Temperature dependence of the magnetic interactions taking place in monodisperse magnetite nanoparticles having different morphologies

The coercivity, at temperatures slightly above the human body euthermia and ac field frequencies ranging from the tens of kHz up to tens of MHz, constitutes, jointly with the saturation magnetization value, the two most important magnetic parameters determining the use of biocompatible nanoparticles (NPs) in the hyperthermia treatment of carcinomata.^1,2 Whereas the saturation magnetization is mostly dependent on the NPs chemical composition,^3–5 their coercive force is an extrinsic quantity⁶ varying with effective anisotropy that can include contributions depending on size and morphology, nature and concentration of dopants, built-in stresses, thermally activated demagnetization and nanoparticles agglomeration and interactions.⁷ This complex origin of the coercivity of the NPs used in hyperthermia defines the playground for the optimization of these materials from the magnetic standpoint.

In Ref. 8, the authors have evidenced how the thermally activated demagnetization processes taking place in magnetite NPs play a major role in deteriorating the dc coercivity values measured at temperatures above 150 K. We aim in the present work at the quantitative evaluation of the inter-particle interactions and, particularly, at the identification of the role played by the nanoparticles morphology.

We have prepared magnetite NPs via oxidative precipitation by controlling their size through the nature of the base and nitrate salt and the ethanol content in the media, and their morphology by changing the final base concentration.⁹ Once the nuclei are formed, surfactants/ligands/ions have a key role in regulating the growth of the particles, their shape evolution, and their stabilization in solution. Ligands dynamically adsorb on certain faces by interaction with the surface cations, which decreases the particle surface energy. The growth rate along those directions is reduced or inhibited and other growth directions are favoured. The key factor to modulate the NPs shape is the ratio of growth rates between [100], [110] and [111] directions, so, for example, faster growth of [100] and [110] directions over [111], in the presence of Na and sulfate ions, leads to octahedral particles, whereas a faster growth without any hampering ion leads to a more isometric shape.¹⁰ Particles sizes, shapes and crystal structures were determined by transmission electron microscopy (TEM), high-resolution electron microscopy (HRTEM), and X-ray diffraction (XRD), respectively. The magnetic properties of the NPs were measured by means of a vibrating sample magnetometer, working in the temperature range from 5 K up to 290 K by applying a maximum field of 1.6 x 10⁶ A/m. The measured samples were pressed powder cylinders having a diameter of 1.4 mm and a length of 9 mm.

The TEM and HRTEM images of the prepared magnetite nanocrystals (Figures 1a) and 1b)), evidenced the exclusive occurrence in the corresponding samples of the S and O morphologies and excellent crystallinity. The octahedra were identified both from the measurement in selected particles of the apical angle and from the preparation parameters. The measured particles maximum transverse dimensions distributions were very similar for both morphologies with average dimensions of 21 nm (volume = 1800 nm³, O NPs) and 22 nm (volume = 5600 nm³, S NPs). The crystal structure of the samples was XRD identified as an inverse spinel structure without any secondary phase.⁸ 

The temperature dependence of the saturation magnetization, M_S (measured under a maximum applied field of 1.6 x 10⁶ A/m) was non-monotonic, with smooth maxima, in both samples, at 70 K. An almost temperature independent difference between the (larger) S NPs saturation magnetization and that of the O NPs, was observed and related to the larger magnitude of the area per mass unit of the O particles (having an average volume smaller by a factor of 3 than that of the S NPs). Table I summarizes the saturation magnetization, coercivity, H_c, and activation volumes, v_ac, measured in the O and S NPs at 5 K, 125 K, 200 K and 295 K.⁸ The activation volumes were estimated from the slow relaxation data obtained at fields near the coercivity following the procedure detailed in Ref. 8.

The evaluation of the interactions was carried out from the measurement of the isothermal and demagnetization remanences. The isothermal remanent magnetization (M_r) measurements started from demagnetized samples by cooling them under zero magnetic field down to the measurement temperature. At that temperature, a field, H, was applied and switched off to measure M_r(H). The field was increased until the remanence saturation M_r(∞) was achieved. The dc demagnetization remanence (M_d) experiments started by initially saturating the samples and then cooling them down to the measurement temperature. At that temperature, a demagnetizing field, H, was applied, and switched off to measure the remanent magnetization M_d(H). The demagnetizing field was increased until the negative saturation was reached. In Figure 1c) we have plotted the internal field evolutions of the isothermal and demagnetization remanences measured in the O sample at 200 K.

The relationship between the M_r(H) and M_d(H) corresponding to randomly oriented, non-interacting, uniaxial, single domain particles is given by M_d(H)=M_r(∞)-2M_r(H).^11,12 Any departure from that behavior is associated to the non-compliance with at least one of the relationship hypotheses.^13,14 Our particles were measured as loosely pressed powder and therefore did not exhibit any preferred crystalline orientation and behaved, due to their reduced sizes, as single domains. Regarding the effective anisotropy, the O NPs had a main uniaxial shape easy axis associated to their longer dimension (from apex to apex of the opposing-by-the-base square pyramids), whereas in the S NPs any deviation from the perfect spherical shape resulted in an affective uniaxial behavior. Thus, in our samples, the departures from the remanences relationship should be ascribed to the occurrence of interactions. The quantity measuring those departures is defined as: δM(H)=m_d(H)–[1-2m_r(H)], m_d and m_r being M_r(H) and M_d(H) normalized to the isothermal remanence saturation value M_r(∞), respectively.^11,12 The negative δM values are associated to the presence of demagnetizing (dipolar) interactions, whereas the positive δM values ones are linked to the occurrence of magnetizing (exchange) interactions.^11,12 Figure 2 shows our results for the internal field dependence of δM obtained in both samples at 5 K, 125 K and 200 K, after correcting for demagnetizing effects (maximum applied field 8 x 10⁵ A/m). From the Figure it is apparent that, at all the considered temperatures and in both samples, demagnetizing interactions exclusively occurred.

Figure 3 compares the results of the measurement of the δM(H) curves in the O and S NPs at 200 K, obtained by considering the applied field and the internal one.

As it is clear from the Figure, there were not relevant differences between both measurements. Thus, the interactions present in our samples cannot be described in terms of a mean field and should be understood as local and linked to the poles present at the particle surfaces.

In Figure 4 we summarize the values obtained for the parameters characterizing the δM(H) curves. Figure 4a) shows the temperature dependencies of the fields at which the maxima, H_{int δM max}, in the δM curves are observed and those of the maxima magnitudes, δM_max. As for H_{int δM max}, the obtained values are similar in both samples and several times larger than the corresponding coercivities, which indicates that the reversal process was not collective but field distributed. The interactions maxima values, δM_max, were clearly larger in the S NPs than in the O ones, which is attributed, firstly, to the S NPs magnetization, being larger than that of the O NPs, and secondly, to the higher order multipolar moment of the O NPs (linked to their edges and dihedra) that results in spatially localized demagnetizing fields. Also, δM_max increased in absolute magnitude with the increase of the temperature (oppositely to M_S⁸), which confirms the presence in the samples of non-uniform poles distributions, and can be correlated to the large increase with the increase of temperature of v_ac (see Table I). That increase results, at high temperatures, on the thermally activated reversal of clusters including tens of NPs, which increases the local magnetization inhomogeneities. Finally, the results plotted in Figure 4b) for the temperature dependence of the full field width at half maxima (FWHM) O and S of the δM(H) curves peaks, show that the occurrence of interactions is distributed in field ranges that span several times the coercivity value, increase with the decrease of the temperature and are larger in the O NPs than in the S NPs, accordingly with the high order multipolar moment of the O NPs that renders difficult to saturate the NPs at the high poles densities sites originating the multipolar moment and promotes the easier local reversal from those sites, thus contributing to the coercivity deterioration.¹⁵ The large difference at 200 K between the O and S δM(H) FWHMs could be related to the differences between the thermally activated demagnetizations taking place packing of both types of NPs: whereas the reversal of particles chains is favored in the O NPs that of equiaxed, flux closed clusters can be expected in that of the S NPs, which supports a broader distribution of interactions in the O NPs.

In summary, the maxima of demagnetizing interactions measured in the S NPs are larger, at all temperatures, than in those obtained in the O NPs, which results from both the higher S NPs magnetization and the higher order O NPs multipolar moments (detectable from their associated larger field range in which their interactions are observable in comparison with the range corresponding to the spherical nanoparticles, exhibiting close-to-dipolar moments).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors gratefully acknowledge the financial support from the Spanish Ministry of Economy and Competitiveness and Spanish Research Agency under grants no. MAT2016-80394-R and MAT2017-88148-R (FEDER EU).