Static and dynamic magnetization models of magnetic nanoparticles: an appraisal

Nowadays, magnetic nanoparticles (MNPs) have been extensively used in biomedical fields such as labels for magnetic biosensors, contrast agents in magnetic imaging, carriers for drug/gene delivery, and heating sources for hyperthermia, among others. They are also utilized in various industries, including data and energy storage and heterogeneous catalysis. Each application exploits one or more physicochemical properties of MNPs, including magnetic moments, magnetophoretic forces, nonlinear dynamic magnetic responses, magnetic hysteresis loops, and others. It is generally accepted that the static and dynamic magnetizations of MNPs can vary due to factors such as material composition, crystal structure, defects, size, shape of the MNP, as well as external conditions like the applied magnetic fields, temperature, carrier fluid, and inter-particle interactions (i.e., MNP concentrations). A subtle change in any of these factors leads to different magnetization responses. In order to optimize the MNP design and external conditions for the best performance in different applications, researchers have been striving to model the macroscopic properties of individual MNPs and MNP ensembles. In this review, we summarize several popular mathematical models that have been used to describe, explain, and predict the static and dynamic magnetization responses of MNPs. These models encompass both individual MNPs and MNP ensembles and include the Stoner-Wohlfarth model, Langevin model, zero/non-zero field Brownian and Néel relaxation models, Debye model, empirical Brownian and Néel relaxation models under AC fields, the Landau-Lifshitz-Gilbert (LLG) equation, and the stochastic Langevin equation for coupled Brownian and Néel relaxations, as well as the Fokker-Planck equations for coupled/decoupled Brownian and Néel relaxations. In addition, we provide our peers with the advantages, disadvantages, as well as suitable conditions for each model introduced in this review. The shrinking size of magnetic materials brings about a significant surface spin canting effect, resulting in higher anisotropy and lower magnetization in MNPs compared to bulk materials. Accurate prediction of static and dynamic magnetizations in MNPs Requires both precise data on their magnetic properties and an accurate mathematical model. Hence, we introduced the spin canting effect and models to estimate anisotropy and saturation magnetization in MNPs.