# The Effect of Magnetically Induced Local Structure and Volume Fraction on the Electromagnetic Properties of Elastomer Samples with Ferrofluid Droplet Inserts

^{1}

^{2}

^{*}

## Abstract

**:**

_{0}, B

_{0}and C

_{0}) were manufactured in the absence of a magnetic field, and the other three samples (A

_{h}, B

_{h}and C

_{h}) were manufactured in the presence of a magnetic field, H = 43 kA/m. The component μ″ of the complex effective magnetic permeability of all samples presents a maximum at a frequency, f

_{max}, that moves to higher values by increasing φ, with this maximum being attributed to Brownian relaxation processes. The conductivity spectrum, σ (f), of all samples follows the Jonscher universal law, which allows for both the determination of the static conductivity, σ

_{DC}, and the barrier energy of the electrical conduction process, W

_{m}. For the same φ, W

_{m}is lower, and σ

_{DC}is higher in the samples A

_{h}, B

_{h}and C

_{h}than in the samples A

_{0}, B

_{0}and C

_{0}. The performed study is useful in manufacturing elastomers with predetermined properties and for possible applications such as magneto-dielectric flexible electronic devices, which can be controlled by the volume fraction of particles or by an external magnetic field.

## 1. Introduction

_{3}O

_{4}, Fe

_{2}O

_{3}, CoFe

_{2}O

_{4}, etc.) dispersed in a basic liquid and stabilized with a surfactant to prevent sedimentation [17]. On the other hand, ferrofluid is considered a composite with magneto-dielectric properties that is influenced by the presence of magnetic fields [18,19].

## 2. Obtaining and Characterizing Samples

_{SR}= 1.3 g/cm

^{3}and a kerosene-based ferrofluid (EFH 1 type from Ferrotech) [21] with magnetite particles stabilized with oleic acid and a density of ρ

_{FM}= 1.21 g/cm

^{3}were manufactured by mixing them. Silicone rubber has two components (A and B) and is a non-toxic elastomer with medium hardness and elasticity that can be used even as bolus material in radiotherapy [22].

_{SR}, of silicone rubber (equal quantities, M

_{SR}/2, of each component, A and B) with a small amount, MFM, of ferrofluid (a few drops), thus obtaining three samples, which differ in the volume fraction, φ, of the ferrofluid in the composite. The mixture thus formed was placed in a parallelepiped mold and pressed continuously for 2–3 min by hand (just like a dough) until it took the shape of the mold, and after 24 h, the composite samples were obtained and took the shape of a square parallelepiped plate with a side measuring 5 cm and a thickness of 0.1 cm. This polymerization/hardening of the sample mixture was performed in the absence of a magnetic field, and the composite samples obtained are denoted by sample A

_{0}, sample B

_{0}and sample C

_{0}. Also, three other composite samples with the same volume fractions, φ, were polymerized in the presence of a magnetic field, H = 43 kA/m, and are denoted as sample A

_{h}, sample B

_{h}and sample C

_{h}. For this, the parallelepiped mold, which contains the mixture corresponding to samples A

_{h}, B

_{h}and C

_{h}, was fixed between the magnetic poles, N and S, of a Weiss-type electromagnet powered by a direct current source. The orientation of the magnetic field, H, was parallel to the surface of the sample, and the value of the magnetic field, H, was measured with the aid of a Hall probe from a Gauss meter.

- -
- For samples A
_{0}and A_{h}: M_{FM}= 0.05 g and M_{SR}= 4 g (2 g of each component of the silicon rubber, A and B); - -
- For samples B
_{0}and B_{h}: M_{FM}= 0.10 g and M_{SR}= 4 g (2 g of each component of the silicon rubber, A and B); - -
- For samples C
_{0}and C_{h}: M_{FM}= 0.15 g and M_{SR}= 4 g (2 g of each component of the silicon rubber, A and B).

_{tot}= V

_{SR}+ V

_{FM}, and then the volume fraction, φ = V

_{FM}/V

_{tot}, of the ferrofluid in the composite samples. We obtained the following values: φ

_{1}= 1.31%, φ

_{2}= 2.59% and φ

_{3}= 3.84%.

_{0}, B

_{0}and C

_{0}in Figure 1a–c) and in the presence of a magnetic field, H (samples A

_{h}, B

_{h}and C

_{h}in Figure 1d–f). As can be seen in Figure 1, when the sample preparation takes place in the presence of a magnetic field (Figure 1d–f), the droplets of the ferrofluid stretch along the direction of the magnetic field lines, as well as the magnetite nanoparticles aggregate, thus forming field-induced structures, while in the absence of the magnetic field (Figure 1a–c), the droplets are approximately spherical in shape, oriented randomly in the entire volume of the composite material.

_{in}= 1.078; the concentration of particles, n = 10.49∙10

^{22}m

^{−3}; and the mean magnetic diameter of the particles, d

_{m}= 11.74 nm.

## 3. Results and Discussion

#### 3.1. Investigation of Complex Effective Magnetic Permeability

_{0}), and the inductive reactance, X (or X

_{0}), of a solenoid with a sample as a magnetic core (or empty) connected to an RLC meter (Agilent type E-4980A) were measured at frequencies between 0.5 kHz and 2 MHz. μ′ and μ″ were determined with Equation (1) [25].

_{h}, B

_{h}and C

_{h}, obtained in the presence of an external magnetic field (see Figure 1d–f), the rolling up of the rectangular samples was made parallel to the field-induced structures, thus obtaining a cylindrical shape for the sample, in which the microstructures were arranged parallel to the axis of the obtained cylinder. In this way, the magnetic probing field of the coil is oriented parallel to the field-induced tubular structures obtained in samples A

_{h}, B

_{h}and C

_{h}via their polymerization in a static magnetic field. The obtained results for the composite samples are presented in Figure 3.

_{h}, B

_{h}and C

_{h}(Figure 3b), obtained in the presence of a magnetic field, H, are higher than those corresponding to samples A

_{0}, B

_{0}and C

_{0}(Figure 3a), obtained in the absence of the magnetic field, H, at all values of the volume fraction, φ. This result shows that preparing such samples by mixing a ferrofluid with silicone rubber in the presence of an external magnetic field, H, leads to the obtainment of composite samples with improved magnetic properties that can be controlled by a magnetic field, H, and a volume fraction, φ.

_{max}, that depends on the volume fraction, φ, for each composite sample. The existence of this local maximum indicates a relaxation process in the composite elastomeric samples in the investigated frequency range, which is characterized by the relaxation time, τ.

_{max}, at which μ″ is at maximum via the following relation:

_{max}τ = 1

_{max}, from Figure 3a,b and using Equation (2), the corresponding values of the relaxation times were computed, resulting in the following values: τ

_{(A0)}= 9.18 μs, τ

_{(B0)}= 7.46 μs and τ

_{(C0)}= 7.05 μs for samples A

_{0}, B

_{0}and C

_{0}and τ

_{(Ah)}= 10.88 μs, τ

_{(Bh)}= 7.96 μs and τ

_{(Ch)}= 7.35 μs for samples A

_{h}, B

_{h}and C

_{h}, respectively. The dependence on the volume fraction, φ, of the obtained relaxation times, τ, is shown in Figure 4.

_{N}, is provided by the relation

_{0}is a constant that can take values between 10

^{−12}s and 10

^{−9}s, depending on the material from which the particles are made. For magnetite, it is usually considered τ

_{0}= 10

^{−9}s [27,28]. T is the absolute temperature; k is Boltzmann’s constant; V

_{m}is the magnetic volume of a particle; and K is the anisotropy constant of particles.

_{B}, which is provided by the following equation:

_{h}is the hydrodynamic diameter of the particle or the aggregate, and η is the dynamic viscosity of the carrier liquid.

_{m}value of the mean magnetic diameter of the particles, if we consider that the relaxation process would be a Néel type, the anisotropy constant, K, of the magnetic particles can be computed with Relation (3). The following values were obtained: K

_{(A0)}= 4.46·10

^{4}J/m

^{3}, K

_{(B0)}= 4.36·10

^{4}J/m

^{3}and K

_{(C0)}= 4.33·10

^{4}J/m

^{3}for samples A

_{0}, B

_{0}and C

_{0}and K

_{(Ah)}= 4.54·10

^{4}J/m

^{3}, K

_{(Bh)}= 4.38·10

^{4}J/m

^{3}and K

_{(Ch)}= 4.35·10

^{4}J/m

^{3}for samples A

_{h}, B

_{h}and C

_{h}. The results thus obtained for the anisotropy constant, K, of the magnetite particles in the composite elastomeric samples far exceed the K values corresponding to magnetite particles (1.1∙10

^{4}– 1.5∙10

^{4}) J/m

^{3}[29,30]. This allows us to draw the conclusion that the relaxation process afferent to the local maxima of μ″ (Figure 3) cannot be considered a Néel relaxation process.

^{−3}Pa·s for the viscosity of the carrier liquid (kerosene) and the constant room temperature, T = 300 K, at which the measurements were made, we can determine the hydrodynamic diameter, D

_{h}, of the particles in the samples. The values obtained are D

_{h,A0}= 27.13 nm, D

_{h,B0}= 25.32 nm and D

_{h,C0}= 24.84 nm for samples A

_{0}, B

_{0}and C

_{0}and D

_{h,Ah}= 28.71 nm, D

_{h,Bh}= 25.87 nm and D

_{h,Ch}= 25.20 nm for samples A

_{h}, B

_{h}and C

_{h}, respectively. The values determined for the hydrodynamic diameter, D

_{h}, show that, in all samples, aggregates of 2–3 particles rotate as a single structure in the carrier liquid of the ferrofluid within the droplet inserts from the composite. So, the maximum of the imaginary component, μ″, from Figure 3a,b is due to the Brownian relaxation process in the composite, and the ferrofluid droplet inserts are still present in the composite after polymerization.

_{0}, B

_{0}and C

_{0}, the droplets are approximately spherical in shape, and for the samples polymerized in the magnetic field, the droplets are elongated along the magnetic field lines.

#### 3.2. Investigation of Complex Dielectric Permittivity

_{0}, and reactance, X

_{0}, in the absence of a sample in the capacitor. Components ε′ and ε″ of the complex dielectric permittivity were determined with the following relations [31,32]:

_{0}, B

_{0}and C

_{0}—see Figure 6a) and from 2.0 to 2.6 (for samples A

_{h}, B

_{h}and C

_{h}—see Figure 6b) by increasing the volume fraction, φ, from 1.31% to 3.84%. The values of ε′ corresponding to samples A

_{h}, B

_{h}and C

_{h}(Figure 6b), obtained in the presence of a magnetic field H, are lower than those corresponding to samples A

_{0}, B

_{0}and C

_{0}(Figure 6a), obtained in the absence of a magnetic field, H, at all values of the volume fraction, φ. This result can be correlated with a decrease in the equivalent electric capacity of the sample holder with samples A

_{h}, B

_{h}and C

_{h}(polymerized in a magnetic field) versus that of the sample holder with samples A

_{0}, B

_{0}and C

_{0}(polymerized in no magnetic field). A similar result for a ferrofluid sample was reported in Ref. [31]. When inserting A

_{h}, B

_{h}and C

_{h}samples between capacitor armatures, the electric field lines will be perpendicular to the microstructures induced by polymerization in a magnetic field. As a result, such a structure leads to a decrease in the equivalent capacity and, therefore, the dielectric permittivity, in accordance with the Wigner limits of the permittivity of composite materials [33].

_{0}, B

_{0}and C

_{0}and for samples A

_{h}, B

_{h}and C

_{h}(see Figure 6a,b). Also, at the same value of the volume fraction, φ, the values of ε″ corresponding to samples A

_{h}, B

_{h}and C

_{h}(Figure 6b) are smaller than those corresponding to samples A

_{0}, B

_{0}and C

_{0}at any given frequency.

#### 3.3. DC and AC Conductivity

_{0}ε″

_{dc}), and (2) a dispersion region, where σ depends on frequency, corresponding to AC-conductivity (σ

_{ac}). In other papers [9,35], a similar conductivity frequency dependence was obtained for other composite samples using a combination of Fe

_{3}O

_{4}nanoparticles or graphite nanoplatelets and a polymer. The frequency behavior of the electrical conductivity of the elastomer composite samples (as seen in Figure 7) agrees with Jonscher’s universal law [36]:

_{DC}, remain approximately constant with frequency, up to about 30 kHz, for each volume fraction, φ, both for composite samples A

_{0}, B

_{0}and C

_{0}(Figure 6a) and for samples A

_{h}, B

_{h}and C

_{h}(Figure 6b); the obtained σ

_{DC}values are listed in Table 1.

_{DC}, of silicone rubber (SR), obtaining the value σ

_{DC}= 1.4 × 10

^{−9}S/m. As a result, by adding ferrofluid to the silicone rubber (SR), the static conductivity, σ

_{DC}, of the elastomeric composite samples was increased compared with the σ

_{DC}value of the silicone rubber, which was all the higher in the volume fraction of the ferrofluid (see Table 1).

_{DC}conductivity increases for all composite samples. Also, the values of σ

_{DC}corresponding to samples manufactured in the presence of a magnetic field (samples A

_{h}, B

_{h}and C

_{h}) are higher than the σ

_{DC}values of samples A

_{0}, B

_{0}and C

_{0}, manufactured in the absence of a magnetic field. Therefore, the σ

_{DC}conductivity of the composite samples is correlated with the sample manufacturing process. When sample preparation takes place in the presence of a magnetic field, the magnetite particles from ferrofluid align in the direction of the magnetic field, forming parallel chains of particles, which leads to an increase in conductivity, σ

_{DC}, with respect to the σ

_{DC}of samples prepared in the absence of a magnetic field when the particles are randomly oriented in the entire volume of the elastomer composite material (see Figure 1a–c).

_{AC}component of conductivity depends on frequency—correlated with dielectric relaxation processes due to localized electric charge carriers from the composite samples—and is provided by the following equation:

_{AC}and lnω, which is shown in Figure 8a for samples A

_{0}, B

_{0}and C

_{0}and in Figure 8b for samples A

_{h}, B

_{h}and C

_{h}. Fitting the experimental dependencies, ln(σ

_{AC})(ln(ω)), from Figure 8 with a straight line, we determined the exponent, n, and the parameter, A, corresponding to all the values of the volume fraction, φ. The values obtained are listed in Table 1. It can be observed that, for the same value of the volume fraction, φ, the values of the exponent, n, corresponding to samples A

_{h}, B

_{h}and C

_{h}(obtained in the presence of a magnetic field H) are lower than the values, n, corresponding to samples A

_{0}, B

_{0}and C

_{0}(obtained in the absence of a magnetic field).

_{m}represents the barrier energy [39,40]. Using Relation (9), and the values of n, we determined the barrier energy of the electrical conduction process of each investigated sample. The obtained results for W

_{m}are shown in Table 1.

_{m}, of all composite samples. Also, the W

_{m}values corresponding to the samples manufactured in the presence of a magnetic field (samples A

_{h}, B

_{h}and C

_{h}) are lower than the W

_{m}values of samples A

_{0}, B

_{0}and C

_{0}, manufactured in the absence of a magnetic field. Therefore, a decrease in the barrier energy, W

_{m}, of samples A

_{h}, B

_{h}and C

_{h}compared with the barrier energy of samples A

_{0}, B

_{0}and C

_{0}will lead to an increase in the number of charge carriers that will be able to participate in the electrical conduction of these samples, which determines an increase in their conductivity, as we achieved experimentally (see Table 1).

## 4. Conclusions

_{0}, B

_{0}and C

_{0}) and for samples manufactured in the presence of a magnetic field, H (samples A

_{h}, B

_{h}and C

_{h}). This maximum is attributed to the Brownian relaxation process within the ferrofluid droplet inserts from the composites. Using the experimental results for the complex dielectric permittivity, the conductivity spectra, σ(f), of all the investigated samples were determined. The spectra, σ(f), obey Jonscher’s universal law, as they have two regions: a region in which σ does not vary with frequency, corresponding to DC-conductivity (σ

_{DC}), and a dispersion region where σ rapidly increases with frequency, corresponding to AC-conductivity (σ

_{AC}). An increase in the volume fraction of particles in the elastomeric composite samples, from φ = 1.31% to x = 3.84%, leads to an increase in σ

_{DC}from 4.26∙10

^{−9}S/m to 1.03∙10

^{−8}S/m for samples A

_{0}, B

_{0}and C

_{0}and from 4.93∙10

^{−9}S/m to 1.86∙10

^{−8}S/m for samples A

_{h}, B

_{h}and C

_{h}. Based on Jonscher’s universal response and the CBH (correlated-barrier-hopping) theoretical model, we evaluated, for all composite samples, the energy barrier of the electrical conduction process, W

_{m}. The results show that the W

_{m}values corresponding to the samples manufactured in the presence of a magnetic field (samples A

_{h}, B

_{h}and C

_{h}) are lower than the W

_{m}values of samples A

_{0}, B

_{0}and C

_{0}, manufactured in the absence of the magnetic field, for all values of the volume fraction, φ (this result agrees with the increase in their conductivity, σ

_{DC}). The results obtained are very useful for the manufacture of elastic composites with predetermined properties that can be tuned by changing the volume fraction of particles inside the composite or by modifying the local structure in the presence of an external magnetic field.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Images of the composite samples consisting of silicone rubber with ferrofluid: samples A

_{0}(

**a**), B

_{0}(

**b**) and C

_{0}(

**c**) obtained in the absence of magnetic field; samples A

_{h}(

**d**), B

_{h}(

**e**) and C

_{h}(

**f**) obtained in the presence of a magnetic field, H = 43 kA/m.

**Figure 3.**Variation in the frequency of the μ′ and μ″ components of complex effective magnetic permeability for the elastomeric composite samples: (

**a**) A

_{0}, B

_{0}and C

_{0}; (

**b**) A

_{h}, B

_{h}and C

_{h}.

**Figure 4.**Volume fraction dependence of the relaxation times, τ(φ), for composite samples: τ

_{(0)}for samples A

_{0}, B

_{0}and C

_{0}and τ

_{(H}

_{)}for the samples A

_{h}, B

_{h}and C

_{h}(the line is a spline interpolation of the experimental points).

**Figure 5.**The frequency dependence of the real, μ′, and imaginary, μ″, components of the complex magnetic permeability of the ferrofluid sample, both in no magnetic field (H = 0) and in the presence of a field (H ≠ 0).

**Figure 6.**The frequency dependence of the real, ε′, and imaginary, ε″, components of the complex dielectric permittivity of (

**a**) composite samples A

_{0}, B

_{0}and C

_{0}and (

**b**) composite samples A

_{h}, B

_{h}and C

_{h}.

**Figure 7.**The frequency dependence of the conductivity, σ, of samples A

_{0}, B

_{0}and C

_{0}(

**a**); samples A

_{h}, B

_{h}and C

_{h}(

**b**); and silicone rubber (

**c**).

**Figure 8.**lnσ

_{ac}(lnω) dependence for composite samples A

_{0}, B

_{0}and C

_{0}(

**a**) and samples A

_{h}, B

_{h}and C

_{h}(

**b**).

Samples | A_{0} | B_{0} | C_{0} | A_{h} | B_{h} | C_{h} |
---|---|---|---|---|---|---|

φ = 1.31% | φ = 2.59% | φ = 3.84% | φ = 1.31% | φ = 2.59% | φ = 3.84% | |

Parameters | H = 0 | H = 43 kA/m | ||||

σ_{DC} (S/m) | 4.26∙10^{−9} | 9.40∙10^{−9} | 1.03∙10^{−8} | 4.93∙10^{−9} | 1.73∙10^{−8} | 1.86∙10^{−8} |

n | 0.897 | 0.915 | 0.938 | 0.751 | 0.807 | 0.872 |

A (S/m) | 5.42∙10^{−13} | 5.72∙10^{−13} | 4.73∙10^{−13} | 28.7∙10^{−13} | 20.4∙10^{−13} | 8.56∙10^{−13} |

W_{m} (eV) | 1.51 | 1.83 | 2.51 | 0.62 | 0.81 | 1.22 |

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## Share and Cite

**MDPI and ACS Style**

Marin, C.N.; Malaescu, I.
The Effect of Magnetically Induced Local Structure and Volume Fraction on the Electromagnetic Properties of Elastomer Samples with Ferrofluid Droplet Inserts. *Magnetochemistry* **2024**, *10*, 4.
https://doi.org/10.3390/magnetochemistry10010004

**AMA Style**

Marin CN, Malaescu I.
The Effect of Magnetically Induced Local Structure and Volume Fraction on the Electromagnetic Properties of Elastomer Samples with Ferrofluid Droplet Inserts. *Magnetochemistry*. 2024; 10(1):4.
https://doi.org/10.3390/magnetochemistry10010004

**Chicago/Turabian Style**

Marin, Catalin N., and Iosif Malaescu.
2024. "The Effect of Magnetically Induced Local Structure and Volume Fraction on the Electromagnetic Properties of Elastomer Samples with Ferrofluid Droplet Inserts" *Magnetochemistry* 10, no. 1: 4.
https://doi.org/10.3390/magnetochemistry10010004