# Complex Magnetization Harmonics of Polydispersive Magnetic Nanoclusters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{2}O

_{3}core particles with an average primary size of about 4 nm are three times smaller than the AOS-Fe

_{3}O

_{4}core particles. We used a typical phase-sensitive detection method to measure the complex third-harmonic signals (Figure 1b). In addition, we preliminarily characterized the static and dynamic magnetization curves of the ferrofluid samples, as well as the field-dependent relaxation time constant of the immobilized particles. During all measurements, the concentration and sample volume were set to 28 mg-Fe mL

^{−1}and 0.1 mL, respectively.

## 3. Results and Discussion

#### 3.1. Magnetism of CMEAD-γFe_{2}O_{3}

_{2}O

_{3}nanoclusters are expected to exhibit neither remanence nor coercivity in their static magnetization at room temperature, as well as the case of hydrodynamic AOS-Fe

_{3}O

_{4}nanostructures. By highlighting the minor hysteresis properties of the dynamic magnetization curves at 1 kHz, Figure 2 further reveals that their colloidal relaxation behaviors appear to depend on the primary size and the respective magnetic interaction of the core particles. In the case of the CMEAD-γFe

_{2}O

_{3}ferrofluid, we observe extremely small coercivities for various field amplitudes. Under ±300 Oe, for instance, it reaches only about 0.4 Oe, much smaller than that spotted in the AOS-Fe

_{3}O

_{4}ferrofluid. For analyzing the magnetism of the respective core particles, both ferrofluid samples were solidified by the hydrocolloid polymer (i.e., agar). We then confirm that the immobilized CMEAD-γFe

_{2}O

_{3}nanoclusters have similar dynamic magnetization curves to those dispersed in water. In contrast, the solidified AOS-Fe

_{3}O

_{4}ferrofluid shows an even larger hysteresis area. The coercivity of the AOS-Fe

_{3}O

_{4}core particles increases from 0.9 Oe (under dc field) to 24 Oe (under ±300 Oe at 1 kHz).

_{3}O

_{4}ferrofluid may contain a mixture of superparamagnetic and ferrimagnetic magnetite nanoparticles [18]. Figure 2 further indicates that their hydrodynamic nanostructures may physically rotate under the oscillatory field to result in a hysteresis area. From Figure 3a, the effective Brownian relaxation frequency (under 50-Oe field amplitude) is observable around 6 kHz, associated with the maximum imaginary part of the fundamental susceptibility ${\chi}_{1}^{\u2033}$. This Brownian peak might be attributed to the physical relaxations of either the minor magnetite particles with diameters above 16.5 nm or the major nanoclusters with strong magnetic interactions between the composing particles (see illustration inset Figure 2). The widely distributive core sizes let them collectively behave in the liquid medium as a typical single-core structure with an effective core size [16]. Moreover, the real parts ${\chi}_{1}^{\prime}$ continuously decrease with frequency. The particle immobilization further results in ${\chi}_{1}^{\prime}$ decaying proportionally to $\mathrm{ln}\left(f\right)$ and the ${\chi}_{1}^{\u2033}$ being small and virtually frequency-independent [19,20,21]. Nevertheless, this typical relaxation behavior for thermally blocked magnetic nanoparticles was not the case for CMEAD-γFe

_{2}O

_{3}nanoclusters.

#### 3.2. Frequency and Field Dependences of Complex Third-Harmonic Magnetization

_{2}O

_{3}nanoclusters are morphologically treated as a multicore particle system containing small superparamagnetic maghemite crystallites, in which the dipolar interaction between individual cores inside a single nanocluster is supposed to be negligible. Their fundamental susceptibility spectra appear to be identical, independent of whether the nanoclusters were suspended or immobilized (Figure 3a). Thus, all core particles within the CMEAD-γFe

_{2}O

_{3}nanoclusters should magnetically relax via the Néel mechanism under the alternating field. There is no peak in both ${\chi}_{1}^{\u2033}\left(f\right)$ plots of the CMEAD-γFe

_{2}O

_{3}liquid and solid samples. However, slightly larger ${\chi}_{1}^{\prime}$ values of the liquid sample at low frequencies might be attributed to the insignificant contribution of the Brownian dynamics. Owing to these relaxation properties, their magnetization harmonics are later found to be characteristically different from those of the AOS-Fe

_{3}O

_{4}nanoclusters.

#### 3.2.1. Spectra of Third-Harmonic Susceptibility

_{3}O

_{4}ferrofluid, accordingly, the ${\chi}_{3}^{\u2033}\left(f\right)$ plot shows such typical spectral peak difference upon ${\chi}_{1}^{\u2033}\left(f\right)$, in which it peaks at about 2 kHz, one-third of the frequency-maximizing ${\chi}_{1}^{\u2033}\left(f\right)$ plot (Figure 3a). Meanwhile, for the CMEAD-γFe

_{2}O

_{3}ferrofluid, it is expected to have identical ${\chi}_{3}^{\prime}$ and ${\chi}_{3}^{\u2033}$ spectra of the solid and liquid samples due to the dominant moment dynamics. Interestingly, we found that their ${\chi}_{3}^{\u2033}$ components have negative values at low frequencies. Regarding this characteristic, the short Néel time constant of each core particle should be responsible for creating a large $\Delta {\phi}_{1}$, thus $\Delta {\phi}_{1}>\pi /3$ rad mathematically leads to a negative ${\chi}_{3}^{\u2033}\propto \mathrm{sin}3\Delta {\phi}_{1}$. Being consistent with our finding, the authors of [22], furthermore, numerically identified that ${\chi}_{3}^{\prime}\propto \mathrm{cos}3\Delta {\phi}_{1}$ may also be negative for this situation.

_{2}O

_{3}ferrofluid are not coincidental, we consider an empirical non-Debye relaxation model to evaluate the spectral magnetization responses [16,24]. As illustrated in Figure 3b, the fundamental frequency-varying magnetization ${M}_{1}\left(f\right)$ of the polydispersive nanoparticle system (under arbitrary field strength) settles at a nonzero value of the infinite-frequency magnetization ${M}_{\infty}$; ${M}_{\infty}=0$ for the linear relaxation response. Thus, the equilibrium (static) magnetization ${M}_{0}$ derived from the Langevin function may not be equivalently treated as the initial magnetization ${M}_{i}$ at a near-zero frequency for inhomogeneous relaxation responses. The parametric ${M}_{1}^{\u2033}\left({M}_{1}^{\prime}\right)$ plot further defines ${M}_{\infty}$ and ${M}_{i}$ as the minimum and maximum values of the in-phase magnetization components ${M}_{1}^{\prime}$ at fundamental frequency, respectively. In the case of ${M}_{3}^{\u2033}\left({M}_{3}^{\prime}\right)$ plot strongly correlated with $3\Delta {\phi}_{1},$ the large ${M}_{\infty}$ of such non-Debye case lets ${M}_{3}^{\u2033}$ be negative for $\pi /3<\Delta {\phi}_{1}<\pi /2$ rad, while ${M}_{3}^{\prime}$ remains positive. For the negative ${M}_{3}^{\u2033}$ attributed to the dominating moment dynamics, one may employ it as an indicator of the viscosity-related tracer immobility for a cellular MPI, in addition to the positive ${M}_{1}^{\u2033}$ as a typical estimate for the local thermal dissipation.

#### 3.2.2. Field-Dependent Third-Harmonic Susceptibility

_{2}O

_{3}samples, Figure 4 shows that ${\chi}_{3}^{\prime}\left({H}_{0}\right)$ peaks at relatively the same field strength regardless of frequency and only has a slight magnitude difference between the liquid and solid samples. Such a parabolic property of ${\chi}_{3}^{\prime}\left({H}_{0}\right)$ seems to agree with the projection of the quasi-static ${\chi}_{3}\left({H}_{0}\right)\propto {H}_{0}^{2}/\left({c}_{0}+{c}_{1}{H}_{0}+{c}_{2}{H}_{0}^{2}+{c}_{3}{H}_{0}^{3}\right)$ derived from polynomial forms of the Langevin function for ${H}_{0}\ge 0$ [22], where ${c}_{0},$ ${c}_{1},$ ${c}_{2}$, and ${c}_{3}$ are constants. However, for the AOS-Fe

_{3}O

_{4}samples, ${\chi}_{3}^{\prime}\left({H}_{0}\right)$ is frequency-dependent and is maximized only for the liquid sample. More interestingly, their ${\chi}_{3}^{\u2033}\left({H}_{0}\right)$ are negatively inverted far above 100 Oe, and the phase-inverting field amplitude appears to have a frequency dependence, as shown in Figure 4 (inset).

_{3}O

_{4}nanoparticles have Néel relaxation time constant ${\tau}_{N}$ that should decrease with increasing field strength. For the time required to rotate a single magnetic moment ${m}_{p}$, Equation (3) defines ${\tau}_{N}$ by emphasizing the dependences on the field strength $H$, particle volume ${V}_{m}$, anisotropy constant ${k}_{U}$, and temperature $T$; ${k}_{B}$ is the Boltzmann constant [24]. The pre-exponential component ${\tau}_{0}$ is an intrinsic time constant augmented by a random field term of the thermal fluctuations, which can be defined by Equation (4) taking the Gilbert damping factor $\alpha $ and electron gyromagnetic ratio $\gamma $ into account [5]. Equation (3) is a general approximation of ${\tau}_{N}\left(H\right)$ formulated by Brown [25] in the case of low field values $H\le 0.8{k}_{U}{V}_{m}{m}_{p}^{-1}$. Treating the Brownian time constant ${\tau}_{B}$ similarly as a field-dependent parameter, the shorter effective time constant above 100 Oe might create $\Delta {\phi}_{1}>\pi /3$ rad to initiate the phase-inversion of the third-harmonic magnetization. In this situation, Brownian relaxation might be no longer dominant, even at frequencies below 10 kHz (Figure 4). We further believe that the negative ${\chi}_{3}^{\u2033}$ becomes a distinguishing parameter of such insignificant particle rotation. However, we still confirmed positive ${\chi}_{3}^{\u2033}$ components below 100 Oe in the case of the immobilized nanoclusters and addressed this issue to the damping behavior of the moment dynamics. Therefore, we emphasized the contribution of the $\alpha $ in correlation with the moment alignment within a single particle [5], instead of questioning the imperfect particle immobilization by the hydrocolloid polymer.

#### 3.3. Damping Effect on Moment Dynamics

_{2}O

_{3}and AOS-Fe

_{3}O

_{4}core particles contributes to their magnetization harmonics, Figure 4 highlights the maximum value of ${\chi}_{3}^{\prime}\left({H}_{0}\right)$ and, importantly, the sign inversion of ${\chi}_{3}^{\u2033}\left({H}_{0}\right)$ for the solid samples. We then investigated the magnetization decay rate of the immobilized particles by pulsating a 10-Oe magnetic field perpendicularly under a varying external dc bias field up to 450 Oe (see illustration in Figure 5); the pulse period, width, and time constant were 10, 10

^{−1}, and 10

^{−7}s, respectively. The $\alpha $ was qualitatively examined from the field-dependence of ${\tau}_{N}$. Experimentally, ${\tau}_{N}$ may not satisfy Equation (3) because of the non-uniform core size distribution and dipolar interparticle interactions. The authors of [24] give an empirical approximation of the effective Néel time-constant ${\tau}_{N}^{*}\left(H\right)$, resembling the field-dependent Brownian time constant [22], but it appears to be limited for low-field regimes. An important issue, however, remains at high-field regimes in which ${\tau}_{N}^{*}$ should be proportional to the effective ${\tau}_{0}^{*}$ due to gyromagnetic precession [26]. Therefore, we optionally provided an empirical equation of ${\tau}_{N}^{*}$ by Equation (5) to include the field-independent time constant ${\tau}_{0}^{*}$; ${a}_{0}$ and ${b}_{0}$ are arbitrary constants.

_{3}O

_{4}nanoparticles exhibit a stronger field-dependence (Figure 5). The larger initial ${\tau}_{N}^{*}$ of the AOS-Fe

_{3}O

_{4}nanoparticles significantly drops at $H=450$ Oe. We claim that this result was due to a stronger damping behavior of the respective moment dynamics. Fitting ${\tau}_{N}^{*}\left(H\right)$ by Equation (5), we obtained ${\tau}_{0}^{*}$ of 166 ns and 50.7 ns for the CMEAD-γFe

_{2}O

_{3}and AOS-Fe

_{3}O

_{4}nanoparticles, respectively. These values appear to overestimate those typically assumed (${\tau}_{0}<1$ ns) to consider the dipolar interactions. However, less measurement data for the extrapolation may be another possible cause of high ${\tau}_{0}^{*}$ values. As a qualitative analysis, nevertheless, we can conclude with Equation (4) that the CMEAD-γFe

_{2}O

_{3}nanoparticles should have a smaller $\alpha $ than the AOS-Fe

_{3}O

_{4}nanoparticles, noting that they have smaller ${m}_{p}$ proportional to the core diameter (Figure 1a) and $\alpha <1$ for both iron oxides [27,28] allows us to simplify $\left(1+{\alpha}^{2}\right)/2\alpha $ into $1/2\alpha $. For $\alpha \propto {m}_{p}/{\tau}_{0}^{*}$, the high damping effect on the AOS-Fe

_{3}O

_{4}nanoparticle moments, as well as that originated from the Brownian torques on their hydrodynamic nanostructures, collectively contribute to the apparent magnetization dynamics in the case of liquid sample. Therefore, it is important to control the superparamagnetic core size of magnetic nanoclusters as the parameter contributing to ${m}_{p},$ $\alpha $, and intrinsic dipolar interactions within the nanoclusters to achieve high complex harmonics for a functional phase-contrast MPI.

## 4. Conclusions

_{2}O

_{3}nanoclusters), the negative quadrature components of the third-harmonic susceptibility were due to the fast moment dynamics creating large phase differences of the magnetization against the applied field. In contrast, for larger core particles (e.g., AOS-Fe

_{3}O

_{4}nanoclusters), the positive quadrature components of the third-harmonic susceptibility were spectrally shifted to lower frequencies as compared with those of the fundamental susceptibility spectra. The sign inversion at higher frequencies and field amplitudes is to address the dominance of Neel relaxation over Brownian relaxation. The damping behavior of moment dynamics later becomes another important parameter to generalize the complex magnetization harmonics. We qualitatively concluded that a small damping factor is responsible for the frequency independence of the maximum in-phase components and the negative sign of quadrature components of the third-harmonic susceptibility. For such distinguishable complex components of magnetization harmonics, the magnetic nanoclusters are of the potential for MPI by suggesting an imaginary (phase) image in addition to spatially plotting the harmonic magnitudes.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Transmission electron microscopy images and dynamic light scattering measurements of the characterized ferrofluid samples; (

**b**) Phase-sensitive detection system to measure the complex magnetization harmonics of a 0.1-mL ferrofluid sample placed within an area with a spatially uniform field-distribution. The signal amplitude and phase differences at the fundamental and third-harmonic frequencies ($\Delta {R}_{1}$, $\Delta {\phi}_{1}$, $\Delta {R}_{3}$, and $\Delta {\phi}_{3}$, respectively) were recorded while varying field strength and frequency of the applied fields.

**Figure 2.**Dynamic magnetization curves of the ferrofluid samples at 1 kHz. The different coercive fields ${H}_{C}$ and remanences ${M}_{R}$ within low-field regimes (e.g., ±50, ±100, ±200, and ±300 Oe) indicate magnetization reversal. The measured magnetization $M$ has been normalized by the saturated magnetization ${M}_{S}$ of the respective samples.

**Figure 3.**(

**a**) Frequency-dependent complex magnetic susceptibilities at 50 Oe. The field-induced cluster rotation is recognized from the spectral shift of the imaginary peaks between the imaginary ${\chi}_{1}^{\u2033}$ and ${\chi}_{3}^{\u2033}$ parts (open circles), in addition the larger real ${\chi}_{1}^{\prime}$’ and ${\chi}_{3}^{\prime}$’ parts (solid circles) than those of the solidified samples (solid triangles). The imaginary parts for the solid samples (open triangles) are later attributed to the moment dynamics; (

**b**) Illustrative ${M}_{n}^{\u2033}\left({M}_{n}^{\prime}\right)$ plot of an empirical relaxation model with $a$ and $b$ fitting parameters [16]. The semicircle Cole-Cole model can be partially fitted to the ${M}_{1}^{\u2033}\left({M}_{1}^{\prime}\right)$ plots of the ferrofluid samples, whereas ${M}_{3}^{\u2033}\left({M}_{3}^{\prime}\right)$ appears to have unique patterns.

**Figure 4.**Field-dependent third-harmonic susceptibility at 1, 2, 5, and 10 kHz. The real ${\chi}_{3}^{\prime}$ and the imaginary ${\chi}_{3}^{\u2033}$ parts of the liquid samples (solid and open circles, respectively), as well as those of the solidified samples (solid and open triangles), distinguish the superparamagnetism of CMEAD-γFe

_{2}O

_{3}nanoclusters from the frequency independence.

**Figure 5.**Field-dependent effective Néel time constant ${\tau}_{N}^{*}\left(H\right)$ of the immobilized CMEAD-γFe

_{2}O

_{3}(solid squares) and AOS-Fe

_{3}O

_{4}(solid circles) nanoparticles representing the damping behavior of the particle moment ${m}_{p}$.

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**MDPI and ACS Style**

Trisnanto, S.B.; Takemura, Y.
Complex Magnetization Harmonics of Polydispersive Magnetic Nanoclusters. *Nanomaterials* **2018**, *8*, 424.
https://doi.org/10.3390/nano8060424

**AMA Style**

Trisnanto SB, Takemura Y.
Complex Magnetization Harmonics of Polydispersive Magnetic Nanoclusters. *Nanomaterials*. 2018; 8(6):424.
https://doi.org/10.3390/nano8060424

**Chicago/Turabian Style**

Trisnanto, Suko Bagus, and Yasushi Takemura.
2018. "Complex Magnetization Harmonics of Polydispersive Magnetic Nanoclusters" *Nanomaterials* 8, no. 6: 424.
https://doi.org/10.3390/nano8060424