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

## TL;DR

The study investigates magnetic nanoparticle behavior in fluids, finding that relaxation processes do not follow expected models, which could impact biomedical applications.

## Contribution

The paper reveals non-Debye behavior in magnetic nanoparticle relaxation, challenging existing theoretical models.

## Key findings

- Below the freezing point of the carrier fluid, the Néel relaxation is well described by the DBF model.
- Above the freezing point, the effective relaxation time does not follow the Rosensweig formula and is better described by a hydrodynamic Brown relaxation model.
- Diluted samples show no Tp2 peaks, indicating that aggregation is inhibited by lower concentration.

## Abstract

We used ac-susceptibility measurements to study the superspin relaxation in Fe3O4/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 (TF = 197 K): χ″ shows peaks at temperatures Tp1 and Tp2 around 75 K and 225 K, respectively. Below TF, 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, τN(Tp1), is well described by the Dorman–Bessais–Fiorani (DBF) model: τNT=τrexp⁡EB+EadkB T. Above TF, both the internal (Néel) and the Brownian superspin relaxation mechanisms are active. Yet, we found evidence that the effective relaxation times, τeff, corresponding to the Tp2 peaks observed in the denser samples do not follow the typical Debye behavior described by the Rosensweig formula 1τeff=1τN+1τB. First, τeff is 5 × 10−5 s at 225 K, almost three orders of magnitude more that its Néel counterpart, τN~8 × 10−8 s, estimated by extrapolating the above-mentioned DBF analysis. Thus, 1τN≫1τeff, which is clearly not consistent with the Rosensweig formula. Second, the observed temperature dependence of the effective relaxation time, τeff(Tp2), is excellently described by τB−1T=Tγ0exp⁡−E′kBT−T0′, a model solely based on the hydrodynamic Brown relaxation, τB(T)=3ηTVHkBT, combined with an activation law for the temperature variation of the viscosity, ηT=η0exp⁡E′/kB(T−T0′. The best fit yields γ0=3ηVHkB = 1.6 × 10−5 s·K, E′/kB = 312 K, and T0′ = 178 K. Finally, the higher temperature Tp2 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.