## 3 Results and Discussion
Further increasing the concentration leads to non-negligible magnetic dipolar interactions among the nanoparticles in the ensemble. This collective relaxation requires more energy for the superspin flips. In the DBS model, this is equivalent to an additional energy barrier to superspin reversal that increases with the concentration and becomes dominant above c = 4 \(\times\) 10\({}^{-4}\)%. Our findings are significant for several reasons. First, they demonstrate that despite the high dilution level (4 \(\times\) 10\({}^{-2}\)% \(\geq\) c \(\geq\) 8.3 \(\times\) 10\({}^{-4}\)%), samples S1-S4 still exhibit a collective relaxation of the superspins, which is typically indicative of interparticle interactions. This is unequivocally confirmed by the existence of the additional energy barrier to magnetization reversal \(\rm E_{ad}\) that increases with the increase in c. The ability to identify the concentration range where this behavior occurs is a new result that is particularly important for applications of magnetic nanoparticles in high-density magnetic recording, where each superspin acts as one recording bit that needs to flip individually. Second, our observation that the nanoparticles in sample S5 relax individually allowed us to determine \(\rm E_{B}\) and, more  
Figure 5: Temperature dependence of the relaxation time, \(\tau\), obtained from ac-susceptibility data collected on samples of different volume concentrations S1, c = 4 \(\times\) 10\({}^{-2}\)% (upright triangles), S2, c = 1.3 \(\times\) 10\({}^{-2}\)% (diamonds), S3, c = 1.7 \(\times\) 10\({}^{-3}\)% (inverted triangles), S4, c = 8.3 \(\times\) 10\({}^{-4}\)% (circles), and S5, c = 4 \(\times\) 10\({}^{-4}\)% (squares). The solid lines are best fits to the DBS model with \(\rm E_{B}/\rm k_{B}=366\) K.  
importantly, separate the E\({}_{\text{B}}\) and E\({}_{\text{ad}}\) components in the denser samples. This is important because it adds to our understanding of the contributions of different factors that influence the superspin relaxation, such as bulk and surface magnetic anisotropy, nanoparticle size and size distribution, and interparticle interactions. Finally, the quantitative relations that we established between the blocking temperature, additional energy barrier to magnetization reversal, and nanoparticle concentration are significant for the current efforts to design functional nanomaterials using a rational approach instead of the common trial and error method. Further work will be aimed at investigating the interplay between the Neel and the Brown superspin relaxation mechanisms at temperatures above the melting point of the carrier fluid. Such studies are important because they enhance the knowledge needed to advance magnetic nanoparticle hyperthermia from its current adjunct-to-other-treatments status to a powerful stand-alone cancer therapy. Our experimental strategy will be to increase the average size of the nanoparticle ensembles so that the superparamagnetic unblocking occurs at temperatures above the freezing point of the carrier fluid. In addition, we will use ensembles of different concentrations to vary the interparticle dipolar interactions, and carrier fluids of different viscosities to vary the energy barrier to physical rotation that governs the Brown relaxation. These experiments are currently underway.