# The Influence of Viscosity on Heat Dissipation under Conditions of the High-Frequency Oscillating Magnetic Field

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{B}in the range of microseconds and can be expressed as follows:

_{B}= 3V

_{H}η/k

_{B}T

_{H}is the hydrodynamic volume of the magnetic particle and η is the dynamic viscosity of the surrounding fluid; k

_{B}—Boltzmann’s constant (1.38 × 10

^{−23}JK

^{−1}), T—absolute temperature. The energy barrier for particle reorientation is determined by rotational friction in the surrounding fluid. Losses induced by the Brownian mechanism are also called viscous losses since heat generation is a consequence of viscous friction between the rotating particles and the surrounding medium. This type of loss is not limited to superparamagnetic particles. In general, particles that can be considered small permanent magnets with remanent magnetization M

_{R}are subjected to a torque T = μ

_{0}M

_{R}H

_{V}when exposed to a changing magnetic field H. In the equilibrium state, the viscous resistance in the fluid 12πηVf acts on the magnetic torque T, and the cycle’s loss energy is given by 2πT [11]. The second mechanism is Néel relaxation [12], in which an external AC magnetic field provides energy that helps the magnetization vector to rotate inside the magnetic core of the particle and overcome the energy barrier E = KV, where K is the particle’s anisotropy constant and V is the particle volume. The probability of this transition is exp(σ), where σ is the ratio of anisotropy energy to thermal energy KV/k

_{B}T. The relaxation time τ

_{N}of magnetic nanoparticles during Néel relaxation ranges from milliseconds (even several nanoseconds) to seconds and is given by the expression:

_{N}= τ

_{0}∙exp(σ) = τ

_{0}∙exp(KV/k

_{B}T)

_{0}is the damping or decay time (approximately 10

^{−8}–10

^{−10}s). For high and low values of the energy barrier, we have:

_{N}= τ

_{0}∙σ

^{−1/2}∙exp(σ), σ > 2

= τ

_{0}∙σ, σ << 1.

_{eff}= τ

_{N}∙τ

_{B}/(τ

_{B}+ τ

_{N})

_{MNP}in grams) [13]:

_{MNP}

_{Fe}∙mL

^{−1}), SAR values measured in water and agar were comparable, but at the concentration of 10 mg

_{Fe}∙mL

^{−1}, SAR values in water were twice as high as in agar. The increase in SAR values in water at high particle concentration was associated with interparticle interaction effects that promote increased heat generation. In the present contribution, we analyzed the heating rate and SAR evolution of magnetic fluids prepared based on mineral oils with different viscosity. Understanding the influence of viscosity on heat dissipation in such types of fluids is important from the perspective of their use in conditions of oscillating magnetic fields in the technical science, as well as in the field of magnetic hyperthermia as a therapeutic tool in biomedicine, where fluids of different viscosities can serve as a model material for a deeper understanding of heat evolution under given conditions.

## 2. Experimental Materials and Methods

#### Magnetic Fluids (MF) Based on MOGUL, ITO 100, and MIDEL Oil

_{3}O

_{4}magnetite nanoparticles coated with a surfactant—oleic acid. The preparation was carried out as follows: magnetite was synthesized by coprecipitation at around 80 °C using aqueous solutions of Fe

^{3+}and Fe

^{2+}ions along with a concentrated NH

_{4}OH solution (25%). Surfactants were bound to the magnetic nanoparticles to provide steric stabilization (chemisorption of oleic acid at about 80 °C). Magnetic decantation and multiple rinsing steps with acetone were used to remove byproducts of the precipitation reaction and obtain stabilized magnetic nanoparticles. The magnetic nanoparticles were then dispersed in a nonpolar liquid carrier at 120–130 °C. Additional steps included magnetic decantation/filtration, flocculation, and redispersion of the magnetic nanoparticles to obtain a nonpolar magnetic fluid. To verify the binding of oleic molecules to the surface of magnetic nanoparticles, Fourier-transform infrared spectroscopy (FTIR) was employed.

^{−1}, utilizing an FTIR spectrometer (ABB, Model FTLA2000-100, Québec, Canada). The spectral resolution was configured to 4 cm

^{−1}.

^{−1}. Differential temperature measurements (of the magnetic fluid and a reference sample, i.e., oil as a liquid carrier) and isolation of the glass vial containing the sample from the coil winding by cooled flowing water were used to eliminate possible errors caused by parasitic heat contributions in the carrier liquid. The probes of the thermometer’s optical fibers (FISO Technologies Inc., Québec, Canada) are not sensitive to the alternating magnetic field.

## 3. Particles, Oils, and Magnetic Fluid Parameters

_{3}O

_{4}; indicated by the black line), and magnetic nanoparticles coated with oleic acid (OA@Fe

_{3}O

_{4}; represented by the red line).

_{2}stretch and the symmetric CH

_{2}stretch vibrations appear at 2855 and 2925 cm

^{−1}, respectively. The intense peak at 1710 cm

^{−1}is attributed to the C=O stretch, and the band at 1284 cm

^{−1}signifies the presence of the C–O stretch [22,23]. The peak at 542 cm

^{−1}in the spectrum of magnetic nanoparticles is related to the Fe-O group [24]. In the spectrum of OA@Fe

_{3}O

_{4}, when compared with the spectrum of pure OA, the ν

_{as}(CH

_{2}) and ν

_{s}(CH

_{2}) stretches shifted to 2842 and 2917 cm

^{−1}, respectively. The band at 1710 cm

^{−1}, corresponding to the stretching of C=O in oleic acid, was absent in the spectrum of OA@Fe

_{3}O

_{4}. This can be explained by the absence of free surfactant in the sample. Instead, two new bands at 1414 and 1516 cm

^{−1}appeared, attributed to the asymmetric and symmetric stretch vibration bands of the carboxylate group, respectively [25]. Moreover, the characteristic peaks of magnetic nanoparticles at 551 cm

^{−1}are observed in the spectrum of OA@Fe

_{3}O

_{4}, confirming the presence of magnetic nanoparticles in the sample. These results show that the surface of magnetic particles was successfully coated with the surfactant.

^{−1}) differed from those reported for their bulk counterparts (92–100 emu∙g

^{−1}) and usually decreased with size (~88 emu∙g

^{−1}for Fe

_{3}O

_{4}nanoparticles) [28,29]. Another reason for the lower magnetic saturation is that uncoated magnetite nanoparticles in powder form constitute a different system compared to those in a magnetic fluid, due to particle coating, stability, and the presence of a carrier liquid. Moreover, particle concentration also plays a significant role, and it is necessary to realize that in magnetite converted into a magnetic fluid, the fraction of diamagnetic components increases, resulting in lower values. The basic parameters of the prepared mineral-oil-based magnetic fluids are listed in Table 1.

## 4. Hyperthermal Properties of Magnetic Fluids

^{n}. The observed H

^{n}-type dependence and the value of the power-law exponent “n”, which was the same for all samples with n = 2, provide information on the presence of superparamagnetic particles [10]. This fact is also supported by the size distribution, where individual magnetite particles can generally be considered superparamagnetic under these conditions [31]. Relaxation processes, namely Brownian and Néel relaxation, were responsible for generating energy losses. Indeed, the value of the exponent n = 2 of the power function (H/a)

^{n}, as well as the course of the magnetization curve M(H), clearly indicate the superparamagnetism of the samples and the lack of magnetic hysteresis. The following graph (Figure 8) shows the dependence of the temperature rise rate on the amplitude of the external magnetic field for magnetic fluids based on MOGUL, ITO 100, and MIDEL oil. The graph indicates that the coefficient of the temperature rise rate was indeed proportional to the square of the applied alternating magnetic field.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**FTIR spectra of pure oleic acid (OA; blue), magnetic nanoparticles (Fe

_{3}O

_{4}; black), and coated magnetic nanoparticles using oleic acid (OA@Fe

_{3}O

_{4}; red).

**Figure 2.**Lognormal distribution of magnetite particle sizes in the liquid carrier obtained from VSM data.

**Figure 8.**The rate of temperature increase (ΔT/Δt)

_{t}

_{= 0}as a function of the amplitude of the alternating magnetic field H at a frequency of f = 500 kHz for samples of mineral-oil-based magnetic fluids with varying viscosities.

**Figure 9.**SAR depending on the intensity of the magnetic field at 500 kHz converted to the amount of magnetite in the sample. Magnetic fluids (MF) with liquid carriers MIDEL (154.68 W·g

^{−1}), ITO 100 (169.84 W·g

^{−1}), and MOGUL (196.26 W·g

^{−1}). The stated MF SAR values were determined at 10 kA·m

^{−1}.

**Table 1.**Comparative characteristics of carrier liquids (mineral oils) vs. magnetic liquids prepared on them (ρ

_{oil}—density of oils, ρ

_{MF}—density of magnetic fluids, η

_{oil}—viscosity of oils, η

_{MF}—viscosity of magnetic fluids) and M

_{S}-saturation magnetization values of magnetic fluids.

ρ_{oil} [g∙cm^{−3}] | ρ_{MF} [g∙cm^{−3}] | M_{S} [emu∙g^{−1}] | η_{oil} [mPa∙s] | η_{MF} [mPa∙s] | |
---|---|---|---|---|---|

MIDEL | 0.975 | 1.202 | 26.10 | 55.48 | 85.51 |

ITO 100 | 0.895 | 1.128 | 25.70 | 16.28 | 52.30 |

MOGUL | 0.791 | 1.031 | 25.90 | 12.51 | 44.54 |

**Table 2.**The SAR values obtained per gram of the sample and after conversion to the amount of magnetite in the sample.

Sample | Φ (Magnetic Volume Fraction) | a (H/a) ^{2} | SAR (Specific Absorption Rate) f = 500 kHz | ||
---|---|---|---|---|---|

5 kA·m^{−1} | 10 kA·m^{−1} | 10 kA·m^{−1} | |||

% | W·g^{−1}_{sample} | W·g^{−1}_{sample} | W·g^{−1}_{Fe3O4} | ||

MF MOGUL | 6.6 | 9935 | 0.45 | 1.81 | 196.26 |

MF ITO 100 | 6.6 | 11,241 | 0.39 | 1.56 | 169.84 |

MF MIDEL | 6.6 | 11,570 | 0.35 | 1.42 | 154.68 |

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**MDPI and ACS Style**

Molcan, M.; Skumiel, A.; Tothova, J.; Paulovicova, K.; Kopcansky, P.; Timko, M.
The Influence of Viscosity on Heat Dissipation under Conditions of the High-Frequency Oscillating Magnetic Field. *Magnetochemistry* **2024**, *10*, 2.
https://doi.org/10.3390/magnetochemistry10010002

**AMA Style**

Molcan M, Skumiel A, Tothova J, Paulovicova K, Kopcansky P, Timko M.
The Influence of Viscosity on Heat Dissipation under Conditions of the High-Frequency Oscillating Magnetic Field. *Magnetochemistry*. 2024; 10(1):2.
https://doi.org/10.3390/magnetochemistry10010002

**Chicago/Turabian Style**

Molcan, Matus, Andrzej Skumiel, Jana Tothova, Katarina Paulovicova, Peter Kopcansky, and Milan Timko.
2024. "The Influence of Viscosity on Heat Dissipation under Conditions of the High-Frequency Oscillating Magnetic Field" *Magnetochemistry* 10, no. 1: 2.
https://doi.org/10.3390/magnetochemistry10010002