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Article

Magnetothermal Energy Conversion of Polydopamine-Coated Iron Oxide Ferrogels Under High-Frequency Rotating Magnetic Fields

1
Chair of Acoustics, Faculty of Physics and Astronomy, Adam Mickiewicz University in Poznań, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland
2
Institute of Chemical Technology and Engineering, Faculty of Chemical Technology, Poznan University of Technology, Berdychowo 4, 60-965 Poznań, Poland
*
Author to whom correspondence should be addressed.
Energies 2025, 18(16), 4291; https://doi.org/10.3390/en18164291
Submission received: 22 June 2025 / Revised: 8 August 2025 / Accepted: 9 August 2025 / Published: 12 August 2025
(This article belongs to the Section J: Thermal Management)

Abstract

This study provides a comparison between magnetic-to-thermal energy conversion efficiency in liquid and gel phases under high-frequency magnetic fields. Magnetite cores (11 ± 2 nm) were tested as water-based ferrofluids and as 5 wt% agar ferrogels, both with and without a biocompatible polydopamine (PDA) shell. A custom two-phase coil switched between rotating (RMF) and alternating (AMF) modes, enabling phase- and coating-dependent effects to be measured at identical field strengths and frequencies (100–300 kHz, 1–4 kA/m). Across all conditions, RMF generated 1.7–2.1 times more specific loss power (SLP) than AMF, and moving from the liquid to the gel phase reduced SLP by 5–8%, indicating that heating is controlled by Néel relaxation with negligible Brownian contribution. SLP rose with magnetic-field amplitude according to a power law, while hysteretic losses remained minimal. PDA improved colloidal stability and biocompatibility without harming the heating performance, lowering SLP by <17%. Within Brezovich limits, the system still exceeded therapeutic hyperthermia thresholds. Thus, in this iron-oxide/PDA system, neither medium viscosity nor the PDA shell’s non-magnetic mass significantly affects thermal energy output, an important finding for translating laboratory calorimetry data into reliable, application-oriented modelling for magnetic hyperthermia.

1. Introduction

Magnetic hyperthermia is an advanced experimental cancer treatment that utilizes magnetic nanoparticles (MNPs) to generate heat under an external magnetic field. As the field interacts with MNPs, the rapid reorientation of their magnetic moments leads to energy dissipation through Néel and Brownian relaxation mechanisms, which collectively generate localized heat. This localized temperature increase enables targeted treatment, making it a promising approach for tumor therapy [1,2,3,4,5,6]. Although AMF is commonly used in practice for magnetic heating, RMF has demonstrated potential both theoretically and practically for superior heating performance, warranting further investigation [7,8,9,10].
Superparamagnetic iron oxide nanoparticles (SPIONs) have garnered significant attention in biomedical research due to their exceptional magnetic properties, biocompatibility, and tunable surface chemistry. Unlike bulk ferromagnetic materials, SPIONs retain magnetization only when subjected to an external magnetic field and show no residual magnetism once the field is removed. This behavior prevents particle aggregation, making them highly suitable for medical applications such as magnetic hyperthermia, targeted drug delivery, and MRI contrast enhancement. Additionally, their biodegradability and low toxicity, especially when surface-functionalized with biocompatible coatings like dextran, polyethylene glycol (PEG), or polydopamine (PDA), further enhance their potential for clinical adoption [11,12,13].
A major challenge in hyperthermia research is understanding how surface modifications influence nanoparticle heating efficiency. PDA coatings, for instance, enhance the biocompatibility and colloidal stability of MNPs, but their impact on magnetic anisotropy, relaxation dynamics, and energy dissipation remains only partially understood [12,14]. In addition to surface effects, external factors such as matrix viscosity, interparticle interactions, and the surrounding medium’s composition also affect relaxation mechanisms [4,5]. These variables complicate the predictability of the thermal response, particularly in biologically relevant conditions. Therefore, experimental models that closely mimic tissue-like conditions are critical for reliable hyperthermia optimization.
In magnetic hyperthermia, heat generation primarily results from hysteresis and relaxational losses (Néel and Brownian) [2,7,15]. Although eddy currents may theoretically contribute to heating, their effect is negligible in our study given the well-dispersed nanoparticle system (<15 nm) [15], where the skin depth substantially exceeds particle dimensions, and the operating conditions (100–300 kHz, 1–4 kA/m) are in compliance with the clinical safety limits [16,17]. Hysteresis losses occur strongly in multidomain MNPs above a certain size, where the hysteresis loop area corresponds to the energy dissipated as heat.
In high-viscosity gel-like media simulating biological tissues where particle movement can be inhibited by the gel matrix, the Néel relaxation mechanism predominates, as it does not require particle movement. This is an Arrhenius-type process, driven by thermal energy overcoming the magnetic anisotropy barrier, and is directly related to the known equation:
τ N = τ 0 e x p K V k B T ,
where τ0 is the characteristic time constant, K is the magnetic anisotropy constant, V is the particle volume, kB is the Boltzmann constant, and T is the temperature.
Brownian relaxation, on the other hand, becomes less significant in such systems, as it highly depends on the medium’s viscosity:
    τ B   = 3 η V H   k B T ,
where η is the dynamic viscosity of the surrounding fluid, VH is the hydrodynamic volume of the nanoparticle (including any surface coating or hydration layer), and T is the absolute temperature.
The effective relaxation time combines both mechanisms:
τ e f f = τ B τ N τ B + τ N .
Based on Rosensweig’s linear response theory, the heat generated by magnetic nanoparticles under an oscillating magnetic field can be quantified through the specific loss power (SLP):
S L P   =   µ 0 π f H 2 χ ( ω )  
where µ0 is the permeability of free space, χ″( ω ) is the imaginary part of the dynamic susceptibility, H is the magnetic field strength, and f is the field frequency.
The imaginary part of the dynamic susceptibility can be further calculated as:
χ ( ω ) = χ 0 ω τ e f f 1 + ω τ e f f 2
where τ e f f is the effective relaxation time of a magnetic nanoparticle. According to Rosensweig’s model, the SLP increases quadratically with the applied field amplitude and exhibits a nonlinear dependence on frequency, reaching its peak at the resonant condition specified by:
ω τ e f f = 1 = > > f = 1 / 2 π τ e f f
The advantage of a rotating magnetic field (RMF) in magnetic hyperthermia is theoretically supported by the Raikher model [7,8] which predicts continuous rotation of the magnetization vector without any idle time. In contrast, an alternating magnetic field (AMF) induces brief pauses during the reversal of the field, which interrupts the magnetization process. In comparison, the continuous magnetization dynamics under an RMF allow for uninterrupted relaxation, resulting in increased energy dissipation per cycle through enhanced relaxation-driven losses, thereby improving heating efficiency.
From a theoretical standpoint, this effect is reflected in an increased imaginary component of the dynamic magnetic susceptibility χ″( ω ) part of Equation (4), which directly corresponds to heat loss. Although magnetic susceptibility is often considered a material property, its dynamic, frequency-dependent form captures how efficiently a given material couples with a time-varying field. In RMF conditions, the continuous angular rotation of the magnetic field promotes more effective reorientation of the magnetic moment, enhancing energy transfer from the field to the particles. Mathematically, this is described either by the increased area of the hysteresis loop, ∮M·dH, or by the larger χ″(ω), as predicted by RMF-specific solutions of the Landau–Lifshitz–Gilbert [18] or Fokker–Planck equations [19].
Our research employs a patented magnetic field generator designed for precise control, enhancing the efficiency and safety of potential MH treatments. This novel generator, described in [10], produces a stable RMF using a two-phase magnetic system enclosed within an external magnetic core. The generator’s two-channel control system delivers sinusoidal signals of identical frequency to the power amplifiers, with a 90-degree phase difference between them. These signals originate from the same source, utilizing digital signal synthesis to generate phase-shifted waveforms. As a result, the magnetizing currents in the coils exhibit the same phase shift, leading to the formation of RMF through a magnetic flux superposition. To ensure the RMF intensity vector follows a circular path rather than an elliptical one, it is essential to maintain equal magnetic fluxes in both channels.
This study explores heat generation in agar-based ferrogels, designed to replicate the viscosity of cancerous tissue, using uncoated (Fe3O4) and polydopamine-coated (Fe3O4@PDA) monodomain nanoparticles. Their heating performance is analyzed under the high-frequency alternating (AMF) and rotating (RMF) magnetic fields, with an emphasis on the Néel relaxation mechanism as the dominant mode of energy dissipation. Under these conditions, where Néel relaxation prevails, the influence of viscosity and surface modifications is inherently minimized. This enables a more accurate assessment of Raikher’s predictions regarding the enhanced efficiency of RMF-driven heating.
Examining field type, fluid viscosity, and nanoparticle surface chemistry simultaneously in a single experiment is a combination rarely addressed in earlier studies. This work offers a clear, integrated picture of how these factors jointly govern heating efficiency under biologically relevant conditions.

2. Setups and Experimental Methods

2.1. Description of the System for Generating a Rotating Magnetic Field

Figure 1 illustrates a measurement system designed to generate a high-frequency RMF. The system operates by producing two magnetic fluxes of identical frequency but with a ±90-degree phase shift between them, arranged spatially at a 90-degree angle. As a result of the superposition of magnetic fluxes, a central region is created in which the RMF is established. The amplitude of the magnetic field intensity remains constant in this region, while its direction continuously rotates either clockwise or counterclockwise, depending on the phase difference between the signals. The technical specifications of this system have been detailed in a previous publication [10]. The system allows for the selective deactivation of one of its branches (A or B), enabling the generation of an AMF with the same intensity and frequency, which allows for direct comparisons between AMF and RMF heating efficiency.
The diagram shown in Figure 2 is used to tune both parallel LC branches (A and B) to the same resonant frequency. In circuit A, there is a parallel connected magnetic coil LA and capacitor CA, and similarly, in circuit B, there is a parallel connected magnetic coil LB and capacitor CB. Both branches exhibit maximum impedance at this frequency, and since each is connected in series with a resistor, a voltage minimum will be observed on each oscilloscope channel at resonance. The resonant frequency is adjusted by soldering an additional capacitor, whose capacitance lowers the circuit’s frequency. In practice, despite using the same number of turns for each coil, LA and LB, and capacitors with nominally identical capacitances, frequency differences of ±1% between the two circuits may occur. Therefore, the purpose of this circuit is to equalize these differences.

2.2. Synthesis of Fe3O4@PDA Nanoparticles and Their Characterization

Magnetite nanoparticles (Fe3O4) were synthesized using a chemical co-precipitation technique. Iron(III) chloride hexahydrate (3 g; 11.09 mM) and iron(II) chloride tetrahydrate (1.5 g; 7.55 mM) were dissolved in 100 mL of MilliQ-quality water. Then, the solution was heated up to 90 °C, and then 20 mL of a 25% aqueous ammonia solution was added dropwise. The synthesis of magnetite nanoparticles was carried out in a nitrogen environment to limit oxygen exposure, which could influence the condition of the oxide form of magnetite. After precipitation of iron oxide nanoparticles, the black suspension was cooled down to ambient temperature.
To obtain the magnetite/polydopamine (Fe3O4@PDA) nanoparticles, 100 mg of previously obtained magnetite nanoparticles were added to 200 mL of TRIS buffer solution (10 mM; pH 8.5). Then, dopamine hydrochloride (DA HCl) (100 mg; 0.65 mM) was added in equal weight to Fe3O4, and the mixture was stirred using a magnetic stirrer for 6 h at ambient temperature. Finally, the Fe3O4@PDA nanoparticles were collected using a magnet, washed three times, and suspended in MilliQ water. This suspension was kept in the refrigerator until use.
Zeta potential analysis was performed to determine the stability of materials in a liquid solvent using a Zetasizer Nano ZS (Malvern Panalytical Ltd., Malvern, UK) with a range of 0.6–6000 nm. Additionally, the polydispersity index (PDI) of the nanomaterials was measured to determine the broadness of the molecular weight distribution.
Fourier transform infrared spectroscopy (FT-IR) was used to determine the functional groups present in the structure of the hybrid materials. A Vertex 70 spectrometer (Bruker Optics Co., Ettlingen, Germany) was used to obtain FT-IR spectra. The materials for the analysis were in tablet form, which were made by mixing 2 mg of the material with 250 mg of anhydrous KBr at a pressure of 10 MPa.
Transmission electron microscopy (TEM) images were collected using an ARM 200F (JEOL Co., Ahima, Japan). The MNP size distribution was described by a log-normal function [20]:
f d = 1 2 π d β e x p l n d d 0 2 2 β 2 ,
where d is the nanoparticle diameter, and d 0 and β are the obtained fitting parameters from the log-normal function to the experimental granulometric data. If d represents the mean diameter of the particles given by d = d 0 · exp β 2 2 , the standard deviation value, σ , can be expressed as σ = d · e x p ( β 2 ) 1 2 .

2.3. Sample Preparation Procedure

The preparation of ferrofluid and ferrogel samples was carried out under ambient laboratory conditions using standard equipment for accuracy and reproducibility, following established protocols [21,22].
The base ferrofluid consisted of Fe3O4 and Fe3O4@PDA nanoparticles, both prepared at a 5% mass concentration of MNPs in a water-based suspension. These ferrofluids served as the foundation for ferrogel formation by incorporating agar as a gelling agent.
Individual component masses, including the vials, were measured using a laboratory scale. The ferrofluid–agar mixture was prepared in larger beakers before being portioned into vials. Agar powder (Plate Count Agar, Sigma-Aldrich, St. Louis, MO, USA) was added to the ferrofluid, followed by ultrasonic homogenization for 15 s to ensure uniform nanoparticle dispersion.
The homogenized liquid was then poured into vials and placed in a boiling water bath for controlled gelation, ensuring uniform solidification and preventing phase separation. After cooling to room temperature within minutes, the solidified ferrogels were stored in a refrigerator until further testing.
Each ferrogel sample had an approximate volume of 1.7 cm3, with the agar weight concentration maintained at 5% to replicate the physicochemical properties of soft tissues. The remaining 95% consisted of the ferrofluid, ensuring uniform nanoparticle distribution, with an MNP mass concentration of 50 mg/mL (~5% of the total mass) to optimize heating efficiency. This composition was chosen based on prior research [21,22], which demonstrated its similarity to human soft tissues in acoustic impedance (~1.5 MRayl) [23] and ultrasound attenuation, enabling the estimation of the viscosity modulus using the relationship [23]:
η   =   c S α ϱ c 2 ω 2
where α [Np/m]—the attenuation coefficient, c [m/s]—speed of ultrasonic wave, ϱ —ferrogel density [kg/m3], and c S —dimensionless calibration factor. The ferrogel used in this study has a viscosity of about 0.02 Pa·s, which aligns with reported values for human brain tissue and certain muscle types [24].
The MNP concentration was optimized to achieve a strong calorimetric response while maintaining colloidal stability. Higher concentrations tend to induce dipole–dipole interactions [21,22], leading to particle aggregation and reducing SLP by suppressing Néel relaxation, the dominant heating mechanism in this study. At lower concentrations, nanoparticles remain well-dispersed, maximizing their ability to absorb energy from the applied magnetic field.

3. Characteristics of Tested Magnetic Materials

The results showing the zeta potential and polydispersity of MNPs used are presented in Table 1.
Figure 3 presents the results from FT-IR for both types of MNPs used further to constitute ferrogels.
The results in Figure 3 show that bare magnetite nanoparticles display a strong Fe–O lattice vibration at around 580 cm−1 [25], responsible for the decrease in transmittance at the right edge of the spectrum. Simultaneously, after PDA coating, additional bands emerge at 1500–1300 cm−1, similar to other works in the literature [26,27]. These spectral changes, together with the results in Table 1, confirm successful deposition of a PDA shell on the Fe3O4 core.
An image-centered analysis based on a TEM image of the sample was performed on the results shown in Figure 4. The sizes of MNPs were fitted subsequently to the function yielded at Origin Pro as presented in Figure 5.
The results from TEM and zeta potential measurements shown in Table 1 and Table 2 collectively confirm the impact of PDA coatings on Fe3O4 nanoparticles. All the techniques indicate increased polydispersity and size after coating, with Fe3O4@PDA showing a broader distribution (β = 0.25 vs. 0.19) and higher polydispersity (30.1% vs. 23.6%) compared to bare Fe3O4. TEM reveals a median size increase from 9.9 nm to 10.7 nm, suggesting partial detection of the PDA layer, although the true thickness is likely 2–3 nm due to limited contrast at the organic–inorganic interface. Faint halos seen around the cores in the TEM images further support the presence of the less electron-dense PDA shell. Zeta potential analysis reinforces these findings: bare Fe3O4 nanoparticles exhibit strong electrostatic stability (−46 mV), while Fe3O4@PDA shows reduced charge (−33.6 mV) due to neutralization of surface –OH groups by PDA. However, steric stabilization from the polymeric coating compensates for this, maintaining colloidal stability. Also, the results from our previous works presenting X-ray diffraction (XRD) and X-ray electron spectroscopy (XPS) studies show and confirm the magnetite structure [27,28].

4. Calorimetric Measurements

The heating effect in both RMF and AMF was measured at frequencies of 100, 200, and 300 kHz, with field amplitudes ranging from 1 to 4 kA/m for each material sample, which resulted in over 100 measurements. The amplitude of the magnetic field strength, H, was measured by recording the peak-to-peak voltage (Upp) induced in a probe coil placed in the center of the torus, where the samples were positioned for calorimetric measurement [10]. For each sample, the experiment was first conducted in AMF mode, after which two branches of the system were reactivated to generate the RMF, allowing for a direct comparison.
The MF generator was calibrated by first determining the resonant frequencies of circuits A and B, ensuring that they matched the desired rotational frequencies, fA and fB [5]. The amplitude of the magnetic field strength, H, was measured by recording the peak-to-peak voltage, Upp, induced in the measuring coil placed in the same region as the ferrofluid/ferrogel sample.
The experiment was conducted under non-adiabatic conditions, a common approach in calorimetric measurements due to its simplicity and rapid execution. Although heat loss causes the temperature curve to decline at higher temperatures, this does not affect our results, as the SLP calculation relies only on the initial temperature rise immediately after the magnetic field is applied. The relevant dT/dt ratios were extracted from the graph in the early phase of heating, as shown in Figure 6.
Before turning on the RMF, the sample’s ambient temperature was stabilized using a thermostat placed at the center of the toroidal system, with the sample positioned in the distillation cooler. A circulating water bath maintained the internal environment at room temperature. Prior to each measurement, the sample was left to equilibrate for approximately 10 min to ensure a stable and repeatable thermal baseline. Temperature readings were recorded over 210 s, directly measured using a high-precision optical fiber temperature sensor. The sensor used in the experiment was the FOT-L-SD model from FISO Technology Inc. (Quebec City, QC, Canada), with a measurement range of −40 °C to +300 °C, a response time of less than 1.5 s, an accuracy of ±0.10 °C, and a resolution of 0.01 °C. The temperature data was collected in real time via computer software linked to the probe, ensuring reliable and accurate measurement throughout the experiment.
To evaluate measurement repeatability and assess potential variability, a trial series of experiments was conducted with multiple repetitions. The observed standard deviation was found to be negligible, around ±2%, which is within acceptable error margins for calorimetric studies. Given this low deviation, it was determined that full repetition of each experimental condition was unnecessary, and further measurements were performed only once per sample.
By structuring the experiment so that each sample was first exposed to AMF and then RMF, a direct one-to-one comparison could be made, eliminating variability caused by sample aging or environmental fluctuations.
The relationship between the initial heating rate, dT/d, and magnetic field strength, H, is expected to follow a power-law function, consistent with Rosensweig’s linear response theory [15]:
d T d t = H a n ,
where a and n are numerical parameters obtained from fitting the experimental data.
The value of exponent n, moreover, provides insight into the dominant heat generation mechanism. A value of n ≈ 2 indicates that heating is primarily driven by magnetic relaxation mechanisms (Néel and Brownian relaxation), whereas n ≈ 3 suggests a significant contribution from hysteresis losses. To further distinguish between these contributions, the equation can be rewritten as a sum of two components [29]:
d T d t = H a 2 + H b 3 ,
where the first term represents the relaxational contribution, and the second term corresponds to hysteresis losses. For the ferrogel sample containing polydopamine-coated Fe3O4 nanoparticles, fitting the experimental data yielded the parameters a = 453 and b = 1199, as shown in Figure 7. These values indicate that within the range of magnetic field amplitudes used in the experiment, relaxation dominates the heating mechanism, and the hysteresis contribution remains relatively small. This trend was consistently observed across all measurements, with minor deviations, confirming that magnetic relaxation is the primary heating mechanism in this system, especially at lower magnetic field amplitudes.
However, according to Equation (10), there exists a threshold field—H0 = b3/a2 = 8.4 kA/m—at which both mechanisms contribute equally to the heating. Since such field strengths were not reached in our measurements, the system remained in the relaxation-dominated regime.
Given the magnetic properties of the tested material, the relatively low impact of hysteresis can be attributed to two key factors. First, nanoparticle aggregation was minimized due to the homogenization process, preventing dipole–dipole interactions that typically enhance hysteresis effects. Second, the zeta potential of the Fe3O4 and Fe3O4@PDA were −46.0 ± 0.8 mV and −33.6 ± 0.5 mV, respectively, indicating good electrokinetic stability of the dispersions. Moreover, a polydispersity index (PDI) was evaluated for both materials (see Table 1), presenting low molecular weight distribution (<0.35), which indicates the monodisperse properties of the materials and suggests good surface stability and minimal aggregation. These features further reduce the likelihood of heat dissipation via hysteresis loop losses, which are characteristic of multidomain materials. Finally, the small nanoparticle size ensured that the system remained within the single-domain regime, where Néel relaxation dominates [30,31].

5. Specific Loss Power as a Function of Field Frequency and Amplitude

In magnetic hyperthermia studies, several output parameters are used to assess the efficiency of heat dissipation. One of these is the specific absorption rate (SAR), as defined in the literature [32]:
S A R = c d T d t t = 0 W k g ,
where c [J/kg·K] represents the specific heat capacity of a given sample, and dT/dt [K/s] is the initial heating rate.
The SAR value quantifies the thermal power generated per gram of sample material, with effective heating of biological tissues typically requiring values above 100 mW/g [33,34]. Particularly relevant in this study is the SLP parameter, as it is normalized to the amount of magnetic material present. To evaluate the thermal power emitted by nanoparticles dispersed in the carrier fluid, the SLP is calculated using the following equation [3]:
S L P = c S m S m N P T t = c S c 0 T t W k g ,
where C S is the specific heat capacity of the phantom sample, very similar to water: C S ≅ 4.18 J/g·K. C 0 = m N P / m S ≅ 0.05 is the specific MNPs mass concentration of the examined samples. Note that for 5 wt% solids, SAR ≈ 0.05 × SLP.
Our use of the SLP parameter allows for direct comparison of intrinsic nanoparticle heating efficiency, independent of dispersion medium or polymer matrix dilution.
When comparing hyperthermia effects across different magnetic field frequencies and amplitudes, an alternative metric, known as intrinsic loss power (ILP), is often employed. ILP is defined as [3,35]:
I L P = S L P H 2 f H · m 2 k g ,
where f is the frequency and H2 is the square of the magnetic field amplitude.
Using the dT/dt measurements obtained from the initial phase of magnetic heating, the SLP values were calculated. These SLP results, dependent on the magnetic field amplitude and frequency, are compiled and summarized graphically in Figure 8 and Figure 9. A correlation can be noticed—the higher the field amplitude, the more SLP advantage can be observed with RMF over AMF.
Since SLP is proportional to the initial slope of the temperature curve, dT/dt, it must also follow the same power-law function:
S L P ( f )   H n .
At low field amplitudes where relaxation mechanisms dominate, we expect SLP(f) ∝ H2, meaning energy dissipation increases quadratically with the applied field. At higher fields, where hysteresis contributions become more significant, the dependence shifts toward SLP(f) ∝ H3, as shown exemplarily in Figure 9.
The relationship between the SLP and frequency given an effective relaxation time, τeff, can be expressed as:
S L P ( f ) f · ( τ e f f ) 1 + ( 2 π f τ e f f ) .
This form, derived from Rosensweig’s linear response theory [15], shows that SLP initially increases with frequency but eventually saturates as f becomes large, reaching its maximum when f = 1/2 π τ e f f . The curve interpolation of the data points, obtained at the tested MF frequencies (Figure 8), is used to illustrate the nonlinear trend. The nonlinearity predicted by Rosensweig’s theory arises from the interplay between the energy input and the system’s relaxation capability: at low frequencies, the response is efficient and SLP increases almost linearly, but as the frequency approaches the inverse of the relaxation time, the system can no longer follow the field oscillations efficiently, leading to diminished returns. This behavior is critical for optimizing hyperthermia protocols, as it defines a frequency window where energy dissipation is maximized.
In systems dominated by Néel relaxation, the τeff is primarily determined by nanoparticle magnetic core diameter and anisotropy. Conservative estimates were made by considering magnetic nanoparticle core diameters falling within the range of 11 ± 2 nm using a standard anisotropy value of K = 15,000 J/m3, which is suitable for theoretical estimations in magnetic hyperthermia studies. For such nanoparticles, the average relaxation times, τ B and τ N , satisfy the relationship with respect to the MF rotation period:
T r o t < < τ N < < τ B ,
where Trot is the field rotation period, meaning that Néel relaxation occurs within the experimental timescale, while Brownian relaxation is significantly slower and negligible in this system. As shown in Figure 10, the estimated τ N and τ B curves, plotted as a function of MNP size, intersect at a particle diameter of over ~20 nm, which is well above the size range of our synthesized nanoparticles. For the estimation of Brownian relaxation time in gel matrices, we assumed dynamic viscosities between 0.02 and 0.2 Pa·s, which are typically observed [23] The plot shows that, regardless of how conservative a viscosity estimate is applied, τ B stays several orders of magnitude longer than τ N . In particular, 11 nm nanoparticles correspond to a resonance frequency of approximately 2 MHz, significantly above the tested frequency range (100–300 kHz).
Following on from Equation (15) and illustrated in Figure 8, the experimental SLP(f) relationship is notably nonlinear, and the peak that was expected at the resonance condition, ω τ e f f = 1 , was not observed. This confirms that the actual relaxation times are far shorter than the inverse of the applied field frequencies, meaning that the nanoparticles respond almost instantaneously to the field without significant lag. As a result, the system does not exhibit energy dissipation due to strong lagging effects, and relaxational losses remain limited.
A key finding here is that the SLP and heating efficiency in RMF were approximately twice as high as in AMF, as illustrated in Figure 6 and Figure 7. This enhancement was observed consistently across all tested ferrogel samples. The improved heating efficiency in RMF is attributed to the continuous rotation of the magnetic moment, which maintains energy absorption throughout the field cycle. In contrast, in AMF, energy dissipation is limited by alignment–realignment dynamics that reset with each field oscillation.

6. Discussion

To evaluate the obtained SLP results across frequency f and magnetic field strength H, and to identify optimal conditions, the ILP parameter (Equation (13)) can be utilized in conjunction with the Brezovich criterion [36,37]:
  H × f 4.85 × 10 8   A / m .
By calculating ILP as a function of the H × f product within this safety limit, it was determined from our experimental data that maximum heating efficiency occurs at f = 101 kHz and H = 4.1 kA/m. The corresponding SLP results under these conditions (Figure 11) highlight distinct trends related to material composition, phase (fluid vs. gel), and field type (RMF vs. AMF).
The experimental data reveal consistent trends in how gel viscosity and PDA surface modification influence heating efficiency. Ferrogels exhibit lower SLP than their aqueous counterparts, indicating that increased viscosity suppresses Brownian relaxation, even though Néel relaxation remains dominant. PDA coating, while enhancing colloidal stability and biocompatibility (essential for biomedical use), adds non-magnetic mass, resulting in a reduction in SLP by less than 17% in gel phase compared to uncoated nanoparticles. If normalized to magnetic material only, the obtained values would show an increase, reflecting the added non-magnetic mass of the PDA coating. This conservative normalization approach aligns with recommendations in the recent literature on SLP estimation and provides a realistic lower-bound estimate for heat generation efficiency [32].
The SLP discrepancy between gel and aqueous phases becomes more pronounced for the PDA-coated samples. This suggests that the PDA layer may modify the local microenvironment of the MNPs, potentially affecting magnetic relaxation dynamics. A plausible explanation is that PDA enhances interparticle interactions within the denser gel matrix, leading to increased dipolar coupling, which can hinder the efficient reorientation of magnetic moments under an external magnetic field.
RMF consistently outperforms AMF across all the tested conditions, achieving an average RMF/AMF SLP enhancement ratio of approximately 1.91 with minimal variability. Despite viscosity-induced partial suppression of Brownian relaxation in gels, Néel relaxation dominates for small, single-domain nanoparticles.
Among all the obtained data, we selected SLP values under RMF (4.51–5.58 W/g) that comply with Brezovich safety limits (at f = 101 kHz and H = 4.1 kA/m), including those for PDA-coated nanoparticles in gels, as representative and clinically relevant for comparison. Notably, in a separate set of measurements conducted at f = 303 kHz and H = 4.1 kA/m, we observed SLP values reaching up to 14 W/g, with a total temperature increase of approximately 16 °C over the course of the experiment (see Figure 2). This result highlights the system’s strong heating performance, especially considering the relatively low field amplitude applied. Assuming a 5 wt% concentration of magnetic material in all the samples, the selected SLP values correspond to effective heating rates consistent with, or even exceeding, those reported in the literature as therapeutically relevant under clinically acceptable field conditions [6,37].
TEM analysis reveals a narrower size distribution for bare Fe3O4 nanoparticles (σ = 2.5 nm) compared to Fe3O4@PDA (σ = 3.4 nm), as shown in Figure 5. Although both systems are smaller than the critical single-domain size (~20–30 nm for Fe3O4), the tighter size distribution of bare particles ensures a higher fraction of nanoparticles operate near the Néel relaxation optimum. In contrast, the broader polydispersity of PDA-coated particles introduces suboptimal sizes, diminishing the average SLP.
Although detailed in vitro biocompatibility tests were beyond the scope of this study, polydopamine is widely recognized in the literature as a biocompatible coating [12,14]. The PDA coating slightly increases polydispersity (from ~23% to ~30%) due to the added polymer shell, and the resulting values remain within the acceptable range for biomedical nanoparticles [30,31]. This increase represents a reasonable trade-off between magnetic uniformity and the surface functionalization necessary for biological compatibility. As shown in Table 2, the PDA-coated nanoparticles maintain colloidal stability (zeta potential ~ −33.6 mV) and benefit from steric stabilization.
While the nanoparticle relaxation times remain significantly shorter than the period of the applied fields at the tested frequencies, increasing the frequency beyond approximately 200 kHz yields fewer improvements in SLP. Conversely, raising the magnetic field amplitude (H) continues to enhance heating performance, suggesting that no magnetic saturation is expressed by the condition:
H = 6 ω η V H / M S V m ,
where η [N·s/m2] is the shear viscosity of the carrier liquid, MS [A/m] is the saturation magnetization, Vm [m3] is the volume of the magnetic core, and VH [m3] is the hydrodynamical volume of the MNP. Consequently, optimizing magnetic field amplitude rather than frequency appears more practical and effective in increasing SLP while maximizing ILP.
While our earlier studies primarily addressed field generation [10] or the behavior of uncoated ferrogels under RMF [22], the current work introduces a comparative evaluation of PDA-coated and uncoated nanoparticles under both AMF and RMF. This is performed in aqueous gel matrices that approximate physiologically relevant environments and is supported by relaxation-based modeling. Together, these aspects extend our previous findings into more application-focused contexts.
Despite its strengths, this study remains limited to in vitro conditions and does not account for complex physiological factors such as blood perfusion, immune response, or nanoparticle biodistribution. The use of agar as a tissue-mimicking material, while representative in terms of viscosity and thermal conduction, cannot fully replicate the heterogeneous and dynamic nature of living tissues. However, this controlled simplification enables precise isolation of key variables—medium viscosity, nanoparticle concentration, and surface modifications—providing quantitative benchmarks for systematic complexity scaling. By establishing that neither gel viscosity nor PDA coating significantly impacts the 1.7–2.1-fold RMF enhancement, we offer clear reference metrics for future biological studies. Based on these limitations and methodological advantages, future research should focus on validating the observed RMF-driven heating effects in more complex biological models, including time-resolved imaging, dynamic flow systems, and eventually clinical studies.

7. Conclusions

This study investigates Fe3O4 nanoparticles (uncoated and PDA-coated) under magnetic fields of H = 1–4 kA/m and f = 100–300 kHz, examining how the phase (water vs. gel) and surface modification influence heating efficiency. Agar-based ferrogels served as biomimetic matrices, modeling semi-solid tumor environments for improved in vitro–in vivo relevance.
The experimental results show that rotating magnetic field (RMF) consistently outperforms alternating magnetic field (AMF), producing up to twice the SLP under all the tested conditions. This is attributed to the continuous rotation of magnetic moments in RMF, enabling more efficient energy dissipation compared to the oscillatory behavior in AMF. As a result, lower nanoparticle concentrations can achieve therapeutic heating, enhancing clinical potential while reducing toxicity risks.
The SLP values (4.51–5.58 W/g) remain within Brezovich limits and exceed the 100 mW/g threshold for effective hyperthermia. Notably, SLP shows a non-linear dependence on frequency, with sharp increases between 100 and 200 kHz but diminishing returns beyond this range. In contrast, increasing magnetic field amplitude (H) consistently improves heating performance, indicating that saturation has not yet been reached and that H-optimization may be more impactful than further frequency increases. While PDA surface modifications reduce heating performance by 12–17% (depending on the phase) due to added non-magnetic mass and interparticle effects, they enhance biocompatibility and dispersion, supporting their application in biomedical contexts where long-term stability is essential.
Building on our earlier studies, this work offers a direct comparison of coated and uncoated nanoparticles in both fluid and gel matrices under RMF and AMF, as the current literature lacks such experimental comparisons. Novel contributions also include the direct demonstration of RMF advantages in biologically mimicking environments, quantitative assessment of SLP enhancement under RMF, and identification of critical parameters for heating optimization.
Future research should explore how variations in nanoparticle size, anisotropy, PDA coating properties, and interparticle spacing influence heating performance. Additionally, computational modeling could provide predictive insights into optimal parameter selection before experimental validation. Further in vivo studies are also necessary to bridge the gap between in vitro findings and clinical applications, refining RMF-based hyperthermia for safer and more effective therapeutic outcomes.

Author Contributions

Conceptualization, J.M.; methodology, J.M. and A.S.; validation, J.M. and A.S.; formal analysis, J.M. and A.S.; investigation, J.M.; resources, A.S. and A.J.; writing—original draft preparation, J.M.; writing—review and editing, R.B. and A.S.; visualization, J.M.; supervision, A.S. and R.B. All authors have read and agreed to the published version of the manuscript.

Funding

The work was partially financed by the Poznan University of Technology research grant no. 0912/SBAD/2506.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Acknowledgments

The authors gratefully acknowledge Grzegorz Nowaczyk from the NanoBioMedical Centre at Adam Mickiewicz University in Poznań for acquiring the TEM images used in this study.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Schematic diagram of the measurement setup for generating RMF in a 2-phase system powered by sinusoidal signals [10]. A and B correspond to two resonant branches of the magnetic circuit.
Figure 1. Schematic diagram of the measurement setup for generating RMF in a 2-phase system powered by sinusoidal signals [10]. A and B correspond to two resonant branches of the magnetic circuit.
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Figure 2. Electrical circuit for tuning both parallel LC branches to the same resonant frequency. A and B correspond two resonant branches of the magnetic circuit.
Figure 2. Electrical circuit for tuning both parallel LC branches to the same resonant frequency. A and B correspond two resonant branches of the magnetic circuit.
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Figure 3. FT-IR spectra for bionanomaterials: Fe3O4 and Fe3O4@PDA.
Figure 3. FT-IR spectra for bionanomaterials: Fe3O4 and Fe3O4@PDA.
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Figure 4. TEM images of (a) uncoated Fe3O4 and (b) particles coated with polydopamine (Fe3O4@PDA).
Figure 4. TEM images of (a) uncoated Fe3O4 and (b) particles coated with polydopamine (Fe3O4@PDA).
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Figure 5. Log-normal distribution fitted to granulometric data obtained from TEM images using the fitting function in Equation (7).
Figure 5. Log-normal distribution fitted to granulometric data obtained from TEM images using the fitting function in Equation (7).
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Figure 6. Temperature increase over time at f = 303 kHz and H = 4.1 kA/m for RMF (red) and AMF (blue), with the dT/dt dotted lines corresponding to the initial phase of heating.
Figure 6. Temperature increase over time at f = 303 kHz and H = 4.1 kA/m for RMF (red) and AMF (blue), with the dT/dt dotted lines corresponding to the initial phase of heating.
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Figure 7. Contribution of relaxation (red) and hysteresis (blue) losses to the total release of thermal energy as a function of the magnetic field strength.
Figure 7. Contribution of relaxation (red) and hysteresis (blue) losses to the total release of thermal energy as a function of the magnetic field strength.
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Figure 8. SLP as a function of MF frequency at the highest obtained MF amplitude of H = 4.1 kA/m. The lines are added as eye guidelines.
Figure 8. SLP as a function of MF frequency at the highest obtained MF amplitude of H = 4.1 kA/m. The lines are added as eye guidelines.
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Figure 9. SLP as a function of MF amplitude at the highest obtained MF frequency of 303 kHz.
Figure 9. SLP as a function of MF amplitude at the highest obtained MF frequency of 303 kHz.
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Figure 10. Calculation of relaxation times τ N and τ B for a tested ferrogel in correspondence with the magnetic field rotation cycle, Trot, with dynamic viscosity ηₛ within the range of 0.02–0.20 Pa·s.
Figure 10. Calculation of relaxation times τ N and τ B for a tested ferrogel in correspondence with the magnetic field rotation cycle, Trot, with dynamic viscosity ηₛ within the range of 0.02–0.20 Pa·s.
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Figure 11. Dependence of specific loss power (SLP) on material composition (polydopamine-coated vs. uncoated Fe3O4) and phase (fluid vs. gel), measured at maximum intrinsic loss power (ILP). The RMF/AMF heating efficiency ratio is quantified as μ = 1.91 ± 0.006 (mean ± standard deviation).
Figure 11. Dependence of specific loss power (SLP) on material composition (polydopamine-coated vs. uncoated Fe3O4) and phase (fluid vs. gel), measured at maximum intrinsic loss power (ILP). The RMF/AMF heating efficiency ratio is quantified as μ = 1.91 ± 0.006 (mean ± standard deviation).
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Table 1. Results of the zeta potential and polydispersity index (PDI) for magnetite nanomaterials (Fe3O4) and polydopamine-coated magnetite (Fe3O4–PDA).
Table 1. Results of the zeta potential and polydispersity index (PDI) for magnetite nanomaterials (Fe3O4) and polydopamine-coated magnetite (Fe3O4–PDA).
SampleZeta Potential (mV)PDI
Fe3O4−46.0 ± 0.80.262 ± 0.012
Fe3O4@PDA−33.6 ± 0.50.338 ± 0.026
Table 2. Nanoparticle size distribution parameters from granulometric analysis of Fe3O4 and Fe3O4@PDA samples.
Table 2. Nanoparticle size distribution parameters from granulometric analysis of Fe3O4 and Fe3O4@PDA samples.
Parameter/SampleFe3O4Fe3O4@PDA
Median (d0)9.9 ± 0.3 nm10.7 ± 0.4 nm
Mean (d)10.4 ± 0.3 nm11.3 ± 0.3 nm
Standard Deviation (σ)2.5 ± 0.2 nm3.4 ± 0.3 nm
Polydispersity (σ/dmean)23.60 ± 2.0%30.1 ± 2.5%
Shape Parameter (β) 0.19 ± 0.02 0.25 ± 0.03
Note: Uncertainties represent 95% confidence intervals derived from log-normal distribution fitting to n = 100 particles per sample.
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Musiał, J.; Jędrzak, A.; Bielas, R.; Skumiel, A. Magnetothermal Energy Conversion of Polydopamine-Coated Iron Oxide Ferrogels Under High-Frequency Rotating Magnetic Fields. Energies 2025, 18, 4291. https://doi.org/10.3390/en18164291

AMA Style

Musiał J, Jędrzak A, Bielas R, Skumiel A. Magnetothermal Energy Conversion of Polydopamine-Coated Iron Oxide Ferrogels Under High-Frequency Rotating Magnetic Fields. Energies. 2025; 18(16):4291. https://doi.org/10.3390/en18164291

Chicago/Turabian Style

Musiał, Jakub, Artur Jędrzak, Rafał Bielas, and Andrzej Skumiel. 2025. "Magnetothermal Energy Conversion of Polydopamine-Coated Iron Oxide Ferrogels Under High-Frequency Rotating Magnetic Fields" Energies 18, no. 16: 4291. https://doi.org/10.3390/en18164291

APA Style

Musiał, J., Jędrzak, A., Bielas, R., & Skumiel, A. (2025). Magnetothermal Energy Conversion of Polydopamine-Coated Iron Oxide Ferrogels Under High-Frequency Rotating Magnetic Fields. Energies, 18(16), 4291. https://doi.org/10.3390/en18164291

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