AC susceptibility as a tool to probe the dipolar interaction in magnetic nanoparticles
Introduction
Magnetic nanoparticles (MNPs) have been an active topic of research for over half a century. Initially, much of this interest was related to the magnetic recording industry, but in the past few decades there has been a shift toward biomedical applications [1]. Examples include the use of MNPs for drug delivery [2], stem cell labeling [3], [4], contrast agents for nuclear magnetic resonance [5] and magnetic hyperthermia [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. The latter, in particular, is a cancer treatment technique that has already entered clinical trials [15] and is now considered the most promising application of MNPs. Great progress has also been made in our theoretical understanding of MNPs, particularly through Brown's Fokker-Planck equation [16], [17], which allows one to make valuable predictions about several dynamic properties [18], [19], [20], [21], [22], [23], [24].
Most of our theoretical understanding about MNPs concerns non-interacting samples. However, MNPs are also strongly influenced by the dipolar interaction [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50]. Indeed, recent papers [10], [11], [12], [42], [43], [50] have shown that the dipolar interaction has a strong influence in magnetic hyperthermia treatments. This means that the heating properties of particles diluted in a fluid will be very different from those of particles packed inside cells or nano-carriers, such as magnetoliposomes [12], [51], [52]. Hence, when tailoring a sample for a specific treatment, one must also take into account the spatial arrangement of the nanoparticles. Recently, several theoretical models [10], [25], [42], [43] and simulations methods [50] have been developed to deal with the dipolar interaction and aid in the design of samples for specific treatments.
However, in samples where the particles have some degree of mobility, as is true for many samples used for hyperthermia, the dipolar interaction may also induce the formation of aggregates (sometimes observed in the form of elongated chains [10], [53]). The strong interaction between particles within a cluster cause them to rotate in order to align their easy axes, therefore modifying their effective magnetic anisotropy [10], [54]. Despite its importance, this effect is seldom taken into account when developing theoretical models.
The modifications in the effective magnetic anisotropy of a given MNP will depend on the size and shape of the aggregate that it resides in, and also on the position and relative orientation of that MNP within the cluster. Consequently, this effect may be modeled by considering a distribution of anisotropy constants, in addition to the distribution of volumes. The distribution of anisotropies already has an intrinsic, concentration-independent, contribution due to fluctuations in the crystallinity, shape and surface roughness. The contribution from particle–particle interactions should therefore enter as an additional effect, which may be small for diluted samples, but may very well be dominant when the particle aggregation is high.
Experimentally accessing and quantifying the degree of aggregation, however, is by no means trivial. This problem has generated much interest lately, with recent proposals involving the use of Lorentz microscopy [55] and small angle X ray scattering [56]. The purpose of this paper is to show that AC susceptibility also yields important information concerning the state of aggregation in a sample.
The most common analysis of AC susceptibility curves is the Arrhenius plot, which looks at the temperature where the imaginary part is a maximum. A plot of as a function of the frequency f of the AC field usually yields a straight line [cf. Eq. (5) to be discussed below], from which one may extract information about the average anisotropy energy barrier and the typical precession time τ0 (whose values may be used to estimate the magnitude of the dipolar interaction [29]). Since one uses only the maximum of the imaginary curve, this analysis clearly underuses the data since from each vs. T dataset, just a single point is taken. Moreover, the effects of the particle size distribution only enter indirectly in the average anisotropy barrier. A more robust approach was introduced by Jonsson et. al. [57], who developed a model for that be fitted to the entire dataset, taking into account the size distribution.
In this paper we show how to expand on the model of Ref. [57] to include both aspects of the dipolar interaction. This is done using a mean-field approximation [42] to model the long-range effect, together with a anisotropy distribution to model the particle aggregation. This approach allows us to extract information which reflects the different levels of particle aggregation within the sample. We apply this model to two samples, a commercial magnetite-based ferrofluid dispersed in oil, and a sample containing PLGA [the poly(D,L-lactide-co-glycolic) acid copolymer] nanospheres with cubic nanoparticles trapped on their surface.
We also introduce a new very simple tool to access the qualitative importance of the dipolar interaction in a given sample. It is based on analyzing the maximum height of the vs. T curves as a function of the frequency f. When the dipolar interaction is negligible in a sample, should not increase with f. Conversely, we show that the presence of a dipolar interaction causes to increase with f. Hence, this serves as a signature of the dipolar interaction. Through a simple visual analysis of the imaginary AC susceptibility curves it is possible to see if the dipolar interaction is relevant in that given sample or not.
Section snippets
AC susceptibility for ideal monodisperse samples
We begin by reviewing the theory of AC susceptibility. The relaxation time of a single-domain magnetic nanoparticle with volume V and uniaxial anisotropy constant K is given approximately by the Néel formula [58]:where andThe quantity , which will be used throughout the text, represents the height of the energy barrier in temperature units. For more information, see supplemental information.
In AC susceptibility experiments one measures the response of a sample to
Experiments
We now apply Eq. (20) to two distinct samples. In Section 3.1 we study a commercial ferrofluid containing spherical magnetite nanoparticles of roughly 6.4 nm in diameter and in Section 3.2 we study a sample of cubic magnetite nanoparticles with about 13 nm loaded on the surface of polymeric nanospheres. Both samples are studied under different dilutions.
Discussion and conclusions
The purpose of this paper was to show how AC susceptibility may be used to extract information concerning the presence of the dipolar interaction in a sample. As we have argued, in samples where the particles are left in fluid suspension or packed inside nano-carriers, the dipolar interaction manifests itself in two separate ways. The first is the direct dipolar effect, which can be modeled, for instance, using the Vogel-Fulcher, mean-field or DBF models, as discussed in Section 2.4. In
Acknowledgements
For their financial support, the authors would like to acknowledge the Brazilian funding agencies FAPESP, CNPq and FAPEG, the funding agencies Spanish MINECO and FEDER, under project MAT2014-53961-R. I. Andreu thanks the Spanish CSIC for her JAE Predoc contract. Finally, we thank Prof. S. M. Carneiro for her help with the TEM images of the spherical magnetite nanoparticles.
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