A Magneto–Mechanical Approach in Biomedicine:
Drawing the Magna Charta of a Novel Setup of
Halbach Arrays
DIMITRIOS PAPADOPOULOS
SCHOOL OF PHYSICS | ARISTOTLE UNIVERSITY OF THESSALONIKI
A Magneto–Mechanical Approach
in Biomedicine: Drawing the
Magna Charta of a Novel Setup of
Halbach Arrays
Dimitrios A. Papadopoulos
School of Physics
Aristotle University of Thessaloniki
A thesis submitted for the degree of
Bachelor of Science
Thessaloniki 2023
A MAGNETOMECHANICAL APPROACH IN BIOMEDICINE
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Περίληψη
Εμφανής είναι η αλματώδης ανάπτυξη του τομέα της βιοϊατρικής τον τελευταίο αιώνα, όπου
η τεχνολογική πρόοδος και η επιστημονική εφευρετικότητα συνεχίζει να επιφέρει νέα
ενδιαφέροντα αποτελέσματα σε παθήσεις που ταλανίζουν την ανθρωπότητα από την απαρχή
της. Εάν και το εγχείρημα εύρεσης θεραπειών για χρόνιες ασθένειες είναι μακριά από
ολοκληρωμένο, η εστιασμένη προσοχή της επιστημονικής κοινότητας σε αυτόν τον κλάδο έχει
παράξει σημαντικά παραπροϊόντα και μάλλον υποσχόμενες τεχνικές βασιζόμενες σε φυσικά
φαινόμενα. Ένα φαινόμενο που πιο πρόσφατα επανήλθε στην προσοχή των ερευνητικών
ομάδων αποτελεί το μαγνητό-μηχανικό, δηλαδή η αξιοποίηση της ενέργειας μαγνητικών
υλικών για την δημιουργία μηχανικής ενέργειας με τη μορφή δυνάμεων και ροπών. Το
φαινόμενο αυτό, μέσω της διάταξης που κατασκευάστηκε, επιχειρείται να τιθασευτεί ώστε να
προσαρμοσθεί στις απαιτήσεις βιοϊατρικών εφαρμογών. Στην παρούσα πτυχιακή εργασία, θα
μελετηθούν οι μαγνητικές ιδιότητες μίας πρωτότυπης διάταξης που αποτελείται από κυκλικές
συστοιχίες Halbach, με στόχο την ανάδειξη των προοπτικών της σε βιοϊατρικές εφαρμογές. Η
ιδέα της διάταξης συλλήφθηκε βάσει μοντελοποίησης με το υπολογιστικό πρόγραμμα
COMSOL 3.5a Multiphysics, οπότε πρώτο βήμα θα είναι η καταγραφή των διαφορετικών
τρόπων λειτουργίας της διάταξης με σκοπό να αναδειχθεί η ευελιξία και η χρηστικότητά της.
Ακολούθως, θα μελετηθούν μεγέθη που είναι θεμελιώδη για τις παθήσεις/ασθένειες που
επιδιώκεται να αντιμετωπισθούν με την βοήθεια της διάταξης. Τελικώς, τα παραγόμενα από
αριθμητική ανάλυση αποτελέσματα θα επιβεβαιωθούν πειραματικά με μετρήσεις μαγνητικού
πεδίου ώστε να αποτιμηθούν τυχόν αποκλίσεις από την υπολογιστική προσομοίωση.
Λέξεις κλειδιά: μαγνητο-μηχανικό φαινόμενο, συστοιχία Halbach, υπολογιστική
μοντελοποίηση, αριθμητική ανάλυση, βιοϊατρικές εφαρμογές, μαγνητο-μηχανική
ενεργοποίηση κυττάρων
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Abstract
Τhe obvious rapid development of biomedicine has been clear, where the technological progress
and the scientific creativity continues to bring about interesting results in conditions that have
been troubling humanity since its very beginning. Even though the venture of finding therapies
for chronic diseases is far from being complete, the focused attention of the scientific
community in this sector has produced significant byproducts as well as rather promising
techniques based on physical phenomena. One phenomenon that has recently recaptured the
attention of scientific groups is the magneto-mechanical, that is the utilization of the energy of
magnetic materials for the creation of magnetic energy in the form of forces and torques. This
phenomenon, through the setup that was built, is attempted to be harnessed in order to be
adjusted to the requirements of biomedical applications. In this bachelor thesis, the magnetic
properties of the novel device, that consists of circular Halbach arrays, will be studied aiming
to highlight its perspective use in biomedical applications. The idea for the setup was conceived
based on modeling with the Multiphysics computational program COMSOL v. 3.5a
Multiphysics, thus the first step will be registering the different operational modes of the
apparatus to demonstrate its flexibility and versatility. Subsequently, a study will be conducted
on the fundamental quantities that are directly related to diseases, that are sought to be treated
with the help of the arrangement. Finally, the numerical analysis results will be experimentally
validated via magnetic field measurements, so as to evaluate potential deviations from the
computational model.
Keywords: magneto-mechanical phenomenon, Halbach array, computational modeling,
numerical analysis, biomedical applications, magneto-mechanical cell actuation
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Abbreviation List
AFM:
Atomic Force Microscopy
AMF:
Alternating Magnetic Field
ELF:
Extremely Low Frequencies
EU:
European Union
FDA:
Food and Drug Administration
FEM:
Finite Element Method
MCT:
Magnetic Cell Triggering
MPH:
Magnetic Particle Hyperthermia
MMA:
Magneto-Mechanical Actuation
MME:
Magneto-Mechanical Effect
MNPs:
Magnetic Nanoparticles
MPI:
Magnetic Particle Imaging
MRI:
Magnetic Resonance Imaging
REC:
Rare Earth Cobalt
RMF:
Rotating Magnetic Field
SPIONs: Super Paramagnetic Iron Oxide Nanoparticles
STM:
Scanning Tunnelling Microscopy
PMF:
Pulsed Magnetic Field
3D:
Three-Dimensional
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Acknowledgements
Completing a senior thesis is a significant accomplishment, and I could not have done
it without the support and guidance of many individuals. I am deeply grateful to everyone who
has helped me along the way, and I would like to take this opportunity to express my heartfelt
thanks.
First and foremost, I would like to express my sincerest gratitude to
Professor
Mavroeidis Angelakeris, of the “Condensed Matter and Materials Physics” department of
Aristotle University of Thessaloniki, for the invaluable guidance and support throughout the
entire process of my senior thesis. His insights, encouragement, and expertise were
instrumental in the successful completion of this project. I am incredibly grateful for the time
and effort they have dedicated to me, and for their unwavering belief in my abilities. They have
not only been my advisor, but also a mentor.
To the group members of the MagnaCharta laboratory, thank you for welcoming like I
was always part of the team. Your presence, whether you were directly involved with my
project or not, made my experience that much better and I am grateful that I took my first steps
in the world of research with you. Specifically, I would like to thank Dr. Makridis Antonios
and Dr. Maniotis Nikolaos for their constant guidance and support, despite the workload they
had. Their insights and expertise in biomedical applications and computational modeling have
been a beam of inspiration that pushed me to expand my knowledge and contributed
significantly to the success of my thesis. Last but definitely not least, I would like to
acknowledge Pavlos Kyriazopoulos, postgraduate of the mechanical engineering department,
who undertook the task of modeling and constructing the device as well as its predecessor. His
enthusiasm about the project and the insight he offered in various stages of my research was
truly motivational, contributing significantly to the completion of this thesis.
I would also like to thank Professors Samaras Theodoros and Assistant Professor
Sarafidis Charalampos for their time and constructive feedback on my work. I am deeply
appreciative of their willingness to share their knowledge and expertise.
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To my friends and family that made me the person I am today, I am extremely grateful
for your presence in my life. Your support and belief in me have been invaluable and have
helped me pursue my academic goals. The unwavering love and support that you have given
me over the years will always be the foundation to making my dreams come true and I will
always remember you contribution to my life journey.
In conclusion, I would like to extend my heartfelt thanks to all of the individuals who
have helped me along the way. Your support and encouragement have been invaluable, and I
could not have completed this thesis without you. Thank you all for your support and
encouragement, and for your contributions to my academic and personal growth.
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Table of Contents
Περίληψη
i
Abstract
iii
Abbreviation List
v
Acknowledgements
vii
Table of Contents
ix
Chapter 1 Theoretical Background and Apparatus Introduction
1
Brief Historical Review of Nanobiotechnology
2
Point of intersection
2
Halbach Arrays
3
Magneto-Mechanical Effect
4
Biomedical Applications
6
Magnetogenetics
6
Cancer Therapy
7
Drug Delivery
8
Device compatibility
8
Forces exerted by MNPs
Setup of the Novel Apparatus
Magnets – Motors
9
11
11
Three-axes Hall probe magnetometer
12
Scope of this thesis
13
Chapter 2 Computational Modeling and Numerical Analysis
Computational Modeling
15
16
Magnetic properties
16
Geometry Rotation
17
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Static Study
18
Magnetic Flux Density B
19
Gradient of Magnetic Field ∇B
22
Magnetic Forces Fm
27
Rotational Study
30
Time-dependent Magnetic Flux Density
30
Time-dependent Gradient of Magnetic Field
32
Time-dependent Magnetic Forces
33
Chapter 3 Experimental Validation
Magnetic field mapping and Reliability testing
36
37
Static Study Evaluation
37
Rotational Study Evaluation
42
Long-term time testing
46
Conclusions
51
References
53
Footnotes
59
Appendix A: Tables
61
Rotational Analysis
62
Single Disk – Static Study
65
Double Disk – Static Study: Planes near the Bottom Disk
66
Double Disk – Static Study: Planes near the Top Disk
68
Appendix B: Supplementary Figures
71
Appendix C: 60 second time test
75
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Chapter 1
Theoretical Background and Apparatus Introduction
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Brief Historical Review of Nanobiotechnology
Nanobiotechnology is an interdisciplinary field that integrates principles from biology,
nanotechnology, and engineering to develop novel materials and technologies for medical, and
biological applications. This field, although having existed for only three decades, has grown
dramatically in recent years impacting a vast range of research areas [1] [2].
The concept of nanotechnology was first introduced by physicist Richard Feynman in
a lecture he gave in 1959, titled “There is plenty of room at the bottom”. In that lecture, he
discussed the possibility of controlling and manipulating individual atoms and molecules.
Notably, in the closing of his speech, he challenged the attending physicists to inscribe the
Britannica encyclopedia on the head of a pin [3]. Even though the seed was sown for thirty
years, it wasn’t until the 1980s and 1990s, when technological advancements, such as the
invention of Atomic Force Microscopy (AFM) and Scanning Tunneling Microscopy (STM),
allowed for the practical establishment of the field of nanotechnology and consequently for
researchers to begin exploring its biomedical applications [4] [5].
One of the most promising aspects of this field, has been observed in the advancements
of nanoparticle technology, with the materialization of highly targeted therapies and
applications, marking a landmark for medicine as a whole. Among these advancements are less
invasive and more precise treatment techniques for diseases, as well as improved imaging
technologies like Magnetic Resonance Imaging (MRI) and Magnetic Particle Imaging (MPI)
and sensors in the nanoscale [6] [7]. It is worth pointing out that due to the size of cells, blood
vessels and other targeted areas, area-specific treatment was not possible before learning to
manipulate matter in the nanoscale, that is, before the beginnings of nanotechnology.
Point of intersection
In recent times, one of the most researched phenomena in nanoparticle technology for
biomedical applications is the magneto-mechanical effect. For the latter to be expressed –
besides the magnetic nanoparticles (MNPs) – it is imperative that an externally applied
magnetic field, static or time-dependent, is present [8]. The proposed setup provides just that,
with a versatile, tunable magnetic behavior, adjusting to the required conditions of the
application. In the next sections, a theoretical background of the emerging properties as well
as their compatibility with applications related to the magneto-mechanical approach in
biomedicine, is provided.
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Halbach Arrays
A permanent magnet construction known as a Halbach array is created by placing
individual magnets with alternating polarities in a predetermined arrangement. This
configuration results in an inhomogeneity in the magnetic field distribution on either side of
the composition, that translates to a significantly stronger magnetic field intensity on one side
of the array than the other. This enhancement is attributed to the constructive and destructive
interference of the magnetic fields in two opposite directions. The magnetic field enhancement
is commonly described by the magnetic flux density, which quantifies the strength of the
magnetic field at a given location [9].
This “one-sided flux” effect was first observed by J. C. Mallinson in 1973
characterizing it as a magnetic curiosity [9]. In 1979, Klaus Halbach proved that the two main
factors affecting the strength of the generated field of such an arrangement are the volume
filling factor of the region surrounding the aperture circle as well as the number of easy-axis
orientations 𝑀 ∈ ℤ+ [10].
Bloch et al. underlined the superiority of cylindrical and spherical Halbach structures
in generating a homogenous magnetic field, a statement that was validated by the numerical
simulations that was done in 2010 by Bjørk et al. [11] [12]. Specifically, in the work of the
latter, it was shown that the required volume of magnetic material for achieving a certain
magnetic field strength was minimized for two concentric Halbach cylinders and for two half
Halbach cylinders. Those designs and their effectiveness are illustrated in Figure 1.1.
The geometry illustrated in Figure 1.1a has since become a blueprint for Halbach
structures, showing up with various alternatives and modifications of the original setup in a
plethora of papers published, especially in the biomedical field. In our work, having considered
the alternative geometric structures and their benefits, we followed a similar to the concentric
cylinder model geometry, in which eight cubic magnets are positioned in a circle-like
arrangement, with four easy axes along the positive and negative -x- and -y- direction. This
composition is then repeated, ending up with 16 total magnets that are hosted in two threedimensional (3D) printed polymer disks, designed to allow for a rotational movement.
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c)
a)
b)
Figure 1. 1: Efficiency of different Halbach structures, a) illustration of the concentric Halbach cylinder
structure, b) schematic of the half Halbach cylinder geometry, c) Volume of magnetic materials required for
achieving a certain magnetic field strength. Clearly the a) and b) geometries are superior to the other proposed
setups ሾ10ሿ.
Magneto-Mechanical Effect
The term Magneto-Mechanical Effect refers to the exploitation of the magnetic energy
of magnetic elements or chemical compounds by transformation to mechanical energy that
manifests as forces and torques [13] [14]. For alternating magnetic fields (AMFs) the range of
optimal frequencies is near the order of hundreds of Hz, since in higher magnitudes, the thermal
effects become dominant (in first approximation heating is directly proportional to the
frequency of the field) [8]. In the biomedical field, this definition is further constrained by the
size of the particles that is required to be lower than the microscale. Usually, frequencies under
100 Hz are used, however some works have worked with orders of kHz.
For a solely magneto-mechanical effect, with negligible local and bulk heating of cells,
Golovin et al. suggested that an AMF with B « 1 T in the range of Extremely Low Frequencies
(ELF), that is, f < 100 Hz, ensures a dominant mechanically driven response [8]. To further
clarify the reason for those constraints and the regions where the magneto-mechanical effect is
manifested optimally, a set of schematics highlighting the differences between the size and
field dependency of the mechanical and thermal mechanisms (magneto-mechanical actuation
and magnetic hyperthermia respectively) is displayed on Figure 1.2 [13].
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In general, the applications of the effect can be divided in three main categories: 1)
diffusion related phenomena, 2) molecule deformation and 3) supramolecular structure
disruption [14]. This paper is mainly focused on the last two, aiming to utilize the setup
described below for applications like magneto-mechanical straining tumor cells and
noninvasive thrombectomy. It is worth pointing out that for applications requiring magnetic
nanoparticles, – which are the majority – identifying the optimal operation area for the size,
and the magnetic field, turns into a more complex task, as more properties like the shape of the
particles are instrumental to determining the magnetic response of the MNPs.
a)
b)
c)
Figure 1. 2: Magnetic hyperthermia versus magneto–
mechanical actuation. a) Optimal frequency conditions for
the two effects. b) Optimal size (RM is the magnetic core
radius) regions for the manifestation of the phenomena. c)
Optimal field strength, i.e., magnetic flux density amplitude
B, for enhancing MPH or MMA [13].
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Biomedical Applications
The implementation of the magneto-mechanical effect in the field of biomedicine is a
relatively new venture, emerging in the late 20th century. In those decades, research groups
have investigated the utilization of the generated forces and/or torques for a wide range of less
invasive and more precise alternatives in Theranostics (Figure 1.3).
Figure 1. 3: Schematic illustration of
theranostics in nanobiotechnology.
Magnetogenetics
From the beginning of the 21st century, the focus of the scientific community has
shifted towards the recently emerged disciplines of magnetogenetics and gene therapy. The
manipulation and behavioral study of genetic material have always been a pressing matter due
to the extent of information that can be derived in consequence for the majority of the biological
functions and expressions in the human body. Until the late 2010s, the technologies available
for neuromodulation entailed the use of light propagation (optogenetics) or chemical agents.
However, the constrained tissue permeability of light, combined with the invasive behavior
(surgical insertion of an optical fiber and an electrode) of the former, as well as the poor
temporal resolution, i.e., the delayed cell regulation (attributed to the tardiness of
pharmacokinetics) of the latter have posed significant challenges, especially for in vivo studies,
hindering the advancement of the field to preclinical trials [15] [16].
In 2014 Stanley et al. reported remote regulation of the blood glucose in mice, by
stimulating the insulin transgene expression. The method involved using a magnetic tip and
iron oxide nanoparticles estimating a prerequisite of ~10 pN for the process to take place [16].
One year later, a proof of concept for intracellular magnetic manipulation through control of
the protein gradient inside cells was presented [17], while in the same year, an in vivo study in
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transgenic Caenorhabditis elegans demonstrated muscle contraction of the worms that were
subjected to a magnetic field strength ranging from 0 to 2.5 mT [18]. More recently, Songfang
et al. reported an in vitro magneto-mechanical actuation (MMA) of the TRPV4 ionic channel,
which was tagged with a His-tag, using MNPs functionalized with the anti-His antibody. The
results were also conducted in vivo (mice) resulting in the activation of two brain regions, when
the injected MNPs were exposed to AMFs with an amplitude of the order of 50 mT and a
frequency of 0,1 Hz (90% duty cycle) [19].
Cancer Therapy
One of the most prominent bodies of work inside the field of nanobiotechnology has
undoubtedly been the pursuit of highly target-accurate therapeutic and preventive techniques
for cancers and tumors. Desiring a less side-effect inducing and more effective approach,
MNPs (functionalized or not) under static or time-dependent magnetic fields very rapidly
became a household name. Although the Food and Drug Administration (FDA) and European
Union (EU) approved products and/or technologies are still very limited, the research done in
MPH (beyond our scope) and in the magneto-mechanical effect (MME), has yielded very
promising results for cancer therapy.
A very popular approach for magneto-mechanical deformation was first presented by
Kim et al. In the paper published in 2009, the group utilized permalloy micro disks (1 μm in
diameter), for an in vitro assay of glioma cancer cell viability. Amazingly, a 90% cell death was
observed for magnetic flux densities of 9 mT and frequencies in the range of 10-20 Hz. The
cause of death was attributed to apoptosis, given that the forces generated in these magnetic
conditions, would not be sufficient to rupture the cell membrane; a hypothesis that was
validated with a DNA fragmentation assay [20]. Zhang et al. reported cancer cell apoptosis via
hydrolase leakage, caused by the compromising the lysosomal membrane. To successfully
disrupt this supramolecular structure, the group used superparamagnetic iron oxide
nanoparticles (SPIONs) of varying size that were subjected to a novel mode of moving
magnetic field named “Dynamic Magnetic Field” (10-20 Hz & ~ 30 mT). Notably the work
maintained the temperature near 21 oC, minimizing the tissue damage from necrosis effects
like inflammation [21]. In another work, cubic iron oxide nanoparticles (62 nm), were
stimulated by a rotating magnetic field (15 Hz, 40 mT) aiming to effectively deform plasma
and lysosomal membranes of cancer cells. The magnetic conditions resulted in the formation
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of elongated aggregates that according to the group that overcame the critical value for
disrupting the membranes and inducing necrosis and apoptosis of glioma cells [22].
The plethora of approaches in this subdomain have amassed a significant database
regarding the responses of various MNPs and magnetic fields for different target sites. This
archive has been summed up and collected in a Table, in a review articles, that will later be
used as reference for the potential of our apparatus in biomedical applications [14].
Drug Delivery
In drug release, similarly to magnetogenetics, the aim of the research conducted in last
decades is focused on the generation of forces and/or torques that can mechanically activate
functions that are deemed of vital importance for drug delivery applications. As of today,
harnessing the magneto-mechanical phenomenon for such operations has proven to be
challenging, mainly because of the MNP size-force relation. Specifically, the force thresholds
for activating cellular activities (~ 1-100 pN) cannot be easily surpassed, due to the attenuating
value of magnetic torques with the decrease in size. In a subsequent work, to a one previously
described, where microdisks – with a spin-vortex ground state – were used, the group
hypothesized that the same effects could be achieved for nano-scaled disk by increasing the
magnetic flux density from 9 mT to approximately 270 mT [23]. Given the convolutions that
emerge in the nanoscale however, this suggestion’s validity remains unclear and further
research needs to be conducted. The same suggestion has been presented in works concerning
enzyme modulation via MMA [24] – [26].
Device compatibility
The versatility of the apparatus designed by our lab becomes evident when one
considers that, in all of the works listed in the previous paragraphs, the mode and strength of
the magnetic field required can be generated by this one device, that will be presented next.
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Forces exerted by MNPs
When studying the magneto-mechanical effect, it is commonplace to assume that the
phenomenon can be described in great accuracy by an examination of the Stokes fluidic drag
force and the magnetic forces and is generally considered an acceptable approximation [27]. In
2007, Furlani et al. came up with a mathematical model for predicting the movement of carriers
inside the vascular system. By assuming the magnetic and viscous drag forces to be dominant,
the group derived mathematical expressions for the magnetic force exerted by an ensemble of
nanoparticles among the rest. The principle of that model was taking the effective dipole
moment approximation, that is, replacing a particle by an equivalent point dipole positioned in
the center of the particle [28]. Since then, this angle of “attack” has gained significant
popularity, with research groups following the avenue proposed in 2007 in papers published as
recently as 2021 [29] ̶ [31]. Besides the wide approval of this method by the research
community, another important factor that influences the decision of following this approach is
that this was the angle followed by our lab for this device’s predecessor. Subsequently,
following the same approach would provide insightful information with regards to the
improvements exhibited in the newer apparatus. In the box below, a brief overview of the
mathematical model that was developed by Furlani et al, specifically related to the calculation
of the magnetic forces, is provided.
Assuming linear magnetization, for spherical particles, the following are true for the
region below saturation:
⃗⃗ = 𝜒𝑝 𝐻
⃗ 𝑖𝑛 & 𝐻
⃗ 𝑖𝑛 = 𝐻
⃗𝑎 −𝐻
⃗ 𝑑𝑒𝑚𝑎𝑔
𝑀
⃗ 𝑑𝑒𝑚𝑎𝑔 =
𝐻
⃗⃗
𝑀
3
(1.1)
(1.2)
where the annotations in, a and demag correspond to the field constituents inside the
material, the applied and the demagnetizing one, respectively. Eq. (1.2) is valid for
spherical uniformly magnetized particles, that are within the scope of this work. From the
equations above magnetization M can be written as:
⃗⃗ =
𝑀
3𝜒𝑝
⃗
𝐻
3 + 𝜒𝑝 𝑎
(1.3)
From here, the magnetic force exerted by one particle after some thorough analysis, is
proven to be given by the expression:
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𝐹𝑚𝑝 = 𝜇0 𝑉𝑝
3𝜒𝑝
⃗ ∙∇
⃗ )𝐻
⃗𝑎
(𝐻
3 + 𝜒𝑝 𝑎
(1.4)
4
,where 𝑉𝑝 = 3 𝜋𝑟𝑝3 is the volume of a spherical particle with 𝑟𝑝 being its radius and χp the
susceptibility of the particle.
Equation (1.4) can be split into three one-dimensional equations along the axes of a
cartesian coordinate system for each one of ⃗⃗⃗⃗⃗⃗⃗
𝐹𝑚𝑝 constituents.
3𝜒𝑝
𝜕𝐻𝑎𝑥
𝜕𝐻𝑎𝑥
𝜕𝐻𝑎𝑥
𝑦
[𝐻𝑎𝑥 (𝑥, 𝑦, 𝑧)
+ 𝛨𝛼 (𝑥, 𝑦, 𝑧)
+ 𝐻𝑎𝑧 (𝑥, 𝑦, 𝑧)
]
3 + 𝜒𝑝
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝑦
𝑦
𝑦
3𝜒𝑝
𝜕𝛨𝛼
𝜕𝛨𝛼
𝜕𝛨𝛼
𝑦
𝑦
𝑥 (𝑥,
𝑧
𝐹𝑚𝑝 = 𝜇0 𝑉𝑝
[𝐻
𝑦, 𝑧)
+ 𝛨𝛼 (𝑥, 𝑦, 𝑧)
+ 𝐻𝑎 (𝑥, 𝑦, 𝑧)
]
3 + 𝜒𝑝 𝑎
𝜕𝑥
𝜕𝑦
𝜕𝑧
3𝜒𝑝
𝜕𝐻𝑎𝑧
𝜕𝐻𝑎𝑧
𝜕𝐻𝑎𝑧
𝑦
𝑧
𝐹𝑚𝑝
= 𝜇0 𝑉𝑝
[𝐻𝑎𝑥 (𝑥, 𝑦, 𝑧)
+ 𝛨𝛼 (𝑥, 𝑦, 𝑧)
+ 𝐻𝑎𝑧 (𝑥, 𝑦, 𝑧)
]
3 + 𝜒𝑝
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝑥
𝐹𝑚𝑝
= 𝜇0 𝑉𝑝
(1.5)
⃗𝑎.
, where the terms in (1.5) are the -x-, -y- and -z- components of the vectors 𝐹𝑚𝑝 & 𝐻
The equations above, can be altered by replacing the fraction that includes the particle’s
susceptibility with a factor of 3. This approximation is only valid for ferromagnetic
particles whose susceptibility is typically two or three orders of magnitude larger than 3
(χp ≫ 1) and therefore:
3𝜒𝑝
≈3
3 + 𝜒𝑝
(1.6)
Based on the assumptions accepted to end up with the expressions for the forces exerted
by one particle, the total force of N non-interacting or negligibly interacting particles will
simply be:
𝐹𝑡𝑜𝑡 = 𝑁 ∙ 𝐹𝑚𝑝
(1.7)
The equations for the magnetic force – with the approximation (1.6) – will be utilized
in the computational model that was designed in COMSOL 3.5a in the following chapter.
Before introducing the process of modelling the apparatus for a numerical analysis, the physical
properties and details of the proposed setup need to be highlighted.
Senior Thesis
Dimitrios Papadopoulos
THEORETICAL BACKGROUND & APPARATUS INTRODUCTION
11
Setup of the Novel Apparatus
In the proposed setup, an arrangement that is inspired by the concentric Halbach
cylinder structure (Figure 1.1) is presented. What differentiates the two configurations is the
utilization of one cylindrical array surrounding the aperture cylinder, while the second array is
of identical radius and is positioned along the vertical axis that intercepts the center of the -xyplane that the first array defines, forming a normal angle. To hold the arrays in place two 3D
printed disks are utilized.
The disks as well as their mounts, were designed in AutoCAD Inventor and were
proceeded to be 3D printed using a typical polymer filament. The disks were designed to host
commercial cubic permanent magnets (the specifications of whom are presented in the
following paragraphs), while two legs were printed on either side of the disk with nooks at
equal distances and near the height of the disk. Those alcoves are utilized to mount a bracketshaped table on top of which a typical petri dish (3.5 cm) can be placed for in vitro testing. The
printed disks are screwed on two flush wood planks that are subsequently connected with four
screw rods, in a manner that makes each disk the mirror of the other. The top wooden platform
is secured with bolt nuts in both directions allowing for an adjustable distance between the
disks, as well as a tunable slope (a feature that won’t be further investigated in this paper). In
the experiments that are described in the third Chapter, the distance of the disks is set to 10.3
cm. The features described in this paragraph can be seen on Figure 1.4 a).
Each disk can host a total of eight cubic magnets arranged in a way that every magnet
is equidistant (5 mm) to its neighboring magnets and that the inner side of the magnets inserted
by the side of the disk is 2,5 cm away from the center. Finally, a circle-shaped hole, with a
radius of 2 cm, that goes as deep as the magnets’ height, is created in the center to offer a wider
range of -z- levels in the single disk setup.
Magnets – Motors
The magnets (Figure 1.4 c) placed in the 2 x 2 x 2 cm3 slots are permanent NdFeB N45
magnets sold commercially by Magnethandel [32]. The magnetic and physical properties are
listed in Table 1.1, that can be found on Appendix A.
The motors operate on DC and their frequency of rotation, can be adjusted by inputting
a different voltage. More specifically, a range of 3 – 12 Volts is operational resulting in a range
of 0 – 12 Hz in rotational motion.
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Dimitrios Papadopoulos
12
CHAPTER 1
a)
b)
d)
c)
Figure 1. 4: The experimental setup in the designing platform and in real life. a) Complete
composition of disk screwed on wooden platforms and connected with metal screw rods. b)
model of one disk in the CAD environment. c) Permanent magnet placed in the cubic host slots.
d) Photograph taken of the setup, where all the parts have been assembled.
Three-axes Hall probe magnetometer
The mapping of the Magnetic field’s flux
density B is executed with the aid of Metrolab’s
three-axis magnetometer, THM1176-MF model
[33] that offers a range of 100 mT extending up to
3 T. The instrument has an accuracy of ± 1% of the
value read, or 0,1 mT depending on which one is of
larger magnitude. For a frequency of rotation
around 6,67 Hz the magnetometer provides more
Figure 1. 5: Axis orientation and morphology of the
Hall sensor. Because of its sensitivity, being an
electric component, it is covered with the black
plastic cover seen on the schematic.
than sufficient “time resolution” given that its acquisition rate can comfortably capture 100
points per period ( 0,15 s). In Figure 1.5, a schematic of the instrument and its axes orientation,
from the model’s manual, is illustrated.
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Dimitrios Papadopoulos
THEORETICAL BACKGROUND & APPARATUS INTRODUCTION
13
Scope of this thesis
In this work, a novel setup of two Halbach arrays in rotatable 3D printed polymer disks
is presented. This effect will be mainly looked into for deformation of sub- or supra- molecular
structures, specifically regarding the required magnetic conditions. Emphasis will be given in
demonstrating the versatility (modes and types of magnetic fields, generated amplitude) of the
proposed apparatus, by means of a numerical analysis with the aid of a FEM-based software,
namely COMSOL v. 3.5a Multiphysics. Following the computational modeling, the
convergence of the materialized device will be investigated to establish that the device
functions soundly. Finally, the potential applications that it could facilitate, according to the
computational model are discussed.
Considering the workflow described, it becomes evident that this thesis aims to
primarily map the magnetic field that the proposed setup generates and secondarily, to establish
its aptness for magneto-mechanical actuation of various mechanically or magnetically sensitive
functions within cells (e.g., ion channel activation), for deforming malignant sub- or supramolecular structures and lastly, for drug delivery applications. Besides the ideal external
magnetic conditions, the appropriate type of MNPs is additionally needed for the magnetomechanical effect to manifest. The process of identifying the size, shape, and chemical
compound of the MNPs is beyond the scope of this paper and the compatibility assessment will
grounded on relevant literature that have already conducted such studies.
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Senior Thesis
Dimitrios Papadopoulos
A MAGNETOMECHANICAL APPROACH IN BIOMEDICINE
15
Chapter 2
Computational Modeling and Numerical Analysis
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Dimitrios Papadopoulos
16
CHAPTER 2
Computational Modeling
Calculating the theoretical values of the magnetic field in the space between the two
disks is an arduous time-consuming venture, if attempted to be done, even with the aid of
computing power. An easier approach is to code the problem at hand into a finite element
method-based software. COMSOL v. 3.5a Multiphysics belongs in this category with the
advantage of having physics libraries that can completely describe the phenomena taking place
in this setup. In order to successfully emulate the behavior of the setup, the magnetic as well
as the mechanical properties of the apparatus need to be described.
Magnetic properties
Utilizing the information provided by the supplier – listed in Table 1.1 – the two eightmagnet arrays are designed based on their geometric properties (20 mm thick cubic magnets)
and subsequently the listed magnetic properties are inputted. The last task is executed by
assigning the specified values to the magnetic parameters. Specifically, the module utilized for
this numerical analysis solves the differential equation below for V m , i.e., the scalar magnetic
potential.
(2.1)
⃗ ∙ (𝜇0 𝜇𝑟 ⃗∇𝑉𝑚 − 𝐵
⃗ 𝑟) = 0
−∇
,where Br is the magnetic field remanence, specified to be approximately 1.345 T, and 𝜇𝑟 the
relative magnetic permeability equal to 1.06, a value taken from the literature. In our setup the
polarities of the magnets are rotated by 90 °. This alternation of the magnetization’s direction
̂ → −𝒚
̂ → −𝒙
̂ →𝒚
̂). With this
translates to a circular alternation of the unitary vector of Br ( 𝒙
information both physical and boundary conditions are described by the following equations.
Boundary conditions
A cylinder surrounding the magnet compositions is set as boundary of magnetic insulation; this
reduces the computing power required to solve the problem. In other words, the cylinder
contains the condition:
(2.2)
⃗ =0
𝑛⃗ ∙ 𝐵
The intersurface between the NdFeB magnets and the environment in the large cylinder is set
with a continuity boundary condition. The condition is expressed by the equation (2.3):
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COMPUTATIONAL MODELING & NUMERICAL ANALYSIS
17
(2.3)
⃗1−𝐵
⃗ 2) = 0
𝑛⃗ ∙ (𝐵
⃗⃗⃗⃗2 are the magnetic flux densities inside and outside the intersurface.
, where ⃗⃗⃗⃗
𝐵1 and 𝐵
Subdomain conditions
The cubes representing the permanent magnets, are set up to contain the magnetic
characteristics listed by EarthMag GmbH. In detail:
(2.4)
⃗ = 𝜇0 𝜇𝑟 𝐻
⃗ +𝐵
⃗𝑟
𝐵
The environment surrounding the magnets is given the magnetic properties of air, which can be
considered identical to a vacuum space in the sense that:
𝜇 𝑣𝑎𝑐=1
⃗ = 𝜇0 𝜇𝑟 𝐻
⃗ ⇒𝑟
𝐵
(2.5)
⃗ = 𝜇0 𝛨
⃗
𝐵
Geometry Rotation
The simulation of the disk rotation requires two COMSOL libraries: “3D –
Magnetostatics, No Currents” and “Moving Mesh (ALE)”. The concept of the Mesh simply has
to do with the finite element aspect of the program. The area is divided in small geometric
shapes that cover the studied area, with each shape being considered as one point. Those
dividents have to be able to follow the instructed motion to simulate the rotation of the modeled
objects. The combination of these two libraries can be found as “Rotating Machinery” inside
the AC/DC Module. Simulating the rotation of the disks is made possible with a prescribed
displacement of the mesh that is expressed by the following set of differential equations:
𝑑𝑧 = 0
𝑑𝑥 = cos(−2𝜋𝑓𝑡) ∗ 𝑋 − sin(−2𝜋𝑓𝑡) ∗ 𝑌 − 𝑋
𝑑𝑦 = sin(−2𝜋𝑓𝑡) ∗ 𝑋 + cos(−2𝜋𝑓𝑡) ∗ 𝑌 − 𝑌
(2.6)
, where X and Y are the initial positions of a geometric shape and f is the frequency of rotation.
The expressions above describe a clockwise rotation with constant angular velocity. The
models designed for one and two disks as well as a qualitative image of the generated slice
plots for different -z- levels are illustrated in Figure 2.1.
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18
CHAPTER 2
Figure 2. 1: Computational models. The first column shows the single and double Halbach array setups.
Permanent magnets are colored purple while the sample holder alcove is represented by the lime-colored
cylinder. On the second and third column different angle views of the slices of magnetic flux density is calculated.
Colors near red coincide with higher values of magnetic field B, while the opposite is true for colors closer to
blue. The large cylinder surrounding the magnets delimits the volume in which the magnetic flux density is
calculated.
Static Study
The model described in the previous section can be used to calculate the magnetic flux
density on any point within the bounded cylinder. The depiction of the field in slice plots of
constant -z- is an instructive graphing approach as it illustrates the intuitionally attenuating
value with increasing distance from the disks, while additionally indicating the distance where
the field generated by the disk further away from the plane, becomes non-negligible. The
analysis will be divided into two main categories based on the existence or lack of rotation,
which will subsequently be divided into a single and double disk study1. For all the groups the
slice plots presented here are at most 3 cm away from one of the two arrays.
In this section we emulate the magnetic field under no motion of the magnet configurations.
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Dimitrios Papadopoulos
COMPUTATIONAL MODELING & NUMERICAL ANALYSIS
19
Magnetic Flux Density B
The generated field by the complete configuration is presented in Figures 2.2 and 2.3.
The plots presented concern planes positioned 1, 2 and 3 cm away from the bottom (Fig. 2.2)
and top (Fig. 2.3) disk. Please note the change of the color bar limits for each graph.
b)
y-distance from the center (m)
a)
y-distance from the center (m)
c)
Figure 2. 2:
Numerical analysis of the
magnetic flux density B for
two disks in a static study
near the bottom disk. b) 1
away from the bottom disk.
a) Schematic illustration of
the hypothetical slices that
are graphed in b), c) & d)
d)
y-distance from the center (m)
cm, c) 2 cm and d) 3 cm
x-distance from the center (m)
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20
b)
y-distance from the center (m)
a)
CHAPTER 2
y-distance from the center (m)
c)
Figure 2. 3:
Numerical analysis of the
magnetic flux density B for
two disks in a static study
near the top disk. b) 1 cm, c)
2 cm and d) 3 cm away from
a) Schematic illustration of
the hypothetical slices that
are graphed in b), c) & d)
d)
y-distance from the center (m)
the top disk.
x-distance from the center (m)
From the graphs in the figures above, a convergence of the homogeneity area (with the
strongest magnetic field) towards the center, as the distance from the disks increases, is
observed. This phenomenon further extends the range of applications, due to the variation of
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Dimitrios Papadopoulos
COMPUTATIONAL MODELING & NUMERICAL ANALYSIS
21
the “red surface” area providing with larger magnetic gradients to cell cultures and/or animals
if positioned properly.
In the single disk study, the magnetic field inside the sample holder (Fig. 2.1) and on
top of the disk2 is demonstrated. The derived graphs can be found on Figures 2.4 and 2.5
z = 0 cm
c)
Figure 2. 4:
Numerical analysis of the
magnetic flux density B for
an individual disk in a static
study inside the sample
holder – hence the smaller
radius – at b) 0 cm, c) 0.5
cm and d) 1 cm from the
bottom
of
the
holder.
a) Schematic illustration of
the
theoretical
planes
graphed in b), c) & d).
y-distance from the center (m)
b)
d)
y-distance from the center (m)
a)
y-distance from the center (m)
respectively.
x-distance from the center (m)
Senior Thesis
Dimitrios Papadopoulos
CHAPTER 2
Figure 2. 5: Numerical analysis of
B (mT) for a single disk study 2
mm above the top surface of the
magnets. Next to the graph, a
schematic of the position of the
slice.
y-distance from the center (m)
22
x-distance from the center (m)
From the morphology of the contour plots in the single disk study the following are
observed:
•
Inside the sample holder (Fig. 2.4) the field is extremely homogenous with the
exception at the corners along the y = x and y = -x direction. These spikes are most
likely due to them having the minimal distance from a magnet compared to rest of the
holder area.
•
Above the top surface of the Halbach array (Fig. 2.2, 2.3 & 2.5) and along the circle
with radius between 3 and 4 cm, the field strength transitions are significantly steeper.
This remark will be verified in the next sections, where this numerical analysis will be
repeated for the gradient of the magnetic flux density.
•
When the two disks are 10.3 cm away from each other, the fields capacity reaches the
maximum value of magnetic strength at around 0.5 T. Thus, the range of magnetic fields
required for the prescribed biomedical applications, on Chapter 1, can most definitely
be achieved, at least on a theoretical level.
Gradient of Magnetic Field ∇B
Besides the magnetic field strength, the gradient of the magnetic flux density contributes
significantly to the magnitude of the generated magnetic forces. This is expressed by Equation
(1.4) and it is true for the effective dipole approach, based on which the numerical analyses in
this work are computed. Following the same road map that was presented in the calculations
⃗⃗ ∙ 𝑩
⃗⃗
of the magnetic flux density B in the previous section, the corresponding slice plots for 𝛁
are illustrated in Figures 2.6 – 2.9 below.
Senior Thesis
Dimitrios Papadopoulos
COMPUTATIONAL MODELING & NUMERICAL ANALYSIS
b)
y-distance from the center (m)
a)
23
y-distance from the center (m)
c)
Figure 2. 6:
Numerical analysis of the
magnetic field gradient ∇
B for two disks in a static
study near the bottom
disk. b) 1 cm, c) 2 cm and
d) 3 cm away from the
a) Schematic illustration
of the hypothetical slices
that are graphed in b), c)
& d).
d)
y-distance from the center (m)
bottom disk.
x-distance from the center (m)
Senior Thesis
Dimitrios Papadopoulos
24
b)
y-distance from the center (m)
a)
CHAPTER 2
y-distance from the center (m)
c)
Figure 2. 7:
Numerical analysis of the
magnetic field gradient ∇ B
for two disks in a static study
near the top disk. b) 1 cm, c)
2 cm and d) 3 cm away from
the top disk.
a) Schematic illustration of
are graphed in b), c) & d).
d)
y-distance from the center (m)
the hypothetical slices that
x-distance from the center (m)
The observation that was made in the magnetic flux density numerical analysis can be now
validated from the larger gradient magnitude areas, that are located mainly between 3 to 4 cm
from the center of the disk. The Single Disk study is portrayed in Figures 2.8 and 2.9.
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COMPUTATIONAL MODELING & NUMERICAL ANALYSIS
25
b)
y-distance from the center (m)
a)
z = 0 cm
y-distance from the center (m)
c)
Figure 2. 8:
Numerical
analysis
of
the
Magnetic Field Gradient ∇ B
for an individual disk in a static
study inside the sample holder –
hence the smaller radius – at b)
0 cm, c) 0.5 cm and d) 1 cm
from the bottom of the holder.
a) Schematic illustration of the
theoretical planes graphed in b),
d)
y-distance from the center (m)
c) & d).
x-distance from the center (m)
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Dimitrios Papadopoulos
CHAPTER 2
Figure 2. 9: Numerical analysis of ∇B
for a single disk study 2 mm above the
top surface of the magnets. Next to the
graph, a schematic of the position of the
slice.
y-distance from the center (m)
26
x-distance from the center (m)
The color distributions in the gradient plots are a very instructive depiction, providing
a more transparent look in the distinct magnetic areas in their “behavioral” patterns.
Particularly the homogeneity of the aperture circle in the center, one of the most characteristic
features of a Halbach configuration, is expressed via the extremely low values (white color) of
the gradient in the respective area. This trait is observed in the middle (both radially and
vertically) of an arrangement and therefore, in this setup, three spaces fulfil this condition: in
the middle of the top and bottom disk and, interestingly, in the plane that marks the dichotomy
of the distance between the two disks. The third area exists due to the cylindrical-like
arrangement of the two arrays. Even though the magnets do not fill the conceivable cylinder
volumetrically, the symmetry of the geometry is approximated by the two disks, resulting in a
homogenous field, yet not as consistent as the other two regions. Those spatially blank areas
should not be confused with the low values near the edge of the bounded environment as they
are associated with near zero magnetic flux densities.
Aside from some smaller petri dishes with a diameter around 30 mm, the magnetic
conditions inside the sample holder cannot be used in practice and are mainly displayed to
exhibit the homogeneity that comes as a result of the constructive interference of the alternating
directions of the NdFeB field.
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COMPUTATIONAL MODELING & NUMERICAL ANALYSIS
27
Magnetic Forces Fm
Having collected datasets for the magnetic field and its gradient, the process of
calculating the magnetic forces is simplified. As Equation 1.5 dictates, the only parameters left
to input, are the volume of the spherical nanoparticle at hand and the magnetic susceptibility
that is correlated to the chemical compound at a given temperature and size. To demonstrate
the potential of the apparatus for magnetic cell triggering (MCT), assuming room temperature,
three sizes (20, 40 & 80 nm in diameter) of the magnetite phase of iron oxide nanoparticles,
where they exhibit a ferromagnetic behavior [34], are inspected; this is a prerequisite set by
(1.6). Indicatively in Figure 2.10, the spatial distribution of the magnetic force per particle, is
presented for a plane 2 mm away from a disk, that is for a plane containing the maximal force
magnitudes. The complete analysis for the different sizes and disk distances can be found in
Appendix B (S1-S3).
Comparison to literature
For a qualitative numerical evaluation, one can look through the literature, where many
in vitro assays have been conducted with different concentrations and core diameters. It can be
said that, for a typical study of MNPs in the size range studied here, typical N values are in the
order of thousands or more. Taking the scenario of an ensemble of 103 MNPs, for the minimal
array distance, the setup can generate from tens up to hundreds of pN depending on the core
diameter of the studied particle. This is easily inferred from the Figure 2.10, as the color bar
conveniently includes a 10-3 multiplier.
The process of gauging the setup’s capabilities for biomedical applications is concluded
with a comparison of the recorded force thresholds for mechano-sensitive cellular functions
and for integrity compromise (membrane lysis, cytoskeleton deformation, etc.). Force
thresholds registered in literature are presented in Table 2.1, which has been created based on
previous works of our colleagues Dr. Makridis and Dr. Maniotis as well as on the collective
efforts of Nikitin et al. in a review article [13] [14] [35].
From the threshold values presented in Table 2.1, it becomes evident that for an
ensemble of 103 magnetite nanoparticles, functions and operations requiring forces of the order
of 100 pN can easily be activated/initiated as the generated forces for this hypothetical scenario
are well over that value. Generating forces in the order of nN however, would require a larger
concentration of MNPs in the targeted site, or alternatively, if it’s physically possible,
nanoparticles of greater size.
Senior Thesis
Dimitrios Papadopoulos
CHAPTER 2
c)
y-distance from the center
b)
y-distance from the center
a)
y-distance from the center
28
x-distance from the center
Figure 2. 10: Spatial distribution of magnetic force Fm per single magnetite nanoparticle,
2 mm above the bottom Halbach array (bottom disk). a) cyan, b) magenta and c) yellow
colorations correspond to a diameter of 20, 40 and 80 nm respectively. The analysis is
executed with both arrays contributing to that plane.
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COMPUTATIONAL MODELING & NUMERICAL ANALYSIS
29
In addition to the exerted forces in a more analytical study the possibility of induced
torques, due to the shape of the nanoparticles themselves, or the agglomerate formed, have to
be considered. Because this subject relies heavily on the choice of MNPs, this will not be
further considered here. However, considering the relevant works on this mechanical
manifestation that were presented in the first chapter, it becomes clear that our device can
undoubtedly be utilized for either or both manifestations of mechanical stress (i.e., forces and
torques).
Table 2.1
Biological effects induced by the magneto-mechanical effect and their threshold forces.
Effects
Force Threshold (pN)
Reference
Diffusion of ions and
biologically relevant
molecules in solutions
102 – 103
[36]
Magnetically assisted cell
migration and positioning
102 – 103
[37]
Endocytosis (magnetically
mediated)
1 – 102
[38]
102 – 103
[39]
Activation of various ionic
channels
0.2–10
[40] [41] [42]
Antibody-antigen interaction
10–100
[43] [44]
Cancer cell-selective
treatment through
cytoskeletal disruption
3
[45]
Lysosomal Membrane
disruption inducing
apoptosis (hydrolase
leakage)
~ 10
[46]
102 – 103
[47]
80
[48]
Change differentiation
pathway and gene
expression
Cell swelling
Remote control of αchymotrypsin activity
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CHAPTER 2
Rotational Study
In the previous section a numerical simulation of the generated magnetic field and its
implications for ferromagnetic nanoparticles with a 50 nm radius were explored. The disks
were studied as one configuration and separately in a single disk assay. The same analysis is
conducted for one or two rotating Halbach arrays. Beyond the obvious differentiations of the
two studies (explained in Computational Modeling), this investigation focuses on points and
the temporal evolution of the magnetic conditions (B, ∇ B, Fm) in those coordinates. Here, we
present the time dependent fields at the center and at the circumference of a conceived cylinder
with 5 mm larger radius than the sample holder, specifically the -y- axis intercept1. The points
will, from now on, denoted as C for the central point and R for the point in the periphery.
Time-dependent Magnetic Flux Density
One of the most significant benefits of the proposed assembly is the ability to produce
a strongly homogenous signal at some areas while simultaneously providing an alternating
⃗⃗ | ≠ 0 2 (Pulsed Magnetic Field (PMF) and Rotating Magnetic Field
mode at points with |𝒓
(RMF) modes can be achieved by distancing the setup from a Halbach array i.e., removing
some of the permanent magnets and modifying their polarity directions; these modes will not
be explored any further in this work). This section aims to demonstrate both aspects of the
setup. For that to be achieved, the analysis of the magnetic field as a function of time will be
executed for the two most informational and useful coordinates of the -xy- plane. Informational,
because they describe both the homogenous and the rotational areas and useful, since they are
inside the aperture circle, in which the biological samples will typically be placed. The
rotational analysis is again split in a single and double array study.
In Figure 2.11, the data from the complete configuration are illustrated, while Figure
2.12 corresponds to the simulated rotation of an individual disk. All graphs are restricted to the
length of one period, calculated by the angular velocity of the motors (T = 0.15 s).
1
2
In other words, the point with coordinates (x, y, z) = (3 cm, 0 cm, z cm)
The magnitude of r refers to the radial component in a spherical coordinate system (r, θ, φ)
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COMPUTATIONAL MODELING & NUMERICAL ANALYSIS
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Figure 2.11: Time evolution of the magnetic flux density for the two-disk setup. The plots denoted
as C (1st column) correspond to simulations at the center of the disk whereas, plots denoted as R (2nd
column) describe B at the periphery of the conceived aperture cylinder (described above). The first
row contains the numerical calculations for 0.2 (black), 1 (gold), 2 (cyan) and 3 (magenta) cm above
the bottom array while the second-row graphs regard planes 0.2 (black), 1 (gold), 2 (cyan) and 3
(magenta) cm below the top disk.
Figure 2.12: Time evolution of the magnetic flux density for the single disk study. The plot
denoted as C (1st graph) corresponds to the simulation at the center of the disk whereas, the
plot denoted as R (2nd graph) describe B at the periphery of the conceived aperture cylinder
(described above). The numerical analysis illustrated is for 0.2 (black), 1 (gold), 2 (cyan)
and 3 (magenta) cm above the array.
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As predicted the behavior of the points C and R align perfectly with a spatially
homogenous and a temporally alternating magnetic field. The characteristics of the displayed
behaviors are listed in Table 2.2, in Appendix A.
Time-dependent Gradient of Magnetic Field
Similarly to the static study, the gradient of the magnetic flux density B is now presented
for the points R and C. In this scenario, the time dependency of those points is computed for
the duration of one period (0,15 s). The curves are presented in Figures 2.13 and 2.14 – for the
double and single array configurations respectively – using the same format that was used in
the previous section.
Figure 2. 11: Time evolution of the magnetic field gradient for the two-disk setup. The plots
denoted as C (1st column) correspond to simulations at the center of the disk whereas, plots
denoted as R (2nd column) describe ∇B at the periphery of the conceived aperture cylinder
(described previously). The first row contains the numerical calculations for 0.2 (black), 1
(gold), 2 (cyan) and 3 (magenta) cm above the bottom array while the second-row graphs
regard planes 0.2 (black), 1 (gold), 2 (cyan) and 3 (magenta) cm below the top disk.
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Figure 2. 12: Time evolution of the magnetic field gradient for the single disk study. The
plot denoted as C (left graph) corresponds to the simulation at the center of the disk
whereas, the plot denoted as R (right graph) describe B at the periphery of the conceived
aperture cylinder (described previously). The numerical analysis illustrated is for 0.2
(black), 1 (gold), 2 (cyan) and 3 (magenta) cm above the array.
The magnetic field strength reaches comfortably the order of 400 mT, an observation
that leads effortlessly to the deduction, that the device can easily participate in the majority of
applications, that aim to harness the magneto-mechanical effect in the biomedical field. It is
important to note that, on the periphery of the conceived cylinder, the data have a sinusoidal
form, but because of the logarithmic scale they are deformed in the previous Figures. In Chapter
3, the same graphs are presented in a linear scale (for experimental validation) rendering this
distortion evident. The characteristics of the displayed behaviors are listed in Table 2.3, in
Appendix A.
Time-dependent magnetic forces
To evaluate the device’s potential for triggering cellular functions, the magnetic forces
exerted on a single ferromagnetic nanoparticle will be calculated. For this assessment we use
the example of the magnetite phase, being one of the most researched iron oxide nanoparticles
in nanobiotechnology. In order to be within the frame of the Furlani model, the nanoparticles
must be in a size range that is not correlated to a superparamagnetic region. For magnetite this
condition is met for nanoparticles with a diameter greater than 20 nm. Doubling as a
demonstration of the magnetic force variation based on size, a numerical analysis is conducted
for diameters equal to 20, 40 and 80 nm. These values are considered to be well inside the size
region where the ferromagnetic behavior emerges (single and multi-domain regions). The
signal is investigated for the previously introduced points C and R, and the Fm(t) curve for the
three sizes and various distances near the bottom disk is drawn on Figure 2.13.
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CHAPTER 2
Figure 2. 13: Time evolution of the Magnetic Force per magnetite nanoparticle Fm for the
complete configuration near the bottom disk. The curves are calculated for three diameters within
the ferromagnetic region of the iron oxide MNPs, as illustrated on the top left corner of each
graph. The annotations R and C correspond to the points with coordinates (3 cm, 0) and (0, 0)
respectively.
The characteristics of the temporal behaviors are listed in Table 2.4, in Appendix A.
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Dimitrios Papadopoulos
A MAGNETOMECHANICAL APPROACH IN BIOMEDICINE
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Chapter 3
Experimental Validation
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Magnetic field mapping and Reliability testing
Before moving up to any biological assays, it is imperative that the reliability of the
setup is assessed. To make this possible, the Hall magnetometer that was presented in Chapter
1, will be positioned at different points on the conceived -xy- planes along the -z- axis, as
illustrated in Figure 2. With the three-dimensional map formulated by the COMSOL-mediated
numerical analysis, the experimental measurements are compared with the corresponding
theoretical values. For a more intuitional depiction of the data – COMSOL convergence, the
dataset is imported on top of the slice plots presented in the previous chapter and their color
follows the respective color bar.
The evaluation of the device is again divided into a static and a rotational study. Besides
the obvious reasons behind the former differentiation, given that the components that make up
the device have been 3D printed inside the lab and are not factory graded, it is crucial that a
single disk setup is explored, to identify any discrepancies that are related to the parts
themselves.
Static Study Evaluation
The Hall probe is placed strategically to nine points in each -xy- plane that demonstrate
the versatile behavior of this Halbach configuration. The distance between the points and
between the horizontal planes is decided based on the active volume of the magnetometer to
avoid measurement overlapping. The experimental values are then placed against the
computationally derived contour plot of the respective plane to visualize the agreement of the
experimental values with the COMSOL model.
Single disk assay
For the individual disk the magnetic field’s flux density is measured at four different
heights, that correspond to the -z- levels shown in Figures 2.4 – 5. Figure 3.1 depicts the
experimental data, with the theoretical magnetic morphology of the plane as background. As
the available surface in the holder is significantly smaller, the field is measured in the center
and in the four intercepts of the -x- and -y- axes along the circumference of the inner circle.
Finally, the size of the data covers the instrument’s effective volume projected on the -xy- plane.
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CHAPTER 3
a)
b)
z = 0 cm
y-distance from the center (m)
38
Figure 3. 1:
the magnetic flux density B
y-distance from the center (m)
Experimental validation of
c)
for a single disk in a static
study inside the sample
holder – hence the smaller
radius – at b) 0 cm, c) 0.5
cm and d) 1 cm from the
bottom
of
the
holder.
a) Schematic illustration of
the
theoretical
planes
semi-transparent circle is
the slice plot derived from
the
previous
numerical
analysis, while the squares
correspond
to
the
experimental points with
the active surface of the
magnetometer taken into
consideration.
d)
y-distance from the center (m)
graphed in b), c) & d). The
x-distance from the center (m)
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y-distance from the center (m)
EXPERIMENTAL VALIDATION
x-distance from the center (m)
Figure 3. 2: Experimental validation of B (mT) for a single disk study 2 mm above the top surface of
the magnets. Next to the graph, a schematic of the position of the slice. The semi-transparent circle is
the slice plot derived from the previous numerical analysis, while the squares correspond to the
experimental points with the active surface of the magnetometer taken into consideration.
In Appendix A, Tables 3.1–3.4, the experimental and computational data, including the
standard deviations, are recorded. When examining the deviations from the computationally
predicted magnetic field strengths the following are observed. Firstly, the data with coordinates
P1(0, 0.03) and P2(0, -0.03) on the -xy- planes commonly show the largest deviations. This is
attributed to the highly transitional magnetic field in those positions, that is, the gradient of the
⃗ is significantly larger than any other points measured on the plane,
magnetic flux density ⃗∇ ∙ 𝐵
a fact that can verified by the respective graphs. As a result, inside the area of the datapoint the
magnetic field values exhibit a greater range and consequently, the value captured by the
instrument’s sensor is more likely to diverge from the expected value. Even with this
discrepancy at play however, the errors are under 10 % .
Double disk assay
For the case of both disks mounted on the screw rods a similar approach to individual
disk study is followed. The nine points that were taken in Figure 2.1(d) are measured one two
and three centimeters above the bottom disk and below the top disk. Once again, using the
COMSOL slice plots for these planes, the data points are placed on top of them to gage the
convergence of the experimental measurements. The results of the complete assembly for a
static environment are shown on Figure 3.2 and Figure 3.3 (for data near the bottom and top
disk respectively).
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c)
Figure 3. 3:
Experimental validation of
the magnetic flux density
B for two disks in a static
study near the bottom disk.
b) 1 cm, c) 2 cm and d) 3
cm away from the bottom
disk.
y-distance from the center (m)
b)
a)
y-distance from the center (m)
40
a) Schematic illustration
of the hypothetical slices
d).
The
semi-transparent
circle is the slice plot
derived from the previous
numerical analysis, while
the squares correspond to
the experimental points
with the active surface of
the magnetometer taken
d)
y-distance from the center (m)
that are graphed in b), c) &
into consideration.
x-distance from the center (m)
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b)
41
y-distance from the center (m)
EXPERIMENTAL VALIDATION
y-distance from the center (m)
c)
Figure 3. 4:
Experimental validation
of the Magnetic Flux
Density B for two disks
in a static study near the
top disk. b) 1 cm, c) 2 cm
and d) 3 cm away from
the top disk.
a) Schematic illustration
of the hypothetical slices
&
d).
The
semi-
transparent circle is the
slice plot derived from
the previous numerical
analysis,
while
the
squares correspond to
the experimental points
with the active surface of
the magnetometer taken
d)
y-distance from the center (m)
that are graphed in b), c)
into consideration.
x-distance from the center (m)
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CHAPTER 3
Once again, a larger error can be observed for the critical points P 1 and P2. It is crucial
to keep in mind that as the probe’s distance from the magnets increases, the magnetic field
amplitude exponentially decreases and consequently, the information provided by the deviation
percentage becomes inaccurate, since a 5 mT difference translates to a greater than 10%
deviation in flux densities of the order of 30 mT. The data that fall within that description, in
Appendix A, Tables 3.5–3.10, are highlighted to mark the “invalidity” of the deviation
percentage. Overlooking the percentages for values of the order of ~ 30 mT, we are once again
within 10% of the values derived by the numerical analysis.
Rotational Study Evaluation
By turning on the motors connected to each disk, it is possible to measure the
alternating signal that is generated at any individual point. Because the volume of the
instrument renders the measurement on top of the magnets impossible, in this evaluation the
experimental data are taken 2 and 3 cm away from the surface of each magnet configuration.
The magnetometer is positioned at the coordinates (x, y) = (0, 0), denoted as C, and at the point
R, which corresponds to the coordinates: (x, y) = (3 cm, 0).
Single Disk Assessment
For the single disk inspection, the time evolution of B is recorded at two -z- levels,
specifically 2 cm and 3 cm higher than the surface of the disk3. In Figure 3.5 the experimental
results are compared with the computationally derived curves. In detail, each graph describes
one of the two points R and C, for all -z- levels that this position was evaluated experimentally.
As per the equivalent numerical analysis graphs, cyan corresponds to the nearest to the disk
point and magenta to the second nearest. These curves have been calculated by COMSOL. On
the other hand, blue and red, mark the experimental data and are assigned to the -z- level that
corresponds to their affinitive color.
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Figure 3. 5: Time evolution of the magnetic flux density B for a single disk setup at the
center of the disk (C) and at 3 cm -x- distance from the center of the disk (R). Blue and red
data correspond to the experimental measurements 2 and 3 cm above the disk, while cyan
and magenta mark the corresponding COMSOL simulated AMFs for the respective
coordinates.
For the coordinates that identify with a homogenous, almost time-independent,
magnetic field, the expected amplitude is approached with excellent accuracy by the
experimental data. However, a temporal mismatch between the experimental and the
computational data can be observed (Appendix B, S4). This divergence translates to a deviation
from the 400 rpm (= 41,89 rad/s) angular velocity ω, and consequently, from the 0,15 s period.
This discrepancy can be attributed to the instrument’s spatial resolution that, as explained later,
is converted to a temporal resolution. Another reason could be the motor’s temporal accuracy
being in the order of seconds, which is 3 orders higher than the observed deviation. In a
following section the angular velocity will be revisited to ensure that the these are the only
factors contributing to the phase difference.
Double Disk Assessment
The time-dependent study for both Halbach arrays in the measuring zone, similarly to
the previous section, is conducted at the points C and R. Additionally, for this study, the
measurements are repeated for the equivalent -z- levels below the top disk. Figures 3.6 contain
the time evolution data retrieved for these two points at -z- levels near the bottom and top disk.
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Figure 3. 6: Time evolution of the magnetic flux density B for the two-disk setup at the center
of the disk (C) and at 3 cm -x- distance from the center (R). Blue and red data correspond to
the experimental measurements 2 and 3 cm away from the respective disk, while cyan and
magenta mark the corresponding COMSOL simulated AMFs for the two points.
For the measurements recorded at the two -z- levels near the bottom disk both the
homogenous and the alternating mode seem to be in great compliance with the theoretically
computed time curves. On some occasions, a peak-to-peak difference was observed during one
period (Appendix B, S5). This effect takes non negligible dimensions at great distances from
the bottom Halbach configuration, inducing a maximum of 5 mT difference at the peak of the
signal.
One potential reason for this mismatch would be the underlying standard errors in the
magnetic field measurements. Although this hypothesis has merit, given the deviations of the
data from the static study, it is more likely that another factor is responsible.
It is hypothesized that this discrepancy stems from an infinitesimal difference in height
between the permanent magnets of the bottom array. Given the exponential attenuation of the
signal’s strength as the distance increases (Beer law), this height difference would result in a
notable mismatch of the two peaks. Despite this potential defect of the bottom disk, its
operational impact is next to zero, as the areas that are of importance for biological testing are
unfazed because of their small distances from the suspected disk.
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Error sources
In terms of errors shown on the graphs above, the two main
mechanisms are related to the instrument’s spatial resolution and
its systemic error.
•
Taking into consideration the instrument’s active volume
(demonstrated in Figures 2.1-3 by the size of the experimental
data) projected onto the -xy- plane, when the disks are
rotating an error on the horizontal axis, that is time, emerges.
This -x- error can be calculated with the help of fundamental
Figure 2. 14: The geometry
of the “time error” calculation
geometry like so:
The projection along the -x- axis of the effective magnetometer surface according to the
manufacturer is 0,4 cm. This is the opposite to the angle 𝝋 side (σ) of the right triangle
portrayed on Figure 2.7. The radius of the disk corresponds to the adjacent side (R) and is
equal to 3 cm. Consequently, by propagating the 𝝈𝝋 is:
𝜎𝜑 =
𝑦=0
𝑑
𝑦
[tan−1 ( )] 𝜎𝑦 ⇒ 𝜎𝜑 = 𝜎𝑦 = ±0,2 𝑟𝑎𝑑 3
𝑑𝑦
𝑅
(3.1)
From there the “t-error” can easily be calculated from the angular velocity that is equal to
400 rpm which is the equivalent of approximately 41,89 rad/s. That is:
𝜎𝑡 =
•
1
𝜎 = ±0,005 𝑠
𝜔 𝜑
(3.2)
Concerning the systemic error, the manufacturer lists an accuracy of  1% of reading or
specified resolution (whichever is greater). For the experimental measurements that were
taken to assess the device’s reliability and to calibrate it for future medical application the
 1% is the larger error. Thus, the vertical error bars that appear in the Figures under the
rotational study, correspond to the latter systemic accuracy of the Hall probe 4.
3
The condition y = 0 is true for the coordinates of the point R measured in this study.
4
Please note that the “t-errors” have not been included in the measurements on the center of the disk to
avoid “overcrowding” the graphs.
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CHAPTER 3
Long-term time testing
Having established the agreement of the COMSOL simulated model with the
experimental data of the previous section, one important step towards ensuring the quality of
this novel configuration of rotating Halbach arrays is to carry out one measurement for a
significantly greater time. So far, the datasets length was approximately seven seconds. For the
time-resilience test, a sixty second measurement will be taken to demonstrate the “perseverance”
of the results in the passage of time. Although a one-minute measurement is not nearly enough
to deduce the setup’s reliability for applications like magneto-mechanical cell straining, it is
assumed that since the build does not have many moving parts, any deficiencies will emerge
during one minute of rotation with 400 rpm. From the sixty second database that is collected
three eight-second partitions will be presented in the following analysis: one in the beginning
of the measurement, one in the middle and one in the end. For readers interested in the complete
experiment, please find the full graph in S6, located in Appendix C.
In those 8 second intervals, the time evolution of the magnetic flux density’s amplitude
and the time dependance of the curve’s period are investigated. The set up will be considered
reliable if the amplitude and period of the B-t curve maintain – within reason – their values
through time. The experiment is carried out near the bottom disk at the already investigated
point R with coordinates: (x, y, z) = (3 cm, 0 cm, 4 cm). The results for the three sections are
depicted in Figures 3.7 (0 ≤ 𝑡 ≤ 8 𝑠), 3.8 (26 ≤ 𝑡 ≤ 34 𝑠) and 3.9 (52 ≤ 𝑡 ≤ 60 𝑠).
To begin with, the datasets are plotted along with horizontal dashed lines that
correspond to the minimum and maximum value of those 8 seconds. The deviation from the
min and max values offers an approximate estimate of the amplitude alteration that can be
translated to a quantitative analysis. Each period is then divided in spans of 0,7 seconds, from
which the first, last and middle segments are illustrated in the following Figures. As the data
per period are significantly less (5), in comparison to the previous rotational study (100), this
step provides insight on the curve’s characteristics. To further improve the reader’s experience,
the data are fitted with a simple sinusoidal function (gray dotted line), providing a better
understanding of the sequence of the points.
Finally, the blue vertical dash lines in the 0,7-second graphs signify one period and its
value for that time is written between them.
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a)
b)
c)
d)
Figure 3.7: Segment of the time-resilience test of the amplitude and the period of the magnetic flux
density B (mT) for the first 8 seconds. a) B-t measurements collected with a frequency of 5 points per
period. Magnetic field strength amplitude remains close to its original value. b), c) & d) depict the first,
middle and last 0.8 second partitions of a) to increase clarity regarding period duration. Behind the
points a sinusoidal function is fitted, aiding in identifying the datapoint sequence.
a)
b)
c)
d)
Figure 3.8: Segment of the time-resilience test of the amplitude and the period of the magnetic flux
density B (mT) for the middle 8 seconds. a) B-t measurements collected with a frequency of 5 points
per period. Magnetic field strength amplitude remains close to its original value. b), c) & d) depict the
first, middle and last 0.8 second partitions of a) to increase clarity regarding period duration. Behind
the points a sinusoidal function is fitted, aiding in identifying the datapoint sequence.
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CHAPTER 3
a)
b)
c)
d)
Figure 3.9: Segment of the time-resilience test of the amplitude and the period of the magnetic flux density B
(mT) for the last 8 seconds. A) B-t measurements collected with a frequency of 5 points per period. Magnetic
field strength amplitude remains close to its original value. b), c) & d) depict the first, middle and last 0.8
second partitions of a) to increase clarity regarding period duration. Behind the points a sinusoidal function is
fitted, aiding in identifying the datapoint sequence.
From the results of the graphs presented above, it becomes clear that the amplitude as
well as the period throughout the minute of measuring maintain their magnitude and no
significant deviations are observed. Although the preservation is evident, the fact that the period
is 5 ms smaller than the expected value requires further investigation since the propagated error
(see Eq . 3.2) may not be the only factor impacting the period’s value. To begin with, a Fourier
transformation is utilized to determine the true value of the frequency of disk rotation, using
the complete length of the measurement (i.e. 60 seconds). Following this approach, the period
is calculated by filming the rotation of the disks, so as to exclude the potential error of the
active volume of the Hall probe.
The Fourier Transformation (FT) is implemented for the measurement, illustrated in
Appendix C, and from it a spectrum of frequencies is obtained. One important detail to keep in
mind when inspecting the FT graph is that during on revolution of the disk, the magnetic field
is maximised three times, meaning that a periodic phenomenon with half the duration of one
disk revolution exists. Consequently, the frequencies derived by FT are expected to be two
times greater in magnitude than the actual frequency of rotation. The Fourier transformation of
the 60 second dataset is portrayed in Figure 3.10.
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Figure 3. 10: Fourier Transformation of the time evolution of the magnetic flux density
B(t) applied to the time resilience test (60 s). The vertical axis corresponds to a measure
of intensity that expresses how often each frequency in the waveform is found. In other
words, the larger the amplitude of the frequency, the more prominent it is in the
measurement.
From Figure 3.10, it becomes evident that the disk is almost exclusively rotating with
a frequency of 14.5/2 = 7.25 Hz (keeping in mind that each revolution contains two periods of
magnetic field fluctuation cycles). This value deviates by the expected 6.67 Hz by 8.7 %.
This deviation could potential be attributed to the temporal resolution that the
instrument’s effective volume imposes. The time error calculated in (3.2) propagates to the
frequency implying an error of:
𝜎𝑓 =
1
𝜎 = ± 2.22 𝐻𝑧
𝑇2 𝑇
(3.3)
, which accounts for the frequency mismatch.
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CHAPTER 3
Despite the deviation being within the expected error range, an unbiased measurement
of the rotational frequency is deemed necessary as a safety net. This measurement is executed
with the use of a 30 frames per second (fps) video camera which is placed next to the setup.
The period was measure for 10 rotations and it was calculated as the mean value, giving us a
period of 0.143 seconds or 7 Hz. The deviation thus becomes 3.6 % which is well within the
accepted error range. The reasons related to this innate divergence from the specified 0.15
second period are likely related to the motor’s temporal accuracy, which as mentioned
previously, is in the order of seconds.
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Conclusions
This thesis has explored a novel setup of Halbach arrays that was designed to harness
the magneto-mechanical effect in biomedicine. The apparatus’ versatility was demonstrated by
inspecting two of the possible magnetic field modes, that can be offered simultaneously. Using
the FEM-based software, COMSOL 3.5a Multiphysics, the magnetic characteristics of the
generated field, spatially and temporally were derived through a numerical analysis. The
outcomes of this study highlighted the setup’s capabilities in achieving the necessary conditions
for the magneto-mechanical effect to manifest, while simultaneously providing a visual
depiction, a Magna C(h)arta, of the varying magnetic field strength in different distances
between the two disks. The analysis was subsequently extended to a calculation of the magnetic
force exerted by a spherical particle when exposed to the described magnetic conditions
following an effective dipole approximation commonly employed in relevant works. Our
computational findings for three scenarios of ferromagnetic magnetite nanoparticles suggested
the generation of forces up to hundreds of pN when an ensemble of 103 MNPs successfully
reaches the targeted site. Comparing the order of forces to various cellular functions and tensile
strengths, we concluded the overcoming of said thresholds, rendering the device a more than
capable tool to mediate magneto-mechanical effects that are investigated for biomedical
applications in recent times. At last, the numerical analysis was confirmed by experimental
measurements using a Hall magnetometer for a static and rotational study. The results proved
the device’s reliability showing a great convergence with the computational model, even in the
passage of time.
Although mitigating interactions between the particles and the environment (e.g., drag
forces when the particles are suspended in a fluid) have not been accounted for here, previous
in vitro studies conducted in our laboratory with a similar 3D printed setup, have yielded
successfully selective apoptosis and cell deformation among other effects. Given that this setup
utilizes the Halbach configuration, an improvement compared to its predecessor, dominant
opposing forces, cannot typically negate the generated forces to a point where the threshold at
hand is not surpassed.
Significantly, this work opens up a vast range of applications from magneto-genetics to
cancer therapy. Having demonstrated its reliability and potency, future research entails assays
with biological samples both in vitro and in vivo to attest our findings in nanobiotechnology.
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Footnotes
1
The reason for the secondary distinction lies in the potential utility of a single disk setup,
for applications that require magnetic fields of lower amplitude.
2
When studies are conducted “on top of the array’s surface”, in reality, there is a 2 mm
distance included to account for the thickness of a typical petri dish’s bottom.
3
The choice to measure the field on the two farther points from the disks, stems from them
being measured experimentally, with the greatest accuracy. In other words, the Hall
probe’s position entails less -z- uncertainty when position further away from the disk (due
to its active volume)
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Appendix A: Tables
Table 1.1
Permanent magnet specifications
General Information
Material
NdFeB
Coating
NiCuNi
Magnetizing
N45
Holding force
 23 kg
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Magnetic Characteristics
Remanence
Br
Coercive force
HcB
Coercive force
HcJ
Energy product
BHmax
1.33 – 1.36 T
> 920 kA/m
> 955 kA/m
342-358 kJ/m3
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Rotational Analysis
Table 2.2
Time dependent magnetic flux density characteristics.
Near disk
Point
Min disk distance (cm)
Bmax (mT)
Bmin (mT)
p-p (mT)
Bottom
C
0.2
181.6
181.4
0.2
Bottom
C
1
181.3
117.4
0.8
Bottom
C
2
62
60.6
1.4
Bottom
C
3
33.1
32.5
0.6
Bottom
R
0.2
489.7
193.3
296.4
Bottom
R
1
146.5
80.3
66.2
Bottom
R
2
60.6
33.9
26.7
Bottom
R
3
26.1
18.2
7.9
Top
C
0.2
181.1
180.1
1
Top
C
1
117.5
117.1
0.4
Top
C
2
62.1
58.8
3.3
Top
C
3
32.5
32.1
0.4
Top
R
0.2
515.4
195.4
320
Top
R
1
146.4
82
64.4
Top
R
2
60.2
34.1
26.1
Top
R
3
25.8
18.8
7
1 Disk
C
0.2
181.2
178.9
2.3
1 Disk
C
1
119.4
115.6
3.9
1 Disk
C
2
59.6
58.8
0.8
1 Disk
C
3
30.1
29.8
0.3
1 Disk
R
0.2
478.9
185.3
293.6
1 Disk
R
1
148
80.7
67.3
1 Disk
R
2
62.7
33.2
29.5
1 Disk
R
3
29.2
16.5
12.7
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Table 2.3
Time dependent magnetic flux density gradient characteristics.
Area
Point
Min disk distance (cm)
∇Bmax (T/m)
∇Bmin (T/m)
p-p (T/m)
Bottom
C
0.2
8.23
8.18
0.05
Bottom
C
1
7.88
7.42
0.45
Bottom
C
2
4.5
4.02
0.48
Bottom
C
3
2.29
1.9
0.39
Bottom
R
0.2
45.29
17.59
27.7
Bottom
R
1
18.31
8.42
9.88
Bottom
R
2
5.49
2.89
2.6
Bottom
R
3
2.48
1.14
1.33
Top
C
0.2
8.23
8.18
0.05
Top
C
1
7.88
7.42
0.45
Top
C
2
4.5
4.02
0.48
Top
C
3
2.29
1.9
0.39
Top
R
0.2
74.42
17.54
56.89
Top
R
1
13.67
8.37
5.3
Top
R
2
5.43
2.84
2.59
Top
R
3
1.82
1.14
0.68
1 Disk
C
0.2
8.29
7.9
0.39
1 Disk
C
1
8.59
7.9
0.69
1 Disk
C
2
4.05
3.39
0.66
1 Disk
C
3
2
1.78
0.22
1 Disk
R
0.2
59.6
13.43
46.17
1 Disk
R
1
13.87
8.49
5.38
1 Disk
R
2
5.48
3.11
2.36
1 Disk
R
3
2.27
1.17
1.1
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Table 2.4
Time dependent magnetic force per magnetite nanoparticle of varying size characteristics.
Diameter (nm)
Area
Point
Min disk distance (cm)
Fmax (pN)
Fmin (pN)
20
Bottom
C
0.2
1.20E-4
1.19E-4
20
Bottom
C
1
7.40E-5
7.02E-5
20
Bottom
C
2
2.23E-5
1.96E-5
20
Bottom
C
3
5.99E-6
5.01E-6
20
Bottom
R
0.2
1.23E-3
2.73E-4
20
Bottom
R
1
2.09E-4
5.57E-5
20
Bottom
R
2
2.66E-5
7.84E-6
20
Bottom
R
3
5.11E-6
1.69E-6
40
Bottom
C
0.2
9.57E-4
9.49E-4
40
Bottom
C
1
5.92E-4
5.62E-4
40
Bottom
C
2
1.78E-4
1.57E-4
40
Bottom
C
3
4.80E-5
4.00E-5
40
Bottom
R
0.2
9.83E-3
2.18E-3
40
Bottom
R
1
1.67E-3
4.53E-4
40
Bottom
R
2
2.12E-4
6.27E-5
40
Bottom
R
3
4.09E-5
1.35E-5
80
Bottom
C
0.2
7.65E-3
7.59E-3
80
Bottom
C
1
4.74E-3
4.49E-3
80
Bottom
C
2
1.42E-3
1.25E-3
80
Bottom
C
3
3.84E-4
3.21E-4
80
Bottom
R
0.2
7.86E-2
1.75E-2
80
Bottom
R
1
1.34E-2
3.56E-3
80
Bottom
R
2
1.70E-3
5.01E-4
80
Bottom
R
3
3.27E-4
1.08E-4
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Single Disk – Static Study
Table 3.1
Measurements at the Bottom of the Sample Holder.
x (m)
0.00
0.03
0.00
-0.03
y (m)
0.03
0.00
-0.03
0.00
z (m)
0.00
0.00
0.00
0.00
Hall Probe (mT)
205.0
262.3
206.6
256.5
COMSOL (mT)
207.9
265.1
210.0
256.0
Deviation (%)
1.40
1.06
1.64
0.18
0.00
0.00
0.00
203.8
197.9
3.00
Table 3.2
Measurements 0.5 cm above the Bottom of the Sample Holder.
x (m)
0.00
0.03
0.00
-0.03
y (m)
0.03
0.00
-0.03
0.00
z (m)
0.005
0.005
0.005
0.005
Hall Probe (mT)
210.4
287.0
208.6
286.4
COMSOL (mT)
221.8
287.3
221.7
273.6
Deviation (%)
5.41
0.10
6.27
4.45
0.00
0.00
0.05
216.4
206.6
4.52
Table 3.3
Measurements 1 cm above the Bottom of the Sample Holder.
x (m)
0.00
0.03
0.00
y (m)
0.03
0.00
-0.03
z (m)
0.01
0.01
0.01
-0.03
0.00
0.00
0.00
Senior Thesis
COMSOL (mT)
208.7
266.6
208.6
267.6
Deviation (%)
7.92
1.51
8.84
0.01
Hall Probe (mT)
193.4
262.7
191.7
264.6
0.01
203.8
196.2
3.74
1.13
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Table 3.4
Measurements at the Top Surface of the Magnets.
x (m)
y (m)
z (m)
Hall Probe (mT)
130.0
COMSOL (mT)
152.2
Deviation (%)
0.00
0.03
0.03
0.00
0.00
-0.03
-0.03
0.00
0.02
0.02
0.02
0.02
196.6
129.8
206.7
213.1
152.1
198.6
7.75
14.64
0.00
0.00
0.00
0.05
0.02
170.0
183.0
0.02
0.02
70.0
112.9
66.7
122.3
7.10
4.92
0.05
0.00
0.00
-0.05
0.02
68.3
63.7
7.72
7.29
-0.05
0.00
0.02
120.7
126.6
4.71
14.58
4.11
Double Disk – Static Study: Planes near the Bottom Disk
Table 3.5
Measurements 1 cm above the Bottom Disk.
x (m)
0.05
y (m)
0.00
Hall Probe (mT)
62.8
COMSOL (mT)
68.8
Deviation (%)
8.69
0.00
-0.05
0.03
0.03
35.9
32.1
11.90
-0.05
0.00
0.00
0.05
0.03
0.03
62.5
35.1
67.8
33.9
7.78
3.38
0.00
0.00
0.03
123.7
118.0
4.83
0.03
0.00
-0.03
0.00
-0.03
0.00
0.03
0.03
0.03
146.4
75.5
135.0
153.9
76.7
145.6
4.90
1.60
7.29
0.00
0.03
0.03
78.8
81.5
3.37
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z (m)
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Table 3.6
Measurements 2 cm above the Bottom Disk.
x (m)
0.05
y (m)
0.00
z (m)
Hall Probe (mT)
25.7
COMSOL (mT)
22.2
Deviation (%)
15.55
0.00
-0.05
0.00
-0.05
0.00
0.05
0.04
0.04
0.04
0.04
12.9
26.6
12.1
14.3
22.1
14.4
10.26
20.58
16.53
0.00
0.00
0.04
68.3
62.9
8.55
0.03
0.00
0.00
-0.03
0.04
0.04
56.4
31.9
55.8
35.6
0.99
10.40
-0.03
0.00
0.04
57.2
57.9
1.11
0.00
0.03
0.04
31.2
36.2
13.90
Hall Probe (mT)
11.9
COMSOL (mT)
12.0
Deviation (%)
0.70
5.7
12.2
5.8
7.2
12.5
7.5
21.02
2.16
23.60
Table 3.7
Measurements 3 cm above the Bottom Disk.
x (m)
0.05
y (m)
0.00
0.00
-0.05
0.00
-0.05
0.00
0.05
0.05
0.05
0.05
0.05
0.00
0.00
0.05
33.1
32.5
1.64
0.03
0.00
0.05
25.8
26.2
1.43
0.00
-0.03
0.05
16.7
18.1
7.95
-0.03
0.00
0.00
0.03
0.05
0.05
24.8
19.0
26.4
18.6
6.31
1.80
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z (m)
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Double Disk – Static Study: Planes near the Top Disk
Table 3.8
Measurements 1 cm below the Top Disk.
x (m)
0.05
y (m)
0.00
z (m)
Hall Probe (mT)
58.6
COMSOL (mT)
56.0
Deviation (%)
4.57
0.00
-0.05
0.113
0.113
32.9
28.3
16.39
-0.05
0.00
0.00
0.05
0.113
0.113
54.4
35.5
57.3
30.2
5.01
17.55
0.00
0.00
0.113
108.3
118.3
8.49
0.03
0.00
-0.03
0.00
-0.03
0.00
0.113
0.113
0.113
135.6
77.4
140.6
148.4
80.9
139.9
8.64
4.28
0.48
0.00
0.03
0.113
80.5
86.0
6.44
Hall Probe (mT)
24.3
COMSOL (mT)
21.6
Deviation (%)
12.71
13.4
11.2
19.86
Table 3.9
Measurements 2 cm below the Top Disk.
x (m)
0.05
y (m)
0.00
0.00
-0.05
0.103
0.103
-0.05
0.00
0.00
0.05
0.103
0.103
21.9
12.2
23.5
10.9
6.91
11.29
0.00
0.00
0.103
62.1
60.8
2.03
0.03
0.00
-0.03
0.00
-0.03
0.00
0.103
0.103
0.103
55.0
33.4
60.3
59.6
34.0
56.1
7.76
2.01
7.43
0.00
0.03
0.103
37.7
36.0
4.62
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z (m)
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Table 3.10
Measurements 3 cm below the Top Disk.
x (m)
0.05
y (m)
0.00
Hall Probe (mT)
13.6
COMSOL (mT)
13.0
Deviation (%)
4.62
0.00
-0.05
0.00
-0.05
0.00
0.05
0.093
0.093
0.093
0.093
7.5
13.7
7.6
7.6
13.1
7.3
0.74
4.25
3.71
0.00
0.00
0.093
27.9
32.0
12.84
0.03
0.00
0.00
-0.03
0.093
0.093
23.5
18.7
24.6
19.1
4.60
1.98
-0.03
0.00
0.093
24.8
26.4
6.03
0.00
0.03
0.093
18.1
18.5
2.16
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z (m)
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Appendix B: Supplementary Figures
Y (m)
Y (m)
Y (m)
Magnetic Forces per magnetite nanoparticle, 20 nm in diameter
X (m)
force per magnetite nanoparticle, 20 nm in
diameter. i), ii) & iii) illustrate -xy- planes 1, 2 and
Y (m)
Figure S1: Computational simulation of magnetic
3 cm above the bottom disk respectively. iv), v),
vi) & vii) portray -xy- planes 1, 2, 3 and 0.2 cm
below the top disk, respectively.
X (m)
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Y (m)
Y (m)
Y (m)
Magnetic Forces per magnetite nanoparticle, 40 nm in diameter
X (m)
force per magnetite nanoparticle, 40 nm in
diameter. i), ii) & iii) illustrate -xy- planes 1, 2 and
Y (m)
Figure S2: Computational simulation of magnetic
3 cm above the bottom disk respectively. iv), v),
vi) & vii) portray -xy- planes 1, 2, 3 and 0.2 cm
below the top disk, respectively.
X (m)
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Y (m)
Y (m)
Y (m)
Magnetic Forces per magnetite nanoparticle, 80 nm in diameter
X (m)
force per magnetite nanoparticle, 80 nm in
diameter. i), ii) & iii) illustrate -xy- planes 1, 2 and
Y (m)
Figure S3: Computational simulation of magnetic
3 cm above the bottom disk respectively. iv), v),
vi) & vii) portray -xy- planes 1, 2, 3 and 0.2 cm
below the top disk, respectively.
X (m)
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Figure S4: Magnification of the experimental validation for the point
C(0,0) in the single disk study.
Figure S5: Magnification of the experimental validation for the point
C(0,0) in the two-disk study, near the bottom disk.
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Magnetic flux density B (mT)
Appendix C: 60 second time test
Time t (s)
Figure S6: 60-second-long measurement of the time-dependent magnetic flux density. The measurement is taken at the point R with coordinates: (x, y, z) = (3 cm, 0 cm, 4
cm) with both disks present.
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