Monte Carlo and Experimental Study of the Magnetic Relaxation of Superparamagnetic Nanoparticle Ensembles

Abstract

Superparamagnetic Iron Oxide Nanoparticles (SPION) are nanosize beads of iron oxides. Their peculiar magnetic behaviour makes them particularly suited for a variety of biomedical applications, ranging from cellular imaging to cancer treatment by hyperthermia. The usual theory used to describe their magnetic behaviour is that developed by Paul Langevin [1], which only applies to idealized (isotropic, monodisperse in size and non-interacting) nanoparticles at high temperatures.Reality however usually deviates from that theoretical framework. First, real samples are polydisperse in size, which impacts the particles' magnetic moments and therefore their behaviour. Second, particles usually have at least one easy magnetization axis, which adds a term to their Hamiltonian. Finally, particulary in biological media, they tend to aggregate, leading to locally high particle volumic fractions and therefore dipolar interaction between their magnetic moments [2]. Moreover, as demonstrated by Néel and Brown [3], at low temperatures the particles’ magnetic moments and anisotropy axises appear effectively blocked at their initial directions. Under the Néel blocking temperature, the particles are unable to reorganize their internal moments to change the resulting moment's direction, and under the Brown blocking temperature they are unable to rotate so as to align their easy axis with their moment and/or the external field.All those phenomena impact the magnetization of particle ensembles in a non-trivial way and are impossible to study simultaneously theoretically. In this work, these deviations from the Langevin law are studied numerically, at thermodynamic equilibrium, using a Metropolis algorithm, and compared with experimental data obtained on a Vibrating Sample Magnetometer.The Metropolis algorithm is adapted to take into account the Néel and Brown blocking of the particles, as those time-dependant effects are not explained by free energy minimization. Its Hamiltonian is also chosen to take into account the particles' magnetic anisotropy and the dipolar interactions between the particles as well as the interaction of their moments with the magnetic field.Thorough tests were led on the simulation program to ensure correct convergence of the algorithm. Notably, the temperature step used in the ZFC curve does not impact the simulation results, as expected from a robust computation. Convergence was tested with respect to all of the simulation parameters, and the impact of the magnetic anisotropy on the MH curves was validated by comparing to previous theoretical results obtained by M. Respaud et al [4]. Various other validation tests were conducted.The interest of using numerical simulation is that it allows to discriminate the effects of each tweak to the theory, leading to a better understanding of their various impacts. Curves studying the effect of varying each parameter were obtained, as can be seen in figure 1, where the effect of the magnetic anisotropy on the MH curve is compared in different situations. The ultimate goal being to replicate by numerical simulation the experimental results, it is important to understand how each effect impacts the shape of the curves, to have a clear sense of which experimental parameters impact the results and which don't.In terms of experimental data, two types of magnetization curves in particular are obtained: so called MH curves, where the magnetization is plotted versus the external magnetic field that is applied, and magnetization vs temperature curves. These can be obtained following different protocols, leading to different types of curves, the most common being the zero field cooled (ZFC) curve and the field cooled (FC) curve. Their shape is, naturally, impacted by the different effects at play, and by the precise experimental protocol, as can be seen in figure 2, where a ZFC curve is compared with two FC curves: one obtained after freezing the sample under no magnetic field and one after freezing it under a 5 mT magnetic field.FC/ZFC curves and MH curves (at various temperatures) of three different particle samples were obtained and compared. In particular, PolyMAG particles, which are commercialized for transfection protocols, were studied, and revealed a very unusual ZFC curve, showing no peak nor the usual bell shape of those graphs. One of the future goals of the numerical simulation will be to explain, at least qualitatively, this peculiar behaviour. **