# Dynamics of Magnetic Fluids in Crossed DC and AC Magnetic Fields

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}and alternating h = a cos ωt) features a number of peculiarities, which provide valuable information on the internal structure of colloidal solutions, including the characteristic sizes of particles and clusters. The source of information is the dependence of the electromotive force (emf) E(H

_{0}), induced in the measuring coil at double frequency (output signal) on the bias field strength H

_{0}. The specific feature of the experiment is that the output signal changes nonmonotonically with increasing H

_{0}. Similar studies were conducted previously [1,2,3] in the low-frequency region corresponding to a quasistatic limit ωτ << 1, where τ is the magnetization relaxation time. In this paper, we derive equations that are valid at any value of ωτ, provided that the frequency of the probe field ω remains small compared with the frequency of ferromagnetic resonance at 10

^{10}Hz [4,5]. These equations are used to analyze the experimental data over a wide frequency range. We focus on the polydispersity of particles and interparticle interactions causing particle aggregation. The influence of the magnetic dipole–dipole interparticle interactions is most dramatic in weak and moderate fields. Depending on the particle concentration, the interparticle interactions may be responsible for the two- or four-fold growth in the initial magnetic susceptibility and increase in the nonlinearity of the magnetization curve [6,7]. The polydispersity of particles (i.e., broad particle-size distribution) affects practically all physical properties of magnetic fluids. Consideration of polydispersity can result in a qualitatively new interpretation of experimental data. In this work, we consider the crossed-field method as the tool for obtaining information about the biggest particles contained in a ferrofluid. The coarse fraction leads to the formation of nanoscale (tens of nanometers [8,9,10]) and drop-like (microns and tens of microns [11,12,13,14]) aggregates, and fluid separation into weakly and strongly concentrated phases.

## 2. Methods and Materials

_{0}(Figure 1A,B). The weak alternating field h

_{0}(t) = a

_{0}cosωt was directed normally to the cylinder axis. The measuring coil enclosed the middle part of the sample and its axis coincided with that of the sample. Simultaneous application of the bias and probing alternating fields caused the vector of the total field

**H**to oscillate in the vertical plane. The magnetization

**M**executed the same oscillations. Although the field projection on the z-axis was constant, the corresponding magnetization projection M

_{z}oscillated with time at double the frequency due to the nonlinear dependence M(H). These oscillations of M

_{z}induced the output signal. According to the Faraday law, the value of emf in the measuring coil is given by:

_{0}= 4π × 10

^{–7}H/m, S is the area of the sample cross-section, and N is the number of turns in the measuring coil. Pshenichnikov et al. [3], in the case of a weak probe field and low frequency (ωτ << 1), showed that the output signal is described by:

_{s}x

^{3}/6, where M

_{s}is the saturation magnetization of the magnetic kernel. So, the contribution of separate fractions to the output signal is proportional to the magnetic moment of the 4th power or the diameter of the 12th power. Then, the form of the function F(x) strongly affects the output signal and the coarse dispersed fractions are the main contributor. Here, saturation magnetization of the magnetic fluids is determined by the average magnetic moment of particles, whereas susceptibility in the Langevin approximation by the mean square of the magnetic moment is:

_{0}varied from 0 to 25 kA/m. The amplitude of the probe alternating current (AC) field varied depending on the experimental conditions, but in all cases did not exceed 2 kA/m. In the experiments, we studied two samples of magnetic fluids (samples No. 1 and 2) of the magnetite–kerosene–oleic acid type obtained by diluting kerosene with base fluids FM1 and FM2, respectively. The base fluid FM1 was obtained following the standard chemical precipitation method [15] in the Institute of Continuous Media Mechanics UB RAS (Perm, Russia), FM2 fluid was prepared in the Ivanovo State Power University (Ivanovo, Russia). They differed mainly in particle size distributions. The desired particle size distribution was obtained by varying the synthesis conditions (concentration of iron and ammonia salts and pH of the solutions, temperature, solution feed rate, and mixing intensity) [16]. The free oleic acid was removed by replacing the dispersion medium [15].

_{f}under the assumption that the density of protective shells inessential differed from the density of kerosene ρ

_{k}= 0.78 g/cm

^{3}:

_{mag}= 5.24 g/cm

^{3}is the bulk magnetite density. The use of a more accurate formula for φ

_{s}was unreasonable due to the lack of reliable information on the effective density of the protective shell.

_{0}and α are the distribution parameters, and 〈x

^{q}〉 is the moment x of order q. In particular, the average core diameter 〈x〉 and the relative distribution width δ

_{x}are described by the following equations:

_{∞}, initial susceptibility χ

_{0}, average magnetic moment <m>, mean square magnetic moment <m

^{2}>, average diameter of the magnetic core of particles <x>, volume fraction φ

_{ρ}of magnetite, and relative distribution width δ

_{x}) are presented in Table 1 for the base magnetic fluids.

_{L}:

_{s}= 480 kA/m) did not exceed 1.7%, which did not diminish the marked contribution of the magneto dipole interactions. The susceptibility due to the interactions increased from 7% for sample No. 1 to 27% for sample No. 2, containing the maximum fraction of coarse particles.

_{4}voltages) by the simple equation:

_{0}and S are the cross-sectional areas of the coil and the sample, respectively. Equation (10) allowed us to calculate the desired susceptibility components in terms of voltages ΔU and U

_{4}and the phase shift between them. The amplitudes and phases of the two voltages were measured with a dual-channel synchronous amplifier eLockIn 203 (Anfatec Instruments AG, Oelsnitz Saxony, Germany). For the experiment conditions (the sample in the form of the long cylinder), the demagnetizing factor of the sample was sufficiently small (κ = 0.0065 ± 0.0005) and was used to compute the correction for the voltage enabled determination of the maximum error of measurement of susceptibility χ′, which is equal to $\pm (0.2+2\chi )\times {10}^{-2}$. The measurement error for χ″ was at the level of 0.01 SI units for diluted magnetic fluids and did not exceed 5% of the static susceptibility value for concentrated solutions. The coupling constant λ in Table 2 is the ratio of dipole–dipole interaction energy to thermal energy and is discussed in detail in Section 3.3.

## 3. Results

#### 3.1. Relaxation Processes

**M**and

**H**was violated (Figure 1B), and Equation (2) became inapplicable. To determine the components of magnetization, it was necessary to solve the relaxation equation, which contains the characteristic relaxation times τ of the magnetic moment taken as parameters. Real magnetic fluids, as a rule, have broad particle-size distributions and wide ranges of magnetization relaxation times. Considering the polydispersity of particles in the dynamic problem makes it unnecessarily cumbersome. Therefore, in this section, we restrict our discussion to the monodisperse ferrofluids, in which all particles are identical.

_{II}and ${\tau}_{\perp}$ for the longitudinal and transverse components of magnetization, respectively, depend on the field strength; in the dilute solution they are described by [20,21]:

_{B}is the Brownian time of rotational diffusion of particles in zero magnetic field, η is the ferrofluid viscosity, and V is the volume of the particle covered with a protective shell. In weak fields (ξ << 1), both relaxation times coincide with the Brownian time τ

_{B}and monotonically decrease with the growth of the magnetic field.

_{L}of the model fluids were considered identical to those of samples No. 1 and 2 in Table 1 and Table 2. Since Equations (19) and (20) do not consider the polydispersity of particles, one good quantitative agreement between the calculated and experimental data in the case of the broad size distribution of particles should not be expected. The main advantage of Equations (18) and (20) is that they consider the dynamic effects, magneto dipole–interparticle interactions, and the demagnetizing field. The last two factors compete with each other. The magneto dipole interactions increase ferrofluid magnetization, whereas the demagnetizing field causes its decrease. Their total effect remains appreciable even for moderately concentrated solutions, such as samples No. 1 and 2. In Figure 3A,B this effect is manifested in the shift of the maximum of the function E* = f(ξ) toward weak fields. For dilute magnetic fluids in which the interparticle interactions and demagnetizing fields are negligible, the Langevin parameter, corresponding to the maximum of the function E* = f(ξ), is equal to ξ* = 1.93 [2,3]. Equation (2) supports the correctness of this value in the case when the fluid magnetization is calculated in the Langevin approximation. However, the curves in Figure 3 constructed with regard for the interparticle interactions demonstrate markedly lower values of ξ*: ξ* ≈ 1.53 for sample No. 1 and ξ* ≈ 1.65 for sample No. 2.

_{B}≥ 1), the output signal decreased and the maximum was smeared. In the strong bias field (ξ

_{0}≥ 10), relaxation times decreased, as predicted by Equation (12), therefore, the dynamic contributions to Equations (18) and (20) decreased by almost two orders of magnitude. The process of magnetization reversal became quasi-static and the output signal was no longer dependent on the frequency, at least at ωτ

_{B}≤ 6.

#### 3.2. Results of Dynamic Susceptibility Experiment

_{m}, where K is the magnetic anisotropy constant and V

_{m}is the volume of the particle magnetic core. In the weak field, this barrier can be overcome due to thermal fluctuations within the particle itself, which corresponds to the Néel relaxation mechanism.

_{N}required to overcome the barrier grows exponentially with decreasing temperature, i.e., τ

_{N}~ τ

_{0}exp σ, where τ

_{0}~ 10

^{−9}s is the damping time of the Larmor precession, and σ is the reduced barrier height (anisotropy parameter):

_{N}= τ

_{B}provides the characteristic magnetic core diameter x

^{*}, which corresponds to “switching” off the relaxation mechanism. If x < x*, then the Néel relaxation mechanism prevails, and if x > x

^{*}, then the Brownian mechanism is predominant. Generally, x

^{*}does not coincide with the limiting size of superparamagnetic particles: the Brownian fraction includes both magnetically hard particles and some superparamagnetic particles with τ

_{N}> τ

_{B}. According to estimates [25], for low-viscous magnetite ferrofluids x

^{*}≈ 16−18 nm, τ

_{N}= 10

^{−10}−10

^{−5}s, and τ

_{B}= 10

^{−5}−10

^{−3}s. Thus, at frequencies up to 10

^{4}Hz, the dispersion of the dynamic susceptibility is specified by the particles with Brownian relaxation mechanism and at frequencies above 10

^{5}Hz by the particles with the Néel magnetic moment relaxation mechanism.

^{5}Hz; for sample No. 2, the maximum dispersion was observed at frequencies of 300–400 Hz. Such difference in the dynamics of magnetization is the direct result of differences in the particle size distributions.

^{*}≈ 16 nm and rather short relaxation times (τ

_{N}< 10

^{−5}s). In the examined frequency range, this sample showed a quasi-static behavior, and the region of susceptibility dispersion was beyond the upper boundary of this range. Conversely, sample No. 2 had a broad particle-size distribution (Table 1) with a long tail, so the main contribution to the dynamic susceptibility was due to the Brownian particles with a magnetic core diameter x > x* and long relaxation times. Notably, the contribution of large particles to the initial susceptibility was disproportionately high; it grew as the squared magnetic moment or as the sixth power of the diameter. The hydrodynamic diameter d of the particles, which contributed the most to the susceptibility dispersion, could be estimated from the condition 2πf*τ

_{B}= 1, where f* is the frequency corresponding to the maximum on the χ″(ω) curve. Substituting f* ≈ 330 Hz into this condition and the Brownian relaxation time from Equation (12) yielded d ≈ 100 nm. This hydrodynamic diameter value is approximately three or four times greater than the maximum possible diameter of the individual particles, which in magnetite ferrofluids is determined with an electron microscope. The existence of the surfactant protective shell with a characteristic thickness slightly higher than 2 nm cannot account for such a large difference in size. This led us to conclude that the multi-particle aggregates (clusters) rather than single particles acted as independent kinetic units, which generated the spectrum of dynamic susceptibility of sample No. 2. This conclusion agrees well with the data, which we obtained earlier in similar experiments [25] and in experiments on the diffusion and magnetophoresis of particles in magnetic fluids containing coarse particles [8,26]. The same results were reported [9] using the dynamic light scattering method.

#### 3.3. Crossed Field Experiment

^{3}> and <x

^{6}> [17]. It may be necessary to use the moments of the ninth order <x

^{9}> when processing experimental data on birefringence in magnetic fluids, because in weak magnetic fields the output signal grows in proportion to the ninth power of the diameter [27,28].

^{3}>, Langevin susceptibility (<x

^{6}>), and output signal in the crossed field experiment (<x

^{12}>). The density of the particle size distribution (curve 1) was calculated by Equation (7) for the Γ-distribution. All curves were normalized to unity. The parameters used in calculations were taken from Table 1 and Table 2. The parameters associated with sample No. 1 can be considered typical of magnetite ferrocolloids, including commercial ferrofluids. Sample No. 2 was specifically chosen to demonstrate the effects associated with polydispersity, and had a very broad particle size distribution.

_{max}≈ 20–25 nm of magnetite particles, which were still observable in an electron microscope in highly stable magnetic fluids and powders obtained by the chemical precipitation method [9,10,29,30,31,32]. In real solutions, particles with magnetic cores of large diameter are absent, since the concentration of iron salts in solutions and the duration of the chemical reaction are limited. Strictly, the Γ-distribution in Equation (7), which suggests the existence of particles with arbitrary large diameters, contradicts the experimental data on the existence of x

_{max}. However, Equation (7) is often used to approximate the size distribution of particles, since the systematic error associated with the tail of the distribution is inessential when calculating the moments of x of low orders, for example, <x>. Figure 6 shows that the Γ-distribution correctly describes the size distribution of the particles in both samples. Only a negligible fraction of particles had magnetic cores with diameter exceeding x

_{max}.

_{x}> 0.4). Figure 6B shows that even when calculating <x

^{6}>, the systematic error associated with the long tail of the distribution reached 40% and became unacceptably large. The results of calculation of <x

^{12}> (curve 4, Figure 6B) are unreliable due to uncertainty about the concentration of large particles (with diameters x ≈ 25 nm and higher). Notably, replacing the Γ-distribution with the lognormal distribution, which is often used to analyze experimental data [27,28], does not resolve the issue. The lognormal distribution has a longer tail than the Γ-distribution and the systematic error in calculating high-order moments will be even higher.

_{e}of particles in the coarse fraction, responsible for the appearance of the maximum on the E*(H) curve and contributing the most to the normalized signal. To this end, we equated the Langevin parameter ξ*, corresponding to the maximum of the signal in Figure 3, to the Langevin parameter determined in terms of the effective magnetic moment m

_{e}and the value of the bias field H* in Figure 5:

_{s}= 480 kA/m is the saturation magnetization of the magnetite. For sample No. 1 we obtained ξ* = 1.53 and m

_{e}= 6.2 × 10

^{−19}A·m

^{2}, which is a value three times higher than the average magnetic moment <m> = 2.08 × 10

^{−19}A·m

^{2}in Table 1. The maximum diameter of the magnetic core of particles was x

_{max}= 13.5 nm, and the hydrodynamic diameter of the particles was d = 18 nm. The corresponding Brownian relaxation time was ${\tau}_{B}=3\eta V/kT=2\times {10}^{-6}s$. This implies that the dynamic effects in weak fields should be observed at frequencies higher than 80 kHz, which is substantiated by the experimental data on the dynamic susceptibility in Figure 4A. The dispersion of dynamic susceptibility at frequencies up to 80 kHz is rather weak, since particles with the diameter of magnetic cores close to the average value contribute the most to the susceptibility, at which the quasi-static condition ωτ

_{B}<< 1 is fulfilled.

_{B}<< 1). With increasing bias field strength, the relaxation times decrease, according to Equation (12), and the signal dispersion should decrease additionally, as shown in Figure 3A. However, in the experiment, we observed an opposite effect. With the growth of the bias field, the signal dispersion also increased. For H

_{0}≥ 5 kA/m the frequency dependence was observed at a frequency of 9 kHz, which implies a three- to four-fold increase in the Brownian relaxation time. In our opinion, this paradoxical behavior of sample No. 1 can only be explained by the formation of short chains in the magnetic fluid at the cost of anisotropic dipole–dipole interparticle interactions and bias field.

_{L}and the hydrodynamic concentration φ of particles, which were determined from the results of independent experiments: $\lambda ={\chi}_{L}/(8\phi )$ [6,19]. The values of the dipolar coupling constant for the examined samples calculated by this formula are provided in Table 2. For sample No. 1, λ = 0.6, and in the weak bias field, the influence of aggregates can be neglected. The application of the stronger field corresponding to the Langevin parameter ξ ≥ 1 stimulated the growth of chains and increased the relaxation time of the magnetization due to an increase in the volume of the chain and its form-factor. So, the dispersion of the signal observed in Figure 5A at frequencies higher than 9 kHz is the consequence of the change in the internal structure of the magnetic fluid.

_{e}= 54 × 10

^{−19}A m

^{2}. This value exceeds the average magnetic moment <m> = 2.31 × 10

^{−19}A m

^{2}already by a factor of 23. The diameter of the magnetic core of such particles should be close to the maximum possible value of x

_{max}. An estimate using Equation (22) provided x

_{max}≈ 28 nm, which is only slightly higher than the maximum possible value corresponding to the dashed line in Figure 6, but is significantly smaller than the diameter x

_{max}≈ 41 nm obtained with the use of the standard Γ-distribution in Equation (7). This result demonstrates once again the need to replace the Γ-distribution by another distribution characterized by the absence of a long tail.

_{max}and the Γ-distribution parameters, which was valid, at least for the magnetite colloids obtained by the chemical precipitation method. According to Aref’ev et al. [34]:

_{max}= 29 nm, which practically coincides with the estimate x

_{max}= 28 nm found by Equation (22). Estimates for sample No. 1 were x

_{max}= 13.6 and 13.5 nm for Equations (30) and (22), respectively. Thus, the two methods for evaluating the maximum size of particles existing in magnetite colloids agree well for both samples, despite these methods being based on different experimental techniques.

## 4. Discussion and Conclusions

_{0}directed along the sample and the weak alternating field normal to its axis act on the sample of magnetic fluid in the form of the long cylinder. The axis of the measuring coil was oriented along the sample and the induction of emf E(H

_{0},ω) was realized at double the frequency. The characteristic feature of the experiments was a strong dependence of the output signal (emf) on the particle size distribution in the weak bias field and the pronounced maximum in the region of the Langevin parameter ξ ≈ 1.5–1.9. Our attention was focused on the dependence of the output signal on the probe field frequency and the use of crossed field experiments for obtaining information about the largest particles in magnetic fluids.

_{B}≈ 1, but the degree of their influence on the output signal depended on the initial susceptibility of the solution. The higher the susceptibility, the stronger the dynamic effects. With an increase in the bias field, the relaxation times decreased according to formula (12), and at ξ

_{0}> 10 the dynamic effects became negligible.

_{B}<< 1). With increasing bias field strength, the relaxation times decreased according to Equation (12), and the signal dispersion should decrease further, as shown in Figure 3A. However, the experiment demonstrated an opposite effect. With increasing bias field, the signal dispersion did not reduce, but increased. Thus, at H

_{0}≥ 5 kA/m, the frequency dependence is already fixed at the frequency of 9 kHz, which implies an increase in the Brownian relaxation time by three to four times. In our opinion, such paradoxical behavior of sample No. 1 can only be explained by the formation of short chains in the magnetic fluid due to anisotropic dipole–dipole interparticle interactions and the bias field.

_{max}of the magnetic core of particles: x

_{max}= 13 nm for sample No. 1 and x

_{max}= 28 nm for sample No. 2. The same values of the maximum diameter (within the experimental error) were obtained in our study using the correlation between x

_{max}and the moments x of the third and sixth orders [34]. These results demonstrate the necessity of re-evaluating the applicability of lognormal and Γ-distributions to computation of high-order moments. In the case of a broad particle size distribution, such calculations are incorrect due to long tails. This problem can be solved by truncation of the tails according to Equation (23). However, this reduction makes simple formulas like Equation (7) inapplicable for arbitrary moments x, requiring a numerical calculation of the corresponding integrals.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Luca, E.; Cotae, C.; Calugaru, G.H. Some aspects of the Procopiu effect in ferrofluids. Rev. Roum. Phys.
**1978**, 23, 1173–1178. [Google Scholar] - Pirozhkov, B.I. Investigation of aggregation phenomena in magnetic fluid by crossed magnetic field method. Bull. Acad. Sci. USSR. Phys. Ser.
**1987**, 51, 1088–1093. [Google Scholar] - Pshenichnikov, A.F.; Fedorenko, A.A. Chain-like aggregates in magnetic fluids. J. Magn. Magn. Mater.
**2005**, 292C, 332–344. [Google Scholar] [CrossRef] - Raikher, Y.L.; Stepanov, V.I. Ferromagnetic resonance in a suspension of single-domain particles. Phys. Rev. B
**1994**, 50, 6250. [Google Scholar] [CrossRef] - Gazeau, F.; Bacri, J.C.; Gendron, F.; Perzynski, R.; Raikher, Y.L.; Stepanov, V.I.; Dubois, E. Magnetic resonance of ferrite nanoparticles: Evidence of surface effects. J. Magn. Magn. Mater.
**1998**, 186, 175–187. [Google Scholar] [CrossRef] - Ivanov, A.O.; Kuznetsova, O.B. Magnetic properties of dense ferrofluids: An influence of interparticle correlations. Phys. Rev. E
**2001**, 64, 041405. [Google Scholar] [CrossRef] - Wang, Z.; Holm, C.; Muller, H.W. Molecular dynamics study on the equilibrium magnetization properties and structure of ferrofluids. Phys. Rev. E
**2002**, 66, 021405. [Google Scholar] [CrossRef] - Pshenichnikov, A.F.; Ivanov, A.S. Magnetophoresis of particles and aggregates in concentrated magnetic fluids. Phys. Rev. E
**2012**, 86, 051401. [Google Scholar] [CrossRef] - Yerin, C.V. Particles size distribution in diluted magnetic fluids. J. Magn. Magn. Mater.
**2017**, 431, 27–29. [Google Scholar] [CrossRef] - Lee, W.K.; Ilavsky, J. Particle size distribution in ferrofluid macro-clusters. J. Magn. Magn. Mater.
**2013**, 330, 31–36. [Google Scholar] [CrossRef] - Shliomis, M.I.; Pshenichnikov, A.F.; Morozov, K.I.; Shurubor, I.Y. Magnetic properties of ferrocolloids. J. Magn. Magn. Mater.
**1990**, 85, 40–46. [Google Scholar] [CrossRef] - Ivanov, A.S. Temperature dependence of the magneto-controllable first-order phase transition in dilute magnetic fluids. J. Magn. Magn. Mater.
**2017**, 441, 620–627. [Google Scholar] [CrossRef] - Hayes, C.F. Observation of association in a ferromagnetic colloid. J. Colloid Interface Sci.
**1975**, 52, 239–243. [Google Scholar] [CrossRef] - Bacri, J.-C.; Perzynski, R.; Cabuil, V.; Massart, R. Phase diagram of an ionic magnetic colloid: Experimental study of the effect of ionic strength. J. Colloid Interface Sci.
**1989**, 132, 43–53. [Google Scholar] [CrossRef] - Rosensweig, R.E. Ferrohydrodynamics; Cambridge University Press: Cambridge, England, 1985. [Google Scholar]
- Gribanov, N.M.; Bibik, E.E.; Buzunov, O.V.; Naumov, V.N. Physico-chemical regularities of obtaining highly dispersed magnetite by the method of chemical condensation. J. Magn. Magn. Mater.
**1990**, 85, 7–10. [Google Scholar] [CrossRef] - Pshenichnikov, A.F.; Lebedev, A.V.; Radionov, A.V.; Efremov, D.V. A magnetic fluid for operation in strong gradient fields. Colloid J.
**2015**, 77, 196–201. [Google Scholar] [CrossRef] - Pshenichnikov, A.F. A Mutual-Inductance Bridge for Analysis of Magnetic Fluids. Instrum. Exp. Tech.
**2007**, 50, 509–514. [Google Scholar] [CrossRef] - Lebedev, A.V.; Stepanov, V.I.; Kuznetsov, A.A.; Ivanov, A.O.; Pshenichnikov, A.F. Dynamic susceptibility of a concentrated ferrofluid: The role of interparticle interactions. Phys. Rev. E
**2019**, 100, 032605. [Google Scholar] [CrossRef] - Martsenyuk, M.A.; Raikher, Y.L.; Shliomis, M.I. On the kinetics of magnetization of suspensions of ferromagnetic particles. Sov. Phys.-JETP
**1974**, 38, 413–416. [Google Scholar] - Blums, E.; Cebers, A.; Maiorov, M.M. Magnetic Fluids; Walter de Gruyter: Berlin, Germany, 1997. [Google Scholar]
- Bean, C.P.; Yacobs, I.S. Magnetic Granulometry and Super-Paramagnetism. J. Appl. Phys.
**1956**, 27, 1448. [Google Scholar] [CrossRef] - Shliomis, M.I. Magnetic fluids. Sov. Phys. Usp.
**1974**, 17, 153–169. [Google Scholar] [CrossRef] - Coffey, W.T.; Cregg, P.J.; Crothers, D.S.F.; Waldron, J.T.; Wickstead, A.W. Simple approximate formulae for the magnetic relaxation time of single domain ferromagnetic particles with uniaxial anisotropy. J. Magn. Magn. Mater.
**1994**, 131, L301–L303. [Google Scholar] [CrossRef] - Pshenichnikov, A.F.; Lebedev, A.V. Dynamic susceptibility of magnetic liquids. Sov. Phys. -JETP
**1989**, 68, 498–502. [Google Scholar] - Buzmakov, V.M.; Pshenichnikov, A.F. On the Structure of Microaggregates in Magnetite Colloids. J. Colloid Interface Sci.
**1996**, 182, 63–70. [Google Scholar] [CrossRef] - Hasmonay, E.; Dubois, E.; Bacri, J.-C.; Perzynski, R.; Raikher, Y.L.; Stepanov, V.I. Static magneto-optical birefringence of size-sorted γ-Fe
_{2}O_{3}nanoparticles. Eur. Phys. J. B**1998**, 5, 859–867. [Google Scholar] [CrossRef] - Buzmakov, V.M.; Pshenichnikov, A.F. Birefringence in concentrated ferrocolloids. Colloid J.
**2001**, 63, 275–282. [Google Scholar] [CrossRef] - Dikansky, Y.I. Experimental investigation of effective magnetic fields in a magnetic fluid. Magnetohydrodynamics
**1982**, 18, 237–240. [Google Scholar] - Sato, T.; Iijima, T.; Seki, M.; Inagaki, N. Magnetic properties of ultrafine ferrite particles. J. Magn. Magn. Mater.
**1987**, 65, 252–256. [Google Scholar] [CrossRef] - Kang, Y.S.; Risbud, S.; Rabolt, J.F.; Stroeve, P. Synthesis and Characterization of Nanometer-Size Fe
_{3}O_{4}and γ-Fe_{2}O_{3}Particles. Chem. Mater.**1996**, 8, 2209–2211. [Google Scholar] [CrossRef] - Bender, P.; Balceris, C.; Ludwig, F.; Posth, O.; Bogart, L.K.; Szczerba, W.; Castro, A.; Nilsson, L.; Costo, R.; Gavilán, H.; et al. Distribution functions of magnetic nanoparticles determined by a numerical inversion method. New J. Phys.
**2017**, 19, 073012. [Google Scholar] [CrossRef] - de Gennes, P.G.; Ρincus, P.Α. Pair correlations in a ferromagnetic colloid. Phys. Kondens. Mater.
**1970**, 11, 189–198. [Google Scholar] [CrossRef] - Aref’ev, I.M.; Lebedev, A.V. Determination of maximum particle size in magnetic fluids. Colloid J.
**2016**, 78, 269–272. [Google Scholar] [CrossRef]

**Figure 1.**Measuring cell in crossed magnetic fields. The black cylinder is the test tube containing magnetic fluid, the light cylinder is the measuring coil. (

**A**) The low frequency limit; (

**B**) the arbitrary frequencies.

**Figure 2.**Scheme of the experimental setup.

**1**, Sine voltage generator;

**2**, temperature sensor;

**3**, test tube with magnetic fluid;

**4**, measuring coil;

**5**, bias coil;

**6**, thermostatic shell;

**7**, single-layer solenoid for generation of alternating current (AC) field.

**Figure 3.**The normalized output signal versus the Langevin parameter for monodisperse ferrofluid and different frequencies of the probe field. (

**A**) The simple No. 1 sample (<m> = 2.08 × 10

^{−19}A m

^{2}, χ

_{L}= 0.29; (

**B**) the simple No. 2 sample (<m> = 2.3 × 10

^{−19}A m

^{2}, χ

_{L}= 0.84).

**Figure 4.**The real (curve 1) and imaginary (curve 2) parts of the dynamic susceptibility versus frequency for (

**A**) sample No. 1 and (

**B**) sample No. 2.

**Figure 5.**The normalized output signal versus the bias field strength for sample (

**A**) No. 1 and (

**B**) No. 2. Points indicate experimental data at different frequencies of the probe field; solid lines indicate spline smoothing.

**Figure 6.**Γ-distribution of particle sizes (curve 1); the relative contribution of particles to the 3rd- <x

^{3}> (curve 2), 6th- <x

^{6}> (curve 3), and 12th-order moment <x

^{12}> (curve 4). The dashed line corresponds to the maximum possible diameter of magnetite particles in stable colloids. (

**A**) sample No. 1; (

**B**) sample No. 2.

Physical Property | FM1 | FM2 |
---|---|---|

T (K) | 291 | 286 |

M_{∞} (kA/m) | 66.5 | 13.3 |

χ_{0} | 4.40 | 4.86 |

χ_{L} | 2.39 | 2.56 |

<m> (10^{−19} A m^{2}) | 2.08 | 2.31 |

<m^{2}> (10^{−38} A^{2} m^{4}) | 7.16 | 41.6 |

<x>, nm | 8.82 | 6.92 |

ρ (g/cm^{3}) | 1.597 | 0.984 |

φ_{ρ} | 0.183 | 0.045 |

δ_{x} | 0.26 | 0.67 |

Physical Property | Sample 1 | Sample 2 |
---|---|---|

T(K) | 297 | 298 |

M_{∞} (kA/m) | 7.9 | 4.4 |

χ | 0.31 | 1.07 |

χ_{L} | 0.29 | 0.84 |

λ | 0.6 | 2.1 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pshenichnikov, A.; Lebedev, A.; Ivanov, A.O.
Dynamics of Magnetic Fluids in Crossed DC and AC Magnetic Fields. *Nanomaterials* **2019**, *9*, 1711.
https://doi.org/10.3390/nano9121711

**AMA Style**

Pshenichnikov A, Lebedev A, Ivanov AO.
Dynamics of Magnetic Fluids in Crossed DC and AC Magnetic Fields. *Nanomaterials*. 2019; 9(12):1711.
https://doi.org/10.3390/nano9121711

**Chicago/Turabian Style**

Pshenichnikov, Alexander, Alexander Lebedev, and Alexey O. Ivanov.
2019. "Dynamics of Magnetic Fluids in Crossed DC and AC Magnetic Fields" *Nanomaterials* 9, no. 12: 1711.
https://doi.org/10.3390/nano9121711