Simple models for dynamic hysteresis loop calculations of magnetic single-domain nanoparticles: Application to magnetic hyperthermia optimization

Abstract

To optimize the heating properties of magnetic nanoparticles (MNPs) in magnetic hyperthermia applications, it is necessary to calculate the area of their hysteresis loops in an alternating magnetic field. The separation between “relaxation losses” and “hysteresis losses” presented in several articles is artificial and criticized here. The three types of theories suitable for describing hysteresis loops of MNPs are presented and compared to numerical simulations: equilibrium functions, Stoner–Wohlfarth model based theories (SWMBTs), and a linear response theory (LRT) using the Néel–Brown relaxation time. The configuration where the easy axis of the MNPs is aligned with respect to the magnetic field and the configuration of a random orientation of the easy axis are both studied. Suitable formulas to calculate the hysteresis areas of major cycles are deduced from SWMBTs and from numerical simulations; the domain of validity of the analytical formula is explicitly studied. In the case of minor cycles, the hysteresis area calculations are based on the LRT. A perfect agreement between the LRT and numerical simulations of hysteresis loops is obtained. The domain of validity of the LRT is explicitly studied. Formulas are proposed to calculate the hysteresis area at low field that are valid for any anisotropy of the MNP. The magnetic field dependence of the area is studied using numerical simulations: it follows power laws with a large range of exponents. Then analytical expressions derived from the LRT and SWMBTs are used in their domains of validity for a theoretical study of magnetic hyperthermia. It is shown that LRT is only pertinent for MNPs with strong anisotropy and that SWMBTs should be used for weakly anisotropic MNPs. The optimum volume of MNPs for magnetic hyperthermia is derived as a function of material and experimental parameters. Formulas are proposed to allow to the calculation of the optimum volume for any anisotropy. The maximum achievable specific absorption rate (SAR) is calculated as a function of the MNP anisotropy. It is shown that an optimum anisotropy increases the SAR and reduces the detrimental effects of the size distribution of the MNPs. The optimum anisotropy is simple to calculate; it depends only on the magnetic field used in the hyperthermia experiments and the MNP magnetization. The theoretical optimum parameters are compared to those of several magnetic materials. A brief review of experimental results as well as a method to analyze them is proposed. This study helps in the determination of suitable and unsuitable materials for magnetic hyperthermia and provides accurate formulas to analyze experimental data. It is also aimed at providing a better understanding of magnetic hyperthermia to researchers working on this subject.