Chemical Manipulation of the Collective Superspin Dynamics in Heat-Generating Superparamagnetic Fluids: An AC-Susceptibility Study

Abstract

We use Co doping to alter the magnetic relaxation dynamics in superparamagnetic nanofluids made of 18 nm average diameter Fe₃O₄ nanoparticles immersed in Isopar M. Ac-susceptibility data recorded at different frequencies and temperatures, χ″vs. T|_f, reveals a major (~100 K) increase in the superspin blocking temperature of the Co_0.2Fe_2.8O₄-based fluid (CFO) compared to its Fe₃O₄ counterpart (FO). We ascribe this behavior to the strengthening of the interparticle magnetic dipole interactions upon Co doping, as demonstrated by the relative χ″-peak temperature variation per frequency decade $\mathsf{\Phi }={\displaystyle \frac{∆\mathrm{T}}{\mathrm{T}·∆\mathrm{l}\mathrm{o}\mathrm{g}\left(\mathrm{f}\right)}}$, which decreases from Φ~0.15 in FO to Φ~0.025 in CFO. In addition, χ″vs. T|_f datasets from the CFO fluid reveal two magnetic events at temperatures T_p1 = 240 K and T_p2 = 275 K, both above the fluid’s freezing point (T_F = 197 K). We demonstrate that the physical rotation of the nanoparticles within the fluid, the Brown mechanism, is entirely responsible for the collective superspin relaxation observed at T_p1, whereas the Néel mechanism, the superspin flip across an energy barrier within the particle, is dominant at T_p2. We confirm this finding through fits of models that describe the temperature dependence of the relaxation time via the two mechanisms: ${\mathsf{\tau }}_{\mathrm{B}}(\mathrm{T})={\displaystyle \frac{3{\mathsf{\eta }}_0{\mathrm{V}}_{\mathrm{H}}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}}\exp \left[{\displaystyle \frac{{\mathrm{E}}^{\prime }}{{\mathrm{k}}_{\mathrm{B}}\left(\mathrm{T}-{{\mathrm{T}}_0}^{\prime }\right)}}\right]$ and ${\mathsf{\tau }}_{\mathrm{N}}\left(\mathrm{T}\right)={\mathsf{\tau }}_0\exp \left[{\displaystyle \frac{{\mathrm{E}}_{\mathrm{B}}}{{\mathrm{k}}_{\mathrm{B}}\left(\mathrm{T}-{\mathrm{T}}_0\right)}}\right]$. The best fits yield ${\mathsf{\gamma }}_0={\displaystyle \frac{3{\mathsf{\eta }}_0{\mathrm{V}}_{\mathrm{H}}}{{\mathrm{k}}_{\mathrm{B}}}}$= 1.5 × 10⁻⁸ s·K, E′/k_B = 7 03 K, and T₀′ = 201 K for the Brown relaxation, and E_B/k_B = 2818 K and T₀ = 143 K for the Néel relaxation.

1. Introduction

Magnetic nanoparticle (MNP) ensembles are known to have a particularly broad range of applications in several important fields ranging from magnetic recording [1,2,3] to energy storage [4] and environmental remediation [5,6,7]. In addition, their role in biomedicine has grown exponentially during the past two decades [8,9,10,11,12]. This was mostly driven by rapid progress in identifying new ways to make use of MNPs’ ability to respond to relatively weak magnetic fields and have their surface functionalized [13]. In particular, magnetic nanoparticle hyperthermia—a method by which tissue heating can be achieved by subjecting MNP ensembles to an alternating magnetic field—has received considerable attention due to its potential to be used in cancer therapy [14,15,16,17,18]. The idea is not new [19], and there are several technical issues such as the ability to drive the nanoparticles to the tumor site and achieve uniform heating. This is important, as malignant cells are known to die at temperatures just a few degrees below the threshold above which normal tissue dies as well, so heating the tumor to temperatures within this window is necessary to selectively kill cancer. Yet, the main barrier to progress towards the use of magnetic hyperthermia cancer therapy as a stand-alone procedure stems from the inability of commonly used materials (e.g., Fe₃O₄) to generate enough heat within the limits on the amplitude of the driving magnetic field (H_m) and its frequency (f), whose product H_m·f needs to be less than 10⁶ Oe·Hz in order for the procedure to be usable in humans.

At the temperatures where MNPs are used in magnetic hyperthermia cancer therapy applications, the nanoparticle ensembles are almost always in their superparamagnetic state as these temperatures are above the blocking temperature of the ensemble, T_B [20]. This is important because heat generation in superparamagnetic MNPs is not hysteretic as in their ferro- or ferrimagnetic counterparts. Instead, the ability of these systems to generate heat, measured by the specific absorption rate (SAR), depends on the collective dynamics of the giant magnetic moments (the superspins) of the nanoparticles [21].

The superspin dynamics in an ensemble of MNPs can occur via two mechanisms. First, the superspin flips over an energy barrier to magnetization reversal, E_B, via thermal activation. This Néel relaxation mechanism is driven by thermal activation and, for a single nanoparticle, its relaxation time (i.e., the time it takes its magnetic moment to complete one full flip), τ_N, is given by

$${\mathsf{\tau }}_{\mathrm{N}}(\mathrm{T})={\mathsf{\tau }}_0\exp \left({\displaystyle \frac{{\mathrm{E}}_{\mathrm{B}}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}}\right)$$

(1)

where τ₀ is the characteristic time (typically of the order of 10⁻¹⁰ to 10⁻¹² s), E_B is the energy barrier to magnetization reversal, and k_B is the Boltzmann constant. E_B depends on the MNP’s material through the anisotropy constant, K, and the nanoparticle’s volume, V, according to E_B = KV. For an ensemble of MNPs that have an average volume of <V> and interact with one another via magnetic dipole forces, an empirical model based on the Vogel–Fulcher law was proposed to describe the Néel relaxation [22]:

$${\mathsf{\tau }}_{\mathrm{N}}\left(\mathrm{T}\right)={\mathsf{\tau }}_0\exp \left[{\displaystyle \frac{{\mathrm{E}}_{\mathrm{B}}}{{\mathrm{k}}_{\mathrm{B}}\left(\mathrm{T}-{\mathrm{T}}_0\right)}}\right]$$

(2)

Here E_B = K <V>, and T₀ is a parameter that accounts for the strength of the interparticle interactions in the ensembles. When the MNP ensemble is immersed in a carrier fluid to form a ferrofluid, a second type of superspin dynamics characterized by the rotation of the whole nanoparticle within the fluid is activated. The relaxation time of this Brown relaxation mechanism is given by

$${\mathsf{\tau }}_{\mathrm{B}}\left(\mathrm{T}\right)={\displaystyle \frac{3\mathsf{\eta }\left(\mathrm{T}\right){\mathrm{V}}_{\mathrm{H}}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}}={\displaystyle \frac{3{\mathsf{\eta }}_0{\mathrm{V}}_{\mathrm{H}}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}}\exp \left[{\displaystyle \frac{{\mathrm{E}}^{\prime }}{{\mathrm{k}}_{\mathrm{B}}\left(\mathrm{T}-{{\mathrm{T}}_0}^{\prime }\right)}}\right]$$

(3)

Here, V_H is the average hydrodynamic volume of the nanoparticles, and η(T) = η₀exp[E′/[k_B(T − T₀′)] is the temperature-dependent fluid viscosity [23], where η₀ is a constant that depends on the material, E′ is the activation energy, and T₀′ is the divergence temperature of the viscosity.

In any ferrofluid, both Néel and Brown relaxation are simultaneously active at any temperature, and their temperature-dependent interplay yields a so-called effective relaxation time, τ_eff, that includes contributions from both mechanisms. Understanding these collective superspin dynamics at the microscopic level is important, as a ferrofluid’s magnetic loss power density (its ability to generate heat under the action of an alternating magnetic field) directly depends on the above-described effective relaxation. A model developed by Rosensweig, where the effective superspin relaxation time is given by ${\mathsf{\tau }}_{eff}\left(\mathrm{T}\right)={\displaystyle \frac{{\tau }_N\left(\mathrm{T}\right)·{\tau }_B\left(\mathrm{T}\right)}{{\tau }_N\left(\mathrm{T}\right)+{\tau }_B\left(\mathrm{T}\right)}}$ [24], has been extensively used to analyze the complex magnetization dynamics of MNP ensembles immersed in carrier fluids.

The ability to alter in a controlled way the interplay between Néel and Brown relaxation is critical for the design of ferrofluids with enhanced heat generation properties, and much research effort has recently been invested towards this end [25,26]. Besides measuring and analyzing the magnetic response of ferrofluids throughout a broad range of temperatures and frequencies of the driving field, many of these studies focused on varying the meso-structural characteristics of the MNP ensemble (e.g., its average nanoparticle volume and size distribution) and the interparticle magnetic interaction strength by progressively diluting the ferrofluid to increase the interparticle distances [27]. Most of these studies used biocompatible Fe₃O₄ nanoparticles immersed in hydrocarbon-based fluids. Significant progress has been made, including observations of multiple magnetic events (as ac-susceptibility peaks) upon heating, which were attributed to a possible decoupling between the Brown and Néel mechanisms in dense systems [25,26].

Here we used chemical modification in an attempt at altering the magnetization dynamics in ferrofluids. Specifically, we doped cobalt into Fe₃O₄/Isopar M (FO) to obtain Co_0.2Fe_2.8O₄/Isopar M (CFO) ferrofluids, and we used temperature-resolved ac-susceptibility data to analyze the superspin relaxation in the two samples, both qualitatively and quantitatively. This approach has at least two advantages over the ones described in the paragraph above. First, it is known that chemical manipulation via doping offers more flexibility, sensitivity, and a substantially broader range of options on how to affect the magnetic behavior of MNPs in general [28]. In our case, for example, replacing just 6.67% of the Fe²⁺ ions with Co²⁺ leads to dramatic changes in the MNP ensemble’s dynamic behavior under the action of an oscillating magnetic field. Second, Co doping increases the blocking temperature of the ensemble by more than 100 K (for the same average volume), thus allowing the Néel relaxation component to be analyzed from powders (obtained by evaporating the carrier fluid) at temperatures close to the ones where the contribution of this mechanism to the effective relaxation is evaluated.

Our out-of-phase magnetic susceptibility data was collected at different temperatures and frequencies on FO and CFO (both fluids and powders). These χ″vs. T|_f datasets exhibit a peak at temperatures where the MNP ensemble’s relaxation time is the same as the observation time, which for this type of measurement is fully determined by the frequency as τ_obs = 1/2πf. This allowed us to calculate, for all samples, the relative χ″-peak temperature variation per frequency decade $\mathsf{\Phi }={\displaystyle \frac{∆\mathrm{T}}{\mathrm{T}·∆\mathrm{l}\mathrm{o}\mathrm{g}\left(\mathrm{f}\right)}}$, a known measure of the interparticle dipole interaction strength in an MNP ensemble [29]. Our first finding is that Φ decreases by nearly one order of magnitude, from Φ~0.15 in FO to Φ~0.025 in CFO, revealing a significant strengthening of the interactions among nanoparticles upon Co doping. In turn, the blocking temperature of the ensemble increases in CFO above the melting point of the carrier fluid, T_F = 197 K, so all the magnetic events in CFO fluids and powders occur above T_F. Our main result comes from the analysis of the two peaks observed in the χ″vs. T|_f datasets from the CFO fluid around T_p1 = 240 K and T_p2 = 275 K. Using Rosensweig’s formula and the temperature dependence of Néel relaxation τ_N(T) obtained from the CFO powder, we managed to unambiguously identify the microscopic relaxation mechanisms responsible for the superspin dynamics observed in the ferrofluid. We found that the Brown mechanism is entirely responsible for the collective superspin relaxation observed at T_p1, whereas the Néel mechanism is dominant at T_p2. Finally, using fits of Equations (2) and (3), we determined the values of the parameters that characterize the Brown relaxation at T_p1, i.e., ${\displaystyle \frac{3{\eta }_0{V}_H}{k_B}}$= 1.5 × 10⁻⁸ s·K, E′/k_B = 703 K, and T₀′ = 201 K, and the ones that characterize the Neel relaxation at T_p2 , i.e., E_B/k_B = 2818 K and T₀ = 143 K. Our results are important because the ability to control the superspin dynamics in ferrofluids via chemical manipulation has the potential to enhance biomedical applications such as magnetic nanoparticle hyperthermia therapy.

2. Materials and Methods

Both the Fe₃O₄ and Co_0.2Fe_2.8O₄ magnetic nanoparticle ensembles used in this study were made by the common co-precipitation method [30]. Briefly, to synthesize Fe₃O₄ nanoparticles, FeCl₃·6H₂O and FeCl₂·4H₂O (2:1 molar ratio) were dissolved in deionized water, and NaOH was subsequently added to promote precipitation. To synthesize Co_0.2Fe_2.8O₄ nanoparticles, the process was the same, with the only exception that a 0.2:2.8 molar ratio mixture of CoCl₂ and FeCl₂ was used (instead of FeCl₂) to achieve the desired Co doping level. To make the Fe₃O₄/Isopar M and the Co_0.2Fe_2.8O₄/Isopar M ferrofluids, the nanoparticles were coated with oleic acid to avoid agglomeration and then redispersed in the carrier fluid. In both cases, the as-prepared ferrofluids had a solid mass/fluid volume concentration of 1 mg/mL.

Transmission Electron Microscopy (TEM) images were collected to assess the quality (monodispersity) of the MNP ensembles and to determine their average size. equipped with a goniometer-tilt stage (Hitachi, Schaumburg, IL, USA) and a CCD camera was used. The operating voltage was 300 kV.

Powder X-ray diffraction (XRD) measurements were performed using a Bruker D8 Quest diffractometer equipped with a Mo source (λ = 0.71 Å) and area detector (Bruker Scientific LLC, Billerica, MA, USA). The transmission geometry was used, and the I vs. 2θ powder diffraction patterns were obtained by integrating over the circular projections of the Debye–Scherrer cones on the plane of the detector.

Ac-magnetic-susceptibility data was collected using a Quantum Design^® Physical Property Measurement System (PPMS) (Quantum Design, San Diego, CA, USA) generating an alternating magnetic field of amplitude H_m = 3Oe and frequencies ranging from 10 Hz to 10,000 Hz. For all measurements, about 1 mL of ferrofluid was sealed in a polycarbonate capsule that was subsequently immobilized within a plastic straw and attached to the end of the instrument’s sample holder rod. For both the CF and the CFO ferrofluid, the out-of-phase (imaginary) susceptibility, χ″, was recorded at different temperatures upon heating from 5 K to 310 K in 5 K steps. At each temperature, χ″ was measured at six frequencies of the driving magnetic field: 10 Hz, 100 Hz, 500 Hz, 1000 Hz, 5000 Hz, and 10,000 Hz. The resulting χ″vs. T|_f datasets were analyzed using fits of the Vogel–Fulcher law (Equation (2)) and a hydrodynamic model (Equation (3)) to describe Néel and Brown superspin relaxation, respectively.

3. Results and Discussion

Figure 1a shows a TEM image of the FO nanopowder obtained from the corresponding ferrofluid. The image demonstrates the quality of the nanoparticle ensemble in terms of its particle shape and size distribution. Moreover, it is important to mention that the same MNP ensemble was present in the ferrofluid and in the powder for both FO and CFO (as the powders were obtained by the evaporation of the Isopar M fluid). Figure 1b presents a higher-magnification image of the same MNP ensemble showing the spherical shape of the nanoparticles and an average diameter of <D> ≅ 18 nm. The crystallinity and purity of the nanoparticles in the ensemble were also checked by X-ray diffraction (XRD). This is important, as studies have shown that even for basic iron oxide nanoparticle ensembles synthesized via well-established techniques, impurities like FeO or α-Fe can be present [31]. This might be due to inherent surface effects that have been observed to induce deviations from the ideal 2:1 Fe³⁺/Fe²⁺ ratio even when stoichiometric amounts of reactants are used [32]. Figure 1c shows the powder XRD pattern from the FO nanoparticle ensemble. All observed peaks correspond to the Bragg reflections from the cubic crystal structure of Fe₃O₄ (space group Fd3m, lattice constant a = 8.36 Å), the angular 2θ positions of which are marked by the vertical bars. The quality of the ensemble is demonstrated by the lack of measurable impurity peaks. For example, the strongest reflection from FeO would be at 2θ = 18.95 deg., but the number of counts detected at this angular position is within the background level (i.e., 0.6 % of the intensity of the strongest Fe₃O₄ peak observed). Finally, Figure 1d presents the size distribution of the nanoparticles in the ensemble and the resulting average diameter. The data reveals a sharp size distribution, which further confirms the morphological quality of the sample.

The temperature-resolved ac-susceptibility data from the undoped FO ferrofluid is shown in Figure 2a. The six χ″vs. T|_f datasets were collected at different frequencies of the driving magnetic field: 10 Hz (purple), 100 Hz (orange), 500 Hz (green), 1000 Hz (red), 5000 Hz (blue), and 10,000 Hz (black).

The most notable feature of these datasets is that they each exhibit a well-defined peak, and the peak temperature T_p increases with an increase in the frequency, f, or equivalently, the observation time τ_obs = 1/2πf. As the MNP ensemble’s superspin relaxation time, τ, is equal to the observation time at T_p, this type of measurement allows us to determine the temperature dependence of the superspin relaxation time, τ(T). In this particular case, all T_p values are below the freezing point of Isopar M, so the only active mechanism is Néel relaxation as the nanoparticles are immobilized in the frozen carrier fluid. Another result is the value of the relative χ″-peak temperature variation per frequency decade, $\mathsf{\Phi }={\displaystyle \frac{∆\mathrm{T}}{\mathrm{T}·∆\mathrm{l}\mathrm{o}\mathrm{g}\left(\mathrm{f}\right)}}$. For the undoped FO ferrofluid, we found Φ = 0.15, which indicates the presence of medium-strength interactions. Figure 2b shows the results of similar ac-susceptibility measurements, this time carried out on FO powders obtained by evaporating the Isopar M from the FO ferrofluid. This led to a weaker χ″ signal (by nearly one order of magnitude), so only the highest four frequencies were used to collect χ″vs. T|_f datasets: 500 Hz (green), 1000 Hz (red), 5000 Hz (blue), and 10,000 Hz (black). The peak temperature shift with the measurement frequency yields the temperature variation of the Néel relaxation time, τ_N(T), and the peak temperature variation per frequency decade is Φ = 0.12, less than its FO ferrofluid counterpart. The analysis of the ac-susceptibility data from the FO ferrofluid and corresponding FO powder is shown in Figure 2c. The open symbols represent the measured dependence of ln (τ_N) on the inverse temperature 1/T for the FO powder, and the filled symbols show the corresponding ln (τ_N) vs. 1/T data from the FO ferrofluid. In both cases, the dashed lines are best fits of Equation (2) to the data, where two parameters—the reduced energy barrier to magnetization reversal, E_B/k_B, and the interparticle interaction strength parameter, T₀—were allowed to vary independently. The best-fit results indicate an increase in both T₀ and E_B/k_B in the powder sample compared to the ferrofluid, from 28 K to 39 K and from 1645 K to 1904 K, respectively. Clearly, this is a result of the increase in the interparticle distances in the ferrofluid, which weakens the dipolar interactions, a fact also confirmed by the observed values of Φ. Moreover, this analysis allows us to quantify the overall change in the Néel relaxation time from the powder to the 1 mg/mL ferrofluid. Within the 155–185 K temperature range, our data shows that τ_N decreases by 3.5 times on average in the ferrofluid compared to the powder. This observation is important for the analysis of the doped Co_0.2Fe_2.8O/Isopar M fluid (CFO) presented below. It is also important to mention that Co-doped Fe₃O₄ magnetic nanoparticles are biocompatible [33].

Figure 3a shows the temperature-dependent ac-susceptibility data collected on the Co_0.2Fe_2.8O₄/Isopar M (CFO) ferrofluid. The effects of Co doping are evident. First, at all measurement frequencies, there are two magnetic events in the form of χ″ peaks at temperatures T_p1 and T_p2, as clearly illustrated in Figure 3b. Second, both peaks exhibit a temperature shift with frequency similar to the one observed in the FO samples, demonstrating the dynamic nature of the microscopic phenomena that underlie the χ″ behavior observed at T_p1 and T_p2. In addition, both peaks are above the freezing point of the carrier fluid, where the Néel and Brown superspin relaxation mechanisms are active. Previous studies did reveal a double peak in χ″vs. T data from nanofluids, but in those cases the lowest temperature peak was below the carrier fluid’s freezing point, which completely inhibited the physical rotation of the MNPs [25].

To uncover the microscopic origin of the observed magnetic behavior of the CFO ferrofluid, the first step was to analyze the characteristics of the χ″-peak temperature variation with the frequency of the applied magnetic field. As shown in Figure 4, we carried out this analysis for both T_p1 and T_p2. The left inset in Figure 4 presents the T_p1 values observed at the six measurement frequencies used. This data leads to two findings. First, it yields a value of the peak temperature variation per frequency decade of Φ = 0.025. This is one order of magnitude less than the value of Φ in the FO nanofluid, confirming the significant strengthening of interparticle interactions induced by Co doping in the CFO sample.

We also used the observed frequency dependence of T_p1 to determine how the superspin relaxation time changes upon heating within the temperature interval corresponding to the lower temperature peak. This was enabled by the fact that, at each peak temperature, τ = τ_obs = 1/2πf (as explained in more detail in the Section 1). The open symbols in Figure 4 (line a) show the resulting ln (τ) vs. 1/T dependence. It is important to mention that τ is, in principle, an effective relaxation time with contributions from both the Néel and Brown mechanisms. Finally, we found that ln (τ) depends linearly on the inverse temperature, as demonstrated by the fit shown by the red dashed line. Figure 4 (line b) presents a similar analysis of the data corresponding to the high-temperature χ″-peak, T_p2. The right inset presents the T_p2 values observed at the six measurement frequencies used. The same type of ln (τ) vs. 1/T is observed (solid symbols) but this time in the context of somewhat weaker interparticle interactions, as indicated by Φ = 0.05.

Our strategy to identify the individual contributions of the Néel and Brown mechanisms to the overall, effective superspin dynamics observed at T_p1 and T_p2 in the CFO fluid was to isolate the Néel relaxation by blocking the physical rotation of the MNPs. We accomplished that by evaporating the Isopar M from the CFO ferrofluid, which resulted in a CFO powder that contained an identical nanoparticle ensemble to the one in the nanofluid. Figure 5a presents the χ″vs. T|_f datasets recorded on the CFO powders at five frequencies, 100 Hz (orange), 500 Hz (green), 1000 Hz (red), 5000 Hz (blue), and 10,000 Hz (black). The data exhibits just one peak at temperatures that shift with the frequency within the 265–295 K temperature range, as shown in the inset. The resulting ln (τ_N) vs. 1/T dependence in the CFO powder is represented by the open symbols in Figure 5b. In addition, the filled symbols here show the variation with the inverse temperature of ln (τ_eff) obtained from the T_p2 peaks observed in the CFO fluid. In both cases, the dashed lines are a guide for the eye. These results are remarkably similar to the ones from the FO fluid and powder shown in Figure 2c. The main difference is that, for FO, Néel relaxation was the only active mechanism (as the fluid was frozen), which allowed us to determine the factor by which Φ increased in the FO 1 mg/mL ferrofluid compared to the corresponding powder. As the concentration of the Co-doped ferrofluid, CFO, was also 1 mg/mL, we used the same analysis to calculate the τ_N contribution to the effective relaxation time at all temperatures corresponding to the T_p2 peaks.

Figure 6 shows the contributions of the Néel and Brown mechanisms to the overall effective superspin relaxation time of the MNP ensemble in the CFP ferrofluid at the six temperatures where the T_p2 peaks occur. Each temperature corresponds to one of the six measurement frequencies used: 10 Hz, 100 Hz, 500 Hz, 1000 Hz, 5000 Hz, and 10000 Hz. At each temperature, the superspin relaxation times τ_eff, τ_N, and τ_B were determined as follows. The overall time was calculated from the measurement frequency according to τ_eff = τ_obs = 1/2πf, and the Néel time was determined from the CFO powder data according to the procedure described above. Once the effective and Néel times were known, τ_B was calculated based on the Rosensweig formula according to 1/τ_B = 1/τ_eff − 1/τ_N. The results clearly indicate that the Néel mechanism is dominant at all T_p2 temperatures. Consequently, we used Equation (2) to quantitatively characterize the magnetization dynamics within this temperature range.

The open symbols in Figure 7 represent the temperature dependence of ln(τ_N) based on the analysis presented in Figure 6. The horizontal error bars are related to the precision of the T_p2 peak temperature measurements. The dashed line is the best fit of the Vogel–Fulcher activation law in Equation (2) to the ln(τ_N) vs. T data. Allowed to vary in the fit were two independent parameters: the reduced energy barrier to superspin reversal, E_B/k_B, and the interparticle interaction strength parameter, T₀. The refinement converged to low residuals (reduced χ² = 0.019), and the best fit yielded E_B/k_B = 2818 K and T₀ = 143 K.

Finally, we further analyzed the Néel and Brown contributions to the relaxation time corresponding to the low-temperature χ″-peak, T_p1. As in the case of its T_p2 counterpart, τ_eff can be directly determined at each of the T_p1 peak temperatures from the corresponding frequencies of the applied magnetic field as τ_eff = τ_obs = 1/2πf. To determine τ_N at these temperatures, however, we cannot directly use the data from the CFO powder (as that falls within a higher temperature range, around T_p2). We therefore extrapolated the CFO powder’s linear ln(τ_N) vs. 1/T dependence to the 235–255 K temperature range, where the T_p1 peaks occur. At T = 250 K, for example, τ_eff = 1.59 × 10⁻⁴ s and τ_N = 2.5 × 10⁻² s, i.e., 1/τ_eff>>1/τ_N. Using Rosensweig’s equation we found τ_B = 1.61 × 10⁻⁴ s, which indicates that the Brown mechanism is almost entirely (99%) responsible for the collective superspin relaxation in the CFO fluid at this temperature. We found similar results at all the other T_p1 temperatures.

Based on these results, we used a hydrodynamic model of Brown relaxation (Equation (3)) to quantitatively characterize the magnetization dynamics within the 235–255 K temperature range. Figure 8 shows the temperature dependence of the observed overall relaxation time, and the horizontal error bars are related to the precision with which the χ″-peak temperatures were determined at the six measurement frequencies used. The dashed line is the best fit of a hydrodynamic model based on the temperature dependence of the nanofluid’s viscosity. The model only accounts for Brown relaxation and has three independent parameters: the prefactor, γ₀ = 3η₀V_H/k_B; the reduced activation energy for the viscosity variation with the temperature, E′/k_B; and the divergence temperature of the viscosity, T₀′. The best fit converged upon the simultaneous variation of these parameters, confirming the Brown relaxation origin of the superspin dynamics event observed at T_p1. In addition, the best fit yielded ${\gamma }_0={\displaystyle \frac{3{\eta }_0{V}_H}{k_B}}$=1.5 × 10⁻⁸ s·K, E′/k_B = 703 K, and T₀′ = 2 01 K.

Overall, our study shows that Co doping of Fe₃O₄ magnetic nanoparticles immersed in a carrier fluid strongly affects the microscopic dynamics that underlie superspin relaxation via the Néel and Brown mechanisms. Our results show that the interparticle interactions strengthen upon Co doping, and the blocking temperature increases by more than 50%. Most importantly, the interplay between Néel and Brown relaxation exhibits a new and very distinct behavior in which the two mechanisms remain coupled, but each is overwhelmingly enhanced within a given temperature range: Brown between 240 K and 250 K and Néel between 270 and 290 K. This behavior is likely due to several factors. First, the increase in the nanoparticles’ magnetic anisotropy known to occur upon Co doping [34] leads to an increase in the energy barrier to magnetization reversal, which directly affects the Néel relaxation time and blocking temperature. Second, the strengthening of the interparticle interactions in the doped ferrite (shown by an increase in the peak shift per frequency decade, ϕ, and the interaction strength parameter, T₀) has an effect not only on the Néel mechanism but also on its Brown counterpart. Indeed, stronger interparticle interactions slow down the reorientation of the superspins via the physical rotation of the nanoparticles within the fluid, thus increasing the Brown relaxation time. In addition, recent studies have suggested that a similar effect can result from an increase in the nanoparticles’ hydrodynamic volume via aggregation driven by strong interparticle interactions [35].

4. Conclusions

We used temperature-resolved ac-susceptibility measurements to carry out an investigation of the magnetization dynamics of Co_0.2Fe_2.8O₄/Isopar M ferrofluid (CFO) at temperatures above the melting point of the carrier fluid, T_F = 197 K. We found that Co doping substantially increased the CFO interparticle ensemble compared to its Fe₃O₄/Isopar M (FO) counterpart. This was indicated by a one-order-of-magnitude decrease in the relative χ″-peak temperature variation per frequency decade, ${\displaystyle \frac{∆\mathrm{T}}{\mathrm{T}·∆\mathrm{l}\mathrm{o}\mathrm{g}\left(\mathrm{f}\right)}}$, from Φ~0.15 in FO to Φ~0.025 in CFO. As a result, the blocking temperature increased in the Co-doped ferrofluid by almost 100 K, so the magnetic events relevant to collective superspin relaxation occurred in CFO above T_F, where both the Néel and Brown mechanisms are active. Our main finding came from χ″vs. T|_f datasets that showed two magnetic events at temperatures T_p1 = 240 K and T_p2 = 275 K. For each of them we identified the microscopic mechanisms that contributed to the observed effective superspin relaxation time, τ_eff. Using Rosensweig’s formula, 1/ τ_eff = 1/τ_N + 1/τ_B, and the temperature dependence of the Néel relaxation, τ_N(T), obtained from the CFO powder, we determined that the Brown mechanism fully describes the collective spin relaxation at 240 K, whereas Néel relaxation becomes dominant at 275 K. Finally, we carried out fits of models that describe the temperature dependence of the relaxation time via these two mechanisms: ${\mathsf{\tau }}_B(\mathrm{T})={\displaystyle \frac{3{\mathsf{\eta }}_0{V}_H}{k_{B}T}}\exp \left[{\displaystyle \frac{{\mathrm{E}}^{\prime }}{{\mathrm{k}}_{\mathrm{B}}\left(\mathrm{T}-{{\mathrm{T}}_0}^{\prime }\right)}}\right]$ and ${\mathsf{\tau }}_N\left(\mathrm{T}\right)={\mathsf{\tau }}_0\exp \left[{\displaystyle \frac{{\mathrm{E}}_{\mathrm{B}}}{{\mathrm{k}}_{\mathrm{B}}\left(\mathrm{T}-{\mathrm{T}}_0\right)}}\right]$. The best fits yielded ${\gamma }_0={\displaystyle \frac{3{\eta }_0{V}_H}{k_B}}$=1.5 × 10⁻⁸ s·K, E′/k_B = 703 K, and T₀′ = 201 K for Brown relaxation and E_B/k_B = 2818 K and T₀ = 143 K for Néel relaxation. These results are important because they have the potential to open up new avenues for the synthesis of magnetic nanoparticle ensembles with enhanced properties for biomedical applications.