Magnetization Dynamics and Energy Dissipation of Interacting

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C: Physical Processes in Nanomaterials and Nanostructures

Magnetization Dynamics and Energy Dissipation of Interacting Magnetic Nanoparticles in Alternating Magnetic Fields with and without A Static Bias Field Zhiyuan Zhao, and Carlos Rinaldi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b04071 • Publication Date (Web): 15 Aug 2018 Downloaded from http://pubs.acs.org on August 21, 2018

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The Journal of Physical Chemistry

Magnetization Dynamics and Energy Dissipation of Interacting Magnetic Nanoparticles in Alternating Magnetic Fields with and without A Static Bias Field Zhiyuan Zhao† and Carlos Rinaldi∗,†,‡ †Department of Chemical Engineering, University of Florida. Gainesville, Florida 32611, United States ‡J. Crayton Pruitt Family Department of Biomedical Engineering, University of Florida. Gainesville, Florida 32611, United States E-mail: [email protected] Phone: 1-352-392-0881. Fax: 1-352-392-9513

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Abstract The effect of inter-particle interactions on the magnetization dynamics and energy dissipation rates of spherical single-domain magnetically-blocked nanoparticles in static and alternating magnetic fields (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 particles 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 interactions. The results show that strong magnetic interactions result in longer chain-like particle aggregates and eventually longer characteristic relaxation time of the particles. 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 particle magnetization and dynamic hysteresis loops, whereas the energy dissipation of the particles was studied in terms of the calculated specific absorption rate. 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-particle interactions on the performance of magnetic nanoparticles for applications in magnetic hyperthermia and magnetic particle imaging.

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1.Introduction In recent decades, magnetic nanoparticles (MNPs) have attracted attention because they can be manipulated using applied external magnetic fields. For example, when an alternating magnetic field (AMF) is applied, MNPs can generate heat due to hysteresis losses. This effect is used to actuate release of a drug, 1 or to deposit heat in cancer tumors, as in magnetic hyperthermia. 2,3 MNPs are also of interest in magnetic particle imaging (MPI), an emerging biomedical imaging technology that maps MNPs tracers in vivo with millimeter to submillimeter spatial resolution. 4,5 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. 6,7 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. 8–12 Experimental 6,13,14 and computational 12,15,16 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 interactions between particles may play an important role in the energy dissipation rate of the nanoparticles in an AMF. 17–21 The magnetization dynamics of chains and clusters of single-domain MNPs in various geometries have also been experimentally studied. 22–24 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 model, 10,25,26 solving the FokkerPlanck equation, 11,27,28 and analysis based on the stochastic Landau-Lifshitz-Gilbert (LLG) equation. 29 The LLG equation has also been applied in simulations, in which magnetically interacting nanoparticles are fixed in a solid matrix, with 30,31 and without 32–35 consideration 3



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 Metropolis 36–38 and kinetic 39 Monte-Carlo algorithms. For example, Ruta et al. 39 studied the effect of a truncated dipolar interaction between a collection of StonerWohlfarth spherical MNPs on energy dissipation rate by tuning the packing fraction of the particles as well as by introducing random distributions of anisotropy constant and particle positions and volumes. Their simulations 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. 40 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 a 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 Rinaldi 41 reported a comparison of the predictions for the energy dissipation rates of MNPs using the linear response theory model by Rosensweig, 42 solution of the magnetization relaxation equations of Shliomis 43 and Martsenyuk, Raikher and Shliomis 44 (MRSh), and results obtained from rotational Brownian dynamics simulations. However, in their work they considered the infinitely dilute regime where there are negligible particle-

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particle interactions and for which particle translation does not contribute to the response of the nanoparticles to the magnetic field. In this contribution, we report a computational study of the magnetization dynamics of spherical single-domain magnetically-blocked nanoparticles in static and alternating magnetic fields, using Brownian dynamics simulations. 45 The term “thermally-blocked”, also referred to “magnetically-blocked”, indicates that the magnetic dipole moment of the particles is fixed in a so-called crystal easy axis, such that the particle physically rotates to align its dipole with the local magnetic field, in what is called the Brownian relaxation mechanism. 46 It should be noted that small-sized MNPs can also be “thermally-blocked” when their magnetocrystalline anisotropy constant is large (e.g., as with cobalt ferrite). The assumed single-domain nature of the particles allows us to neglect heating due to internal dipole rotation, 47 such that the heat dissipated by the particles is only due to their rotational motion. The algorithm takes into account translation and rotation of the nanoparticles, hydrodynamic drag, thermal fluctuations, magnetic 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 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 specific absorption rate (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

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rates. 20,26,32

2. Computational Simulation Methods 2.1. Brownian Dynamics Simulations Considering that thermally-blocked MNPs respond to a change in an external magnetic field through physical rotation, we assume their internal dipole moment is always “saturated”, with magnitude ms = Md Vp

(1)

where Md represents the saturation magnetization of the material and Vp represents the particle volume. The simulation box is set up in a Cartesian coordinate system and fixed in free space (i.e., the laboratory coordinates). A magnetic dipole moment in the particle coordinates has the form of ˆ0 m0 = µ0 ms m

(2)

where the prime indicates a vector is in particle coordinates. The vacuum permeability is ˆ is a unit vector specifying the orientation of the magnetic dipole µ0 = 4π × 10−7 N/A2 and m moment. As all magnetic dipole moments are assumed to have uniform magnitude and fixed ˆ0 orientation along the z-axis of their individual particle coordinates, we can transform m into the laboratory space through ˆ = A−1 · m ˆ0 m

(3)

The transformation matrix A is in the form of   2 2 2 2 2(ζχ − ηξ) 2(ζη + ξχ)  −ζ + η − ξ + χ    2 2 2 2 A= −2(ηχ + ζχ) −ζ − η + ξ + χ 2(ηχ − ζξ)     2(ζη − ξχ) −2(ζξ + ηχ) ζ 2 − η 2 − ξ 2 + χ2

6


(4)

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where ζ, η, ξ and χ are the quaternion parameters satisfying the condition ζ 2 +η 2 +ξ 2 +χ2 = 1. 48 Each particle in the simulation experiences forces and torques including those due to hydrodynamic drag, Fh and Th , those due to external magnetic fields, Tm and Tm , those due to magnetic dipole-dipole interactions, Fdd and Tdd , those due to other particle-particle interactions, represented here through a repulsive hard-core Yukawa potential, FYkw , and those due to thermal agitation resulting from collisions of the particles with solvent molecules, FB and TB . When applying the stochastic linear and angular momentum equations to nanoparticles, the assumption of negligible inertia is justified 49 and the resulting force and torque balances are given by             0  Fh   Fm   Fdd  FYkw   FB  +  +  = + + TB 0 Tdd Tm Th 0

(5)

As in our previous work, 45 particle-particle van der Waals forces and hydrodynamic interactions are neglected relative to the long-range magnetic interactions, whereas inter-particle excluded volume interactions are included in order to preclude the overlap of particles with neighbors. In a quiescent fluid, a particle is always subject to the hydrodynamic force Fh and torque Th , which are related to the particle’s velocity U and angular velocity ω through the mobility matrix M, according to     U  Fh   =M·  ω Th

(6)

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



UF MUT  M M=  ωF ωT M M

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(7)


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where MUF , MUT , MωF and MωT are the mobility matrices for the infinite boundary condition. In a medium without time-varying electric fields or currents, the magnetic force acting on the nanoparticles is given by Fm = µ0 m · ∇H

(8)

where H denotes the magnetic field vector. In our work, magnetic fields are all applied in the +z direction of the laboratory coordinates, having the forms of Hz,dc = Hdc iz with strength Hdc for the static magnetic fields and Hz,ac = Hac cos(Ωt)iz 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 Hz,bias = Hbias iz with strength Hbias . The magnetic force is zero for a uniform magnetic field. However, the nanoparticles experience a magnetic torque when their dipole moment is not aligned with the local magnetic field, which is given by

T m = µ0 m × H

(9)

The magnetic dipole-dipole force and torque exerted by particle j on particle i are given by 50,51

Fdd,ji =

3µ0 mj mi ˆj ·m ˆ i) + m ˆ j (ˆrji · m ˆ i) + m ˆ i (ˆrji · m ˆ j ) − 5ˆrji (ˆrji · m ˆ j )(ˆrji · m ˆ i )] (10) [ˆrji (m 4 4πrji Tdd,ji =

µ0 mj mi ˆ j · ˆrji )(m ˆ i × ˆrji ) + (m ˆj ×m ˆ i )] [3(m 3 4πrji

(11)

where mj and mi represent the magnitudes of magnetic dipole moment j and i, respectively, rji represents the center distance between particle j and i, ˆrji denotes the unit vector of ˆ j and m ˆ i represent the unit vectors of dipole moment j and center-to-center distance, and m i, respectively. The inter-particle repulsion due to electrostatic interactions is modeled using a repulsive 8


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hard-core Yukawa potential, truncated at σ ≤ rji ≤ λρ−1/3 and which has the form 52

uYkw,ji = ε

exp[−κ(rji − σ)] rji /σ

(12)

where σ is the hard-core diameter, λρ−1/3 denotes the cut-off distance, λ is a pre-factor to modulate the cut-off distance, ρ−1/3 is proportional to the average inter-particle distance, ε represents the pair potential, and κ represents the inverse Debye screening length. The force due to the repulsive Yukawa potential is obtained from

FYkw,ji = −∇uYkw,ji

(13)

Dimensionless variables are introduced according to ω ˜ ˜ = U , ω ˜ = U , ∇ = a∇, r˜ji = rji /a, κ ˜ = aκ aDr Dr

(14)

where a is the radius of the uniform-size particles, Dr = kB T (8πη0 a3 )−1 is the rotational diffusivity for a spherical particle, kB is the Boltzmann constant, T represents the absolute temperature, and η0 is the viscosity of the carrier fluid. The mobility matrix components of particle i are non-dimensionalized according to MUF i aDr MUT i aDr MωF i Dr MωT i Dr

4 a mUF 3 kB T i 4 1 = mUT 3 kB T i 4 a = mωF 3 kB T i 4 1 mωT = 3 kB T i =

(15a) (15b) (15c) (15d)

UT ωF where mUF and mωT are the dimensionless components of the mobility matrix i , mi , mi i UT ωF ˜ i as the corresponding to MUF and MωT xi and dΦ i , Mi , Mi i , respectively. By setting d˜

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infinitesimal translation and rotation vectors of the particle i, their relationship with velocity and angular velocity are given by 







˜i xi  U  d˜  =     dt˜ ˜i ˜i dΦ ω

(16)

where t˜ is non-dimensionalized according to t˜ = tDr . Integrating from t˜ to t˜ + ∆t˜ based on the first-order forward Euler method, and applying the fluctuation-dissipation theorem 53 to the Brownian terms, the motion equation results in   · eTm ,i  xi  α 43 mUT i  ∆˜ ˜   =  ∆t 4 ωT ˜ α 3 mi · eTm ,i ∆Φi  ! P P e e Fdd ,i Tdd ,i · · + mUT βdd mUF  4 i i r˜ji r˜3   j6=i j6=i ji !  ∆t˜ +  P eFdd ,i P eTdd ,i    ωT · · βdd mωF + m 4 i i r˜ r˜3 



j6=i

ji

j6=i

(17)

ji



  i  P 8 h −˜κ(˜rji −2) 1 κ ˜ UF ˆ βYkw 3 e · r + m 2 ji i ˜ i ∆t˜ r˜ji r˜ji   X  j6=i  ∆t˜ +  + h i     P 8 −˜κ(˜rji −2) 1 κ ˜ ωF ˜ ˜ ˆ βYkw 3 e + m · r W ∆ t ji i i r˜ji r˜2 j6=i

ji

As the time step ∆t˜ is assumed longer than the relaxation time for particle momentum, the particle is assumed to have negligible acceleration. The first term on the right side of eq. 17 calculates the translational and rotational variance of particles due to the external magnetic field at each time step, where we have defined

ˆ i × iz eTm ,i = m

(18)

and the Langevin parameter α = µ0 ms H0 /(kB T ). It should be noted that H0 is replaced by the intensity Hdc or Hbias for static fields and amplitude Hac for AMFs. Similarly, the second term and the third term on the right side of eq. 17 calculates the change in both 10


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position and orientation of particles due to magnetic dipole-dipole interactions and repulsive hard-core Yukawa potential, respectively, by defining

ˆj ·m ˆ i) + m ˆ j (ˆrji · m ˆ i) + m ˆ i (ˆrji · m ˆ j ) − 5ˆrji (ˆrji · m ˆ j )(ˆrji · m ˆ i) eFdd ,i = ˆrji (m

(19)

1 ˆ j · ˆrji )(m ˆ i × ˆrji ) + (m ˆj ×m ˆ i) eTdd ,i = (m 3

(20)

and by setting the parameter of magnetic dipole-dipole interactions as βdd = µ0 m2s /(πa3 kB T ) ˜ i (∆t˜) and the parameter of hard-core Yukawa repulsion as βYkw = ε/(kB T ). In eq. 17, X ˜ i (∆t˜) are random vectors characterized by a Gaussian distribution with mean and and W covariance as

˜ i (∆t˜)i = 0 hX ˜ i (∆t˜) · X ˜ i (∆t˜)i = 8 (mUF + mUT )∆t˜ hX i 3 i ˜ i (∆t˜)i = 0 hW ˜ i (∆t˜) · W ˜ i (∆t˜)i = 8 (mωF + mωT )∆t˜ hW i 3 i

(21a) (21b) (21c) (21d)

respectively.

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 φ = 1% (particle number N =3375) in a cubic simulation box with side length 1.12 µm and periodic boundaries. 54 We note that a volume fraction of 1% for magnetite corresponds to ∼ 50 mgFe3 O4 /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. 11



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Runs were executed starting from random particle configurations, by using a minimum time interval of ∆t˜ = 0.01 for cases of static magnetic fields (homogeneous and constant in the ˜ for cases of AMFs, where Ω ˜ = Ω/Dr . The cut-off +z direction) and ∆t˜ = 2π/(8000Ω) distance corresponding to a pre-factor of λ = 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 ≤ βdd ≤ 500. For the repulsive hard-core Yukawa potential we used a radius-scaled inverse Debye screening length of κ = 3 and a scaled interaction energy of βYkw = 3 (βYkw = 0 for the case of βdd = 0). The Langevin parameter of the static magnetic field for magnetorelaxometry was considered to be αdc =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 ≤ αac ≤ 100 (0.44 kA/m ≤ Hac ≤ ˜ ≤ 10 (corresponding to field frequency 3.07 kHz ≤ f ≤ 307.35 176.35 kA/m) and 0.1 ≤ Ω kHz), respectively, whereas the Langevin parameter of the bias static field was varied in the range of 0 ≤ αbias ≤ 14 (0 kA/m ≤ Hbias ≤ 24.69 kA/m).

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. 55 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, then removing the magnetic field. Due to the uniform particle dimensions and material, the average magnetization of the collection of nanoparticles can be represented by the average normalized z-direction magnitude of the magnetic dipole moments N

X mz,i ˜z = 1 M N i=1 ms

12


(22)

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where mz,n represents the z-direction magnitude of magnetic dipole moment of the nth particle. To analyze the results of these simulations we draw from simple theoretical models obtained by solving the phenomenological magnetization relaxation equation of Shliomis 43 ˜z dM 1 ˜ ˜ = (M z − Mz,t ) dt τ

(23)

˜ z represents the z-component average magnetization of the particles at any given where M ˜ z,t represents the magnetization at equilibrium with the instantaneous magnetic instant, M field, and τ 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 z-direction, the solution to the magnetization relaxation equation, in dimensionless form, becomes

ln(1 −

˜z t˜∗ M )=− ˜ z,eq τ˜ M

(24)

˜ z,eq is scaled by the saturation magnetization of the collection of particles (i.e. where M the saturated magnitude of magnetic moment), and t˜∗ and τ˜ are scaled by the Brownian relaxation time τB =

3η0 Vp kB T

(25)

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

ln

˜z t˜∗ M =− ˜ z,eq τ˜ M

13


(26)


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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 as 41 ∞ ∞ X X ˜ t˜) + ˜ t˜)] ˜ z = 1 α[ χ0 sin(nΩ χ00n cos(nΩ M 3 n=1 n n=1

(27)

where χ0n and χ00n are the nth-order in-phase and out-of-phase components of the complex susceptibility. When n=1, the fundamental in-phase and out-of-phase susceptibilities can be calculated through 2π

3 χ = πα

Z

3 χ = πα

Z

0

00

˜ t˜)d(Ω ˜ t˜) Mz (t˜) cos(Ω

(28)

˜ t˜)d(Ω ˜ t˜) Mz (t˜) sin(Ω

(29)

0 2π

0

2.5. Calculation of Energy Dissipation Rates 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 equation 41,46 ˙ = − 1 µ0 hQi 2p

Z

2p

M 0

dH dt dt

(30)

where the period of the cycle 2p = 2π/Ω. By substituting a sinusoidal AMF and re-writing variables in dimensionless form, eq. 30 becomes ˙ = µ0 Hac φMd Ω hQi 2˜ p

Z

2˜ p

˜ z sin(Ω ˜ t˜)d(t˜) M

(31)

0

where p˜ = pDr . For cases in which there is a bias field, the rate of energy dissipation has ˜ z , which is significantly influenced the same mathematical expression of eq. 31 but with a M by the bias field. Additionally, below we express the rate of energy dissipation using the

14


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specific absorption rate, or SAR, which is given by

SAR =

˙ hQi φρ

(32)

where ρ denotes the mass density of the particles.

3. Results and Discussion 3.1. Equilibrium Response Simulations were carried out for various diameters of MNPs, various intensities of applied static magnetic field and uniform strength of repulsive Yukawa potential. By assuming that all the particles 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 M = bL(αfit ) = b coth(αfit ) + αfit

(33)

3 where b is a constant, αfit = µ0 Md Vp,fit Hdc /(kB T ), and Vp,fit = πDp,fit /6. The results are

shown in Figure 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 1 compares the diameters used in the simulations (Dp ) to the corresponding fitted diameters (Dp,fit ), obtained by fitting the Langevin function to the simulation results. As seen from the comparison, 15



for Dp ≤ 40 nm the particle diameter obtained by fitting the Langevin function is larger than the diameter used in the simulations, whereas for Dp ≥ 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. 1 0.8 "#,%&' !

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0.6 0.4 0.2 0

0

5

15 10 (dc , [kA/m]

+, = 10 nm +, = 20 nm +, = 30 nm

20

+, = 40 nm +, = 50 nm +, = 60 nm

Figure 1: Equilibrium magnetization of magnetic nanoparticle suspension in an applied static magnetic field as a function of intensity of the field, for various particle diameters.

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

3.2. Simulations of Magnetorelaxometry Figure 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 interparticle interactions. In the figure, it is seen that introducing magnetic interactions not only increases the relaxation time of particles, but also leads to deviation from single-exponential 16


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response. This suggests a distribution of relaxation times in the response. In detail, (a)-(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)-(f) show that increasing the intensity of the field leads to more pronounced deviation from single exponential response for the interacting particles. Table 2 summarizes the characteristic relaxation times obtained from the linear relaxation region for Figure 2 (a)-(c) and from the initial relaxation region for (d)-(f). According to Table 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.

17



Figure 2: Time-dependent magnetization relaxation curves for cases where the external dc field is suddenly applied with intensity (a) αdc = 1 (Hdc = 1.76 kA/m), (b) αdc = 10 (Hdc = 17.63 kA/m) and (c) αdc = 100 (Hdc = 176.35 kA/m), and for cases where the field is suddenly suppressed for (d) αdc = 1 (Hdc = 1.76 kA/m), (e) αdc = 10 (Hdc = 17.63 kA/m) and (f) αdc = 100 (Hdc = 176.35 kA/m), and under various strengths of inter-particle interactions.

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Table 2: Magnetic relaxation time for the cases that an applied static magnetic field is suddenly applied and suppressed, and for various Langevin parameters and strengths of inter-particle interactions. Parameters of interactions βdd βdd βdd βdd βdd βdd

= 0, βYkw = 0 = 100, βYkw = 3 = 200, βYkw = 3 = 300, βYkw = 3 = 400, βYkw = 3 = 500, βYkw = 3

Magnetic field is applied Magnetic field is suppressed αdc = 1 αdc = 10 αdc = 100 αdc = 1 αdc = 10 αdc = 100 0.64 0.11 0.01 0.80 0.77 0.77 1.11 0.13 0.01 1.84 1.88 1.86 2.85 0.15 0.01 42.86 30.20 20.42 2.93 0.16 0.01 157.95 55.94 46.50 2.92 0.16 0.01 200.14 60.40 46.96 2.78 0.16 0.01 203.49 84.57 49.69

To gain further insight into the effect of inter-particle interactions on particle dynamics, we generated snapshots of particle 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 and the south pole is shown in white. Videos S1 and S2 in the Supporting Information show the evolution of configurations for non-interacting particles and interacting particles (βdd = 500 and βYkw = 3), respectively, for the intensity of the static magnetic field αdc = 10. 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 +z direction to respond to the applied magnetic field. However, the particles 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 particles with strong interactions have longer magnetic relaxation times because of the significant local fields as well as formation of particle aggregates which restrain the rotation of individual particles. 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 shows the length distributions of particle chains and the corresponding particle configurations where the particles are in equilibrium with an applied static field, for interaction parameters βdd = 500 19



and βYkw = 3, and various intensities of the static magnetic field. It is observed that the particles 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 Lavg,c , which is in the unit of particles. By inspecting the snapshots of particle 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 particles and particle chains, but suppress the rotation due to inter-particle interactions and then shorten the length of chains. Figures S1-S3 in the Supporting Information show the equilibrium length distributions of particle chains and the corresponding particle 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 αdc , one can observed that for βdd ≥ 300, increasing the field intensity shortens the average chain length.

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Figure 3: Number of particle chains as a function of the length of particle chain, and corresponding representative snapshots of particle configuration, for interaction parameters βdd = 500 and βYkw = 3, Langevin parameter of static magnetic field (a) αdc = 1 (Hac = 1.76 kA/m), (b) αdc = 10 (Hac = 17.63 kA/m) and (c) αdc = 100 (Hac = 176.35 kA/m).

3.3. Energy Dissipation Rate in An Alternating Magnetic Field Figure 4 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 αac = 10 and various field frequencies. In Figure 4(a)-(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 ˜ = 0.1 and 1 increasing βdd significantly affects harmonic spectra and leads the area for Ω 21



˜ = 10, as βdd increases, the of the hysteresis loop to increase first and then decrease. For Ω 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 particles 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 particles 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 particles to respond to the field and result in longer particle chains, such that the particles experience a relaxation time that is longer than the cycle time and cannot achieve the maximum response as for the weak- or non-interacting particles. 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 particles. As a result, the particles 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. Further insight into the effect of magnetic interactions on particle dynamics can also be obtained from Videos S3 and S4 in the Supporting Information. As seen in the videos, non-interacting particles physically rotate to respond to the change of the AMF, whereas the particles with strong interactions form chain-like aggregates and then hardly rotate with the field. In addition, the formed particle chains in Video S4 are randomly distributed and do not rotate significantly. This suggests that for the particles with strong dipole-dipole interactions, the applied AMF contributes to form random-distributed particle chains. Figure 5 shows the representative snapshots of particle configurations that are extracted from the videos, for various interaction parameters and time points. In the figure, it is observed that for the non-interacting particles 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

22


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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 5.

23



(d) 0.15

Signal

01 = 0.1 34,567 2

0.5 0 -0.5 -1 (b)

0.1 0.05 0

100 150 200 250 300 350

Signal

01 = 1 34,567 2

-1

1

5

0 -0.5

15

20

25

30

1 0.5 0

35

(f)

1

1

5

-0.5 6

6.5 .̃

7

7.5

6

0

4

0

8

1

1

-1

2 5.5

0 3 8

9 13 17 21 25 29 33

34,567 2

Signal

0

-1

0

-1

8 0.5

1

1

-1

10

0 3 8

9 13 17 21 25 29 33

(e) 1.5

0.5

(c)

0

-1

1

-1

1 34,567 2

1

34,567 2

(a)

01 = 10 34,567 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 43

-1

1

!"" = 200,!&'( = 3 !"" = 0, !&'( = 0 !"" = 300,!&'( = 3 !"" = 100,!&'( = 3 Applied magnetic alternatingfield magnetic field

5

0 3 8

1

9 13 17 21 25 29 33 Harmonic number !"" = 400,!&'( = 3 !"" = 500,!&'( = 3

Figure 4: Scaled z-direction magnetization as a function of dimensionless time for Langevin ˜ = 0.1 parameter αac = 10 (Hac = 17.63 kA/m) and dimensionless angular frequency (a) Ω ˜ = 1 (f = 30.73 kHz) and (c) Ω ˜ = 10 (f = 307.35 kHz). The (f = 3.07 kHz), (b) Ω corresponding harmonic spectra of magnetization signal and dynamics hysteresis loops (in inset) are in (d), (e) and (f).

24


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Applied alternating magnetic field !"" = 0,!&'( = 0 !"" = 500,!&'( = 3

Figure 5: Magnetization curve and representative snapshots of particle configuration for ˜ = 10 Langevin parameter αac = 10 (Hac = 17.63 kA/m), dimensionless angular frequency Ω (f = 307.35 kHz) and various interaction parameters.

By applying eqs. 28 and 29, we obtained the real and imaginary components of the complex susceptibility as a function of dimensionless angular frequency for Langevin parameter αac = 10 and various strengths of inter-particle interactions, as shown in Figure 6. In the figure, one can observe that increasing βdd 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, 40 which agrees with the above observations. 25



Figure 6: (a) Real and (b) imaginary components of the complex susceptibility as a function of dimensionless angular frequency for Langevin parameter αac = 10 (Hac = 17.63 kA/m) and various strengths of inter-particle interactions.

In order to consider the effect of magnetic interactions on the heating efficiency of the particles, the SAR is plotted in Figure 7 as a function of amplitude of the AMF for various field frequencies and strengths of inter-particle interactions. At the lowest frequency (f = 3.07 kHz), it is observed that increasing βdd leads first to an increase and then a decrease in SAR for all field amplitudes. The βdd for which SAR peaks is dependent on the amplitude of the AMF. At the intermediate frequency (f = 30.73 kHz), increasing βdd leads the SAR to increase first and then decrease for small field amplitudes, and to increase monotonically for

26


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large amplitudes. At the highest frequency (f = 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, particles 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.

27



Figure 7: Specific absorption rates as a function of the amplitude of alternating magnetic ˜ = 0.1 (f = 3.07 kHz), (b) Ω ˜ = 1 (f = 30.73 kHz) and (c) Ω ˜ = 10 field for frequency (a) Ω (f = 307.35 kHz), and various strengths of inter-particle interactions.

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3.4. Effect of static bias magnetic field on energy dissipation rate Figure 8 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 param˜ = 10, and for various strengths of bias field and magnetic eter αac = 10 and frequency Ω interactions. The corresponding harmonic spectra of the magnetization, and dynamic hysteresis loops are also included. In Figure 8(a)-(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 +z direction, the bias field contributes to enhance the total magnetic field exerted on particles, 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 -z 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 +z section of the plot. In all cases it is shown that increasing the strength of magnetic interactions reduces the ability of the particles 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).

29



(d) 8

0.5

6

0

6

6.5

7

7.5

0

8

1

(e) 8

0.5

6

0

(c)

5.5

6

6.5

7

7.5

(f)

0.5 0 -0.5

0 6 :

1

9 13 17 21 25 29 33 1

0

-1 -1

1

5

4 2

0 6 :

1

9 13 17 21 25 29 33

8 6

Signal

67,389 5

5

4

0

8

1

-1

1

1 67,389 5

-1

-1

2

-0.5

0

-1

67,389 5

5.5

Signal

67,389 5

01234 = 8

(b)

4 2

-0.5 -1

1 67,389 5

1

Signal

67,389 5

01234 = 4

(a)

01234 = 12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 43

0

-1 -1

0 6 :

1

0 6.5 7 7.5 8 1 5 9 13 17 21 25 29 33 ̃. Harmonic number !"" = 200,!&'( = 3 !"" = 0, !&'( = 0 !"" = 400,!&'( = 3 !"" = 500,!&'( = 3 !"" = 300,!&'( = 3 !"" = 100,!&'( = 3 Applied alternating magnetic field Applied magnetic field 5.5

6

Figure 8: Scaled z-direction magnetization as a function of dimensionless time for Langevin ˜ = 10 parameter of alternating magnetic field αac = 10 (Hac = 17.63 kA/m), frequency Ω (f = 307.35 kHz), Langevin parameters of bias field (a) αbias = 4 (Hbias = 7.05 kA/m), (b) αbias = 8 (Hbias = 14.11 kA/m) and (c) αbias = 12 (Hbias = 21.16 kA/m). The corresponding harmonic spectra of magnetization signal and dynamic hysteresis loops (in insets) are in (d), (e) and (f).

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Figure 9 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. f = 3.07 kHz) and of a field frequency typical of MPI (i.e. f = 30.73 kHz), increasing the strength of magnetic interactions results in the value of SAR to increase first and then decrease. The value of βdd for achieving the highest SAR varies with the strength of the bias field. On the other hand, for a typical hyperthermia frequency (i.e. f = 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. The effect of magnetic interactions on magnetization dynamics with superimposed AMF and bias fields can be seen in Videos S5 and S6 in the Supporting Information. In the videos, it is observed that the bias field is so strong that particles hardly rotate to respond to the AMF. Compared with the non-interacting particles, the particles with strong interactions form chain-like aggregates, which better align along the direction of the bias field and have shorter chain length as compared to Videos S3 and S4. This result suggests that for strong magnetic interactions the application of a bias field contributes to better align both particles and particle chains, but reduces the average length of the particle chains.

31



SAR, [W/g]

(a)

14 12

SAR, [W/g]

! = 3.07 kHz

10 8 6 4 2 0

(b) 140

0

5

10

15

20

25

! = 30.73 kHz

120 100 80 60 40 20 0

(c) 1400 1200 1000

SAR, [W/g]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0

5

10

15

20

25

! = 307.35 kHz

800 600 400 200 0

0 5 25 10 15 20 Strength of bias magnetic field, [kA/m] !"" = 0, !&'( = 0 !"" = 100, !&'( = 3 !"" = 200, !&'( = 3 !"" = 300, !&'( = 3 !"" = 400, !&'( = 3 !"" = 500, !&'( = 3

Figure 9: Specific absorption rates as a function of strength of bias field for Langevin pa˜ = 0.1 rameter of alternating magnetic field αac = 10 (Hac = 17.63 kA/m), frequency (a) Ω ˜ = 1 (f = 30.73 kHz) and (c) Ω ˜ = 10 (f = 307.35 kHz), and various (f = 3.07 kHz), (b) Ω strengths of inter-particle interactions.

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4. Conclusions In the present work, we report a computational study of 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 particles 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 particles. 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, 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 33



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size distribution of MNPs, should also be taken into account, because of their non-negligible effects on particle dynamics and energy dissipation.

Acknowledgement This work was supported in part by the U.S. National Science Foundation, Grant CBET1511113.

Supporting Information Available Figure S1. Number of particle chains as a function of the length of particle chain, and corresponding snapshots of particle configuration at t˜ = 19.80, for the Langevin parameter of static magnetic field αdc = 1 (Hdc = 1.76 kA/m) and various strengths of magnetic dipoledipole interactions. Lavg,c represents the average length of particle chains. (PDF) Figure S2. Number of particle chains as a function of the length of particle chain, and corresponding snapshots of particle configuration at t˜ = 19.80, for the Langevin parameter of static magnetic field αdc = 10 (Hdc = 17.63 kA/m) and various strengths of magnetic dipole-dipole interactions. Lavg,c represents the average length of particle chains. (PDF) Figure S3. Number of particle chains as a function of the length of particle chain, and corresponding snapshots of particle configuration at t˜ = 19.80, for the Langevin parameter of static magnetic field αdc = 100 (Hdc = 176.35 kA/m) and various strengths of magnetic dipole-dipole interactions. Lavg,c represents the average length of particle chains. (PDF) Video S1. Evolution of magnetic nanoparticle configurations for interaction parameters βdd = 0 and βYkw = 0, Langevin parameter αdc = 10 (Hdc = 17.63 kA/m), and for the case that a static magnetic field is suddenly applied for half simulation time and then suddenly switched off. (AVI) Video S2. Evolution of magnetic nanoparticle configurations for interaction parameters βdd = 500 and βYkw = 3, Langevin parameter αdc = 10, (Hdc = 17.63 kA/m), and for 34


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the case that a static magnetic field is suddenly applied for half simulation time and then suddenly switched off. (AVI) Video S3. Evolution of magnetic nanoparticle configurations for interaction parameters βdd = 0 and βYkw = 0, Langevin parameter αac = 10 (Hac = 17.63 kA/m), field frequency ˜ = 10 (f = 307.35 kHz), and for the case of an applied alternating magnetic field without Ω a bias field. (AVI) Video S4. Evolution of magnetic nanoparticle configurations for interaction parameters βdd = 500 and βYkw = 3, Langevin parameter αac = 10 (Hac = 17.63 kA/m), field frequency ˜ = 10 (f = 307.35 kHz), and for the case of an applied alternating magnetic field without Ω a bias field. (AVI) Video S5. Evolution of magnetic nanoparticle configurations for interaction parameters βdd = 0 and βYkw = 0, Langevin parameter αac = 10 (Hac = 17.63 kA/m), field frequency ˜ = 10 (f = 307.35 kHz), and for the case of an applied alternating magnetic field with a Ω static bias field of αbias = 8 (Hbias = 14.11 kA/m). (AVI) Video S6. Evolution of magnetic nanoparticle configurations for interaction parameters βdd = 500 and βYkw = 3, Langevin parameter αac = 10 (Hac = 17.63 kA/m), field frequency ˜ = 10 (f = 307.35 kHz), and for the case of an applied alternating magnetic field with a Ω static bias field of αbias = 8 (Hbias = 14.11 kA/m). (AVI) This material is available free of charge via the Internet at http://pubs.acs.org/.

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advances, challenges, and opportunity. International Journal of Hyperthermia 2013, 29, 706–714. (3) Singh, A.; Sahoo, S. K. Magnetic nanoparticles: a novel platform for cancer theranostics. Drug Discovery Today 2014, 19, 474–481. (4) Gleich, B.; Weizenecker, J. Tomographic imaging using the nonlinear response of magnetic particles. Nature 2005, 435, 1214–1217. (5) Goodwill, P. W.; Scott, G. C.; Stang, P. P.; Conolly, S. M. Narrowband magnetic particle imaging. IEEE transactions on medical imaging 2009, 28, 1231–1237. (6) Hensley, D.; Tay, Z. W.; Dhavalikar, R.; Zheng, B.; Goodwill, P.; Rinaldi, C.; Conolly, S. Combining magnetic particle imaging and magnetic fluid hyperthermia in a theranostic platform. Physics in medicine and biology 2017, 62, 3483. (7) Tay, Z. W.; Chandrasekharan, P.; Chiu-Lam, A.; Hensley, D. W.; Dhavalikar, R.; Zhou, X. Y.; Yu, E. Y.; Goodwill, P. W.; Zheng, B.; Rinaldi, C. Magnetic Particle Imaging Guided Heating In Vivo using Gradient Fields For Arbitrary Localization of Magnetic Hyperthermia Therapy. ACS nano 2018, (8) Vallejo-Fernandez, G.; Whear, O.; Roca, A.; Hussain, S.; Timmis, J.; Patel, V.; O’Grady, K. Mechanisms of hyperthermia in magnetic nanoparticles. Journal of Physics D: Applied Physics 2013, 46, 312001. (9) Alvarez-Berrios, M. P.; Castillo, A.; Merida, F.; Mendez, J.; Rinaldi, C.; TorresLugo, M. Enhanced proteotoxic stress: one of the contributors for hyperthermic potentiation of the proteasome inhibitor bortezomib using magnetic nanoparticles. Biomaterials science 2015, 3, 391–400. (10) Chen, B.; Ho, C.; Kao, L.; Wu, W.; Tsai, Y.; Ma, C. Hysteresis loss-induced tem-

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Magnetization Dynamics and Energy Dissipation of Interacting

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