# Non-Debye Behavior of the Néel and Brown Relaxation in Interacting Magnetic Nanoparticle Ensembles

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We used ac-susceptibility measurements to study the superspin relaxation in Fe_{3}O_{4}/Isopar M nanomagnetic fluids of different concentrations. Temperature-resolved data collected at different frequencies, χ″ vs. T|_{f}, reveal magnetic events both below and above the freezing point of the carrier fluid (T_{F} = 197 K): χ″ shows peaks at temperatures T_{p1} and T_{p2} around 75 K and 225 K, respectively. Below T_{F}, the Néel mechanism is entirely responsible for the superspin relaxation (as the carrier fluid is frozen), and we found that the temperature dependence of the relaxation time, τ_{N}(T_{p1}), is well described by the Dorman–Bessais–Fiorani (DBF) model: $\tau NT=\tau r\mathrm{exp}EB+EadkBT$. Above T_{F}, both the internal (Néel) and the Brownian superspin relaxation mechanisms are active. Yet, we found evidence that the effective relaxation times, τ_{eff}, corresponding to the T_{p2} peaks observed in the denser samples *do not* follow the typical Debye behavior described by the Rosensweig formula $1\tau eff=1\tau N+1\tau B$. First, τ_{eff} is 5 × 10^{−5} s at 225 K, almost three orders of magnitude more that its Néel counterpart, τ_{N}~8 × 10^{−8} s, estimated by extrapolating the above-mentioned DBF analysis. Thus, $1\tau N\gg 1\tau eff$, which is clearly not consistent with the Rosensweig formula. Second, the observed temperature dependence of the effective relaxation time, τ_{eff}(T_{p2}), is excellently described by $\tau B-1T=T\gamma 0\mathrm{exp}-E\prime kBT-T0\prime $, a model solely based on the hydrodynamic Brown relaxation, $\tau B(T)=3\eta TVHkBT$, combined with an activation law for the temperature variation of the viscosity, $\eta T=\eta 0\mathrm{exp}E\prime /kB(T-T0\prime $. The best fit yields $\gamma 0=3\eta VHkB$ = 1.6 × 10^{−5} s·K, E′/k_{B} = 312 K, and T_{0}′ = 178 K. Finally, the higher temperature T_{p2} peaks vanish in the more diluted samples (δ ≤ 0.02). This indicates that the formation of larger hydrodynamic particles via aggregation, which is responsible for the observed Brownian relaxation in dense samples, is inhibited by dilution. Our findings, corroborating previous results from Monte Carlo calculations, are important because they might lead to new strategies to synthesize functional magnetic ferrofluids for biomedical applications.