# The Role of Anisotropy in Distinguishing Domination of Néel or Brownian Relaxation Contribution to Magnetic Inductive Heating: Orientations for Biomedical Applications

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## Abstract

**:**

_{max}), optimal nanoparticle diameter (D

_{c}) and its width (ΔD

_{c}) are considered as being dependent on magnetic nanoparticle anisotropy (K). The calculated results suggest 3 different Néel-domination (N), overlapped Néel/Brownian (NB), and Brownian-domination (B) regions. The transition from NB- to B-region changes abruptly around critical anisotropy K

_{c}. For magnetic nanoparticles with low K (K < K

_{c}), the feature of SLP peaks is determined by a high value of D

_{c}and small ΔD

_{c}while those of the high K (K > K

_{c}) are opposite. The decreases of the SLP

_{max}when increasing polydispersity and viscosity are characterized by different rates of d(SLP

_{max})/dσ and d(SLP

_{max})/dη depending on each domination region. The critical anisotropy K

_{c}varies with the frequency of an alternating magnetic field. A possibility to improve heating power via increasing anisotropy is analyzed and deduced for Fe

_{3}O

_{4}magnetic nanoparticles. For MIH application, the monodispersity requirement for magnetic nanoparticles in the B-region is less stringent, while materials in the N- and/or NB-regions are much more favorable in high viscous media. Experimental results on viscosity dependence of SLP for CoFe

_{2}O

_{4}and MnFe

_{2}O

_{4}ferrofluids are in good agreement with the calculations. These results indicated that magnetic nanoparticles in the N- and/or NB-regions are in general better for application in elevated viscosity media.

## 1. Introduction

_{max}) at a characteristic particle diameter D

_{c}for different material parameters and field conditions [21,22,23,24,25,26]. From the studies of various Fe

_{3}O

_{4}(FO) MNPs with different size at frequency f = 376 kHz, the maximum SLP

_{max}was found for a 16 nm (size standard deviation, σ = 0.175) sample [27,28,29]; for iron oxide nanocubes at f = 520 kHz, it is 19 ± 3 nm [30,31]. An analysis of various data after taking a normalization of field factor gave an elevation of SLP in the diameter range from 14–18 nm [25]. For γ-Fe

_{2}O

_{3}, elevation of SLP for 5 sample sizes from 5.3 nm to 16.5 nm (σ = 0.19–0.43) is in agreement with the theoretical prediction for the peak appearance at D

_{c}= 14.5 nm [22]. Although these results have confirmed the existence of optimal particle size D

_{c}of MNPs in MIH, finding the value of D

_{c}is not a simple task in experimental works because of the impact of other parameters such as the magnetic anisotropy, viscosity of fluid, and the size distribution.

_{max}and D

_{c}decreases monotonically with increasing anisotropy [23,24]. It is noted that, due to the difference in morphology, the synthesized MNPs of the same material can have quite different polydispersity, anisotropy K values (e.g., FO MNPs having K ranging from about 20 to 550 kJ/m

^{3}[32,33]) which can influence the SLP

_{max}and D

_{c}parameters. Secondly, the value of SLP

_{max}was found to reduce with expansion of size distribution [22]. In practice, there will be some size distribution of the MNPs regardless of the synthesis method and producing MNPs with a sufficiently narrow size distribution is a difficult task. Thus far there have been few reports on experimental verification of calculated optimal D which would require the ability to synthesize MNPs with high monodispersity and to tune D precisely [22,25,28,34]. Thirdly, there might be some relationships between the value of D

_{c}and SAR and ferrofluid viscosity. The size dependences of heating power with some variations of D and ferrofluid viscosity was studied experimentally and theoretically which showed for example for γ-Fe

_{2}O

_{3}and CoFe

_{2}O

_{4}MNPs SAR decreased about 20% and 80% with an increase of the viscosity, respectively [22]. To be able to obtain MNPs with optimal heating power, it is desirable to develop a complete correlation of SLP versus D and to consider impacts of different parameters, e.g., size distribution and shape of the NPs [20,21,22].

_{max})/dσ and d(SLP

_{max})/dη for FO MNPs with various anisotropy have been studied and explained as a result of the competition between Néel and Brownian dissipation processes. Here, the experimental results of MNPs of manganese ferrite (MFO), and cobalt ferrite (CFO) are considered as representative of low K (Néel domination) N type, and high K (Brownian domination) B type, which have confirmed the theoretical predictions.

## 2. Theoretical Basis

^{3}) and P the loss power density (W/m

^{3}). The calculations were carried out for particles with size standard deviation (or size polydispersity) parameter σ = 0–0.5, and viscosity η = 1–10 mPa·s, assuming the volume fraction Φ = 0.1% and the surface ligand layer thickness of 1 nm.

_{C}[19]. The correlation between SLP (based on LRT), the field amplitude (H), frequency (f) and the MNPs imaginary susceptibility (χ”) is ref. [20,24]:

_{0}being the permeability of free space; where

_{N}) and Brownian (τ

_{B}) processes [20,24]:

_{H}are the volume of the core magnetic, and the whole capped particle, respectively.

## 3. Results and Discussion

#### 3.1. Characteristics of Optimal Parameters in Domination Regions of Néel or Brownian Relaxations

^{3}[32,33]. The calculations are, therefore, made for different K up to 50 kJ/m

^{3}and the results of SLP versus particle diameter at K equal to 9 kJ/m

^{3}, 20 kJ/m

^{3}, 41 kJ/m

^{3}are presented in Figure 1.

_{c}for describing the peaks. The D

_{c}, ΔD

_{c}and SLP

_{max}obtained at f = 100 kHz, H = 5.18 kA/m (65 Oe), σ = 0, η = 1 mPa⋅s for monodisperse FO ferrofluids with different anisotropy K are listed in Table 1.

^{3}) FO sample (18.5 nm) has much smaller value of D

_{c}compared with that of the low K (3 kJ/m

^{3}) one (28.5 nm). In contrast, there is a large increase in the value of FWHM ΔD

_{c}when the value of K goes from 3 kJ/m

^{3}(3.5 nm) to 50 kJ/m

^{3}(16.5 nm). The fact that the SLP peak is narrow for low K but broad for high K can be attributed to Néel or Brownian domination, respectively [9,20,22,24].

^{3}) [25]. We calculated the value of relaxation times at D

_{c}for all samples. As can be seen in Table 1, the value of effective relaxation time is approximately equal to the Brownian relaxation time at D

_{c}when the value of K is higher than 19 kJ/m

^{3}. The calculations indicated that the Brownian relaxation dominated at D

_{c}when the value of K is higher than 19 kJ/m

^{3}. Therefore, the difference in peak behaviors of the SLP against D depends on the domination of Néel or Brownian relaxations. As can be seen in Table 1, the value of D

_{c}changes with an increase of K from 3 kJ/m

^{3}to 20 kJ/m

^{3}, because the Néel relaxation loss still affects this parameter. When the value of K for FO is higher than 20 kJ/m

^{3}, D

_{c}and SLP

_{max}are unchanged due to the domination of Brownian relaxation loss. Therefore, it changes abruptly at some critical anisotropy K

_{c}= 20 kJ/m

^{3}.

_{max}, D

_{c}and ΔD

_{c}as a function of K for f = 100 kHz, η = 1 mPa⋅s are presented in Figure 2, where it suggests 3 different regions. In the lowest K (region I: K ≤ 5 kJ/m

^{3}): SLP

_{max}, D

_{c}and ΔD

_{c}decrease with increasing K. The middle region (region II: 5 kJ/m

^{3}≤ K ≤ 20 kJ/m

^{3}) is characterized by a tendency of the decrease of D

_{c}to slow down while ΔD

_{c}starts to increase with increasing K. In the high K region (region III: K > 20 kJ/m

^{3}), all the 3 parameters become almost constant. Relating to the dissipation mechanisms, the regions I, II and III can be correspondingly assigned as Néel-domination (N-region), Néel/Brownian overlap (NB-region) and Brownian-domination (B-region). While the transition from N- to NB-region is quite smooth, the transition to the B region is distinctly of the first order and in case of FO MNPs it corresponds to a critical anisotropy value K

_{c}of about 20 kJ/m

^{3}.

#### 3.2. Dependence of Critical Anisotropy K_{c} on Frequency (f)

_{max}at the optimal particle size D

_{c}when the condition ωτ = 1 is satisfied. Therefore, the anisotropy boundary of the transition from NB- to B-region might depend on the frequency of AMF.

_{c}versus K are presented in Figure S1a. For frequencies up to 1000 kHz can be seen an abrupt change of D

_{c}at K

_{c}associated with the transition to the Brownian domination region (for example with f = 500 kHz, the value of K

_{c}is about 60 kJ/m

^{3}, and the three N, NB and B regions are presented in Figure S1b. The dependence of the critical anisotropy K

_{c}on the field frequency is presented in Figure 3, which follows the function:

^{3}, B = 0.0000814 ms, and f

_{0}= 81.27 kHz are fitting constants.

_{c}at different frequencies for fluids with various viscosities are listed in Table 2. For MFO fluid with K = 3 kJ/m

^{3}, the Brownian relaxation dominates when η = 1 mPa⋅s and f = 100 kHz (low frequency) but it becomes the Néel relaxation domination when η ≥ 1 mPa⋅s or f ≥ 100 kHz. For CFO fluid, the Néel relaxation dominates when η ≥ 4 mPa⋅s or f ≥ 500 kHz, but it is the Brownian relaxation domination when η ≤ 2 mPa⋅s or f ≤ 1000 kHz.

#### 3.3. Orientations to Choose Proper Regions in Biomedical Applications

_{2}O

_{3}with σ = 0.43, which could be reduced to 0.19 (d = 5.3 nm) after size sorting by successive phase separations [22]. M.P. Pileni et al. synthesized 2–5 nm CoFe

_{2}O

_{4}with 30% polydispersity in size distribution by using functionalized surfactants [35]. When A. K. Gupta et al. compared the different characteristic features of the iron oxide NPs fabricated through different methods, they were always polydisperse NPs with narrow or broad size distributions [3]. Therefore, the assumption (σ = 0) appears not to be realized. In fact, in the synthesis of MNPs using seed mediated-growth or size sorting route, the common value of σ was reported in between 0.3–0.5 but sometime it can reduce to 0.15–0.2 [2,22,36,37]. The polydispersity characterized by the σ of MNPs can give rise to the reduction of SLP [20]. For γ-Fe

_{2}O

_{3}MNPs, σ in the range of 0.08–0.15 could result in a decrease of SLP

_{max}to half of the monodispersive value [22,24]. Calculations were carried out for various anisotropy (K) and standard deviation (σ) of FO MNPs at f = 100 kHz, H = 5.18 kA/m, η = 1 mPa⋅s and the results of relative SLP

_{max}(σ)/SLP

_{max}(σ = 0) are represented in Figure 4.

_{max}decreases with increasing σ and there is a clear trend of increasing relative SLP

_{max}when the value of K changes from the NB-region (9 kJ/m

^{3}) to K

_{c}(20 kJ/m

^{3}) and then the B-region (41 kJ/m

^{3}). In other words, for the similar polydispersity degree σ, the MNPs in the B-region would have higher heating power than that of those in the NB-region. These results indicated that the monodispersity requirement for getting the same heating power is much less strict for the MNPs in B-region as compared with the case of MNPs in N- or NB-region. Quantitatively, the polydispersity-caused SLP reduction for iso-dispersity of σ = 0.2 FO MNPs and σ

_{50}(defined as at the position of ½ SLP

_{max}) with various anisotropies at f = 100 kHz, H = 5.18 kA/m, η = 1 mPa⋅s are listed in Table 3. The parameter σ

_{50}is seen to increase from 0.21 to 0.58 with K from 9 to 32 kJ/m

^{3}, then becomes constant for K ≥ 36 kJ/m

^{3}indicating that the monodispersity requirement for obtaining the same heating power is much less strict for the high K as compared with the case of low K MNPs. The results in Table 3 also show that the polydispersity-caused SLP reduction decreased with an increase of magnetic anisotropy. From the slope of the SLP graph in Figure S2, it is estimated that an increase of K by about 4 times can result in saving as much as 35% of the heating power. These results seem to be consistent with the observation on cubic-shaped FO MNPs of about 20 nm (K = 180 kJ/m

^{3}) having higher heating power than spherical ones (K = 20 kJ/m

^{3}) [32]. The improvement of heating power by increasing K is an interesting topic and it has been recently reported for different materials, while the elevation of K was possible by various approaches including surface morphology, exchange-bias or stacking of particles in chains [32,33,36,37,38,39].

_{max}in the viscosity range from 1 mPa⋅s to 10 mPa⋅s for MNPs with various K. The results of the relative SLP

_{max}(η)/SLP

_{max}(η = 1 mPa⋅s) for the fluid of FO MNPs are presented in Figure 5. As can be seen in Figure 5, for FO MNPs with the value of K in NB-region (9 kJ/m

^{3}), SLP

_{max}decreases very little with increasing η. In contrast, there is a large decrease in the SLP

_{max}when the value of η is going from 1 kJ/m

^{3}to 4 mPa⋅s for 20 kJ/m

^{3}, and 41 kJ/m

^{3}FO MNPs. It is noted that the value of critical anisotropy at 4 mPa⋅s for the field frequency of 100 kHz is 60 kJ/m

^{3}(Table 2). Therefore, the decrease of SLP

_{max}with increasing η might occur when the value of anisotropy is smaller than K

_{c}. In other words, this behavior happens when the K of FO MNPs is in the B-region. These calculations indicated that the FO MNPs with the K in the N- or NB-regions are much better for MIH applications in highly viscous conditions. The observation of weak and strong decrease of the heating power with increasing viscosity, will be discussed as compared with experimental results in the following section.

#### 3.4. A Comparison with Experimental Results

_{2}O

_{4}and 15 nm high-K CoFe

_{2}O

_{4}MNPs coated with chitosan. They were prepared by co-precipitation. To create the magnetic fluids with different viscosity, the MNPs suspension was mixed with various agar concentrations. The value of viscosity extended up to about 8 mPa⋅s by adding more agar amounts. The fabrication of samples with various viscosities as well as the method of viscosity measurement were described earlier [40] as were the experimental details of the synthesis and the coating procedure by chitosan [41]. As discussed in our previous reports, chitosan has been widely used in biomedicine due to its biocompatibility and biodegradability [41]. Based on the measured magnetization curves shown in Figure S3, the thickness of chitosan coating layer was estimated to be around 4 nm. The value of M

_{S}, H

_{C}and K

_{eff}for MFO and CFO NPs (shown details in Supplementary Section S3) are presented in Table 4.

^{3}and H

_{c}were close to or smaller than the AFM amplitudes, the contribution of ${\mathrm{SAR}}_{\mathrm{exp}}^{\mathrm{hys}}$ is less than 2.5% of the whole heating power. However, for the CFO samples with K = 62 kJ/m

^{3}and H

_{c}much larger than the field amplitude used, the maximum ${\mathrm{SAR}}_{\mathrm{exp}}^{\mathrm{hys}}$ contributions were 17%, 16% and 9% for 178, 340 and 145 kHz, respectively.

_{2}O

_{3}(D = 7.1 nm, σ = 0.37) and CoFe

_{2}O

_{4}(D = 9.7 nm, σ = 0.35) MNPs where SAR decreased, respectively, about 20% and 80% when the viscosity increased from 0.75 to 335 mPa⋅s [22]. In fact, it was found that SLP value from MNPs dispersed in water is higher than in glycerol for the same type of MNPs, MNP dose, and AMF condition [43]. A similar phenomenon was observed in the cellular environment as well, in which the measured SLP value decreased by half associated with attenuation of the Brownian relaxation in cellular conditions [44]. Thirdly, while the MFO ferrofluids remain almost not impacted by the field frequency used, it is interesting to observe for CFO samples a clear shift from high K toward low-K characteristic behavior when the used field frequency increases from 178 kHz to 450 kHz. In particular, Figure 7c for the case of f = 450 kHz indicates that the relative SAR versus η for the CFO MNPs approaches to the type of very weak decrease of the heating power with increasing the media viscosities, similarly to that of MFO MNPs. This observation is in good agreement with the scheme shown in Figure S7, according to which the MFO MNPs of K = 11 kJ/m

^{3}are predicted to exhibit the SAR contributed mainly from the Neel or overlapped Neel-Brownian relaxation losses depending not on the used field frequency, and the dissipation mechanism contributing to the SAR for the CFO MNPs with K = 62 kJ/m

^{3}belongs to the pure Brownian region for AMF with f below about 460 kHz and changes to the Neel-Brown or Neel type above this frequency.

## 4. Conclusions

_{c}and its amplitude SLP

_{max}, we have introduced a peak width ΔD

_{c}and performed systematic analyses of the three parameters against the magnetic anisotropy of MNPs, fluid viscosities and the frequency of the alternating magnetic field. High K MNPs have been shown to exhibit domination of Brownian loss while it becomes that of Néel loss in low K ones. To the best of our knowledge, our results have demonstrated for the first time the transition from Néel to Brownian loss region not to occur in a continuous way, but at critical anisotropy K

_{c}which increases with the frequency of the alternating magnetic field and the viscosity of the ferrofluid. We have pointed out that, for a given material, fabrication of MNPs with higher anisotropy up to K

_{c}can improve the heating power as much as 35% thanks to the offsetting of the polydispersity-caused reduction. From the calculation and experimental results, it appears that low K MNPs are in general better for MIH applications in highly viscous conditions than those of high K ones.

## Supplementary Materials

_{c}versus K for various AMF frequencies. (b) The width ΔD

_{c}, and Dc versus K at f = 500 kHz showing 3 characteristic N, NB and B regions; Figure S2: Polydispersity-caused SLP reduction calculated at f = 100 kHz for iso-dispersity σ = 0.2 FO MNPs as a function of anisotropy K, Figure S3: Magnetization curves measured for as-synthesized and chitosan coated (a) MFO, and (b) CFO MNPs; Figure S4: The initial magnetization curves of MFO and CFO MNPs. The solid lines represent the fitting curve assuming “the law of approach to saturation”; Figure S5: Hyperthermia curves measured at fields of frequency f = 340 kHz, H = 15.9 kA/m (200 Oe) for (a) MFO and (b) CFO ferrofluids of various viscosities; Figure S6: Hyperthermia curves measured at fields of frequency f = 450 kHz, H = 15.9 kA/m for (a) MFO and (b) CFO ferrofluids of various viscosities; Figure S7: Illustration scheme for the MIH experiments for CFO and MFO MNPs, Table S1: Values of ${\mathrm{SAR}}_{\mathrm{exp}}$, ${\mathrm{SAR}}_{\mathrm{exp}}^{\mathrm{hys}}$, ${\mathrm{SAR}}_{\mathrm{exp}}^{\mathrm{LRT}}$, and $\frac{{\mathrm{SAR}}_{\mathrm{exp}}^{\mathrm{LRT}}}{{\mathrm{SAR}}_{\mathrm{exp}}^{\mathrm{LRT}}\text{}\left(\mathsf{\eta}=1\text{}\mathrm{mPa}\xb7\mathrm{s}\right)}$ at 5.18 kA/m, 178 kHz; Table S2: Values of ${\mathrm{SAR}}_{\mathrm{exp}}$, ${\mathrm{SAR}}_{\mathrm{exp}}^{\mathrm{hys}}$, ${\mathrm{SAR}}_{\mathrm{exp}}^{\mathrm{LRT}}$, and ${\mathrm{SLP}}^{\mathrm{LRT}}$ at 15.9 kA/m, 340 kHz; Table S3: Values of ${\mathrm{SAR}}_{\mathrm{exp}}$, ${\mathrm{SAR}}_{\mathrm{exp}}^{\mathrm{hys}}$, ${\mathrm{SAR}}_{\mathrm{exp}}^{\mathrm{LRT}}$, and ${\mathrm{SLP}}^{\mathrm{LRT}}$ at 15.9 kA/m, 450 kHz.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**SLP versus D, calculated at f = 100 kHz, H = 5.18 kA/m, σ = 0, η = 1 mPa⋅s for monodispersive FO MNPs with 3 different K of 9, 20, and 41 kJ/m

^{3}. The bars denote the (full-width-half-maximum (FWHM)) ΔD

_{c}.

**Figure 2.**Changes of characteristic parameters, SLP

_{max}, D

_{c}and ΔD

_{c}from Néel-domination to Brownian-domination region calculated for frequency f = 100 kHz, viscosity η = 1 mPa⋅s.

**Figure 4.**SLP

_{max}(σ)/SLP

_{max}(σ = 0) calculated at f = 100 kHz, H = 5.18 kA/m, η = 1 mPa⋅s for FO samples with K = 9, 20, and 41 kJ/m

^{3}.

**Figure 5.**SLP

_{max}(η)/SLP

_{max}(η = 1 mPa⋅s) calculated at f = 100 kHz, H = 5.18 kA/m for FO samples with K = 9, 20, and 41 kJ/m

^{3}.

**Figure 6.**Hyperthermia curves measured at field frequency f = 178 kHz, H = 5.18 kA/m for (

**a**) MFO and (

**b**) CFO ferrofluids of various viscosities.

**Figure 7.**Relative ${\mathrm{SAR}}_{\mathrm{exp}}$ and ${\mathrm{SAR}}_{\mathrm{exp}}^{\mathrm{LRT}}$ measured at field of (

**a**) 5.18 kA/m, 178 kHz; (

**b**) 15.9 kA/m, 340 kHz; and (

**c**) 15.9 kA/m, 450 kHz.

**Table 1.**D

_{c}, ΔD

_{c}, SLP

_{max}, τ

_{N}, τ

_{B}, and τ obtained for monodisperse FO ferrofluids with different anisotropy constant K. (f = 100 kHz, H = 5.18 kA/m, σ = 0, η = 1 mPa⋅s).

K (kJ/m^{3}) | D_{c} (nm) | ΔD_{c} (nm) | SLP_{max} (W/g) | τ_{N} (s) | τ_{B} (s) | τ (s) |
---|---|---|---|---|---|---|

3 | 28.5 | 3.5 | 76.9 | $1.95\times {10}^{-6}$ | $10.77\times {10}^{-6}$ | $1.65\times {10}^{-6}$ |

5 | 24 | 3 | 71.4 | $1.87\times {10}^{-6}$ | $6.67\times {10}^{-6}$ | $1.47\times {10}^{-6}$ |

7 | 21.5 | 4 | 66.2 | $1.98\times {10}^{-6}$ | $4.92\times {10}^{-6}$ | $1.42\times {10}^{-6}$ |

9 | 20 | 6.5 | 62.3 | $2.65\times {10}^{-6}$ | $4\times {10}^{-6}$ | $1.6\times {10}^{-6}$ |

12 | 18.5 | 9.5 | 58.3 | $4.26\times {10}^{-6}$ | $3.27\times {10}^{-6}$ | $1.85\times {10}^{-6}$ |

15 | 17.5 | 12.5 | 50.9 | $3.24\times {10}^{-6}$ | $2.81\times {10}^{-6}$ | $2.02\times {10}^{-6}$ |

19 | 17 | 14 | 46.9 | $3.46\times {10}^{-6}$ | $2.6\times {10}^{-6}$ | $1.52\times {10}^{-6}$ |

20 | 17 | 14.5 | 45.9 | $2\times {10}^{-3}$ | $2.6\times {10}^{-6}$ | $2.6\times {10}^{-6}$ |

21 | 18.5 | 14.5 | 43.6 | $4\times {10}^{-3}$ | $3.27\times {10}^{-6}$ | $3.27\times {10}^{-6}$ |

30 | 18.5 | 15 | 43.6 | $4.903$ | $3.27\times {10}^{-6}$ | $3.27\times {10}^{-6}$ |

41 | 18.5 | 16 | 43.6 | $2.8\times {10}^{4}$ | $3.27\times {10}^{-6}$ | $3.27\times {10}^{-6}$ |

50 | 18.5 | 16.5 | 43.6 | $3.4\times {10}^{7}$ | $3.27\times {10}^{-6}$ | $3.27\times {10}^{-6}$ |

f (kHz) | K_{c} (kJ/m^{3}) | ||||
---|---|---|---|---|---|

η = 1 | η = 2 | η = 4 | η = 6 | η = 8 | |

100 | 20 | 33 | 60 | 83 | 102 |

200 | 31 | 56 | 103 | 147 | 205 |

300 | 41 | 74 | 141 | 207 | 261 |

400 | 51 | 95 | 188 | 252 | 322 |

500 | 59 | 112 | 227 | 308 | 399 |

750 | 100 | 153 | 292 | >400 | >400 |

1000 | 102 | 205 | 364 | >400 | >400 |

**Table 3.**Polydispersity-caused SLP reduction for iso-dispersity σ = 0.2 FO MNPs and σ

_{50,}defined as at the position of ½ SLP

_{max}, with various anisotropy K (f = 100 kHz, H = 5.18 kA/m, η = 1 mPa⋅s).

K (kJ/m^{3}) | SLP Reduction (%) | σ_{50} |
---|---|---|

9 | 48.6 | 0.21 |

12 | 38.8 | 0.3 |

16 | 25.3 | 0.43 |

20 | 17 | 0.52 |

24 | 15.6 | 0.54 |

28 | 14.8 | 0.57 |

32 | 14.3 | 0.58 |

36 | 13.9 | 0.59 |

40 | 13.9 | 0.59 |

Sample | M_{S} (emu/g) | H_{C} (Oe) | K_{eff} (kJ/m^{3}) |
---|---|---|---|

MFO (uncoated) | 60.9 | 72 | - |

MFO (coated) | 51.9 | 72 | 11 |

CFO (uncoated) | 67.2 | 875 | - |

CFO (coated) | 57.4 | 875 | 62 |

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**MDPI and ACS Style**

Nguyen, L.H.; Phong, P.T.; Nam, P.H.; Manh, D.H.; Thanh, N.T.K.; Tung, L.D.; Phuc, N.X.
The Role of Anisotropy in Distinguishing Domination of Néel or Brownian Relaxation Contribution to Magnetic Inductive Heating: Orientations for Biomedical Applications. *Materials* **2021**, *14*, 1875.
https://doi.org/10.3390/ma14081875

**AMA Style**

Nguyen LH, Phong PT, Nam PH, Manh DH, Thanh NTK, Tung LD, Phuc NX.
The Role of Anisotropy in Distinguishing Domination of Néel or Brownian Relaxation Contribution to Magnetic Inductive Heating: Orientations for Biomedical Applications. *Materials*. 2021; 14(8):1875.
https://doi.org/10.3390/ma14081875

**Chicago/Turabian Style**

Nguyen, Luu Huu, Pham Thanh Phong, Pham Hong Nam, Do Hung Manh, Nguyen Thi Kim Thanh, Le Duc Tung, and Nguyen Xuan Phuc.
2021. "The Role of Anisotropy in Distinguishing Domination of Néel or Brownian Relaxation Contribution to Magnetic Inductive Heating: Orientations for Biomedical Applications" *Materials* 14, no. 8: 1875.
https://doi.org/10.3390/ma14081875