# Study of Static Magnetic Properties of Transformer Oil Based Magnetic Fluids for Various Technical Applications Using Demagnetizing Field Correction

Authors: Oana Maria Marinica
Journal: Journal of Nanomaterials
 (2017)

---

## 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.


---

## Body

## 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 factorN 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, χ.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 γ≥10, Chen has calculated the fluxmetric demagnetization factor Nf, for -1≤χ<∞, and the magnetometric demagnetization factor Nm, for χ→∞. Also, with a bidimensional model, he obtained the values of the demagnetization factors Nf,m for 0.01≤γ≤50 and -1≤χ<∞. The case of χ=0 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 0.00001≤γ≤1000. More precise values were tabulated for cylindrical samples. They were obtained by numerical calculation of Nf,m, as a function of the magnetic susceptibility and the length-diameter ratio, with -1≤χ<∞ and 0.01≤γ≤500 in axial direction and -1≤χ<∞ and 0.01≤γ≤1 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 (Table1), 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 φFe3O4 were calculated (Table 1).Table 1
Physical properties of the transformer oil based magnetic fluid samples.
NMFsamples
M (kA/m)
ρ
2
5
°
C (g/cm3)
φ
F
e
3
O
4 (%)
P1
9.01
0.99
2.75
P2
17.42
1.10
5.24
P3
26.29
1.19
7.30
P4
31.16
1.28
9.56
P5
39.44
1.39
12.08
P6
43.55
1.46
13.72
P7
65.06
1.72
19.54
P8
82.48
2.00
26.03
M corresponds to the maximum value of the applied magnetic field (950 kA/m).


### 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.


## 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 (Table1), 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 φFe3O4 were calculated (Table 1).Table 1
Physical properties of the transformer oil based magnetic fluid samples.
NMFsamples
M (kA/m)
ρ
2
5
°
C (g/cm3)
φ
F
e
3
O
4 (%)
P1
9.01
0.99
2.75
P2
17.42
1.10
5.24
P3
26.29
1.19
7.30
P4
31.16
1.28
9.56
P5
39.44
1.39
12.08
P6
43.55
1.46
13.72
P7
65.06
1.72
19.54
P8
82.48
2.00
26.03
M corresponds to the maximum value of the applied magnetic field (950 kA/m).


## 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 fieldHd values from those of the applied magnetic field Ha, at fixed values of the sample magnetization M, to obtain the true field acting on the sample. The process, schematically presented in Figure 1, is accomplished by a counterclockwise rotation, around the originO, of the linear dependence Hd=f(M) (OD line) until it coincides with theOY axis, simultaneously with the displacement of the experimental values of M (OA curve) parallel withOX axis, keeping the distance between the lineOD and the value of M fixed [20].Figure 1
Schematic presentation of the demagnetizing field correction process. The process is accomplished by a counterclockwise rotation about the originO of the linear dependence Hd=f(M) (OD line) until it coincides with theOY axis, simultaneously with the displacement of the experimental values of M (OA curve) [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:(1)H=Ha+Hd=Ha-NM,where N 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 fieldHa and average magnetization of the entire samples Mm are obtained. Knowing that the magnetic susceptibility due to the applied magnetic field is(2)χa=MmHaand using (1), the magnetic susceptibility of the sample will be(3)1χ=1χa-Nm.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 ofl=2.7mm and diameter of d=5.5mm. 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 γ and magnetic susceptibility χ, Nmr=f(γ,χ).The tabulated values ofNmr=f(γ,χ), calculated by Chen et al. [21], for thin disks 0<γ≤1 at different values of the magnetic susceptibility, were plotted in Figure 2. The intersection of Nmr=f(γ,χ) curves with γ=l/d=0.49 line, specific to VSM880 magnetometer, gives the dependence Nmr=f(χ) in logarithmic scale (Figure 3).Figure 2
Radial magnetometric demagnetization factor versus length-diameter ratio, at different values of the magnetic susceptibility, calculated for thin disks by Chen et al. [21]. The intersection with γ=l/d=0.49 line, specific to VSM880 magnetometer, gives the dependence Nmr=f(χ).Figure 3
The radial magnetometric demagnetization factor versus magnetic susceptibility of the samples,Nmr=f(χ), obtained for a constant value of the length-diameter ratio γ=l/d=0.49 (specific dimension of VSM880 sample holder).The fit function which approximates this dependence is given by(4)Nmr=a+b·χ1+c·χ,with a,b,c obtained as fit parameters: a=0.26012±0.00237;b=0.12805±0.01324;c=0.54353±0.05404R2=0.9879.Using in (4) the values of the differential magnetic susceptibility, χ=dM/dH, 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 Nm1 for a certain value of the magnetic susceptibility χa1, the magnetic susceptibility of the sample after the 1st iteration was obtained, χ1. The value of χ1 replaces the value of χ in (4) to get Nm2. χ2 is obtained after the 2nd iteration and so on.Figure 4
Differential magnetic susceptibility versus magnetic field intensity for sample P8, calculated for each measured point of static magnetization curve.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 Figures5(a) and 5(b) where the measured magnetization curve and the first iteration are presented.Figure 5
Demagnetizing field correction applied to the experimental data of samples P1 (a) and P8 (b). In the low field region (<200 kA/m), the corrected magnetization curves are moved upwards, proportional to the saturation magnetization of the samples.
(a)
(b)In the high field region, the sample magnetization reaches the saturation, which corresponds to a magnetic susceptibilityχ≅0. In this situation, it is obtained that M≪Ha for an applied magnetic field (3), which is equivalent with Hd≪Ha (2) or, in other words, the magnetic field in the sample will be about equal to the applied magnetic field H=Ha+Hd≈Ha. 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 curvesH<1kA/m;M=χ·H, and, respectively, the saturation magnetization of the samples, in the quasi-saturation region H>700kA/m. 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 1/H, according to the Langevin model [23]:(5)MH≅Ms-nkBTμ0H,where H is the intensity of the applied magnetic field, Ms is the saturation magnetization of the samples, T is the absolute temperature at which magnetic measurements were made, and n represents the number of magnetic particles per unit of volume. Intersection of M=f1/H curves with ordinate axis gives the saturation magnetization of the NMF samples (Table 2).Table 2
The initial magnetic susceptibility and saturation magnetization of NMF samples, obtained from the experimental (_exp) and corrected (_cor) static magnetization data.
NMFsamples
φ
F
e
3
O
4 (%)
χ
i
_
e
x
p
χ
i
_
c
o
r
M
s
_
e
x
p (kA/m)
M
s
_
c
o
r  (kA/m)
P1
2.75
0.26
0.28
9.44
9.44
P2
5.24
0.52
0.60
18.33
18.33
P3
7.30
0.77
0.96
27.72
27.71
P4
9.56
0.85
1.08
32.68
32.66
P5
12.08
1.09
1.48
41.57
41.55
P6
13.72
1.21
1.71
45.84
45.81
P7
19.54
1.59
2.60
68.78
68.70
P8
26.03
1.96
3.73
86.79
86.67It 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 (Figure6).Figure 6
Variation of the initial magnetic susceptibility of transformer oil based magnetic fluid samples with the magnetite volume fraction and due to the application of the demagnetizing field correction. The demagnetizing field correction modifies the initial magnetic susceptibility variation curve, emphasizing the interparticle interactions, especially forφFe3O4>10% (7).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],(6)χiL=πμ0φmMd2Dm318kBT,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, Dm (11), and the standard deviation, σ (12), by evaluating the dimensional distribution parameters D0,S. In (6), φm represents the solid volume fraction of the dispersed magnetic material and Md=480kA/m 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 dependence(7)χi=χiL1+13χiL+1144χiL2,where χiL represents the initial magnetic susceptibility corresponding to the polydisperse Langevin model:(8)χiL=μ0π2nMd2D06108kBTexp⁡36S22,and n is magnetic particles density.In Figure6, 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 φFe3O4>10%. 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 m(x) 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.Figure 7
Reduced magnetization curves of the NMF samples. The non-overlapping curves evidence the magnetic particles interactions.According to Ivanov’s model, the magnetization of such ferrofluid is given by(9)M=n∫0∞mxLmxHekBTfxdxwith L(ξ) being Langevin function and He having the meaning of an effective magnetic field that acts on each monodomenic particle. f(x) 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 particles(10)fx=1xS2πexp⁡-ln2⁡x/D02S2,with the dimensional distribution parameters: D0 is defined as ln⁡(D0)=ln⁡(x), S is deviation of ln⁡(x) from ln⁡(D0) value, and, by evaluating the integral in (9), the mean magnetic diameter and the standard deviation are obtained:(11)Dm=D0exp⁡S22,(12)σ=D0exp⁡S22exp⁡S2-11/2.It was noticed that the demagnetizing field correction smooths the dimensional distribution parameters valueD0_cor,S_cor, which have a small variation with the increasing volume fraction, that is, the saturation magnetization. Moreover, the correction moves D0_cor parameter to smaller values D0_cor<6nm and expands slightly the range of variation of the magnetic diameters distribution, by increasing S_cor 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 Dm_cor, is lower than that calculated from experimental data Dm_exp 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)).Figure 8
Mean magnetic diameter and standard deviation of studied NMF samples, obtained with (_cor) and without (_exp) demagnetizing field correction. The demagnetizing field correction tends to smooth the magnetogranulometric analysis results variation with magnetic content of the samples.
(a)
(b)In Figure9, 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).Figure 9
Size distribution functions over the magnetic diameters of the particles,f(x), obtained in terms of Ivanov’s model (2nd-order modified mean field theory), with (_cor) and without (_exp) demagnetizing field correction, applied to static magnetization data of P1 (a) and P8 (b) samples.
(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 samplesχi_cor was observed, compared to that of the mean magnetic diameter of the dispersed particles Dm_cor, for the same dimensional distribution parameters D0_cor,S_cor. 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.


## 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 fieldHd values from those of the applied magnetic field Ha, at fixed values of the sample magnetization M, to obtain the true field acting on the sample. The process, schematically presented in Figure 1, is accomplished by a counterclockwise rotation, around the originO, of the linear dependence Hd=f(M) (OD line) until it coincides with theOY axis, simultaneously with the displacement of the experimental values of M (OA curve) parallel withOX axis, keeping the distance between the lineOD and the value of M fixed [20].Figure 1
Schematic presentation of the demagnetizing field correction process. The process is accomplished by a counterclockwise rotation about the originO of the linear dependence Hd=f(M) (OD line) until it coincides with theOY axis, simultaneously with the displacement of the experimental values of M (OA curve) [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:(1)H=Ha+Hd=Ha-NM,where N 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 fieldHa and average magnetization of the entire samples Mm are obtained. Knowing that the magnetic susceptibility due to the applied magnetic field is(2)χa=MmHaand using (1), the magnetic susceptibility of the sample will be(3)1χ=1χa-Nm.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 ofl=2.7mm and diameter of d=5.5mm. 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 γ and magnetic susceptibility χ, Nmr=f(γ,χ).The tabulated values ofNmr=f(γ,χ), calculated by Chen et al. [21], for thin disks 0<γ≤1 at different values of the magnetic susceptibility, were plotted in Figure 2. The intersection of Nmr=f(γ,χ) curves with γ=l/d=0.49 line, specific to VSM880 magnetometer, gives the dependence Nmr=f(χ) in logarithmic scale (Figure 3).Figure 2
Radial magnetometric demagnetization factor versus length-diameter ratio, at different values of the magnetic susceptibility, calculated for thin disks by Chen et al. [21]. The intersection with γ=l/d=0.49 line, specific to VSM880 magnetometer, gives the dependence Nmr=f(χ).Figure 3
The radial magnetometric demagnetization factor versus magnetic susceptibility of the samples,Nmr=f(χ), obtained for a constant value of the length-diameter ratio γ=l/d=0.49 (specific dimension of VSM880 sample holder).The fit function which approximates this dependence is given by(4)Nmr=a+b·χ1+c·χ,with a,b,c obtained as fit parameters: a=0.26012±0.00237;b=0.12805±0.01324;c=0.54353±0.05404R2=0.9879.Using in (4) the values of the differential magnetic susceptibility, χ=dM/dH, 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 Nm1 for a certain value of the magnetic susceptibility χa1, the magnetic susceptibility of the sample after the 1st iteration was obtained, χ1. The value of χ1 replaces the value of χ in (4) to get Nm2. χ2 is obtained after the 2nd iteration and so on.Figure 4
Differential magnetic susceptibility versus magnetic field intensity for sample P8, calculated for each measured point of static magnetization curve.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 Figures5(a) and 5(b) where the measured magnetization curve and the first iteration are presented.Figure 5
Demagnetizing field correction applied to the experimental data of samples P1 (a) and P8 (b). In the low field region (<200 kA/m), the corrected magnetization curves are moved upwards, proportional to the saturation magnetization of the samples.
(a)
(b)In the high field region, the sample magnetization reaches the saturation, which corresponds to a magnetic susceptibilityχ≅0. In this situation, it is obtained that M≪Ha for an applied magnetic field (3), which is equivalent with Hd≪Ha (2) or, in other words, the magnetic field in the sample will be about equal to the applied magnetic field H=Ha+Hd≈Ha. 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 curvesH<1kA/m;M=χ·H, and, respectively, the saturation magnetization of the samples, in the quasi-saturation region H>700kA/m. 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 1/H, according to the Langevin model [23]:(5)MH≅Ms-nkBTμ0H,where H is the intensity of the applied magnetic field, Ms is the saturation magnetization of the samples, T is the absolute temperature at which magnetic measurements were made, and n represents the number of magnetic particles per unit of volume. Intersection of M=f1/H curves with ordinate axis gives the saturation magnetization of the NMF samples (Table 2).Table 2
The initial magnetic susceptibility and saturation magnetization of NMF samples, obtained from the experimental (_exp) and corrected (_cor) static magnetization data.
NMFsamples
φ
F
e
3
O
4 (%)
χ
i
_
e
x
p
χ
i
_
c
o
r
M
s
_
e
x
p (kA/m)
M
s
_
c
o
r  (kA/m)
P1
2.75
0.26
0.28
9.44
9.44
P2
5.24
0.52
0.60
18.33
18.33
P3
7.30
0.77
0.96
27.72
27.71
P4
9.56
0.85
1.08
32.68
32.66
P5
12.08
1.09
1.48
41.57
41.55
P6
13.72
1.21
1.71
45.84
45.81
P7
19.54
1.59
2.60
68.78
68.70
P8
26.03
1.96
3.73
86.79
86.67It 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 (Figure6).Figure 6
Variation of the initial magnetic susceptibility of transformer oil based magnetic fluid samples with the magnetite volume fraction and due to the application of the demagnetizing field correction. The demagnetizing field correction modifies the initial magnetic susceptibility variation curve, emphasizing the interparticle interactions, especially forφFe3O4>10% (7).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],(6)χiL=πμ0φmMd2Dm318kBT,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, Dm (11), and the standard deviation, σ (12), by evaluating the dimensional distribution parameters D0,S. In (6), φm represents the solid volume fraction of the dispersed magnetic material and Md=480kA/m 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 dependence(7)χi=χiL1+13χiL+1144χiL2,where χiL represents the initial magnetic susceptibility corresponding to the polydisperse Langevin model:(8)χiL=μ0π2nMd2D06108kBTexp⁡36S22,and n is magnetic particles density.In Figure6, 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 φFe3O4>10%. 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 m(x) 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.Figure 7
Reduced magnetization curves of the NMF samples. The non-overlapping curves evidence the magnetic particles interactions.According to Ivanov’s model, the magnetization of such ferrofluid is given by(9)M=n∫0∞mxLmxHekBTfxdxwith L(ξ) being Langevin function and He having the meaning of an effective magnetic field that acts on each monodomenic particle. f(x) 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 particles(10)fx=1xS2πexp⁡-ln2⁡x/D02S2,with the dimensional distribution parameters: D0 is defined as ln⁡(D0)=ln⁡(x), S is deviation of ln⁡(x) from ln⁡(D0) value, and, by evaluating the integral in (9), the mean magnetic diameter and the standard deviation are obtained:(11)Dm=D0exp⁡S22,(12)σ=D0exp⁡S22exp⁡S2-11/2.It was noticed that the demagnetizing field correction smooths the dimensional distribution parameters valueD0_cor,S_cor, which have a small variation with the increasing volume fraction, that is, the saturation magnetization. Moreover, the correction moves D0_cor parameter to smaller values D0_cor<6nm and expands slightly the range of variation of the magnetic diameters distribution, by increasing S_cor 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 Dm_cor, is lower than that calculated from experimental data Dm_exp 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)).Figure 8
Mean magnetic diameter and standard deviation of studied NMF samples, obtained with (_cor) and without (_exp) demagnetizing field correction. The demagnetizing field correction tends to smooth the magnetogranulometric analysis results variation with magnetic content of the samples.
(a)
(b)In Figure9, 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).Figure 9
Size distribution functions over the magnetic diameters of the particles,f(x), obtained in terms of Ivanov’s model (2nd-order modified mean field theory), with (_cor) and without (_exp) demagnetizing field correction, applied to static magnetization data of P1 (a) and P8 (b) samples.
(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 samplesχi_cor was observed, compared to that of the mean magnetic diameter of the dispersed particles Dm_cor, for the same dimensional distribution parameters D0_cor,S_cor. 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 Nmr=f(γ,χ) curves with the particular case of a sample with the length-diameter ratio of γ=l/d=0.49 (VSM880) gives the dependence Nmr=f(χ). Using (4) and the values of the differential magnetic susceptibility χ=dM/dH, 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 (Figures5(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 susceptibilityχi, in the quasi-linear region of the static magnetization curve, and the saturation magnetization of the magnetic fluid samples Ms, 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, φFe3O4≅0.03) up to 29%, for P8 sample with φFe3O4≅0.26 (Figure 6).It was noticed that the demagnetizing field correction smooths the dimensional distribution parameters valueD0_cor,S_cor, 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.



---

*Source: 5407679-2017-06-15.xml*