μSR-Study of a 3% CoFe2O4 Nanoparticle Concentration Ferrofluid

Magnetic fluids based on single-domain magnetic spinel ferrite nanoparticles dispersed in various liquid media are of particular practical and scientific interest. This paper presents a muon spectroscopy study of a ferrofluid based on magnetic nanoparticles of CoFe2O4 molecules dispersed in water (H2O) with a nanoparticle concentration of 3%. In this study, it was determined that the structure and magnitude of the magnetization of a ferrofluid depend on the viscosity of the liquid itself. It was shown that, at room temperature (290 K) and under an external magnetic field of 527 G, the observed additional magnetization was ~20 G. In a small fraction of the sample under study (~20%), negative magnetization (diamagnetism) was observed. At low temperatures (~30 K), the sample acted as a paramagnet in a magnetic field. For the first time, the magnetic field inside and in the immediate vicinity of a CoFe2O4 nanoparticle has been measured experimentally using the μSR method: the value was 1.96 ± 0.44 kG; thus, direct measurement of the magnetization of a nanoscale object was performed.

For many decades, ferrofluids (FF)-ultrastable dispersions of single-domain nanomagnetic particles in various liquids, stabilized by different methods-have been continuously studied by scientists and engineers [15].
Ferrofluids have been used in an increasingly diverse spectrum of applications in research and/or development, such as seals, lubricants, adjustable dampers, clutches, inkjet printers, cooling for electronic devices, heat transfer, mass density separation, optics applications, various sensors or biosensors, cell separation, the controlled targeting of drugs with magnetic resonance imaging, hyperthermia, etc., to mention a few [3,5,16].
The developments for novel applications are directly connected with active research on the production of new types of ferrofluids, with particles of different sizes, shapes, compositions, and concentrations [17,18], using innovative synthesis and stabilization methods and compounds, as well as conducting studies on their properties. Currently, liquids based on single-domain magnetic nanoparticles of spinel ferrites, MeFe 2 O 4 (where Me are Cr, Mg, Fe, Co, and Zn), dispersed in various liquid media are widely studied.
The stability of magnetic fluids is achieved by the use of various surfactants that prevent nanoparticles from sticking together [19][20][21]. The magnetic fluid in water was stabilized at low concentrations (<5-7%) of nanoparticles. At temperatures below the Curie temperature, each nanoparticle had a magnetic moment close to the total moment of molecules in the particle.
Among various solid-state nuclear techniques, muon spin research (µSR) is a relatively new technique [34,35]. However, as in all fields of science, in applied muon research new methods and techniques are also being developed, and capabilities of existing methods are continuously being expanded to meet challenges posed by new objects of investigation. The positive-muon rotation technique has proven to be an excellent method for investigating static and dynamic local magnetic fields in condensed matter. Due to its positive charge, the muon usually resides at an interstitial site in a crystal; therefore, these regions are normally inaccessible with other more conventional techniques (magnetic resonance, Mössbauer, etc.). Particular attention has been devoted to magnetic oxides and especially to magnetite [36,37].
The Curie temperature for cobalt ferrite is 793 K, and among the spinel ferrites, CoFe 2 O 4 has the highest magnetic crystallographic anisotropy constant [38]: about 20 times higher than that of magnetite Fe 3 O 4 , and equal to 1.84 × 10 6 erg/cm 3 at T = 300 K for cobaltite (CoFe 2 O 4 ) nanoparticles of about 32 nm in size.
The µSR method can be used to help understand magnetic phenomena peculiarities in systems containing nanomagnetic particles by determining local magnetic fields in the sample [39].

Materials
For these studies, a ferrofluid sample composed of a 3% volume concentration of magnetic nanoparticles in water, stabilized using sodium dodecyl sulfate, CH 3 (CH 2 ) 11 SO 4 Na, with a density ρ = 1.01 g/cm 3 , and lauric acid, C 11 H 23 COOH, with a density ρ = 0.88 g/cm 3 , was used as a double surfactant layer.
One milliliter of ferrofluid contained 0.17 g of cobalt ferrite, and for every 1 g of CoFe 2 O 4 nanoparticles, 0.25 g of surfactant was used.
The investigated sample was synthesized at the Institute of Technical Chemistry, Ural Branch of the Russian Academy of Sciences [18,40,41]. Preliminarily, a double separation of nanoparticles by size was carried out on a Biofuge 15R centrifuge for 60 min at 6000 rpm. The particle size distribution was studied at the Center for Advanced Technologies (Moscow State University, www.nanoscopy.ru) using a LEO 912 AB OMEGA high-resolution transmission electron microscope with an accelerating voltage of 120 kV. The size distribution of nanoparticles was approximated by the lognormal distribution function, and the following parameter values were determined: D 0 = 7.8 ± 0.1 nm, σ = 0.40 ± 0.01 nm, with the mathematical expectation value D m = D 0 exp(σ 2 /2) = 8.5 nm.

Muon Spin Rotation (µSR) Method
The sample was studied using the µSR setup [42] installed at the output of the muon channel of the SC-1000 synchrocyclotron of the NRC "Kurchatov Institute", PNPI. The principles of µSR and uses of various relaxation (depolarization) functions of the muon spins to describe the time spectra of the µSR signals are presented in studies which have become references in the field [34,43]. At this facility, we have previously conducted studies of ferrofluids based on Fe 3 O 4 [39]. The beam of positively charged muons with an average momentum p µ = 90 MeV/s and a momentum dispersion ∆p µ /p µ = 0.02 (FWHM) had a longitudinal polarization p µ = 0.90 ÷ 0.95. The ferrofluid sample, packed in a copper cell 80 mm in diameter and 10 mm in height, was placed in a blowdown cryostat, which made it possible to set the temperature with an accuracy of~0.1 K in the temperature range 10-290 K. An external magnetic field on the sample, transverse to the direction of the muon spin, was created by Helmholtz coils with a stability of~10 −3 . The uniformity of the magnetic field in the central region of the magnetic system was no worse than 10 −4 . The degree of uniformity of the magnetic field was measured both by the standard method using Hall sensors and by the rate of attenuation of the polarization precession in a copper sample with a size of Ø 40 × 5 mm (λ = 0.0053 ± 0.0031 µs −1 ); such uniformity of the field is acceptable for conducting µSR experiments. The Earth's magnetic field and external scattered magnetic fields were compensated for by additional Helmholtz coils to a level of~0.05 G.
The time spectra of the positrons from the muons' decay were measured simultaneously in two ranges (0 ÷ 10 µs and 0 ÷ 1.1 µs) with channel widths of 4.9 and 0.8 ns/channel, respectively.
The measurement procedure consisted of cooling the sample in a zero external magnetic field (ZFC) to a temperature of 25 K. After that, a magnetic field of 527 G, transverse to the magnetic moment of the muon, was switched on, and ZFC measurements were recorded as the sample was heated to room temperature (290 K). Then, the sample was again cooled to a temperature of 30 K in the same transverse magnetic field of 527 G, and FC measurements were performed.
The experimental data (time spectra) were approximated using the function: where N 0 is a normalization constant; τ µ is the muon lifetime (2.197 µs); B is the background of random coincidences, determined by the form of the time spectrum in the part before the beginning of the decay spectrum, t 0 ; and a s and G s (t) are the asymmetry and relaxation function of the observed muon component of the muon spin precession in the sample, respectively. The term a b · G b (t) corresponds to the contribution of the constructive background to the observed asymmetry. The constructive background itself is mainly associated with the decay of muons stopped in the copper walls of the container with ferrofluid, and its contribution can be described as follows: The parameters F b and λ b , the precession frequency and relaxation rate, respectively, are determined from the processing of the time spectrum measured on a copper sample. The a b parameter was obtained from the joint processing of two spectra measured on the ferrofluid sample in an external magnetic field (H = 0) and in a zero magnetic field (H = 0): In this case, the following requirement is imposed: The amplitude a 2 of a weakly damped exponent is chosen as a b . The indices (H, L) on the parameters (a, λ) correspond to the observed frequencies of the precession of the muon spin F H and F L , where F H > F Cu > F L . The value F Cu corresponds to the rotation frequency of the muon spin in the pure copper sample.
Using the relaxation function (Equation (1)), and taking into account Equations (2) and (3), made it possible to accurately describe the experimental data of FC and ZFC measurements in a wide range of sample temperatures.
In the temperature range of 26-100 K, the time spectra obtained in ZFC measurements are fairly well described using the following single-frequency relaxation function:

Zero Field Measurements
Even in the absence of an external magnetic field, µ + will precess around the internal dipolar field. At a sample temperature of 290 K, two measurements were performed in a zero external magnetic field. One was performed before turning on the external field (round light points), as at T = 41 K, and the second measurement was carried out immediately after turning on the magnetic field with a value of H = 527 G. In both measurements, the relaxation functions did not differ from each other within the error limits. Figure 1 displays the behavior of the relaxation function G(t) at sample temperatures of 290 K and 41 K in a zero external magnetic field. This indicated that the external magnetic field did not lead to the spatial displacement of nanoparticles.
The amplitude 2 a of a weakly damped exponent is chosen as b a .
The indices ( H , L) on the parameters (a, λ) correspond to the observed frequencies of the precession of the muon spin H F and L F , where H Cu L F F F > > . The value Cu F corresponds to the rotation frequency of the muon spin in the pure copper sample. Using the relaxation function (Equation (1)), and taking into account Equations (2) and (3), made it possible to accurately describe the experimental data of FC and ZFC measurements in a wide range of sample temperatures.
In the temperature range of 26-100 K, the time spectra obtained in ZFC measurements are fairly well described using the following single-frequency relaxation function:

Zero Field Measurements
Even in the absence of an external magnetic field, μ + will precess around the internal dipolar field. At a sample temperature of 290 K, two measurements were performed in a zero external magnetic field. One was performed before turning on the external field (round light points), as at T = 41 K, and the second measurement was carried out immediately after turning on the magnetic field with a value of H = 527 G. In both measurements, the relaxation functions did not differ from each other within the error limits. The relaxation function at a temperature of 41 K had two components, which sharply differed in terms of the decay rates 1 λ = 0.41 ± 0.02 μs −1 and 2 λ = 23 ± 11 μs −1 . The rapidly decaying component was apparently associated with the presence of muon stops in the The relaxation function at a temperature of 41 K had two components, which sharply differed in terms of the decay rates λ 1 = 0.41 ± 0.02 µs −1 and λ 2 = 23 ± 11 µs −1 . The rapidly decaying component was apparently associated with the presence of muon stops in the near zone of the scattered fields of nanoparticles, where the inhomogeneity of the magnetic field was high. The separation of the sample volume into near and far zones occurred only at low temperatures when the rotation or oscillation of nanoparticles stopped. At a temperature of 290 K, the entire sample volume was homogeneous in terms of the relaxation rate. This behavior of the relaxation functions was associated with the presence of major anisotropy of the magnetic properties of the nanoparticle itself.

The Field Scan at the Room Temperature
Further, the sample was measured at room temperature for magnetic field values in the range of up to~600 G. Figures 2-4 demonstrate the behavior of frequencies, decomposition rates, and populations as a function of the external magnetic field at the sample temperature of 290 K. tion rate. This behavior of the relaxation functions was associated with the presence of major anisotropy of the magnetic properties of the nanoparticle itself.

The Field Scan at the Room Temperature
Further, the sample was measured at room temperature for magnetic field values in the range of up to ~600 G. Figures 2-4 demonstrate the behavior of frequencies, decomposition rates, and populations as a function of the external magnetic field at the sample temperature of 290 K.

The Field Scan at the Room Temperature
Further, the sample was measured at room temperature for magnetic field values in the range of up to ~600 G. Figures 2-4 demonstrate the behavior of frequencies, decomposition rates, and populations as a function of the external magnetic field at the sample temperature of 290 K.  For all external magnetic fields in the range of 50-527 G, the sample under study exhibited muon spin precession at two frequencies, F H and F L , in the order F H > F Cu > F L (Figure 2). The value F Cu corresponds to the rotation frequency of the muon spin in the pure copper sample.
Additional magnetization was observed in 80% of the sample volume (Figure 4), which, at room temperature (~300 K), corresponded to an additional ∆H = 20 ± 1 G when the external magnetic field was more than 300 G (Figure 2). In addition, there was a much less noticeable diamagnetic contribution ∆H = 6.7 ± 0.2 G in 20% of the sample volume (Figures 2 and 4). The relaxation rates λ H and λ L , as well as the populations a H and a L , practically do not depend on the value of the external magnetic field (Figures 3 and 4). The values of λ L in Figure 3 are in the range of 0.09-0.12.
For all external magnetic fields in the range of 50-527 G, the sample under study exhibited muon spin precession at two frequencies, Additional magnetization was observed in 80% of the sample volume ( Figure 4), which, at room temperature (~300 K), corresponded to an additional H Δ = 20 ± 1 G when the external magnetic field was more than 300 G ( Figure 2). In addition, there was a much less noticeable diamagnetic contribution H Δ = 6.7 ± 0.2 G in 20% of the sample volume ( Figures 2 and 4). The relaxation rates H λ and L λ , as well as the populations H a and L a , practically do not depend on the value of the external magnetic field (Figures 3 and   4). The values of L λ in Figure 3 are in the range of 0.09-0.12.

FC and ZFC Measurements
The behavior of systems with nanoparticles depends on the conditions in which they were cooled. At the FC procedure, the sample is cooled in the magnetic field, whereas the ZFC sample is cooled first in the absence of magnetic field and then the magnetic field is switched on.

FC and ZFC Measurements
The behavior of systems with nanoparticles depends on the conditions in which they were cooled. At the FC procedure, the sample is cooled in the magnetic field, whereas the ZFC sample is cooled first in the absence of magnetic field and then the magnetic field is switched on. Figures 5-8 present the results of processing the time spectra measured on a ferrofluid sample in an external magnetic field H = 527 G in the sample temperature range of 26 K to 290 K. In the FC measurement mode, two muon spin precession frequencies, F H and F L , were observed over the entire temperature range ( Figure 5).    A significant excess of H a over L a was observed over the entire temperature range. This difference was especially large at room temperature, when the solution was a real ferrofluid, and not ice. Definite structures of the temperature dependence of the quantities H a and L a were observed in FC measurements. The sum of the partial amplitudes at all temperature points was almost 20% less than the amplitude of the precession obtained in water (H2O) (Figure 9). This difference can be explained by the fact that some of the muons stop in places with large inhomogeneity in the field. This leads to their rapid depolarization, and the oscillation of their spins was not observed. The behavior of positive muons in water has been well-studied previously [44]. Nevertheless, we conducted our own measurements in our cryostat for confirmation. In the ZFC measurement mode, in the temperature range of 26-100 K, one precession frequency of muon spin was observed. With an increase in the sample temperature, starting from T = 100 K, two muon spin precession frequencies, F H and F L , appeared; moreover, the F H frequency began to increase noticeably. For comparison, the same figure ( Figure 5) shows the behavior of the precession frequency of muons in a sample of H 2 O and Cu. The relationship F H > F Cu,H 2 O > F L is maintained throughout the entire temperature regime. Figure 6 shows the behavior of the parameters λ H , λ L , and λ 0 in the sample temperature range of 26 K to 290 K. Here, λ 0 is the relaxation rate of muon polarization in a zero external magnetic field. A noticeable difference of the parameter λ 0 at T = 290 K from the value at T = 41 K is associated with the dynamics of nanoparticles in the medium at T = 290 K. In the FC measurement mode, the values of λ H and λ L were practically independent within the error limits at the sample temperature. In the ZFC measurement mode in the temperature range 26-125 K, λ H ≈ λ 0 (Figure 6). At these temperatures, the sample behavior was paramagnetic ( Figures 5 and 6). With a further increase in the temperature of the sample, the parameter λ H began to increase noticeably, and was practically comparable with the value of λ 0 at T = 290 K. Figures 7 and 8 display the behavior of the populations of the upper F H and lower F L frequencies (a H and a L ) versus sample temperature in FC and ZFC measurements.
A significant excess of a H over a L was observed over the entire temperature range. This difference was especially large at room temperature, when the solution was a real ferrofluid, and not ice. Definite structures of the temperature dependence of the quantities a H and a L were observed in FC measurements.
The sum of the partial amplitudes at all temperature points was almost 20% less than the amplitude of the precession obtained in water (H 2 O) ( Figure 9). This difference can be explained by the fact that some of the muons stop in places with large inhomogeneity in the field. This leads to their rapid depolarization, and the oscillation of their spins was not observed. The behavior of positive muons in water has been well-studied previously [44]. Nevertheless, we conducted our own measurements in our cryostat for confirmation.

The Estimation of the Magnitude of the Magnetic Field Near the Nanoparticle
The relaxation function ( ) G t for the sum of spectra in the temperature range 26-100 K, obtained in the ZFC mode in an external magnetic field H = 527 G, and its satisfactory fitting curve obtained within the framework of the two-frequency hypothesis presented in Equation (3), are depicted in Figure 10.

The Estimation of the Magnitude of the Magnetic Field Near the Nanoparticle
The relaxation function G(t) for the sum of spectra in the temperature range 26-100 K, obtained in the ZFC mode in an external magnetic field H = 527 G, and its satisfactory fitting curve obtained within the framework of the two-frequency hypothesis presented in Equation (3), are depicted in Figure 10.

The Estimation of the Magnitude of the Magnetic Field Near the Nanoparticle
The relaxation function ( ) G t for the sum of spectra in the temperature range 26-100 K, obtained in the ZFC mode in an external magnetic field H = 527 G, and its satisfactory fitting curve obtained within the framework of the two-frequency hypothesis presented in Equation (3), are depicted in Figure 10. Therefore, in addition to the main contribution characterized by the amplitude a L = 0.099 ± 0.001, the relaxation rate λ L = 0.35 ± 0.03 µs −1 , and the frequency F L = 7.167 ± 0.002 MHz, corresponding to a magnetic field value of H = 528.76 ± 0.14 G, a small supplementary contribution is determined with the parameters a H = 0.027 ± 0.007, λ H = 24 ± 6 µs −1 , and F H = 26.5 ± 0.6 MHz, corresponding to a magnetic field value of H = 1.96 ± 0.44 kG.
This contribution is associated with the muon stopping in the immediate vicinity of nanoparticles and inside nanoparticles. Thus, direct measurement of the magnetization of the nanoscale object was carried out.

Conclusions
A muon spectroscopy study of a ferrofluid with a 3% volume concentration of CoFe 2 O 4 magnetic nanoparticles dispersed in H 2 O has been reported.
New information on the magnetic properties of the system have been determined or confirmed using a new method: (i) The structure and value of the magnetization of a ferrofluid with CoFe 2 O 4 nanoparticles depend on the viscosity of the liquid itself. (ii) At room temperature (~290 K), when the sample was a superparamagnetic system, from the experimentally determined magnetization an average magnetic field of 20 G was obtained. (iii) A small fraction of the sample with negative magnetization, characteristic for diamagnetic systems, was also determined. (iv) At low temperatures (~30 K), the sample exhibits paramagnetic behavior in a magnetic field.
(v) The magnetic field inside and in the immediate vicinity of the CoFe 2 O 4 nanoparticles was first measured by the µSR method, and its value was B = 1.96 ± 0.44 kG.
Although the presented results belong primarily to the category of fundamental knowledge, with the development of technologies for nanoparticle manipulation in technical and bio-medical applications, such information may turn out to be valuable.