1 Introduction

Magnetic nanoparticles (MNPs) have received significant interest recently, mainly owing to their size of being able to interact with their environment at the nanometer scale, and their high signal-to-noise ratio of detection using the magnetic method. MNPs have been utilized in the in vivo imaging as a contrast agent in magnetic resonance imaging (MRI) technique [1] and recently, as a tracer in an emerging tomographic imaging modality called magnetic particle imaging (MPI) technique [2,3,4,5,6,7]. Since their non-linear magnetic signal is one type of transparent to biological tissues, MPI shows a promising clinical application owing to its high temporal and spatial resolutions compared to other imaging techniques. In fact, MNPs’ intrinsic magnetic characteristics and their environmental interaction are commonly utilized in biomedical applications such as exploiting their heat generation at a high frequency in magnetic hyperthermia [8, 9] and magnetic relaxation in magnetic immunoassays [10]. Moreover, it was also reported that MNPs enable a sensitive viscosity determination of their suspension liquid [11]. On the whole, MNPs are preferable and have been employed in a wide range of applications through the exploitation of their dynamics magnetization which is strongly depended by the effects of the particle size, morphology, intrinsic magnetic properties, and environmental interaction (e.g., viscosity and hydrodynamic size). Therefore, clarification on these factors is essential so that MNPs can be tailored for intended applications [12].

The magnetic characterization of the MNPs can be achieved from the DC and AC susceptibility [13] methods apart from magnetic relaxation [14] and remanence methods. The AC susceptibility measurement can provide the information of size distribution, harmonics, and magnetic anisotropy with a fast response to the measurement by observing Neel and Brownian relaxations [15]. The DC susceptibility could also provide the size distribution of the MNP’s core [16, 17]. To evaluate and serve as one of the methods for the standardization of MNPs, we have reported specially designed DC and AC magnetometers [18, 19]. In this work, using the developed magnetometers, the investigation on the dynamics of a commercial ferromagnetic multi-core iron oxide nanoparticles at a different viscosity of carrier liquids is performed. Later, the distribution of magnetic moment is constructed based on the measured magnetization curve. The hydrodynamic size of the particles is determined from the AC susceptibility model proposed by Shliomis and Stepanov [20], which includes an inter-potential- and intra-potential-well contribution to the magnetic response. The measurements on the harmonic profile of MNPs at the different intensity of AC magnetic fields are presented. Also, the viscosity of carrier liquids that is estimated from the imaginary part of the dynamic magnetization, ratio, and lagging phase of the odd harmonics even at the different intensity of the AC excitation fields are also demonstrated.

2 Methodology

2.1 AC Magnetometer

The AC magnetometer used in this work has been specially developed for characterization of MNPs [19]. Briefly, the AC magnetometer is mainly composed of the excitation and detection units. In the excitation unit, an excitation coil is resonated by a capacitor network to reduce the impedance of the excitation coil at the high-frequency region. The capacitor network is fabricated to resonate the excitation coil at discrete frequencies on a logarithmic scale. The 300-turn excitation coil is fabricated from a Litz wire and designed to achieve a high homogeneity and efficiency of the excitation field [21]. In the detection unit, a first-order axial differential coil is used as the detection sensor. The first-order differential coil is composed of two identical 1000-turn coils, whose inner diameter and baseline are 15 mm and 23 mm, respectively. The attenuation of the direct feedthrough is achieved by manually tuning the position of the detection coil relative to the excitation coil. The signal from the detection coil is lock-in detected using a lock-in amplifier (LI5640, NF Corporation) for a phase-sensitive detection.

2.2 AC Susceptibility Model

Shliomis and Stepanov proposed that for an ensemble of MNPs with randomly distributed easy axes, the complex AC susceptibility in response to the applied AC magnetic field is given by [20, 22, 23]:

$$ \chi =\frac{1}{3}\left(\frac{\chi_{\parallel }}{1+ i\omega {\tau}_{\parallel }}+2{\chi}_{\perp}\right). $$
(1)

Here, ω = 2πf, χ∥, χ⊥, and τ∥ are the angular frequency, the susceptibilities for particles with easy axes parallel and perpendicular to the applied AC field, and the parallel component of relaxation time. The parallel and perpendicular components of susceptibilities are expressed by \( {\chi}_{\parallel }=\frac{\mu_0{M_s}^2{V}_c}{k_BT}\frac{R^{\prime }}{R} \) and \( {\chi}_{\perp }=\frac{\mu_0{M_s}^2{V}_c}{k_BT}\frac{R-{R}^{\prime }}{2R} \) with \( R\left(\sigma \right)=\underset{0}{\overset{1}{\int }}{e}^{\sigma {x}^2} dx \) and \( {R}^{\prime}\left(\sigma \right)=\underset{0}{\overset{1}{\int }}{x}^2{e}^{\sigma {x}^2} dx \). Here, μ0 is the vacuum permeability, Ms is the saturation magnetization of MNP, Vc is the core volume of MNP, and σ = KVc/kBT is the dimensionless ratio between the anisotropy energy KVc and thermal energy kBT. In the large barrier range where the magnetic moment is thermally blocked, σ ≫ 1, χ⊥ reaches the independent temperature value μ0Ms2/2K whereas χ∥ ≈ μ0Ms2Vc/kBT − μ0Ms2/K [22]. In this large barrier, Eq. (1) can be rewritten by,

$$ \chi ={\chi}_0\left(\frac{\sigma -1}{\sigma}\frac{1}{1+ i\omega {\tau}_{\parallel }}+\frac{1}{\sigma}\right), $$
(2)

where static susceptibility χ0 = μ0Ms2Vc/3kBT. It should be noted that the expression given in Eq. (2) is similar to equation in the reference [24] where the second term in the right-hand side stands for the intra-potential-well contribution, which is in contrast to the standard Debye model given by χ = χ0/(1 + iωτ).

Furthermore, τ∥−1 = τl−1 + τB−1 represents the effective relaxation time of longitudinal Neel and Brownian relaxation times, where they are given by \( {\tau}_l={\tau}_0\sqrt{\pi }{e}^{\sigma }/2{\sigma}^{3/2}\ \left(\sigma \ge 2\right) \) and τB = 3VHη/kBT, respectively. Here, τ0 is the intrinsic relaxation time related to intra-potential-well dynamics (typically of the order 10−9 s [24]), VH is the hydrodynamic volume of MNPs, and η is the viscosity of carrier liquid. In the case where τB ≪ τl, i.e., thermally blocked MNPs, and the hydrodynamic diameter DH described by a log-normal distribution \( f\left({D}_H\right)=\exp \left(-{\left(\ln \left({D}_H/{D}_0\right)\right)}^2/2{\delta_D}^2\right)/\sqrt{2\pi }{D}_H{\delta}_D \), the dynamic susceptibility of multi-core MNPs can be expressed by,

$$ \chi ={\chi}_0\left(\frac{\sigma -1}{\sigma}\underset{0}{\overset{\infty }{\int }}\frac{f\left({D}_H\right)d{D}_H}{1+ i\omega {\tau}_B}+\frac{1}{\sigma}\right), $$
(3)

where D0 and δD are the median and dimensionless parameter related to the standard deviation. The average diameter and standard deviation of DH are given by \( {D}_0\exp \left({\delta}_D^2/2\right) \) and \( {D}_0\exp \left({\delta}_D^2/2\right)\sqrt{\exp \left({\delta}_D^2-1\right)} \). In this work, we assume that σ and χ0 can be approximated by the average ratio of anisotropy and thermal energies 〈σ〉, and average static susceptibility 〈χ0〉 for multi-core particles [15]. Then, the real χ′ and imaginary parts χ″of the susceptibility can be expressed by,

$$ {\chi}^{\prime }=\left\langle {\chi}_0\right\rangle \left(\frac{\left\langle \sigma \right\rangle -1}{\left\langle \sigma \right\rangle}\underset{0}{\overset{\infty }{\int }}\frac{f\left({D}_H\right)d{D}_H}{1+{\left(\omega {\tau}_B\right)}^2}+\frac{1}{\left\langle \sigma \right\rangle}\right), $$
(4)
$$ {\chi}^{{\prime\prime} }=\left\langle {\chi}_0\right\rangle \left(\frac{\left\langle \sigma \right\rangle -1}{\left\langle \sigma \right\rangle}\underset{0}{\overset{\infty }{\int }}\frac{\omega {\tau}_Bf\left({D}_H\right)d{D}_H}{1+{\left(\omega {\tau}_B\right)}^2}\right). $$
(5)

It should be noted that Eqs. (4) and (5) are almost similar to equations derived in references [24, 25]; however, 〈σ〉 and 〈χ0〉 were used instead of individual values of particles and the high frequency susceptibility χ∞ in reference [25] could be estimated using the relation between the first and second terms of Eq. (4) even in a narrowband frequency measurement range.

2.3 Sample Preparation

The commercial multi-core iron oxide nanoparticle used in this study was nanomag®-D (Micromod Partikeltechnologie GmbH) with a typical diameter of 130 nm. The five sets of the different carrier liquid viscosity were prepared by varying the weight of glycerol mixed in a constant volume of purified water from 0 to 80 wt/V %. Then, 200 μl of stock suspension of the particles was diluted in 1 ml of the prepared carrier liquids. The dynamic magnetization with respect to excitation frequency was measured using the developed AC magnetometer [19] from 3 Hz to 10.48 kHz at a constant amplitude of 2.6 mT. The response characteristic at different amplitudes of the excitation field was measured at 30 Hz from 1.3 to 7.8 mT.

3 Result and Discussion

3.1 The Distribution of Core and Hydrodynamic Sizes

Fig. 1 a shows the initial magnetization curve measured by the high-Tc SQUID magnetometer from 0.1 to 240 mT. From the magnetization curve, the distribution of magnetic moment is constructed as shown in Fig. 1 b. Details on the distribution construction can be found in references [18, 26]. The highest peak corresponding to a core size of 11.7 nm, was in good agreement with the size determined from transmission electron microscopy image (see the superimposed image in Fig. 1a). The saturation magnetization of Ms = 295 kA/m was determined from the measured magnetization curve to calculate the core size and reasonably agreed to the value reported in reference [7] for a 12-nm iron oxide particle. Compared to the Ms of bulk magnetite (from 470 to 515 kA/m), the determined Ms was smaller and could be thought due to the existence of mixed phase of magnetite and maghemite. The existence of the second peak corresponded to a particle size of 6.1 nm was assumed to occur due to the substantial inter-core interaction effect that was not accounted in the static magnetization model.

Fig. 1
figure 1

a The static magnetization of the multi-core iron oxide nanoparticles. b The constructed distribution of magnetic moment

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Figure 2 a depicts the dynamic magnetizations of the iron oxide nanoparticles in the carrier liquids having different wt/V % of glycerol. The peaks of the imaginary part of the dynamic magnetization reflected the Brownian relaxation process of the thermally blocked iron oxide nanoparticle where their positions shifted to a lower frequency as the wt/V % of the glycerol increased in the carrier liquids. The thermally blocked particle could be resulted from the presence of the hysteresis in the magnetization [18] where it was also reported in [7] for a 12-nm iron oxide particle. The response showed that the viscosity of the carrier liquid was increased due to the Brownian relaxation mechanism. The peak position of the imaginary part around 400 Hz agreed with the result reported in reference [27]. Using 0.89 mPa s as the viscosity of the purified water (0 wt/V % solution) at 25 °C, the distribution of DH was determined based on the nonlinear curve fitting of Eq. (4) in the least-squares sense. The solid line shows the constructed hydrodynamic size distribution in Fig. 2 b, where an average diameter of 130 nm, the standard deviation of 76 nm, and 〈σ〉 of 3.1 were obtained. The comparison between hydrodynamic diameter obtained from the dynamic magnetization and intensity weighted distribution measured using the dynamic light scattering (DLS) method (Zetasizer Nano S90, Malvern Instruments, Worcestershire, UK) is shown as the solid and dashed lines as indicated in Fig. 2 b. The DLS measurement had resulted in a Z-average of 128.7 nm; however, it should be noted that the intensity weighted distribution may overestimate the size due to domination of a few large particles, i.e., the intensity is proportional to DH6 [15].

Fig. 2
figure 2

a The real and imaginary parts of dynamic magnetization in different viscosity of carrier liquids. b The constructed distributions of hydrodynamic diameter from AC susceptibility and DLS methods

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3.2 The Effect of Viscosity on the Dynamics of Particles

The dashed lines in Fig. 2 a show the real and imaginary parts derived from the hydrodynamic diameter distribution. The response indicates a good agreement with the measured data. Since the viscosity of the carrier liquid reflects the peak position of the imaginary part, we estimated the viscosity of the 20 to 80 wt/V % solutions by fitting their imaginary parts with Eq. (5). Here, the previously determined hydrodynamic diameter distribution of the 0 wt/V % solution was used in the nonlinear curve fitting. The dashed lines in Fig. 2 a show the derived for both real and imaginary parts with their corresponding viscosities are plotted in Fig. 3.

Fig. 3
figure 3

Estimated viscosity of the carrier liquids with respect to wt/V % of glycerol in the carrier liquids

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The derivation results obtained from imaginary parts are in good agreement with the measured data while a slight deviation of the real parts was observed, particularly for the 80 wt/V % solution. This deviation was thought due to the existence of the σ distribution and further clarification on the effect should be investigated.

In practice, measurements that requiring sweeping of a wide range of frequency region result to increment of measuring time. The results indicated that for the change of the real and imaginary parts, a single-frequency measurement could be used to obtained the concentration of MNPs and environmental factors that affected the dynamic magnetization of MNP such as the viscosity of the carrier liquid [28]. The selection of the measurement frequency is crucial, depending on the types of relaxation mechanism. In this work, we selected a frequency of 30 Hz since a lower frequency can capture the information of the Brownian relaxation process occurred at a higher frequency owing to the distribution of the particle size. Figure 4 shows the intensity profile of the first, third, and fifth harmonics of the iron oxide nanoparticle with respect to the viscosity at different intensities of the AC excitation field. Generally, the intensity of the harmonics decreased with the increase of viscosity, which was consistent with the results indicated in Fig. 2 a. Moreover, the intensity of the harmonics increased with the magnitude of the excitation field. However, it should be noted that the increment in the harmonic intensity profile is correlated with the increase in the MNPs concentration, i.e., a slight difference in concentrations of MNPs may result to a negligible difference in harmonic intensity, resulting to an estimation error of the viscosity. This can be shown by the first harmonic during the excitation field of 7.8 mT. On the other hand, the phase angle of the harmonics reflects the phase change of the Brownian relaxation process. In this case, Fig. 5 shows the phase angle change of the harmonics referenced at the phase angle of 0 wt/V % solution (viscosity of 0.89 mPa s). The phase delay of the harmonics is increased with the increase of the viscosity, which was consistent with the increasing magnitude of the imaginary part in Fig. 2 a.

Fig. 4
figure 4

a The intensity of the first f1, b third f3, and c fifth f5 harmonics concerning viscosity and excitation field

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Fig. 5
figure 5

a The phase angle of the first f1, b third f3, and c fifth f5 harmonics concerning viscosity and excitation field

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However, the phase delay was reduced at a higher magnitude of the excitation field. The phenomenon can be interpreted as the higher intensity of magnetic field has increased the magnetic torque of the thermally blocked particles where it reduces the relaxation time, thus resulting in the reduced phase delay. At this point, the viscosity of liquid carrier could be determined quantitatively based on the MNP solution phase angle profile using a known distribution of the hydrodynamic diameter. However, it must be noted that the phase delay is sensitive to the noise when the harmonics intensity is relatively small and comparable to the noise. The condition is shown by the phase angle of the fifth harmonic at the excitation field of 1.3 mTpp in Fig. 5 c. It is known that the intensity of the harmonics is proportional to the concentration of MNPs. By taking a ratio between harmonics, the effect of concentration can be normalized, thus providing a rather simple determination of viscosity using harmonics ratio compared to determination of phase delay. Figure 6 shows the third-to-first, fifth-to-first, and fifth-to-third ratios as a function of the viscosity. The harmonics ratio decreased with the increase of the viscosity. Since high magnetic field will produce a higher harmonics intensity, the harmonics ratio will also increase when the intensity of the excitation field is increased. However, it shall be noted that the phase delay and harmonic ratio were not linear and depended on the strength of the AC magnetic field. Therefore, it can be proved that harmonics ratio can be used as a concentration-independent parameter in the determination of the MNP’s dynamics.

Fig. 6
figure 6

a The harmonics ratio of the f3/f1, bf5/f1, and cf5/f3 with respect to viscosity and excitation field

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4 Conclusion

The static and dynamic magnetizations of the multi-core iron oxide nanoparticles have been characterized by the developed systems. The core size of 11.7 nm and its hydrodynamic diameter profile with an average diameter of 130 nm derived from their static and dynamic magnetizations were consistent with the results achieved by TEM analysis and DLS measurement. It is indicated that the environmental factors such as viscosity could affect the dynamic behavior of MNPs where the peaks of the imaginary components shifted to a lower frequency region when the solution viscosity was increased from 0.89 to 8.11 mPa s. Also, it was shown that the harmonic ratio decreased and the phase angle delay increased when the viscosity was increased. The harmonics ratio and phase angle delay could be used as the concentration-independent parameter to estimate the viscosity of the liquid carrier where it can be implemented in certain applications such as in the magnetic immunoassay and magnetic particle imaging (MPI) technique.