University of Florida Thesis Or Dissertation Formatting

DYNAMICS AND MAGNETIZATION DYNAMICS OF MAGNETIC NANOPARTICLES IN APPLIED MAGNETIC FIELDS

ZHIYUAN ZHAO

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2019

To my parents Wenzhong Zhao and Mei Zhu

ACKNOWLEDGMENTS

At first, I would like to express my deepest gratitude to my adviser Dr. Carlos

Rinaldi for giving me this opportunity to pursue doctoral studies and guiding me on the research scopes and skills. I will also appreciate for his advice and encouragement during my doctoral study, for his teaching in Continuum Basis class that has a great influence on my life, and for his help on my professional presenting and writing skills.

I would like to thank Dr. David P. Arnold, Dr. Jason Butler and Dr. Ranga

Narayanan, for their guidance, suggestions and support on my doctoral research and dissertation writing.

I would like to give a special thank to Dr. Isaac Torres-Díaz for his patient guidance and help on my learning of coding and computational simulations. The impressive and encouraging talks with him not only contributed to my research, but also motivated me to do better in both work and life.

I would like to thank Camilo Velez Cuervo and Nicolas Garraud for their hard work and contributions to my research.

I would like to thank all the members in the research group, for their helps and supports in my daily life.

I would like to thank my parents for supporting me to pursue my dream, and for their understanding and encourage when I got frustrated.

I would like to thank the Department of Chemical Engineering for giving me the opportunity to study in the University of Florida. The advanced research facilities and inspiring academic atmosphere provided me with higher perspectives and pave me way towards further study.

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS ...... 4

LIST OF TABLES ...... 7

LIST OF FIGURES ...... 8

LIST OF ABBREVIATIONS ...... 11

ABSTRACT ...... 12

CHAPTER

1 SCOPE OF THE DISSERTATION ...... 14

2 BROWNIAN DYNAMICS SIMULATIONS OF MAGNETIC NANOPARTICLES CAPTURED IN STRONG MAGNETIC FIELD GRADIENTS ...... 18

2.1 Background and Motivation ...... 18 2.2 Simulation Method ...... 22 2.2.1 Model of Magnetic Dipole for Different Relaxation Mechanisms ...... 22 2.2.2 Motion Equation ...... 25 2.2.3 Methods to Identify and Quantify Magnetically Captured Nanoparticles .. 32 2.2.4 Simulation Parameters and Conditions ...... 33 2.3 Results ...... 34 2.3.1 Particle Motion ...... 34 2.3.2 Magnetic Capture Rates for Different Relaxation Mechanisms ...... 35 2.3.3 Magnetic Capture Rates for Various Strengths of The External Magnetic Field Gradient...... 36 2.3.4 Magnetic Capture Rates for Various Nanoparticle Volume Fractions ...... 37 2.3.5 Shape of Magnetic Nanoparticle Aggregates ...... 38 2.4 Conclusions ...... 39

3 MAGNETIZATION DYNAMICS AND ENERGY DISSIPATION OF INTERACTING MAGNETIC NANOPARTICLES IN ALTERNATING MAGNETIC FIELDS WITH AND WITHOUT A STATIC BIAS FIELD ...... 47

3.1 Background and Motivation ...... 48 3.2 Simulation Method ...... 51 3.2.1 Brownian Dynamics Simulations ...... 51 3.2.2 Simulation Parameters and Conditions ...... 52 3.2.3 Simulations of Magnetorelaxometry ...... 53 3.2.4 Simulations of Dynamic Magnetic Susceptibility ...... 55 3.2.5 Calculation of Energy Dissipation Rate ...... 56

3.3 Results ...... 56 3.3.1 Equilibrium Response of Magnetization ...... 56 3.3.2 Simulations of Magnetorelaxometry ...... 57 3.3.3 Energy Dissipation Rate in An Alternating Magnetic Field ...... 60 3.3.4 Effect of Static Bias Magnetic Field on Energy Dissipation Rate ...... 63 3.4 Conclusions ...... 64

4 EFFECTS OF PARTICLE DIAMETER AND MAGNETOCRYSTALLINE ANISOTROPY ON MAGNETIC RELAXATOIN AND MAGNETIC PARTICLE IMAGING PERFORMACE OF MAGNETIC NANOPARTICLES ...... 79

4.1 Background and Motivation ...... 80 4.2 Simulation Method ...... 85 4.2.1 The Landau-Lifshitz-Gilbert (LLG) Equation ...... 85 4.2.2 Simulation Parameters and Conditions ...... 88 4.2.3 Simulations of Magnetorelaxometry ...... 89 4.2.4 Simulation Parameters and Conditions ...... 91 4.3 Results ...... 93 4.3.1 Equilibrium Response of Magnetization ...... 93 4.3.2 Simulations of Magnetorelaxometry ...... 94 4.3.3 Magnetization Signal in An Alternating Magnetic Field ...... 99 4.4 Conclusions ...... 102

5 CONCLUSION REMARKS ...... 118

LIST OF REFERENCES ...... 123

BIOGRAPHICAL SKETCH ...... 133

LIST OF TABLES

Table page

3-1 Diameters used in simulations (퐷p) and corresponding diameters (퐷p,fit) obtained by applying a nonlinear fit to the Langevin function...... 67

3-2 Magnetic relaxation time (휏̃) for the case that an applied static magnetic field is suddenly turned on and off for various Langevin parameters and strengths of inter-particle interactions...... 69

4-1 Scaled anisotropy energy ∆퐸ani⁄푘B푇 for a representative anisotropy constant value of 퐾 = 13.5 kJ⁄m3 and various nanoparticle diameters...... 104

LIST OF FIGURES

Figure page

2-1 Magnetic field gradients in simulations...... 41

2-2 Zoomed-in 3D configurations snapshots close to the capture line at various times for magnetic nanoparticles that relax by different mechanisms...... 42

2-3 Trajectories of representative magnetic nanoparticles that respond to the applied magnetic field by different mechanisms...... 43

2-4 Number of captured magnetic nanoparticles as a function of capture time for magnetic nanoparticles that relax by the Brownian and Néel relaxation mechanisms...... 43

2-5 Number of captured magnetic nanoparticles as a function of capture time for various maximum Langevin parameters and magnetic nanoparticles that relax by different mechanisms...... 44

2-6 Number percentage of captured magnetic nanoparticles as a function of capture time for various particle volume fractions and magnetic nanoparticles that relax by different mechanisms...... 45

2-7 Average height and width of magnetic nanoparticle aggregates as a function of capture time for magnetic nanoparticles that relax by different mechanisms...... 45

2-8 Average height and width of magnetic nanoparticle aggregates as a function of capture time for various ratios of dipole-dipole interaction parameter to hard-core Yukawa repulsion parameter and magnetic nanoparticles that relax by different mechanisms...... 46

3-1 Equilibrium magnetization of magnetic nanoparticles suspension in an applied static magnetic field as a function of intensity of the field for various particle diameters...... 67

3-2 Time-dependent magnetization relaxation curves for cases where the external static magnetic field is suddenly turned on and off and under various strengths of inter-particle interactions...... 68

3-3 Number of particle chains as a function of length of the particle chain and corresponding representative snapshots of particle configuration for various intensities of static magnetic field...... 69

3-4 Number of particle chains as a function of length of the particle chain and corresponding representative snapshots of particle configuration at 푡̃ = 19.80, for various strengths of magnetic dipole-dipole interactions...... 70

3-5 Number of particle chains as a function of length of the particle chain and corresponding representative snapshots of particle configuration at 푡̃ = 19.80, for various strengths of magnetic dipole-dipole interactions...... 71

3-6 Number of particle chains as a function of length of the particle chain and corresponding representative snapshots of particle configuration at 푡̃ = 19.80, for various strengths of magnetic dipole-dipole interactions...... 72

3-7 Scaled time-dependent 푧-direction magnetization and corresponding harmonic spectra and dynamic hysteresis loops for various dimensionless angular frequencies...... 73

3-8 Magnetization curve and representative snapshots of particle configuration for various strengths of inter-particle interactions...... 74

3-9 Real and imaginary components of the complex susceptibility as a function of dimensionless angular frequency for various strengths of inter-particle interactions...... 75

3-10 Specific absorption rate as a function of the amplitude of alternating magnetic field for various angular frequencies and strengths of inter-particle interactions...... 76

3-11 Scaled time-dependent 푧-direction magnetization and corresponding harmonic spectra and dynamic hysteresis loops for various Langevin parameters of bias field...... 77

3-12 Specific absorption rate as a function of the strength of bias field for various angular frequencies and strengths of inter-particle interactions...... 78

4-1 Equilibrium average magnetization of a collection of magnetic nanoparticles in an applied static magnetic field as a function of intensity of the field for various nanoparticle diameters...... 104

4-2 Magnetization curve and corresponding magnetic relaxation curves for collections of immobilized magnetic nanoparticles with different anisotropy symmetries...... 105

4-3 Representative magnetization curves and corresponding magnetic relaxation curves for collections of immobilized magnetic nanoparticles, for various magnetic field intensities, nanoparticle diameters, values of anisotropy constant, and types of anisotropy symmetry...... 106

4-4 Magnetization curve and corresponding representative orientation distributions of magnetic dipoles of a collection of magnetic nanoparticles with uniaxial anisotropy symmetry...... 107

4-5 Magnetization curve and corresponding representative orientation distributions of magnetic dipoles for a collection of magnetic nanoparticles with cubic anisotropy symmetry...... 108

4-6 Characteristic magnetic relaxation times 휏34 or 휏3 and 휏4 as a function of intensity of the applied static magnetic field for various nanoparticle diameters, values of anisotropy constant, and types of anisotropy symmetry. ..109

4-7 Scaled characteristic magnetic relaxation times 휏34, or 휏3 and 휏4 as a function of dimensionless anisotropy constant for various values of the magnetocrystalline anisotropy constant, and for nanoparticles with different anisotropy symmetries...... 110

4-8 Scaled characteristic magnetic relaxation times 휏34, or 휏3 and 휏4 as a function of dimensionless anisotropy constant for uniaxial-anisotropy nanoparticles, various values of the magnetocrystalline anisotropy constant and damping parameters...... 111

4-9 Characteristic magnetic relaxation times 휏12 or 휏1 and 휏2 as a function of intensity of the applied static magnetic field for various values of anisotropy constant and different anisotropy symmetries...... 112

4-10 Characteristic magnetic relaxation times 휏12 or 휏1 and 휏2 as a function of intensity of the applied static magnetic field for various nanoparticle diameters, values of anisotropy constant, and types of anisotropy symmetry. ..113

4-11 Magnetization curves and corresponding dynamic hysteresis loops for collections of magnetite nanoparticles for various nanoparticle diameters and for different magnetocrystalline anisotropy symmetries...... 114

4-12 Harmonic spectrum of magnetization signal of a collection of magnetite nanoparticles for various nanoparticle diameters and for different magnetocrystalline anisotropy symmetries...... 115

4-13 Positive and negative scan tracer response of a collection of magnetite nanoparticles as a function of intensity of applied alternating magnetic field for cases of uniaxial and cubic anisotropy symmetries...... 116

4-14 Intensity, full-width-at-half-maximum (FWHM) and peak deviation of magnetization signal of a collection of magnetite nanoparticles as a function of nanoparticle diameter for cases of uniaxial and cubic anisotropy symmetries...... 117

LIST OF ABBREVIATIONS

AMF Alternating magnetic field

FWHM Full width at half maximum

LLG Landau-Lifshitz-Gilbert

MNP Magnetic nanoparticle

MPI Magnetic particle imaging

PSF Point spread function

SAR Specific absorption rate

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

DYNAMICS AND MAGNETIZATION DYNAMICS OF MAGNETIC NANOPARTICLES IN APPLIED MAGNETIC FIELDS

Zhiyuan Zhao

December 2019

Chair: Carlos Rinaldi Major: Chemical Engineering

Magnetic nanoparticles (MNPs) can align their magnetic dipole with the direction of externally applied magnetic field through internal rotation of the magnetic dipole, i.e.

Néel relaxation mechanism, or physical rotation of the nanoparticle, i.e. Brownian relaxation mechanism. Based on this property, MNPs has been exploited to drive magnetic assembly of particles in patterned static magnetic field gradients, to generate heat that can be employed to actuate release of a drug or magnetic hyperthermia in applied uniform alternating magnetic fields (AMFs), and to act as tracers in magnetic particle imaging (MPI) technology. However, a literature review suggests that no prior work explicitly compared the dynamic capture process and aggregate size of MNPs with

Néel and Brownian relaxation mechanisms for the application of magnetic assembly.

Some prior computational work has studied the role of inter-particle interactions in heat dissipation of MNPs but has not considered the potential role of field-induced particle aggregation on energy dissipation rate. Additionally, no prior computational work has studied the effects of nature of particle diameter, magnetic anisotropy energy and magnetocrystalline anisotropy symmetry on x-space MPI performance of MNPs undergoing the Néel relaxation.

In this work, Brownian dynamics simulations are carried out to computationally study the magnetic capture rates and evolution of aggregate shapes of MNPs with different relaxation mechanisms in static magnetic field gradients, and the magnetization dynamics and energy dissipation rates of Brownian-relaxation MNPs in

AMFs. The results suggest that the relaxation mechanism and strength of inter-particle interaction have great influence on the capture rate and aggregate shape of MNPs. The effect of the strength of inter-particle interaction on energy dissipation rates of the

Brownian-relaxation MNPs in AMFs is also significant. Moreover, simulations based on the Landau-Lifshitz-Gilbert (LLG) equation was performed for a collection of Néel- relaxation MNPs that are fixed in space, the results of which suggest that particle diameter and magnetocrystalline anisotropy (both symmetry and energy) play an important role in the magnetization dynamics and MPI performance of the nanoparticles. In summary, these computational studies provide theoretical insight into the dynamics and magnetic dynamics of MNPs in applied magnetic fields.

CHAPTER 1 SCOPE OF THE DISSERTATION

The focus of this dissertation is to study the dynamics and magnetization dynamics of spherical single-domain magnetic nanoparticles in different types of externally applied magnetic fields, by carrying out computational simulations. Here the term “single-domain” indicates that the MNPs are so small that the magnetization of particle material (both the magnitude and direction) is uniform throughout the particle. In such case, each MNP can be treated to occupy a magnetic dipole moment, which can responds to changes of external magnetic field and re-align to the field direction through two mechanisms: Néel relaxation mechanism and Brownian relaxation mechanism.1 In the Néel relaxation mechanism, the internal magnetic dipole undergoes a fast rotation to align with the local magnetic field, whereas the nanoparticle is motionless. For the

Brownian relaxation mechanism, the nanoparticles have their magnetic dipoles

“thermally-blocked” in a so-called crystal easy axis due to high magnetocrystalline anisotropy barriers, such that they must physically rotate to align their dipoles with the local magnetic field1. The different relaxation mechanisms finally lead MNPs to behave in very distinct manners in applied magnetic fields.

To conduct the study on the dynamics and magnetization dynamics of MNPs, it is significant to develop algorithms that describe the translational and rotational motion of the nanoparticles and their magnetization evolution governed by Néel and Brownian relaxation mechanisms. When carrying out simulations, it is also necessary to take into account various conditions and parameters, such as type and magnitude of magnetic field, property of nanoparticles and interactions between nanoparticles, because of the limitations or requirements of actual applications of the nanoparticles. In this

dissertation, computational studies were conducted for some representative and popular applications of MNPs, including magnetic assembly, heat dissipation and biomedical imaging. In the simulations for each application, algorithm was customized, magnetic field and condition were simulated and wider range of parameters were discussed. The dissertation structure is shown as follow.

Chapter 2 presents a computational study of the capture of MNPs with Brownian or Néel relaxation mechanisms in a magnetic field gradient that is generated by a reversal in perpendicular magnetization of a substrate. Brownian dynamics simulations are carried out by taking into account the effects of hydrodynamic drag, force due to the applied magnetic field, magnetic dipole-dipole interactions, repulsive hard-core Yukawa potential, hydrodynamic particle-wall interactions and thermal agitations. For the nanoparticles that relax with Brownian mechanism, additional magnetic torque and thermal rotation are taken into account. In results, the effects of mechanism of magnetic relaxation, magnitude of the magnetic field gradient, volume fraction of the nanoparticles, and strength of particle-particle interactions are explored in relation to the rate of magnetic capture, particle capture trajectories, and shape (average width and height) of the collection of captured nanoparticles.2

Chapter 3 presents a computational study of the magnetization dynamics of spherical single-domain magnetically-blocked nanoparticles in static and AMFs. The same algorithm of Brownian dynamics simulation as in Chapter 2 is employed but improved to include AMFs. In the case of an applied static magnetic field, we investigated the effect of magnetic dipole-dipole interactions on the method of calculating the effective magnetic diameter of MNPs by fitting their equilibrium

magnetization to the Langevin function. Then, we study the magnetic relaxation time of the nanoparticles for cases where a static magnetic field is suddenly turned on or turned off and for various values of the magnetic interaction strength parameter. In cases where an AMF is applied, with or without a static bias field, the particle response is analyzed in terms of the evolution and harmonic spectrum of average magnetization, dynamic hysteresis loops, and calculated SAR as a function of the amplitude and frequency of the AMF, value of the magnetic interaction strength parameter, and the magnitude of an applied bias field.3

Finally, Chapter 4 presents a computational study of the effect of particle diameter and magnetic anisotropy (considering both type of symmetry and barrier energy magnitude) on the magnetization dynamics of immobilized spherical single- domain MNPs in static and AMFs. Simulations based on the LLG equation are carried out to account for the damped precession of internal magnetic dipoles due to the applied magnetic field, magnetocrystalline anisotropy energy barrier and thermal agitations. In the case of static magnetic fields, a comparison was made between the equilibrium magnetization response of the nanoparticles with uniaxial and cubic anisotropy and the predictions of the Langevin function. Then, we investigated the effects of particle diameter, magnetic anisotropy symmetry and energy on the magnetic relaxation time of the nanoparticles as well as the dynamics of the magnetic dipole moments for cases where a static magnetic field is suddenly turned on and off. For the case of an applied AMF, the intrinsic MPI performance of magnetite nanoparticles that are typical of MPI applications was studied in terms of the evolution and harmonic spectrum of ensemble magnetization, hysteresis loops, and signal PSFs for various

nanoparticle sizes. The focus of this work is on comparing the magnetization dynamics and MPI performance of MNPs with different sizes and magnetic anisotropies.

CHAPTER 2 BROWNIAN DYNAMICS SIMULATIONS OF MAGNETIC NANOPARTICLES CAPTURED IN STRONG MAGNETIC FIELD GRADIENTS

In Chapter 2, the behavior of spherical single-domain MNPs in strong inhomogeneous magnetic fields is investigated through Brownian dynamics simulations, by taking into account magnetic dipole-dipole interactions, repulsive hard-core Yukawa potential, hydrodynamic particle-wall interactions and the mechanism of magnetic dipole rotation in the presence of a magnetic field. The magnetic capture process of nanoparticles in prototypical magnetic field gradients generated by a sudden reversal in perpendicular magnetization of a flat substrate (defining a “capture line”) is studied as a function of strength of the magnetic field and volume fraction of the MNPs. Capture curves show a regime where capture follows a power law model and suggest that nanoparticles with the Brownian relaxation mechanism are captured at a slightly faster rate than nanoparticles with the Néel relaxation mechanism, under similar conditions of the field gradient. Additionally, evaluation of the shape of the aggregates of captured nanoparticles suggests that greater dipole-dipole interactions result in aggregate structures that are flatter/wider than in the case of negligible dipole-dipole interactions.

These results can help guide the design of systems for magnetically-directed assembly of nanoparticles into complex shapes at a substrate.

2.1 Background and Motivation

Magnetic particles can be manipulated by applied magnetic fields to drive organization of magnetic and non-magnetic particles into a variety of structures through

Figures reproduced with permission from Zhao, Z. Y.; Torres-Diaz, I.; Velez, C.; Arnold, D.; Rinaldi, C., Brownian Dynamics Simulations of Magnetic Nanoparticles Captured in Strong Magnetic Field Gradients. J Phys Chem C 2017, 121 (1), 801-810.

directed magnetic assembly at a substrate with patterned magnetic field gradients4-6. As examples, MNPs have been assembled to generate diffraction gratings7, to produce geometric patterns such as lines, triangles, squares and circles8,9, and to produce structures with more complex shapes that can be crosslinked and released, generating free-floating magnetic microstructures10.

On par with experimental studies of magnetic capture of particles at a magnetically-patterned substrate, several groups have developed computational models to study the mechanism of magnetic particle capture and to provide a theoretical basis for the rational design of magnetic patterns to obtain structures of interest rapidly and reproducibly. The magnetic capture of particles in a variety of applications has motivated computational study of the effects of gravity and buoyancy and of fluid flow on particle trajectories11,12, distributions13 and capture efficiency14-16. However, in most of these cases the effect of Brownian motion of the particles has been neglected because the focus has been on the behavior of micron-sized particles. This is not the case for capturing nanoparticles using magnetically-patterned substrates, where the nanoscale size of the particles can make the effect of translational and rotational Brownian motion significant, especially at moderate and large distances from the magnetic patterns due to spatial decay of the magnetic field gradient.

Some prior computational work has been done to study nanoparticle assembly in quiescent fluids, with an emphasis on studying the shapes adopted by the magnetically captured nanoparticles. For example, Xue et al.17 developed a simulation method to study the assembly of magnetic-dielectric core-shell nanoparticles into extended monolayer geometric patterns with nanoscale precision. In their work, the suspension

was modeled as containing a low number density of nanoparticles, which responded to the action of forces generated by external fields, viscous drag, Brownian motion, magnetic dipole-dipole interactions, and other inter-particle interactions. The influence of these interactions on configurations adopted by the captured nanoparticles was investigated in situations with varying shapes of the magnetic pattern, under different conditions of nanoparticle volume fraction and properties. More recently, using a model combining the Langevin equation and the Monte Carlo method, Xue et al.18 provided insight into the self-assembly of mono- and polydisperse magnetic dielectric core-shell nanoparticles into multilayer structures. Again, the focus was on the configurations adopted by the captured nanoparticles and not on the dynamics of the capture process.

Although the prior computational work cited above has provided important insights into the magnetic particle capture process, these studies assume that the particles respond to the applied magnetic fields through the Néel relaxation mechanism of fast internal dipole rotation to align with the local magnetic field. While this is certainly a valid assumption for small spherical magnetic nanoparticles consisting of iron oxide, nanoparticles with other compositions, such as cobalt ferrite, can have their magnetic dipoles “thermally-blocked” and relax through the Brownian mechanism. For MNPs with the Brownian relaxation mechanism the process of magnetic capture could be influenced by both their translational and rotational Brownian motion, as well as by magnetic torques exerted on their thermally-blocked magnetic dipoles. The addition of thermally-blocked nanoparticles, such as cobalt ferrite, to magnetically-assembled structures could yield free-floating magnetic microstructures with mixed magnetic relaxation modes, which in turn could be used in their external manipulation.10 However,

we are not aware of any theoretical or computational studies that explicitly compare the magnetic capture of nanoparticles with Néel and Brownian relaxation mechanisms.

In Chapter 2, we report a computational study of the capture of MNPs with

Brownian or Néel relaxation mechanisms in a magnetic field gradient that is generated by a reversal in perpendicular magnetization of a substrate. This situation, illustrated in

Figure 2-1, is representative of the magnetic field gradients generated for magnetic particle capture by using magnetically patterned perpendicular recording media. The magnetic patterns used here represent those obtained by using selective reversal magnetization10. Considering that the region with maximum magnetic field gradient lies along the interface of two regions with opposed magnetic poles, we define this as the

“magnetic capture line”. A simulation algorithm was developed to account for translational and rotational Brownian motion of the MNPs. For the case of nanoparticles with Néel relaxation the algorithm takes into account the dependence of the nanoparticle’s magnetic dipole on the magnitude of the local magnetic field, as well as the effects of magnetic force, magnetic dipole-dipole interactions, hydrodynamic drag, hydrodynamic particle-wall interactions, repulsive hard-core Yukawa potential, and thermal agitations. For the case of nanoparticles with Brownian relaxation mechanism the algorithm also takes into account the coupling of translational and rotational motion of the nanoparticles, as well as the effects of magnetic torque and thermal rotation, in addition to those already mentioned for the Néel-relaxation nanoparticles. The magnetic fields generated by the magnetically-patterned substrate are calculated using COMSOL

Multiphysics® and used in the algorithm to calculate the magnetic forces and torques

(for the case of Brownian-relaxation nanoparticles) on the nanoparticles. For simplicity,

here we consider the case of a long “magnetic capture line”, although we point out that the same algorithm can be employed to simulate capture by more complex magnetically-patterned substrates. The effects of mechanism of magnetic relaxation, magnitude of the magnetic field gradient, volume fraction of the nanoparticles, and strength of particle-particle interactions are explored in relation to the rate of magnetic capture, particle capture trajectories, and shape (average width and height) of the collection of captured nanoparticles.

2.2 Simulation Method

2.2.1 Model of Magnetic Dipole for Different Relaxation Mechanisms

Most work11,13,19 assumes that the magnetic dipole moment of each MNP is

“saturated” along the direction of the local magnetic field, that is, it is commonly assumed that the magnetic dipole moment 퐦 of nanoparticle is given by an expression of the form:

퐦 = 휇퐇̂ (2-1) where 휇 is a constant and 퐇̂ represents a unit vector pointing along the local direction of the magnetic field 퐇. While this simple assumption captures the expectation that the magnetic dipole instantaneously aligns with the local magnetic field, it does not capture the dependence of the strength of the particle’s magnetic dipole on the magnitude of the local magnetic field. For nanoparticles that respond to changes of magnetic field through the Néel relaxation mechanism, the magnetic dipoles are continuously

−9 20 changing direction due to thermal agitations at characteristic time scales (휏N~10 s) that are much shorter than the characteristic time scales for particle translation and rotation of interest in magnetic capture. As such, the magnetic dipole is capable of

sampling configurations where it is not aligned with the local magnetic field, leading to reduced magnetic dipole strength when one averages over a time longer than the Néel relaxation time but shorter than the characteristic time for particle translation. This reduction in the strength of the magnetic dipole is expected to result in a reduction in the strength of the magnetic forces experienced by the particles, which in turn is expected to influence the rate of magnetic particle capture.

In our work, MNPs that relax by the Brownian relaxation mechanism are assumed to always have a “saturated” dipole moment, of magnitude given by

푚s = 푀d푉p (2-2) where 푀d represents the saturation magnetization of the material and 푉p represents the nanoparticle volume. The direction of the dipole moment varies due to the balance of rotational Brownian motion and magnetic/hydrodynamic torques on the nanoparticle.

Past work21,22 has demonstrated that such a model accurately accounts for the field- dependent ensemble average magnetization of a suspension of MNPs. On the other hand, for nanoparticles that relax by the Néel mechanism, it is assumed that the direction of the dipole instantaneously aligns with the direction of the local magnetic field while, instead of a “saturated” dipole moment, the magnitude of the dipole 푚eff is given by the Langevin function 퐿(훼):

푚 1 eff = coth 훼 − ≡ 퐿(훼) (2-3) 푚s 훼 where the Langevin parameter is

휇 푚 퐻 훼 = 0 s 0 (2-4) 푘B푇

−7 2 In Equation (2-4) the vacuum permeability is 휇0 = 4휋 × 10 N⁄A , 퐻0 represents the magnitude of the magnetic field, 푘B is the Boltzmann constant and 푇 represents the absolute temperature. It should be noted that the assumption of “effective” dipole magnitude is only used in Chapter 2, because the relaxation time of Néel-relaxation nanoparticles is much shorter as compared with the characteristic time for particle translation and rotation. In Chapter 4, the assumption of “saturated” dipole moment will be re-employed to account for the Néel relaxation mechanism of immobilized MNPs but in a different simulation model. It will be talked later.

The simulation box is set up in a Cartesian coordinate system and fixed to free space (i.e., the laboratory coordinates). As a result, a magnetic dipole moment in the particle coordinates is given by

퐦′ = 휇0푚퐦̂ (2-5) where the prime indicates a vector is in particle coordinates, 푚 can be replaced by 푚s and 푚eff for Brownian- and Néel-relaxation nanoparticles, respectively, and 퐦̂ is a unit vector specifying the orientation of the magnetic dipole moment. By assuming that all

“saturated” magnetic dipole moments have uniform magnitude and always point along the +푧 direction of the particle coordinates, 퐦̂ can be transformed into the laboratory space through

퐦̂ = 퐀−1 ∙ 퐦̂′ (2-6) where 퐀 is the transformation matrix in the form of

−휁2 + 휂2 − 휉2 + 휒2 2(휁휒 − 휂휉) 2(휁휂 + 휉휒) 퐀 = [ −2(휂휉 + 휁휒) −휁2 − 휂2 + 휉2 + 휒2 2(휂휒 − 휁휉) ] (2-7) 2(휁휂 − 휉휒) −2(휁휉 + 휂휒) 휁2 − 휂2 − 휉2 + 휒2

In Equation (2-7), 휁, 휂, 휉 and 휒 are the quaternion parameters, which satisfy the condition 휁2 + 휂2 + 휉2 + 휒2 = 1.23

2.2.2 Motion Equation

When external magnetic fields are applied to a suspension of MNPs, the forces and torques acting on each dispersed nanoparticle include those due to hydrodynamic drag 퐅h and 퐓h, those due to external magnetic fields 퐅m and 퐓m, those due to magnetic dipole-dipole interactions 퐅dd and 퐓dd, those due to other particle-particle interactions, represented here through a repulsive hard-core Yukawa potential for the case of charge-stabilized nanoparticles 퐅Ykw, and those due to thermal agitation resulting from collisions of the nanoparticles with solvent molecules 퐅B and 퐓B. For a rigid and nanosized particle, its motion can be governed by stochastic linear and angular momentum equations and the assumption of negligible inertia is justified24. Thus, the linear and angular momentum balance equations reduce to force and torque balances, expressed as

ퟎ 퐅 퐅 퐅 퐅 퐅 ( ) = ( h ) + ( m ) + ( dd) + ( Ykw) + ( B) (2-8) ퟎ 퐓h 퐓m 퐓dd ퟎ 퐓B

Since in a dilute suspensions of MNPs long-range magnetic interactions are expected to be more significant than inter-particle hydrodynamic interactions, we explicitly neglect particle-particle hydrodynamic interactions in our simulations.25 Because our interest here is to simulate particle capture during magnetically-guided assembly, wherein the goal is to obtain well defined patterns of nanoparticles at a substrate, our simulations focus on a system where the nanoparticles are colloidally stable against aggregation in solution. This implies conditions where repulsive interactions, such as electrostatic repulsion, are dominant over attractive interactions, such as van der Waals interactions.

As such, in our simulations we take into account particle-particle interactions through a repulsive Yukawa potential, as an approximation for long-range electrostatic repulsion, and long-range magnetic dipole-dipole interactions. We explicitly neglect the effect of van der Waals interactions as these are of much shorter range and, if dominant, would result in particle aggregations in solution. Furthermore, the strength of the magnetic dipole-dipole interactions is non-negligible, but not high enough to cause significant chain formation. Finally, a hard-sphere repulsion potential is used to prevent particle- particle and particle-wall overlap, which can be a problem especially close to the magnetic capture line and for high values of the magnetic field gradient.

In a quiescent fluid, a particle is always subject to the hydrodynamic force 퐅h and torque 퐓h, which are related to the velocity 퐔 and angular velocity 훚 of the particle through the mobility matrix 퓜, in the manner of

퐔 퐅 ( ) = −퓜 ∙ ( h) (2-9) 훚 퐓h

The symmetric and positive-definite mobility matrix can be written as

UF UT 퓜 = (퐌 퐌 ) (2-10) 퐌ωF 퐌ωT where 퐌UF, 퐌UT, 퐌ωF and 퐌ωT are mobility components that vary with the fluid viscosity and particle position relative to the wall, and relate hydrodynamic forces and torques to particle translational and rotational velocity.

In our work, the magnetic field results from the reversal perpendicular magnetization in a planar substrate, with one region magnetized upwards (+푧 direction) and the other magnetized downwards (−푧 direction). Such a situation is found, for example, in perpendicular recording media, where specific areas of a thin magnetic

layer (e.g. films, substrates, tapes, hard drives) are magnetized out of plane, in opposite directions of the original magnetization of the layer. To understand this behavior, a 2D magneto-static finite-element simulation26 was used to estimate the magnetic field at a magnetic pole boundary in a magnetic substrate (Hi8MP video cassette tape).10 This simulation was carried out using the AC/DC module in the COMSOL Multiphysics® software (version 5.1), using “Magnetic Fields, No Current” (mfnc) as the physics. The out of plane magnetization of the tape material was defined using demag-corrected magnetization curves measured using a vibrating sample magnetometer (VSM-EV9

ADE technologies) showing a remanence (휇0푀r) of ~50 mT and a coercivity (퐻c) of

~65 kA/m. A 2D cross section simulation was recreated with an air domain (1 μm height and 4 μm length) on top of the magnetic substrate (1.75 μm thick and 15 μm length), as illustrated in Figure 2-1, in which 퐻max represents the maximum magnetic field in the section.

In a medium without time-varying electric fields or currents, the magnetic force acting on the MNPs is given by

퐅m = 휇0퐦 ∙ 훁퐇 (2-11) where 퐇 is a vector that represents the magnetic field and 훁퐇 is a matrix that represents the magnetic field gradient. Nanoparticles that relax by the Brownian mechanism also experience a magnetic torque when their dipole moment is not aligned with the local magnetic field, given by

퐓m = 휇0퐦 × 퐇 (2-12)

The magnetic torque is zero for the Néel-relaxation nanoparticles because we consider their magnetization is always collinear with the local magnetic field.

The simulation algorithm included the magnetic forces and torques resulting from magnetic dipole-dipole interactions between MNPs. The magnetic dipole-dipole force and torque exerted by nanoparticle 푗 on nanoparticle 푖 are given by27,28

3휇 푚 푚 퐅 = 0 푗 푖 [퐫̂ (퐦̂ ∙ 퐦̂ ) + 퐦̂ (퐫̂ ∙ 퐦̂ ) + 퐦̂ (퐫̂ ∙ 퐦̂ ) dd,푗푖 4휋푟 4 푗푖 푗 푖 푗 푗푖 푖 푖 푗푖 푗 푗푖 (2-13)

− 5퐫̂푗푖(퐫̂푗푖 ∙ 퐦̂푗)(퐫̂푗푖 ∙ 퐦̂ 푖)]

휇0푚푗푚푖 퐓dd,푗푖 = 3 [3(퐦̂푗 ∙ 퐫̂푗푖)(퐦̂ 푖 × 퐫̂푗푖) + (퐦̂푗 × 퐦̂푖)] (2-14) 4휋푟푗푖 where 푚푗 and 푚푖 represent the magnitudes of magnetic dipole moment 푗 and 푖, respectively, 푟푗푖 represents the center distance between nanoparticle 푗 and 푖, 퐫̂푗푖 represents the unit vector of center distance, and 퐦̂푗 and 퐦̂ 푖 represent the unit vectors of dipole moment 푗 and 푖, respectively.

The algorithm also takes into account repulsion between the nanoparticles due to electrostatic interactions, modeled using a repulsive hard-core Yukawa potential,

−1⁄3 29 truncated at 휎 ≤ 푟푗푖 ≤ 휆휌 and which has the form

exp[−휅(푟푗푖 − 휎)] 푢Ykw,푗푖 = 휀 (2-15) 푟푗푖⁄휎 where 휎 is the hard-core diameter, 휆휌−1⁄3 denotes the cut-off distance, 휆 is a pre-factor to modulate the cut-off distance, 휌−1⁄3 is proportional to the average inter-particle distance, 휀 represents the pair potential, and 휅 represents the inverse Debye screening length. The force due to the repulsive hard-core Yukawa potential is obtained from

퐅Ykw,푗푖 = −훁푢Ykw,푗푖 (2-16)

After substituting the forces and torques due to different effects, the motion equation of nanoparticle 푖 becomes

∑ 퐅dd,푗푖 ∑ 퐅 퐔푖 퐅m,푖 푗≠푖 Ykw,푗푖 퐅B,푖 ( ) = 퓜푖 ∙ ( ) + 퓜푖 ∙ + 퓜푖 ∙ (푗≠푖 ) + 퓜푖 ∙ ( ) (2-17) 훚푖 퐓m,푖 퐓B,푖 ∑ 퐓dd,푗푖 ퟎ (푗≠푖 )

Dimensionless variables are introduced according to

퐔푖 훚 퐇 푟푗푖 퐔̃푖 = , 훚̃ 푖 = , 훁̃ = 푎훁, 퐇̃ = , 푟푗푖̃ = 휅̃ = 푎휅 (2-18) 푎퐷r 퐷r 퐻max 푎

3 −1 where 푎 is the radius of the uniform-sized nanoparticles, 퐷r = 푘B푇(8휋휂0푎 ) is the rotational diffusivity for a spherical particle and 휂0 is the viscosity of the carrier fluid. The mobility matrix components are non-dimensionalized according to

UF 퐌 4 푎 UF = 퐦푖 (2-19) 푎퐷r 3 푘B푇

UT 퐌 4 1 UT = 퐦푖 (2-20) 푎퐷r 3 푘B푇

ωF 퐌 4 푎 ωF = 퐦푖 (2-21) 퐷r 3 푘B푇

ωT 퐌 4 1 ωT = 퐦푖 (2-22) 퐷r 3 푘B푇

UF UT ωF ωT where 퐦푖 , 퐦푖 , 퐦푖 and 퐦푖 are the dimensionless components of the mobility

UF UT ωF ωT matrix corresponding to 퐌 , 퐌 , 퐌 and 퐌 , respectively. By setting 푑퐱̃푖 and 푑횽̃ 푖 as the infinitesimal translation and rotation vectors of the nanoparticle 푖, their relationship with velocity and angular velocity are given by

푑퐱̃푖 퐔̃ ( ) = ( 푖 ) 푑푡̃ (2-23) 푑횽̃ 푖 훚̃ 푖 where 푡̃ is non-dimensionalized according to 푡̃ = 푡퐷r. Integrating from 푡̃ to 푡̃ + ∆푡̃ based on the first-order forward Euler method, and applying the fluctuation-dissipation theorem30 to the Brownian terms, the motion equation results in

4 훼 (퐦UF ∙ 퐞 + 퐦UT ∙ 퐞 ) 푑퐱̃ max 푖 퐅m,푖 푖 퐓m,푖 ( 푖 ) = ( 3 ) Δ푡̃ 푑횽̃ 4 푖 훼 (퐦ωF ∙ 퐞 + 퐦ωT ∙ 퐞 ) max 3 푖 퐅m,푖 푖 퐓m,푖

퐞퐅 퐞퐓 훽 (퐦UF ∙ ∑ dd,푖 + 퐦UT ∙ ∑ dd,푖) dd 푖 푟̃ 4 푖 푟̃ 3 푗≠푖 푗푖 푗≠푖 푗푖 + Δ푡̃ 퐞퐅 퐞퐓 훽 (퐦ωF ∙ ∑ dd,푖 + 퐦ωT ∙ ∑ dd,푖) dd 푖 푟̃ 4 푖 푟̃ 3 (2-24) ( 푗≠푖 푗푖 푗≠푖 푗푖 )

8 1 휅̃ −휅̃(푟̃푗푖−2) UF 훽Ykw ∑ 푒 ( 2 + ) (퐦푖 ∙ 퐫̂푗푖) 3 푟̃ 푟푗푖̃ 푗≠푖 푗푖 4 + Δ푡̃ + 훁̃ ∙ 퓜̃∆푡̃ 8 1 휅̃ 3 −휅̃(푟̃푗푖−2) ωF 훽Ykw ∑ 푒 ( 2 + ) (퐦푖 ∙ 퐫̂푗푖) 3 푟̃ 푟푗푖̃ ( 푗≠푖 푗푖 )

퐗̃ (Δ푡̃) + ( 푖 ) 퐖̃푖(Δ푡̃)

As the time step Δ푡̃ is assumed longer than the relaxation time for particle momentum, the nanoparticle has null acceleration. The first term on the right side of Equation (2-24) calculates the translational and rotational variance of nanoparticles due to the external magnetic field at each time step, where we have defined

̃ ̃ 퐞퐅m,푖 = 퐦̂ 푖 ∙ 훁퐇 (2-25)

̃ 퐞퐓m,푖 = 퐦̂ 푖 × 퐇 (2-26) and the maximum Langevin parameter 훼max = 휇0푚퐻max⁄푘B푇. Similarly, the second term and the third term on the right side of Equation (2-24) calculates the one-step variance in both position and orientation of nanoparticles due to magnetic dipole-dipole interactions and repulsive hard-core Yukawa potential, respectively, by defining

퐞퐅dd,푖 = 퐫̂푗푖(퐦̂푗 ∙ 퐦̂ 푖) + 퐦̂푗(퐫̂푗푖 ∙ 퐦̂푖) + 퐦̂푖(퐫̂푗푖 ∙ 퐦̂푗) − 5퐫̂푗푖(퐫̂푗푖 ∙ 퐦̂푗)(퐫̂푗푖 ∙ 퐦̂푖) (2-27)

1 퐞 = (퐦̂ ∙ 퐫̂ )(퐦̂ × 퐫̂ ) + (퐦̂ × 퐦̂ ) (2-28) 퐓dd,푖 푗 푗푖 푖 푗푖 3 푗 푖

2 3 and by setting the parameter of magnetic dipole-dipole interaction 훽dd = 휇0푚 ⁄휋푎 푘B푇 and the parameter of hard-core Yukawa repulsion 훽Ykw = 휀⁄푘B푇. In Equation (2-

24), 4훁̃ ∙ 퓜̃∆푡̃⁄3 is the term due to Brownian drifta, where the normalized mobility matrix of particle 푖 is given by

UF UT ̃ 퐦푖 퐦푖 퓜푖 = ( ωF ωT) (2-29) 퐦푖 퐦푖

퐗̃푖(Δ푡̃) and 퐖̃푖(Δ푡̃) are random vectors characterized by a Gaussian distribution with mean and covariance asb

〈퐗̃푖(Δ푡̃)〉 = ퟎ (2-30)

〈퐖̃푖(Δ푡̃)〉 = ퟎ (2-31)

퐗̃푖(Δ푡̃) 퐗̃푖(Δ푡̃) 8 〈( ) ( )〉 = 퓜̃ 푖Δ푡̃ (2-32) 퐖̃푖(Δ푡̃) 퐖̃푖(Δ푡̃) 3 respectively. To solve Equation (2-32), Cholesky decomposition was introduced.

To preclude the overlap of MNPs with neighbors and take into account confinement due to the boundary wall, excluded volume interactions between nanoparticles and between nanoparticles and the wall are taken into account by applying a hard sphere interaction.

a The Brownian drift term was not shown in Reference 2 because of typographical error, but it was included in the algorithms of all the simulations in Reference 2. b Equation (2-32) is different from the one shown in Reference 2, which is incorrect due to a mistake in calculation. The correct expression is shown in Equation (2-32). All the results in Chapter 2 are obtained by carrying out simulations with the correct expression.

The calculation for far field hydrodynamic interactions between single nanoparticles and the wall is included in the normalized mobility terms31

1 −1 −3 −5 푚UF = − (9ℎ̃ − 2ℎ̃ + ℎ̃ ) (훿 − 훿 훿 ) 푖,푝푞 16 푖 푖 푖 푝푞 푝3 푞3 (2-33) 1 −1 −3 −5 − (9ℎ̃ − 4ℎ̃ + ℎ̃ ) 훿 훿 8 푖 푖 푖 푝3 푞3

3 −4 푚ωF = − ℎ̃ 휀 (2-34) 푖,푝푞 32 푖 3푝푞

15 −3 3 −3 푚ωF = − ℎ̃ (훿 − 훿 훿 ) − ℎ̃ 훿 훿 (2-35) 푖,푝푞 64 푖 푝푞 푝3 푞3 32 푖 푝3 푞3 ̃ where ℎ푖 denotes the radius-scaled distance between nanoparticle 푖 and the wall, 훿푝푞 is the Kronecker delta function, and 휀3푝푞 is the Levi-Civita symbol. Due to the symmetry of

UT ωF the mobility matrix, the matrix component 퐦푖 can be obtained by transposing 퐦푖 .

2.2.3 Methods to Identify and Quantify Magnetically Captured Nanoparticles

An aggregate-box method was used to count the number of MNPs that are captured by the patterned magnetic field. In this method, a box with size

0.3 μm × 4 μm × 0.2 μm was considered immediately above the capture line. The number of magnetically captured nanoparticles at a given time point was calculated by subtracting the initial number of nanoparticles from the current number of nanoparticles in the aggregate box. Although this method is appropriate for arbitrary strength of external magnetic fields, it cannot evaluate the configuration of the captured nanoparticles and is unable to track particle position. In order to track changes in the shape of the aggregates of captured nanoparticles, we applied another method where nanoparticles that reach the capture line are treated as being magnetically captured.

This approach is supported by observations of nanoparticle trajectories in simulations,

where we observed that captured nanoparticles can continue to move within the aggregates. We found that the capture-line method was only suitable for cases with strong applied fields (훼max ≥ 70), but it is more appropriate to observe the shape of aggregates based on the positions of captured MNPs.

2.2.4 Simulation Parameters and Conditions

Simulations were made for spherical MNPs with uniform magnetic radius of

10 nm, dispersed in a solvent with the viscosity of water and in a simulation box with size of 4 μm × 4 μm × 1 μm. The bottom surface of the simulation box corresponds to an impermeable wall, whereas the vertical surfaces are modeled as periodic boundaries.

Nanoparticles that leave the simulation box through the top surface are re-introduced at random positions in the 푥- and 푦- directions. The temperature in the simulations corresponds to 300 K. Runs were executed starting from random particle configurations, and using a minimum time interval of Δ푡̃ = 0.01. The cut-off distance corresponding to a pre-factor of 휆 = 1 was taken into account for both magnetic dipole-dipole interactions and repulsive hard-core Yukawa potential. In addition, for the repulsive hard-core

Yukawa potential we used a radius-scaled inverse Debye screening length of 휅̃ = 3, and a scaled interaction energy of 5 ≤ 훽Ykw ≤ 20. The scaled dipole-dipole interaction was 훽dd = 100. The maximum Langevin parameter in the simulation box was varied in the range of 10 ≤ 훼max ≤ 500, which for a nanoparticle with diameter of 10 nm and a temperature of 300 K corresponds to maximum magnetic field 17.63 kA⁄m ≤ 퐻max ≤

881.73 kA⁄m. Nanoparticle volume fractions of 0.001 % ≤ 휙 ≤ 0.07 % were considered.

2.3 Results

Simulation runs for MNP capture under a wide range of conditions were analyzed with respect to nanoparticle configurations and representative particle trajectories over time, magnetic capture rates by evaluating the number of captured MNPs as a function of time, and temporal variation of the width and height of the collection of captured nanoparticles. These results are presented and discussed below.

2.3.1 Particle Motion

Figure 2-2 shows representative zoomed-in 3D configuration snapshots of MNPs that are close to the capture line at different time steps during the magnetic capture process, for scaled parameter of Yukawa repulsion 훽Ykw = 5, maximum Langevin parameter of 훼max = 100, particle volume fraction 휙 = 0.05 %, and for nanoparticles relaxing through the Brownian and Néel relaxation mechanisms. These snapshots were plotted through POV-Ray, using a top-view camera perspective, setting the left side and right side of the capture line as the south pole and north pole of the magnetic substrate, and specifying the north pole of the magnetic dipole moment in red color and the south pole in white. In such 3D configurations, all dimensions are scaled by the uniform nanoparticle radius. As seen in Figure 2-2, MNPs that are initially dispersed in random positions and orientations are attracted by the strong magnetic field gradients and translate towards the capture line (푥 = 푧 = 0), where the magnetic field strength is highest. Some nanoparticles look larger than others because they are closer to the camera plane in constructing the image. With the increase in the number of captured nanoparticles and due to the balance of excluded volume interactions, electrostatic repulsion and dipole-dipole interactions, the size of the nanoparticle aggregates along the capture line grows over time.

Figure 2-3 shows trajectories of four representative MNPs that relax by the different relaxation mechanisms. The simulations were executed at a particle volume fraction of 휙 = 0.05% and with the same magnetic field and condition of inter-particle interaction strength. Although all the nanoparticles shown in Figure 2-3 are eventually captured by the magnetic field gradient, it is observed that the Néel-relaxation nanoparticles undergo more complex paths as compared with the Brownian-relaxation nanoparticles. Together, Figures 2-2 and 2-3 suggest that MNPs that relax by the

Brownian mechanism are captured at a faster rate than nanoparticles relaxing by the

Néel mechanism.

2.3.2 Magnetic Capture Rates for Different Relaxation Mechanisms

Figure 2-4 shows the number of captured nanoparticles as a function of capture time, i.e. the magnetic capture curves, for MNPs that relax by the Brownian and Néel mechanisms, under identical conditions of inter-particle interaction strength, magnetic field gradient, and particle volume fraction. It should be noted that the y-intercepts of the magnetic capture curves vary with simulation conditions because the capture curves are plotted starting from the second time step. In Figure 2-4, the curves initially have almost constant slopes. After some time, the slope of the capture curve decays as the number of captured nanoparticles saturates. We thus separate the capture process into two periods. In the first period, the dispersed nanoparticles surge towards the capture line due to the magnetic force and assemble along the capture line. The slope of the capture curve during this period is considered to follow a power law between the number of magnetically captured nanoparticles 푁c and capture time 푡:

ln 푁c = 훾 ln 푡 + ln 퐶 (2-36)

where 훾 is the power law exponent (i.e., the slope of the capture curve in log-log coordinates), a measure of the magnetic capture rate, and 퐶 is a constant depending on the simulation conditions. During the second period, the rate of capture slows down due to the decreasing number of free nanoparticles in the simulation box, eventually becoming zero when all the particles are captured.

Making a comparison between the two curves in Figure 2-4, it is again evident that Brownian-relaxation MNPs are more rapidly captured than Néel-relaxation nanoparticles, but apparently the difference stems from rapid capture in the initial time steps. As a result, the Brownian-relaxation nanoparticles are more rapidly depleted from the simulation box, hence the number of captured nanoparticles begins to asymptote earlier. Similar results are obtained for other conditions of the magnetic field gradient and volume fraction of the nanoparticles.

2.3.3 Magnetic Capture Rates for Various Strengths of The External Magnetic Field Gradient

Figure 2-5 A) and B) compare magnetic capture curves for various maximum

Langevin parameters, for MNPs that relax by the Brownian and Néel relaxation mechanisms. As shown, when 훼max ≥ 30 for both Brownian- and Néel-relaxation nanoparticles, the capture curves look smooth with a wide range of power-law behavior.

On the other hand, for smaller values of the maximum Langevin parameter, the magnetic capture curves fluctuate significantly. These distinguishing behaviors are explained by the Brownian agitation of the nanosized particles, which becomes significant as the magnetic force and torque exerted on the nanoparticles decreases due to decreasing magnitude of the magnetic field gradient. Focusing on the capture period that follows the power law model, we fitted the magnetic capture rates in order to

compare the slopes as a function of the maximum Langevin parameters. The results are shown in Figure 2-5 C), which illustrates that with the same maximum Langevin parameter, the capture rates for Brownian-relaxation particles is slightly higher than the rates for nanoparticles relaxing by the Néel mechanism. Additionally, as the maximum

Langevin parameter 훼max ≥ 50 (i.e. the maximum magnitude of applied magnetic field

퐻max ≥ 88.17 kA⁄m) for both Brownian- and Néel-relaxation nanoparticles, the magnetic capture rates change very little with respect to the variation of field strength.

2.3.4 Magnetic Capture Rates for Various Nanoparticle Volume Fractions

Figure 2-6 shows the magnetic capture curves for various particle volume fractions, for MNPs that relax by the Brownian and Néel relaxation mechanisms. For all the capture curves, the scaled parameter of Yukawa repulsion 훽Ykw = 5 and the maximum Langevin parameter was 훼max = 100. As seen in Figure 2-6, besides the curves at particle volume fraction 휙 = 0.001 %, all other curves under the same relaxation mechanism have similar shapes and slopes. The abnormal behavior of the curves can be explained by the discontinuous supply of nanoparticles during the magnetic capture process at very low particle volume fraction. This result suggests that, when 휙 ≥ 0.005 % for both Brownian- and Néel-relaxation nanoparticles, the magnetic capture rate is no more a function of particle volume fraction. These results appear to be at odds with experimental and simulation work by Faraudo et al.32, who studied rates of particle capture for separation applications and observed that the time to capture particles decreased with increasing particle concentration. In their work, Faraudo et al.32 considered particles with a diameter of 410 nm (compared to 10 nm considered here) and were interested in so-called cooperative magnetophoresis, where a uniform applied

field is used to cause chain formation, which leads to faster separation of the particles in a given magnetic field gradient. However, in our study we were interested in the capture of individual nanoparticles so as to generate a pattern on a substrate. In that case, chain formation in solution would lead to low fidelity pattern formation and as such is undesirable. Thus, our simulations were performed under conditions that do not lead to significant chain formation and we do not see the enhancement in particle separation with increasing concentration reported by Faraudo et al.32

2.3.5 Shape of Magnetic Nanoparticle Aggregates

The size evolution of MNP aggregates on the capture line under different relaxation mechanisms is shown in Figure 2-7, by plotting the average height and width of the aggregates as a function of capture time based on the capture-line method. From

Figure 2-7, one can observe that the aggregates assembled through two different relaxation mechanisms grow in height at roughly equal rates. However, the aggregates assembled by Brownian-relaxation nanoparticles grow faster in width than the aggregates assembled by nanoparticles relaxing by the Néel relaxation mechanism.

That’s because the Brownian-relaxation nanoparticles that aggregate on the capture line preferentially form nanoparticle strings because of magnetic dipole-dipole interactions. This phenomenon can also be observed in Figure 2-2.

Figure 2-8 presents a comparison of the size evolution of MNP aggregates for different ratios of magnetic dipole-dipole interaction parameter to Yukawa repulsion parameter, with other conditions and parameters uniform. For nanoparticles relaxing by the Brownian mechanism, the aggregates assembled with magnetic dipole-dipole interactions and hard-sphere Yukawa repulsion undergo greater growth in width than aggregates assembled without such interactions. At 훽Ykw = 5, the aggregates have a

lower height and a larger width than aggregates obtained using other parameters. This indicates that, by generating plentiful nanoparticle strings, dominant magnetic dipole- dipole interactions (훽dd⁄훽Ykw = 20) lead to flat-shaped nanoparticle aggregates. When

훽Ykw increases to 10, we observed that the nanoparticle aggregates increase in height and have narrower width. This behavior is explained by the enhanced Yukawa repulsion, which inhibits the formation of particle strings by increasing the inter-particle distance at equilibrium. As a result, magnetically captured nanoparticles have to accumulate on aggregates in the height dimension. At 훽Ykw = 20 for 훽dd⁄훽Ykw = 5, the repulsive Yukawa interaction becomes so strong that the aggregates exhibit a looser structure as compared with those at smaller 훽Ykw. In this case, both the height and width of the aggregates rises, as shown in Figure 2-8 A) and B). For nanoparticles relaxing by the Néel mechanism, the introduction of magnetic dipole-dipole interactions generates little influence on the aggregate size. However, an increase in repulsive

Yukawa interactions still results in an increase in the height and width of nanoparticle aggregates, resulting in loose structures.

2.4 Conclusions

In Chapter 2, magnetic capture rate and evolution of the size of nanoparticle aggregate at a capture line generated by a magnetic pole reversal are investigated, using Brownian dynamics simulations of a suspension of MNPs in strong external magnetic field gradients that are generated at a solid substrate. The simulation results suggest that under identical conditions of nanoparticle size, volume fraction, and magnetic fields nanoparticles that relax through Brownian relaxation mechanism are captured at a faster rate than nanoparticles that relax through Néel relaxation

mechanism. We also observed that, when 휙 = 0.05 % and the maximum Langevin parameter 훼max ≥ 50 (퐻max ≥ 88.17 kA⁄m) for both Brownian- and Néel-relaxation nanoparticles, increasing the intensity of magnetic field results in little change on the power-law dependence of number of captured nanoparticles with capture time. Similar observations were made for the influence of particle volume fraction. When 훼max = 100 and particle volume fraction 휙 ≥ 0.005 % for both Brownian- and Néel-relaxation nanoparticles, the magnetic capture rate is not a function of particle volume fraction.

Additionally, strong magnetic dipole-dipole interactions are verified to assist on building tight aggregate structures and Brownian-relaxation nanoparticles contribute wider- shaped aggregates.

In summary, the study in Chapter 2 provides a theoretical understanding of magnetic capture mechanisms and the behaviors of the dispersed MNPs, relaxing by the Brownian and Néel relaxation mechanisms. These theoretical predictions of magnetic capture in a quiescent fluid can help in controlling the size of nanoparticle aggregations formed during magnetic assembly and in designing better devices for

MNP separation.

Figure 2-1. Magnetic field gradients in simulations. A) Lateral view of device with simulated area and two ideal magnetic poles magnetized with opposite directions. B) COMSOL simulation of normalized magnetic field in the simulated area.

Figure 2-2. Zoomed-in 3D configurations snapshots close to the capture line at various times for interaction parameters 훽dd = 100 and 훽Ykw = 5, maximum Langevin parameter 훼max = 100, particle volume fraction 휙 = 0.05 %, for magnetic nanoparticles that relax by different mechanisms. A) and B) are representative for Brownian- and Néel-relaxation nanoparticles, respectively. The north pole of magnetic dipole moment is specified in red color and south pole in white. Coordinate dimensions are scaled by nanoparticle radius.

Figure 2-3. Trajectories of representative magnetic nanoparticles that respond to the applied magnetic field by different mechanisms, for interaction parameters 훽dd = 100 and 훽Ykw = 5, maximum Langevin parameter 훼max = 100, and particle volume fraction 휙 = 0.05 %. A) and B) are representative for Brownian- and Néel-relaxation nanoparticles, respectively.

Figure 2-4. Number of captured magnetic nanoparticles as a function of capture time for interaction parameters 훽dd = 100 and 훽Ykw = 5, maximum Langevin parameter 훼max = 100, particle volume fraction 휙 = 0.05 %, and with magnetic nanoparticles that relax by the Brownian and Néel relaxation mechanisms.

Figure 2-5. Number of captured magnetic nanoparticles as a function of capture time for interaction parameters 훽dd = 100 and 훽Ykw = 5, particle volume fraction 휙 = 0.05 %, various maximum Langevin parameters, and with magnetic nanoparticles that relax by different mechanisms. A) and B) are representative for Brownian- and Néel-relaxation nanoparticles, respectively. C) Magnetic capture rates fitted based on the data in A) and B).

Figure 2-6. Number percentage of captured magnetic nanoparticles as a function of capture time for interaction parameters 훽dd = 100 and 훽Ykw = 5, maximum Langevin parameter 훼max = 100, various particle volume fractions, and with magnetic nanoparticles that relax by different mechanisms. A) and B) are representative for Brownian- and Néel-relaxation nanoparticles, respectively.

Figure 2-7. Average height and width of magnetic nanoparticle aggregates as a function of capture time for interaction parameters 훽dd = 100 and 훽Ykw = 5, maximum Langevin parameter 훼max = 100, particle volume fractions 휙 = 0.05 %, and with magnetic nanoparticles that relax by different mechanisms. A) and B) are representative for the height and width evolutions, respectively.

Figure 2-8. Average height and width of magnetic nanoparticle aggregates as a function of capture time for maximum Langevin parameter 훼max = 100, particle volume fractions 휙 = 0.05 %, various ratios of dipole-dipole interaction parameter to hard-core Yukawa repulsion parameter, and with magnetic nanoparticles that relax by different mechanisms. A) and B) are representative for Brownian- relaxation magnetic nanoparticles and C) and D) are representative for Néel- relaxation magnetic nanoparticles.

CHAPTER 3 MAGNETIZATION DYNAMICS AND ENERGY DISSIPATION OF INTERACTING MAGNETIC NANOPARTICLES IN ALTERNATING MAGNETIC FIELDS WITH AND WITHOUT A STATIC BIAS FIELD

In Chapter 3, the effect of inter-particle interactions on the magnetization dynamics and energy dissipation rates of spherical single-domain magnetically-blocked

MNPs in static and AMFs was studied using Brownian dynamics simulations. For the case of an applied static magnetic field, simulation results suggest that the effective magnetic diameter of interacting nanoparticles determined by fitting the equilibrium magnetization of the nanoparticles to the Langevin function differs from the actual magnetic diameter used in the simulations. Parametrically, magnetorelaxometry was studied in simulations where a static magnetic field was suddenly applied or suppressed, for various strengths of magnetic dipole-dipole interactions. The results show that strong dipole-dipole interactions result in longer chain-like nanoparticle aggregates and eventually longer characteristic relaxation time of the nanoparticles. For the case of applied AMF with and without a static bias magnetic field, the magnetic response of interacting nanoparticles was analyzed in terms of the harmonic spectrum of nanoparticle magnetization and dynamic hysteresis loops, whereas the energy dissipation of the particles was studied in terms of the calculated specific absorption rate (SAR). Results suggest that the effect of magnetic interactions on the SAR varies significantly depending on the amplitude and frequency of the AMF and the intensity of the bias field. These computational studies provide insight into the role of particle-

Figures reproduced with permission from Zhao, Z. Y.; Rinaldi, C., Magnetization Dynamics and Energy Dissipation of Interacting Magnetic Nanoparticles in Alternating Magnetic Fields with and without a Static Bias Field. J Phys Chem C 2018, 122 (36), 21018-21030.

particle interactions on the performance of MNPs for applications in magnetic hyperthermia and magnetic particle imaging (MPI).

3.1 Background and Motivation

In an externally applied AMF, MNPs can generate heat because of hysteresis losses. This effect is used to actuate release of a drug33, or to deposit heat in cancer tumors, as in magnetic hyperthermia34,35. MNPs are also of interest in MPI, an emerging biomedical imaging technology that maps MNPs tracers in vivo with millimeter to sub- millimeter spatial resolution.36,37 In MPI, a static bias magnetic field gradient is superimposed with a uniform AMF that is of lower frequency than that used in hyperthermia applications. This generates a small “field free region” where the particles are able to respond to the AMF, resulting in a signal in a pick-up coil which is then used to generate a quantitative image of the distribution of MNPs in a field of view. Recently, it has been reported that MPI can be combined with higher frequency AMF to achieve image guided, spatially controlled heating using MNPs.38,39

Prior work has investigated the dependence of heating efficiency of MNPs on the properties of the particles and the amplitude and frequency of the AMF.40-44

Experimental38,45,46 and computational44,47,48 work has investigated the energy dissipation rate of non-interacting MNPs in AMFs, with and without superimposed static magnetic fields. However, several recent experiments suggest that magnetic dipole- dipole interactions may play an important role in the energy dissipation rate of the nanoparticles in an AMF.49-53 The magnetization dynamics of chains and clusters of single-domain MNPs in various geometries have also been experimentally studied.54-56

In order to better understand these effects and make predictions for experiments, theoretical studies have been performed based on various models and methods,

including the well-known Stoner-Wohlfarth model42,57,58, solving the Fokker- Planck equation43,59,60, and analysis based on the stochastic Landau-Lifshitz-Gilbert (LLG) equation61. The LLG equation has also been applied in simulations, in which magnetically interacting nanoparticles are fixed in a solid matrix, with62,63 and without64-

67 consideration of the rotational degrees of freedom of the nanoparticles. In addition, some groups have studied the dependence of energy dissipation rate on magnetic dipolar interactions by using the standard Metropolis68-70 and kinetic71 Monte-Carlo algorithms. For example, Ruta et al.71 studied the effect of a truncated dipolar interaction between a collection of Stoner-Wohlfarth spherical MNPs on energy dissipation rate by tuning the packing fraction of the nanoparticles as well as by introducing random distributions of anisotropy constant and particle positions and volumes. Their simulation results suggest that dipolar interactions lead to important and complex effects on the energy dissipation rate and that the effects are dependent on intrinsic statistical properties of the particles. However, the Monte-Carlo algorithms are based on generating random states in a system, which are independent of time.

Furthermore, the studies above did not describe the time-dependent translational dynamics of the particles, which are expected to be relevant in many applications where

MNPs are able to move towards or away from each other due to their interactions.

Langevin dynamics simulations, which can consider the translational and rotational degrees of freedom of suspended nanoparticles through time-dependent equations of motion, have been applied previously to investigate the influence of magnetic interactions on the relaxation dynamics of MNPs. For example, Berkov et al.72 applied the analytical second order virial expansion and computational Langevin

dynamics simulations for moderately concentrated suspensions of MNPs (ferrofluids).

Their results suggest that strong magnetic interactions increase the magnetization relaxation time for the case of an AMF and the case when a dc field changes significantly. However, in their simulations the ferrofluid was assumed to be colloidally stable, without forming particle aggregates. In another study, Soto-Aquino and Rinaldi73 reported a comparison of the predictions for the energy dissipation rates of MNPs using the linear response theory model by Rosensweig74, solution of the magnetization relaxation equations of Shliomis75 and Martsenyuk, Raikher and Shliomis76 (MRSh), and results obtained from rotational Brownian dynamics simulations. However, in their work they considered the infinitely dilute regime where there are negligible particle-particle interactions and for which particle translation does not contribute to the response of the nanoparticles to the magnetic field.

In Chapter 3, we report a computational study of the magnetization dynamics of spherical single-domain magnetically-blocked nanoparticles in static and AMFs, using

Brownian dynamics simulations.2 The term “thermally-blocked”, also referred to

“magnetically-blocked”, indicates that the nanoparticle aligns its magnetic dipole with the local magnetic field by undergoing Brownian relaxation mechanism.77 The assumed single-domain nature of the nanoparticles allows us to neglect heating due to internal dipole rotation,78 such that the heat dissipated by the nanoparticles is only due to their rotational motion. The algorithm takes into account translation and rotation of the nanoparticles, hydrodynamic drag, thermal fluctuations, magnetic dipole-dipole interactions, and a repulsive interaction potential. In the case of a static magnetic field, we investigate the effect of magnetic interactions on the method of calculating MNPs’

effective magnetic diameter by fitting their equilibrium magnetization to the Langevin function. Then we study the magnetic relaxation time of the nanoparticles for cases where a static magnetic field is suddenly turned on and off, and for various values of the magnetic interaction strength parameter. In cases where an AMF is applied, with or without a static bias field, the particle response is analyzed in terms of the evolution and harmonic spectrum of average magnetization, dynamic hysteresis loops, and calculated

SAR as a function of the amplitude and frequency of the AMF, value of the magnetic interaction strength parameter, and the magnitude of an applied bias field. It should be noted that the main point of our work focuses on studying the effect of inter-particle interactions on the magnetization dynamics and energy dissipation of MNPs. Therefore, the size distribution of the MNPs is not considered, although it has significant effects on the particle behavior, such as in energy dissipation rates.52,58,64

3.2 Simulation Method

3.2.1 Brownian Dynamics Simulations

Considering that thermally blocked MNPs respond to a change in an external magnetic field through physical rotation, we assume that their internal dipole moment is always “saturated”, with magnitude given by Equation (2-2). The motion equation of nanoparticle is given by Equation (2-8), by taking into account the same forces and torques on the nanoparticle as in Chapter 2.

In Chapter 3,we assume that magnetic fields are homogeneous and all applied in the +z direction of the laboratory coordinates, having the forms of 퐇dc = Hdc퐢푧 with strength Hdc for the static magnetic fields and 퐇ac = Hac cos(Ω푡) 퐢푧 with amplitude Hac and angular frequency Ω for the AMFs, respectively. For the case that the AMFs are superimposed with static magnetic fields, we named such static fields as static bias

fields or bias fields, which have the form of 퐇bias = Hbias퐢푧 with strength Hbias.

Therefore, the MNPs experience a zero force but a non-zero torque when their dipole is not aligned with the local magnetic field.

By doing the same non-dimensionalizations and transformations as in Chapter 2, the motion equation finally becomes

4 훼 (퐦UT ∙ 퐞 ) 푑퐱̃ 푖 퐓m,푖 ( 푖 ) = ( 3 ) Δ푡̃ 푑횽̃ 4 푖 훼 (퐦ωT ∙ 퐞 ) 3 푖 퐓m,푖

퐞퐅 퐞퐓 훽 (퐦UF ∙ ∑ dd,푖 + 퐦UT ∙ ∑ dd,푖) dd 푖 푟̃ 4 푖 푟̃ 3 푗≠푖 푗푖 푗≠푖 푗푖 + Δ푡̃ (3-1) 퐞퐅 퐞퐓 훽 (퐦ωF ∙ ∑ dd,푖 + 퐦ωT ∙ ∑ dd,푖) dd 푖 푟̃ 4 푖 푟̃ 3 ( 푗≠푖 푗푖 푗≠푖 푗푖 )

8 1 휅̃ −휅̃(푟̃푗푖−2) UF 훽Ykw ∑ 푒 ( 2 + ) (퐦푖 ∙ 퐫̂푗푖) 3 푟푗푖̃ 푟푗푖̃ 푗≠푖 퐗̃푖(Δ푡̃) + Δ푡̃ + ( ) 8 1 휅̃ 퐖̃ (Δ푡̃) −휅̃(푟̃푗푖−2) ωF 푖 훽Ykw ∑ 푒 ( 2 + ) (퐦푖 ∙ 퐫̂푗푖) 3 푟̃ 푟푗푖̃ ( 푗≠푖 푗푖 ) where the dimensionless mobility matrix was calculated for the condition of infinite boundary and the Brownian drift term equals zero. It should be noted that 훼

(correspondint to 퐻0) is replaced by the intensity 훼dc (corresponding to 퐻dc) or 훼bias

(corresponding to 퐻bias) for static fields and 훼ac (corresponding to 퐻ac) for AMFs.

3.2.2 Simulation Parameters and Conditions

Unless otherwise noted, simulations were made for spherical MNPs with uniform magnetic radius of 10 nm and dispersed in a solvent with the viscosity of water at a volume fraction 휙 = 1 % (particle number 푁 = 3375) in a cubic simulation box with side length 1.12 μm and periodic boundaries.79 We note that a volume fraction of 1 % for

magnetite corresponds to ∼ 50 mgFe3O4/mL. The temperature in the simulations corresponds to 300 K. For simulations in which the particles dissipate heat, holding the temperature constant at 300 K effectively corresponds to assuming fast heat transfer to the surroundings. In such a case the rate of energy dissipation is calculated from the thermodynamic relations given below. Runs were executed starting from random particle configurations, by using a minimum time interval of ∆푡̃ = 0.01 for cases of static magnetic fields (homogeneous and constant in the +푧 direction) and ∆푡̃ = 2휋⁄8000Ω̃ for cases of AMFs, where Ω̃ = Ω⁄퐷r. The cut-off distance corresponding to a pre-factor of

휆 = 1 was taken into account for both magnetic dipole-dipole interactions and the repulsive hard-core Yukawa potential. In addition, the scaled dipole-dipole interaction parameter was varied in the range of 0 ≤ 훽dd ≤ 500. For the repulsive hard-core

Yukawa potential, we used a radius-scaled inverse Debye screening length of 휅̃ = 3 and a scaled interaction energy of 훽Ykw = 3 (훽Ykw = 0 for the case of 훽dd = 0). The

Langevin parameter of the static magnetic field for magnetorelaxometry was considered to be 훼dc = 1, 10 and 100 (corresponding to 1.76 kA/m, 17.63 kA/m and 176.35 kA/m for the 10 nm radius particles). The Langevin parameter and dimensionless angular frequency of the AMF was varied in the range of 0.25 ≤ 훼ac ≤ 100 (0.44 kA/m ≤ 퐻ac ≤

176.35 kA/m) and 0.1 ≤ Ω̃ ≤ 10 (corresponding to field frequency 3.07 kHz ≤ 푓 ≤

307.35 kHz), respectively, whereas the Langevin parameter of the bias static field was varied in the range of 0 ≤ 훼bias ≤ 14 (0 kA/m ≤ 퐻bias ≤ 24.69 kA/m).

3.2.3 Simulations of Magnetorelaxometry

In magnetorelaxometry, a collection of MNPs is subjected to sudden changes in the magnitude of an applied magnetic field and their dynamic magnetization response is

monitored.80 Here, we model this situation by taking a collection of randomly distributed nanoparticles, applying a magnetic field of prescribed magnitude for enough time steps to ensure that equilibrium is reached, and then removing the magnetic field. Because of the uniform particle dimensions and material, the average magnetization of the collection of nanoparticles at any given instant can be represented by the average normalized 푧-direction magnitude of the magnetic dipole moments

푁 1 푚푧,푖 푀̃푧 = ∑ (3-2) 푁 푚s 푖=1 where 푚푧,푖 represents the 푧-direction magnitude of magnetic dipole moment of the 푖th nanoparticle. To analyze the results of these simulations, we draw from simple theoretical models obtained by solving the phenomenological magnetization relaxation equation of Shliomis75

푑푀̃ 1 푧 = (푀̃ − 푀̃ ) (3-3) 푑푡 휏 푧 푧,t where 푀̃푧,t represents the magnetization at equilibrium with the instantaneous magnetic field, and 휏 represents the characteristic magnetic relaxation time. This equation can be solved for the two situations modeled here, suddenly turning on and off a magnetic field.

For the case where a collection of MNPs is at equilibrium at zero field and then a magnetic field is suddenly applied in the 푧-direction, the solution to the magnetization relaxation equation, in dimensionless form, becomes

푀̃ 푡̃∗ ln (1 − 푧 ) = (3-4) 푀̃푧,eq 휏̃

where 푀̃푧,eq is scaled by the saturation magnetization of the collection of particles (i.e., the saturated magnitude of magnetic moment), simulation time 푡̃∗ and characteristic magnetic relaxation time 휏̃ are scaled by the Brownian relaxation time

3휂0푉p 휏B = (3-5) 푘B푇

For the case where a collection of MNPs is at equilibrium with an applied static magnetic field and the field is suddenly switched off, the solution to the magnetization relaxation equation, in dimensionless form, is

푀̃ 푡̃∗ ln 푧 = (3-6) 푀̃푧,eq 휏̃

3.2.4 Simulations of Dynamic Magnetic Susceptibility

For a suspension of MNPs subjected to an AMF in the z direction, their dynamic magnetization can be expressed in the form of Fourier series as73

∞ ∞ 1 푀̃ (푡̃) = 훼 [∑ 휒′ sin(푛Ω̃푡̃) + ∑ 휒′′ cos(푛Ω̃푡̃)] (3-7) 푧 3 ac 푛 푛 푛=1 푛=1

′ ′′ where 휒푛 and 휒푛 are the nth-order in-phase and out-of-phase components of the complex susceptibility, respectively. When 푛 = 1, the fundamental in-phase and out-of- phase susceptibilities can be calculated through

2휋 ′ 3 휒푛 = ∫ 푀̃푧(푡̃) cos(Ω̃푡̃)푑(Ω̃푡̃) (3-8) 휋훼ac 0

2휋 ′′ 3 휒푛 = ∫ 푀̃푧(푡̃) sin(Ω̃푡̃)푑(Ω̃푡̃) (3-9) 휋훼ac 0

3.2.5 Calculation of Energy Dissipation Rate

In simulations where an AMF is applied (with or without a static bias magnetic field), the average rate of energy dissipation in a cycle of magnetic field can be calculated using the equation73,77

1 2푝 푑퐻 〈푄̇ 〉 = − 휇0 ∫ 푀 푑푡 (3-10) 2푝 0 푑푡 where the period of the cycle 2푝 = 2휋⁄Ω. By substituting a sinusoidal AMF and rewriting variables in dimensionless form, Equation (3-10) becomes

2푝̃ 휇0퐻ac휙푀dΩ 〈푄̇ 〉 = ∫ 푀̃푧 sin(Ω̃푡̃)푑푡̃ (3-11) 2푝̃ 0 where 푝̃ = 푝퐷r. For cases in which there is a bias field, the rate of energy dissipation has the same mathematical expression of Equation (3-11) but with a 푀̃푧, which is significantly influenced by the bias field. Additionally, below we express the rate of energy dissipation using the specific absorption rate or SAR, which is given by

〈푄̇ 〉 푆퐴푅 = (3-12) 휙휌 where 휌 denotes the mass density of the nanoparticles.

3.3 Results

3.3.1 Equilibrium Response of Magnetization

Simulations were carried out for uniform strength of repulsive Yukawa potential and various particle diameters and intensities of applied static magnetic field. By assuming that all the nanoparticles have the same domain magnetization, i.e. in the same material, the strength of magnetic interactions was calculated based on the particle diameter. The scaled equilibrium magnetization of particles was plotted as a function of the field intensity and then fitted to the Langevin function

1 푀 = 푏퐿(훼fit)푏 (coth(훼fit) + ) 훼fit (3-13)

3 where 푏 is a constant, 훼fit = 휇0푀d푉p,fit퐻dc⁄푘B푇, and 푉p,fit = 휋퐷p,fit ⁄6. The results are shown in Figure 3-1. As expected, all curves follow the typical shape expected based on the Langevin function, where at small fields the response is linear and there is a monotonic approach to saturation. As the diameter of the particles increases the approach to magnetic saturation occurs at smaller applied fields. This is because increasing the diameter of the particles results in an increase in the magnetic dipole moment of the particles, which results in greater magnetic torques, even for small fields.

As a result, the degree of alignment of the dipoles increases at any given field as particle diameter increases. Table 3-1 compares the diameters used in the simulations

(퐷p) to the corresponding fitted diameters (퐷p,fit), obtained by fitting the Langevin function to the simulation results. As seen from the comparison, for 퐷p ≤ 40 nm the particle diameter obtained by fitting the Langevin function is larger than the diameter used in the simulations, whereas for 퐷p ≥ 50 nm the particle diameter obtained by fitting the Langevin function is smaller than the diameter used in the simulations. This underscores the importance of ensuring negligible particle-particle interactions when applying magnetogranulometry to estimating the magnetic properties of MNPs.

3.3.2 Simulations of Magnetorelaxometry

Figure 3-2 shows the change of scaled particle magnetization as a function of dimensionless simulation time, i.e. the magnetization relaxation curves, for various strengths of inter-particle interactions. In Figure 3-2, it is seen that introducing magnetic interactions not only increases the relaxation time of particles, but also leads to deviation from single-exponential response. This suggests a distribution of relaxation

times in the response. In detail, Figure 3-2 A) to C) show that as a static magnetic field is suddenly applied to a collection of MNPs that is at equilibrium at zero field, increasing the intensity of the field reduces the relaxation time and suppresses the effect of magnetic interactions. For the case where the initial equilibrium field is suddenly switched off, D) to F) show that increasing the intensity of the field leads to more pronounced deviation from single exponential response for the interacting particles.

Table 3-2 summarizes the characteristic relaxation times obtained from the linear relaxation region for Figure 3-2 A) to C) and from the initial relaxation region for D) to F).

According to Table 3-2, for the case where the magnetic field is suddenly applied increasing the intensity of the applied field leads to a reduction in relaxation time for all magnetic interaction strengths. On the other hand, at a fixed applied field strength increasing the strength of magnetic interactions leads to an increase in the effective relaxation time, with the effect being reduced as the strength of the applied magnetic field increases. Similar observations can be made for the case where the field is suddenly suppressed, with the difference being that in all cases the effective relaxation times are much longer than in the case where the magnetic field is suddenly applied.

These observations can be rationalized as follows. In all simulations magnetic particle- particle interactions tend to promote particle alignment relative to each other, slowing down the approach to a completely random state. For the case where the magnetic field is suddenly applied, the external magnetic field results in a magnetic torque that aligns the particles (and their chain-like aggregates) in the direction of the field. As the relative strength of the applied magnetic field increases the torque increases, making the effects of particle-particle interactions negligible. In contrast, for the case where the applied

magnetic field is suddenly suppressed the external magnetic torque vanishes during the relaxation process, resulting in longer effective relaxation times. However, the effective relaxation times are still a function of the strength of the initial applied field because as this increases the degree of initial particle alignment and chaining increases.

To gain further insight into the effect of inter-particle interactions on nanoparticle dynamics, we generated snapshots of nanoparticle configurations using the Persistence of Vision Ray Tracer (POV-Ray), which is a ray tracing program for generating images.

In the snapshots, the north pole of the magnetic dipole moment is shown in red color and the south pole is shown in white. It should be noted that the static magnetic field is suddenly applied at the beginning of the videos and then suddenly switched of at the half time of the videos. As seen, the magnetic dipole moments of the particles in both cases rotate to align in the +푧 direction to respond to the applied magnetic field.

However, the nanoparticles with strong magnetic interactions form chain-like aggregates under the applied field and relax still in the aggregates as the field is suddenly switched off. This difference suggests that nanoparticles with strong interactions have longer magnetic relaxation times because of the significant local fields as well as formation of nanoparticle aggregates which restrain the rotation of individual nanoparticles.

To further study the above observations, we developed an algorithm to calculate the length distribution of the particle chains for MNPs with interactions. Figure 3-3 shows the length distributions of particle chains and the corresponding nanoparticle configurations where the nanoparticles are in equilibrium with an applied static field, for interaction parameters 훽dd = 500 and 훽Ykw = 3, and various intensities of the static

magnetic field. It is observed that the nanoparticles in the strong magnetic field have better alignment along the field direction. Interestingly, the result suggests that increasing the field intensity shortens the average chain length 퐿avg,c, which is in the unit of nanoparticles. By inspecting the snapshots of nanoparticle configurations, we observed that for weak magnetic fields the particle chains are long and curled, and neither chains nor in-chain particles are well aligned along the field direction. In contrast, as the field intensity increases, particle chains become short but better aligned in the static magnetic field direction. This indicates that strong static magnetic fields contribute to align both nanoparticles and particle chains, but suppress the rotation due to inter-particle interactions and then shorten the length of chains. Figures 3-4 to 3-6 show the equilibrium length distributions of particle chains and the corresponding nanoparticle configurations for various strengths of magnetic interactions and various intensities of the static magnetic field. As expected, stronger magnetic interactions lead to an increase in the average length of particle chains. Similarly, by comparing the results for different values of 훼dc, one can observed that for 훽dd ≥ 300, increasing the field intensity shortens the average chain length.

3.3.3 Energy Dissipation Rate in An Alternating Magnetic Field

Figure 3-7 shows the dynamic magnetization of suspensions of MNPs in an AMF as a function of time, corresponding harmonic spectra of magnetization, and dynamic hysteresis loops, for 훼ac = 10 and various field frequencies. In Figure 3-7 A) to C), we observe that increasing the field frequency results in changes in the shape of the magnetization curve and a lag with respect to the applied field. As the strength of magnetic interactions increases, the magnitude of the magnetization signal decreases.

Simultaneously, the magnetization lag increases first and then slightly decreases for the low and intermediate frequencies, and decreases monotonically for the highest frequency considered. By taking the Fast Fourier Transform (FFT) of the magnetization signals and by plotting hysteresis loops (in insets), it can be observed that for Ω̃ = 0.1 and 1 increasing 훽dd significantly affects harmonic spectra and leads the area of the hysteresis loop to increase first and then decrease. For Ω̃ = 10, as 훽dd increases, the effect on the harmonic spectra is less pronounced but the loop area contracts monotonically. This can be explained as follows. Although the relaxation time of weak- interacting nanoparticles slightly increases due to the effect of local attractive interactions on the free rotation of the particles to the field, it is still shorter than the cycle time of the AMFs in the low and intermediate frequencies. Accordingly, the nanoparticles are able to achieve the maximum response to the applied field. This is not the case for strong magnetic interactions, which significantly suppress the rotation of the nanoparticles to respond to the field and result in longer particle chains, such that the nanoparticles experience a relaxation time that is longer than the cycle time and cannot achieve the maximum response as for the weak- or non-interacting nanoparticles. On the other hand, when the field frequency is high, the cycle time of the AMF is even shorter than the relaxation time of the weak-interacting nanoparticles. As a result, the nanoparticles with any interaction strengths cannot attain the maximum response to the field, which results in a decrease in the area of the dynamic hysteresis loops.

Figure 3-8 shows the representative snapshots of particle configurations that are extracted from the videos, for various interaction parameters and time points. It is observed that for the non-interacting nanoparticles the dipole moments at time C align

better in the field direction than those at time B, where the magnetic field have larger magnitude. Similarly, better dipole alignment is observed at time D as compared with that at time C, although the direction of the field at D is opposite to the field at C.

However, the above observation does not apply to the strong-interacting MNPs, for which strong magnetic interactions result in long particle chains and inhibit the particle rotation due to the field change. As a result, particle chains are neither well aligned along the magnetic field nor respond obviously to the field change. All the results of particle dynamics show a good agreement with the magnetization curve in Figure 3-8.

By applying Equations (3-8) and (3-9), we obtained the real and imaginary components of the complex susceptibility as a function of dimensionless angular frequency for Langevin parameter 훼ac = 10 and various strengths of inter-particle interactions, as shown in Figure 3-9, where one can observe that increasing 훽dd results in flattening the in-phase susceptibility curve and shifting the peak of the out-of-phase susceptibility curve towards lower frequencies. This indicates that increasing the strength of magnetic interactions increases the effective relaxation time of the suspension,72 which agrees with the above observations.

In order to consider the effect of magnetic interactions on the heating efficiency of the nanoparticles, the SAR is plotted in Figure 3-10 as a function of amplitude of the

AMF for various field frequencies and strengths of inter-particle interactions. At the lowest frequency (푓 = 3.07 kHz), it is observed that increasing 훽dd leads first to an increase and then a decrease in SAR for all field amplitudes. The 훽dd for which SAR peaks is dependent on the amplitude of the AMF. At the intermediate frequency (푓 =

30.73 kHz), increasing 훽dd leads the SAR to increase first and then decrease for small

field amplitudes, and to increase monotonically for large amplitudes. At the highest frequency (푓 = 307.35 kHz), the SAR is seen to decrease monotonically with increasing strength of magnetic interactions, except for very high field amplitude, where the SAR increases monotonically. These observations correlate with the effects of interactions on the areas of the hysteresis loops and the accompanying discussions above.

Additionally, at very strong field amplitudes, MNPs with various interaction strengths all relax much faster and can better align with the field direction. In this case, nanoparticles with stronger magnetic interactions have longer relaxation times, which results in greater magnetization lag and as a result greater energy dissipation rate, i.e. dissipated energy per unit time. These results suggest that the SAR value of a suspension of

MNPs is decided by the relative strengths of inter-particle interactions, and the amplitude and frequency of the applied AMF.

3.3.4 Effect of Static Bias Magnetic Field on Energy Dissipation Rate

Figure 3-11 shows the response of magnetization of the MNPs in suspension to the combination of an applied AMF and a superimposed static bias field, for the AMF with Langevin parameter 훼ac = 10 and frequency Ω̃ = 10, and for various strengths of bias field and magnetic interactions. The corresponding harmonic spectra of the magnetization, and dynamic hysteresis loops are also included. In Figure 3-11 A) to C), it is observed that large strength of the bias field reduces the amplitude of the magnetization curve, by slightly increasing the peak value and significantly increasing the valley value. This can be attributed to the fact that when the AMF is in the +푧 direction, the bias field contributes to enhance the total magnetic field exerted on nanoparticles, which leads to a slight increase in the alignment of magnetic dipoles and

eventually a slight increase in the suspension magnetization. However, when the AMF is in the −푧 direction, the bias field strongly opposites the AMF, which reduces the negative amplitude of magnetization curve. When the bias field is dominant, the antiparallel AMF cannot change the direction of suspension magnetization, so that the curve valley moves into the +푧 section of the plot. In all cases it is shown that increasing the strength of magnetic interactions reduces the ability of the nanoparticles to respond to the AMF. This is evident in a decrease in the value of the harmonics and in the loop area in the insets of D), E) and F).

Figure 3-12 shows predictions for the SAR of the MNPs in the combination of an

AMF and a superimposed static bias field, for various frequencies of the AMF, strengths of the bias field, and strengths of magnetic interactions. As seen, at the low field frequency (i.e. 푓 = 3.07 kHz) and of a field frequency typical of MPI (i.e. 푓 = 30.73 kHz), increasing the strength of magnetic interactions results in the value of SAR to increase first and then decrease. The value of 훽dd for achieving the highest SAR varies with the strength of the bias field. On the other hand, for a typical hyperthermia frequency (i.e.

푓 = 307.35 kHz), increasing the strength of magnetic interactions leads the SAR value to decrease for all intensities of the bias field. Thus, the effect depends on the relative strengths of the interactions and bias field.

3.4 Conclusions

In Chapter 3, we computationally studied the effect of inter-particle interactions on the magnetization dynamics and energy dissipation of spherical single-domain magnetically-blocked nanoparticles in static and AMFs, by carrying out Brownian dynamics simulations that account for translation and rotation of the nanoparticles,

hydrodynamic drag, thermal fluctuation, magnetic dipole-dipole interactions, and repulsive Yukawa potential. Our simulation results suggest that the magnetic diameter of interacting MNPs determined by fitting the Langevin function is larger than the actual particle size for particle sizes equal and less than 40 nm, and is smaller than the actual size for particle sizes equal and larger than 50 nm. The effect of particle-particle interactions and the formation of particle chains on the behavior and performance of the nanoparticles were investigated by parametrically tuning the strength of magnetic dipole-dipole interactions. The results of magnetorelaxometry show that increasing the strength of magnetic interactions increases the average length of chain-like particle aggregates and as a result increases the characteristic relaxation time of the nanoparticles. For an applied AMF without a bias field, we observed that for small and intermediate frequencies of the AMF, increasing the strength of magnetic interactions increases the SAR first and then decreases. For the high AMF frequency considered, increasing the strength of magnetic interactions decreases the SAR monotonically.

However, exceptions are observed for high field amplitudes, where increasing the strength of magnetic interactions enhances the SAR monotonically for all considered field frequencies. When a static bias magnetic field is superimposed to the AMF, the simulations suggest that increasing the strength of magnetic interactions leads the SAR to increase first and then decrease for the low frequency and the frequency typical of

MPI, and to decrease only for the frequency typical of hyperthermia. Moreover, the SAR decreases as the intensity of bias static field increases.

In summary, the study in Chapter 3 provides a theoretical insight into the role of particle-particle interactions on the performance of MNPs for applications in magnetic

hyperthermia and MPI. However, to predict for real systems some other factors, such as the size distribution of MNPs, should also be taken into account, because of their non- negligible effects on particle dynamics and energy dissipation.

Figure 3-1. Equilibrium magnetization of magnetic nanoparticles suspension in an applied static magnetic field as a function of intensity of the field for various particle diameters.

Table 3-1. Diameters used in simulations (퐷p) and corresponding diameters (퐷p,fit) obtained by applying a nonlinear fit to the Langevin function. 퐷p, [nm] 10 20 30 40 50 60 퐷p,fit, [nm] 10.74 30.60 37.89 45.07 46.82 48.22

Figure 3-2. Time-dependent magnetization relaxation curves for cases where the external static magnetic field is suddenly turned on and off and under various strengths of inter-particle interactions. For the field-on case: A) 훼dc = 1 (퐻dc = 1.76 kA⁄m), B) 훼dc = 10 (퐻dc = 17.63 kA⁄m), and C) 훼dc = 100 (퐻dc = 176.35 kA⁄m). For the field-off case: D) 훼dc = 1 (퐻dc = 1.76 kA⁄m), E) 훼dc = 10 (퐻dc = 17.63 kA⁄m), and F) 훼dc = 100 (퐻dc = 176.35 kA⁄m).

Table 3-2. Magnetic relaxation time (휏̃) for the case that an applied static magnetic field is suddenly turned on and off for various Langevin parameters and strengths of inter-particle interactions. Interactions Magnetic field is turned on Magnetic field is turned off parameters 훼dc = 1 훼dc = 10 훼dc = 100 훼dc = 1 훼dc = 10 훼dc = 100 훽 = 0, dd 0.64 0.11 0.01 0.80 0.77 0.77 훽Ykw = 0 훽 = 100, dd 1.11 0.13 0.01 1.84 1.88 1.86 훽Ykw = 3 훽 = 200, dd 2.85 0.15 0.01 42.86 30.20 20.42 훽Ykw = 3 훽 = 300, dd 2.93 0.16 0.01 157.95 55.94 46.50 훽Ykw = 3 훽 = 400, dd 2.92 0.16 0.01 200.14 60.40 46.96 훽Ykw = 3 훽 = 500, dd 2.78 0.16 0.01 203.49 84.57 46.69 훽Ykw = 3

Figure 3-3. Number of particle chains as a function of length of the particle chain and corresponding representative snapshots of particle configuration for interaction parameters 훽dd = 500 and 훽Ykw = 3 and various intensities of static magnetic field. The field intensities are A) 훼dc = 1 (퐻dc = 1.76 kA⁄m), B) 훼dc = 10 (퐻dc = 17.63 kA⁄m), and C) 훼dc = 100 (퐻dc = 176.35 kA⁄m).

Figure 3-4. Number of particle chains as a function of length of the particle chain and corresponding representative snapshots of particle configuration at 푡̃ = 19.80, for Langevin parameter 훼dc = 1 (퐻dc = 1.76 kA⁄m) and various strengths of magnetic dipole-dipole interactions. 퐿avg,c represents the average length of the particle chains.

Figure 3-5. Number of particle chains as a function of length of the particle chain and corresponding representative snapshots of particle configuration at 푡̃ = 19.80, for Langevin parameter 훼dc = 10 (퐻dc = 17.63 kA⁄m) and various strengths of magnetic dipole-dipole interactions. 퐿avg,c represents the average length of the particle chains.

Figure 3-6. Number of particle chains as a function of length of the particle chain and corresponding representative snapshots of particle configuration at 푡̃ = 19.80, for Langevin parameter 훼dc = 100 (퐻dc = 176.35 kA⁄m) and various strengths of magnetic dipole-dipole interactions. 퐿avg,c represents the average length of the particle chains.

Figure 3-7. Scaled time-dependent 푧-direction magnetization and corresponding harmonic spectra and dynamic hysteresis loops for Langevin parameter 훼ac = 10 (퐻ac = 17.63 kA⁄m) and various dimensionless angular frequencies. Magnetization curves are shown in A), B) and C). The corresponding harmonic spectra of magnetization signal and dynamic hysteresis loops (in inset) are shown in D), E) and F). A) and D) are for Ω̃ = 0.1 (푓 = 3.07 kHz), B) and E) are for Ω̃ = 1 (푓 = 30.73 kHz), and C) and F) are for Ω̃ = 10 (푓 = 307.35 kHz).

Figure 3-8. Magnetization curve and representative snapshots of particle configuration for Langevin parameter 훼ac = 10 (퐻ac = 17.63 kA⁄m), dimensionless angular frequency Ω̃ = 10 (푓 = 307.35 kHz), and various strengths of inter-particle interactions.

Figure 3-9. Real and imaginary components of the complex susceptibility as a function of dimensionless angular frequency for Langevin parameter 훼ac = 10 (퐻ac = 17.63 kA⁄m) and various strengths of inter-particle interactions. A) and B) are representative for the real and imaginary components of the complex susceptibility, respectively.

Figure 3-10. Specific absorption rate as a function of the amplitude of alternating magnetic field for various angular frequencies and strengths of inter-particle interactions. The angular frequencies are A) Ω̃ = 0.1 (푓 = 3.07 kHz), B) Ω̃ = 1 (푓 = 30.73 kHz), and C) Ω̃ = 10 (푓 = 307.35 kHz).

Figure 3-11. Scaled time-dependent 푧-direction magnetization and corresponding harmonic spectra and dynamic hysteresis loops for Langevin parameter 훼ac = 10 (퐻ac = 17.63 kA⁄m), dimensionless angular frequency Ω̃ = 10 (푓 = 307.35 kHz), and various Langevin parameters of bias field. Magnetization curves are shown in A), B) and C). The corresponding harmonic spectra of magnetization signal and dynamic hysteresis loops (in inset) are shown in D), E) and F). A) and D) are for 훼bias = 4 (퐻bias = 7.05 kA⁄m), B) and E) are for 훼bias = 8 (퐻bias = 14.11 kA⁄m), and C) and F) are for 훼bias = 12 (퐻bias = 21.16 kA⁄m).

Figure 3-12. Specific absorption rate as a function of the strength of bias field for Langevin parameter of alternating magnetic field 훼ac = 10 (퐻ac = 17.63 kA⁄m), various angular frequencies and strengths of inter-particle interactions. The angular frequencies are A) Ω̃ = 0.1 (푓 = 3.07 kHz), B) Ω̃ = 1 (푓 = 30.73 kHz), and C) Ω̃ = 10 (푓 = 307.35 kHz).

CHAPTER 4 EFFECTS OF PARTICLE DIAMETER AND MAGNETOCRYSTALLINE ANISOTROPY ON MAGNETIC RELAXATOIN AND MAGNETIC PARTICLE IMAGING PERFORMACE OF MAGNETIC NANOPARTICLES

In Chapter 4, the dynamic magnetization of immobilized spherical single-domain

MNPs with uniaxial or cubic magnetocrystalline anisotropy were studied computationally by executing simulations based on the LLG equation. For situations when a static magnetic field was suddenly applied and then removed, the effects of particle diameter and anisotropy (considering both type of symmetry and characteristic energy) on the characteristic magnetic relaxation time were studied parametrically. The results, for both anisotropy symmetries, show that when a static magnetic field is suddenly turned on or off the MNPs undergo a successive two-step or combined one-step relaxation. Whether a MNP relaxes with one or two steps when the field is turned on is determined by the competition between the energy of the applied magnetic field, the magnetic anisotropy energy, and thermal energy. When the applied magnetic field is suddenly turned off, our results show good agreement with theoretical predictions for the cases of ∆퐸ani⁄푘B푇 ≤ 1 and ∆퐸ani⁄푘B푇 ≫ 1, where ∆퐸ani represents the magnetic anisotropy energy barrier, 푘B is the Boltzmann constant and 푇 represents the absolute temperature. For the case of an applied AMF that is typical of magnetic particle imaging applications, the effects of particle diameter and anisotropy symmetry were studied in terms of time-domain magnetization dynamics, dynamic hysteresis loops, harmonic spectra, and the x-space point spread function (PSF). Results illustrate that the type of magnetocrystalline anisotropy (uniaxial vs cubic) has a significant effect on the MPI performance of the nanoparticles. These computational studies provide insight into the role of particle

diameter and magnetic anisotropy on the performance of MNPs for applications in magnetorelaxometry and MPI.

4.1 Background and Motivation

Over the past decade MNPs have attracted increasing attention for application in MPI, an emerging biomedical imaging technology that relies on the non-linear dynamic magnetization response of MNPs. Compared to other molecular imaging modalities, such as computed tomography (CT) and magnetic resonance imaging (MRI), MPI has the advantages of high sensitivity, not requiring ionizing radiation and having high image contrast (zero signal from the tissue background).36 In MPI, a static magnetic field gradient is superimposed with a uniform AMF to generate a small “field-free region”, where the nanoparticles are able to fully respond to the AMF and generate a signal.

This signal is then used to reconstruct a quantitative image of the distribution of MNPs in a field of view. While the underlying hardware and physics are similar, there are two approaches of image reconstruction and tracer characterization in common use for MPI: harmonic-space MPI and x-space MPI. In harmonic-space MPI, the response signal due to the MNPs dynamic magnetization is characterized by the harmonic spectrum that is obtained by taking the Fourier transform. In this modality, good MPI performance is typically indicated by strong signal of the third harmonic and slow signal decay for increasing harmonics.81-83 On the other hand, in x-space MPI the MNP signal is represented using a PSF describing the variation of a signal intensity with distance from a point source in the imaging volume.84 Compared with harmonic-space MPI, x-space

MPI offers the advantages of linearity and shift invariance, real-time imaging, and providing a simple means to estimate MPI tracer resolution from the PSF. Recent work has reported applications of different MPI techniques in tracking nanoparticle

accumulation in cancer85, 86 in evaluating brain injury or gut bleeds87, 88, in quantifying pulmonary drug delivery89, 90 and in monitoring transplanted stem cells91 or islets92.

In order to enhance the performance of MPI technology, prior experimental work81, 82, 93 has studied the effect of nanoparticle properties on the intensity and resolution of MPI signals. On par with the experimental work, some computational work has also studied the effects of nanoparticle properties on MPI performance, by carrying out simulations based on the LLG equation. For examples, Weizenecker et al.83 studied the dependence of MPI signal on particle size and AMF frequency for spherical and prolate ellipsoidal particles, and suggested that smaller anisotropy constants can increase the MPI performance of the particles. However, they assumed that MNPs have uniaxial-shaped anisotropy and their easy axes were fixed in the direction of the field.

This is typically not the case for spherical MNPs that are used in MPI, such as magnetite nanoparticles, which have cubic magnetic anisotropy and randomly distributed orientations. By using the MNPs and AMFs that are typical in MPI and numerically solving the stochastic differential equation that incorporates the LLG equation, Shah et al.94 studied the magnetic susceptibility and dynamic hysteresis loops of a collection of mobile and immobile spherical magnetite nanoparticles with cubic magnetic anisotropy. Good agreement was observed between computational and experimental results. However, since assumption was made that the orientations of easy axes of the nanoparticles were partially aligned in the field instead of random distribution, the cubic anisotropy of the nanoparticles was finally replaced by an effective uniaxial anisotropy, which didn’t actually describe characteristic of cubic anisotropy. Moreover, Shah et al.94 only studied magnetite nanoparticles of specific size

and anisotropy constant that were representative of those in their experiments, the results of which cannot be generalized for different particles. By performing simulations of the LLG equation for interacting superparamagnetic iron oxide nanoparticles, some prior work95,96 has also demonstrated the effect of magnetic dipole-dipole interactions on the harmonic spectrum of the nanoparticles. But this work is limited to uniaxial anisotropy symmetry and unique nanoparticle and AMF conditions. In summary of above work that focuses on the MPI performance, most of them only considered unique or narrow range of simulation parameters such as particle diameter and magnetic anisotropy constant. None of them have modeled the magnetic anisotropy with a real cubic symmetry that is compatible with randomly distributed easy axes. Furthermore, all of the above work focused on calculating the performance of MNPs for harmonic-space

MPI, and we are not aware of any reports of calculated x-space MPI performance.

Therefore, while the previous literature has contributed understanding the effect of internal dipole reorientation on MPI performance of nanoparticles, there remains a need for further work to systematically evaluate the effects of different types of magnetocrystalline anisotropy and a wide range of magnetocrystalline anisotropy constant and nanoparticle size on the non-linear dynamic magnetization and MPI performance of spherical particles.

For the case of x-space MPI, the PSF and corresponding resolution of MNPs have been studied by Goodwill and Conolly, using the Langevin function.84 However, the Langevin model assumes that MNPs respond to applied magnetic fields through instantaneous dipole alignment with the field, which is not applicable for nanoparticles that have non-negligible magnetic relaxation time. By considering the relaxation of

MNPs through a theoretical magnetization relaxation equation, Croft et al.97 obtained good agreement between their theoretical and experimental results. But the relaxation time is dependent on the equilibrium magnetization of the nanoparticles, which is predicted through the Langevin function and failed to account for particle magnetocrystalline anisotropy. Dhavarikar and Rinaldi98 studied both the magnetization harmonics and PSFs for MNPs that relax by the Brownian mechanism, using rotational

Brownian dynamic simulations and ferrohydrodynamic magnetization equations. In their work, the nanoparticles were assumed to be “thermally-blocked” and respond to the changing magnetic field by physical rotation. More recently, Shasha et al.99 studied the x-space MPI performance of MNPs by coupling the internal rotation of magnetic dipole and physical rotation of the particle and then concluded that 28 nm in core diameter is an optimal size for single-core monodisperse magnetite nanoparticles. However, most nanoparticle tracers of interest for MPI are not magnetically blocked or physically rotating, such as would often be the case for nanoparticles that accumulate inside cells.

These nanoparticles respond to changes in the magnetic field by only internal magnetic dipole rotation, i.e. through the Néel relaxation mechanism. Moreover, in Shasha and coauthor’s work99 magnetite nanoparticles were modeled with uniaxial anisotropy and the MPI tracer resolution was calculated based on the Langevin function and experimentally-determined parameters instead of applying x-space formulism to raw magnetization signal of the particles. A review of the recent literature suggests that no prior computational work has studied the effects of different magnetocrystalline anisotropy shapes and then nanoparticle diameter and anisotropy energy barrier on x- space MPI performance of MNPs undergoing the Néel relaxation.

The most commonly employed approaches to model the magnetization evolution of nanoparticles that respond by the Néel mechanism are based on the stochastic

Landau-Lifshitz (LL) equation62, 100, 101 or the LLG equation94, 102 that consider large damping of the magnetization field. For example, Berkov et al.101 applied the LL equation and computational Langevin dynamics simulations to study magnetorelaxometry of MNP suspensions in a range of nanoparticle concentrations, for cases where a static magnetic field was suddenly attenuated or switched off. By coupling the LLG equation to rotational Brownian simulations, Ilg et al.103 studied the effects of size of MNP cluster and value of uniaxial anisotropy constant on magnetorelaxometry of the cluster as a static magnetic field was suddenly switched off.

Their results suggest that a fast initial decay due to the Néel relaxation mechanism is followed by long-time relaxation that is due to the Brownian mechanism. However, the above work did not account for the effect of nanoparticle size and magnetic anisotropy symmetry on magnetorelaxometry, nor provided insight into the dipole dynamics for nanoparticles that are physically fixed in a matrix. Thus, further work is needed to fully understand how nanoparticle properties influence non-linear magnetization dynamics of

MNPs.

In Chapter 4, we report a computational study of the effect of particle diameter and magnetic anisotropy (considering both type of symmetry and barrier energy magnitude) on the magnetization dynamics of immobilized spherical single-domain

MNPs in static and AMFs, by employing the LLG equation accounting for the precession of internal magnetic dipoles. In the case of static magnetic fields, a comparison was made between the equilibrium magnetization response of the nanoparticles with uniaxial

and cubic anisotropy and the predictions of the Langevin function. Then, we investigated the effects of particle diameter, magnetic anisotropy symmetry and energy on the magnetic relaxation time of the nanoparticles as well as the dynamics of the magnetic dipole moments for cases where a static magnetic field is suddenly turned on and off. For the case of an applied AMF, the intrinsic MPI performance of magnetite nanoparticles that are typical of MPI applications was studied in terms of the evolution and harmonic spectrum of ensemble magnetization, hysteresis loops, and signal PSFs for various nanoparticle sizes. Since the focus of this work is on comparing the magnetization dynamics and MPI performance of MNPs with different sizes and magnetic anisotropies, the effects of magnetic dipole-dipole interactions, and nanoparticle size and anisotropy distributions81 are left to future studies.

4.2 Simulation Method

4.2.1 The Landau-Lifshitz-Gilbert (LLG) Equation

The magnetic dipole moment of single-domain MNPs, which have uniform magnetization throughout the nanoparticle volume is expressed by

퐦 = 푚 퐦̂ = 푀 푉 퐦̂ s d p (4-1) where 푚s represents the magnitude of the magnetic dipole moment, 푀d represents the domain magnetization of the material, and 퐦̂ is a unit vector specifying the orientation of the magnetic dipole moment. The damped precession of the magnetic dipole moment due to effective fields can be described by the LLG equation104

푑퐦′ 훾0 ′ 휆 ′ ′ = − 2 [퐦 × 퐁′eff + 퐦 × (퐦 × 퐁′eff)] 푑푡 1 + 휆 푚s (4-2) where the prime denotes vectors in particle coordinates (as opposed to laboratory coordinates), 푡 represents time, 훾0 is the gyromagnetic ratio, 휆 is the damping constant,

′ and 퐁eff represents the effective fields, which contains different energy contributions that are able to provoke changes in the orientation of the magnetic dipole. In our work,

′ we consider the energy contributions of 퐁ext, due to due to Zeeman energy (which is

′ generated by the external applied magnetic field), 퐁ani, due to magnetocrystalline

′ anisotropy energy, and 퐁therm, due to thermal agitations. On the right-hand side of

Equation (4-2), the first term accounts for the moment precession in the total effective field, whereas the second term accounts for the convergence of this precession trajectory to the direction of the effective field due to the magnetic energy dissipation.

External magnetic fields were applied in the +z direction in laboratory coordinates, represented by 퐁ext = 퐵dc퐢푧 with strength 퐵dc for the static magnetic fields and 퐁ext = 퐵ac sin(2휋푓푡) 퐢푧 with amplitude 퐵ac and frequency 푓 for AMFs, respectively.

The magnetic fields can be transformed into the coordinates of nanoparticle 푖 through

퐁′ = 퐁 ∙ 퐀 ext,푖 ext 푖 (4-3) where 퐀푖 is the transformation matrix of nanoparticle 푖.

For MNPs with uniaxial anisotropy, the corresponding effective field is of the form of105

′ 퐾u푉p ′ ′ ′ 퐁ani,푖 = 2 (퐦̂ 푖 ∙ 퐮̂푖)퐮̂푖 푚s (4-4)

′ where 퐾u is the (positive) uniaxial magnetocrystalline anisotropy constant and 퐮̂푖 is a unit vector that determines the orientation of uniaxial easy axis of nanoparticle 푖.

For MNPs that have cubic anisotropy, there are two typical easy axis constructions: six easy axes, with orientations along the center of the cubic surface, and eight easy axes, with orientations along the cubic vertices. Since our objective is to predict the MPI performance of magnetite nanoparticles, the eight-easy-axes

construction is employed and only the first cubic magnetocrystalline anisotropy constant

105 퐾c (with negative values) is considered. Thus the corresponding effective field is given by105

′ 2퐾c푉p ′ ′ 2 ′ ′ 2 ′ ′ ′ 퐁ani,푖 = − {[(퐦̂ 푖 ∙ 퐞2,푖) + (퐦̂푖 ∙ 퐞3,푖) ] (퐦̂푖 ∙ 퐞1,푖)퐞1,푖 푚s

′ ′ 2 ′ ′ 2 ′ ′ ′ + [(퐦̂ 푖 ∙ 퐞1,푖) + (퐦̂푖 ∙ 퐞3,푖) ] (퐦̂푖 ∙ 퐞2,푖)퐞2,푖 (4-5)

′ ′ 2 ′ ′ 2 ′ ′ ′ + [(퐦̂ 푖 ∙ 퐞1,푖) + (퐦̂푖 ∙ 퐞2,푖) ] (퐦̂푖 ∙ 퐞3,푖)퐞3,푖}

′ where 퐞푖 is a unit vector that determines three orthogonal directions of nanoparticle 푖. In

′ Equations. (4-4) and (4-5), the magnitude of 퐁ani indicates the ratio of the total magnetic anisotropy energy barrier ∆퐸ani to the total magnetic dipole moment 푚s, both of which are proportional to the particle volume. For the purpose of convenience, we use a unique notation 퐾 instead of 퐾u and 퐾c. The height of anisotropy energy barrier ∆퐸ani is then given by 퐾푉p for the uniaxial-anisotropy nanoparticles and |퐾|푉p⁄12 for the eight- easy-axes cubic-anisotropy nanoparticles.106

By applying the fluctuation-dissipation theorem, the field due to thermal

′ 107 fluctuation 퐁therm is characterized by

〈퐁′ (푡)〉 = ퟎ therm,푖 (4-6)

′ ′ 2푘B푇휆 〈퐁therm,푖(푡)퐁therm,푖(푡 + ∆푡)〉 = 훿(∆푡)퐈 훾0푚s (4-7) where 푘B is the Boltzmann constant, 푇 represents the absolute temperature, ∆푡 represent the discrete time interval, the Dirac delta function 훿 implies the white noise field and 퐈 is the identity matrix.

By introducing the dimensionless variables

퐦 푘 푇 푓 B ̃ 퐦̂ = , 휐 = , 푡̃ = 휇0푀s훾0푡, 푓 = 푚s 휇0푚s푀s 휇0푀s훾0 (4-8) and integrating from 푡̃ to 푡̃ + ∆푡̃, Equation (4-2) becomes

휐 2휆 ∆퐦̂ ′ = − [퐦̂ ′ × (훼퐞′ + 2퐾̃퐞′ + √ 퐖′) (1 + 휆2) 퐇ext,푖 퐇ani,푖 ∆푡̃ 푖

(4-9) 2휆 + 휆퐦̂ ′ × (퐦̂ ′ × (훼퐞′ + 2퐾̃퐞′ + √ 퐖′))] ∆푡̃ 퐇ext,푖 퐇ani,푖 ∆푡̃ 푖 where the Langevin parameter ⁄ and ′ for the static 훼 = 푚s퐵dc 푘B푇 퐞퐇ext,푖 = 퐢푧 ∙ 퐀푖 magnetic fields, ⁄ and ′ ̃ ̃ for the AMFs, dimensionless 훼 = 푚s퐵ac 푘B푇 퐞퐇ext,푖 = sin(Ω푡) 퐢푧 ∙ 퐀푖

̃ ′ angular frequency Ω̃ = 2휋푓, dimensionless anisotropy constant 퐾̃ = |퐾|푉p⁄(푘B푇), 퐖푖 is a random vector with zero mean and unit standard deviation, ′ ( ̂ ′ ̂′)̂′ for the 퐞퐇ani,푖 = 퐦푖 ∙ 퐮푖 퐮푖 uniaxial anisotropy symmetry, and

2 2 ′ 퐦′ ∙ 퐞′ + 퐦′ ∙ 퐞′ 퐦′ ∙ 퐞′ 퐞′ 퐞퐇ani,푖 = [( ̂ 푖 2,푖) ( ̂ 푖 3,푖) ] ( ̂ 푖 1,푖) 1,푖

2 2 + [(퐦̂ ′ ∙ 퐞′ ) + (퐦̂ ′ ∙ 퐞′ ) ] (퐦̂ ′ ∙ 퐞′ )퐞′ 푖 1,푖 푖 3,푖 푖 2,푖 2,푖 (4-10)

2 2 ′ ′ ′ ′ ′ ′ ′ + [(퐦̂ 푖 ∙ 퐞1,푖) + (퐦̂ 푖 ∙ 퐞2,푖) ] (퐦̂ 푖 ∙ 퐞3,푖)퐞3,푖 for the cubic anisotropy symmetry.

4.2.2 Simulation Parameters and Conditions

Simulations were carried out for 푁 = 3375 uniform immobilized spherical MNPs

5 (푀d = 4.46 × 10 A/m) at a temperature of 298.15 K. Since the nanoparticles are motionless and magnetic dipole-dipole interactions are not considered, the nanoparticle number is only significant for calculating the average magnetization of the

nanoparticles. Runs were executed starting from random particle configurations using a minimum time interval of ∆푡̃ = 0.1 for the case of static magnetic fields and ∆푡̃ = 1 for the case of AMFs. For the case of static magnetic fields, we varied the nanoparticle diameter and magnetic anisotropy constant in the range of 5 nm ≤ 퐷p ≤ 40 nm and

3 kJ⁄m3 ≤ |퐾| ≤ 30 kJ⁄m3, respectively. The intensity of the static magnetic field was varied in the range of 10 mT ≤ 퐵dc ≤ 200 mT. For the case of AMFs, the particle diameter was varied in the range of 5 nm ≤ 퐷p ≤ 40 nm, whereas the anisotropy constant was fixed at |퐾| = 13.5 kJ⁄m3 for magnetite nanoparticles.108 The AMF amplitude and frequency were 20 mT and 25 kHz, typical of MPI systems in the literature.36 Since the value of damping parameter applied in current literatures is in the range of 0.1 to 1 and no rigorous experimental results have shown the exact value, we applied the damping parameter 휆 = 1 for all simulations, which accounts for the fastest reversal of magnetic dipoles and has been used in many studies.94,106,109-111

4.2.3 Simulations of Magnetorelaxometry

In magnetorelaxometry, a collection of immobilized MNPs is subjected to sudden changes of magnitude of an applied static magnetic field, and their dynamic magnetization response is monitored. This situation was modeled by putting the nanoparticles in a zero magnetic field, then applying a static magnetic field of prescribed intensity for a certain time, and finally removing the magnetic field and monitoring relaxation of the nanoparticles towards equilibrium. The dimensionless average 푧- direction magnetization of the nanoparticles is given by Equation (3-2) and the evolution of the dimensionless average magnetization can be described by the phenomenological magnetization relaxation equation of Shliomis75, i.e. Equation (3-3).

Equation (3-3) can be solved for the two cases that were modeled to study magnetorelaxometry. For the case where the collection of MNPs is in equilibrium at zero field and then suddenly subjected to a static magnetic field in the +푧 direction, the solution to the magnetization relaxation equation, in dimensionless form, is

푀̃ 푡̃ ln (1 − 푧 ) = − 푀̃푧,eq 휏̃ (4-13) where 휏̃ = 휇0푀d훾0휏. For the case where the nanoparticles are in equilibrium with an applied magnetic field and the field is suddenly turned off, the solution to the magnetization relaxation equation is given by

푀̃ 푡̃ ln ( 푧 ) = − 푀̃푧,eq 휏̃ (4-14)

It should be noted that Equations (3-3), (4-13) and (4-14) describe an exponential decay of magnetization, in which particles respond to field changes with a constant (in time) relaxation time. In our simulations for the case of magnetorelaxometry the applied magnetic field is either suddenly turned on or suddenly turned off. Hence, when the magnetic field is on the particles are subjected to a constant magnetic field. A simple assumption would be that under such condition particle magnetization evolves with a constant relaxation time and a phenomenological model such as the ones in Equations

(3-3), (4-13) and (4-14) should capture the dominant behavior and allow for calculation of a relaxation time.

On the other hand, some prior work has also been done to predict the relaxation time of single-domain magnetic particles for various magnetic anisotropy energy and symmetries, by solving the Fokker-Planck differential equation. For example, Aharoni110

calculated the field-off relaxation time for MNPs with uniaxial and cubic anisotropy that are under the condition of ∆퐸ani⁄푘B푇 ≤ 1

푀d푉p 휏A1973 = 훾0푘B푇 (4-15)

For the condition of ∆퐸ani⁄푘B푇 ≫ 1, the field-off relaxation time has also been studied by

Brown107

푀 퐾푉 d p 휏B1963 = 1⁄2 exp ( ) 2퐾훾0(퐾푉⁄휋푘B푇) 푘B푇 (4-16) for particles with uniaxial anisotropy symmetry, and by Eisenstein and Aharoni111

3휋푀 |퐾|푉p 휏 ~ d exp ( ) EA1977 (4-17) √2훾0|퐾| 12푘B푇 for particles with cubic anisotropy symmetry. It should be noted that Equations (4-15) to

(4-17) were all obtained for the precondition of 휆 = 1. In our work, we compare simulation results of field-off relaxation time with the predictions of Equations (4-16) to

(4-17), as a means to validate our algorithm.

4.2.4 Simulation Parameters and Conditions

For MPI applications, the performance of MNPs can be assessed using the so- called tracer response (TR), given by112

̇ ̇ 푀푧(푡) 푇푅 = 휎Fe푀̃푧(퐻app) = 퐶Fe퐻̇ app(푡) (4-18)

2 where 휎Fe represents the specific magnetization in A ∙ m ⁄kgFe, 퐶Fe represents the iron concentration in mgFe⁄ml, and 퐻app is the time-dependent applied magnetic field. In

Equation (4-18), 푇푅 is normalized by the particle concentration of iron and the rate of field change (which corresponds to the FFP velocity compensation used in x-space MPI

reconstruction), and expressed as a function of the applied field instead of time (which accounts for FFP position compensation).112

In our work, 퐻app = 퐵ac sin(2휋푓푡)⁄휇0 because the z-direction AMF is applied without a bias static magnetic field. Signal intensity is then determined by 푇푅 ∙ 퐶Fe, from which the full-width-at-half-maximum (FWHM), deviation between envelop peaks, and peak intensity can be obtained. The projected intrinsic resolution Δ푥 of a MPI system is calculated112

FWHM Δ푥 = 퐺 (4-20) where 퐺 represents the constant field gradient.

Since in our simulations the minimum time step is too small relative to a cycle of the AMF and thermal agitations are non-negligible, direct numerical differentiation of the magnetization signal results in significant noise. In order to reduce this effect, we applied a moving average of the magnetization signal over a range of 푝 data points.

This moving average was only applied when calculating the PSF for a given simulation.

Because applying a moving average over too large a time window will remove fast magnetization dynamics generated by the AMF, we sought to determine suitable values of 푝 systematically. To do this, we reasoned that simulations of dynamic magnetization using the Langevin function would yield the fastest dynamics and determined, for each nanoparticle diameter (since the Langevin function is a function of particle diameter), the largest value of 푝 that would result in no more than 5 % deviation in the calculated values of the 푇푅 peak magnitude and FWHM.

4.3 Results

4.3.1 Equilibrium Response of Magnetization

Figure 4-1 shows the dimensionless equilibrium average magnetization of a collection of MNPs in a static magnetic field as a function of intensity of the field for anisotropy constant value of |퐾| = 13.5 kJ⁄m3, and uniaxial and cubic anisotropy symmetries. To define the arrival of magnetization to an equilibrium, we plotted the dimensionless z-direction magnetization 푀̃푧 of the nanoparticles as a function of the dimensionless time 푡̃ and computed the curve slope close to the end time of applying the field. When the magnitude of slope is less than 10−6, the magnetization is regarded as equilibrium. Accompanied with the scaled anisotropy energy shown in Table 4-1, we observe that in Figure 4-1 the equilibrium magnetization for both anisotropy symmetries have good agreement with the prediction of the Langevin model for ∆퐸ani⁄푘B푇 < 1.

However, for ∆퐸ani⁄푘B푇 > 1 increasing the value of ∆퐸ani leads to a divergence of the equilibrium magnetization from the prediction. These results have been shown by prior work113, and here give a validation to our algorithms. In addition, Figure 4-1 shows that for the same value of |퐾|, the nanoparticles with uniaxial anisotropy symmetry have a larger divergence of equilibrium magnetization from the Langevin model than those with cubic anisotropy symmetry. This is because the anisotropy energy barrier of the latter nanoparticles is one-twelfth of the anisotropy energy barrier of the former, which results in better alignment of the magnetic dipoles in the field direction and then larger equilibrium average magnetization of the cubic-anisotropy nanoparticles.

4.3.2 Simulations of Magnetorelaxometry

Figure 4-2 shows representative dimensionless average magnetizations of collections of MNPs as a function of dimensionless time, for nanoparticle diameter 퐷p =

3 15 nm, anisotropy constant value of |퐾| = 13.5 kJ⁄m (퐸ani⁄푘B푇 = 5.80 for uniaxial symmetry and 퐸ani⁄푘B푇 = 0.48 for cubic symmetry) and magnetic field intensity 퐵dc =

20 mT (훼 = 3.83 for both uniaxial and cubic symmetries). The corresponding magnetic relaxation curves were also plotted in terms of Equations. (4-13) and (4-14) for the field- on and off cases, respectively. In the magnetic relaxation curves, the slope is equal to

− 1⁄휏̃. As seen in Figure 4-2, for both types of anisotropy symmetries the average magnetization is zero before a magnetic field is applied. When the external field is suddenly turned on, the magnetization changes to achieve a new equilibrium with the field, through a successive two-step process (characterized by relaxation time 휏1 and 휏2) for the uniaxial-anisotropy nanoparticles and a combined one-step process

(characterized by relaxation time 휏12) for the cubic-anisotropy nanoparticles. Similarly, when the field is suddenly turned off, the decrease in the magnetization can also occur through two different processes: a successive two-step relaxation (characterized by relaxation time 휏3 and 휏4) and a combined one-step relaxation (characterized by relaxation time 휏34) for the uniaxial- and cubic-anisotropy nanoparticles, respectively.

Moreover, we observed that in the successive two-step relaxation processes, the first step relaxation was always faster than the second step relaxation, i.e. 휏1 < 휏2 and 휏3 <

휏4. Combined with more representative magnetization curves and magnetic relaxation curves shown in Figure 4-3, the results suggest that the successive two-step and combined one-step relaxations happen for both MNPs with uniaxial and cubic

anisotropy symmetries. More explicitly, we only observed the successive two-step relation when ∆퐸ani > 푘B푇 and ∆퐸ani⁄푘B푇 > 훼 for the field-on case and ∆퐸ani > 푘B푇 for the field-off case. Otherwise, the combined one-step relaxation takes place. It should be noted that the two-step relaxation processes observed here is different from that observed by Ilg103. In their work the second step was due to Brownian relaxation, whereas in our work the nanoparticles are unable to rotate or translate so that the

Brownian relaxation mechanism is abrogated.

To explain the above relaxation behaviors, we focused on the successive two- step process first. When the suddenly applied magnetic field is weak and cannot provide enough energy to overcome the anisotropy energy barrier, the magnetic dipoles which initially align in the easy axes antiparallel to the field direction cannot immediately flip to align with the field. Instead, they instantaneously re-align to some preferable direction nearby the easy axes (corresponding to relaxation step 휏1), and then due to thermal agitations undergo slow flips until an equilibrium state is reached

(corresponding to relaxation step 휏2). It is noted that the preferable direction is always located between the directions of the applied field and easy axis that the individual magnetic dipole is initially aligned with. The precise orientation is determined by competition between magnetic field energy and anisotropy barrier energy. When the field is suddenly turned off, since the anisotropy energy is greater than the thermal energy, magnetic dipoles rapidly snap back to their vicinal easy axes first

(corresponding to relaxation step 휏3) and then undergo slow flips until a new equilibrium is achieved (corresponding to relaxation step 휏4). For the combined one-step relaxation, since the anisotropy energy barrier is low, the dipoles that are antiparallel to the field

direction can be easily flipped by the suddenly applied field and only experience a one- step relaxation 휏12. Similarly, when the field is suddenly turned off, the dominant thermal agitations easily randomize the magnetic dipoles into easy axes with a one-step relaxation time 휏34.

Further insight into the successive two-step relaxation process can be obtained from Figure 4-4, which shows magnetization curves and orientation distributions of magnetic dipoles at representative time points, for magnetic field intensity 퐵dc = 30 mT

(훼 = 13.62) and a collection of uniaxial-anisotropy MNPs with diameter 퐷p = 20 nm and

3 anisotropy constant |퐾| = 13.5 kJ⁄m (퐸ani⁄푘B푇 = 13.74). It should be noted that in

Figure 4-4, the +푧′ direction of the particle coordinates represents the easy axis which points to the particle hemisphere in the +푧 direction of the laboratory coordinates. The representative time points are also noted in the magnetization curves by red solid circles. One can observe in Figure 4-4 that starting from a random configuration and subjected to a zero magnetic field, the magnetic dipoles align in vicinal easy axes (+푧′ or −푧′ direction), which acts as a random re-distribution in the laboratory coordinates.

When a static magnetic field is suddenly applied in the +푧 direction, the dipoles slant from the easy axes and align in more preferable directions, through relaxation step 휏1, and then slowly flip to align around the field, through relaxation step 휏2. When the field is suddenly turned off, the magnetic dipoles snap back to the vicinal easy axes, through relaxation step 휏3, and then undergo thermal randomization through relaxation step 휏4.

Here relaxation step 휏4 is not obvious because the MNPs are thermally blocked under the simulation conditions. Figure 4-5 shows the representative orientation distributions of magnetic dipoles for nanoparticles with cubic anisotropy symmetry and under the

same condition as in Figure 4-4. Both observations in Figures 4-4 and 4-5 verify the explanations that we made before for the different relaxation processes of a collection of immobilized MNPs.

We calculated the field-off relaxation times (휏34 or 휏3 and 휏4) and plotted them as a function of intensity of the magnetic field for a range of values of magnetic anisotropy constant and types of anisotropy symmetry. The results are shown in Figure 4-6, where some data points are missing because the magnetization of MNPs did not reach equilibrium with the applied field or the nanoparticles are effectively thermally blocked

(i.e. with an extremely long relaxation). It is observed that 휏34 and 휏4 are independent of the field intensity, whereas increasing the field intensity leads 휏3 to decrease first and then plateau. This can be explained because of the small angle between the dipole’s preferable direction and the direction of the nearest easy axis under small field intensities. However, for large field intensities (퐵dc ≥ 100 mT) the anisotropy energy barrier is negligible, which leads to better alignment of the dipoles. Additionally, the field intensity does not affect 휏34 and 휏4 because of the dominant thermal agitations and irrelevance of initial divergent angle of the magnetic dipole, respectively.

Figure 4-7 shows the magnetic relaxation time 휏34, or 휏3 and 휏4 as a function of dimensionless anisotropy constant 퐾̃ for a range of values of the magnetic anisotropy constant and for the two types of anisotropy symmetry. It should be noted that in Figure

4-7, 휏3 is the average value for cases of 퐵dc ≥ 100 mT. The relaxation times were scaled by 휏B1963 and 휏EA1977 for uniaxial- and cubic-anisotropy nanoparticles, respectively. For both types of anisotropy symmetry, one can observe good agreement between the combined one-step relaxation time 휏34 and the prediction of 휏A1973 for small

values of 퐾̃ (as seen in Figure 4-7 B) and D)). Increasing the value of 퐾̃ leads to a transition of particle relaxation from the combined one-step process to the successive two-step process, where the scaled 휏3 increases logarithmically and the scaled 휏4 increases linearly. However, the scaled 휏3 is always smaller than the scaled 휏4. For the two types of anisotropy symmetry, Figure 4-7 A) and C) show that simulation results of

휏3 are in good agreement with the predictions of 휏B1963 and 휏EA1977 for the uniaxial- and cubic-anisotropy nanoparticles, respectively. These results serve to validate our algorithms from the perspective of predicting dynamic magnetization response.

Furthermore, we also carried out simulation by apply the damping parameter 휆 = 0.1 and compared the simulation results to the theoretical prediction, as shown in Figure 4-

8. As observed in the figure, the relaxation time for 휆 = 0.1 diverge from the theoretical prediction. One reason is that Equations (4-15) to (4-17) were derived based on the precondition that 휆 = 1.

Figure 4-9 shows the representative magnetic relaxation time 휏12 or 휏1 and 휏2 as a function of magnetic field intensity for nanoparticle diameter 퐷p = 15 nm, and a range of values of magnetic anisotropy constant for the two types of symmetry. As seen, increasing the field intensity leads to a decrease in the relaxation time. Meanwhile, the relaxation of the uniaxial-anisotropy nanoparticles transitions from the successive two- step process to the combined one-step process. The transition field intensity increases with the value of the anisotropy constant. This phenomenon is justified because the successive two-step relaxation only happens for the case where the anisotropy energy is dominant over the energy of the magnetic field, and larger transition field intensity is required as the height of the anisotropy energy barrier increases. Our results suggest

that the nanoparticles with uniaxial anisotropy symmetry undergo a successive two-step relaxation when 훼푘B푇⁄∆퐸ani < 1.34. However, for nanoparticles with cubic anisotropy symmetry the energy of the magnetic field is always dominant over the anisotropy energy under the conditions studied here, so that only the combined one-step relaxation is observed. In addition, our results suggest that 휏12 is not a function of the value of the anisotropy constant. This is explained because for the case of combined one-step relaxation the anisotropy energy barrier is too low as compared to other energies

(thermal and magnetic). More field-on relaxation times for various particle diameters are shown in Figure 4-10, where one can observe that the transition field intensity is also independent of the particle diameter for the case of uniaxial anisotropy symmetry.

4.3.3 Magnetization Signal in An Alternating Magnetic Field

Figure 4-11 shows the magnetization curves and corresponding dynamic hysteresis loops of a collection of immobilized magnetite nanoparticles in an AMF typical of MPI, for various particle diameters and anisotropy symmetries. In Figure 4-11

A) and C), one can observe that increasing the particle diameter results in a change in the shape of the magnetization curve for both uniaxial- and cubic-anisotropy nanoparticles. Simultaneously, a significant response lag with respect to the applied field is observed only for the nanoparticles with uniaxial anisotropy symmetry. The magnetization curve of 40 nm uniaxial-anisotropy nanoparticles has a triangular shape because the anisotropy energy barrier is so high that the magnetic dipoles cannot be flipped by the applied field or thermal agitation. Furthermore, Figure 4-11 B) and D) show that increasing the particle diameter leads first to an increase and then to a decrease in the area of the dynamic hysteresis loop for the uniaxial-anisotropy

nanoparticles, whereas under the conditions studied this is seen to increase monotonically for the cubic-anisotropy nanoparticles. These behaviors are explained because for the same |퐾| value the uniaxial-anisotropy nanoparticles have a higher anisotropy energy barrier so that they experience a longer magnetic relaxation than the cubic-anisotropy nanoparticles.

By taking the fast Fourier transform of the magnetization signals in Figure 4-11,

Figure 4-12 shows the corresponding harmonic spectra of magnetization for the first 51 harmonics. The signals are normalized by the intensity of the first harmonic. It is seen that increasing particle diameter leads the decay rate of signals from the third harmonic to decrease first and then increase for the uniaxial-anisotropy nanoparticles and decrease monotonically for the cubic-anisotropy nanoparticles, under the conditions studied here. According to the analysis method that is typical of harmonic-space MPI, good MPI performance can be determined by the slow decay of harmonic signal from the third harmonic until the plateau of signal amplitude. The plateau above a certain harmonic is caused by thermal fluctuations which make the estimate of the harmonic unreliable. Therefore, our result suggests that for 퐷p ≥ 25 nm the magnetite nanoparticles that are modeled with cubic anisotropy symmetry have better MPI performance than those modeled with uniaxial anisotropy symmetry. Within the range of particle diameter considered here, the optimal performance for harmonic-space MPI is achieved at 퐷p = 20 nm and 퐷p = 40 nm for the uniaxial- and cubic-anisotropy nanoparticles, respectively.

Figure 4-13 shows representative PSFs for collections of magnetite nanoparticles in an AMF that is typical for MPI, for nanoparticle diameter 퐷p = 20 nm and the two

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types of magnetocrystalline anisotropy symmetry. As observed in Figure 4-13, the signal intensities for uniaxial- and cubic anisotropy nanoparticles are similar and agree with the prediction of the Langevin model. However, the nanoparticles with uniaxial anisotropy symmetry have smaller FWHM (i.e. finer intrinsic resolution) and larger peak- to-center deviation as compared with the nanoparticles with cubic anisotropy symmetry.

Figure 4-14 shows the corresponding signal intensity, FWHM, and peak deviation for the PSFs calculated for a range of nanoparticle diameters and for the two types of magnetocrystalline anisotropy symmetry. It should be noted that for uniaxial-anisotropy nanoparticles with diameter 퐷p ≥ 25 nm, the PSF calculation algorithm broke down because the peak shift is beyond the range of applied magnetic field amplitude. As observed in Figure 4-14, for 퐷p ≤ 25 nm the uniaxial-anisotropy nanoparticles have the same signal intensity, smaller FWHM/expected resolution, and much greater peak deviation as compared to the cubic-anisotropy nanoparticles. For the cubic anisotropy symmetry that magnetite nanoparticles should have, we compared the simulation results with the prediction of the Langevin model and observed good agreement for

퐷p ≤ 30 nm. For these nanoparticles, increasing the diameter leads to an increase and then slight decrease in signal intensity and a decrease and then slight increase in the

FWHM/expected resolution. Both the strongest signal intensity and the smallest

FWHM/expected resolution are obtained at 퐷p = 35 nm. Moreover, the peak deviation of the cubic-anisotropy nanoparticles hardly changes with diameter for 퐷p ≤ 30 nm and slightly increases for 퐷p ≥ 35 nm. According to the analysis method that is typical of x- space MPI, our results suggest that the optimal performance would be expected for the nanoparticles with 퐷p = 35 nm.

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4.4 Conclusions

In Chapter 4, we reported a computational study of the effects of particle diameter and magnetocrystalline anisotropy (considering both type of symmetry and barrier energy magnitude) on the magnetization dynamics and MPI performance of a collection of immobilized spherical single-domain MNPs that relax in Néel relaxation mechanism, by carrying out simulations based on the LLG equation. Our LLG simulation algorithm was validated in two ways. First, we demonstrated that for nanoparticles in equilibrium with an applied static magnetic field the average magnetization agreed with the predictions of the Langevin model for ∆퐸ani⁄푘B푇 < 1.

Second, we demonstrated that for magnetic dipoles undergoing thermal randomization in the absence of an applied magnetic field the decay in magnetization follows an exponential model with characteristic time in agreement with the predictions of prior works for a wide range of nanoparticle diameters and values of magnetic anisotropy constant, and for uniaxial and cubic anisotropy symmetries.

Our results suggest that for both types of anisotropy symmetry and both cases where a static magnetic field is suddenly turned on or off, MNPs may undergo a successive two-step (periods 휏1 and 휏2 for field turned on, periods 휏3 and 휏4 for field turned off) or combined one-step (period 휏12 for field turned on and 휏34 for field turned off) relaxation. Whether a nanoparticle relaxes with one or two periods when the field is turned on is determined by the competition between the energy of the applied magnetic field, the magnetic anisotropy energy, and thermal energy. For the case of an applied

AMF that is typical of MPI applications, the results suggest that different anisotropy symmetries lead to different MPI performance. Within the parameters studied, the

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optimal x-space MPI performance (signal and resolution) of the cubic-anisotropy magnetite nanoparticles was observed for 퐷p = 35 nm. In summary, the study in

Chapter 4 provides insight into the role of nanoparticle diameter and magnetic anisotropy energy and type of symmetry on the nonlinear dynamic magnetization response and performance of MNPs for applications in MPI.

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Figure 4-1. Equilibrium average magnetization of a collection of magnetic nanoparticles in an applied static magnetic field as a function of intensity of the field, for a representative anisotropy constant value of |퐾| = 13.5 kJ⁄m3 and various nanoparticle diameters.

Table 4-1. Scaled anisotropy energy ∆퐸ani⁄푘B푇 for a representative anisotropy constant value of |퐾| = 13.5 kJ⁄m3 and various nanoparticle diameters. 퐷p, [nm] 5 10 15 20 Uniaxial anisotropy 0.21 1.72 5.80 109.95 Cubic anisotropy 0.02 0.14 0.48 9.16

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Figure 4-2. Magnetization curve and corresponding magnetic relaxation curves for collections of immobilized magnetic nanoparticles with different anisotropy symmetries, nanoparticle diameter 퐷p = 15 nm, anisotropy constant |퐾| = 3 13.5 kJ⁄m (∆퐸ani⁄푘B푇 = 5.80 for uniaxial symmetry and ∆퐸ani⁄푘B푇 = 0.48 for cubic symmetry), and magnetic field intensity 퐵dc = 20 mT (훼dc = 3.83 for both uniaxial and cubic symmetries). A) and B) are representative for uniaxial and cubic anisotropy symmetries, respectively.

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Figure 4-3. Representative magnetization curves and corresponding magnetic relaxation curves for collections of immobilized magnetic nanoparticles, for various magnetic field intensities, nanoparticle diameters, values of anisotropy constant, and types of anisotropy symmetry.

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Figure 4-4. Magnetization curve and corresponding representative orientation distributions of magnetic dipoles of a collection of magnetic nanoparticles with uniaxial anisotropy symmetry, for nanoparticle diameter 퐷p = 20 nm and 3 magnetic anisotropy constant |퐾| = 13.5 kJ⁄m (∆퐸ani⁄푘B푇 = 13.74) and magnetic field intensity 퐵dc = 30 mT (훼dc = 13.62). A) and B) show the magnetization curve and corresponding representative orientation distributions of magnetic dipoles, respectively.

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Figure 4-5. Magnetization curve and corresponding representative orientation distributions of magnetic dipoles for a collection of magnetic nanoparticles with cubic anisotropy symmetry, for nanoparticle diameter 퐷p = 20 nm, 3 magnetic anisotropy constant |퐾| = 13.5 kJ⁄m (∆퐸ani⁄푘B푇 = 1.15), and magnetic field intensity 퐵dc = 30 mT (훼dc = 13.62). A) and B) show the magnetization curve and corresponding representative orientation distributions of magnetic dipoles, respectively.

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Figure 4-6. Characteristic magnetic relaxation times 휏34 or 휏3 and 휏4 as a function of intensity of the applied static magnetic field for various nanoparticle diameters, values of anisotropy constant, and types of anisotropy symmetry.

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Figure 4-7. Scaled characteristic magnetic relaxation times 휏34, or 휏3 and 휏4 as a function of dimensionless anisotropy constant for various values of the magnetocrystalline anisotropy constant, and for nanoparticles with different anisotropy symmetries. A) and C) are representative for uniaxial- and cubic- anisotropy nanoparticles, respectively. B) and D) provide zoom-in views for small dimensionless magnetic anisotropy constants in A) and C), respectively.

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Figure 4-8. Scaled characteristic magnetic relaxation times 휏34, or 휏3 and 휏4 as a function of dimensionless anisotropy constant for uniaxial-anisotropy nanoparticles, various values of the magnetocrystalline anisotropy constant and damping parameters. In the figure, A) and C) are for damping parameters 휆 = 1 and 휆 = 0.1, respectively. B) and D) provide zoom-in views for small dimensionless magnetic anisotropy constants in A) and C), respectively.

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Figure 4-9. Characteristic magnetic relaxation times 휏12 or 휏1 and 휏2 as a function of intensity of the applied static magnetic field for nanoparticle diameter 퐷p = 15 nm, various values of anisotropy constant, and different anisotropy symmetries. A) and B) are representative for uniaxial- and cubic-anisotropy magnetic nanoparticles, respectively.

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Figure 4-10. Characteristic magnetic relaxation times 휏12 or 휏1 and 휏2 as a function of intensity of the applied static magnetic field for various nanoparticle diameters, values of anisotropy constant, and types of anisotropy symmetry.

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Figure 4-11. Magnetization curves and corresponding dynamic hysteresis loops for collections of magnetite nanoparticles in an alternating magnetic field with amplitude 퐵ac = 20 mT and frequency 푓 = 25 kHz, for various nanoparticle diameters and for different magnetocrystalline anisotropy symmetries. A) and C) show the magnetization curves for uniaxial- and cubic-anisotropy nanoparticles, respectively. The corresponding dynamic hysteresis loops are in B) and D).

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Figure 4-12. Harmonic spectrum of magnetization signal of a collection of magnetite nanoparticles for alternating magnetic field amplitude 퐵ac = 20 mT and frequency 푓 = 25 kHz, various nanoparticle diameters, and for different magnetocrystalline anisotropy symmetries. A) and B) are representative for uniaxial- and cubic-anisotropy nanoparticles, respectively.

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Figure 4-13. Positive and negative scan tracer response of a collection of magnetite nanoparticles as a function of intensity of applied alternating magnetic field for the field amplitude 퐵ac = 20 mT, frequency 푓 = 25 kHz, nanoparticle diameter 퐷p = 20 nm and cases of uniaxial and cubic anisotropy symmetries. In the Figure, the corresponding full-width-at-half-maximum (FWHM) and intrinsic resolution of the nanoparticles (∆푥) are noted.

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Figure 4-14. Intensity, full-width-at-half-maximum (FWHM) and peak deviation of magnetization signal of a collection of magnetite nanoparticles as a function of nanoparticle diameter for the field amplitude 퐵ac = 20 mT and frequency 푓 = 25 kHz, and cases of uniaxial and cubic anisotropy symmetries.

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CHAPTER 5 CONCLUSION REMARKS

In this research, the focus is to develop algorithms and then execute computational simulations to describe the dynamics and magnetization dynamics of spherical single-domain MNPs in externally applied magnetic fields. The simulation algorithm was developed based on the linear and angular momentum balance equations or the LLG equation, for different relaxation models of magnetic dipole.

According to different applications of MNPs, external magnetic fields such as strong static magnetic field gradients, static homogeneous magnetic field and alternating homogeneous magnetic field were applied to a collection of suspended or fixed nanoparticles.

At first, we studied the magnetic capture rate and evolution of size of nanoparticle aggregate at a capture line generated by a magnetic pole reversal, using

Brownian dynamics simulations of a suspension of MNPs in strong external magnetic field gradients that are generated at a solid substrate. The simulation results suggest that under identical conditions of nanoparticle size, volume fraction, and magnetic fields nanoparticles that relax through Brownian relaxation mechanism are captured at a faster rate than nanoparticles that relax through Néel relaxation mechanism. We also observed that, when 휙 = 0.05 % and the maximum Langevin parameter 훼max ≥ 50

(퐻max ≥ 110.80 mT) for both Brownian- and Néel-relaxation nanoparticles, increasing the intensity of magnetic field results in little change on the power-law dependence of number of captured nanoparticles with capture time. Similar observations were made for the influence of particle volume fraction. When 훼max = 100 (퐻max = 221.60 mT) and particle volume fraction 휙 ≥ 0.005 % for both Brownian- and Néel-relaxation

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nanoparticles, the magnetic capture rate is not a function of particle volume fraction.

Additionally, strong magnetic dipole-dipole interactions are verified to assist on building tight aggregate structures and Brownian-relaxation nanoparticles contribute wider- shaped aggregates. This study can provide a theoretical understanding of magnetic capture mechanisms and the behaviors of the dispersed MNPs, relaxing by the

Brownian and Néel relaxation mechanisms. However, the simulation time in current work is very short (~0.01 s) as compared with the minimum practical operation time of magnetic assembly (~2 s). In order to theoretically predict the practical size of nanoparticle aggregations formed during magnetic assembly and design better devices for MNP separation, it is necessary to execute simulations with much longer time and and larger amount of MNPs.

Then, we studied the effect of inter-particle interactions on the magnetization dynamics and energy dissipation of a suspension of magnetically-blocked nanoparticles in static and AMFs, by carrying out Brownian dynamics simulations. Our simulation results suggest that the magnetic diameter of interacting MNPs determined by fitting the

Langevin function is larger than the actual particle size for particle sizes equal and less than 40 nm, and is smaller than the actual size for particle sizes equal and larger than

50 nm. The effect of particle-particle interactions and the formation of particle chains on the behavior and performance of the nanoparticles were investigated by parametrically tuning the strength of magnetic dipole-dipole interactions. The results of magnetorelaxometry show that increasing the strength of magnetic interactions increases the average length of chain-like particle aggregates and as a result increases the characteristic relaxation time of the nanoparticles. For an applied AMF without a

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bias field, we observed that for small and intermediate frequencies of the AMF, increasing the strength of magnetic interactions increases the SAR first and then decreases. For the high AMF frequency considered, increasing the strength of magnetic interactions decreases the SAR monotonically. However, exceptions are observed for high field amplitudes, where increasing the strength of magnetic interactions enhances the SAR monotonically for all considered field frequencies. When a static bias magnetic field is superimposed to the AMF, the simulations suggest that increasing the strength of magnetic interactions leads the SAR to increase first and then decrease for the low frequency and the frequency typical of MPI, and to decrease only for the frequency typical of hyperthermia. Moreover, the SAR decreases as the intensity of bias static field increases. This study provides a theoretical insight into the role of particle-particle interactions on the performance of MNPs for applications in magnetic hyperthermia and

MPI. However, to predict for real systems some other factors, such as the size distribution of MNPs, should also be taken into account, because of their non-negligible effects on particle dynamics and energy dissipation.

Finally, we studied the effects of particle diameter and magnetocrystalline anisotropy (considering both type of symmetry and barrier energy magnitude) on the magnetization dynamics and MPI performance of a collection of immobilized MNPs that relax in Néel relaxation mechanism, by carrying out simulations based on the LLG equation. Our LLG simulation algorithm was validated in two ways. First, we demonstrated that for nanoparticles in equilibrium with an applied static magnetic field the average magnetization agreed with the predictions of the Langevin model for

∆퐸ani⁄푘B푇 < 1. Second, we demonstrated that for magnetic dipoles undergoing thermal

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randomization in the absence of an applied magnetic field the decay in magnetization follows an exponential model with characteristic time in agreement with the predictions of prior works for a wide range of nanoparticle diameters and values of magnetic anisotropy constant, and for uniaxial and cubic anisotropy symmetries. Our results suggest that for both types of anisotropy symmetry and both cases where a static magnetic field is suddenly turned on or off, MNPs may undergo a successive two-step

(periods 휏1 and 휏2 for field turned on, periods 휏3 and 휏4 for field turned off) or combined one-step (period 휏12 for field turned on and 휏34 for field turned off) relaxation. Whether a nanoparticle relaxes with one or two periods when the field is turned on is determined by the competition between the energy of the applied magnetic field, the magnetic anisotropy energy, and thermal energy. For the case of an applied AMF that is typical of

MPI applications, the results suggest that different anisotropy symmetries lead to different MPI performance. Within the parameters studied, the optimal x-space MPI performance (signal and resolution) of the cubic-anisotropy magnetite nanoparticles was observed for 퐷p = 35 nm. This study provides insight into the role of nanoparticle diameter and magnetic anisotropy energy and type of symmetry on the nonlinear dynamic magnetization response and performance of MNPs for applications in MPI.

Moreover, since the focus is on understanding the individual effect of each particle property (such as size, relaxation mechanism, anisotropy energy and shape) on the dynamics and magnetization dynamics of the particles, above work didn’t take into account heterogeneous particle properties, such as distribution of particle size and material types. However, these heterogeneous particle properties are non-negligible in practical experiments and applicaions. In such cases, the distribution of particle size

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and particle material property should be introduced to above simulations by constructing an information matrix of particles, where each particle (or particle number) is bound with the corresponding particle size and material type (characterized through domain magnetization, magnetocrystalline anisotropy constant and symmetry). Then the magnetic dipole moments and anisotropy energy barriers of the particles will be re- calculated in the simulations. The other algorithms, such as those accounting for inter- particle interactions and motion equation, will not change.

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BIOGRAPHICAL SKETCH

Zhiyuan Zhao received his degrees in Bachelor of Engineering in chemical engineering and technology at Beijing University of Chemical Technology, China in

June 2012. His undergraduate thesis “Preparation of Biocompatible Small-size

Microcapsule and Characterization” was awarded as the department outstanding thesis in Beijing University of Chemical Technology. During the following one year, he worked as a research assistant in Tianjin Institute of Industrial Biotechnology, Chinese

Academy of Science. In fall 2015, he attended the Department of Chemical Engineering at University of Florida and then joined in the Rinaldi lab and worked on the Brownian dynamics simulations of magnetic nanoparticles. He received the degree in Master of

Science in May 2015, with a thesis titled “Brownian Dynamics Simulation of Magnetic

Nanoparticle Captured by A Strong Magnetic Field Gradient”. After graduation, he continued working as a research assistant in the Rinaldi lab for one year and re- admitted to enroll the Ph.D. program in Department of Chemical Engineering at

University of Florida in August 2016. His doctoral research was to computationally study the dynamics and magnetic dynamics of magnetic nanoparticles in externally applied magnetic fields, and experimentally study the rheology of ferrofluid in constant rotating magnetic fields.

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