Evaluation of interparticle interactions between magnetic nanoparticles using first order reversal curves and Weiss temperature

Two conventional methods for evaluating interparticle magnetic interactions are applied to magnetic nanoparticle assemblies with various interparticle distances that are controlled by a silica coating. According to Weiss temperatures derived from superparamagnetic response analysis, the mean values of the interparticle magnetic interactions are relatively small and seem to be independent of the interparticle distance. The interaction fields in the first-order reversal curve diagram for narrow interparticle distances are widely distributed. However, the interaction fields disappear when the interparticle distance is sufficiently large. Analysis of these two contrasting results indicates that ferromagnetic-like and antiferromagnetic-like magnetic couplings coexist and cancel each other, as in atomic spin glasses.


Introduction
Recently, magnetic nanoparticles have attracted significant attention for their excellent potential for applications in magnetic recording sensors [1,2], magnetic refrigeration [3], and biomedical applications [4]. However, many issues remain to be solved regarding the design of magnetic nanoparticle assemblies for each application. In particular, there is little consensus regarding the effects of dipole interactions between nanoparticles, even though they are expected to induce useful cooperative magnetic responses [5,6]. 3 where d is the diameter of the nanoparticle, M 0 is the spontaneous magnetisation, and r is the distance between nanoparticles. In other words, the amplitude of interactions can be controlled by varying r. Therefore, many efforts have been made to prepare nanoparticles with various r values by changing the particle number density or controlling the nonmagnetic layer thickness t between nanoparticles [7][8][9][10][11][12][13]. However, little attention has been devoted to the experimental evaluation of actual interparticle interaction strengths in such samples. In studies on randomly assembled nanoparticles, mixtures of ferromagnetic-like and antiferromagnetic-like interparticle interactions are assumed. Based on this assumption, superspin glasses have been intensively discussed in terms of their intriguing collective phenomena for decades, similar to atomic spin glasses. However, the mixture of interparticle interactions has never been experimentally confirmed [5][6][7][8][9][10][11][12][13]. In this study, two conventional evaluation methods, namely first-order reversal curve (FORC) analysis [14][15][16], which is often employed in applied magnetics, and Curie-Weiss law fitting, which is used in fundamental magnetism, were applied to nonmagneticlayer thickness-controlled magnetic nanoparticle assemblies. The actual probability distributions of ferromagnetic-like and antiferromagnetic-like interparticle interaction strengths were investigated.

Experiments
Magnetite nanoparticles were synthesised as follows. First, 11.0 mmol of goethite (FeO(OH)) was dissolved in 31.7 ml of 1-octadecene. Next, 14.1 ml of oleic acid was added and the solution was heated at 393 K for 1 h and at Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 590 K for 2 h. The resulting black powder was washed several times to remove excess surfactants and solvents. X-ray diffractometry revealed that the powder had a spinel-type structure. Magnetometry indicated that M 0 at 2.0 K is 53 Am 2 /kg. This magnitude of M 0 is roughly consistent with the values previously reported for iron oxide magnetic nanoparticles [17,18], although it is smaller than the values obtained for bulk magnetite or maghemite. This study aimed to evaluate interparticle interactions, so the origin of the relatively small M 0 value is left unexplored for following studies.
Next, coating with a silica shell [19] was conducted as follows. Igepal CO-520 was dispersed in cyclohexane and sonicated. Then, the powder synthesised above was added to the solution under continuous stirring. Subsequently, 1.6, 1.6, and 3.2 ml of ammonium hydroxide and 1.12, 2.24, and 4.43 ml of tetraethyl orthosilicate were added to generate three different shell thicknesses. Transmission electron microscopy (TEM) revealed that the magnetite cores with an average diameter d of 7.8 nm were well coated with silica shells with thicknesses t of 6.5, 13.7, and 20.8 nm for samples A7, A14, and A21, respectively, as shown in figure 1. The nanoparticles before silica coating in sample A0 have a surfactant layer with t=1−2 nm. Therefore, we succeeded in controlling the interparticle distance.
The magnetisation M of all samples was measured under the following thermal and field conidtions using a superconducting quantum interference device magnetometer (MPMS, Quantum Design). Zero-field-cooled magnetisation M ZFC was measured under heating in a magnetic field H of 0.8 kA m −1 after zero-field cooling to 2.0 K, whereas field-cooled magnetisation M FC was measured under cooling in a magnetic field H of 0.8 kA m −1 . The measurements of FORC were performed as follows. From a positive saturation state at an H values of 0.8 MA/m, H was decreased to a reversal field H r . A partial hysteresis curve M(H, H r ) was then measured as H increased from H r back to saturation. This measurement procedure was repeated for several different H r values. Figure 2 presents the temperature dependencies of M ZFC and M FC . As the temperature T increases, M ZFC first increases, then peaks at T m , and finally decreases. M FC is almost the same as M ZFC in the high-temperature range of T ? T m , whereas M FC branches off from M ZFC at T<T m . These results have been conventionally interpreted as follows. The system is equilibrated by the thermal fluctuation of μ at T ? T m . In contrast, magnetic hysteresis occurs based on the suppression of fluctuations at T<T m . The boundary temperature indicated by T m , which is the so-called blocking temperature or freezing temperature, is reduced with an increasing coating thickness t, as shown in the inset in figure 2. This type of variation in T m with a changing t has been attributed to the effects of interparticle magnetic interactions because the magnetic properties of individual nanoparticles should be invariant with t [5][6][7][8][9][10]. However, previous studies have not contained experimental evaluations of interparticle interaction strengths. Therefore, detailed discussion remains difficult. For this reason, two evaluation methods of FORC analysis and Curie-Weiss law fitting were applied in this study.

Results and discussion
In the paramagnetic phases of atomic spin systems, the magnetisation at small H values is given by the Curie-Weiss law as follows: where n is the number density of magnetic moments μ, k B is the Boltzmann constant, T w is the Weiss temperature, and M s is the saturation magnetisation. Figure 3   proportional to T in the anhysteretic regime at T ? T m . Based on the slopes of the fitting lines, the magnitudes of μ can be roughly estimated to be 7×10 3 μ B for A0 and A21, and 8×10 3 μ B for A7 and A14. These values correspond to the total magnetic moments contained in each iron oxide nanoparticle with a size of 8 nm at an M 0 of 53 Am 2 kg −1 . This size is consistent with the diameters observed via TEM. Therefore, we can confirm that collinearly aligned spins in each nanoparticle thermally fluctuate in the form of superspins μ. In contrast, the values of T w estimated from the T-axis intercepts are less than 10 K and are almost independent of t, as shown in the inset in figure 2. In the molecular field approximation, T w is expressed as / a m M k 3 . 0 s B Therefore, aM s can be estimated to be less than 5 kA m −1 , regardless of t, where α is the molecular field coefficient. The magnitude and t-dependence of T w contrast with those of T m . Therefore, one may doubt the explanation above that the elevation of T m is caused by interparticle interactions.
We turn to FORC analysis to resolve this conflict. To avoid the influence of thermal fluctuations, the FORC curves at 2.0 K, which is much lower than T m , were examined, where each nanoparticle with a switching field of H sw was assumed to be under a constant interaction field of H u . In this case, the reversal of magnetisation occurs at (−H sw +H u ) or (+H sw +H u ). Therefore, the distributions of these fields, FORC diagrams, can be obtained as ( ) where H sw =(H−H r )/2 and H u =(H+H r )/2. Figure 4 presents the FORC diagrams of the samples. One can see that H u for A0 is widely distributed across the range of tens of kA/m. This result indicates the existence of significant interparticle interactions, although their details cannot be analysed based solely on FORC diagrams because positive and negative values of H u are not associated with ferromagnetic and antiferromagnetic natures, respectively. As t increases, the width of the distribution of H u narrows and eventually becomes negligible compared to the experimental resolution of 8 kA m −1 at t=20.8 nm for A21. This result is reasonable because dipolar interactions weaken at larger distances, as stated above.  At this stage, it should be noted that the width of the H u distribution is much larger than the molecular field aM s estimated from T w for A0. It should also be noted that the molecular field represents the mean value of all magnetic couplings. Therefore, we can say that ferromagnetic-like and antiferromagnetic-like couplings coexist in almost equal quantities and cancel each other because the average magnetic interaction estimated from the Weiss temperature is much smaller than the distribution width of local interaction fields in the FORC diagrams. This result is consistent with a simple theoretical consideration. The dipolar magnetic field operates in a ferromagnetic-like manner on the neighbouring μ in the axial direction of the source μ, whereas it operates in an antiferromagnetic-like manner on μ in the equatorial direction. The faraway dipolar fields from a distant μ are cancelled by demagnetising fields when the shape of the boundary between a neighbouring and distant μ is set to be the same as the outward form of the powder sample. This interpretation based on small molecular fields is supported by the additional information acquired from the observed FORC diagrams, which lack wishboneshaped ridges caused by the molecular fields from a distant μ [15], as shown in figure 4.
The actual phenomena may be relatively complicated because the width of the distribution of H u is evidently larger than the dipolar field from the single nearest neighboring μ. This field was calculated to be 13. 3 In other words, H u is cooperatively generated by multiple nanoparticles. This interpretation agrees with the spin-glass model, where the competition of ferromagnetic and antiferromagnetic interactions causes cooperative freezing of thermal fluctuations at the temperature corresponding not to T w but to the distribution width of coupling strengths between different μ [20]. We cannot discuss the nature of such superspin-glass-like behaviour further from the perspective of FORC diagrams because cooperative phenomena are entirely beyond the scope of application of the Preisach model [14,15]. However, the coexistence of ferromagnetic and antiferromagnetic interactions is experimentally confirmed by currently comparing the results based on two conventional evaluation methods. This information is vital for advancing the study of superspin glasses.

Summary and prospects
Two conventional methods for evaluating interparticle magnetic interactions were applied to interparticle-distancecontrolled magnetic nanoparticle assemblies. According to the Weiss temperatures of the superparamagnetic responses, the mean values of the interparticle magnetic interactions were relatively low compared to the blocking temperature and seemed to be independent of the interparticle distance. In contrast, the interaction fields in the FORC diagrams were widely distributed for samples with narrow interparticle distances. However, the width of the interaction field distribution narrows with an increase in the non-magnetic layer thickness. Therefore, when comparing the small value of the mean field to larger local field effects, we can say that ferromagnetic-like and antiferromagnetic-like magnetic couplings are mixed in approximately equal quantities in nanoparticle assemblies, similar to atomic spin glasses. It should be noted that this evaluation could not be performed using only one of the two methods. For example, interparticle interactions may be misinterpreted as weak in all samples if only the Weiss temperature is considered. As demonstrated in this study, the complementary use of two methods is essential for characterising the magnetic interactions in nanoparticle assemblies.