## 3 Results and Discussion
On the other hand, we cannot exclude a certain level of particle agglomeration upon the freezing of the carrier fluid in our studies, which is the  
Figure 2: Temperature dependence of the in-phase susceptibility, \(\chi^{\prime}\) vs. T, measured at f = 100 Hz of the five samples of different volume concentrations S1, c = 4 \(\times\) 10\({}^{-2}\)% (open circles), S2, c = 1.3 \(\times\) 10\({}^{-2}\)% (filed squares), S3, c = 1.7 \(\times\) 10\({}^{-3}\)% (open squares), S4, c = 8.3 \(\times\) 10\({}^{-4}\)% (filled circles), and S5, c = 4 \(\times\) 10\({}^{-4}\)% (open diamonds). The shift of the \(\chi^{\prime}\) peak temperature upon dilution allows the T\({}_{\rm B}\) vs. c dependence to be determined. main reason for which we report our results in terms of the volume concentration at synthesis and not in terms of the interparticle distance calculated assuming a uniform distribution of the nanoparticles. We then investigated the quantitative aspects of the T\({}_{\rm B}\) vs. c behavior. Figure 3 shows the blocking temperature measured as a function of the ferrofluid's volume concentration (solid symbols) obtained from the ac-susceptibility data in Figure 2. We made several attempts to fit these data using basic functions and found that that a power law:  
\[\rm T_{B}=A\cdot c^{\gamma} \tag{3}\]  
excellently describes the observed behavior of the blocking temperature. Indeed, a least-squares fit of this newly determined equation to the T\({}_{\rm B}\) vs. c data converges to low residuals upon the simultaneous variation of parameters A and \(\gamma\). The solid line shows the best fit that yields A = 64 K and \(\gamma\) = 0.056. Next, we focus our analysis on the temperature dependence of superspin relaxation time, \(\rm\tau(T)\), for samples of different concentrations. We will then use those data to gain quantitative information on the energy barrier to magnetization reversal and finally establish how the barrier changes with the concentration. Figure 4a shows the shift with frequency of the \(\chi^{\prime\prime}\) vs. \(\rm T\)\(\rm T\)\(\rm t\) curve peak temperature for the S3 sample. As the system's relaxation time, \(\rm\tau\), is the same as the observation time, \(\rm\tau_{obs}\), (i.e., \(\rm\tau=\tau_{obs}=1/2\pi f\)) at the peak temperature, these curves allow us to determine the \(\rm\tau(T)\) dependence for the S3 sample. This is shown by the solid symbols in Figure 4b, where the natural logarithm of \(\rm\tau\) is plotted as a function of inverse temperature 1/T. We then analyzed theses data in the framework of the DBF theory. We used Equation (2), where the relaxation time is calculated based on three quantities: the pre-factor \(\rm\tau_{r}\), the energy barrier for the individual nanoparticle relaxation according to the Neel and Brown model E\({}_{\rm B}\), and the additional energy barrier, E\({}_{\rm ad}\), which corresponds to the collective superspin relaxation due to interparticle interactions. Two parameters, \(\rm\tau_{r}\) and E\({}_{\rm ad}\), were allowed to vary simultaneously in the fit of the DBF equation to the data, while E\({}_{\rm B}\)/k\({}_{\rm B}\) was kept at 366 K for reasons explained below. The best fit, shown by the solid line, converges to low residuals and yields \(\rm\tau_{r}\) = 1.1 \(\times\) 10\({}^{-12}\) s and E\({}_{\rm ad}\)/k\({}_{\rm B}\) = 289 K.  
Figure 3: Blocking temperature, T\({}_{\rm B}\), dependence on the concentration, c, obtained from the peak shift of the ac-susceptibility data in Figure 2 (solid symbols). The solid line shows the best fit of a power law T\({}_{\rm B}\) = A\(\cdot\)c\({}^{\gamma}\), with A = 64 K and \(\gamma\) = 0.056. We determined the individual energy barrier to magnetization reversal, EB, as follows.