Dynamic magnetic response of ferrofluids under a static electric field

Ferrofluids are well-known colloidal suspensions of magnetic, single-domain nanoparticles in a non-magnetic carrier liquid.¹ One of the defining characteristics of ferrofluids is the ability to control their physical properties by the application of an external magnetic field.^2–6 The interaction between nanoparticles and the applied field may lead to a significant change in the structural organization of the nanoparticles, and this often has a measurable effect on a particular physical property of the ferrofluid. Magnetic nanoparticles can create complex chains in uniform fields while inhomogeneous fields apply attractive forces to nanoparticles that pull them toward the strongest field intensity region.⁷ The resulting decrease in interparticle distances may lead to dipole–dipole interactions or exchange interactions with significant consequences to magnetic properties.⁸ Among numerous experimental examples, one can mention a magnetically directed assembly of magnetite nanoparticles into arrays of magnetic wires with a remarkable impact on their saturation magnetization, coercivity, and remanence.⁹ The effect of externally applied magnetic field was utilized in a magnetic particle assembly during inkjet printing for suppression of the unfavorable coffee-ring effect.¹⁰ Magnetic field-induced aggregations of magnetic nanoparticles are also interesting from a magnetic separation and magnetophoretic application point of view.¹¹ Indeed, several experimental studies on magnetic field-driven structural changes of ferrofluids under various conditions have been published.^12–17 

Recently, it was shown that magnetic nanoparticle assembling and structural changes in ferrofluids based on transformer oils can be induced even by external electric fields.¹⁸ The formed structures are reversible and their presence influences rheological properties (electro-viscous effect)¹⁹ and dielectric properties of the ferrofluids.^20,21 Interestingly, the applied static electric field may cause formation of percolative nanoparticle chains and the ferrofluid then exhibits seemingly negative low-frequency permittivity.²² This area of ferrofluid research is not extensive yet, as the electric field effect on the structure of colloids is investigated on colloids with non-magnetic particles, e.g., polymethylmethacrylate²³ or polystyrene particles.²⁴ However, electric field-driven assembly and reconfiguration of ferrofluid nanoparticles raise further research issues and open potential applications. One of the arising questions is related to an electro-magnetic coupling in ferrofluids. Can external electric fields control magnetic properties of ferrofluids the way magnetic fields can control their dielectric properties (e.g., dielectric permittivity of ferrofluids is easily controlled by magnetic fields^25–27)? To the best of our knowledge, this issue has not received much experimental attention yet. On the other hand, the electric field control of magnetism is well documented for solid materials, e.g., at oxide interfaces.²⁸ Thus, it is the aim of the present study to contribute to this issue via experimental investigation of alternating current (AC) magnetic susceptibility of transformer oil-based ferrofluid exposed to a direct current (DC) electric field.

Initial magnetic susceptibility $χ=∂M/∂H$ is one of the most important properties of ferrofluids, describing the response of magnetization M to a weak magnetic field H. Its frequency dependence is a well-known complex function represented by the real $χ′ω$ and imaginary $χ″ω$ components as follows:

where $ω=2πf$ is the angular frequency given by the frequency $f$ of the AC magnetic field. The use of AC susceptibility measurements in ferrofluids has been pioneered by Fannin.^29–33 The obtained AC susceptibility spectra enable the determination of the Brownian and Néel relaxation times, which are strongly associated with the nanoparticle size. For this reason, AC susceptibility measurements have been used for probing nanoparticle coatings³⁴ and for the determination of core and hydrodynamic size distributions of ferrofluid nanoparticles.^35–37 Frequency-dependent susceptibility characteristics were also introduced as a suitable tool for monitoring colloidal stability,³⁸ an alternative magnetic particle imaging strategy,³⁹ or as a method of detection of light-induced agglomeration in ferrofluids.⁴⁰ Special theoretical attention has been given to the influence of dipolar interactions and correlations between nanoparticles on the dynamic (AC) susceptibility of ferrofluids.^41–43 It was found that at low concentrations, self-assembled nanoparticles may reduce the susceptibility,⁴⁴ while at higher concentrations, dipolar correlations have the opposite effect.⁴⁵ It is known that interparticle dipole–dipole interactions do not affect the Néel mechanism significantly, but they do slow down the Brownian mechanism. Various theoretical approaches to the dynamical response of interacting ferrofluid nanoparticles and distinction between Brownian and Néel relaxation in the interacting system are available.^46,47 Recently, a theory for the frequency-dependent magnetic susceptibility of a ferrofluid in a static uniform magnetic field, including dipolar interactions, was developed.⁴⁸ 

In this paper, a study of AC magnetic susceptibility of transformer oil-based ferrofluids in a static electric field is presented. Namely, we demonstrate how an applied voltage causes variations in values of the susceptibility spectrum. The effect of the electric field acting parallel and perpendicular to the probing AC magnetic field is demonstrated.

The ferrofluid investigated in this study is based on inhibited transformer oil TO 40 A MOL, with a density of 0.86 g/cm³, pour point 228 K, flash point 413 K, and kinematic viscosity 22 mm²/s. The dispersed phase is composed of co-precipitated iron oxide nanoparticles stabilized with oleic acid in a well-proven way.^49,50 Eight ferrofluid samples, labeled from FF1 to FF8, were prepared by diluting the stock concentrated ferrofluid in the base oil. Selected physical parameters of the ferrofluids are summarized in Table I.

Magnetization of the ferrofluids was measured by means of a vibrating sample magnetometer installed on a cryogen-free superconducting magnet (Cryogenic Ltd.). The magnetization curves measured at 295 K and magnetic field up to 6 T are shown in Fig. 1(a). Each curve exhibits zero hysteresis, thus pointing out the superparamagnetic state of the nanoparticles. The increase in magnetization of saturation reflects an increase in nanoparticle concentration. The magnetic mass fraction was determined as a ratio of the magnetizations of saturation of the particular ferrofluid and the powder nanoparticles (73 Am²/kg). Nanoparticle size distribution from the magnetization data (superposition of Langevin fitting functions applied to the magnetization curve) is presented in Fig. 1(b), revealing the mean magnetic diameter of 11.08 nm. Temperature-dependent magnetization measurements are carried out in the zero-field-cooling (ZFC) and field-cooling (FC) regime, in a static magnetic field of 10 mT [Fig. 1(c)]. The ZFC curve shows a pronounced maximum around 60 K corresponding to the area of blocking temperatures of the nanoparticles. A kink at 200 K is associated with the solid–liquid phase transition of the carrier oil.

The solid volume fraction of the FF samples is determined according to the well-known formula $φ=(ρFF−ρTO)/(ρNP−ρTO)$⁠, where $ρFF$⁠, $ρTO$⁠, and $ρNP$ are the densities of the ferrofluid, transformer oil, and magnetic nanoparticles, respectively. The bulk density of magnetite (5.17 g/cm³) was considered in the calculations. Dielectric permittivity of the FF samples is determined from capacity measurements by using an LCR meter Agilent E4980A at 1 kHz frequency and room temperature. The permittivity of the pure transformer oil is 2.12. For each FF sample, no on-shelf sedimentation was observed even after a year of rest.

The AC magnetic susceptibility was measured by a commercial AC susceptometer (IMEGO, DynoMag, SE), working at laboratory temperature and frequencies from 1 Hz up to 250 kHz, with a volume susceptibility resolution of 4 × 10⁻⁷. The amplitude of the excitation field is 0.5 mT. The AC susceptibility of FF samples was measured in a glass vial with an inner diameter of 6.2 mm and sample volume of 200 μl [Fig. 2(a)]. The period of one measurement (whole spectrum) is approximately 17 min. Before the measurements, the background calibration was performed with the empty glass vial, while the gain and phase were calibrated by using a Dy₂O₃ powder sample. Thus, the susceptibility measurements of FF samples are free from the glass vial diamagnetic contribution.

In other to perform measurements of AC magnetic susceptibility under a static electric field, the experimental vial was equipped with a pair of electrodes. To supply the electrodes with a DC voltage, a signal generator (33210A, Agilent, USA) and a high-voltage amplifier (40/15, TREK, USA) were employed. Disk-shaped copper electrodes were fixed at the bottom of the vial [Fig. 2(b)]. Upon application of an electric voltage, an electric field is generated between the electrodes, with intensity parallel to the AC magnetic field intensity. The gap between the electrodes is filled with a ferrofluid [Fig. 2(c)], while the electrode separation distance corresponds to the column height of the calibration powder sample (0.5 cm). A perpendicular configuration of the probing AC magnetic field and the static electric field was achieved by inserting a pair of U-shaped brass electrodes into the vial [Figs. 2(d) and 2(e)]. Again, the height of these electrodes and the ferrofluid column matches the height of the calibration standard (0.5 cm). Moreover, the U-shaped electrodes' location in the vial allows the insertion of a thin thermometer [Fig. 2(d)].

The electric field intensity distribution between the electrodes holding 2400 V with the transformer oil as a medium is depicted in Fig. 3. The numerical simulation was performed using the Finite Element Method Magnetics (FEMM 4.2) software. According to the color scale, a quasi-uniform electric field is generated between the disk–disk electrodes, however with noticeable gradients at the electrode edges. A less uniform electric field is found between the U-shaped electrodes with stronger gradients at the edges.

In addition to the built-in thermometer of the AC susceptometer, two external fiber optic temperature sensors (FPI-HR module equipped with the fiber optic temperature sensors, model FOT-L-SD-C1-F2-M2-R5-SCAI, FISO Technologies; Canada) were employed to monitor the ferrofluid temperature during the experiment. Temperature resolution of the sensors is ±0.1 K. It is verified that the temperature sensors are not affected by the applied electric field. Thus, the whole experimental setup consists of three parts. As Fig. 4 shows, the first measurement circuit is focused on the measurement of electric current flowing in the circuit with the ferrofluid exposed to high voltage (HV). The current I is determined according to Ohm's law from the voltage drop on the connected serial resistor with a resistance of 3.09 MΩ. The voltage is measured using data acquisition recorder HIOKI MR8880–20. The middle of the scheme represents the sample chamber axially placed in the excitation and detection coils. This circuit is devoted to the accurate measurement of the AC magnetic susceptibility. On the right side, the temperature monitoring circuit is depicted with two thermometers connected to the channels of the FISO module. One thermometer is immersed in the tested ferrofluid, the other is placed near the excitation coil.

The whole spectra of real and imaginary AC magnetic susceptibility measured on each FF sample are presented in Fig. 5. The real component of susceptibility exhibits a quasi-constant behavior in the frequency range up to 10 kHz. Above this frequency, a moderate decrease in real susceptibility with increasing frequency is observed, especially for ferrofluids with higher nanoparticle volume fraction φ. Clearly, the increase in φ results in the increase in magnitude of real susceptibility, as indicated in Fig. 5 by the arrow. The constant behavior of real susceptibility up to 10 kHz is accompanied by zero magnetic losses (imaginary component), while the moderate real susceptibility decrease above 10 kHz is reflected in a moderate increase in the imaginary susceptibility.

The absence of any remarkable real susceptibility dispersion or a maximum in the imaginary component confirms the small nanoparticle size and relatively narrow size distribution. The individual nanoparticles apparently prefer the Néel relaxation mechanism, which is associated with magnetization reversal from one easy direction of magnetization to another by overcoming an energy barrier. The switching time (Néel relaxation time) is generally given by the following expression:⁵¹ 

where $f0$ is a constant with an often-quoted approximate value of 10⁹ s⁻¹, KV is the energy barrier given by K—the anisotropy constant (a typical value for magnetite nanoparticles is 2–5 × 10⁴ J m⁻³) and V—the nanoparticle volume. The thermal energy is determined by kT (Boltzmann constant and absolute temperature). Considering the obtained mean magnetic diameter 11.08 nm [Fig. 1(b)] and calculating the Néel relaxation time according to Eq. (2), one gets a value of 33.78 × 10⁻⁹ s. Thus, the relaxation maximum would appear at a frequency 4.7 MHz. However, this frequency region is not covered in our experiment. It can be shown that the Néel relaxation mechanism is faster than the Brownian relaxation mechanism, which relies on nanoparticle rotation as a whole. In ferrofluids, the Brownian relaxation time has a hydrodynamic origin and is determined as follows:¹ 

with $η$ and $kB$ denoting the dynamic viscosity of the carrier liquid (transformer oil) and the Boltzmann constant, respectively. Considering an approximate hydrodynamic diameter of 17 nm (including the magnetic dead layer and the surfactant thickness), the Brownian relaxation time is 3.65 × 10⁻⁵ s and the corresponding relaxation maximum in the frequency domain would appear around 4.3 kHz. Clearly, the faster Néel relaxation mechanism is preferred in our samples.

In further experiments, it is intended to observe the susceptibility spectra under the influence of a static electric field. A few FF samples with low, middle, and higher nanoparticle concentration have been chosen for this study. In Fig. 6, the real susceptibility spectra of three samples (FF1, FF4, and FF7) under various voltages applied on the disk–disk electrodes are presented. In this electrode configuration, the electric field intensity is parallel to the probing magnetic field intensity. However, the disk–disk electrodes do not allow insertion of the temperature sensor directly into the ferrofluid. Instead, the temperature is read from the built-in thermometer near the excitation coil. As the figure shows, the real susceptibility of FF1 under 0 kV has constant values at low frequencies and from 1 kHz, a dramatic susceptibility drop is observed. This drop is caused by the diamagnetic contribution of the copper electrodes, because the calibration was performed for the empty glass vial background. One should therefore realize that both the ferrofluid and the electrodes (especially at higher frequencies) contribute to the measured susceptibility. In further analysis, we will focus on real susceptibility below 1 kHz, as the signal from electrodes dominates the spectrum above 1 kHz. In this frequency range, the imaginary susceptibility exhibits zero or very low values and therefore only the real component is presented. It is seen that the susceptibility spectrum of FF1 under 0 kV shows values about 7 × 10⁻³. However, this level decreases upon application of each voltage value (from 0.6 kV up to 2.4 kV), but the profile of the spectrum does not change. At 2.4 kV, the susceptibility spectrum is again quasi-constant about the value 4.5 × 10⁻³. In comparison to the initial susceptibility value under zero voltage, the susceptibility of FF1 under 2.4 kV is lowered by about 36%. A similar trend of decreasing susceptibility of unchanged profile with increasing voltage is found for FF4 and FF7. However, the percentual decrease in both cases is about 1%. The measurement of FF1 at 0 kV started at the coil temperature 298.05 K (measured by the device built-in thermometer). At the end of the last measurement of FF1 at 2.4 kV, the coil temperature was 298.55 K. During the five measurements of FF1, the coil temperature thus rose by 0.5 K. During the five measurements of FF4 and FF7, the coil temperature rose by 0.8 K (from 297.55 to 298.35 K) and 0.2 K (from 298.65 to 298.85 K), respectively. Following the slight increase in temperature of the excitation coil, one can assume that the generated heat may be transferred to the ferrofluid. However, under limited conditions of heat exchange (the air gap between the coil and the sample holder, Teflon holder of the glass vial), the increase in sample temperature due to the heated coil may be considered to be insignificant. Without direct measurement of the sample temperature, one cannot draw a strong conclusion from the observed drop of the susceptibility spectrum.

As the imaginary susceptibility in the considered frequencies is near zero, the heating of ferrofluids due to magnetic relaxations (magnetic losses) is also negligible. On the other hand, the Joule heating due to electric current flowing through the ferrofluid may be considered as another source of the ferrofluid temperature increase. As the ferrofluid susceptibility is strongly coupled with temperature (Curie–Weiss law), these aspects must be considered in the interpretation of susceptibility decrease under the applied voltage.

An unambiguous explanation of the moderate ferrofluid susceptibility decrease requires a precise measurement of the ferrofluid temperature during susceptibility measurements. This is possible in the U-shaped electrode geometry, when the electric field acts perpendicular to the probing magnetic field. The obtained results are presented in Fig. 7. On the left side, the real susceptibility spectra at various voltages are presented for the three samples, while the development of the sample and coil temperature during measurements is shown in the graphs on the right side.

As Fig. 7 shows, there are negligible variations in the values of FF1 susceptibility spectra with applied voltage. This behavior can be associated with negligible temperature variations of FF1 at various voltages, as shown in the graph on the right side. On the other hand, a gradual decrease in the whole susceptibility spectrum of FF4 and FF7 with increasing voltage is revealed. However, in these cases, the sample temperature noticeably increases upon increasing the voltage level. The temperature peaks emergent at the voltage step change may be ascribed to two reasons. The first is related to the current impulse, which may result in Joule heating of the ferrofluid. The other reason may be associated with the sample holder moving out of the coil (sample chamber) at the end of each spectrum measurement, resulting in the change of heat transfer conditions. To better understand the susceptibility variations, we performed its measurement on FF8 (giving the strongest signal) at various voltages, rising and decreasing in a hysteresis-like manner (Fig. 8).

From Fig. 8(a) one can clearly see that the susceptibility spectrum drops down upon application of the voltage step increase. However, when the voltage decreases back to 0 kV [Fig. 8(b)], the spectrum retains the previously achieved level except for a small drop measured at 0 kV, which is remarkably below the susceptibility level measured initially at 0 kV. By application of the negative voltage values [Fig. 8(c)], the susceptibility spectrum slightly increases, but again up to the level measured previously at 4 kV. This susceptibility level and the whole spectrum are further retained also during the reverse voltage change back to 0 kV [Fig. 8(d)] and during the repeated increase to 4 kV [Fig. 8(e)]. It can be therefore said that the final susceptibility level was reached during the initial voltage increase [Fig. 8(a)], while during the following variations in voltage, the susceptibility retained the decreased values and did not return to the initial value. This behavior is highlighted in Fig. 8(f), where the susceptibility at a selected frequency (206 Hz) is drawn in dependence on the applied voltage. Again, this behavior must be analyzed with respect to the sample temperature. The time-dependent temperature variations during the susceptibility measurements shown in Fig. 8 are presented in Fig. 9.

The graph in Fig. 9 shows that the temperature of the sample and the coil environment in the beginning of the experiment (0 kV) is slightly below 297 K. Upon step changes of the voltage up to 4 kV, the sample temperature exhibits step-like increases, while the coil temperature increases steadily. At 4 kV, one can see the highest sample temperature increment that apparently caused further increase in temperature of the coil due to heat transfer. During the decrease in voltage from 4 to 0 kV, a step-like decrease in the sample temperature is found. However, at 0 kV, the sample temperature does not drop to the value measured at the initial 0 kV state. This is associated with inherent thermal inertia. Then, the temperature behavior is repeated when the voltage of opposite polarity increases and decreases back to 0 kV, and finally increases to 4 kV. During the action of these voltages, the sample temperature exhibits step-like changes around 298.2 K. One can notice that this temperature level was achieved during the initial voltage step increase. Thus, the observed sample temperature behavior apparently corresponds with the susceptibility changes presented in Fig. 8. From Fig. 9 it is seen that the sample temperature increases to a greater measure than the coil temperature. This implies that besides the heating of the coil, there are other sources of heat causing the increase in sample temperature. The magnetic losses due to the relaxation processes are negligible (below the measurable limit) at the presented frequencies, and therefore any heating due to magnetic relaxations is also considered to be negligible. The sample temperature is remarkably sensitive to the voltage step change, which results in a current impulse. The current impulses appearing as a response to the voltage step change are presented in Fig. 10. Upon application of the voltage step change, the current impulse is produced and during the measurement at a particular voltage level, the current exhibits continuous decrease (charging behavior at both voltage polarities).

The increase in sample temperature due to Joule heating can be also quantitatively estimated from the measured current values. For ferrofluid FF8, the maximal current reaches values up to 4 μA. Clearly, this current value is temporal upon application of the voltage level (let us consider 4 kV). The flow of this current (4 μA) through the ferrofluid during 10 s corresponds to the generated Joule heat of 0.16 J. Then, considering the sample mass (0.236 g) and the specific heat of the ferrofluid (2.273 kJ·kg⁻¹·K⁻¹, calculated according to Ref. 52), one finds that such a heat can cause increase in temperature by about 0.3 K. This estimation agrees with the observed temperature variations caused by the applied voltage step change.

However, according to the simple Debye's theory, the complex susceptibility $χω$ of ferrofluids is given by the following equation:

where $χ∞$ and $χ0$ are the real susceptibilities at high- and low-frequency limits, respectively.^22,24 From the presented susceptibility spectra, it is clear that, $χ0$ is the parameter most strongly influenced by the applied voltage. This parameter is dependent on the nanoparticle number density n, magnetic moment of the nanoparticles m, and temperature T as follows:

with $μ0$ denoting the vacuum permeability. The presented experiments showed a decrease in $χ0$ with increasing voltage. According to (5), the decrease in $χ0$ may reflect a decrease in n and m or increase in T. The effect of increasing T has been discussed above and the effect of varying n is seen from Fig. 5. However, if a fraction of the magnetic nanoparticles with extrinsic superparamgnetism would be trapped in the gradient of the static electric field (unable to respond to the AC magnetic field), the real magnetic susceptibility would noticeably decrease. The trapping of nanoparticles at the electrodes may be facilitated whether by electrophoresis or dielectrophoresis. The former effect occurs due to the action of electric field on the fixed, net charge of the nanoparticle, while dielectrophoresis only occurs when there are induced charges, and only results in motion in a non-uniform field (this can be a static or an alternating field).⁵³ The presence of a charge on the nanoparticles can be verified by measuring the zeta potential (electrokinetic potential). We have therefore undertaken measurements of the zeta potential on the diluted ferrofluid by using Zetasizer Nano ZS (Malvern Instruments Ltd., Malvern, UK). However, the experiment confirmed unmeasurable zeta potential. This is quite clear because the ferrofluid is based on a non-polar liquid. On the other hand, this experimental result does not necessarily mean the nanoparticles experience no electrophoresis. Realizing that the iron oxide nanoparticles have significant conductivity and may carry an uncompensated electric charge, the electrophoretic effects cannot be ruled out. Nevertheless, along with electrophoresis, dielectrophoresis can also be a significant mechanism accounting for nanoparticle trapping in the field gradient, resulting in the decreased nanoparticle number density n. Owing to the dielectric contrast between the non-polar carrier liquid and the iron oxide nanoparticles, the nanoparticles can be polarized in a static, non-uniform electric field and the induced electric dipoles of greater nanoparticles may be pulled toward the electrodes. The magnitude of the dielectrophoretic force F can be estimated according to the well-known effective dipole moment approach, which for a spherical nanoparticle is expressed as follows:⁵⁴ 

where a is the radius of the nanoparticle; $ε1$ and $ε2$ are the relative permittivities of the fluid medium and the nanoparticle, respectively; and E is the electric field intensity. Based on the simulated gradient of the squared electric field intensity, which reaches values up to 3 × 10¹³ near the electrodes, and considering the size (a = 5.54 nm) and permittivity of the nanoparticles (80) and the oil (2.1), one finds the maximal dielectrophoretic force acting on nanoparticles near the electrodes to be 6.5 × 10⁻¹¹ N. It is clear that with increasing distance from the electrode toward the center between the electrodes, this force rapidly decreases. However, the estimated dielectrophoretic force is sufficiently high to overcome the thermal energy or the force of gravity (around 3 × 10⁻²⁰ N, neglecting the viscous forces) acting on nanoparticles. The competitive magnetic drag force is also neglected in this consideration due to the absence of remarkable magnetic field gradients, as the ferrofluid sample is inserted in the center of the long solenoid.

While the static electric field is applied, the dielectrophoretically and electrophoretically trapped nanoparticles are not capable of responding to the AC magnetic field. As a result, the nanoparticle number density n decreases, and so decreases the low-frequency magnetic susceptibility $χ0$⁠. On the other hand, one has to bear in mind that the magnetic nanoparticles in ferrofluid forced by the static electric field into geometrical clusters¹⁸ can exhibit magnetic interactions resulting in an increased or reduced net magnetic moment m. According to (5), the reduction of m can be another reason for the observed decrease in low-frequency magnetic susceptibility $χ0$⁠. This hypothesis requires further experimental verification, e.g., by polarized neutron scattering or magnetic resonance.

Based on the presented susceptibility decreases due to the acting static electric field, it can be shown that increased sample temperature is not the only reason for the decreased susceptibility. If one considers Eq. (5), the temperature should not contribute to that much decrease in the susceptibility as observed in Fig. 6 and Fig. 7. From Fig. 6, it is seen that the susceptibility of FF1 under 0 V is 7 × 10⁻³ at temperature 298.05 K. Under 2.4 kV, the temperature is 298.55 K and the measured susceptibility is 4.5 × 10⁻³. Such a dramatic drop cannot be caused by such a small temperature increase, because according to (5), the calculated susceptibility is 6.98 × 10⁻³. Similar differences in the measured and calculated susceptibility at the maximal applied voltage are presented in Table II for FF4 and FF7 from Fig. 7 too. As the calculated temperature-dependent susceptibility values are remarkably greater than those measured in the experiment, it is reasonable to consider the above-mentioned mechanisms (nanoparticle trapping in the gradient electric field and magnetic moment reduction in the electric field-induced nanoparticle clusters) as physical reasons for susceptibility decrease under the action of the static electric field.

In this study, the potential electro-magnetic coupling in transformer oil-based ferrofluid has been experimentally verified via AC magnetic susceptibility measurements. It is stressed that such experimental measurements must be accompanied by simultaneous and direct measurements of the ferrofluid's temperature. It is found that ferrofluid's real susceptibility decreases with increasing electric field (voltage). Even though the observed susceptibility decrease corresponds with the increasing trend of measured temperature, the temperature effect cannot be the only reason for the susceptibility change. In addition, other mechanisms resulting from the nanoparticle–electric field interactions must contribute to the observed magnetic susceptibility decrease. Dielectrophoretic nanoparticle migration and trapping in the static electric field gradient (near the electrodes) are considered to be crucial in the interpretation of the electric field-dependent magnetic susceptibility of the ferrofluid. The trapped nanoparticles are not capable of responding to the alternating magnetic field, and the effect thus resembles the decreasing nanoparticle concentration. As the electric field-induced decrease in magnetic susceptibility is observable even in the interrupted (zero-voltage) electric field, the third reason disabling the return of the ferrofluid's susceptibility to initial values should be considered. The reason may be associated with magnetic interactions of the nanoparticle in the clusters previously induced by the static electric field. The interactions can result in a reduced magnetic moment and subsequently in decreased magnetic susceptibility. These results and discussion stimulate the application of other experimental methods, like in situ magnetic resonance or polarized neutron scattering techniques. In this way, the established ferrofluid research may be directed to a new field of electro-magnetic study of ferrofluids and broaden the potential applications of ferrofluids.

This research was funded by the Slovak Academy of Sciences and Ministry of Education in the framework of Project Nos. VEGA 2/0011/20 and 1/0154/21; Slovak Research and Development Agency under Contract No. APVV-18–0160; and the Cultural and Educational Grant Agency of the Ministry of Education, Science, Research and Sport of the Slovak Republic (KEGA) under Project No. 008TUKE-4/2019. The work also received support from the NATO Science for Peace and Security Programme (No. G5683); European Regional Development Fund, Project No. ITMS 26220220186 (PROMATECH); and Structural Funds of EU, Ministry of Education, Slovakia, project MODEX (No. ITMS 313011T548).

The data that support the findings of this study are available from the corresponding author upon reasonable request.