## 3 Results and Discussion  
Figure 1a shows the temperature dependence of the in-phase susceptibility measured on the S3 sample (1.7 \(\times\) 10\({}^{-3}\)% _v/v_) at five different frequencies: 100 Hz, 300 Hz, 1000 Hz, 3000 Hz, and 10,000 Hz. Figure 1: In-phase (**a**) and out-of-phase (**b**) components of the ac-susceptibility collected on the S3 (c = 1.7 \(\times\) 10\({}^{-3}\) % _v/v_) sample within the 20 K–100 K temperature range at five different frequencies of the driving magnetic field: 100 Hz (squares), 300 Hz (inverted triangles), 1000 Hz (diamonds), 3000 Hz (upright triangles), and 10,000 Hz (circles). The main feature of these \(\chi^{\prime}\) vs. T \(\mid\)\({}_{\rm f}\) curves is that they exhibit robust peaks, whose temperature increases with the increase in the measurement frequency. This allows the calculation of the relative peak temperature variation per frequency decade, \(\phi=\frac{\Delta\Gamma}{\rm T\cdot\Delta\log(f)}\). We found \(\phi\) = 0.15 for the S3 sample, a value that shows the superparamagnetic nature of the transition undergone by the nanoparticle ensemble upon heating above a blocking temperature T\({}_{\rm B}\) that depends on the measurement frequency. Similar \(\chi^{\prime}\) vs. T \(\mid\)\({}_{\rm f}\) data and analyses on samples S1, S2, S4, and S5 yielded \(\phi\) values within the 0.1-0.2 range. This confirms that for all the concentrations used here, the magnetic relaxation occurs in the presence of weak interparticle interactions, and the ensembles exhibit superspin blocking-unblocking transitions. The out-of-phase susceptibility curves \(\chi^{\prime\prime}\) vs. T \(\mid\)\({}_{\rm f}\) measured on sample S3 are shown in Figure 1b. The magnitudes of these curves are reversed compared to their in-phase counterparts (i.e., the \(\chi^{\prime\prime}\) vs. T \(\mid\)\({}_{\rm 10,000\;Hz}\) peak has the largest magnitude), but the peak temperature still increases with the increase in the frequency f or, equivalently, the observation time \(\tau_{\rm obs}\) = 1/2\(\pi\)f. For this reason, \(\chi^{\prime\prime}\) vs. T \(\mid\)\({}_{\rm f}\) curves will be used to determine the observed \(\tau\)(T) dependence in each of the five nanoparticle ensembles used in this study. To determine which model to use to analyze the observed temperature dependence of the superspin relaxation, we first studied the variation of the blocking temperature with the nanoparticle ensemble concentration, T\({}_{\rm B}\) vs c. Figure 2 shows the \(\chi^{\prime}\) vs. T \(\mid\)\({}_{\rm 100\;Hz}\) curves measured on the five samples of different concentrations c.  
As indicated above, these curves peak at the blocking temperature corresponding to the transition from the blocked to the superparamagntic state of the nanoparticle ensemble. Therefore, the data demonstrate that T\({}_{\rm B}\) increases monotonically with the increase in c, from T\({}_{\rm B}\) = 39 K at c = 4 \(\times\) 10\({}^{-4}\)% (S5) to T\({}_{\rm B}\) = 53 K at c = 4 \(\times\) 10\({}^{-2}\)% (S1). This is the type of behavior predicted by the DBF model, which might seem somewhat unexpected given the high level of dilution of the samples used in our work, where the concentrations are below 4 \(\times\) 10\({}^{-2}\)% \(v\)/v. It is worth noting, however, that denser Fe\({}_{3}\)O\({}_{4}\)/hexane ferrofluids of \(w\)/\(v\) concentration 20 mg/mL (i.e., 4 \(\times\) 10\({}^{-1}\)% \(v\)/v) have been shown by previous studies [27] to correspond to strong interactions that lead to a collective superspin-glass-like freezing upon cooling below a critical temperature. This is markedly different from the blocking-unblocking superparamagnetic transition observed by us and cannot be analyzed in the framework of the DFB model.