ABSTRACT

Magnetic nanoparticle assemblies are difficult systems to model due to the interplay between intrinsic and collective effects. The first ones are associated with the magnetic properties of an individual particle and require considering the atomic spins of the magnetic ions, their mutual exchange interactions, and magnetocrytalline anisotropy. Due to its finite size, an individual particle has magnetic properties different from the bulk counterpart material. Moreover, the high proportion of surface spins with reduced coordination influences the equilibrium magnetic configuration of an individual particle, with spin noncollinearities being a consequence of the distinct surface

anisotropy. The collective effects have to do with interactions among the nanoparticles in an ensemble, such as long-range dipole-dipole interactions, and can be tackled more easily by considering each nanoparticle as one effective spin having the magnitude of the total magnization of the particle. Therefore, at this level of description, nanoparticle ensembles can be modeled by a collection of macrospins having their own anisotropy axes and interacting through dipolar interactions, neglecting their internal structure, which in turn is equivalent to assuming that the interactomic exchange coupling is strong enough to keep atomic magnetic ions aligned along the global magnetization direction. Whereas in atomic magnetic materials the exchange interaction usually dominates over dipolar interactions, the opposite happens in many nanoscale particle or clustered magnetic systems, for which interparticle interactions are mainly of dipolar origin. Therefore, one-spin (OSP) models should, in principle, provide a correct description of noninteracting systems and, to a first approximation, be valid also to account for the main features observed in more concentrated samples where interactions cannot be neglected. It has been shown also recently [17, 54] that spin noncollinearities due to surface anisotropy can even be incorporated within the OSP approach if an effective cubic anisotropy term is added to the original uniaxial anisotropy energy. However, incorporation of dipolar interactions along these lines does not seem feasible within the present theoretical frameworks. While we have a valid theoretical framework to compute equilibrium magnetic properties for noninteracting systems (such as thermal dependence M(T), isothermal field dependence M(H), and low-temperature configurations), analytically or numerically within the scope of OSP models [19] models including dipolar interactions can only compute these quantities using perturbative thermodynamic theory and, even so, anaytical expressions can only be obtained under certain limits and approximations. In contrast, dynamic properties [such as hysteresis loops, field-cooledzero-field-cooled (FC-ZFC) processes, susceptibility, or magnetic relaxation] are nonequilibrium phenomena for which a unique theoretical framework covering the wide range of time scales involved is not available, even for noninteracting systems. Therefore, most studies on dynamics revert to numerical simulations of ensembles of macrospins [3, 14, 15, 18, 26, 46, 50, 52] based on

Monte Carlo (MC) methods, since simulations based on the LandauLifschitz equation cannot access the long time scales involved in these phenomena. The main difficulty in modeling the long-time dynamics of magnetic nanoparticle ensembles is the calculation of the relaxation rates between metastable states as they depend on the energy barriers that have to be overcome by thermal fluctuations and, consequently, they depend on the orientation of the nanoparticle easy axis with respect to the field axis. At the same time, the presence of interparticle interactions modifies in a complex manner the energy landscape due to the long-range character of the dipolar interactions, and several escape paths out of a metastable minimum may coexist. Therefore, in general, the energy barriers responsible for the thermal relaxation of the nanoparticle ensemble toward equilibrium cannot be computed analytically and numerical simulations have to be used. While dilute systems are well understood, experimental results for dense systems are still a matter of controversy. Some of their peculiar magnetic properties have been attributed to dipolar interactions, although many of the issues are still controversial. Different experimental results measuring the same physical quantities give contradictory results, and theoretical explanations are many times inconclusive or unclear. In the following section, we briefly outline some of the issues that are still under debate:

1. The complexity of dipolar interactions and the frustration provided by the randomness in particle positions and anisotropy axes directions present in highly concentrated ferrofluids seem enough ingredients to create a collective glassy dynamics in these kinds of systems. Experiments probing the relaxation of the thermoremanent magnetization [23, 24, 35] have evidenced magnetic aging. Studies of the dynamic and nonlinear susceptibilities [10, 24, 27] have also found evidence of a critical behavior typical of spin-glass-like freezing. All these studies have attributed this collective spin-glass behavior to dipolar interactions, although surface exchange may also be at the origin of this phenomenon. However, MC simulations of a system of interacting monodomain particles [18] show that, while the dependence of ZFC/FC curves on interaction and cooling rates are reminiscent of a spin-glass

transition at TB, the relaxational behavior is not in accordance with the picture of cooperative freezing.