# Experimental Investigations on the Ferromagnetic Resonance and Absorbing Properties of a Ferrofluid in the Microwave Range

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{w}(f,H), at the normal incidence of the wave, for a ferrofluid of thickness d, deposited on a totally reflective support, following multiple internal reflections of the electromagnetic wave in the material. The results show that by increasing both, H, and d, the parameter, R

_{w}(f,H) presents a minimum that decreases from 0.90 (for d = 2 mm) to 0.64 (for d = 10 mm), at frequency f = 5 GHz, which indicates an increase in the absorption of the electromagnetic wave by the ferrofluid. The obtained results are useful in the study of some materials that could be used as electromagnetic absorbers in the microwave range, by the determination of the overall reflection coefficient, R

_{w}(f,H), controlled both by the thickness, d, of the absorber and by the external applied field, H.

## 1. Introduction

_{S}V

_{m}, where M

_{S}is the spontaneous magnetization of the bulk material from which the particles come, and V

_{m}is the magnetic volume of the particle.

_{eff}, is the result of the combination of magnetocrystalline, shape or surface anisotropy [2,3]. For the single-domain particle with uniaxial anisotropy, the anisotropy field, H

_{A}is given by the relation [2,4]:

_{0}is the magnetic permeability of free space.

_{A}, and other magnetic parameters of nanoparticles in ferrofluids. Starting from the Landau–Lifshitz equation [4], which describes the movement of the magnetic moment m of the particle, in a magnetic field H, for a strong polarizing magnetic field (H >> H

_{A}), the resonance condition is written in the form [8]:

_{res}and H, whose slope is γ, (called the gyromagnetic ratio of the particle), and is given by the following relation:

_{e}= 8.791 × 10

^{10}s

^{−1}·T

^{−1}is the electronic gyromagnetic ratio and α

_{L}is the damping parameter of the Landau–Lifshitz equation [4]. In the ferromagnetic resonance range, knowing these parameters of the nanoparticles in the ferrofluid is very important in some microwave applications [9]. Among the practical applications based on microwaves, we can mention the wireless connection, through radio waves, global positioning system (GPS), radar and mobile telephony, and with the progress made in these applications, the problem of electromagnetic interference appears, thus requiring the finding of materials with electromagnetic absorbing properties [10], which are very necessary in the field of electromagnetic shielding [11]. At the same time, the increased use of electronic devices leads to significant pollution with electromagnetic waves, so that the study of electromagnetic wave absorbers (EMAs) is appropriate [12,13,14]. Recently, studies were conducted on microwave absorbers (MAs), based on metamaterials [15], which play an important role in reducing the size and thickness of MAs as a function of operating frequency. These absorbers (MAs) are required in applications to reduce electromagnetic wave interference (EMI), to improve electromagnetic shielding or potential military applications. Magnetic nanomaterials can be of interest for microwave absorbers [16], due to high magnetic saturation and low coercivity. Among the magnetic nanomaterials used in recent years, can be mentioned magnetic metals [17]; magnetic alloys [18] or magnetic oxides [19]. The design of potential microwave absorbers based on magnetic nanomaterials presents high reflection losses (R

_{L}), thin thickness and wide bandwidth, which leads to improved efficiency in electromagnetic absorption (EM) [16]. In paper [20], a flexible absorber for microwaves based on a nanocomposite of nickel ferrite in a natural rubber matrix was analyzed, using the complex dielectric permittivity and magnetic permeability measurements in S-band (2–4) GHz and X-band (8–12) GHz. The reflection loss was estimated using the model of single layer absorber deposited on a perfect conductor. Also, the results reported in [21] clearly show that composite materials are effective in reducing electromagnetic interference (EMI) pollution, with special attention being paid to the development of new nanocomposite materials with high electromagnetic absorbing properties.

_{0}e

^{−αz}and (E

_{0}/Z

_{m})e

^{−αz}are the amplitudes of the electric and magnetic components of the electromagnetic wave, where α and Z

_{m}represent the attenuation constant and the intrinsic impedance of the propagation medium, respectively, which are given by the relations [25]:

^{8}m/s is the speed of electromagnetic waves in free space and Z

_{0}= (μ

_{0}/ε

_{0})

^{1/2}= 377 Ω is the intrinsic impedance of free space (ε

_{0}being the dielectric permittivity of free space). Considering the complex form of the relative magnetic permeability, ${\mu}_{r}={\mu}_{r}^{\prime}-i{\mu}_{r}^{\u2033}$ as well of the relative dielectric permittivity, ${\epsilon}_{r}={\epsilon}_{r}^{\prime}-i{\epsilon}_{r}^{\u2033}$, the following expression for calculating the intrinsic impedance of the medium is obtained from Equation (7):

_{m}

_{1}and Z

_{m}

_{2}are the intrinsic impedances of the medium 1 and medium 2, respectively.

_{w}(f,H) of a plane wave absorber with a single layer supported by a perfect conductor, following multiple internal reflections, at the normal incidence of the wave, valid for any material/composite system that has absorbing properties in microwave range, applied to the investigated ferrofluid sample. The proposed theoretical model assumes that the ferrofluid is homogeneous in its thickness and local structure when subjected to external magnetic fields.

## 2. Theoretical Model of the Overall Reflection Coefficient

_{w}, of a material with thickness d, through which an electromagnetic wave propagates. To define R

_{w}, consider a thin layer of an absorbent material (having the attenuation constant α and reflection coefficient R), deposited on a metal support (total reflective support) (Figure 2). In the case of the electromagnetic field perpendicular to the plane of incidence, we consider that the incident wave (E

_{i}) comes from medium 1 (air) and on the air-material interface, at point A, is reflected (R

_{1}wave) and simultaneously, is transmitted (E

_{t}wave) in the material (medium 2).

_{w}, of the absorbing material.

_{R1(A)}and E

_{t(A)}, are the reflected and transmitted wave, respectively, at point A (see Figure 2).

_{t(A)}wave travels through the absorbent material and will arrive, attenuated at point B on the total reflective metallic support, and is then reflected. As a result,

_{r}

_{2(B)}= E

_{i}

_{2(B)}, will then travel through the absorbing material (medium 2) and will arrive attenuated at point C (E

_{i}

_{3(C)}wave), on the air-absorbing material incident surface (see Figure 2); E

_{i}

_{3(C)}being expressed by the relation:

_{i}

_{3(C)}will be reflected in medium 2 and at the same time will be transmitted in medium 1 (air) (Figure 2). On incident surface at the point C, the transmission coefficient T, will be the same as at point A, so that the following relation can be written for the transmitted wave E

_{t}

_{3(C)}:

_{w}, at normal incidence is given by the following equation:

_{w}, at the normal incidence, which takes into account the multiple total reflections of the electromagnetic wave on the reflective metal support, when the wave passes through absorbent material, being valid for any material/composite system that has electromagnetic wave absorption properties.

## 3. Sample and Experimental Setup

_{sat}= 22.47 kA/m; the mean magnetic diameter of a particle, d

_{m}= 8.03 nm; the particles concentration, n = 12.44·10

^{22}m

^{−3}and initial susceptibility χ

_{in}= 1.048.

## 4. Results and Discussion

_{res}[8,33]. By increasing the polarizing magnetic field H, the f

_{res}frequency moves to higher values from 1.245 GHz (for H = 0) to 6.170 GHz (at H = 135.45 kA/m). At the same time, from Figure 5 it is observed that the imaginary component, μ″(f), has a maximum at the frequency f

_{max}, very close to the f

_{res}frequency, named the maximum absorption frequency, which moves to higher values with increasing H, from 0.942 GHz (for H = 0) to 6.048 GHz (for H = 135.45 kA/m).

_{res}, corresponding to each H value, from Figure 5, we were able to obtain the dependence of f

_{res}on H, which is linear, as can be seen in Figure 6, being in accordance with Equation (2).

_{res}(H) from Figure 6, and using Equation (2), we were able to determine the anisotropy field H

_{A}and the gyromagnetic ratio of the particle γ; the values obtained being: H

_{A}= 38.75 kA/m and γ = 2.25 × 10

^{5}s

^{−1}A

^{−1}m. From Equation (1), we determine that the effective anisotropy constant, K

_{eff}= 1.16 × 10

^{4}J/m

^{3}, which agrees with the values obtained in the literature [36,37] for magnetite particles. Taking into account both the value obtained for γ, and the relation (3) valid for high values of the magnetic field H > H

_{A}, we were able to determine the damping parameter of the Landau–Lifshitz equation α

_{L}, and using the value g = 2, specific to a kerosene-based ferrofluid with magnetite particles [5,8,33], we obtained the value, α

_{L}= 0.194. As shown in the paper [38], high values obtained for the α

_{L}parameter (α

_{L}> 0.1) can be due either to the polydispersity of the nanoparticle systems [39], or to the interactions between the particles [40], in presence of a high polarizing magnetic field.

_{m}

_{,1}, …, f

_{m}

_{,10}, for each value of H, which shifts from 1.764 GHz (for H = 0) to 6.164 GHz (for H = 135.45 kA/m). For frequencies less than 1.764 GHz, the attenuation constant α decreases with increasing H. For values of the magnetic field H between 0 and 135.45 kA/m, for a frequency greater than 6 GHz, α increases with increasing H. For other H values of the magnetic field, another frequency range can be established, for which α increases with increasing H.

_{max}, which moves to higher values from 40 kA/m (at f = 2 GHz) to 109 kA/m (at f = 5 GHz), over the frequency range of (1.764–6) GHz. Also, for H > H

_{max}, α decreases with increasing H (Figure 8). This result is very useful in magnetically controlled electromagnetic absorbers applications because it provides information on the maximum limit of the magnetic field applied for control of α, at different frequencies.

_{w}(f,H) for 3 values of the thickness d of the ferrofluid sample: 2 mm, 5 mm and 10 mm. For the third term in Equation (22), in R

_{eff}’s calculation, we considered only the terms corresponding to k = 1 and k = 2, because for $k\ge 3$, the calculated values decrease very quickly, tending towards zero, being thus negligible. The calculation relations for the third term in Equation (22) corresponding to the values k = 1, 2, 3, 4 are as follows:

_{3(3)}and T

_{3(4)}, corresponding to k = 3 and k = 4, respectively, are much smaller than the terms T

_{3(1)}and T

_{3(2)}, corresponding to k = 1 and, respectively k = 2, for all frequencies f and all thicknesses d considered, of the sample, both in the zero field and in the presence of the H field. At the same time, as the thickness of the sample d increases, the value of the terms T

_{3(3)}and T

_{3(4)}, becomes very small close to zero, so that the contribution to the calculation of the R

_{eff}parameter, with Equation (22) of all terms starting with k ≥ 3, was neglected.

_{w}, for 3 values of the thickness d of the ferrofluid sample, 2 mm, 5 mm and 10 mm, the frequency and magnetic field dependencies of the overall reflection coefficient, R

_{w}(f,H), of the ferrofluid deposited on a totally reflective support, for normal incidence, are presented in Figure 10a–c.

_{w}(f,H) decreases with the increase in the magnetic field at frequencies higher than 5.9 GHz, while at frequencies below approximately 1 GHz, the overall reflection coefficient, R

_{w}(f,H) increases with increasing H.

_{w}, on the magnetic field H, at three frequencies, f, of the electromagnetic field, in the range (1–6) GHz, for the three considered thicknesses d of the ferrofluid sample deposited on a totally reflective support, for normal incidence.

_{w}presents a minimum that is accentuated by increasing the thickness d, of the ferrofluid sample. Thus, at f = 2 GHz, the minimum of R

_{w}appears at the same value of the polarizing magnetic field, H

_{min(2GHz)}= 40 kA/m, decreasing from the value 0.93 (for d = 2 mm) to 0.87 (for d = 10 mm), while at f = 5 GHz the minimum of R

_{w}appears at the constant value of the field, H

_{min(5GHz)}= 109 kA/m, decreasing from the value 0.90 (for d = 2 mm) to 0.64 (for d = 10 mm) (see Figure 11). At the same time, for H > H

_{min}, but very close to H

_{min}, R

_{w}increases with increasing H (Figure 11)), for any value d of the sample thickness. As a result, obtaining a value as low as possible for the overall reflection coefficient R

_{w}, indicates that the investigated material is a good absorber of the electromagnetic wave. The obtained result is useful in the study of some materials/composite systems with possible absorbing properties in microwaves, by determination of the overall reflection coefficient R

_{w}(f,H) controlled both by the thickness d of the absorber and by the external field H applied, at different frequencies from the range (1–6) GHz.

## 5. Conclusions

_{max}, which moves to higher values from 40 kA/m (at f = 2 GHz) to 109 kA/m (at f = 5 GHz), while for H > H

_{max}, α decreases with increasing H. The obtained result is very important in applications of electromagnetic absorbers because it provides information on the maximum limit of the magnetic field H, for control of attenuation constant α, at different frequencies. Also, in this paper, we defined another parameter, namely the overall reflection coefficient, R

_{w}, of a ferrofluid with thickness d and the attenuation constant α, through which the electromagnetic wave propagates, perpendicular to the incidence plane. For this, we proposed a theoretical model for computing the overall reflection coefficient, R

_{w}(f,H) of a plane wave absorber with a single layer deposited on a perfect conductor, following multiple internal reflections, at the normal incidence of the wave, valid for any material/composite system that has absorbing properties in the microwave range. Using this model, we established for the first time an equation for the computation of R

_{w}(f,H) for 3 values of the thickness d of the ferrofluid sample: 2 mm, 5 mm and 10 mm, which takes into account, the multiple total reflections of the electromagnetic wave on the reflective metal support, when the wave passes through the ferrofluid. The dependence of R

_{w}(H) at different frequencies, f, in the range (1–6) GHz shows that the parameter R

_{w}presents a minimum at a value of polarizing magnetic field, H

_{min}, which moves to higher values from 40 kA/m (at f = 2 GHz) to 113 kA/m (at f = 5 GHz), for all values d of thickness of sample, while for H > H

_{min}, R

_{w}increases with increasing H. On the other hand, at a fixed frequency, such as f = 5 GHz, the minimum of R

_{w}decreases from 0.90 (for d = 2 mm) to 0.64 (for d = 10 mm) which shows an increase in the absorption of the electromagnetic wave in the ferrofluid. The obtained results could be applied to any system of superparamagnetic particles located in a solid dielectric matrix, not only to the ferrofluid, being very useful in the use of these materials as electromagnetic absorbers in the microwave range.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic representation of the reflected and transmitted wave, on the separation surface between two electromagnetic media.

**Figure 2.**Schematic representation of the electromagnetic wave absorption and reflection through an absorbent material deposited on a total reflective support.

**Figure 4.**The frequency dependence of the complex dielectric permittivity components of the ferrofluid sample, for 10 different values of polarizing magnetic field, H.

**Figure 5.**The frequency dependence of the complex magnetic permeability components of the ferrofluid sample, for 10 different values of polarizing field, H, corresponding to, (1) H = 0; (2) H = 15.24 kA/m; (3) H = 35.28 kA/m; (4) H = 57.07 kA/m; (5) H = 79.33 kA/m; (6) H = 90.66 kA/m; (7) H = 102.40 kA/m; (8) H = 113.43 kA/m; (9) H = 124.47 kA/m; (10) H = 135.45 kA/m.

**Figure 6.**The polarizing magnetic field dependence, of the resonance frequency f

_{res}(H), for ferrofluid sample.

**Figure 7.**Frequency dependence of the attenuation constant α at different values of polarizing magnetic field, H.

**Figure 8.**A plot of the attenuation constant α of microwaves in ferrofluid, against the polarizing magnetic field, H, at different frequencies.

**Figure 9.**Frequency dependence of the reflection coefficient R at the interface air–ferrofluid for normal incidence at various magnetic field values: (1) 0 kA/m, (2) 15.24 kA/m, (3) 35.29 kA/m, (4) 57.07 kA/m, (5) 79.33 kA/m, (6) 90.66 kA/m, (7) 102.4 kA/m, (8) 113.43 kA/m, (9) 124.47 kA/m, (10) 135.45 kA/m.

**Figure 10.**Frequency dependence of the overall reflection coefficient, R

_{w}, at various magnetic field H values for a thickness of ferrofluid, d = 2 mm (

**a**), d = 5 mm (

**b**) and d = 10 mm (

**c**).

**Figure 11.**The plot of the overall reflection coefficient, R

_{w}(f,H), against the polarizing magnetic field H, at different frequencies f.

**Table 1.**The values computed with Equations (23)–(26) of the terms T

_{3(1)}, T

_{3(2)}, T

_{3(3)}and T

_{3(4)}.

Thickness d | 2 mm | 5 mm | 10 mm | ||||||
---|---|---|---|---|---|---|---|---|---|

Frequency f [GHz] | 2 | 4 | 5 | 2 | 4 | 5 | 2 | 4 | 5 |

T_{3(1)}for H = 0 | 0.13418 | 0.13597 | 0.13582 | 0.12687 | 0.13329 | 0.13483 | 0.11557 | 0.12895 | 0.1332 |

T_{3(1)} for H = 135.45 kA/m | 0.12291 | 0.10942 | 0.10053 | 0.11862 | 0.09676 | 0.07818 | 0.1118 | 0.07884 | 0.05141 |

T_{3(2)}for H = 0 | 0.03203 | 0.03189 | 0.03123 | 0.02945 | 0.03096 | 0.03089 | 0.0256 | 0.02946 | 0.03033 |

T_{3(2)} for H = 135.45 kA/m | 0.0235 | 0.01829 | 0.01612 | 0.02228 | 0.01521 | 0.01106 | 0.02039 | 0.01118 | 0.0059 |

T_{3(3)}for H = 0 | 0.00764 | 0.00748 | 0.00718 | 0.00683 | 0.00719 | 0.00708 | 0.00567 | 0.00673 | 0.00691 |

T_{3(3)} for H = 135.45 kA/m | 0.00449 | 0.00306 | 0.00259 | 0.00419 | 0.00239 | 0.00156 | 0.00372 | 0.00159 | 0.00067 |

T_{3(4)}for H = 0 | 0.00182 | 0.00175 | 0.00165 | 0.00159 | 0.00167 | 0.00162 | 0.00126 | 0.00154 | 0.00157 |

T_{3(4)} for H = 135.45 kA/m | 0.00444 | 0.00293 | 0.00238 | 0.00406 | 0.00216 | 0.00127 | 0.0035 | 0.00129 | 0.00044 |

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Malaescu, I.; Marin, C.N.; Fannin, P.C.
Experimental Investigations on the Ferromagnetic Resonance and Absorbing Properties of a Ferrofluid in the Microwave Range. *Magnetochemistry* **2024**, *10*, 7.
https://doi.org/10.3390/magnetochemistry10020007

**AMA Style**

Malaescu I, Marin CN, Fannin PC.
Experimental Investigations on the Ferromagnetic Resonance and Absorbing Properties of a Ferrofluid in the Microwave Range. *Magnetochemistry*. 2024; 10(2):7.
https://doi.org/10.3390/magnetochemistry10020007

**Chicago/Turabian Style**

Malaescu, Iosif, Catalin N. Marin, and Paul C. Fannin.
2024. "Experimental Investigations on the Ferromagnetic Resonance and Absorbing Properties of a Ferrofluid in the Microwave Range" *Magnetochemistry* 10, no. 2: 7.
https://doi.org/10.3390/magnetochemistry10020007