Dynamics of clustered magnetic nanoparticles in biomedical applications

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biomedical applications

Academic year 2019-2020

Master of Science in Electromechanical Engineering

Master's dissertation submitted in order to obtain the academic degree of

Counsellors: Dr. ir. Annelies Coene, Dr. Jonathan Leliaert

Supervisors: Prof. dr. ir. Luc Dupré, Dr. ir. Annelies Coene

Student number: 01000932

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Universiteitsbibliotheek Gent, 2021.

This page is not available because it contains personal information.

Ghent University, Library, 2021.

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Preface and thanks

This thesis could not have been completed without the work of some important and inspiring people. It was quite the ride and it took more time than initially in mind. But as I am typing this text right here, the end of this work and my studies is near.

First, thanks to my mentors Annelies Coene and Jonathan Leliaert to guide me through this thesis and for showing a lot of patience towards my work. The knowledge of scientific papers and the discussion about chapter titles are some things to remember.

I would also like to thank Luc Dupr´e, the promotor of this thesis. Although we did not communicate much, I received some good feedback after my intermediate presentation.

The next 3 people have been of great importance to welcome me when I started my engineer-ing studies. Wito Plas for always beengineer-ing the positive, fun factor. I will always cherish our conversations in the student restaurants and it is nice to get to know people whom you can trust. Arne Decadt for being who you are and grabbing drinks together and our conversa-tions about mathematical topics. Joeri Roels for being the quiet dude, but always being a binding factor for our friendship. The four of us should really begin a bowling team by the way!

Special thanks to Rutger Jonghmans for listening to my complaints about my thesis work and always being there when I was down. We will certainly celebrate the end of this thesis work by drinking until we (almost) pass out!

The people from my band, the Nealions1_{. It is always nice to let off some steam while playing}

the drums and being in good company at the same time.

I could not have written this thesis without my mother, as otherwise I would not be here on planet Earth. Thanks to take care of me the past 27 years, that includes cleaning up my vomit and helping me getting down the stairs when I was too drunk to walk.

Zazou and Bert to help me through the night and keeping me warm and safe. My pet Kurt, although you sometimes have an explosive personality.

Last but not least, thank you to my girlfriend Rani Moret. You are my muse, or should I say Musa, that keeps inspiring me every day. I know that I have a complex personality and I sometimes was a pain in the ass, but you kept supporting me through the good and bad times. You cause a flood of happiness, vleermuisje!

To those that I forgot to mention: my sincere apologies, I hope you all know who you are!

1_{https://vi.be/thenealions}

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Abstract

This dissertation offers theoretical insight in the effect of clustering of magnetic nanoparticles on biomedical applications through the dynamical behavior of these particles. The dynamical behavior is a result of the magnetic properties of the particles that cause interacting forces and the forces exerted by fluids surrounding the particles. Two vector properties are directly influenced by this dynamical behavior: the magnetization vector and the anisotropy vector, the latter describing the mechanical rotation of particles.

First some theoretical background concerning the basics of magnetic nanoparticles, their en-ergy properties and possible biomedical applications are introduced.

Ensembles of particles that do not interact are simulated to proof that the software offers reliable results compared to established theory of magnetic nanoparticles.

The obtained results are compared to simulations for interacting particles in simple configu-rations, i.e. chains of 2 and 3 magnetic nanoparticles. The effect of clustering on relaxation of magnetization vectors and on the rotation of particles in clusters is consequently described in a qualitative and quantitative way, ultimately linking these results to consequences for biomed-ical applications. A small outlook chapter summarizes the topics that were not treated in this dissertation, but offer a framework for future research.

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Dynamics of clustered magnetic nanoparticles in

biomedical applications

Nicolas Vanden Berghe

Supervisors: prof. dr. ir.Luc Dupr´e, dr. ir. Annelies Coene, dr. Jonathan Leliaert

Abstract— This dissertation offers theoretical insight in the effect of clustering of magnetic nanoparticles on biomedical applications through the dynamical behavior of these par-ticles. The dynamical behavior is a result of the magnetic properties of the particles that cause interacting forces and the forces exerted by fluids surrounding the particles. Two vector properties are directly influenced by this dynamical behavior: the magnetization vector and the anisotropy vec-tor, the latter describing the mechanical rotation of parti-cles. First some theoretical background concerning the ba-sics of magnetic nanoparticles, their energy properties and possible biomedical applications are introduced. Ensembles of particles that do not interact are simulated to proof that the software offers reliable results compared to established theory of magnetic nanoparticles. The obtained results are compared to simulations for interacting particles in simple configurations, i.e. chains of 2 and 3 magnetic nanoparti-cles. The effect of clustering on relaxation of magnetization vectors and on the rotation of particles in clusters is con-sequently described in a qualitative and quantitative way, ultimately linking these results to consequences for biomed-ical applications. A small outlook chapter summarizes the topics that were not treated in this dissertation, but offer a framework for future research.

Keywords— Magnetic nanoparticles, simulations, hyper-thermia, magnetorelaxometry, magnetic dynamics

I. Introduction

M

AGNETIC nanoparticles [1] have a promising possi-bility to be used in biomedical applications. These applications include hyperthermia useful for cancer ther-apy [2], magnetic particle imaging which can be combined with traditional imaging techniques to enhance resolution of images [3] and as a contrast agent in MRI [4]. The main problem with these applications is that the clustering effect of magnetic nanoparticles is not taken into account but this has an influence on the experimental procedures at hand [5], [6]. Simple models with non-interacting particles are used, often predicting false results and thus causing the in-ability to create an optimal procedure for the biomedical applications.

Effects of clustering have already been described in litera-ture [7], [8] but the exact relaxation paths of the particles have not been explained and simulated yet. These paths offer physical insight in the processes that drive the relax-ation process of magnetic nanoparticles in a cluster, i.e. when their dynamics are influenced by the interaction be-tween each other.These results offer a theoretical insight that can be extrapolated towards more complex cluster types like fractals.

The clustered structures that are described in this disser-tation are chains consisting of 2 and 3 nanoparticles. The dynamical interaction between the particles in a cluster

is the consequence of the Landau-Lifshitz-Gilbert (LLG) equation governing the magnetization and the physical ro-tation of the particles as a result of the surrounding fluid and thermal field [9].

Both processes are coupled and will influence the total dy-namical behavior of clustered structures.

II. Magnetic nanoparticles A. General description

Magnetic nanoparticles are particles with a length scale reaching from 1-100 nm in each dimension. They show superparamagnetic properties, which is a hybrid of param-agnetism and ferromparam-agnetism: the magnetization reaches an asymptotic value MSfor higher fields, but no hysteresis

loop is found as for ferromagnetic materials. B. Magnetic dynamics

During simulations in Vinamax [10], the dynamics of magnetic nanoparticles, i.e. the dynamics of the magneti-zation vector m and anisotropy vvector u, are solved using the LLG-equation [11] and the equation for spatial rotation Θ∂ω ∂t = µ0MSVh γ0 ∂m ∂t +µ0MSm×(Hext+Hdemag)−ξω+τth (1) With Θ the moment of inertia, ω the angular velocity, µ0 the permeability of vacuum 1, Vh the total

hydrody-namic volume, Hext the external magnetic field, Hdemag

the magnetostatic field, ξ a damping factor2_{and τ}

tha

ther-modynamical torque.

These particles have an uniaxial magnetocrystalline anisotropy, leading to an anisotropy energy

Eanis= KVdsin2θ (2)

with K the anisotropy constant in J/m3_{, V}

dthe magnetic

volume in m3_{and θ the angle between m and u.}

Due to the thermal field, m can switch its direction along the u axis. This type of switching is called N´eel switching [12]. The average switching time for a particle is given by

τN = τ0exp

_∆E kBT

(3) with kB the Boltzmann constant and τ0 a prefactor

de-pendent on K, MSand the temperature T [13]. The

sec-ond possible relaxation mode of a magnetic nanoparticle is

1_µ

0= 4π· 107Tm/A

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through Brownian relaxation [14]. This relaxation is de-scribed by the physical rotation of the particles, i.e. the rotation of u relative to a surrounding fluid. The relax-ation time is given by

τB=

3ηVh

kBT

(4) with η the viscosity of the surrounding fluid and Vh the

total, hydrodynamic volume of the particle. The total re-laxation time for non-interacting particles is then given by

τef f = 1 τN + 1 τB −1 (5) When the particles are subdued to an alternating exter-nal magnetic field, hysteresis loops will occur. This seems in contradiction to the superparamagnetic properties, but it should be noted that these superparamagnetic proper-ties are obtained for slowly varying fields. Each measuring point is then an equilibrium position. When a fast alter-nating field is introduced, such an equilibrium is not ob-tained. The hysteresis loops are a function of ωτef f, in

which ω = 2πf is the frequency of the AC field [15].

Fig. 1. Hysteresis loop as a function of the relaxation times multiplied with external field frequency and τR = τef f, figure taken from

[15].

The hysteresis loop area has a maximum when ωτef f =

1, which is why knowledge of clustering effect is important to optimize hyperthermia experiments which rely on the hysteresis loop areas. If clustering has a severe effect on the time constant τef f it has an immediate effect on the

efficiency of the therapy. C. Clustering of MNPs

The magnetostatic field Hdemag

Hdemag= 1 4π X i Vd,iMS,i " 3(mi· ri)ri r5 i − mi r3 i # (6) is a field that is of short range, as seen by the _r13

i

depen-dence in which riis the distance of particle i to the point

in space in which the field is calculated.

When particles are in each others vicinity they will start

to attract and form clusters. The magnetostatic field of eq. (6) will influence the LLG-equation and as a conse-quence, the dynamics of the magnetic nanoparticles will change. Because of the altered dynamics, the relaxation constants change too and eq. (5) will not hold anymore as seen in the simulations for chains of 2 and 3 magnetic nanoparticles forming a chain.

III. Simulations of non-interacting nanoparticles

N´eel and Brownian relaxation processes are investigated for non-interacting magnetic nanoparticles. N´eel relaxation is obtained for particles in a fluid with η₋_{→ ∞ while the} Brownian relaxation is obtained for particles that satisfy τN > 100τB, i.e. when the anisotropy energy results in

a negligible N´eel switching rate. Based on the results, pa-rameters for a mixed relaxation with τN ≈ τBare obtained.

The standard parameters for the performed simulations are summarized in table I.

TABLE I

Standard simulation parameters for MRX curves.

Parameter Value #particles 10000 K [kJ/m3_] ₁₀ α 0.01 Ms[kA/m] 400 η [mPa.s] 0.86 T [K] 300 A. N´eel relaxation

The results for the simulations of N´eel relaxation follow the theory nicely, proving that Vinamax calculates the dy-namics correctly. The values for τ0in eq. (3) are calculated

and compared to theoretical values from [13].

4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 5 6 7 8 9 10 τ0 (ns) radius (nm) τ 0 τ_0,theo

Fig. 2. τ0for N´eel relaxation of non-interacting particles

From fig. 2 it is visible that τ0 follows the theoretical

value nicely for radii larger than 7 nm. For smaller radii, there is some anomaly visible between theory and simula-tion.

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B. Brownian relaxation

The Brownian relaxation follows the theoretical pre-dicted values of eq. (4) correctly. During the relaxation processes, there is always a first thermalisation process due to T _{6= 0. This thermalisation process leads to a relaxation} curve of_hmzi not starting from 1 but a value given by

hmzi0= A = Rπ/2 0 cos θ exp − KV sin2_θ kBT ! sin θdθ Rπ/2 0 exp − KV sin2_θ kBT ! sin θdθ (7)

During N´eel relaxation, this initial value was not always found as the thermalisation process and relaxation process are of the same time order. For Brownian relaxation how-ever, the thermalisation process occurs during a negligible time. This thermalisation is illustrated in fig. 3

0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 0 0.05 0.1 0.15 0.2 0.25 0.3 <m z > (-) t (µs) <mz> A = 0.9695

Fig. 3. The theoretical value of A (here for r = 12 nm) denotes the ultimate mean value ofhmzi when Brownian relaxation and N´eel

relaxation are turned off.

C. Mixed relaxation

Mixed relaxation is caused when τN ≈ τB. fig. 4 shows

the theoretical relaxation times for particles with the pa-rameters of table I. τef f is a combination of both relaxation

processes when r_{≈ 8nm: mixed relaxation occurs for these} particle sizes. 0 1 2 3 4 5 6 7 8 6 8 10 12 14 τ ( µ s) radius (nm) τN τB τeff

Fig. 4. The N´eel, Brownian and total relaxation times for MNPs as a function of their radii.

IV. Clustering of magnetic nanoparticles in chains

When magnetic nanoparticles cluster, the magnetostatic interaction will cause the relaxation times to increase. The total magnetic field in the LLG-equation and eq. (1) change the dynamics of the particles.

A. Chains of 2 particles

A chain consisting of 2 MNPs has an infinite possibilities to relax. The N´eel relaxation with the lowest energy thresh-old however is when both magnetization vectors, that are initially aligned along the length of the chain (i.e. the z-direction), rotate oppositely in the xz-plane. The resulting energy barrier is

∆E = E(π/2)− E(0) = 2V K + 1 48µ0M

2 S

! (8) in which the angles π/2 and 0 denote the antiparallel con-figuration with the magnetization vectors along the x-axis and the initial aligned configuration respectively.

When pure Brownian relaxation is considered, the magneti-zation vectors are pinned along the anisotropy axis and the only energy barrier that needs to be crossed is the one due to the magnetostatic field, i.e. the second term in eq. (8). A mixed relaxation is possible when both relaxation times are of the same order, similar to the case of non-interacting particles. The relaxation path of this configuration is rather complex due to this combination.

The relaxation times increase significantly compared to the case of non-interacting particles.

An example of such a curve for Brownian relaxation of a chain consisting of particles with radius 10 nm is given in fig. 5. 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 <m z > (-) t (µs) <mz> 0.923exp((-t/173.42639)1.01) non-interacting Radius = 10.0 nm

Fig. 5. Relaxation curve of hmzi for a chain consisting of 2

MNPs with radius 10 nm together with the relaxation for non-interacting particles.

The relaxation curves of_hmzi and huzi show a stretched

exponential behavior of the shape y = A exp−t_τ c. Values for chains of 2 MNPs and with parameters as in table I are given in table II

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TABLE II

Relaxation constants for a chain of 2 MNPs with different radii. r (nm) τef f(µs) τN(µs) τB(µs) c cB 6 0.103 0.576 0.669 1.41 0.96 7 0.359 11.668 1.529 1.42 0.84 8 1.329 1319.83 5.331 1.41 0.60 9 9.179 > 105 _21.266 _0.98 _0.82 10 173.426 > 105 _141.749 _1.01 _1.04 12 138_{· 10}3 _{> 10}5 _61.8 · 103 _1.11 _1.05

It is clear that the relaxation times have a huge depen-dence on the particle size. The exponential values show odd behavior as c seems to have a transition from values around 1.4 to values around 1 when the radius crosses 8 nm.

B. Chains of 3 MNPs

Possible relaxation paths for a chain of 3 MNPs are not easily deduced as for a chain of 2 MNPs. A possible energy barrier is given in eq. (9) when all 3 particles have a fixed anisotropy axis along the chain length and the magneti-zation vectors rotate in the xz-plane with m of the outer particles rotating clockwise and m of the middle particle rotating counter clockwise.

∆E = E(π/2)_{− E(0) = 3KV +} 19 192µ0V M

2

S (9)

The problem is that there exist relaxation processes that follow a different path, but with an even lower energy bar-rier. The coherence between the particles is strong, even for the outer particles. This is a result of the coherence between neighboring particles leading to a literally chained coherence between the outer particles.

The values of τef f, τB, c and cBfor chains of 3 MNPs and

with parameters as in table I are given in table III

TABLE III

τ , τB, c and cBfor a chain of 3 MNPs.

radius (nm) τ τB c cB

6 0.201 0.752 1.53 0.94 7 0.980 2.142 1.66 0.72 8 11.517 8.372 1.01 0.93 9 1299.426 48.307 0.94 0.96 Comparing table III to table II, the relaxation times not only rise as a function of the radius but also as a function of the chain length.

V. Consequences on biomedical applications The relaxation times have an influence on the hysteresis loop areas as given in [15]. This area is proportional to

1

√_1+ω2_τ2 ef f

. A large error in the relaxation times, with a fixed frequency ω will lead to lower hysteresis loop areas

as the relaxation time increases when clustering effects ap-pear. This effect is countered by adding a non-magnetic coating to the particles. When these particles cluster, the magnetostatic influence decreases as a function of coating thickness. This result is depicted in fig. 6.

0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 <m z > (-) t (µs) d=2r d=2.2r d=2.4r d=3.0r d=4r non-inter Radius = 7.0 nm

Fig. 6. Relaxation ofhmzi in time for 2 MNPs with varying distance

d between the particle centers.

A distance between the particle centers d = 4r with r the radius of the magnetic core is sufficient to counteract the effects of clustering by approximately 6%.

VI. Outlook on clustering effects To conclude, some interesting topics that were not inves-tigated during the dissertation are shortly mentioned here. For chains with the amount of particles N −→ ∞ the re-laxation times and exponential factors cB asymptotically

reach an end value.

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 <u z > (-) t (µs) 1 2 3 4 7 10 12 15 20 Radius = 6.0 nm

Fig. 7. Asymptotic behavior of the relaxation curves ofhuzi for

chains of different lengths and MNPs with r = 6 nm.

The asymptotic behavior was not investigated for N´eel relaxation.

Ring structures are possible clusters that manifest them-selves in 2-dimensional space. The magnetostatic interac-tion will alter the relaxainterac-tion curves as well. These relax-ation curves are harder to express as the equilibrium mag-netization of the complete chain equals 0.

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The complete rotation of clusters like chains was not in-vestigated. Only the rotation of the separate particles was included in the Brownian relaxation time. The complete rotation is also an interesting future topic to include in Vinamax.

Acknowledgements— The computational resources and ser-vices used in this work were provided by the VSC (Flemish Supercomputer Center), funded by Ghent University, FWO and the Flemish Government – department EWI

References

[1] J.M.D. Coey. Magnetism and magnetic materials. Cambridge, 2010.

[2] E.A. P´erigo et al. Fundamentals and advances in hyperthermia. Applied physics reviews, 2015.

[3] H.C. Bryant E.R. Flynn. A biomagnetic system for in vivo cancer imaging. NIH, 2007.

[4] Yi-Xiang J. Wang. Superparamagnetic iron oxide based mri contrast agents: current status of clinical application. Quant imaging med surg, 2011.

[5] J.G. Ovejero. Effects of inter- and intra-aggregate magnetic dipolar interactions on the magnetic heating effeciency of iron oxide nanoparticles. Journal of materials chemistry, 2016. [6] Michael L´evy et al. Modeling magnetic dipole-dipole interactions

inside living cells. Physical review B, 2011.

[7] O. Laslett et al. Interaction effects enhancing magnetic parti-cle detection based on magneto-relaxometry. Applied physics letters, 2015.

[8] B. Fischer et al. Brownian relaxation of magnetic colloids. Jour-nal of magnetism and magnetic materials, 2004.

[9] C. Usadel K. D. Usadel. Dynamics of magnetic single domain particles embedded in a viscous liquid. AIP, 2015.

[10] Arne vansteenkiste et al Jonathan Leliaert. Vinamax: a macrospin simulation tool for magnetic nanoparticles. Med Biol Eng Comput, 2014.

[11] Daniel B. Reeves and John B. Weaver. Combined n´eel and brown rotational langevin dynamics in magnetic particle imaging, sens-ing, and therapy. Applied Physics letters, 2015.

[12] A. Coene et al J. Leliaert. Regarding the n´eel relaxation time constant in magnetorelaxometry. AIP, 2014.

[13] W.T. Coffey et al. Thermally activated relaxation time of a single domain ferromagnetic particle subjected to a uniform field at an oblique angle to the easy axis: comparison with experimental observations. Phys. Rev. Letters, 1998.

[14] E.N. Ivanov. Theory of rotational brownian motion. Soviet physics JETP, 1964.

[15] J. Carrey et al. Simple models for dynamic hysteresis loop cal-culations of magnetic single-domain nanoparticles: application to magnetic hyperthermia optimization. AIP, 2011.

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Glossary

MNP magnetic nanoparticle

SPM superparamagnetic

MRX magnetorelaxometry

MPI magnetic particle imaging

MRI magnetic resonance imaging

FFP field free point

SNR signal to noise ratio

SW Stoner Wohlfarth

LRT linear response theory

SAR specific absorption rate

LLG Landau-Lifshitz-Gilbert

S/N Signal to noise ratio

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List of Figures

2.1 Ferromagnetism compared to superparamagnetism. H (magnetic field) and M (magnetization) are both expressed in A/m. Hc is the coercive field for

which the magnetization M is zero. Mris the remanent magnetization for zero

magnetic field H. MSis the saturation magnetization, the maximal obtainable

value of M for the considered material. The cross in the middle of the figure denotes the origin for H and M . Figure taken from Ref.[1]. . . 3 2.2 Qualitative graphs of the coercive field Hc(top) and the hysteresis loops

(bot-tom) of MNPs as a function of the particle size. The red lines describe the SPM state. Figure taken from Ref.[2]. . . 4 2.3 Magnetite (a) and maghemite (b) as powder materials. Figure taken from Ref.[3]. 5 2.4 Physical comparison between the N´eel relaxation time τN and Brownian

relax-ation time τB. The blue arrow denotes the magnetization vector m while the

green dashed line is the anisotropy axis, which is fixed to the particle. . . 8 2.5 A magnetic particle represented by an ellipse with the longest side in the

di-rection of the easy axis: θ is the angle between MS and the easy axis, φ the

angle between external field H and the easy axis, figure from [4]. . . 11 2.6 The magnetization in function of ξ and σ defined by equilibrium functions,

figure from [4] . . . 12 2.7 Hysteresis loop as a function of the relaxation times multiplied with external

field frequency and τR= τef f, figure taken from [4]. . . 14

2.8 The z component of the magnetization in a hysteresis loop for a particle with T = 0. As the magnetic field B increases, a transition is clearly visible (green and blue curve), figure taken from [5]. . . 16 2.9 The relaxation processes of a proton as observed in MRI: (a) With only a

constant external field, the magnetization of the proton will precess around this field with a frequency ω0, (b) With a strong time-varying field perpendicular to

B0 and with frequency ω0 the magnetization vector will lie in the xy-plane. (c)

and (d) mxy and mz when the time-varying field disappears. This is natural

precession. . . 18 2.10 Comparison of the magnetization signal obtained for a FFP (a) and a point in

saturation (b). Figure taken from Ref.[6]. . . 20 2.11 The principle of blurring of an image in MPI in the scanning direction.

Adia-batic here means the signal without blurring, i.e. when the relaxation time is negligible. . . 21

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2.12 A MRX spectrum: the particles are initially magnetized in the z-direction for 1 s by using an external magnetic field. The field is switched off and after a time τD the MRX spectrum is measured for 0.5 s. Figure taken from Ref.[7]. 22

2.13 Different possible clustering configurations for MNPs, figure adapted from [8] and [9] . . . 24 2.14 Comparison of the hysteresis loops for MNPs dispersed in water or cells between

MNPs with a radius of 11 nm (left) and 21 nm (right), figure taken from Ref.[10] 25 2.15 Example of fractal structure of N = 100 with Df = 2.25 and kf = 1.0, the

λ-values are principal axes as defined in Appendix A, figure taken from Ref.[8]. 26 2.16 Effect of clustering on the relaxation constants for 100 randomly distributed

MNPs: the relaxation time decreases for higher fractal dimensions but is smeared out over time, the blue curve illustrates the case for non-interacting MNPs, figure taken from Ref.[8]. . . 27 2.17 The third harmonic of the magnetization normalized to the non-interacting

case for 100 MNPs in function of the fractal dimension, the actual value for a biomedical application will lie between curves hˆz and hˆx which are both decreasing functions lower than 1, figure taken from Ref.[8]. . . 28

3.1 MRX curve with fit for a MNP with radius 7 nm, the prefactor is clearly different from the theoretical prediction . . . 33 3.2 τ0 in function of the particle radius, with error bars together with the

theoret-ical values obtained by Eq. (3.11). . . 35 3.3 MRX curve for a MNP with radius 10 nm, the deviation between theory and

simulation due to the thermal field is clear here for larger times t > 2 µs. . . . 36 3.4 MRX curve for a particle with radius 12 nm. . . 37 3.5 The theoretical value of A denotes the ultimate mean value of hmzi when

Brownian relaxation and N´eel relaxation are turned off. . . 37 3.6 Brownian relaxation in a constant external magnetic field along the z-axis for

fields between 0.01 and 0.1 T. . . 39 3.7 Brownian relaxation in a constant external magnetic field along the z-axis for

fields between 0.001 and 0.01 T. . . 39 3.8 Values of _hmzi∞ in function of the external field Bext together with a fit

ac-cording to a Langevin function. . . 41 3.9 Notations for the angles and vectors of a nanoparticle, ~u denotes the anisotropy

axis, ~m the magnetization vector. . . 41 3.10 Anisotropy axis relaxation in a constant external magnetic field along the z-axis

for fields between 0.01 and 0.1 T. . . 43 3.11 Anisotropy axis relaxation in a constant external magnetic field along the z-axis

for fields between 0.001 and 0.01 T. . . 43 3.12 Values of_huzi∞ in function of the external field strength Bexttogether with a

Langevin fit. . . 44 3.13 The relaxation times τB and τu as a function of the external magnetic field

strength, a log-log scale is used here . . . 45 3.14 The N´eel, Brownian and total relaxation times for MNPs as a function of their

radii. . . 46 3.15 The deviation from the N´eel and Brownian relaxation times as compared to

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3.16 MRX curve for a MNP of radius 8 nm with mixed relaxation processes . . . . 48

4.1 The initial configuration of 2 particles that form a chain. . . 50 4.2 The energy function (here without dimension, for simplicity) as a function of

the angle of the magnetization vector m and the anisotropy axis for the case that b < 2K (left) and b > 2K (right). . . 52 4.3 The energy for 1 particle switching in a two particle chain, in which the other

particle has a fixed magnetization along the positive z-direction. . . 53 4.4 The energy of a system of 2 particles along the z-axis in which one particle has

its magnetization vector along the positive z-direction and the other particle has a free magnetization vector. θ is the angle between the magnetization vector and the z-axis, d denotes the distance between both particles. . . 54 4.5 Evolution of the magnetization vectors m for N´eel relaxation, the anisotropy

axes are in the vertical direction, the radius of both particles is 6 nm. . . 55 4.6 The magnetic energy in function of the angle θ between the magnetization

vector and the anisotropy axis. . . 56 4.7 Pure N´eel relaxation for 2 interacting MNPs of radius 6 nm forming a chain,

together with the exponential fit. . . 58 4.8 The sum of the magnetostatic energy and anisotropy energy for a chain of 2

particles with a radius of 6 nm. Energy transition 1 corresponds to consecutive separate switching events, energy transition 2 to the simultaneous switching event. The silver boxes denote all switching events as seen in Fig. 4.10 and the gold boxes indicate occurring switching events that are viewed in more detail. 59 4.9 The distribution of the sum of the anisotropy energy and the magnetostatic

energy for a chain consisting of 2 MNPs that are free to rotate. . . 59 4.10 mz for both particles of radius 6 nm in a chain of 2 particles, the boxes are the

same as in Fig. 4.8. . . 61 4.11 The sum of the magnetostatic energy and the anisotropy energy for a chain of

2 particles with radius 6 nm, focused on 2 switching events occurring around 8 and 9.8 µs. . . 61 4.12 mz for particle both particles of radius 6 nm in a chain of 2 particles, focused

on 2 transitions . . . 63 4.13 The configuration of the magnetization vectors in time for a chain consisting

of 2 MNPs and radius 6 nm. A switching event occurs. . . 65 4.14 The energy of the chain consisting of 2 MNPs with radius 6 nm in a time

interval between 9.53 and 9.59µs. The grey boxes highlight the figures from Fig. 4.13. . . 66 4.15 The configuration of the magnetization vectors in time for a chain consisting

of 2 MNPs with radius 6nm. No switching event occurs. . . 68 4.16 The energy of the chain consisting of 2 MNPs with radius 6 nm in a time

interval between 4.96 and 5.03µs. The grey boxes highlight the figures from Fig. 4.15. . . 68 4.17 Autocorrelation function of mz for both particles with radius 6 nm in a chain. 69

4.18 Cross-correlation function between the magnetization mzof both particles with

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4.19 Autocorrelation function for the magnetization of a chain consisting of particles with radius 6 nm with a distance between the centers equal to 2, 2.5, 3 and 4 times the radius. . . 71 4.20 Cross-correlation function for the magnetization of a chain consisting of

par-ticles with radius 6 nm with a distance between the centers equal to 2, 2.5, 3 and 4 times the radius. . . 71 4.21 Brownian relaxation behavior for 2 MNPs of radius 6 nm forming a chain, the

fit is a stretched exoponential function. . . 72 4.22 The magnetostatic energy for a chain consisting of 2 MNPs with a radius of

6 nm where Brownian relaxation is included, the energy threshold levels are included. The grey boxes denote switching events while the green box denotes a local energy maximum that does not lead to a switching event, as explained further on in Fig. 4.31. . . 74 4.23 The distribution of the magnetostatic energy for a chain consisting of 2 MNPs

that are free to rotate. . . 74 4.24 uz for both particles of radius 6 nm in a chain consisting of 2 particles with

Brownian rotation possible. The gold boxes denote the switching events be-tween which a detail picture is offered in Fig. 4.26 while the green box denotes an incomplete switching event, i.e. a rotation of the anisotropy vector of 360◦. 75 4.25 The magnetostatic energy for a chain consisting of 2 MNPs with a radius of 6

nm where Brownian relaxation is included. This figure is Fig. 4.22 focused on the interval between 8 and 19 µs. . . 76 4.26 uz for both particles of radius 6 nm in a chain of 2 particles which are free to

rotate, with focus on the interval between 8 and 19 µs . . . 77 4.27 The configuration of the anisotropy vectors in time for a chain consisting of 2

MNPs with radius 6nm. A switching event occurs. . . 79 4.28 The energy of the chain consisting of 2 MNPs with radius 6 nm in a time interval

between 8 and 9.8µs. The grey boxes highlight the figures from Fig. 4.30. . . 79 4.29 uz for both particles during an event in which the anisotropy vectors rotate

over 360◦, denoted by the green box in Fig. 4.24. . . 80 4.30 The configuration of the anisotropy vectors in time for a chain consisting of 2

MNPs with radius 6nm. A switching event does not occur here. . . 81 4.31 The energy of the chain consisting of 2 MNPs with radius 6 nm in a time

interval between 26.5 and 28.5µs. The grey boxes highlight the figures from Fig. 4.30. . . 82 4.32 Autocorrelation function for both particles with radius 6 nm in a configuration

of 2 particles forming a chain, the anisotropy constant k = 107_J/m3 _{for both}

particles. . . 82 4.33 Cross-correlation function for both particles with radius 6 nm in a configuration

of 2 particles forming a chain, the anisotropy constant k = 107_J/m3 _{for both}

particles, the cross-correlation is given in both directions. . . 83 4.34 Relaxation behavior of mz for a chain consisting of 2 MNPs with radius 6 nm

together with a(n) (stretched) exponential fit. . . 84 4.35 Relaxation behavior of uz for a chain consisting of 2 MNPs with radius 6 nm

together with a stretched exponential fit. . . 84 4.36 Evolution of the magnetization vectors m and anisotropy axes (green color)

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4.37 The sum of the anisotropy energy and the magnetostatic energy for a chain of 2 MNPs with a radius of 6 nm, the gray box denotes a switching event of the system consisting of a N´eel relaxation followed by a Brownian relaxation. . . 86 4.38 The distribution of the sum of the anisotropy energy and the magnetostatic

energy for a chain consisting of 2 MNPs that are free to rotate. . . 86 4.39 mz for both particles of radius 6 nm in a chain, the gray box highlights the

switching events according to the energies in Fig. 4.37. . . 87 4.40 uz for both particles of radius 6 nm in a chain, the gray box highlights the

switching events according to the energies in Fig. 4.37. . . 87 4.41 mz for both particles of radius 6 nm in a chain in a time interval 32.5-33.5 µs. 88

4.42 energy of the system for a chain of 2 MNPs of radius 6 nm in a time interval 32.5-33.5µs. The highlighted boxes denote the events in Fig. 4.43. . . 88 4.43 The configuration of the anisotropy and magnetization vectors in time for a

chain consisting of 2 MNPs with radius 6nm. A N´eel switching event occurs here. . . 89 4.44 uz for both particles of radius 6 nm in a chain in a time interval 36-38 µs. . . 90

4.45 energy of the system for a chain of 2 MNPs of radius 6 nm in a time interval 36-38µs. The highlighted boxes denote the events in Fig. 4.46. . . 90 4.46 The configuration of the anisotropy and magnetization vectors in time for a

chain consisting of 2 MNPs with radius 6nm. A N´eel switching event occurs here. . . 91 4.47 Autocorrelation function of mzfor a particle with radius 6 nm in a configuration

of 2 MNPs forming a chain. The anisotropy constant K = 4_{· 10}4J/m3 for both particles. . . 92 4.48 Autocorrelation function of uzfor a particle with radius 6 nm in a configuration

of 2 MNPs forming a chain. The anisotropy constant K = 4· 104_J/m3 _{for both}

particles . . . 92 4.49 Cross-correlation function of mz between both particles with radius 6 nm in

a configuration of 2 particles forming a chain. The anisotropy constant k = 4_{· 10}4_J/m3 _{for both particles. . . .} ₉₃

4.50 Cross-correlation function of uz for both particles with radius 6 nm in a

config-uration of 2 particles forming a chain. The anisotropy constant k = 4·104_J/m3

for both particles. . . 93 4.51 Cross-correlation function between uz and mz for a particle with radius 6 nm

in a configuration of 2 particles forming a chain. The anisotropy constant K = 4_{· 10}4J/m3 for both particles. . . 94 4.52 Cross-correlation function between uz and mz of two different particles with

radius 6 nm in a configuration of 2 particles forming a chain. The anisotropy constant K = 4_{· 10}4J/m3 for both particles. . . 94 4.53 Pure N´eel relaxation for 2 MNPs of radius 7 nm forming a chain, together with

the exponential fit and comparison with the case for non-interacting particles. 95 4.54 τ0 (or a) for a chain of 2 MNPs, there is a clear decrease of τ0 in function of

the diameter. . . 97 4.55 f0 (or 1/2a) for a chain of 2 MNPs, there is a clear increase of f0 in function

of the diameter although a fit is difficult to make . . . 98 4.56 Relaxation curve of _huzi for a chain consisting of 2 MNPs with radius 10 nm

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4.57 Relaxation curve of hmzi for a chain consisting of 2 MNPs with radius 10 nm

together with the relaxation for non-interacting particles. . . 99

4.58 The configuration of a chain consisting of 3 MNPs directed along the z-axis. . 101

4.59 N´eel relaxation curve ofhmzi together with fit for a chain consisting of 3 MNPs with radius 6 nm. . . 102

4.60 Evolution of the magnetization vectors m for N´eel relaxation of a chain con-sisting of 3 MNPs, the anisotropy axes are in the vertical direction. . . 102

4.61 The sum of the anisotropy energy and the magnetostatic energy for a chain of 3 MNPs with a radius of 6 nm. . . 103

4.62 mz for all 3 particles of radius 6 nm in a chain. . . 104

4.63 Autocorrelation function for a particle of radius 6 nm in a chain consisting of 3 MNPs. . . 104

4.64 Cross-correlation between an outer particle and the middle particle in a chain consisting of 3 MNPs. . . 105

4.65 Cross-correlation between both outer particles in a chain consisting of 3 MNPs. 105 4.66 The magnetostatic energy for a chain of 3 MNPs with a radius of 6 nm. K = 107J/m3 for these particles. . . 106

4.67 uz for all 3 particles of radius 6 nm in a chain . . . 107

4.68 Autocorrelation function of uzfor a particle of radius 6 nm in a chain consisting of 3 MNPs. . . 107

4.69 Cross-correlation function of uz between the inner particle and one of the outer particles of radius 6 nm in a chain consisting of 3 MNPs. . . 108

4.70 Cross-correlation function of uz between the outer particles of radius 6 nm in a chain consisting of 3 MNPs. . . 108

4.71 Relaxation behavior for 3 MNPs of radius 7 nm forming a chain, with stretched exponential fit. . . 109

5.1 Relaxation of _hmzi in time for 2 MNPs with varying distance d between the particle centers. . . 114

5.2 A chain of 2 MNPs with a non-magnetic shell to avoid clustering . . . 117

6.1 Asymptotic behavior of the relaxation curves of _huzi for chains of different lengths and MNPs with r = 6nm . . . 120

6.2 τB as a function of amount of particles with radius 6 nm. . . 121

6.3 τB as a function of amount of particles with radius 6 nm. . . 122

6.4 Configuration of the magnetization vectors in a ring consisting of 6 MNPs. . 122

6.5 mz values for all 6 particles in a ring consisting of 6 MNPs, each with a radius equal to 6 nm. . . 123

6.6 uz values for all 6 particles in a ring consisting of 6 MNPs, each with a radius equal to 6 nm. . . 124

C.1 The imaginary error function erfi(x). . . 136

C.2 _hmzi as a function of y = _kKV_B_T. . . 136

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List of Tables

3.1 Standard simulation parameters for MRX curves. . . 30 3.2 Simulated values for A and τN with standard error. . . 33

3.3 Dependence of the N´eel relaxation times τN and prefactor τ0 in function of

changing material parameters. . . 34 3.4 τ0 with error in function of the particle radii. . . 34

3.5 Brownian relaxation rates, theoretical values are compared to simulated values. 36 3.6 Dependence of the Brownian relaxation time τBon changing material parameters. 38

3.7 The initial thermalisation values ofhmzi, the final values hmzi∞and the

relax-ation times of_hmzi for particles of radius 12 nm placed in an external magnetic

field. . . 40 3.8 The final valueshuzi∞ and the relaxation times ofhuzi for particles of radius

12 nm placed in an external magnetic field. . . 44 3.9 Time constants for MNPs of 8 nm. . . 48

4.1 τN for a chain of 2 MNPs of interacting and non-interacting particles. ∆E is

the energy threshold according to Eq. (4.14) and a the prefactor in Eq. (4.16). 96 4.2 τN with theoretical prefactors a for a chain of 2 MNPs. . . 96

4.3 Relaxation constants for a chain of 2 MNPs as a function of particle size. . . 100 4.4 τ , τB, c and cB according to Eq. (4.21) and Eq. (4.22). . . 109

5.1 Error on the total relaxation time for a chain of 2 MNPs compared to non-interacting MNPs. . . 115 5.2 Hysteresis loop area ratios with the used frequency ω for non-interacting particles.116

6.1 τB and cB from Eq. (4.22) as a function of the amount of particles in a chain. 121

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Introduction

1.1 Motivation

The biomedical world is one in which innovation is always at the center. Easier and more effective techniques with lesser side effects are desirable for the comfort of the patient and the quality of information of the human body.

Magnetic nanoparticles have a promising future in biomedical applications. The magnetic properties of the particles are used to function as contrast agents in MRI, but also as an agent in cancer treatments opposed to more traditional techniques such as chemotherapy. Nanoparticles are easily implemented into the human body, as they are non-toxic and an ex-ternal magnetic field is able to centralize them into a specific area that is desired for therapy. As these particles are easily taken up by the liver, there are no side effects as these are present in chemotherapy which attacks the whole body.

The physics of nanoparticles as separate particles is already well understood and well docu-mented. The problem that arises is when clustering of these particles is considered.

1.2 Problem statement

The particles will start to form clusters together because of the magnetic behavior and fol-lowing attraction towards each other.

This has an influence on the magnetic dynamical behavior of the particles. This change in behavior will lead to a direct consequence in therapy as blurring in imaging techniques arises and for hyperthermic therapies the frequency of the external magnetic field changes for an optimal heating of tissue.

The clustering behavior of magnetic nanoparticles is not yet widely understood and experi-ments using these nanoparticles assume non-interacting particles, leading to theoretical results that do not match the real situation.

As it is of utter importance that a therapy is successful, the influence of clustering on rel-evant therapeutic parameters should be investigated to have a better understanding on the severeness of this effect in a qualitative and quantitative way.

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1.3 Outline

To have a better understanding of these clustering effects, this thesis is divided in the follow-ing chapters.

First off, the principles of magnetic nanoparticles are described to have a better understand-ing about the basic properties these particles describe based on physical models.

In the following chapter a deeper explanation of the physics exhibited by these particles is presented. The energy and dynamic behavior of these particles is further described here, as well as an introduction to clustering of nanoparticles in general.

The following chapters include simulations of the magnetic nanoparticles. The software pack-age Vinamax is used to obtain the results.

The simulations are performed for particles with different volumes but keeping the other pa-rameters fixed. In this way, the clustering is mapped in function of the size of the magnetic nanoparticles. The parameter that quantifies this clustering effect is the relaxation time for the average magnetization of the particles.

The first simulations chapter describes the non-interacting particles to see the accordance with the literature and as a consequence, a verification model is obtained for the software used.

The follow up of this chapter includes the simplest form of clustering: a chain of 2 magnetic nanoparticles, in which the energy of the system and relaxation parameters are thoroughly described. A larger relaxation time reflects a stronger bonding between particles and vice versa.

To conclude, simulations are also run for chains consisting of 3 particles and for planar structures consisting of 3 particles, albeit in lesser detail than for a cluster of 2 particles. An outlook chapter concludes this work, with more complex forms of clustering yet to be investigated.

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Magnetic nanoparticles

2.1 General description

2.1.1 Basic physical properties

A nanoparticle is defined as an object where the length scales in all dimensions range from 1 nm to 100 nm. The atomic-scale structure of the object can be ignored but the mesoscopic dimensions of the object in the aforementioned range result in different physical properties as compared to bulk material, see Ref.[11].

When considering magnetic nanoparticles (MNPs) a superparamagnetic state is reached, which differs from the ferromagnetic state as found in the bulk material.

The difference in this behavior is shown in Fig. 2.11

Figure 2.1: Ferromagnetism compared to superparamagnetism. H (magnetic field) and M (magne-tization) are both expressed in A/m. Hc is the coercive field for which the magnetization

M is zero. Mris the remanent magnetization for zero magnetic field H. MS is the

sat-uration magnetization, the maximal obtainable value of M for the considered material. The cross in the middle of the figure denotes the origin for H and M . Figure taken from Ref.[1].

The graph shown in Fig. 2.1 is obtained after averaging out M , as a single nanoparticle will still show a net magnetization even in the absence of an external magnetic field.

1_{Further information concerning principles of magnetism and basic notations can be found in Refs.[11, 12].}

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Superparamagnetism is a combination of paramagnetism and ferromagnetism: no remanent magnetization or coercive field is observed, but for higher magnetic fields the MNP reaches the saturation magnetization MS as is the case for the ferromagnetic bulk material.

Fig. 2.2 explains the transition between ferromagnetic and superparamagnetic states in a more thorough, yet still qualitative way.

The coercive field Hc is given as a function of the particle size: when Hc reaches zero, the

MNP becomes a superparamagnetic (SPM) particle. The hysteresis loops as function of the particle size are also illustrated to be complete.

Figure 2.2: Qualitative graphs of the coercive field Hc (top) and the hysteresis loops (bottom) of

MNPs as a function of the particle size. The red lines describe the SPM state. Figure taken from Ref.[2].

When starting from a large size of the particle, the particle will be a multi-domain particle [13]. Several magnetic domains, each represented by a single magnetization vector will con-stitute the particle.

For decreasing size, the amount of domains will decrease and Hc will increase until a

single-domain state is reached. Hcwill reach a maximal value here and so the hysteresis loop area

will also reach a maximal value.

For even smaller sizes, Hcwill decrease again and eventually reach zero. This is a consequence

of the decreasing anisotropy energy, compared to the thermal energy, which is further ex-plained in section 2.2 which describes the magnetic dynamics of MNPs.

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Because of this superparamagnetic behavior, they have superior characteristics when com-pared to ferromagnetic nanoparticles as they have a higher chemical stability, see Ref.[14] which is interesting for applications in bio-medicine and they are easier to simulate as well, as the particle can be represented by a simple magnetization vector m.

2.1.2 Materials

In general the materials of the MNPs must possess following properties for in vivo applications in bio-medicine, see Ref.[15]:

• Non-toxic and non-immunogenic

• Small enough to remain in the blood vessels and pass capillaries • High magnetization to be immobilized easily by an external field

The materials that abide to these criteria in biomedicine are Fe3O4 and γ− Fe2O3 also known

as magnetite and maghemite respectively.

Figure 2.3: Magnetite (a) and maghemite (b) as powder materials. Figure taken from Ref.[3].

The saturation magnetization of magnetite and maghemite as bulk materials are 4.8× 105

A/m and 4.2_{× 10}5 A/m respectively, see Ref.[16]. These magnetization values are however a function of the size of the MNPs [17]. Higher volumes will yield higher magnetization values and so an optimal volume for a maximal area of the hysteresis loop areas (see section on hysteresis loops) could be determined. For simulations, a value of MS= 4× 105 A/m will be

assumed.

Several techniques for the synthesis of MNPs can be found in the literature, see Refs. [18–20]. Some commercial formulations are already available, of which Resovist and Feridex are the most commonly used in biomedical applications. These MNPs are coated with a layer of dextran to ensure binding with the cellular environment.

After using the MNPs in the human body for several applications, they will eventually be taken up in the liver and spleen where they are degraded in the course of 2 weeks, depending on the size of the particles, see Ref.[21].

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2.2 Magnetic dynamics of nanoparticles

2.2.1 Magnetic anisotropy

The first feature about MNPs that determines their dynamical behavior is the magnetic anisotropy. This is the property that the (ferro)magnetic axes lie in certain fixed directions in a magnetic sample. In the absence of an external magnetic field and with T = 0 K, the magnetization of the MNP will lie along one of these axes 2. Only uniaxial anisotropy is considered in this text: a single direction for the easy axis exists and the anisotropy is quantified by an anisotropy constant K with dimension J/m3. Two physical contributions to the anisotropy can be defined:

• Shape anisotropy: the shape of the sample determines the easy axis of magnetization. For a prolate ellipsoid shape, the easy axis will be along the long axis. Contrary, a spherical particle will not show shape anisotropy.

• Magnetocrystalline anisotropy: the crystal lattice of the material determines the easy magnetization direction(s). When spherical particles show magnetic anisotropy, it’s a consequence of this magnetocrystalline anisotropy.

The anisotropy for uniaxial MNPs is mathematically described by Eq. (2.1)

Eanis= KVdsin2θ (2.1)

with Vd the (magnetic) volume3 of the particle and θ the angle between the magnetization

M and the easy axis (see also Fig. 2.5). In the absence of an external field, θ = 0 results in the minimal energy configuration, from which the definition of the easy axis follows.

In general, the temperature of the MNP is not equal to zero and from statistical physics it is known that physical properties will have a fluctuating behavior, see Refs.[22–24]. For a single MNP, in an external field H = 0, with energy barrier ∆E = KVd and thermal energy kBT

(with kB the Boltzmann constant4), this leads to the following behavior.

2.2.2 Relaxation time constants

For a higher temperature, the switching of the magnetization along the easy axis will occur faster than for a lower temperature. This is a result of the higher thermal energy compared to the energy barrier ∆E, the energy threshold is overcome easier through the thermal fluc-tuations.

The term ‘faster’ is quantified through a critical measuring time τm. When the switching

time is lower than this τm, the average magnetization that will be measured equals zero in

the absence of an external field.

2_{i.e. multiple so called ”easy axes” are possible}

3_{A magnetic nanoparticle can have a non-magnetic shell and thus the total volume differs from the magnetic}

volume. The total volume is denoted by Vhwhile the magnetic volume is Vd 4_k

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If a hysteresis loop would be measured, with τm per separate measurement, a SPM curve

would be observed because the average magnetization of the MNP would have already re-laxed to a unique state corresponding to the minimal energy for a certain external field H.

When an assembly of MNPs are introduced together, the situation changes. First it is as-sumed that the particles do not interact with each other. The magnetization of each particle continuously switches along the easy axis as a result of the thermal energy in the system. The total contribution of all these switching processes leads to an exponential decay of the mean magnetization, for which the time constant is given by

τN = τ0exp ∆E kBT (2.2)

which is called the N´eel relaxation time. τ0 is a rather complex function of magnetic

param-eters such as the anisotropy constant K and the saturation magnetization MS as found in

Ref.[25] and has typical ranges between 10−12 and 10−8 s, see Refs.[4, 7]. The (qualitative) dependence on these parameters is also deduced during the simulations in chapter 3.

From Eq. (2.2) and the definition of τm, a critical volume Vcrit is calculated as

Vcrit= 1 K ln τm τ0 kBT (2.3)

A higher temperature will lead to more thermal fluctuations and thus a faster switching of the magnetic moment, hence leading to a higher Vcrit. The critical volume is linearly dependent

on the temperature.

The relaxation process described thus far is the relaxation of the magnetization relative to the anisotropy axis of the particle.

Another relaxation process exists due to the physical rotation of the particles relative to the surrounding fluid, called Brownian relaxation, see Ref.[26]. The time constant for Brownian relaxation (τB) is given by Eq. (2.4)

τB =

3ηVh

kBT

(2.4)

η is the viscosity of the surrounding fluid and Vh is the hydrodynamic volume, i.e. the total

volume of the MNP. When there is addition of a shell or when several particles form a con-glomerate together, the physical volume increases. Hence Vh > Vd is always true.

As both effects occur simultaneously, the effective relaxation time constant is given by

τef f = 1 τN + 1 τB −1 (2.5)

The effective relaxation time will always be shorter than either relaxation time of the separate processes.

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Figure 2.4: Physical comparison between the N´eel relaxation time τN and Brownian relaxation time

τB. The blue arrow denotes the magnetization vector m while the green dashed line is

the anisotropy axis, which is fixed to the particle.

Fig. 2.4 shows both possible relaxation processes. In a N´eel relaxation process, the magneti-zation of the MNP will align with the easy axis. Due to thermal fluctuations, when T _{6= 0,} the magnetization vector will move away from this axis. When the thermal energy is high enough, the magnetization vector can switch along the easy axis as it makes a turn of 180 degrees.

In the case of Brownian relaxation, the particle rotates as a whole due to the thermal en-ergy of the surroundings. Of course, a combination of both processes is possible, leading to Eq. (2.5).

The relaxation processes described in this subsection consist of those where the particles lie in a constant external field, independent of time (this comprises the situation in the absence of an external field too).

2.2.3 Dynamical equations

The Landau-Lifshitz-Gilbert (LLG) equation describes the dynamics of the magnetization vector m, see Refs.[27, 28]

∂m ∂t =− γ0 1 + α2 m× Hef f + αm× (m × Hef f) + α γ0 m× (ω × m) +α 2 γ0 m× ω (2.6) m is the magnetic moment scaled to a unit vector, further called magnetization vector. Hef f

is the total magnetic field and consists of

• external field Hext

• magnetostatic (magnetostatic) field Hdemag

• anisotropy field Hanis

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External field

The magnetic field that is externally applied, this can be a time-dependent field.

Magnetostatic (magnetostatic) field

The magnetic field that is caused by other MNPs, given by Eq. (2.7)

Hdemag = 1 4π X i Vd,iMS,i " 3(mi· ri)ri r5 i −mi r3 i # (2.7)

with: i the particle number, Vd,i its volume, MS,ithe saturation magnetization, mi the

mag-netization vector and ri the distance vector between particle i and the point in which the

magnetostatic field is calculated. ri is the norm of this vector.

Anisotropy field

This field reflects the anisotropy energy associated with a MNP and thus the tendency of m to align with the anisotropy axis.

Hanis= 2K

µ0Msat

(m· u)u (2.8)

with u the anisotropy axis unit vector.

Thermal field

To include thermal switching of the magnetic moment in the SPM particles, a thermal mag-netic field is introduced

Htherm(t) = η(t) µ0 s 2kBT α γ0MsatV ∆t (2.9)

η(t) defines a random vector with mean value 0 and uncorrelated in time and space as this reflects the total random characteristics of a thermal field.

Stochastic switching is also a possibility to incorporate the thermal field as this is a faster method than the stochastic field during simulations, but this only works when no external field is applied. With f = f0exp −∆E kBT ! (2.10)

which is similar to the N´eel relaxation time, switching occurs with a probability

P = 1− exp(−f∆t) and so P can be generated as a random variable, uniformly distributed between 0 and 1 leading to a switching time t =_{−(1/f) ln(1 − P ).}

In Eq. (2.6), α denotes a damping factor and ω denotes the angular velocity of the MNP as the particles are free to rotate in space.

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The first term leads to precession of the magnetization vector around the effective field vec-tor, the second term describes damping towards the effective field vector. The third and the fourth term describe the magnetization vector with respect to the rotation of the particle. The third term now describes some kind of precession while the last term shows that the magnetization will eventually rotate together with the particle.

The spatial rotation can explicitly be described by the exerted torque, which results in the second dynamical equation of the MNP [5]

Θ∂ω ∂t = µ0MSV γ0 ∂m ∂t + µ0MSm× (Hext+ Hdemag)− ξω + τth (2.11)

With Θ the moment of inertia, ω the angular velocity, ξ a damping factor5 and τtha

thermo-dynamical torque, see Ref.[29] with

hτth,i(t)i = 0 (2.12)

hτth,i(t), τth,i(t0)i = qδi,jδ(t− t0) (2.13)

q = 2kBT ξ (2.14)

The mass moment of inertia term of the particles can be neglected in most cases as this leads to a small inertia torque compared to the other terms in Eq. (2.11), so Θ = 0.

The dynamical equations form the basis for the simulations run in chapters 3, 4 and 5. The software used to solve these equations numerically is Vinamax, see Ref.[29].

Each MNP is considered as a macrospin with an anisotropy axis so both N´eel and Brownian relaxation processes are simulated. Vinamax is written in the language Go6

2.2.4 Hysteresis loops

In this part, hysteresis loops for MNPs are explained in further detail as these loops were only shortly mentioned in the general description of MNPs. The hysteresis loops are a result of the magnetic dynamics of MNPs in a time varying external magnetic field. The area of these loops is a function of particle size and external field frequency as discussed in this subsection.

3 theories are used to describe the hysteresis phenomena of single-domain MNPs with uniaxial anisotropy, see Ref.[4]

• Equilibrium functions

• Stoner-Wohlfarth model based theories, see Ref.[30]

• Linear response theory using N´eel and Brownian relaxation times

5_{ξ = 6ηV for spherical particles} 6_{http://golang.org.}

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The energy of a single MNP with volume Vdin an external time-varying field with magnitude

H is given by

E(θ, φ) = KVdsin2(θ)− µ0MSVdH cos(θ− φ) (2.15)

with the angles θ and φ as defined in Fig. 2.5.

Figure 2.5: A magnetic particle represented by an ellipse with the longest side in the direction of the easy axis: θ is the angle between MS and the easy axis, φ the angle between external

field H and the easy axis, figure from [4].

The energy equation is now formulated in a simpler representation with only 4 dimensionless variables

• σ = KVd/KBT , which denotes the dimensionless anisotropy energy

• ξ = µ0MSVdH/kBT , which denotes the dimensionless energy in the magnetic field

• θ, the angle between the magnetization m and the easy axis • φ, the angle between the magnetic field and the easy axis leading to Eq. (2.16)

E θ, φ kBT

= σ sin2(θ)_{− ξ cos(θ − φ)} (2.16) Defining the critical anisotropy field as

µ0HK = 2K/MS (2.17)

Two situations arise: if H > HK, the energy will have one minimum value with θ = φ. In

the other case, when H < HK, two energy minima arise that are separated by a saddle point.

In the extreme case that H = 0, the two minima correspond to θ = 0 and θ = π, i.e. a magnetization along the easy axis.

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A first way to describe the magnetic behavior of the MNPs is by the use of equilibrium functions that are based on the probability of a particle to occupy a certain energy state given by f θ, φ= exp E(θ,φ) kBT R θexp E(θ,φ) kBT dθ (2.18)

For σ−→ 0 the magnetization in the direction of the external field will approach the Langevin function L(ξ) = coth ξ₋ 1_ξ.

For values of σ−→ ∞, the magnetization will be along the easy axis. Using Eq. (2.18) and assuming φ = 0, the magnetization becomes

M = MSexp(σ + ξ)− MSexp(σ− ξ) MSexp(σ + ξ) + MSexp(σ− ξ)

= MStanh(ξ) (2.19)

For randomly distributed easy axes, the situation is different: the Langevin function is the upper boundary for the magnetization in function of ξ. For higher values of σ, the curves will be of the shape presented in Fig. 2.6

Figure 2.6: The magnetization in function of ξ and σ defined by equilibrium functions, figure from [4]

From this figure, it is clear that a higher anisotropy, thus a higher value for σ will cause a lower hysteresis loop area for randomly oriented easy axes when the MNPs are placed in the same field.

The Stoner-Wohlfarth (SW) model is based on the critical anisotropy field from Eq. (2.17). When the external field reaches this value, switching of the magnetization becomes possible as a consequence of the transition of two energy minima to only one value. For a particle with easy axis in the direction of the external field, this leads to a hysteresis loop with rectangular shape and area A = 4µ0HKMS = 8K.

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For randomly distributed particles, this becomes A = 1.92K. The hysteresis loop is not rect-angular in this case, as also presented in the paper by Carrey.

Thus far thermal effects were ignored, although they have a significant effect for the energy terms and the relaxation constants, Eq. (2.2) and 2.4 and will consequently change the area of the hysteresis loops. In the energy model this leads to the probability of jumping over the energy barrier into the other equilibrium position when there are two energy minima. The magnetization is now expressed as follows: assume p1 the probability to find the particle in

minimum 1 and p2 the probability to find the particle in minimum 2, then

M = MS(p1cos θ1+ (1− p1) cos θ2) (2.20)

The evolution of p1 and p2 = 1− p1 with ν1 and ν2 the jump frequency from state 1 to state

2 and vice versa is

∂p1

∂t = (1− p1)ν2− p1ν1 (2.21) In this way, the coercive field HC becomes dependent on the temperature and the hysteresis

loop area is (implicitly) given by

A(T ) = a_{· µ}0HC(T )MS (2.22)

with a a prefactor dependent on the orientation of the easy axis (1.92 for random orientation, 4 for orientation in the field direction), see also Carrey.

A more complete theory to describe minor hysteresis loops7, i.e. the loops that are created when the maximal magnetic field Hmax < Hc, is the linear response theory (LRT). The

relaxation time described by N´eel and Brownian relaxation is explicitly used in this theory. As the name suggests, a linear relationship between M and H is proposed as

M (t) =χH(t)_e (2.23) With e χ = χ0 1 1 + iωτef f (2.24)

the dynamic susceptibility of the MNP8 and τef f is the relaxation time given by Eq. (2.5).

Together with

H(t) = Hmaxcos(ωt) (2.25)

these equations lead to the time dependent magnetization

M (t) =|χ|Hmaxcos(ωt− φ) (2.26) in which|χ| = χ0 q 1+ω2_τ2 ef f and tan φ = ωτef f.

7_{These minor hysteresis loops are of importance for hyperthermia applications} 8_χ

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Eq. (2.25) and Eq. (2.26) represent an ellipse in the (H, M )-plane with area

A = πH_max2 |χ| sin φ (2.27) which can easily be found by applying Green’s theorem (proof in Appendix B) and the angle γ between the large axis and the abscise (proof also in Appendix B)

tan 2γ = 2H 2 max|χ| cos φ H2 max− Hmax2 |χ|2 (2.28)

An example of such hysteresis loops is given in Fig. 2.7

Figure 2.7: Hysteresis loop as a function of the relaxation times multiplied with external field fre-quency and τR= τef f, figure taken from [4].

The hysteresis loop area is maximal for ωτef f = 1 and thus for each τef f, which is dependent

on material parameters,there exists an optimal frequency to maximize the area.

When comparing numerical simulations that generate hysteresis loops to the analytic solu-tions of LRT, the results are in accordance to each other for ξ < 0.5 when the MNPs are randomly distributed, as is the case in biomedical applications. The linear response theory is most suitable for MNPs with high anisotropy, i.e. when H/HK < 1, as a higher field will

result in non-linear behavior as already mentioned in the part on equilibrium functions.

The area of the hysteresis loops show different qualitative behavior in function of the radius of the MNPs. For smaller MNPs of Fe3O4, with a radius of 3nm, the area shows a quadratic

dependence on the external field amplitude. For larger radii, the exponential dependence on the external field decreases until 0.6 and then rapidly increases when the radius r > 9nm. For the rising part of the curve, an exponent as high as 6 is observed for the area of the loop in function of external magnetic field strength when the radius equals 12nm.

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After introducing a factor κ = kBT

KV ln

kBT

4µ0HmaxMSV f τ0

!

, that describes the ratio between

thermal field and anisotropy field, the utilization of the 3 models can be summarized as follows:

• Small radii (κ > 1.6): reversible equilibrium functions, the hysteresis loops are reversible due to a fast thermal field

• Moderate radii (κ < 1.6 and ξ < 1): linear response theory (LRT), suitable for hyper-thermia applications (see section ...)

• Larger radii (κ < 0.7 and saturated MNPs): Stoner-Wohlfarth (SW) model

If the particles become too large, reversal and multiple domains will occur in the particle and these models are not valid anymore.

The viscosity of the embedding fluid can be taken in account more explicitly as described in Ref.[5].

First, low field amplitudes are described. The hysteresis loops correspond to the ellipsoids in Fig. 2.8.

It is also important to note that however the absorbed power does not explicitly depend on particle size, thermal effects will introduce certain absorption peaks because these thermal fluctuations are dependent on particle size. For small sizes, the thermal fluctuations act so fast that the power absorption is almost equal to zero as already described earlier. For too large sizes, the switching process is blocked. In between, the switching will have a similar period as the driving field, yielding higher absorption values. The phenomena for bigger sizes correspond to Brownian relaxation processes because the magnetization vectors are fixed to the easy axis when the external field is assumed to be low enough.

The value of the absorbed power is taken over a large number of field cycles to average the thermal fluctuations out. For smaller particles, the hysteresis loop will be reversible and will result in an area equal to zero. When the particles become larger, N´eel relaxation sets in and will increase this area. After reaching a maximal value, the relaxation process will be blocked and only Brownian relaxation occurs. The absorption peak, i.e. the maximal area of hysteresis loop, occurs when the switching of the magnetization with respect to the easy axis has the same frequency of the driving field.

The averaged absorbed power pav is then proportional to

pav ∼

ωτef f

1 + (ωτef f)2

(2.29)

This is true for small MNPs in the LRT model (see also Eq. (2.27)) as for larger sizes this value will reach zero and the real absorbed power goes to the limit of T = 0, which is given by the SW model and not equal to zero.