in Figure A1a. This idealized magnet
is a hollow cylinder of infinite length and continuously varying magnetization direc-
tion. The angle of the magnetization direction, ϕ, depends on the position angle, θ, (cf.
Figure A1a) as
ϕ = (k + 1)θ with k ∈ Z. (A4)
Figure A1b–g illustrates how the modulus of k determines the polarity, p = 2|k|, and
its sign the location of the produced magnetic field. For k > 0 the field is exclusively inside
Cells 2021, 10, 2708 22 of 31
⇀
the cylinder, and for k < 0 exclusively on its outside. The magnetization M [A/m] of the
hollow cylinder changes with r∣esp∣ect to its pos∣ition as
⇀ ⇀ ∣
M(R) = ∣∣⇀ ∣∣∣ ∣ ∣⇀ ∣M exp(i ϕ) = ∣∣M∣∣ exp(i[k + 1]θ)
[ ]
cos([k + 1]θ)
= BRµ exp(i[k + 1]θ) =
BR
0 µ0 [sin k
(A5)
([ +]1]θ)
⇀ cos θ
with R = R exp(iθ) = R ,
sin θ
where BR [T] is the remanence (remanent flux density) of the used permanent magnet
material and µ0 ≈ 4π × 10−7 kg m/(A s)2 is the permeability of vacuum. Please note
that the vectors in the magnet plane are represented in complex and Cartesian notation,
because both notations are used in the literature. For this work, only magnetic structures
⇀
are relevant, which encase the magnetic flux density, B , they produce. This limits the
discussion to k > 0 with the following general expression (the asterisk indicates the complex
conjugate to ensure Gauss’ law) (Equation (21) in[[25]) ]
⇀∗ ⇀ ⇀ ⇀ B ⇀k−1
B ( r ) = Bx( r )− iBy( r ) = x− [ ]= f (k) rBy (A6)
⇀ x
with r = x + iy = .
y
Figure A1. Conceptual idea of cylindrical ideal Halbach multipoles: (a) The magnet consists of a hollow cylinder of
⇀ ⇀
permanent magnet material∣∣∣(g∣∣∣ray). Its magnetization M (red arrow) is continually changing with position R (defined ona central circle with radius ∣⇀R∣ ⇀= R). If the position makes an angle θ with the x-axis, M is rotated by an angle ϕ. This
angle ϕ is an integer multiple of θ depending on the polarity and (inside/outside) location of the magnetic field of the
final arrangement (Equation (A4)) as given by the index k. The polarity of such cylinders is then increased from left to
right: (b,e) display dipolar (|k| = 1); (c,f) quadrupolar (|k| = 2); and (d,g) hexapolar fields (|k| = 3). The magnetic field
is completely inside the cylinder for k > 0, i.e., in (b–d) and zero outside. The opposite is the case for k < 0. For k = 0, the
magnetization has radial orientation and no transverse field (only axial, z-direction) is produced.
Cells 2021, 10, 2708 23 of 31
This results in multipolar magnetic fields of amplitude f (k), which is given by BR ln RoR for k = 1i
f (k) ≡ f ideal(k) =  ( ) . (A7)B k 1 1R k−1 k−1 − k−1 for k > 1Ri Ro
where Ri and Ro are the inner and the outer radius of the hollow cylinder (cf. Figure A1a).
As illustrated in Figure A2, several steps will be introduced in the following to break
down these ideal Halbach multipoles to more realistic structures. This will be included in
Equation (A6) by extending the amplitude, f, by further factors.
The efficiency of magnets is defined as the ratio of the energy stored in the accessible
region to the most storable in the magnetic material (see Equation (13.8) in [51]). Halbach
magnets are extremely efficient magnets with a maximum for dipoles with Ro = 2.21 Ri.
Figure A2. Illustration of transferring the ideal Halbach in (a), which has continuously changing magnetization and is
infinitely long, into a real system. (b) First the magnet is discretized into N cylinder-segments with one, homogeneous
magnetization direction. (c) The cylinder segments can be replaced by identical magnets that are rotated to the correct
magnetization direction. (d) Finally the arrangement is truncated to finite length.
Appendix B.2. Permeability of the Permanent Magnet Material
So far the relative permeability, µr, of the permanent magnet material was neglected
or set to that of vacuum µr = 1. This is not completely off scale, because modern rare
earth permanent magnets have values very close to that (“neodymium magnets” or
Nd2Fe14B typically have µr = 1.05, and “samarium-cobalt magnets” SmCo5 or Sm2Co17
have µr = 1.03–1.11).
If this is treated like a cylindrical shield, this should give an inner field reduced by a
factor (Equation (72) in [53])
4µ
fµ = r ( )2 . (A8)
(µ + 1)2r − (µr − 1)2 RiRo
If we imagine the magnetized hollow cylinder as a superposition of very thin cylindri-
cal shells, the field generated by the outermost shell is shielded by all the inner shells, while
the innermost is not shielded at all. The solution for this scenario is given by (Equation (31)
in [27])
µ (1−Λ)Λ R2o ≡ µr + 1f = 2 with Λ − and µr 6= 1. (A9)Ri −Λ2R2o µr 1
Simulations for the relevant range of the permeability show that Equation (A9) can be
approximated by (the error is smaller than 1% for µr < 1.35)
µ 1f = √ . (A10)
µr
If more than one magnet-cylinder is used to construct a (nested) system, the shielding
of the field originating from the outer rings by the permeability of the inner rings must also
be taken into account (e.g., by using Equation (A8)).
Cells 2021, 10, 2708 24 of 31
Appendix B.3. Segmented Cylinders
Since permanent magnets with the shape of hollow cylinders with continuously
varying magnetization direction as suggested by Equation (A5) cannot be made yet, the
typical way to approximate the ideal Halbach multipole is to construct it from N segments
with a single magnetization direction (cf. Figure A2b). This reduces the inner field by
(Equation (24b) in [25])
seg sin([k + 1]π/N)f (k) = . (A11)
[k + 1]π/N
This correction becomes mainly important for small N and high k (e.g., for k = 1 and N = 8,
f seg = 0.6366, but for N = 16, f seg = 0.9745).
Appendix B.4. Non-Cylindrical Segments (Mandhalas)
It can be advantageous to replace the cylindrical segments of Figure A2b by N iden-
tical magnets with a polygonal, round, or trapezoidal footprint (as shown for squares in
Figure A2c). This arrangement is also named Mandhalas [54] and densely packed geome-
tries for differently shaped sub-pieces are discussed in [55]. The main advantage of using
identically magnetized sub-pieces is to determine their individual strength (remanence),
sort, and orient them to optimize the homogeneity of the resulting magnet [56–58]. If
the footprint of the sub-pieces covers an area AM, the stren(gth of th)e resulting magnet is
reduced by the ratio to the area of the ideal Halbach (i.e., π R2o − R2i ).
M ( N Af = M 2) . (A12)π R2o − Ri
Appendix B.5. Finite Length
All magnets discussed so far were assumed to be two-dimensional, assuming infinite
length in the third or z-dimension. Simplifying the Halbach-multipoles by a ring of dipoles
gives the following decay along this third dimension (for k = 1 see Equation (8) in [56]) and
for x and y close to zero
B (x, y, z) = R
2k+3
xy k+3/2 B2 xy(x, y, 0) for k ≥ 1 ,(R2+(z−z0) )
(A13)
and for x, y ' 0 , and with R = Ri+Ro2 .
If this expression is integrated over z0 = ± L/2 (cf. Figure A2d) and set in relation
to the integral for the infinitely long ideal cylinder (z0 = ±∞), the reduction factor for
the length L is found, which can be normalized to the value at z = 0. Unfortunately, the
general solution of these integrals contains hypergeometric functions, but for the two cases
relevant here (k = 1 and k = 2), the following simple expressions can be found (for k = 1 see
Equation (A.5) in [46])
L∫/2  L(6R2 2Bxy(k, z +L )= 0, z0) dz0 for k = 1∫ (4R2 L2
3/2
− + )f L L/2(k) = ∞ =  . (A14)
Bxy(k, z = 0, z ) dz  L(L4+10L2R2+30R4)0 0
− 4R2 L2 5/2
for k = 2
∞ ( + )
An alternative solution is presente(d in Equation (49) in)[59] for Halbach dipoles only
f L = 1 + 1 L L L+Oln(Ro/R −√i) 2I 2O
− ln L+I
√ (A15)
with I = L2 + 4R2 and O = L2i + 4R
2
o .
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The decay ~z−5 along this direction can be avoided by placing multiple rings with
gaps in between them to homogenize the field in this direction [56] (see Appendix E). Note
that it is still the field components in the xy-plane that decay here.
Appendix B.6. Summary and Numerical Example
For this work only the cases k = + 1 (inner dipole, p = 2) and k = +2 (inner quadrupole,
p = 4) are of relevance, and the treatment above is summarized for each of them.
Dipole: (k = 1) 
⇀
BD(x, y, 0) = 

Bx
0  with B = B ln Ro fµ f segx R R (1) f M f L(1) ,i
0
(A16)
⇀ 2 2
BD(x, y, 0) = √
BR
µ ln
Ro sin(2π/N) N AM L(6R +L ) ê ;
r Ri (2π/N) π(R2o−R2 2 2 3/2 xi ) (4R +L )
Quadrupole: (k = 2) 
⇀
BQ(x, y, 0) = 

Bx
−By  with Bx = GQx and By = GQy ,
( 0 ) (A17)
G = √2BR 1 − 1 sin(3π/N) N AM L(L
4+10L2R2+30R4)
Q µr Ri Ro (3π/N
.
) π(R2−R2) 4R2 L2 5/2o i ( + )
Finally, a short numerical example is given for a Halbach dipole with BR = 1.3 T,
Ri = 5 cm, and Ro = 10 cm (simulated values using COMSOL Multiphysics 5.5 are shown
for comparison in squared brackets). This gives Bideal = 0.901 T for an ideal dipole sys-
tem (Equation (A7)). If this is segmented in N = 16 pieces, the flux density reduces to
Bseg = 0.878 T [0.878 T] (Equation (A11)), which reduces further to Bµ = 0.857 T [0.856 T]
(either Equation (A9) or Equation (A10) with µr = 1.05). If the magnet should be as-
sembled from permanent magnets with a squared footprint of side length a = 2.3 cm,
the field drops to BM = 0.308 T [0.304 T] (Equation (A12)), and if the magnets are finally
cut to a length of L = 6 cm, the field decreases to BT = 0.164 T (Equation (A14)) and
0.172 T (Equation (A15)) [0.164 T]. With the same indices, the following values are obtained
by repeating this procedure for a quadrupole: Gideal = 26.000 T/m, Gseg = 24.522 T/m
[24.522 T/m], Gµ = 23.928 T/m [23.925 T/m], GM = 8.595 T/m [8.101 T/m], and
GT = 5.458 T/m (Equation (A14)) [5.244 T/m]. So except for the last two values the devia-
tions between analytical calculation and simulation are less than a percent.
Equations (A16) and (A17) are very helpful for first estimations of fields of simple
geometries. In [60], some program codes can be downloaded to calculate such cylindrical
multipoles. Nevertheless, simulations are necessary to fine-tune the design of magnets,
estimate their homogeneity, and possible demagnetization effects [28,29], for instance, by
checking if inside the magnets the intrinsic coercivity is exceeded. For k = 1 that will be the
case for the magnets at θ = 90◦ and 270◦ towards Ro, and at θ = 0◦ and 180◦ towards Ri. For
k = 2 the magnets at θ = 45◦,135◦,225◦, and 315◦ (towards Ro) as well as at θ = 0◦,90◦,180◦,
and 270◦ (towards Ri) are the ones most affected.
The homogeneity of the magnetic field produced by real dipoles (possibly also
quadrupoles) can be further improved by using optimized angular distributions, which are
slightly off the theoretical positions [58].
If one has to use wires instead of permanent magnets, the field of a Halbach dipole
can be produced by a cos θ-coil [61], and that of a quadrupole by a cos 2θ-coil.
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Appendix C. Rotation of Vectors and Vector Fields
The discussion can be limited to two dimensions and therefore rotation by a single
angle, α. Then the rotation matrix is giv[en by ]
cosα − sinα
R(α) = . (A18)
sinα cosα
⇀
If the magnetic field, B , can be described by a single vector over the entire region of
interest (e.g., like the fie[ld of a dipole), its]r[otation] by[α is described by′ ]⇀ ⇀ cosα − sinα Bx Bx cosα− By sinαB = R(α) B = = . (A19)
sinα cosα By Bx sinα+ By cosα
However, the situation becomes quite different if a spatially-dependent magnetic field
⇀
is rotated (e.g., the field of multipoles, which depends on r , see Equation (A6)). This is
because both field- and space-coordinates have to be turned. The rotation of such a vector
field is then given by ( )
′ ⇀ ⇀ ⇀
G ( r ) = R(α) G( r ) · R−1(α) r
[ ] [ ] ([ ][ ])
cosα − sinα 1 0 cosα sinα x
= G
[sinα cosα ] (0[ −
·
1 − sinα ])cosα y
cosα sinα · x cosα+ y sinα= G
[ sinα − cosα −x sinα+ y cosα ] (A20)
cosα(x cosα+ y sinα) + sinα(−x sinα+ y cosα
= G )
[ si(nα(x cosα+ y s)inα)− cosα(−x sinα+ y cosα)
x cos2 α− sin2 α +
= G ( ) ]y(2 sinα cosα)
[ x(2 sinα cosα)− y ]cos
2 α− sin2 α
x cos 2α+ y sin 2α
= G .
x sin 2α− y cos 2α
If the magnetic dipole field of strength, BD, is combined with a the gradient field of a
quadrupole with gradien[t strength, G]Q, the lo[cal m] agneti[c field is ][ ]
⇀ B (x, y) 1 1 0 x
B(x, y) = x = B + G . (A21)
By(x, y) D 0 Q 0 −1 y
Now the quadrupole[is rotated by]an a[ngle α relative to the dipole, Equa]tion (A20) gives
⇀′ B′x(x, y) BD + GQ(x cos 2α+ y sin 2αB (x ), y) = B′y(x, y
= . (A22)
) GQ(x sin 2α− y cos 2α)
The gradient of both components of this resultant field is then given by calculating its
tensor (see Appendix F)
  [ ]  ′ ∂B′x ∂By [ ]
⇀ ⇀′ ∂x
∇⊗   cos 2α ⇀ ⇀′  ∂x ∇⊗  sin 2αB x = = GQ and By = = GQ − . (A23)∂B′ sin 2α ∂B′ cos 2αx y
∂y ∂y
This means that by a rotation of the quadrupole by an angle α, the resulting field gradient
and thus the magnetic force is rotated by 2α, see also Equation (A36).
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From simple symmetry considerations, it is clear that the magnetic field of any Halbach
multipole (see Appendix B) must rotate by k-times the angle at which the multipole itself is
rotated, because after rotation by 360◦/k every multipole is mapped on itself.
Appendix D. Two Dipoles and Two Quadrupoles
A magnet system, which can serve all possible requirements in magnetic guiding,
consists of two (Halbach) dipoles and two (Halbach) quadrupoles. Each of them is part of
a nested, concentric structure, and rotated by an individual angle. Then the total field is
given by
⇀ ⇀ ⇀
B tot = B1R(α) êx + B2R(β) êx + G1R(2γ) G( r ) + G2R(2δ) G( r ). (A24)
where B1 is the field strength (cf. Equation (A16)) of the first dipole, which is rotated
by an angle α, and B2 is the second dipole at an angle β (both point along êx the unit
vector in x-direction when not rotated). Then G1 and G2 are the gradient strengths of the
quadrupoles as given by Equation (A17); they are rotated by γ and δ respectively. This can
[ then be written as (using Equations (A19) and (A23)) ]
⇀ B1 cosα+ B2 cosβ+ G1(x cos(2γ) + y sin(2γ)) + G2(x cos(2δ) + y sin(2δ)B )tot = − − − − . (A25)B1 sinα B2 sinβ+ G1(x sin(2γ) y cos(2γ)) + G2(x sin(2δ) y cos(2δ))
⇀
A magnetic dipole with moment m orients then along the direction of this local field
∣
m ∣∣⇀m∣∣∣ ∣∣ ∣ ⇀⇀ B= êBtot = ∣m∣∣ ∣∣∣ tot∣ ∣ , (A26)⇀ ∣B tot∣∣
∣∣ ∣∣ and in this case∣∣⇀ √B tot∣∣ = X2 + Y2,
with X = B1 cosα+ B2 cosβ+ x(G1 cos(2γ) + G2 cos(2δ)) + y(G1 sin(2γ) + G2 sin
(A27)
(2δ))
and Y = B1 sinα+ B2 sinβ− x(G1 sin(2γ) + G2 sin(2δ)) + y(G1 cos(2γ) + G2 cos(2δ)).
The magnetic force is calculated by Equation (A1) or (A2)
⇀ ∣ ∣
∣∣⇀m∣∣∣ ∣
[ ( )
∣∣ (G21 + G22)x + 2G1G2 cos(2(γ− δ))xF m = ∣∣⇀ ∣∣ + . . .B 2tot G1 + G22 y + 2G1G2 cos(2(γ− δ))y ] (A28)
B1(G1 cos(α+ 2γ) + G2 cos(α+ 2δ)) + B2(G1 cos(β+ 2γ) + G2 cos(β+ 2δ))
+ .
B1(G1 sin(α+ 2γ) + G2 sin(α+ 2δ)) + B2(G1 sin(β+ 2γ) + G2 sin(β+ 2δ))
A lengthy equation but general and easy to compute (cf. Figure 5).
Appendix E. Gradient in the Third Dimension
As already stated at the end of Appendix B, it is advantageous to separate the Halbach-
cylinders into differently spaced rings. Very much like in Helmholtz coils or solenoids
with denser pitch towards their ends, this improves the homogeneity in the third (z) di-
mension [56]. The field of a Halbach-dipole in this direction is given by Equation (A13). To
homogenize the central field produced by two dipole rings placed at distances± z0, the sec-
ond derivative has to become zero at the center z = 0. Later different strength of the dipoles
will become important, so the ring at –z0 should have a strength B(0, 0,−z ) = [B10 x, 0, 0
T
]
Cells 2021, 10, 2708 28 of 31
and the one at +z0 also produces a homogeneous field in x-direction with a strength B2x,
then the field along the z-axis is given by
R5 R5
Bx(z) = ( ) 1 25/2 Bx + ( )5/2 Bx, (A29)
R2 + (z + z0)
2 R2 + (z− z 20)
which has the first derivate (later of importance) 
∂Bx(z) z + z z− z
= 5R5−( 0 )7/2 B1 0 2x + ( )5/2 Bx, (A30)∂z R2 + (z + z 20) R2 + (z− z 20)
and the second derivative
∂B 2x (z) 5R5
 
( 6(z + z0)− R)2 2= B1 ( 6(z− z0)− R 2
∂z2 9/2 x
+ )9/2 Bx. (A31)
R2 + (z + z 20) R2 + (z− z )20
In the center (z = 0) and initially for B1x = B2x both derivatives have to become zero for
a flat maximum.
∂B 2x (0)
2 = 0 ⇒ z0 = ±√
R
. (A32)
∂z 6
√
Generally this is fulfilled at z0 = ±R/ 2k + 4 for k ≥ 1. This is the distance where
no gradient ∂Bx/∂z is present and hence ideal working conditions are provided to move
the object in the central xy-plane by using the quadrupole(s) as described above.
As suggested by Figure 7d, a gradient of this x-component of the magnetic field along
the third dimension must be introduced to move the object truly in three dimensions.
This is achieved by changing the sign of one of the two rings (cf. Figure 7c,d). Then
Equation (A29) reads
Bx(z) = − ( ( R5 ) ) 15/2 Bx + ( ( R5 ) ) B2x, (A33)
R 2 R 2
5/2
R2 + z + √ R2 + z− √
6 6
and the first derivative (cf. Equation (A30) now both terms are positive) at z = 0 is given by
∂B (0) 5 · 63 B1 2 1 2x
= √ x + Bx ≈ B + B1.1901 x x . (A34)
∂z 73 7 R R
Hence, the gradient in the z-direction in the center of the apparatus in Figure 7c,d is
proportional to the sum of the field strength of the two dipoles with opposite direction. If
these fields can be made adjustable, so can the gradient strength and velocity in z-direction.
Appendix F. Some Mathematical Definitions
The gradient tensor is a second rank tensor given by ∂B B x ∂ x
∂Bx
 ∂x ∂y ∂z ⇀ ⇀T 
G ≡ ∇⊗ B  ∂By ∂B = y ∂By ∂x . (A35)∂y ∂z 
∂Bz ∂Bz ∂Bz
∂x ∂y ∂z
Cells 2021, 10, 2708 29 of 31
If we apply this to Equation (A22), it can immediately be seen that the gradient field is
rotated by 2α  ∂Bx ∂B x⇀ ⇀TG ≡ ∇⊗ B =  ∂x ∂y  [ ]cos 2α sin 2α= GQ
∂By ∂By sin 2α − cos 2α
∂x ∂y[ ] (A36)
1 0
= GQR(2α) .0 −1
Appendix G. Some Practical Considerations for Building Such Magnet Systems
To plan the construction of such a magnetic guiding instrument, the author suggests to
start with an estimation of the necessary gradient strength using the equations in Section 5.
The overall size together with Equation (7) then determines the necessary strength for
the homogeneous field. The size of the various magnets should then be estimated from
the equations in Appendix B (some general programming code for this purpose is found
here [60]) before optimizing them by finite element or boundary software packages.
When planning the size of the apparatus, it is the strong belief of the author that one
should first build a small mockup system (cf. Figure 8c,d) to investigate if the magnetic
forces are sufficient to move the object with the desired speed. Such a mockup system can
easily be constructed on a centimeter-scale using 3D-printed supports and inexpensive
magnets, which are available from manifold sources on the internet. It is good advice to
keep the magnets as small as possible for the desired effect, because
(a) even with such a small system, the hydrodynamics can readily be tested;
(b) magneto-mechanical forces can be estimated, which might become a real problem
during construction or operation; and
(c) the magnets are less expensive and much safer to handle!
Scaling the system to the application size, the designer should always be aware that
the force between two magnets (each with a surface A) scales as [36]
F ≈ BR A , (A37)
2µ0
for small distances and is in the range of 100 N for ca. 1 cm2 and already 1000 N for 10 cm2
size FeNdB magnets. This is just the force between two small magnets; two completely
assembled rings can easily generate forces of about 10 kN. Therefore, scaling such magnetic
designs up often becomes an engineering problem in calculating and managing forces,
using suitable materials of sufficient thickness, constructing tailored tools, and establishing
safe procedures for placing and fixing the permanent magnets. A process which requires a
lot of experience, which one can only acquire safely by first tinkering with smaller systems.
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