3.1. Investigation of Complex Effective Magnetic Permeability
The frequency dependencies of the real (
μ′) and imaginary (
μ″) components of the complex effective magnetic permeability of each sample were measured with the complex impedance method. For this, both the resistance,
R (or
R0), and the inductive reactance,
X (or
X0), 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].
The elastomeric composite samples were obtained in the form of a rectangular plate (see
Figure 1) and were rolled up like a scroll, thus obtaining a cylindrical shape for the sample, which was then inserted into the suitable measuring coil to fill the entire inner space of the coil as well as possible. In this way, what was measured represented the effective permeability of the sample. In the case of samples A
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.
In
Figure 3a,b, it can be observed that, at a constant value of the volume fraction,
φ, the real component,
μ′, of the complex effective magnetic permeability remains approximately constant with the frequency change. Also, for all samples,
μ′ increases by increasing the volume fraction,
φ, of the magnetite particles dispersed in the composite. It should be noted that the values of
μ′ corresponding to samples A
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.
The imaginary component,
μ″, of complex effective magnetic permeability has a local maximum (
Figure 3) at a frequency,
fmax, 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.
From Debye’s theory [
26], it is known that the relaxation time,
τ, is related to the frequency,
fmax, at which
μ″ is at maximum via the following relation:
Considering the experimental values,
fmax, 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.
The obtained maxima of
μ″ in
Figure 3a,b could be attributed to either the Néel relaxation process or the Brownian relaxation process. In the case of the Néel relaxation process, the magnetic movements of the particles rotate inside the particles, and the particles remain fixed in the composite [
27]; the relaxation time,
τN, is provided by the relation
where
τ0 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
τ0 = 10
−9 s [
27,
28].
T is the absolute temperature;
k is Boltzmann’s constant;
Vm is the magnetic volume of a particle; and
K is the anisotropy constant of particles.
The Brownian relaxation process is correlated to the particle’s rotation, or the rotation of particle aggregates, in the carrier liquid [
27], as characterized by Brownian relaxation time,
τB, which is provided by the following equation:
Here, Dh is the hydrodynamic diameter of the particle or the aggregate, and η is the dynamic viscosity of the carrier liquid.
Taking into account the values of the relaxation times obtained for the investigated composite elastomeric samples and the
dm 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.
If we assume that the relaxation process is Brownian, by replacing the values of the relaxation time corresponding to all the investigated composite samples in Equation (4) and considering the value
η = 1.2·10
−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,
Dh, of the particles in the samples. The values obtained are
Dh,A0 = 27.13 nm,
Dh,B0 = 25.32 nm and
Dh,C0 = 24.84 nm for samples A
0, B
0 and C
0 and
Dh,Ah = 28.71 nm,
Dh,Bh = 25.87 nm and
Dh,Ch = 25.20 nm for samples A
h, B
h and C
h, respectively. The values determined for the hydrodynamic diameter,
Dh, 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.
To support the statement that the relaxation maximum of the elastomeric composite samples is the Brownian relaxation maximum (given the rotation of aggregates in the carrier liquid of the ferrofluid), we performed complex magnetic permeability measurements for the ferrofluid as well. The obtained result is presented in
Figure 5. In
Figure 5, it can be observed that at a frequency of 13.76 kHz,
μ″ has a shoulder similar to the elastomeric composite samples. Moreover, when applying a low-intensity field,
H = 4 kA/m, can be observed that the shoulder turns into a well-defined relaxation maximum at a frequency of 11.39 kHz. The behavior of the ferrofluid confirms the fact that the
μ″ maximum in the composite elastomer samples is a Brownian relaxation maximum in the ferrofluid.
Also,
Figure 1 shows that ferrofluid droplet inserts are present in all samples. For samples A
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
The real component,
ε′, and imaginary component,
ε″, of the complex dielectric permittivity were determined over a frequency range of 500 Hz–2 MHz. For this, each composite sample was placed in a planar capacitor with circular plates with a diameter of 4 cm and a distance between plates of
d = 1 mm. The capacitor with a composite sample was connected to an RLC meter, and the electric field between the plates of the capacitor was perpendicular to the sample. In the case of samples Ah, Bh and Ch, obtained in the presence of an external magnetic field (see
Figure 1d–f), the electric field between the plates of the capacitor was perpendicular to the field-induced tubular structures. For a fixed frequency,
f, the RLC meter indicated resistance,
R, and reactance,
X, in the presence of a composite sample within the capacitor and resistance,
R0, and reactance,
X0, 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]:
Figure 6 shows the frequency dependence of the real (
ε′) and imaginary (
ε″) components in the frequency range 500 Hz–2 MHz at different values of volume fraction,
φ, for the magnetite particles.
As can be observed in
Figure 6a,b, at a constant value of the volume fraction,
φ, the real component, ε′, of the complex dielectric permittivity remains approximately constant with the frequency change. Also, one can observe that
ε′ increases from 2.3 to 2.9 (for samples A
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].
For a constant value of the volume fraction,
φ, the imaginary component,
ε″, of the complex dielectric permittivity decreases with increasing frequency,
f, both for samples A
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.
This result shows that preparing such samples by mixing a ferrofluid with silicone rubber in the presence of an external magnetic field leads to the obtainment of composite samples with different dielectric properties that can be controlled by a magnetic field, H and by a volume fraction, φ.
3.3. DC and AC Conductivity
It is known that, for the study of composite materials, an important parameter is electrical conductivity, σ, which can be determined from the dielectric data of permittivity [
33,
34] with the following relation:
Taking into account the experimental values of ε″ obtained for the composite elastomeric samples (
Figure 6), with Equation (6), we computed the conductivity,
σ, whose frequency dependence is shown in
Figure 7 for all investigated samples. Knowing the conductivity,
σ is very useful in understanding the transport of electric charges in the studied material [
34] and for its applications.
In
Figure 7, it can be observed that the conductivity spectrum,
σ(f), presents two regions: (1) a region in which
σ remains constant with the frequency, corresponding to DC-conductivity (
σdc), and (2) a dispersion region, where
σ depends on frequency, corresponding to AC-conductivity (
σac). In other papers [
9,
35], a similar conductivity frequency dependence was obtained for other composite samples using a combination of Fe
3O
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]:
The values of static conductivity,
σDC, 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
σDC values are listed in
Table 1.
Also, as seen in
Figure 7c, we determined the static conductivity,
σDC, of silicone rubber (SR), obtaining the value
σDC = 1.4 × 10
−9 S/m. As a result, by adding ferrofluid to the silicone rubber (SR), the static conductivity,
σDC, of the elastomeric composite samples was increased compared with the
σDC value of the silicone rubber, which was all the higher in the volume fraction of the ferrofluid (see
Table 1).
In
Table 1, it can be observed that, by increasing the volume fraction, φ, of the particles, the
σDC conductivity increases for all composite samples. Also, the values of
σDC corresponding to samples manufactured in the presence of a magnetic field (samples A
h, B
h and C
h) are higher than the
σDC values of samples A
0, B
0 and C
0, manufactured in the absence of a magnetic field. Therefore, the
σDC 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,
σDC, with respect to the
σDC 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).
The
σAC 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:
Here,
n is an exponent that is dependent on both frequency and temperature (0 <
n < 1), and
A is a pre-exponential factor [
37].
The logarithmation of Equation (8) leads to a linear dependence between
lnσAC 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(σAC)(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).
To investigate the electrical conduction mechanism in the elastomeric composite samples, several theoretical models [
38,
39] can be applied, such as the commonly used correlated-barrier-hop** (CBH) theoretical model [
39]. According to the CBH model, the exponent,
n, can be written in a first approximation as [
39]
In Equation (9),
Wm 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
Wm are shown in
Table 1.
As can be seen from
Table 1, an increase in the volume fraction,
φ, of the particles leads to an increase in the barrier energy,
Wm, of all composite samples. Also, the
Wm 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
Wm 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,
Wm, 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).