Abstract
Static magnetization data of eight transformer oil based magnetic fluid samples, with saturation magnetization ranging in a large interval from 9 kA/m to 90 kA/m, have been subjected to the demagnetizing field correction. Using the tabulated demagnetization factors and the differential magnetic susceptibility of the samples, the values of the radial magnetometric demagnetization factor were obtained in the particular case of VSM880 magnetometer. It was found that the demagnetizing field correction keeps the saturation magnetization values unchanged, but instead the initial magnetic susceptibility of the magnetic fluid samples varies widely. The mean magnetic diameter, obtained through magnetogranulometry from the measured data, is higher than that obtained from the corrected ones and the variation rate increases with the magnetic particle volume fraction growth.
1. Introduction
The saturation magnetization, the initial magnetic susceptibility, magnetization curves or the reduced ones, and magnetogranulometric analysis of the magnetic fluids with different concentrations of the magnetic particle content allow the study of various microstructural characteristics. For example, the magnetic susceptibility, an intrinsic property of the magnetic materials, depends on the magnetic diameter of the particles and the dimensional distribution. The presence of the agglomerates, formed during the chemical synthesis of the samples and also because of the application of an external magnetic field, and interparticle interactions are factors which influence the macroscopic magnetic properties [1–3].
The study of demagnetization factors represents a research subject for about 150 years. Thus, in 1870, by using a ballistic galvanometer to measure the magnetic properties of the iron, the first experimental determinations of the demagnetization factor of ellipsoids and some particular forms were made (spheres, rods, and discs). The demagnetization factor was obtained from hysteresis curves [4] or by measuring the magnetization and field on the faces of a magnetized finite cylinder [5], neglecting its dependence of the material susceptibility, .
Throughout the 20th century, several theoretical models have been proposed and developed regarding the calculation factors demagnetization [6–8]. At the beginning of 90s, Chen et al. presented new results for the demagnetization factors, calculated for cylindrical shaped samples with different dimensions and for magnetic susceptibilities with values ranging in a large scale [9]. Using an unidimensional model for long cylinders, with a ratio of length-diameter , Chen has calculated the fluxmetric demagnetization factor , for , and the magnetometric demagnetization factor , for . Also, with a bidimensional model, he obtained the values of the demagnetization factors for and . The case of was treated by using the expression of mutual inductance of two solenoids with the same diameter and having a large range of length-diameter ratio values . More precise values were tabulated for cylindrical samples. They were obtained by numerical calculation of , as a function of the magnetic susceptibility and the length-diameter ratio, with and in axial direction and and in radial direction [10].
As it is known, there is a strong connection between the microstructural properties of the magnetic fluids and their macroscopic behavior, which qualifies them to be used in various technical applications [11], magnetofluidic devices such as rotating seals [12, 13], bearings [14], sensors, and transducers [15, 16], or even in thermal applications [17]. A proper design of a magnetofluidic device implies detailed data on various properties of magnetic fluids, including the static magnetic ones, studied by applying the demagnetizing field correction.
2. Experimental
2.1. Samples
A series of eight transformer oil based magnetic fluids (NMF), with the saturation magnetization varying between 9 kA/m and 90 kA/m (Table 1), were obtained at the Research Center for Engineering of Systems with Complex Fluids, Politehnica University of Timisoara, following the chemical procedure described in [18, 19]. The mentioned procedure is a complex process that involves two main steps: obtaining of monolayer coated magnetite nanoparticles and dispersion of the coated magnetite particles in a carrier liquid. The magnetic particles are produced by chemical coprecipitation at high temperature (80°C) of hydroxides from aqueous solutions of Fe3+/Fe2+ ions, with molar ratio of 1.7 : 1, using an excess of NH4OH in concentrated solution of 25%, which provides a pH of 11, favorable to obtain magnetite. The monolayer coated magnetic nanoparticles are dispersed in a hydrocarbon carrier (kerosene, toluene, cyclohexane, transformer oil, etc.), at temperature ranging between 60° and 130° C, followed by an advanced purification process, which aims to eliminate the free oleic acid and to obtain a purified magnetic fluid based on nonpolar organic solvent and with a certain saturation magnetization. Using samples density, measured at room temperature (25°C), the dispersed magnetite volume fractions were calculated (Table 1).
2.2. Samples
The full magnetization curves of the samples were measured by using a vibrating sample magnetometer, VSM880, ADE Technologies, USA, in the field range of 0 kA/m to 950 kA/m. The curves were obtained at room temperature, using a cylindrical shape sample holder and by applying the magnetic field in the radial direction, that is, perpendicular to the generatrix of the cylinder.
3. Results and Discussion
3.1. Demagnetizing Field Correction
Magnetic measurements performed with any type of magnetometer (VSM, AGM, or SQUID) or a fluxmeter are magnetic measurements in open circuits and involve the application of the demagnetizing field correction.
In practice, the demagnetizing field correction is important in low fields, where the values of the magnetic permeability or susceptibility and remnant magnetization can be determined, and consists in the extraction of the demagnetizing field values from those of the applied magnetic field , at fixed values of the sample magnetization , to obtain the true field acting on the sample. The process, schematically presented in Figure 1, is accomplished by a counterclockwise rotation, around the origin O, of the linear dependence (OD line) until it coincides with the OY axis, simultaneously with the displacement of the experimental values of (OA curve) parallel with OX axis, keeping the distance between the line OD and the value of fixed [20].
For a magnetic material with constant magnetic susceptibility, subjected to a uniform magnetic field, the demagnetizing field correction can be written as it follows:where is the fluxmetric or magnetometric demagnetization factor, depending on how the magnetization of the sample is measured, by using a fluxmeter or a magnetometer.
From magnetic measurements, the values of the applied magnetic field and average magnetization of the entire samples are obtained. Knowing that the magnetic susceptibility due to the applied magnetic field isand using (1), the magnetic susceptibility of the sample will be
The influence of the demagnetizing field correction on microstructural properties of the transformer oil based magnetic fluid samples was studied. Static magnetic properties of the samples were measured at room temperature, using a cylindrical shaped sample holder with length of and diameter of . Because of the sample holder dimensions, the magnetic field applied in radial direction, that is, perpendicular to the generatrix of cylinder, can be considered uniform inside the sample. So, the appropriate demagnetization factor to use is the magnetometric demagnetization factor in radial direction as a function of length-diameter ratio and magnetic susceptibility , .
The tabulated values of , calculated by Chen et al. [21], for thin disks at different values of the magnetic susceptibility, were plotted in Figure 2. The intersection of curves with line, specific to VSM880 magnetometer, gives the dependence in logarithmic scale (Figure 3).
The fit function which approximates this dependence is given bywith obtained as fit parameters: .
Using in (4) the values of the differential magnetic susceptibility, , calculated in each measured point of static magnetization curves (Figure 4), the values of the radial magnetometric demagnetization factor have been found. According to Chen et al. [21] and considering the magnetic susceptibility due to the applied magnetic field (2) instead of the samples magnetic susceptibility (3), an iterative demagnetizing field correction was applied as follows: with the value of the demagnetization factor for a certain value of the magnetic susceptibility , the magnetic susceptibility of the sample after the 1st iteration was obtained, . The value of replaces the value of in (4) to get . is obtained after the 2nd iteration and so on.
Six iterations were made for each sample. It was noted that the corrected values overlap after the first iteration but, compared to the initial curve, they are moved upwards in the low field region (<200 kA/m), proportional to the saturation magnetization of the samples, as in Figures 5(a) and 5(b) where the measured magnetization curve and the first iteration are presented.
(a)
(b)
In the high field region, the sample magnetization reaches the saturation, which corresponds to a magnetic susceptibility . In this situation, it is obtained that for an applied magnetic field (3), which is equivalent with (2) or, in other words, the magnetic field in the sample will be about equal to the applied magnetic field . So, in the saturation region the contribution of the demagnetizing field can be neglected.
The demagnetizing field correction is an interesting subject, especially in the low field region where the initial susceptibility is measured. It is known that in a magnetic colloid the initial magnetic susceptibility is small when the magnetic diameter of the colloidal particles is small, but growing rapidly with the particles diameter increase, for samples with the same magnetic volume fraction [22].
3.2. Magnetic Properties
For all the samples in the NMF series, the initial magnetic susceptibility has been evaluated from the linear part of the experimental and corrected magnetization curves , and, respectively, the saturation magnetization of the samples, in the quasi-saturation region . In this region, the magnetization of the magnetic fluid samples depends mainly on interactions between magnetic nanoparticles and magnetic field, so the interparticle interactions can be neglected, even for the most concentrated ones. As a result, the magnetization of a NMF sample at saturation has a linear dependence of , according to the Langevin model [23]:where is the intensity of the applied magnetic field, is the saturation magnetization of the samples, is the absolute temperature at which magnetic measurements were made, and represents the number of magnetic particles per unit of volume. Intersection of curves with ordinate axis gives the saturation magnetization of the NMF samples (Table 2).
It was noticed that the saturation magnetization values remain unchanged by applying the demagnetizing field correction. In contrast, the magnetic susceptibility varies widely with the increase of the dispersed magnetite volume fraction: whereas the volume fraction varies from 3% (P1) to 26% (P8), the corrected values of the magnetic susceptibilities increase by an order of magnitude relative to the experimental ones, from 8% for sample P1, to 90% for P8 sample (Figure 6).
According to Shliomis [24], the static magnetic properties of a diluted magnetic fluid are well described by uniparticle model. The energy of dipolar interactions is considered lower than the thermal energy and the equilibrium magnetization is written as a superposition of Langevin functions. Also, the linear dependence of the initial magnetic susceptibility on magnetic nanoparticles concentration, in low fields [23],and the samples magnetization in high magnetic field region (5) form the basic instruments of magnetogranulometric analysis in order to obtain the mean magnetic diameter of the dispersed magnetite particles, (11), and the standard deviation, (12), by evaluating the dimensional distribution parameters . In (6), represents the solid volume fraction of the dispersed magnetic material and is the monodomenial magnetization of magnetite.
Magnetic properties of medium and highly concentrated magnetic fluids showed important differences from Langevin behavior, especially because of dipole-dipole interactions. The initial magnetic susceptibility increases faster than the linear dependencewhere represents the initial magnetic susceptibility corresponding to the polydisperse Langevin model:and is magnetic particles density.
In Figure 6, the corrected values of the initial magnetic susceptibility are described by a nonlinear variation (dashed line) with the volume fraction of the dispersed magnetic particles and not by a linear one, as it is suggested by the experimental curve (solid line). In conclusion, the demagnetizing field correction of static magnetization data “improves” the initial magnetic susceptibility variation curve with solid volume fraction (in this case), highlighting the interparticle interactions, especially for . In addition, because the initial magnetic susceptibility has a large variation with particles diameter, neglecting the demagnetizing field correction inevitably leads to errors in dimensional analysis of magnetic particles stably dispersed in a NMF, even at low levels of solid volume fractions [25].
Several theoretical models are used to analyze the static magnetic properties of NMF with different concentrations of magnetic particles, taking into account the dipole-dipole interactions. Theoretical and experimental studies demonstrated that one appropriate model is Ivanov’s second-order modified mean field theory [25–27]. For this purpose, the transformer oil based magnetic fluid samples, with different concentrations of magnetite particles, were analyzed, considering that in real NMFs the magnetite nanoparticles are polydisperse [28] and the magnetic moment depends on their magnetic diameter [29, 30]. The interparticle interactions were also considered. In Figure 7, an overlap of the reduced magnetization curves would indicate a Langevin (monodisperse) behavior; otherwise, they show the influence of the interparticles interactions on static magnetization curves.
According to Ivanov’s model, the magnetization of such ferrofluid is given bywith being Langevin function and having the meaning of an effective magnetic field that acts on each monodomenic particle. is the continuous distribution function over the magnetic diameters of the ferromagnetic particles.
Considering a log-normal distribution of the magnetic diameters of the dispersed particleswith the dimensional distribution parameters: is defined as , is deviation of from value, and, by evaluating the integral in (9), the mean magnetic diameter and the standard deviation are obtained:
It was noticed that the demagnetizing field correction smooths the dimensional distribution parameters value , which have a small variation with the increasing volume fraction, that is, the saturation magnetization. Moreover, the correction moves parameter to smaller values and expands slightly the range of variation of the magnetic diameters distribution, by increasing parameter values. This trend of broadening the size distribution with increasing solid volume fraction is due to the formation of new agglomerates and the increase of those preformed in the synthesis process.
Using the dimensional distribution parameters value in (11) and (12), the mean magnetic diameter and standard deviation were calculated. Note that the mean magnetic diameter, obtained with the corrected magnetic data , is lower than that calculated from experimental data and the variation rate increases with the magnetic particle volume fraction growth, from 1.03%, for sample, P1 to 6.37% for P8 (Figure 8(a)). Simultaneously, it was noticed that the standard deviation, as a measure of dispersion degree or variance of data sets, tends to a quasi-uniform variation with the dispersed magnetic material volume fraction. In other words, by applying the demagnetizing field correction the dispersion of magnetic diameter values around the mean is not significantly influenced by an increasing volume fraction of magnetic particles (Figure 8(b)).
(a)
(b)
In Figure 9, size distributions of magnetic diameters of the dispersed particles in NMF samples (P1 (a) and P8 (b)) are showed. They were also obtained from experimental and corrected static magnetization data, by considering a log-normal distribution function (10) with Ivanov’s model (9). The same growth trend of the mean magnetic diameter of particles with the magnetic volume fraction is observed. Moreover, the demagnetizing field correction moves the values of the mean magnetic diameter towards lower values, proportional with the increase of the solid volume fraction. Also, the correction slightly enlarges the distribution function in the region of high diameters (10–15 nm).
(a)
(b)
As we might expect, an increase of the initial magnetic susceptibility is related to the mean magnetic diameter increase, but considering the demagnetizing field correction on static magnetization data of a transformer oil based NMF, a stronger increase of the initial magnetic susceptibility of the samples was observed, compared to that of the mean magnetic diameter of the dispersed particles , for the same dimensional distribution parameters . In conclusion, the magnetogranulometric analysis of the studied samples must consider the demagnetizing field correction and the “corrected” values of the distribution parameters, whose variation defines the values of initial magnetic susceptibility and of the mean magnetic diameter.
4. Conclusion
The experimental data, obtained by measuring the static magnetic properties of eight transformer oil based magnetic fluids, with magnetite particles covered with oleic acid, have been subjected to the demagnetizing field correction, to emphasize its influence on various microstructural characteristics such as dimensional distribution of the colloidal particles or on some macroscopic magnetic properties like initial susceptibility.
The tabulated values of the radial magnetometric demagnetization factor, calculated by Chen et al. [1], were plotted in Figure 2 and the intersection of curves with the particular case of a sample with the length-diameter ratio of (VSM880) gives the dependence . Using (4) and the values of the differential magnetic susceptibility , calculated for each measured point of static magnetization curves (Figure 4), the values of the radial magnetometric demagnetization factor were found.
An iterative demagnetizing field correction has been applied and the results were obtained after 6 iterations. The corrected values overlap after the first iteration but, compared to the experimental curve, in the low field region (<200 kA/m) they are moved upwards, proportional with the saturation magnetization of the samples. In the saturation region the contribution of the demagnetizing field can be neglected (Figures 5(a) and 5(b)).
Through the magnetogranulometric analysis of the experimental and corrected magnetization data, using Ivanov’s second-order modified mean field theory, the dimensional distribution parameters were obtained, by evaluating the initial susceptibility , in the quasi-linear region of the static magnetization curve, and the saturation magnetization of the magnetic fluid samples , in the saturation region. As it was expected, the application of the demagnetizing field correction lets the saturation magnetization values unchanged, but instead, the corrected values of the initial magnetic susceptibilities increase by an order of magnitude relative to the experimental ones, from 8% for the sample with the lowest concentration of magnetic particles (P1, ) up to 29%, for P8 sample with (Figure 6).
It was noticed that the demagnetizing field correction smooths the dimensional distribution parameters value , which have a small variation with increasing volume fraction, that is, the saturation magnetization [31]. With (11) and (12), the mean magnetic diameter and standard deviation were obtained (Figure 8). The mean magnetic diameter, obtained with corrected magnetic data, is lower than that calculated from experimental ones and the variation rate increases with the magnetic particle volume fraction growth, from 1.03% for sample P1 to 6.37% for P8 (Figure 8(a)). Moreover, by applying the demagnetizing field correction, the dispersion of magnetic diameter values around the mean (standard deviation) is not significantly influenced by an increasing volume fraction of magnetic particles (Figure 8(b)) [31].
The magnetic diameter of the particles and the dimensional distribution are microstructural parameters which influence the macroscopic magnetic properties of the magnetic colloids, such as the magnetic susceptibility, an intrinsic property of the magnetic materials. Also, the presence of the agglomerates and interparticle interactions (in concentrated magnetic fluids) are factors which influence the macroscopic magnetic behavior, making them suitable for use in a certain application. The magnetogranulometric analysis, as an instrument for microstructural characterization of NMF, must consider the demagnetizing field correction and the “corrected” values of the distribution parameters, whose variation establish the values of the initial magnetic susceptibility of the magnetic fluid samples and mean magnetic diameter of the dispersed particles.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the research projects NanoMagSecurityPaper (Contract no. 280/2014, PN-II-PT-PCCA-2013-4-0560) and ASEMEMS HARVEST (Contract no. 63/2014, PN-II-PT-PCCA-2013-4-0486), financed by the Romanian Executive Unit for Financing Higher Education, Research, Development and Innovation (UEFISCDI). The author is much indebted to Mrs. Florica Balanean for preparing the transformer oil based magnetic fluid samples.