Influence of the Polarizing Magnetic Field and Volume Fraction of Nanoparticles in a Ferrofluid on the Specific Absorption Rate (SAR) in the Microwave Range

Abstract

For the study, we used four kerosene-based ferrofluid samples containing magnetite nanoparticles stabilized with oleic acid. Starting from the initial sample (A0), the other three samples were obtained by dilution with kerosene. The complex magnetic permeability measurements were performed in the microwave region (0.5–6) GHz, for different H values of the polarizing magnetic field, between (0–115) kA/m. These measurements revealed the ferromagnetic resonance phenomenon for each sample, allowing the determination of the anisotropy field (H_A) and the effective anisotropy constant (K_eff) of nanoparticles, depending on the volume fraction of particles (φ). At the same time, the measurements allowed the determination of the specific magnetic loss power (p_m), effective heating rate (HR_eff), intrinsic loss power (ILP), and specific absorption rate (SAR) as functions of the frequency (f) and magnetic field (H), of all investigated samples, using newly proposed equations for their calculation. For the first time, this study evaluates the maximum limit of the applied polarizing magnetic field (H_max ≈ 80 kA/m) and the minimum limit volume fraction of nanoparticles (φ_min ≈ 3.5%) at which microwave heating of the ferrofluid remains efficient. At the same time, the results obtained show that the temperature increase of the ferrofluid samples, upon interaction with a microwave field, can be controlled by varying both H and φ, pointing to possible applications in magnetic hyperthermia.

1. Introduction

In the last years, it has been found that the development of biomedicine is based on nanotechnology, which represents one of the important and promising approaches for the diagnosis and treatment of cancer [1,2]. In this regard, researchers have been increasingly interested in obtaining magnetic nanoparticles [3,4], intending to use them in both technological and biomedical applications [5,6,7]. One of the medical applications that has begun to be studied and developed is magnetic hyperthermia, which leads to local heating of cancerous tumor cells or tissue while leaving the surrounding healthy tissue unharmed [8]. In general, it is based on the use of magnetic nanoparticles to generate heat when exposed to an external electromagnetic field [3,9]. In magnetic hyperthermia therapy, the most widely used magnetic nanoparticles are iron oxide nanoparticles, such as magnetite (Fe₃O₄) [9]. These present positive potential in this field due to their unique magnetic and physical properties, highlighted by the absorption of electromagnetic radiation [9,10] expressed by the frequency dependence of the imaginary component (μ″) of the complex magnetic permeability [11].

Several studies show that in the case of magnetic nanoparticles placed in a radiofrequency magnetic field, heat generation can be due to the following two mechanisms: (i) hysteresis losses, which occur in large magnetic nanoparticles with sizes over 100 nm, due to the displacement of the Weiss domain walls [12]; (ii) Brownian and Neel relaxation losses, respectively, for superparamagnetic particles (with sizes below 15 nm), in which the magnetic hysteresis is canceled [13,14], and the heating is due to the magnetic moment of the particle. In Brownian relaxation, the particle rotates as a whole, and in Neel relaxation, the magnetic moment rotates with respect to the particle considered fixed in the crystal [12,13,14].

In addition to the study and obtaining of nanoparticles with the aim of being used in various thermal applications, the researchers also focused on obtaining composite materials by dispersing nano-/microparticles in either a solid or liquid matrix, intending to use these composites in practical applications [15,16]. One such composite material, which has been intensively studied, is ferrofluid, which is a colloidal system of single-domain magnetic nanoparticles dispersed in a carrier liquid [17], which can be used in biomedical applications [18]. In the case of the composite system, such as a ferrofluid placed in a microwave field, the generation of heat in the ferrofluid is due to another mechanism, the phenomenon of ferromagnetic resonance. This phenomenon can be highlighted for a ferrofluid by performing measurements of the complex magnetic permeability (μ(f)) in the microwave frequency range [19]. The energy dissipated in the form of heat will be taken up by the composite system (ferrofluid), meaning both nanoparticles and the dispersion medium. In view of this, in our previous papers [20,21], we demonstrated that the effective heating rate of the ferrofluid (HR_eff) and the effective specific absorption rate (SAR) due to interaction of ferrofluid with the microwave field can be computed with the following relations:

$$H{R}_{eff}={\displaystyle \frac{\Delta T}{\Delta t}}={\displaystyle \frac{p_m}{(1-\phi ){\rho }_L{c}_L+\phi {\rho }_S{c}_S}}$$

(1)

$$SAR={\displaystyle \frac{p_m}{(1-\phi ){\rho }_L+\phi {\rho }_S}}={\displaystyle \frac{p_m}{{\rho }_F}}$$

(2)

From Equations (1) and (2), the following relationship results for determining the effective heating rate (HR_eff) of the ferrofluid:

$$H{R}_{eff}={\displaystyle \frac{\Delta T}{\Delta t}}={\displaystyle \frac{{\rho }_F\hspace{0.17em}SAR}{(1-\phi ){\rho }_L{c}_L+\phi {\rho }_S{c}_S}}$$

(3)

In these relations, ΔT represents the temperature variation over the heating time interval (Δt); c_L, ρ_L and c_S, ρ_S are the specific heat and the density of the carrier liquid, and of the solid fraction from ferrofluid, respectively; ρ_F is the density of ferrofluid; φ is the volume fraction of nanoparticles from ferrofluid; and p_m, representing the specific magnetic power loss of the ferrofluid, can be calculated with the following relation [20]:

$$p_m=\pi {\mu }_0{\mu }^{″}f{H}_0^2$$

(4)

where f is the frequency, H₀ is the amplitude of the variable magnetic field, and μ₀ is the magnetic permeability of free space.

For the calculation of both the effective heating rate of the ferrofluid (HR_eff) and the effective specific absorption rate (SAR), using the newly proposed Equations (1)–(3), we used the following values for the specific heat and density corresponding to both the solid phase and the liquid phase: the specific heat of the carrier liquid (kerosene), c_L = 2010 J/kgK, and the density of the kerosene, ρ_L = 800 kg/m³ [22]; the specific heat of the solid fraction (magnetite), c_S = 670 J/kgK [23], and the density of the magnetite, ρ_S = 5080 kg/m³ [4].

In this paper, complex magnetic permeability measurements were performed in the microwave frequency range (0.5–6 GHz) for four ferrofluid samples with different volume fractions (φ) of nanoparticles, and at various values of the applied polarizing magnetic field (H). Based on the results obtained from these measurements, and using the newly proposed Equations (1)–(3) for a biphasic composite system, this study aimed to determine the effective heating rate (HR_eff) and the specific absorption rate (SAR) of the investigated ferrofluid samples.

Furthermore, this study examines the dependence of HR_eff and SAR on the frequency (f), the applied polarizing magnetic field (H), the nanoparticle volume fraction (φ), and the effective anisotropy constant (K_eff) of the nanoparticles, within the microwave range. The results allowed, for the first time, the determination of the maximum applied polarizing field (H_max) and the minimum nanoparticle volume fraction (φ_min) in the ferrofluid, at which microwave heating efficiency is significant.

2. Samples Characterization and Experimental

The initial ferrofluid used for measurements (denoted as A0) was obtained by a method called the hydrophobization technique in the absence of a dispersion medium [24,25], the nanoparticles being coated with a layer of organic molecules (oleic acid) that act as a stabilizer. Next, the resulting material was dispersed in the carrier liquid (kerosene), heated slightly to remove any traces of water, and then filtered in a magnetic field gradient to remove coarse particles from the ferrofluid [25]. From the initial sample A0, three other samples were obtained by dilution with kerosene and were involved in the measurements. The density of samples was as follows: ρ_F(A0) = 1300 kg/m³, ρ_F(A1) = 1137 kg/m³, ρ_F(A3) = 950 kg/m³; and ρ_F(A5) = 867 kg/m³. It is known that the density ρ_F of a ferrofluid can be written in the following form [19]:

$${\rho }_F=(1-\phi ){\rho }_L+\phi {\rho }_S$$

(5)

where ρ_L is the density of the carrier liquid (kerosene), and ρ_S is the density of the solid fraction of the ferrofluid (magnetite). With Equation (5), the volume fraction (φ) of the magnetite nanoparticles, corresponding to each ferrofluid sample, was calculated. The following values were obtained: φ(A0) = 11.52%; φ(A1) = 7.68%; φ(A3) = 3.41%; and φ(A5) = 1.53%.

In Figure 1, the static magnetization curve (M(H)) of the initial ferrofluid sample (sample A0) is presented, obtained using the hysteresis graph method, at a low frequency field (50 Hz) [26].

As can be seen in Figure 1, the M(H) dependence of the A0 sample obeys a Langevin-type law, which indicates that the initial A0 sample has a superparamagnetic behavior [17]. Considering the magneto-granulometry analysis of Chantrell [27] and the M(H) dependence from Figure 1, the saturation magnetization (M_sat), the mean magnetic diameter of particles (d_m), the particle concentration (n), and the initial susceptibility (χ_in) were determined, and the obtained values are M_sat = 20.84 kA/m; d_m = 10.47 nm; n = 7.26 × 10²² m⁻³; and χ_in = 0.904.

It is known that [17] the saturation magnetization of a ferrofluid (M_sat) and the spontaneous magnetization of the magnetite nanoparticles (M_S) are correlated by the following relationship:

$$M_{sat}={\phi }_m{M}_S$$

(6)

In Equation (6), φ_m represents the magnetic volume fraction of magnetite nanoparticles in the ferrofluid, which is correlated with the volume fraction (φ) by the following relationship [17]:

$${\phi }_m={\displaystyle \frac{\phi }{{\left(1+\frac{2\delta }{d_m}\right)}^3}}$$

(7)

where δ ≈ 2 nm is the thickness of the nonmagnetic layer (surfactant) [17] and d_m = 10.47 nm is the mean magnetic diameter of magnetite nanoparticle, from samples. Using Equation (7) and the obtained values for the volume fraction (φ) from ferrofluid samples, the following values of the magnetic volume fraction (φ)_m were obtained: φ_m(A0) = 4.36%; φ_m(A1) = 2.91%; φ_m(A3) = 1.29%; and φ_m(A5) = 0.58%.

The measurements of the real μ’(f,H) and imaginary μ”(f, H) components of the complex magnetic permeability of all ferrofluid samples were made with the short-circuited (SC) coaxial line method [21], over the frequency range (0.5–6) GHz and at values of H between 0 to 115 kA/m. For the measurements, the ferrofluid sample was loaded into a Hewlett–Packard (HP) 50 Ω coaxial cell in conjunction with an HP 8753C network analyzer [21,28]. The coaxial cell containing the ferrofluid was positioned between the poles of an electromagnet, ensuring that the applied polarizing magnetic field was oriented perpendicular to the cell axis.

3. Results and Discussions

3.1. Complex Magnetic Permeability in Microwave Range

In Figure 2a–d, the frequency dependencies of the real (μ’) and imaginary (μ”) components of complex magnetic permeability are shown, for different values of the applied polarizing magnetic field (H), and different values of the volume fraction (φ) of the nanoparticles from the ferrofluid samples.

From Figure 2, it is observed that for all ferrofluid samples, the phenomenon of ferromagnetic resonance is present, highlighted by the transition of the real component (μ′) from a value above unity to a subunit value at a frequency (f_res) called the resonance frequency [28]. By increasing the polarizing magnetic field H, f_res shifts towards higher values for all samples. At the same time, by decreasing the volume fraction (φ), the amplitude of the real component (μ′) decreases, becoming gradually smaller with the increase in H for all samples. The imaginary component (μ″) has a maximum for all samples, at a frequency f_max which moves towards higher values when H increases. The amplitude of the component (μ″) decreases as the volume fraction (φ) of the nanoparticles in the samples decreases (see Figure 2).

It is known that [21,29] in a strong polarizing magnetic field (H >> H_A), where H_A is the anisotropy field, the resonance condition can be written as follows [21]:

$$2\pi f_{res}=\gamma \left(H+H_A\right)$$

(8)

which shows a linear dependence between f_res and H, whose slope is γ, called the gyromagnetic ratio of the particle [21,30]. Considering the f_res values corresponding to each H value from Figure 2, of all samples with different volume fractions (φ), in Figure 3, the dependence of f_res(H) is presented, which is linear, in accordance with Equation (8).

Fitting the experimental dependencies f_res(H) from Figure 3 with a straight line and considering Equation (8), the slope of the line, γ (the gyromagnetic ratio of the particle), and the anisotropy field, H_A, respectively, immediately result, and the obtained values are indicated in

Table 1. Magnetic parameters determined from ferromagnetic resonance measurements.

Samples	HA (kA/m)	γ (s−1A−1m)	Keff (J/m3)	g
A0 (φ = 11.52%)	47.92	2.27 × 105	1.44 × 104	2.04
A1 (φ = 7.68%)	45.00	2.28 × 105	1.35 × 104	2.05
A3 (φ = 3.41%)	39.75	2.20 × 105	1.19 × 104	2.00
A5 (φ = 1.53%)	35.54	2.24 × 105	1.07 × 104	2.00

.

Considering that for single-domain nanoparticles with uniaxial anisotropy [30,31], the anisotropy field is given by the following relation:

$$H_A={\displaystyle \frac{2K_{eff}}{{\mu }_0{M}_S}}$$

(9)

we were able to determine the effective anisotropy constant (K_eff) of the nanoparticles in the ferrofluid samples. The values obtained are shown in Table 1.

As can be seen in Table 1, by decreasing the volume fraction (φ) of the magnetite nanoparticles in the ferrofluid samples from 11.52% (sample A0) to 1.53% (sample A5), the effective anisotropy constant (K_eff) decreases from the value of 1.44 × 10⁴ J/m³ to 1.07 × 10⁴ J/m³. The observed decrease in the effective anisotropy constant (K_eff) with decreasing particle concentration is attributed to a reduction in interparticle magnetic interactions, caused by the increased interparticle distance upon dilution of the ferrofluid samples. This behavior agrees with both the theoretical models [32,33] and the experimental data [34,35], which demonstrate that the large value of the effective anisotropy constant K_eff is due to the increase in the concentration of nanoparticles in the ferrofluid (i.e., of the large volume fraction). From Table 1, it is also observed that in the case of samples A0 and A1, having the volume fraction larger than that of samples A3 and A5, the experimental value of K_eff is in agreement with the value obtained by other authors [31,36,37] for similar samples, being larger than the value of the anisotropy constant measured for a magnetite crystal, K = 1.1 × 10⁴ J/m³ [38]. The discrepancy between the K value measured for bulk magnetite crystals and the experimentally determined K_eff values for magnetite nanoparticles in ferrofluids arises from the fact that K_eff represents an effective anisotropy that accounts for shape, interparticle interactions, surface effects, and magnetocrystalline contributions.

As is known, the generation of heat in a ferrofluid placed in a variable magnetic field can be due to either the magnetic relaxation mechanism or ferromagnetic resonance [21,39]. It can be evaluated by measurements of the complex magnetic permeability (μ(f)) over a wide range of frequencies. At low frequencies (tens, hundreds of kHz), the rotation of the particles in the carrier fluid determines the Brownian relaxation process [13], and at higher frequencies (units, tens of MHz), the rotation of the magnetic moment inside the particles [14,39] determines the Néel relaxation process. These relaxation processes are characterized by the corresponding relaxation times, τ_B and τ_N, given by the following relations:

$${\tau }_B={\displaystyle \frac{\pi \eta D^3}{kT}}$$

(10)

$${\tau }_N={\tau }_0\exp \left({\displaystyle \frac{K_{eff}{V}_m}{kT}}\right)$$

(11)

In these relationships, η represents the viscosity of the carrier liquid (kerosene) having the value η = 1.2 × 10⁻³ Pa·s; D = d_m + 2δ [17] is the hydrodynamic diameter of the particles, resulting in the value D = 14.47 nm (for our samples); K_eff is the effective anisotropy constant of the particles from the ferrofluid samples (see Table 1); V_m = π(d_m)³/6 is the mean magnetic volume of the particles, considered spherical; k = 1.38 × 10⁻²³ J/K is the Boltzmann constant; T = 298 K is the room temperature at which the measurements were performed and τ₀ = 10⁻⁹ s is a constant [17]. Considering these values of the parameters involved in the mathematical expressions (10) and (11) corresponding to the Brownian (τ_B) and Neel (τ_N) relaxation times, respectively, we were able to evaluate the value of these relaxation times for the investigated ferrofluid samples, for the mean magnetic diameter of the particles, d_m = 10.47 nm. The following values were obtained: τ_B = 1.383 μs, τ_N = 8.164 ns (for sample A0), τ_N = 7.169 ns (for sample A1), τ_N = 5.684 ns (for sample A3), and τ_N = 4.767 ns (for sample A5). Considering that, between the relaxation time τ and the frequency f_max at which the imaginary component (μ″) has a maximum, the Debye equation [39] is valid, 2πf_maxτ = 1, and we determined the f_max frequencies corresponding to both Brownian and Neel relaxation peaks. The following values were obtained: f_max(Brownian) = 115.1 kHz; f_max(Neel) = 19.5 MHz (for sample A0) and f_max(Neel) = 33.4 MHz (for sample A5), being much lower than the frequencies in the microwave range (0.5–6) GHz, where the phenomenon of ferromagnetic resonance was highlighted. This result confirms that Brownian and Neel relaxation losses are negligible in the microwave range compared to the ferromagnetic resonance loss for the investigated ferrofluid samples.

The gyromagnetic ratio of the particle (γ), corresponding to samples (see Table 1) in a strong polarizing field (i.e., H ≫ H_A) [40], is given by the following relation:

$$\gamma =g{\gamma }_e{\mu }_0{\left({\displaystyle \frac{2}{1+{\left(f_{\max }/f_{res}\right)}^2}}\right)}^{1/2}$$

(12)

where μ₀ = 4π × 10⁻⁷ H/m is the magnetic permeability of free space; γ_e = 8.791 × 10¹⁰ s⁻¹T⁻¹ is the gyromagnetic ratio of the electron, and g is the spectroscopic splitting factor [21,31].

The ratio (f_max/f_res) corresponding to high values of the polarizing magnetic field (H >> H_A) was determined from Figure 4, in which we presented the dependence of the f_max/f_res ratio on the magnetic field (H), for the four ferrofluid samples with different volume fractions (φ) of nanoparticles.

From Figure 4, it can be seen that for a high polarizing field (H >> H_A), the f_max/f_res ratio tends to become constant, for all the investigated samples, resulting in the following values: 0.988 for sample A0 (with φ = 11.52%); 0.986 for sample A1 (with φ = 7.68%); 0.975 for sample A3 (with φ = 3.41%); and 0.973 for sample A5 (with φ = 1.53%). Considering these values of the f_max/f_res ratio for samples, using Equation (10), the spectroscopic splitting factor (g) was determined, and the obtained values are listed in Table 1. It is observed that deviations from the value 2 of g appear in the case of samples A0 and A1, whose volume fraction (φ) is higher (samples with higher concentrations of nanoparticles), which may be due to interactions between particles, or the effects of shape, surface, and magnetocrystalline anisotropy, in the case of these concentrated ferrofluid samples [34,37].

3.2. Specific Absorption Rate (SAR) in Microwave Range

It is known that through the interaction between the microwave field and the ferrofluid, energy absorption occurs, leading to heating of the ferrofluid [41]. In magnetic hyperthermia, the following equation is used to calculate the heating rate (HR):

$$HR={\displaystyle \frac{\Delta T}{\Delta t}}={\displaystyle \frac{p_m}{{\rho }_F{c}_F}}$$

(13)

where ρ_F represents the density of the ferrofluid, and c_F the specific heat of the ferrofluid [41,42,43]. In evaluating HR using Equation (13), many authors generally use the specific heat of the carrier liquid (c_L) [41,43], so Equation (13) becomes the following:

$$H{R}_{clasic}={\displaystyle \frac{p_m}{{\rho }_F\cdot c_L}}={\displaystyle \frac{p_m}{[(1-\phi ){\rho }_L+\phi {\rho }_S]c_L}}$$

(14)

It can be said that Equation (14) represents the usual equation used by many authors for calculating the heating rate HR, where it was considered that the density of the ferrofluid is given by Relation (5), ρ_F = (1-φ)ρ_L + φρ_S. Also, for determining the specific absorption rate (SAR) of the ferrofluid, the following equation is used:

$$SAR=c_S{\displaystyle \frac{\Delta T}{\Delta t}}$$

(15)

In this equation, for the evaluation of SAR, the specific heat of the solid material from which the nanoparticles come (c_S) [44,45,46] is considered.

The novelty of our paper consists in treating the ferrofluid as a composite system and proposing new equations (Equations (1)–(3)) for the calculation of HR_eff and SAR, which assume that all losses are concentrated both in the magnetic solid phase and in the carrier liquid. In this way, as we have shown in a previous paper [22], the effective specific heat (c_eff) of the ferrofluid as a biphasic composite system is given by the following relation:

$$c_{eff}={\displaystyle \frac{(1-\phi ){\rho }_L{c}_L+\phi {\rho }_S{c}_S}{(1-\phi ){\rho }_L+\phi {\rho }_S}}$$

(16)

As a result, the new equation for the specific heating rate, by analogy with Equation (14), will be as follows:

$$H{R}_{new}={\displaystyle \frac{p_m}{{\rho }_F\cdot c_{eff}}}={\displaystyle \frac{p_m}{(1-\phi ){\rho }_L{c}_L+\phi {\rho }_S{c}_S}}$$

(17)

which represents the new Equation (1) proposed for calculating the specific heating rate (HR_eff).

The specific absorption rate SAR, by analogy with Equation (15), is defined as follows:

$$SAR=c_{eff}{\displaystyle \frac{\Delta T}{\Delta t}}$$

(18)

From Equations (1) and (18), the new Equation (2) proposed for calculating the specific absorption rate (SAR) of the ferrofluid is finally obtained.

To explicitly compare the HR and SAR values resulting from Equations (1)–(3) proposed by us with those calculated using the classic Formulas (14) and (15), in Figure 5, we have graphically represented the frequency dependence of the heating rate (HR) and the specific absorption rate (SAR) for all samples and in a zero magnetic field (H = 0).

From Figure 5, it is observed that the HR and SAR values obtained with the new proposed equations are higher than those calculated classically, with Relations (13) and (14), for all ferrofluid samples. The comparative result obtained is very important and demonstrates the extent of the improvement offered by the new approach of the ferrofluid as a composite system, in determining the HR and SAR essential in hyperthermia applications.

The specific magnetic power loss of the ferrofluid (p_m) corresponds to the absorbed energy per unit of volume (W/m³), and the specific absorption rate (SAR) corresponds to the absorbed energy per unit of mass (W/kg). The use of energy dissipated in ferrofluid in the form of heat, under the action of a microwave field, is relevant in magnetic hyperthermia applications [47,48,49,50].

The magnetic heating efficiency of some composite nanosystems such as ferrofluids, is generally evaluated by the specific energy absorption rate (SAR) or by an equivalent parameter, the intrinsic loss power (ILP) [Hm²/kg], obtained by eliminating the extrinsic factors [45,51] (frequency (f) and the square of the variable magnetic field amplitude (H₀), being defined by the following equation:

$$ILP\hspace{0.17em}={\displaystyle \frac{SAR\hspace{0.17em}\hspace{0.17em}}{f\hspace{0.17em}{H}_0^2}}={\displaystyle \frac{\pi {\mu }_0}{{\rho }_F}}{\mu }^{″}$$

(19)

In the study carried out, we determined both the SAR and ILP parameters for H₀ values that satisfy the H₀ << H_A condition, corresponding to the four ferrofluid samples with different nanoparticle volume fractions (φ) and at different H values of the applied polarizing magnetic field in the range (0–115) kA/m. ILP parameters are often preferred to SAR parameters, especially in situations of magnetic hyperthermia heating corresponding to ensembles of magnetic nanoparticles [46].

Using Equations (2), (4) and (17) and obtained values (μ″) of the complex magnetic permeability (see Figure 2a–d), the SAR and ILP values of the ferrofluid samples were determined for a value of the amplitude of the variable magnetic field (H₀ = 5 A/m). In Figure 6, the frequency dependencies of SAR and ILP for different values of volume fractions (φ) of nanoparticles from samples are shown.

As can be seen in Figure 6a,c,e,g, the SAR shows a maximum approximately in the range (1.8–5.25) GHz depending on the value of the polarization magnetic field (H) for all ferrofluid samples. The SAR amplitude increases with the increase in H and by increasing the volume fraction (φ) of the nanoparticles in the ferrofluid samples. The decrease in the maximum SAR value can also be correlated with the decrease in the effective anisotropy constant (K_eff) of the samples (see Table 1), at the decrease in the volume fraction (φ) [52] of the nanoparticles in the ferrofluid samples. The result can be attributed to the ratio of the amplitude of the variable magnetic field (H₀) (in this paper, H₀ = 5 A/m) and the anisotropy field (H_A) (of the order of kA, see Table 1). If the condition H₀ << H_A is met, the maximum value of the specific absorption rate SAR increases when the effective anisotropy constant K_eff decreases [52,53], with the result obtained agreeing with the linear response theory (LRT) [54].

From Figure 6, it is observed that in the frequency range between (2–5) GHz, the variation of both the specific absorption rate (SAR) and the intrinsic loss power (ILP) with the polarizing magnetic field (H) is non-uniform for all samples. Also, from Figure 6a,c,e,g, it is observed that at frequencies lower than 1.8 GHz, SAR decreases with increasing H, and for frequencies higher than approximately 5 GHz, SAR increases with increasing H. This behavior of the specific absorption rate SAR can be due to both the specific magnetic loss power (p_m) of microwaves and the density (ρ_F) of the ferrofluid sample (see Equation (4)).

Figure 6b,d,f,h present a plot of the frequency dependence (f) in the microwave range (0.5–6) GHz of the ILP of ferrofluid samples with different volume fractions (φ) and at different values of the polarizing magnetic field (H). From the plots, it is seen that for all samples, the ILP has a maximum at frequencies approximately in the range of 1.2 GHz–5 GHz, and the amplitude of ILP gradually decreases with increasing H. As in the case of SAR, at frequencies lower than 1.2 GHz, the ILP decreases with increasing H, and for a frequency higher than approximately 5 GHz, the ILP increases with increasing H. The obtained results indicate the possibility of controlling the SAR or ILP of a ferrofluid having different volume fractions (φ) of nanoparticles, by means of a polarizing magnetic field (H), and could be used in magnetic hyperthermia applications [55,56].

In the paper [57], theoretical calculations performed by the authors allowed the determination of the size of Zn-Cu ferrite nanoparticles, for which the effective anisotropy energy of the nanoparticles (K_effV) is greater than the thermal fluctuation energy (k_BT) in the biologically acceptable range (41–46) °C, important in biomedical applications of heating by magnetic hyperthermia. Starting with this result, in Figure 7, we showed the dependence of SAR on the anisotropy parameter (σ = K_effV_m/k_BT), where k_B is Boltzmann’s constant and V_m is the magnetic volume of the nanoparticles. This parameter was determined at room temperature (25 °C), considering the K_eff values of the ferrofluid samples from Table 1, as well as the magnetic diameter (d_m) of the nanoparticles determined from magnetic measurements.

From Figure 7, it is observed that for ferrofluid samples with different volume fractions (φ), the anisotropy parameter σ >> 1, indicating that the anisotropy energy is higher than the thermal fluctuation energy. This suggests good heating efficiency of the samples. It is also observed that as the volume fraction of the samples increases, the specific absorption rate (SAR) increases, with higher values obtained for larger applied polarizing magnetic fields (H). This result is important for microwave heating applications, as the heating efficiency (via SAR) can be controlled through the polarizing magnetic field (H), the frequency (f), and the volume fraction (φ) of the ferrofluid sample.

In Figure 8a, the dependence of the specific absorption rate (SAR) on the polarizing magnetic field (H) is shown, and in Figure 8b, the dependence of the intrinsic loss power (ILP) at three frequencies, 2.5 GHz, 3 GHz, and 4 GHz, for the four values of the volume fraction (φ) of the nanoparticles in the ferrofluid samples.

From Figure 8a,b, it is observed that both SAR and ILP present a maximum at a value of H_max of the magnetic field, for each ferrofluid sample with a different volume fraction (φ), at all three frequencies investigated (2.5 GHz, 3 GHz, and 4 GHz). At the constant frequency (f), as the fraction (φ) decreases, the SAR and ILP values decrease, and the corresponding SAR and ILP maximums shift to higher H_max values.

Table 2. The parameters H_max, SAR_max, and ILP_max of ferrofluid samples, determined at three frequencies.

Samples	A0 (φ = 11.52%)			A1 (φ = 7.68%)			A3 (φ = 3.41%)			A5 (φ = 1.53%)
Frequency (GHz)	2.5	3	4	2.5	3	4	2.5	3	4	2.5	3	4
Hmax(SAR) (kA/m)	23.18	37.45	67.20	24.14	37.78	67.50	32.40	48.20	83.69	46.21	62.01	87.33
SARmax (W/kg)	8.24	8.74	11.52	7.86	8.28	11.17	2.69	3.05	4.16	1.08	1.28	1.79
Hmax(ILP) (kA/m)	24.40	39.41	66.84	25.06	39.88	67.79	32.00	52.74	85.36	38.32	63.11	90.72
ILPmax (nHm2kg−1)	0.129	0.119	0.105	0.123	0.114	0.100	0.043	0.042	0.041	0.018	0.017	0.017

shows the H_max values for both SAR and ILP, as well as the corresponding maximum SAR_max and ILP_max values, respectively, at all three investigated frequencies in the (2–5) GHz range.

At the same time, from Figure 8a,b, it is observed that for values of the polarizing magnetic field (H > H_max), both SAR and ILP decrease with increasing H. The results obtained for both SAR and ILP are significant for applications such as magnetic hyperthermia, providing useful information on the maximum applied magnetic field required to control SAR or ILP at different frequencies in the microwave range. Also, for the same volume fraction (φ), from Table 2 and Figure 8a, it is observed that the SAR value increases with increasing frequency, reaching a maximum value (SAR_max) at an increasingly higher H_max(SAR) field. Simultaneously, for the same volume fraction (φ), from Table 2 and Figure 8b, it is observed that the ILP value decreases with increasing frequency, reaching a maximum ILP_max at an increasingly higher H_max(ILP) field.

Next, we investigated the dependence of both the specific absorption rate (SAR) and the intrinsic loss power (ILP) on the volume fraction (φ) at the three microwave frequencies (2.5 GHz, 3 GHz, and 4 GHz) under varying magnitudes of the applied polarizing magnetic field, H. The first two field values, H₁ = 0 and H₂ = 15.23 kA/m, were selected to satisfy the condition H < H_max, and the third, H₃ = 79.32 kA/m, was chosen such that H > H_max. These conditions were applied consistently across the investigated samples and frequencies, both for SAR (see Figure 8a) and for ILP (see Figure 8b).

In Figure 9a–c, the dependences of SAR and ILP on the volume fraction (φ) for the three chosen values of the applied polarizing magnetic field (H) and at the three microwave frequencies (f) are presented.

As shown in Figure 9, both SAR and ILP exhibit an increasing trend with the volume fraction (φ), reaching an approximately constant value for φ greater than 7.68% across all examined field strengths (H) and frequencies (f). These results are particularly relevant for magnetic hyperthermia applications, as they provide valuable insight into the control of SAR or ILP through the application of a polarizing magnetic field (H) to ferrofluid samples with different volume fractions (φ).

Based on Equation (3), the effective heating rate (HR_eff = ΔT/Δt) was determined for all ferrofluid samples. Figure 10 shows the frequency dependence of HR_eff for ferrofluid samples with different volume fractions for two values of the magnetic polarizing field (H = 0 and H = 79.32 kA/m).

From Figure 10, it can be observed that HR_eff exhibits a maximum both in the absence of a polarizing magnetic field, at a frequency of approximately 2 GHz, and in the presence of a field H = 79.32 kA/m at a frequency of approximately 5 GHz. The amplitude of these maxima decreases with decreasing volume fraction (φ). For sample A0 (φ = 11.52%), the maximum value of HR_eff increases from approximately 0.005 to 0.0096 as the polarizing magnetic field H increases from 0 to 79.32 kA/m. This result indicates that the effective heating rate of the investigated ferrofluid samples (HR_eff) can be controlled both by the applied polarizing magnetic field H and by the nanoparticle volume fraction (φ). Using Relation (3), the heating rate of the samples (ΔT/Δt) was determined. Figure 11a–d shows the time dependence (Δt) of the temperature increase (ΔT) for various values of the polarizing magnetic field H, as well as the corresponding specific absorption rate (SAR) and intrinsic loss power (ILP), for H ≪ H_max (see Figure 7) at a constant frequency f = 4 GHz.

As shown in Figure 11a–d, for the four ferrofluid samples with different volume fractions of nanoparticles (φ) at a frequency of 4 GHz, the temperature increase (ΔT) exhibits a linear dependence on time (Δt) for all values of H. These curves are theoretical dependencies in which we consider that the ferrofluid is adiabatically isolated. In the case of the experimental dependence ΔT(Δt), the curve tends asymptotically to a saturation temperature value. Saturation is explained by the balance between the heat absorbed by the ferrofluid from the alternating magnetic field and the heat dissipated by the ferrofluid to the environment [58].

The slope of each curve in Figure 11 depends on the polarizing magnetic field (H) and the nanoparticle volume fraction (φ). Therefore, at a given microwave frequency, such as f = 4 GHz, the slope of the temperature-time dependence increases with the magnitude of the polarizing magnetic field (H) for all samples. However, as the volume fraction (φ) decreases from 11.52% (sample A0, Figure 11a) to 1.53% (sample A5, Figure 11d), the slope correspondingly diminishes. This obtained result is consistent with the result in Figure 6a,c,e,g, which shows that the specific absorption rate (SAR) increases with (H) field at f = 4 GHz for all samples.

From Figure 11a–d, it can be observed that for the same application time of the microwave field, Δt = 600 s, at f = 4 GHz, by increasing the polarizing magnetic field H between the same values, (0–68.49) kA/m for all samples, an increase in temperature is obtained from the value ΔT₁ (for H = 0) to the value ΔT₄ (for the field H = 68.49 kA/m). The ΔT values thus obtained for all samples are indicated in

Table 3. Variation with the polarizing magnetic field H and the volume fraction (φ) of the temperature increase (ΔT) for a microwave field application time (Δt = const.), respectively, of the values obtained for the time (Δt) when increasing the sample’s temperature by ΔT = const.

Condition	for ΔT = 1.5 K = const.				for Δt = 600 s = const.
H (kA/m)	0	24.47	46.18	68.49	0	24.47	46.18	68.49
time (Δt (s))	Δt1	Δt2	Δt3	Δt4	-	-	-	-
temperature (ΔT (K))	-	-	-	-	ΔT1	ΔT2	ΔT3	ΔT4
A0 (φ = 11.52%)	750	416	215	170	1.1	2.0	4.0	5.0
A1 (φ = 7.68%)	874	513	250	200	0.9	1.7	3.6	4.5
A3 (φ = 3.41%)	1509	1245	896	664	0.59	0.72	1.0	1.3
A5 (φ = 1.53%)	~3600	~3000	~2400	1732	0.25	0.31	0.38	0.51

.

From Table 3, it can also be observed that in the case of samples A0 (φ = 11.52%) and A1 (φ = 7.68%), which have relatively high-volume fractions (φ > 7%), the temperature increase (ΔT) is significant. Specifically, for sample A0, ΔT increases from 1.1 K (for H = 0) to 5 K (for H = 68.49 kA/m), and for sample A1, ΔT increases from 0.9 K (for H = 0) to 4.5 K (for H = 68.49 kA/m). At the same time, Table 3 shows that in the case of samples A3 (φ = 3.41%) and A5 (φ = 1.53%), having low volume fractions (φ < 3.5%), the increase in temperature ΔT is very small. Thus, for sample A3, ΔT increases from 0.59 K (for H = 0) to 1.3 K (for H = 68.49 kA/m), and for sample A5, ΔT increases from 0.25 K (for H = 0) to 0.51 K (for H = 68.49 kA/m). Furthermore, as shown in Figure 11d for sample A5 (with the lowest volume fraction), increasing the polarizing magnetic field (H) from 68.49 kA/m to a value greater than 90.66 kA/m under the same microwave exposure time of 600 s results in only a negligible increase in temperature, from 0.51 K to 0.57 K, which offers no practical benefit. Therefore, these results allow us to estimate for the first time the minimum volume fraction threshold of nanoparticles in the investigated ferrofluid samples (φ_min < 3.5%), below which the heating efficiency of the ferrofluid in the microwave range is very low.

From Figure 11a–d, it can be observed that under the condition of a constant temperature increase (ΔT = 1.5 K = const.) for all samples during the microwave field application, the time required (Δt) to reach this temperature depends strongly on both the polarizing magnetic field (H) and the volume fraction (φ) of ferrofluid samples. The corresponding values are presented in Table 3. For samples A0 (φ = 11.52%) and A1 (φ = 7.68%), which possess relatively high-volume fractions (φ > 7%), the time interval required to reach the temperature ΔT = 1.5 K decreases as both the applied field (H) and the volume fraction (φ) increase (see Table 3). At the same time, from Table 3, it is observed that in the case of samples A3 (φ = 3.41%) and A5 (φ = 1.53%), having small volume fractions (φ < 3.5%), the time interval (Δt) of applying the microwave field increases greatly, reaching (30–60) minutes, even when the polarizing magnetic field (H) is increased (see Figure 11c,d). This behavior is unfavorable for practical applications, as it indicates low heating efficiency of the ferrofluid samples in the microwave frequency range.

Therefore, the results obtained in this study demonstrate that the temperature increase (ΔT) of a ferrofluid sample under microwave field exposure can be effectively controlled by adjusting the nanoparticle volume fraction within the ferrofluid (φ) and by varying the applied polarizing magnetic field (H). This finding has potential relevance for various practical applications, including thermal management, electromagnetic shielding, and magnetic hyperthermia.

4. Conclusions

For four ferrofluid samples with different nanoparticle volume fractions (φ), measurements of complex magnetic permeability were performed in the microwave range (0.5–6) GHz under various polarizing magnetic fields (H) ranging from (0–115) kA/m.

Assuming that the ferrofluid behaves as a biphasic composite system, new calculation formulas were proposed for the more accurate determination of the specific magnetic loss power (p_m), the effective heating rate (HR_eff), the intrinsic loss power (ILP), and the specific absorption rate (SAR) as functions of frequency (f) and polarizing magnetic field (H). These equations can also be applied to other biphasic composite systems.

The dependence of SAR on the polarizing magnetic field (H) at three microwave frequencies (2.5, 3, and 4 GHz) for samples with different φ values showed a clear SAR maximum at a specific field (H_max), which shifted toward higher H as the frequency increased. These findings identified, for the first time, a maximum effective limit of the applied field (H_max ≈ 80 kA/m), up to which SAR increases, ensuring efficient and controllable heating of ferrofluid samples under microwave excitation.

The dependence of SAR and ILP on the nanoparticle volume fraction (ϕ) at the same three frequencies and at different fields below H_max (0, 15.23, and 79.32 kA/m) showed a marked increase that leveled off at approximately constant values for ϕ > 3.5%, regardless of H or f. This result made it possible to identify, for the first time, a lower threshold of nanoparticle concentration (ϕ_min ≈ 3.5%), below which the heating efficiency of the ferrofluid decreases substantially.

Overall, the results demonstrate that during ferrofluid–microwave interaction, the temperature variation (ΔT) depends on both the microwave frequency (f) and the application time (Δt). Furthermore, ΔT can be effectively controlled by adjusting the polarizing magnetic field (H) and the nanoparticle volume fraction (ϕ). These findings are relevant for applications such as thermal management, electromagnetic shielding, and magnetic hyperthermia.