#
Magnetocrystalline and Surface Anisotropy in CoFe_{2}O_{4} Nanoparticles

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

_{ann}on the magnetic properties of cobalt ferrite nanoparticles embedded in an amorphous silica matrix (CoFe

_{2}O

_{4}/SiO

_{2}), synthesized by a sol-gel auto-combustion method, was investigated by magnetization and AC susceptibility measurements. For samples with 15% w/w nanoparticle concentration, the particle size increases from ~2.5 to ~7 nm, increasing T

_{ann}from 700 to 900 °C. The effective magnetic anisotropy constant (K

_{eff}) increases with decreasing T

_{ann}, due to the increase in the surface contribution. For a 5% w/w sample annealed at 900 °C, K

_{eff}is much larger (1.7 × 10

^{6}J/m

^{3}) than that of the 15% w/w sample (7.5 × 10

^{5}J/m

^{3}) annealed at 700 °C and showing comparable particle size. This indicates that the effect of the annealing temperature on the anisotropy is not only the control of the particle size but also on the core structure (i.e., cation distribution between the two spinel sublattices and degree of spin canting), strongly affecting the magnetocrystalline anisotropy. The results provide evidence that the magnetic anisotropy comes from a complex balance between core and surface contributions that can be controlled by thermal treatments.

## 1. Introduction

_{2}O

_{4}, where M

^{2+}can be any divalent metal (e.g., M

^{2+}= Fe

^{2+}, Co

^{2+}, Zn

^{2+}, Ni

^{2+}, Mn

^{2+}, etc.). The atomic arrangement corresponds to a face-centered cubic structure of the oxygen atoms, with Fe

^{3+}and M

^{2+}occupying the tetrahedral (T

_{d}) and octahedral (O

_{h}) sites [7]. Such a structure makes magnetic spinel nanoparticles particularly attractive. It provides a tool to tailor their magnetic properties (e.g., magnetic crystalline anisotropy and saturation magnetization) by the variation of the cation distribution between the two sublattices. This can be done by changing the chemical composition, the preparation method, and thermal treatments [8,9,10].

^{57}Fe Mössbauer experiments [14,15]. This symmetry breaking induces changes in the topology of the surface magnetic moments and, consequently, in the exchange integrals (through super-exchange angles and/or distances between moments), thus leading to a change in the surface anisotropy [15]. Therefore, the magnetic properties of ferrite nanoparticles with a spinel structure are due to a complex interplay of several effects, among which surface disorder, cationic distribution, and spin canting are dominant [14,16].

_{2}O

_{4}nanoparticles dispersed in a silica matrix (CoFe

_{2}O

_{4}/SiO

_{2}). The results show that the thermal treatment plays an important role, along with the particle size, in controlling the surface and core contributions to the magnetic anisotropy and saturation magnetization.

## 2. Materials and Methods

_{2}O

_{4}nanoparticles uniformly embedded in a silica matrix with 15% (w/w) concentration of the magnetic phase were synthesized by a sol-gel auto-combustion method and treated afterward at three different annealing temperatures (T

_{ann}= 700, 800 and 900 °C). Synthesis and morpho-structural characterization of all the samples was already described in detail elsewhere [8,14,17,18]

_{3})

_{3}∙9H

_{2}O (Sigma Aldrich 98%, Darmstadt, Germany), Co(NO

_{3})

_{2}∙6H

_{2}O (Sigma Aldrich 98%, Darmstadt, Germany), citric acid (Sigma Aldrich 99.5%, Darmstadt, Germany) and of 25% ammonia solution (Carlo Erba Reagenti SpA, Cornaredo, Italy) were used without further purification. In this process, 1-molar iron and cobalt nitrate aqueous solutions in a 2:1 ratio, respectively, and citric acid (CA) with 1:1 molar ratio of metals to CA were prepared, and pH-adjusted to ~2 by aqueous ammonia addition. Tetraethoxysilane (TEOS, Sigma Aldrich 98%, Darmstadt, Germany) in ethanol was used as a silica precursor and, after its addition and vigorous stirring for 30 min, the sols were placed in an oven to gel in static air at 40 °C for 24 h. The gels underwent successively a thermal treatment at 300 °C for 15 min, where the auto-combustion reaction took place.

_{TEM}> is the median of the variable “diameter” (Table 1) and σ is the standard deviation. An increase in particle size with the increase in annealing temperature is observed.

## 3. Results and Discussions

_{max}, (Table 1) increases with the annealing temperature. According to Gittleman et al. [22], T

_{max}is related to the average blocking temperature <T

_{B}> through the equation:

_{irr}), and it corresponds to the blocking temperature of the particles with the maximum anisotropy. As expected, both T

_{irr}and T

_{max}grow with increasing size (i.e., increasing temperature). The difference between T

_{irr}and T

_{max}reflects the width of the blocking temperature distribution in the absence of magnetic interparticle interactions and it is correlated to the volume distribution. In our samples, such difference is weakly dependent on the annealing temperature, indicating that the thermal treatment does not significantly affect the distribution of the blocking temperatures. This is confirmed by the thermoremanent magnetization (TRM) curves [21] (Figure 2b, see supplementary information for details). Indeed, the shape of the energy barrier distribution is similar for the three samples, confirming that the sources of anisotropy are basically the same and that the interparticle interactions are weak. Two different models have been proposed to determine the blocking temperature distribution, yielding to its mean value and standard deviation. Starting from the model proposed by Chantrell and co-workers [23], the distribution of the anisotropy energy barriers was fitted by a log-normal function to determine the mean value of the blocking temperature (<T

_{B}>

_{CH}), reported in Table 1 [24,25,26]. We give details of the fit and values of the standard deviation (σ

_{TRM}) in the supporting information (Figures S3 and S4).

_{B}>

_{H.M.}) and its standard deviation (σ

_{H.M.}) [27] for a log-normal distribution of the particles volume, and negligible interparticle interactions. They found that <T

_{B}>

_{H.M.}and σ

_{H.M}can be expressed with known values of T

_{irr}and T

_{max}from <T

_{B}>

_{H.M.}= T

_{max}[1.792 + 0.186·ln(T

_{irr}/T

_{max}− 0.918)]

^{−1}+ 0.0039·T

_{irr}and σ

_{H.M.}= 0.624 + 0.397 ln(T

_{irr}/T

_{max}− 0.665). <T

_{B}>

_{H.M.}values are given in 16(1), 25(2) and 31(2) K for samples (Table 1), their standard deviation values being 0.73, 0.61, and 0.57 for N15T700, N15T800 and N15T900, respectively.

_{CH.}and PD

_{H.M.}) decreases with increasing particle size, although this trend is more evident for the Chantrell model.

_{irr}) as a function of the annealing temperature (T

_{ann}). These results indicate a strong increase in the ferrimagnetic phase between 700 and 900 °C, which can be ascribed to the rise in the particle volume [28].

_{N}= τ

_{0}exp(K

_{eff}V/k

_{B}T). Since T = T

_{B}when τ

_{m}= 1/υ

_{m}, a linear relation between ln(τm) and 1/T

_{B}can be derived:

_{B}is reported for the three samples. The values of the effective magnetic anisotropy constant, K

_{eff}, and the characteristic relaxation times, τ

_{0}, obtained from the linear fitting of Equation (4), are given in Table 2.

_{0}value has a coherent physical meaning (1.9 × 10

^{−9}s), confirming the absence of interparticle interactions. On the other hand, for samples N15T800 and N15T900, the τ

_{0}value is much smaller. This fact indicates that the Néel–Arrhenius model is not appropriate to describe the dynamical behavior of these samples, suggesting that weak interparticle interactions are present.

_{0}[29,30,31]:

_{0}and K

_{eff}(Table 2) have been obtained from the fitting of Equation (5) by fixing the specific relaxation time τ

_{0}equal to 10

^{−10}s for all the samples [14,26]. In sample N15T700, T

_{0}is almost zero, consistent with the absence of interparticle interactions. Then, T

_{0}rises with the annealing temperature, indicating an increase in the dipolar interactions due to the enhancement of the particle magnetic moment. It is worth underlining that the value of K

_{eff}obtained by Néel–Arrhenius and Vogel–Fulcher models are similar for sample N15T700 where the interactions can be considered negligible. A difference in the K

_{eff}values deduced from the two models is observed for N15T800 and N15T900 due to magnetic interactions.

_{eff}increases with a decreasing particle size (i.e., decreasing annealing temperature). We measured a rise of ~30% when the diameter goes from 6.6 (N15T900) to 5.6 nm (N15T800), while a much higher growth of ~70% is observed when it goes from 6.6 (N15T800) to 2.5 nm (N15T700). This result indicates that the surface anisotropy increases with a decrease in the particle size, but its role becomes dominant in tiny particles (e.g., N15T700). This idea is also confirmed by the fact that the K

_{eff}values of N15T800 and N15T900, which are lower than the value of the bulk material (3 × 10

^{5}J/m

^{3}[14,32]), indicating that the magnetic structure also plays a crucial role. The smaller anisotropy in CoFe

_{2}O

_{4}nanoparticles compared to the bulk value can be related to a change in the cation distribution with the size, induced by the annealing treatment. This phenomenon was already observed in CoFe

_{2}O

_{4}particles [14,33] and explained by a modification of the cation distribution leading to a change in the magneto-crystalline anisotropy mainly determined by the distribution of the Co

^{2+}ions between O

_{h}and T

_{d}sites. Indeed, here the cause can be a lower fraction of Co

^{2+}ions in the octahedral sites, having larger anisotropy (+850 × 10

^{−24}J/ion) (due to the orbital contribution in the crystal field

^{4}T

_{1}ground energy term) than Co

^{2+}ions in a tetrahedral site (−79 × 10

^{−24}J/ion;

^{4}A

_{2}term) [8].

_{eff}value of an additional sample consisting of CoFe

_{2}O

_{4}nanoparticles embedded in a silica matrix with a 5% (w/w) concentration of magnetic phase annealed at 900 °C (hereafter named N5T900). For this sample, the average particle size (2.8 ± 0.3 nm [14]) is very close to that of the N15T700 (2.5 ± 0.5 nm), with the same percentual polydispersity (see Figure S5 and Table S1 in the supporting information). It is important to underline that the interparticle interactions in both N5T900 and N15T700 samples are negligible, as indicated by their corresponding ZFC-FC and δM-plots [34] (Figures S6 and S7, respectively, in the supporting information).

_{eff}is much larger for N5T900, which could be related to the cation distribution change caused by the annealing. The highest temperature produces a larger occupancy of O

_{h}sites by the Co

^{2+}ions in sample N5T900 [14].

_{S}) has been estimated by fitting the high field range of the curves to the equation [35]:

_{SAT}is the “non-saturated” magnetic susceptibility (for high applied fields). The latter is strongly related to the non-collinear spin structure due to competing interactions between sublattices, and to the symmetry breaking at the particle surface [36,37].

_{S}increases with particle size (i.e., annealing temperature), as expected. In the same figure, we plot M

_{S}for sample N5T900 (2.8 ± 0.3 nm particle size). Despite N5T900 and N15T700 having the same particle size, M

_{S}for N5T900 is almost twice than for N15T700. Considering that the magnetic interparticle interactions are negligible in both samples, this difference can be ascribed to the combined effect of cation distribution, spin-canting, and surface anisotropy [14,16]. The non-saturated susceptibility (Figure 4b) increases with decreasing particle size (i.e., decreasing annealing temperature) [31]. The trend of χ

_{SAT}indicates, as expected, the more substantial contribution of the surface magnetic anisotropy for smaller particles. It is worth emphasizing that N15T700 has a higher value of χ

_{SAT}, indicating that the surface contribution to the effective magnetic anisotropy is higher in N15T700 than in N15T900. The energy barrier distribution can confirm this. In fact, despite N5T900 and N15T700 having the same PD% of the TEM diameter, the PD% for T

_{B}calculated by H.M. model is much higher for N15T700 (PD% T

_{B}4.56) than for N5T900 (PD% T

_{B}2.45).

_{SAT}and the anisotropy energy barrier distribution indicate a more significant contribution of the surface component to the anisotropy in N15T700, the value K

_{eff}obtained by AC measurements is higher in N5T900. This could be associated with an increase in the magneto-crystalline component of the anisotropy.

_{DCD}is only sensitive to the irreversible component of the magnetization and only the blocked particles contribute to the remanent magnetization. The curve shape is linked to the switching field distribution, which, in turn, is related to the energy barrier distribution; the value of the field at which the remanent magnetization is equal to zero (called remanent coercivity, H

_{Cr}) corresponds to the mean switching field. Although the two samples have different coercivity, the remanent coercivities are close (N15T700: ~2.4 T, N5T900: ~2.1 T). This result is in line with the similar anisotropy fields (N15T700: 5.8(5) T, N5T900: 5.9(6) T) estimated by the Stoner–Wohlfarth model (H

_{K}= 2K

_{eff}/M

_{S}).

_{Cr}and H

_{K}are equal within the experimental error, the coercivities of the two samples are different. Since both systems are non-interacting, such differences can be associated with a larger fraction of very small particles that probably are not well crystallized due to the low treatment temperature. We have confirmed this by the trend of χT

_{irr}(inset of Figure 1a) and the lower value of the remanent and saturation magnetizations [40].

## 4. Conclusions

_{2}O

_{4}nanoparticles embedded in a silica matrix can be controlled by the annealing temperature T

_{ann}. For samples with 15% w/w of nanoparticles, the value of the effective magnetic anisotropy constant K

_{eff}increases and the saturation magnetization decreases by decreasing T

_{ann}from 900 to 700 °C, with a decrease in particle size, showing a dominant role of the disordered surface. On the other hand, the comparison between the 15% w/w sample annealed at 700 °C and a 5% w/w sample annealed at 900 °C, with comparable particle size, (2.5 and 2.8 nm with the same size distribution) shows a much larger saturation magnetization and K

_{eff}values for the latter one, for which the χ

_{SAT}value, related to non-collinear core spin structure and surface disorder, is much lower. The comparison indicates that for the 5% w/w with T

_{ann}= 900 °C sample the major contribution to the anisotropy comes from the core, despite its very small particle size. This should be due to a better crystallinity and change in a core structure (e.g., different cation distribution and degree of spin canting) with larger magnetocrystalline anisotropy induced by the higher T

_{ann}. In conclusion, the results indicate that the effect of the annealing temperature on the anisotropy and saturation magnetization is not limited to the change in the particle size, increasing with T

_{ann}. Besides the decrease in surface disorder, the core structure is also affected by the thermal treatment, which can significantly modify the magnetocrystalline anisotropy and the saturation magnetization.

## Supplementary Materials

_{B}>) and standard deviation (σ) are reported in the graph; Figure S5: Particle size distribution extracted by TEM images of the sample N15T700 (left side) and N5T900 (right side); Figure S6: ZFC-FC curves for the N15T700 (a) and N5T900 (b); Figure S7: δM-plot at 5 K for N15T700 sample and reference sample N5T900 [14]; Figure S8: Hysteresis loop recorded at 5 K for N15T900 (a), N15T800 (b) and N15T700 (c); Table S1: Mean particle size (D

_{TEM}) σ the standard deviation of the natural logarithm of the variable D and percentual polydispersity determined by Equation (7).

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Gomes, J.D.A.; Sousa, M.H.; Tourinho, F.A.; Aquino, R.; Da Silva, G.J.; Depeyrot, J.; Dubois, E.; Perzynski, R. Synthesis of core-shell ferrite nanoparticles for ferrofluids: Chemical and magnetic analysis. J. Phys. Chem. C
**2008**, 112, 6220–6227. [Google Scholar] [CrossRef] - Pardavi-Horvath, M. Microwave applications of soft ferrites. J. Magn. Magn. Mater.
**2000**, 216, 171–183. [Google Scholar] [CrossRef] - Song, G.; Kenney, M.; Chen, Y.S.; Zheng, X.; Deng, Y.; Chen, Z.; Wang, S.X.; Gambhir, S.S.; Dai, H.; Rao, J. Carbon-coated FeCo nanoparticles as sensitive magnetic-particle-imaging tracers with photothermal and magnetothermal properties. Nat. Biomed. Eng.
**2020**, 4, 325–334. [Google Scholar] [CrossRef] - Cardoso, V.F.; Francesko, A.; Ribeiro, C.; Bañobre-López, M.; Martins, P.; Lanceros-Mendez, S. Advances in Magnetic Nanoparticles for Biomedical Applications. Adv. Healthc. Mater.
**2018**, 7, 1700845. [Google Scholar] [CrossRef] [PubMed] - Waag, F.; Gökce, B.; Kalapu, C.; Bendt, G.; Salamon, S.; Landers, J.; Hagemann, U.; Heidelmann, M.; Schulz, S.; Wende, H.; et al. Adjusting the catalytic properties of cobalt ferrite nanoparticles by pulsed laser fragmentation in water with defined energy dose. Sci. Rep.
**2017**, 7, 1–13. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rizzuti, A.; Dassisti, M.; Mastrorilli, P.; Sportelli, M.C.; Cioffi, N.; Picca, R.A.; Agostinelli, E.; Varvaro, G.; Caliandro, R. Shape-control by microwave-assisted hydrothermal method for the synthesis of magnetite nanoparticles using organic additives. J. Nanoparticle Res.
**2015**, 17, 1–16. [Google Scholar] [CrossRef] - da Silva, F.G.; Depeyrot, J.; Campos, A.F.C.; Aquino, R.; Fiorani, D.; Peddis, D. Structural and Magnetic Properties of Spinel Ferrite Nanoparticles. J. Nanosci. Nanotechnol.
**2019**, 19, 4888–4902. [Google Scholar] [CrossRef] - Cannas, C.; Musinu, A.; Piccaluga, G.; Fiorani, D.; Peddis, D.; Rasmussen, H.K.; Mørup, S. Magnetic properties of cobalt ferrite-silica nanocomposites prepared by a sol-gel autocombustion technique. J. Chem. Phys.
**2006**, 125, 1–11. [Google Scholar] [CrossRef] [PubMed] - Jovanović, S.; Spreitzer, M.; Otoničar, M.; Jeon, J.-H.; Suvorov, D. pH control of magnetic properties in precipitation-hydrothermal-derived CoFe
_{2}O_{4}. J. Alloys Compd.**2014**, 589, 271–277. [Google Scholar] [CrossRef] - Albino, M.; Fantechi, E.; Innocenti, C.; López-Ortega, A.; Bonanni, V.; Campo, G.; Pineider, F.; Gurioli, M.; Arosio, P.; Orlando, T.; et al. Role of Zn
^{2+}Substitution on the Magnetic, Hyperthermic, and Relaxometric Properties of Cobalt Ferrite Nanoparticles. J. Phys. Chem. C**2019**, 123, 6148–6157. [Google Scholar] [CrossRef] [Green Version] - Martínez, B.; Obradors, X.; Balcells, L.; Rouanet, A.; Monty, C. Low Temperature Surface Spin-Glass Transition in -gFe
_{2}O_{3}Nanoparticles. Phys. Rev. Lett.**1998**, 80, 181–184. [Google Scholar] [CrossRef] - Coey, J.M.D. Non-collinear Spin Arrangement in Ultrafine Ferrimagnetic Crystallites. Phys. Rev. Lett.
**1971**, 27, 1140. [Google Scholar] [CrossRef] [Green Version] - Lin, D.; Nunes, A.C.; Majkrzak, C.F.; Berkowitz, A.E. Polarized neutron study of the magnetization density distribution within a CoFe
_{2}O_{4}colloidal particle II. J. Magn. Magn. Mater.**1995**, 145, 343–348. [Google Scholar] [CrossRef] - Peddis, D.; Mansilla, M.V.; Mørup, S.; Cannas, C.; Musinu, A.; Piccaluga, G.; D’Orazio, F.; Lucari, F.; Fiorani, D. Spin-canting and magnetic anisotropy in ultrasmall CoFe
_{2}O_{4}nanoparticles. J. Phys. Chem. B**2008**, 112, 8507–8513. [Google Scholar] [CrossRef] - Peddis, D.; Yaacoub, N.; Ferretti, M.; Martinelli, A.; Piccaluga, G.; Musinu, A.; Cannas, C.; Navarra, G.; Greneche, J.M.; Fiorani, D. Cationic distribution and spin canting in CoFe
_{2}O_{4}nanoparticles. J. Phys. Condens. Matter**2011**, 23, 426004. [Google Scholar] [CrossRef] - Peddis, D. Magnetic Properties of Spinel Ferrite Nanoparticles: Influence of the Magnetic Structure. In Magnetic Nanoparticle Assemblies; Trohidou, K.N., Ed.; Pan Stanford Publishing: Singapore, 2014; Volume 7, pp. 978–981. ISBN 9789814411967. [Google Scholar]
- Cannas, C.; Musinu, A.; Peddis, D.; Piccaluga, G. Synthesis and Characterization of CoFe
_{2}O_{4}Nanoparticles Dispersed in a Silica Matrix by a Sol−Gel Autocombustion Method. Chem. Mater.**2006**, 18, 3835–3842. [Google Scholar] [CrossRef] - Muscas, G.; Singh, G.; Glomm, W.R.; Mathieu, R.; Kumar, P.A.; Concas, G.; Agostinelli, E.; Peddis, D. Tuning the size and shape of oxide nanoparticles by controlling oxygen content in the reaction environment: Morphological analysis by aspect maps. Chem. Mater.
**2015**, 27, 1982–1990. [Google Scholar] [CrossRef] - Peddis, D.; Jönsson, P.E.; Laureti, S.; Varvaro, G. Magnetic Interactions: A Tool to Modify the Magnetic Properties of Materials Based on Nanoparticles; Elsevier: Amsterdam, The Netherlands, 2014; Volume 6. [Google Scholar]
- Knobel, M.; Nunes, W.C.; Socolovsky, L.M.; De Biasi, E.; Vargas, J.M.; Denardin, J.C. Superparamagnetism and other magnetic features in granular materials: A review on ideal and real systems. J. Nanosci. Nanotechnol.
**2008**, 8, 2836–2857. [Google Scholar] [CrossRef] - Dormann, J.L.; Fiorani, D.; Tronc, E. Magnetic Relaxation in Fine-Particle Systems. Adv. Chem. Phys.
**1997**, 98, 283–494. [Google Scholar] - Gittleman, J.I.; Abeles, B.; Bozowski, S. Superparamagnetism and relaxation effects in granular Ni-SiO
_{2}and Ni-Al_{2}O_{3}films. Phys. Rev. B**1974**, 9, 3891–3897. [Google Scholar] [CrossRef] - Chantrell, R.W.; El-Hilo, M.; O’Grady, K. Spin-Glass behaviour in fine particle system. IEEE Trans. Magn.
**1991**, 27, 3570–3578. [Google Scholar] [CrossRef] - Lavorato, G.C.; Peddis, D.; Lima, E.; Troiani, H.E.; Agostinelli, E.; Fiorani, D.; Zysler, R.D.; Winkler, E.L. Magnetic Interactions and Energy Barrier Enhancement in Core/Shell Bimagnetic Nanoparticles. J. Phys. Chem. C
**2015**, 119, 15755–15762. [Google Scholar] [CrossRef] [Green Version] - Liu, C.; Zou, B.; Rondinone, A.J.; Zhang, Z.J. Chemical Control of Superparamagnetic Properties of Magnesium and Cobalt Spinel Ferrite Nanoparticles through Atomic Level Magnetic Couplings. J. Am. Chem. Soc.
**2000**, 122, 6263–6267. [Google Scholar] [CrossRef] - Rondinone, A.J.; Liu, C.; Zhang, Z.J. Determination of Magnetic Anisotropy Distribution and Anisotropy Constant of Manganese Spinel Ferrite Nanoparticles. J. Phys. Chem. B
**2001**, 105, 7967–7971. [Google Scholar] [CrossRef] - Hansen, M.F.; Mørup, S. Estimation of blocking temperatures from ZFC/FC curves. J. Magn. Magn. Mater.
**1999**, 203, 214–216. [Google Scholar] [CrossRef] - Cannas, C.; Gatteschi, D.; Musinu, A.; Piccaluga, G.; Sangregorio, C. Structural and Magnetic Properties of Fe
_{2}O_{3}Nanoparticles Dispersed over a Silica Matrix. J. Phys. Chem. B**1998**, 102, 7721–7726. [Google Scholar] [CrossRef] - Pacakova, B.; Kubickova, S.; Reznickova, A.; Niznansky, D.; Vejpravova, J. Spinel Ferrite Nanoparticles: Correlation of Structure and Magnetism. In Magnetic Spinels—Synthesis, Properties and Applications; InTech: London, UK, 2017; ISBN 9789537619824. [Google Scholar]
- Dormann, J.L.; Bessais, L.; Fiorani, D. A dynamic study of small interacting particles: Superparamagnetic model and spin-glass laws. J. Phys. C Solid State Phys.
**1988**, 21, 2015. [Google Scholar] [CrossRef] - del Castillo, V.L.C.D.; Rinaldi, C. Effect of sample concentration on the determination of the anisotropy constant of magnetic nanoparticles. IEEE Trans. Magn.
**2010**, 46, 852–859. [Google Scholar] [CrossRef] [Green Version] - Sharifi, I.; Shokrollahi, H.; Amiri, S. Ferrite-based magnetic nanofluids used in hyperthermia applications. J. Magn. Magn. Mater.
**2012**, 324, 903–915. [Google Scholar] [CrossRef] - Sharifi, I. Magnetic and structural studies on CoFe
_{2}O_{4}nanoparticles synthesized by co-precipitation, normal micelles and reverse micelles methods. J. Magn. Magn. Mater.**2012**, 324, 1854–1861. [Google Scholar] [CrossRef] - Omelyanchik, A.; Knezevic, N.; Rodionova, V.; Salvador, M.; Peddis, D.; Varvaro, G.; Laureti, S.; Mrakovic, A.; Kusigerski, V.; Illes, E. Experimental Protocols for Measuring Properties of Nanoparticles Dispersed in Fluids. In Proceedings of the 2018 IEEE 8th International Conference Nanomaterials: Application & Properties (NAP), Zatoka, Ukraine, 9–14 September 2018; pp. 1–5. [Google Scholar]
- Morrish, A.H. The Physical Principles of Magnetism; Wiley: Hoboken, NJ, USA, 1965; Volume 1, ISBN 0-7803-6029-X. [Google Scholar]
- Muscas, G.; Concas, G.; Cannas, C.; Musinu, A.; Ardu, A.; Orru, F.; Fiorani, D.; Laureti, S.; Rinaldi, D.; Piccaluga, G.; et al. Magnetic Properties of Small Magnetite Nanocrystals. J. Phisical Chem. C
**2013**, 114, 23378–23384. [Google Scholar] [CrossRef] - Peddis, D.; Cannas, C.; Piccaluga, G.; Agostinelli, E.; Fiorani, D. Spin-glass-like freezing and enhanced magnetization in ultra-small CoFe
_{2}O_{4}nanoparticles. Nanotechnology**2010**, 21, 125705. [Google Scholar] [CrossRef] [PubMed] - Pérez, N.; Guardia, P.; Roca, A.G.; Morales, M.P.; Serna, C.J.; Iglesias, O.; Bartolomé, F.; García, L.M.; Batlle, X.; Labarta, A. Surface anisotropy broadening of the energy barrier distribution in magnetic nanoparticles. Nanotechnology
**2008**, 19, 475704. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chantrell, R.W.; O’Grady, K. The Magnetic Properties of Fine Particles. In Applied Magnetism; Springer: Dordrecht, The Netherlands, 1994; pp. 113–164. [Google Scholar]
- El-Hilo, M.; Bsoul, I. Interaction effects on the coercivity and fluctuation field in granular powder magnetic systems. Phys. B Condens. Matter
**2007**, 389, 311–316. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Zero-field-cooled (ZFC) (empty symbols) and field-cooled (FC) (solid symbols) magnetization curves. Inset: product of the magnetic susceptibility times the irreversibility temperature as a function of the annealing temperature. (

**b**) Energy barrier distribution obtained from the first derivative of the thermoremanent magnetization M

_{TRM}(T) versus temperature.

**Figure 2.**(

**a**) Logarithm of the measurement time ${\tau}_{m}$ versus $1/{T}_{B}$ and its linear fit (dashed line); (

**b**) effective anisotropy constant K

_{eff}of N15T700, N15T800 and N15T900 obtained from fitting $ln\left({\tau}_{m}\right)$ versus $1/{T}_{B}$ by Neél–Arrhenius model (empty circles). The value from N5T900 was taken from reference [14] and that of bulk cobalt ferrite from reference [7].

**Figure 3.**Field-dependence of magnetization and direct current demagnetization (DCD) curves measured at 5 K for (

**a**) sample N15T700 and (

**b**) reference sample N5T900 with the same particle size [14].

**Figure 4.**(

**a**) Saturation magnetization, ${M}_{\mathrm{S}}$, and (

**b**) non-saturated susceptibility, ${\chi}_{\mathrm{SAT}}$, obtained by fitting Equation (6). The dashed line represents a guide for the eye. The square in both graphs corresponds to the reference sample N5T900.

Sample | d_{TEM} | T_{max} | T_{irr} | <T_{B}>_{CH.}^{2} | PD_{CH.} | <T_{B}>_{H.M.} ^{2} | PD_{H.M.} |
---|---|---|---|---|---|---|---|

(nm) | (K) | (K) | (K) | (%) | (K) | (%) | |

N15T700 | 2.5(2) ^{1} | 29(1) | 57(5) | 18(1) | 3.26 | 16(1) | 4.56 |

N15T800 | 5.3(5) | 43(1) | 70(5) | 22(2) | 2.86 | 25(2) | 2.44 |

N15T900 | 6.6(5) | 53(1) | 82(3) | 29(1) | 2.41 | 31(2) | 2.45 |

^{1}Uncertainties in the last digits are given in parenthesis;

^{2}Average blocking temperature extracted from thermoremanent magnetization (TRM) (<T

_{B}>

_{CH.}) and (<T

_{B}>

_{H.M.}) from Hansen and Mørup method are reported with their corresponding percentual polydispersity index.

Sample | Néel−Arrhenius ^{1} | Vogel−Fulcher ^{2} | ||
---|---|---|---|---|

${\mathbf{K}}_{\mathbf{e}\mathbf{f}\mathbf{f}}\text{}$ | ${\mathsf{\tau}}_{0}\text{}$ | ${\mathbf{K}}_{\mathbf{e}\mathbf{f}\mathbf{f}}\text{}$ | ${\mathbf{T}}_{0}$ | |

(J m^{−3}) | (s) | (J m^{−3}) | (K) | |

N15T700 | 7.9(4) × 10^{5} | 1.9 × 10^{−9} | 11(1) × 10^{5} | −1(3) |

N15T800 | 2.3(2) × 10^{5} | 8.2 × 10^{−14} | 1.3(1) × 10^{5} | 14(2) |

N15T900 | 1.9(2) × 10^{5} | 1.5 × 10^{−14} | 0.92(1) × 10^{5} | 32(3) |

^{1}Effective magnetic anisotropy constant (K

_{eff}) and characteristic relaxation time (τ

_{0}) obtained from the fitting to Néel−Arrhenius law (Equation (4));

^{2}K

_{eff}and the interaction temperature term, T

_{0}, assuming τ

_{0}= 10

^{−10}s, from Vogel−Fulcher law (Equation (5)).

© 2020 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**

Omelyanchik, A.; Salvador, M.; D’Orazio, F.; Mameli, V.; Cannas, C.; Fiorani, D.; Musinu, A.; Rivas, M.; Rodionova, V.; Varvaro, G.;
et al. Magnetocrystalline and Surface Anisotropy in CoFe_{2}O_{4} Nanoparticles. *Nanomaterials* **2020**, *10*, 1288.
https://doi.org/10.3390/nano10071288

**AMA Style**

Omelyanchik A, Salvador M, D’Orazio F, Mameli V, Cannas C, Fiorani D, Musinu A, Rivas M, Rodionova V, Varvaro G,
et al. Magnetocrystalline and Surface Anisotropy in CoFe_{2}O_{4} Nanoparticles. *Nanomaterials*. 2020; 10(7):1288.
https://doi.org/10.3390/nano10071288

**Chicago/Turabian Style**

Omelyanchik, Alexander, María Salvador, Franco D’Orazio, Valentina Mameli, Carla Cannas, Dino Fiorani, Anna Musinu, Montserrat Rivas, Valeria Rodionova, Gaspare Varvaro,
and et al. 2020. "Magnetocrystalline and Surface Anisotropy in CoFe_{2}O_{4} Nanoparticles" *Nanomaterials* 10, no. 7: 1288.
https://doi.org/10.3390/nano10071288