This study investigates the direct current behavior of a kerosene-based ferrofluid. Experimental data reveal linear current–voltage (⁠ I– V⁠) characteristics, measured using slab-shaped cells with platinum electrodes. A polarization voltage of a few millivolts is observed, which is attributed to electric polarization effects. This phenomenon is interpreted through a model based on adsorption processes. The I– V characteristics are explained by the dissociation of surface ligands on the nanoparticles, generating ionic charges responsible for conduction. The dissociation process follows a chemical equilibrium described as A ⇌ B + + C −⁠, governed by dissociation and recombination coefficients consistent with standard kinetic models.

Topics
Recombination reactions, Current-voltage characteristic, Kerosene, Chemical equilibrium, Electrical conductivity, Nanoparticle, Magnetic fluids

Ferrofluids (FFs) are colloidal dispersions of magnetic nanoparticles (NPs) in a liquid carrier. From the rheological point of view, they behave as isotropic liquids, but at the same time, they exhibit magnetic properties.1 Oil/kerosene-based ferrofluids constitute a large class of surfacted organic-based magnetic colloids. In these systems, nanoparticles of around 10 nm in size are coated with organic molecules to avoid agglomeration. Usually, NPs are grafted with oleic acid molecules such that the colloidal stabilization occurs simply by steric hindrance provided by the long carbon chains. These kinds of FFs are expected to be good dielectrics because kerosene and/or mineral oil are usually free of mobile ions. The presence of free ions inside them has already been reported.2 

In the literature, many papers deal with electrical properties of kerosene-based FFs. Most of these papers refer to magneto-dielectric studies.3,4 The author’s method consists of applying a voltage of sinusoidal type and measuring the system’s response as a function of the frequency. Studies using DC voltage are rare. When a cell filled with this type of FF is subjected to an electric voltage, an electric current flows inside the sample. In this work, we analyze the electric current response I of a kerosene-based FF sample when it is subjected to a DC electric potential V⁠.

The analysis of the I × V curve is relevant from a theoretical point of view, since it can give information on the conduction mechanism in these systems and on the adsorption mechanisms taking place on the electrodes.

Considering the technological aspect, this type of FF has a broad range of applications, such as magnetic sealing, sensors, and energy transport.5,6 Moreover, oil-based ferrofluids are particularly useful as cooling agents for electric transformers.7,8 The FF sample used in this study is a typical kerosene-based ferrofluid, composed of core@shell nanoparticles (⁠ CoFe 2 O 4@ γ − Fe 2 O 3⁠) coated with oleic acid. These NPs are highly attractive for various applications, as they combine a strongly responsive magnetic core of CoFe 2 O 4 with a chemically stable and highly tunable surface of γ- Fe 2 O 3⁠.

The kerosene-based ferrofluid used in this study was prepared through an adapted co-precipitation method9,10 consisting of four main steps, as shown schematically in Fig. 1. First, cobalt ferrite NPs (⁠ CoFe 2 O 4⁠) were obtained by alkalinizing 1:2 mixtures of 0.5M Co ( NO 3 ) 2 ⋅ 6 H 2 O and 0.5M FeCl 3 ⋅ 6 H 2 O with 2M NaOH under vigorous stirring at boiling temperature for 30 min. Next, the NPs were washed two times with de-ionized water and once with 2M HNO 3 to remove any unwanted less soluble by-product formed during co-precipitation. In the third step, the precipitate was hydrothermally treated with a 1M Fe ( NO 3 ) 3 ⋅ 9 H 2 O solution at boiling temperature for 15 min, leading to the core@shell nanoparticles (⁠ CoFe 2 O 4@ γ − Fe 2 O 3⁠), with typical spherical-like shape and a mean diameter of 10 ± 1 nm. The NPs were then removed by magnetic decantation and washed with acetone. Finally, after evaporating the acetone, the NPs were surfacted by adding oleic acid under stirring and then dispersed in kerosene.11 The system was centrifuged at 10 000 rpm for 15 min to remove the aggregates and the final ferrofluid sample (volume fraction 1%) was labeled COK. The resulting NP concentration is N 0 = 1.1 × 10 22 m − 3⁠.

FIG. 1.
Schematic representation of the synthesis of the ferrofluid used in this work.
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Schematic representation of the synthesis of the ferrofluid used in this work.

FIG. 1.
Schematic representation of the synthesis of the ferrofluid used in this work.
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Schematic representation of the synthesis of the ferrofluid used in this work.

Close modal

The electric current was measured employing a Keithley picoammeter model 6485. The electric potential was supplied by a very stable homemade source based on an electrochemical battery. By setting a specific electric potential value, the source can provide an electric current that randomly oscillates within a range of 0.7% around a mean electric current value for at least 30 min.

The experimental setup is straightforward. A schematic representation is shown in Fig. 2. It consisted of the electric source, the picoammeter, and the cell filled with the FF under investigation, all connected in series. The cell comprises two parallel electrodes made of pure platinum in the form of a disk of radius 1 cm separated by d = 127 μ m⁠. The distance between the electrodes is controlled by a spacer in the form of a ring made of polyimide (kapton). The internal radius of the spacer is ∼ 0.85 cm such that the area of the electrodes in contact with the FF is S = 2.3 cm 2⁠.

FIG. 2.
Schematic representation of the experimental setup and the cell used in the measurements.
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Schematic representation of the experimental setup and the cell used in the measurements.

FIG. 2.
Schematic representation of the experimental setup and the cell used in the measurements.
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Schematic representation of the experimental setup and the cell used in the measurements.

Close modal

Various voltage values ranging from 0.12 to 80.0 mV were applied to the sample. We chose small values of electric potential to prevent nonlinear effects as much as possible.12 The electric current response I ( t ) was then measured as a function of time t for about 20 min. Let us call t M this measurement time.

Consider a cell in the shape of a slab of thickness d very large concerning the Debye length of the solution under consideration. In this framework, far from the electrodes, the system is locally neutral. Two identical plane-parallel electrodes limit the cell. The medium is assumed to be an insulating liquid, of dielectric constant ε⁠, containing impurities (NPs). The impurities can dissociate according to the chemical reaction A ⇌ B + + C −⁠, where A indicates the neutral species and B + and C − indicate the positive and negative species generated from the dissociation of A⁠. In our ferrofluid system, A represents the nanoparticles, while B + and C − correspond to the positively charged nanoparticles and oleate anions after dissociation, respectively. The kinetic dissociation and recombination constants are denoted as k d and k a⁠, respectively. The charged species are supposed to be monovalent, with an electrical charge q = 1.6 × 10 − 19 A s⁠. We indicate by n p⁠, n m⁠, and n n the time dependent, out of thermodynamical equilibrium, bulk density of positive, negative, and neutral species. According to the kinetic theory of chemical reactions, the rate of production of ions or the recombination of them is given by
(1)
(2)
(3)
In an infinite sample (⁠ d → ∞⁠) and in the absence of an external power supply, we are in the conditions of thermodynamical equilibrium, where the bulk density of neutral, positive, and negative ions is position and time independent. This means that d n n / d t = d n p / d t = d n m / d t = 0⁠. If we indicate by N 0 the bulk density of dissociable particles, and by N n and N = N p = N m the bulk densities of neutral and charged particles in thermodynamical equilibrium, from Eqs. (1)–(3) we have
(4)
The bulk density of ions, in thermodynamical equilibrium, is given by
(5)
as discussed in the  Appendix. Equation (5) shows that for small N 0⁠, i.e., for N 0 ≪ k d / ( 4 k a )⁠, N → N 0⁠. This means that for diluted systems the dissociation is complete.
From Eq. (5), it follows that for k d = 0⁠, N = 0⁠, i.e., there are no dissociated particles (ions). Since the coating layer is tightly attached to the NPs surface,13,14 it is interesting to consider the case of k d / k a → 0⁠. In this case from (5), we get
(6)
indicating that in this limit N → 0 and N n / N 0 → 1⁠, as expected. In the opposite limit of where k d / k a → ∞⁠, we obtain
(7)
i.e., in the limit of large dissociation constant N → N 0 and N n → 0⁠. See Fig. 3. Derivation of (6) and (7) is reported in the  Appendix.
FIG. 3.
Theoretical dependence of N / N 0, solid line, and N n / N 0, dashed line, on k d / k a.
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Theoretical dependence of N / N 0⁠, solid line, and N n / N 0⁠, dashed line, on k d / k a⁠.

FIG. 3.
Theoretical dependence of N / N 0, solid line, and N n / N 0, dashed line, on k d / k a.
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Theoretical dependence of N / N 0⁠, solid line, and N n / N 0⁠, dashed line, on k d / k a⁠.

Close modal
From Eq. (5), it is possible to evaluate the dissociation coefficient ρ of the chemical reaction A → B + + C − in the framework of the proposed model. It is defined by N = ρ N 0⁠. A simple calculation gives
(8)
where we have introduced the dimensionless parameter related to the dissociation chemical reaction γ defined by
(9)
depending on the dissociation and recombination coefficients (⁠ k d and k a⁠) and the density of dissociable particles. We note that the parameter N / N 0 represents the degree of dissociation, since it is defined as the ratio between the bulk density of dissociated particles and the density of dissociable particles. It depends not only on k d and k a⁠, related to the chemical reaction A → B + + C −⁠, but also on N 0⁠. In Fig. 4, it is shown the dependence of ρ on γ⁠. Since for γ → 0⁠, ρ → γ⁠, and for γ → ∞⁠, ρ → 1 − 1 / γ⁠, it follows that ρ is very sensitive to the variation of γ for small γ⁠, whereas it is practically independent of it for large γ⁠, as it is evident from Fig. 4.
The electrical current through the cell submitted to Δ V for a time t M is
(10)
where σ e is an effective conductivity. It must be noted that σ e is different of the conductivity of the FF itself. This is because the applied direct current (DC) voltage causes the ions to migrate toward the electrode surfaces. Consequently, the electric field within the sample is screened by the ionic layer.
FIG. 4.
Dependence of the dissociation coefficient ρ on the dimensionless parameter γ = k d / ( k a N 0 ).
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Dependence of the dissociation coefficient ρ on the dimensionless parameter γ = k d / ( k a N 0 )⁠.

FIG. 4.
Dependence of the dissociation coefficient ρ on the dimensionless parameter γ = k d / ( k a N 0 ).
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Dependence of the dissociation coefficient ρ on the dimensionless parameter γ = k d / ( k a N 0 )⁠.

Close modal

The presence of a polarization voltage could be interpreted using the adsorption phenomenon. We consider the simple case in which the solution contains just one type of mobile ions. This situation well corresponds to our solution, in which NPs are so large that their mobility is very small with respect to that of the ions originating by dissociation. In this case, the ions present in the solution are attracted to the electrode. The attraction could have a chemical origin or be a consequence of the electric interaction between the ions and their images in the electrodes. In both cases, the adsorption energy depends on the dimensions of the particles. In the case of van der Waals interactions, the interaction energy of the spherical particle with the flat substrate depends on r − 3⁠, where r is the dimension of the adsorbed particle. If the interaction is of the particle with its image in the metal, the adsorption energy depends on r − 1⁠. Since the geometrical dimensions of NPs are very large with respect to those of the mobile ions, in a first approximation it is possible to neglect the adsorption of NPs.15 For this reason in the following, we limit our considerations to the adsorption of the small mobile ions. In the bulk, the solution is locally neutral. Close to the electrode, it is locally charged. The ions adsorbed from the surface give rise to a surface electric field, responsible for a difference in potential between the electrode and the bulk. If the sample is a slab, the effective polarization potential is the difference between the differences in potential due to the two electrodes with respect to the bulk. In a perfectly symmetric cell, the effective difference in potential vanishes due to the adsorption effect. However, since a perfect symmetry is difficult to reach, a polarization voltage across the cell is expected, usually of the order of a fraction of the thermal voltage. In the following, we evaluate the difference in potential between the electrode and the bulk, assuming that the adsorption phenomenon is well described by the Langmuir isotherm.

Let us assume that the sample is rather thick, in the sense the thickness is very large compared to the Debye length of the solution. The evolution of the bulk density of ions and the actual electric potential across the cell are described by the equation of continuity and the equation of Poisson
(11)
(12)
where N is the bulk density of ions in the equilibrium state (in the bulk), q is the electric charge of the mobile ion, E is the electric field, and the current density is
(13)
In (13), the first contribution refers to the diffusion current, the second to the conduction current, and D is the diffusion coefficient of the ions in the solution. In the equilibrium state, and for our one-dimensional problem, in the Einstein–Smoluchowski approximation (⁠ μ / D = q / ( k B T )⁠, where k B is the Boltzmann constant and T is the absolute temperature) from (12) and (11), taking into account (13) we get
(14)
(15)
where ε is the dielectric constant of the medium in which NPs are dispersed (kerosene), and the electric potential V is related to E by E = − ∇ V⁠. We define
(16)
representing the electric potential measured in k B T / q = v t h⁠, and the length of Debye of the system, repectively, and rewrite (14) and (15) as follows:
(17)
(18)
Assuming the sample as a half-space with the adsorbing surface at z = 0⁠, the reduced potential is such that
(19)
In this case, a standard calculation gives for the ionic density
(20)
and for the reduced electric field
(21)
We suppose that the adsorption phenomenon is described by Langmuir’s isotherm. In this case,
(22)
where Σ is the surface density of adsorbed particles, n ( 0 ) is the bulk density of adsorbable particle just in front of the surface, and κ and τ are the adsorbing coefficient and the desorption time. In the equilibrium state, d Σ / d t = 0 and
(23)
Taking into account that the surface electric field and the surface density of adsorbed particles are related by the equation
(24)
using expressions (20) and (21) we get
(25)
Equation (25) determines the surface reduced potential u p⁠. In the limit of small u p⁠, i.e., V p < k B T / q⁠, from (25) we get
(26)
In Fig. 5, it is reported the numerical solution of Eq. (25), in solid line, and its approximated solution, in dashed line, given by (26), vs the adsorption parameter κ τ / Λ⁠. The horizontal line corresponds to the reduced surface potential u p experimentally found for our system (22 mV). As it is evident, the approximated solution works well only for very small u p⁠. Already for u p ∼ 1 the agreement is very poor.
If the sample is a slab, limited by electrodes we indicate by the subscripts 1 and 2, the effective polarization voltage, due to the adsorption phenomenon, is
(27)
FIG. 5.
Reduced surface potential u p vs the adsorption parameter κ τ / Λ determined by solving numerically Eq. (25) (solid line), and the approximated solution (26), valid for small u p (dashed line). The horizontal line corresponds to the value of the surface potential experimentally detected for our system (22 mV). The corresponding value of the adsorbing parameter is κ τ / Λ ∼ 1.85, rather different from that given by the approximated solution ( κ τ / Λ ∼ 7.3).
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Reduced surface potential u p vs the adsorption parameter κ τ / Λ determined by solving numerically Eq. (25) (solid line), and the approximated solution (26), valid for small u p (dashed line). The horizontal line corresponds to the value of the surface potential experimentally detected for our system (22 mV). The corresponding value of the adsorbing parameter is κ τ / Λ ∼ 1.85⁠, rather different from that given by the approximated solution (⁠ κ τ / Λ ∼ 7.3⁠).

FIG. 5.
Reduced surface potential u p vs the adsorption parameter κ τ / Λ determined by solving numerically Eq. (25) (solid line), and the approximated solution (26), valid for small u p (dashed line). The horizontal line corresponds to the value of the surface potential experimentally detected for our system (22 mV). The corresponding value of the adsorbing parameter is κ τ / Λ ∼ 1.85, rather different from that given by the approximated solution ( κ τ / Λ ∼ 7.3).
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Reduced surface potential u p vs the adsorption parameter κ τ / Λ determined by solving numerically Eq. (25) (solid line), and the approximated solution (26), valid for small u p (dashed line). The horizontal line corresponds to the value of the surface potential experimentally detected for our system (22 mV). The corresponding value of the adsorbing parameter is κ τ / Λ ∼ 1.85⁠, rather different from that given by the approximated solution (⁠ κ τ / Λ ∼ 7.3⁠).

Close modal

In the case of two free surfaces, the potential difference between them due to the adsorption phenomenon is rather small with respect to the thermal voltage for similar electrodes. However, in the case in which one of the electrodes is grounded, the difference of potential, in k B T / q units, of interest is given by Eq. (25). In our case, the measured polarization voltage is V p = 22 mV, and from Eq. (25) we get κ τ / Λ ∼ 1.85⁠, see Fig. 5. This means that the adsorption length for the system under consideration κ τ is of the order of 2 Λ⁠.

Figure 6 shows the time evolution of the electric current I ( t ) for two DC voltages V = 40.09 and 60.47 mV. As V is setted up, the I ( t ) decreases initially fast and is followed by a much slower decay toward zero. This second decay regime is exceedingly slow, requiring several days or even weeks to approach zero. Consequently, the measured electric current I t M remains nearly constant during the initial hours of observation. A detailed analysis of the time evolution of I ( t )⁠, however, falls outside the scope of this study.

FIG. 6.
Time evolution of the electric current I ( t ) for a few applied DC voltage.
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Time evolution of the electric current I ( t ) for a few applied DC voltage.

FIG. 6.
Time evolution of the electric current I ( t ) for a few applied DC voltage.
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Time evolution of the electric current I ( t ) for a few applied DC voltage.

Close modal

The current values I t M were determined as the average of the electric current I ( t ) measured from the last two minutes to the end of the measurement period. The error associated with I t M was taken as the standard deviation of the mean ⟨ I ( t ) ⟩ for each current value. The error in the electric voltage V corresponds to the accuracy of the source, approximately 0.01 mV. Both sets of errors’ magnitudes are relatively small, with error bars smaller than the size of the data points on the graphs.

The dependence of I t M on V is shown in Fig. 7. In the figure, the straight line represents a fit indicating Ohm’s law behavior. At voltages less than 30 mV, the data points are more scattered around the line and exhibit negative values for I t M⁠. These negative currents are attributed to static electric charges accumulated on the electrode surfaces. The fit intersects the I t M = 0 line at approximately 22 mV. This is the polarization voltage V p⁠. The net charge accumulation on the electrodes can be estimated using Gauss’s law. Considering the dielectric constant ϵ = 2⁠, a typical value for kerosene, we get from Eq. (24), the surface charge density q Σ = ε E ( 0 )⁠. Assuming E ( 0 ) ∼ V p / Λ⁠, where V p = 22 mV, and Λ ∼ 5 × 10 − 8 m, we have q Σ ∼ 8.5 × 10 − 6 A s / m 2⁠.

FIG. 7.
The electric current I t M as a function of the DC voltage V. The solid line is the best fit made with Eq. (28).
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The electric current I t M as a function of the DC voltage V⁠. The solid line is the best fit made with Eq. (28).

FIG. 7.
The electric current I t M as a function of the DC voltage V. The solid line is the best fit made with Eq. (28).
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The electric current I t M as a function of the DC voltage V⁠. The solid line is the best fit made with Eq. (28).

Close modal
To fit the data relevant to I t M vs the electric potential, Eq. (10) has to be rewritten in order to take into account the polarization voltage due to the adsorption phenomenon, according to which Δ V = V − V p⁠. In this case, we have
(28)
By means of an impedance spectroscopy measurement (not shown), it was possible to find N = 6.9 × 10 20 m − 3 and D = 1.7 × 10 − 12 m 2 / s (see Ref. 16). With the Einstein–Smoluchowski relation μ / D = q / ( k B T )⁠, we can find the ionic mobility μ = 6.6 × 10 − 11 m 2 / V s⁠. By fitting the data of Fig. 7 with Eq. (28) we find σ e = 2.1 × 10 − 10 S / m⁠. Finally, with N and N 0 at hand, we can obtain k d / k a = 4.6 × 10 19 m − 3 from Eq. (5).

The resistance R of the ferrofluid sample was obtained as the inverse of the angular coefficient of the linear fit in Fig. 6, yielding R = 2.53 G Ω⁠. Using R and the geometric dimensions of the sample. The macroscopic conductivity of the cell was calculated as cell conductivity σ = 0.22 nS / m⁠. Furthermore, with S⁠, d⁠, and ϵ⁠, the effective capacitance of the cell was determined to be C = 27.9 pF⁠, associated in parallel to R⁠.

The results indicate that neither pure kerosene nor a mixture of kerosene and 1% vol of oleic acid exhibits any conductivity significantly greater than the noise. This suggests that the flow of electric charges is primarily associated with the NPs. The number density of ions responsible for the electric conduction depends on N 0⁠. This finding implies that the ions contributing to the observed electric current are linked to a dissociation process involving the NPs, probably due to their surface ligands.

Our model for electric current within the ferrofluid sample operates as follows: oleate species attached to the NPs may dissociate, as described by Eqs. (1)–(3). The resulting anions migrate under the influence of the externally applied electric field contributing to the electric conduction. At the electrode surfaces, this conduction is facilitated by the NPs, which, as recently reported, play a key role in the conduction process at the electrode–FF interfaces.17 A precise description of how the charges are exchanged between the metal surfaces and the NP layers cannot be given by the technique we have used.

A kerosene-based ferrofluid was studied in a cell subjected to an externally applied DC voltage. The cell consisted of two disk-like platinum electrodes separated by a fixed distance. Initially, in the absence of applied voltage, an intrinsic electric potential was detected between the electrodes, likely arising from the adsorption of a small quantity of free ions present in the ferrofluid on the electrode surfaces.

Upon applying an external DC voltage, a time-dependent electric current was observed, which gradually reached a quasi-steady-state value. This quasi-steady-state current was found to be directly proportional to the applied DC voltage and dependent on the nanoparticle concentration, consistent with the predictions of the dissociation-recombination model. The time dependence of the electric current is a two-relaxation time phenomenon. The short relaxation time is related to the usual dielectric relaxation, the long one to the chemical reaction responsible for the ionic production, whose description is out of the scope of our investigation.

This work was supported by the INCT/CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico; Grant No. 465259/2014-6), INCT/FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo; Grant No. 14/50983-3), INCT/CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior; Grant No. 88887.136373/2017-00), FAPESP (Thematic Project; Grant No. 2016/24531-3), and INCT-FCx (Instituto Nacional de Ciência e Tecnologia de Fluidos Complexos).

The authors have no conflicts to disclose.

B. M. Oliveira: Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). F. Batalioto: Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). M. Chand: Data curation (equal); Validation (equal); Visualization (equal). A. F. C. Campos: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). G. Barbero: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). A. M. Figueiredo Neto: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

The goal of the appendix is the derivation of the results (4)–(6). From the first equation of (4), representing the conservation of the number of particles, the number of neutral particles, in thermodynamical equilibrium, N n is
(A1)
Substituting (A1) into the second equation of (4) we get that the number of dissociated particles N is solution of the second degree equation
(A2)
whose solutions are
(A3)
Since N represents a density of dissociated particles, it is expected to be a positive quantity, and the sign to be chosen into Eq. (A3) is +⁠, i.e., N is given by (5) of the paper.
In the case in which y = k d / k a ≪ 1 expression (5) for N can be rewritten as
(A4)
at the lowest order in y = k d / k a ≪ 1⁠. Equation (A4) coincides with Eq. (6) of the paper.
In the opposite limit where y = k d / k a → ∞ from (5) we get
(A5)
Taking into account that for y → ∞⁠,
(A6)
from (A5) we get
(A7)
from which it follows Eq. (7) of the paper.
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