This paper presents a novel method for estimating the temperature of magnetic nanoparticles (MNPs) based on AC magnetization harmonics of MNPs dominated by Brownian relaxation. The disparity in the AC magnetization response and magnetization harmonic between the Fokker-Planck equation and the Langevin function was analyzed, and we studied the relationship between the magnetization harmonic and the key factors, such as Brownian relaxation time, temperature, magnetic field strength, core size and hydrodynamic size of MNPs, and excitation frequency. We proposed a compensation function for AC magnetization harmonic with consideration of the key factors and the disparity between the Fokker-Planck equation and the Langevin function. Then a temperature estimation model based on the compensation function and the analytical harmonic expression of the Langevin function was established. By employing the least squares algorithm, the temperature was successfully calculated. The experimental results show that the temperature error is less than 0.035 K in the temperature range from 310 K to 320 K. The temperature estimation model is expected to improve the performance of the magnetic nanoparticle thermometer and be applied to magnetic nanoparticle-mediated hyperthermia.

A method to estimate the effective anisotropy constant distribution of magnetic nanoparticles is proposed, enabling accurate prediction of their AC magnetization behavior for biomedical applications.

This work investigated the chainlike magnetic nanoparticle (CMNP), which is a type of magnetic nanoparticle with dipole-dipole interaction in which individual nanoparticles are connected to form a chain-like structure. We numerically analyzed the AC magnetization characteristics of the CMNP and the single-core magnetic nanoparticle (SMNP) using the Landau–Lifshitz–Gilbert equation. Owing to magnetic diploe–dipole interaction, the magnetization of the CMNP is approximately 10 times that of the SMNP under a certain excitation field. MNPs with a chain-like structure are thus expected to have enhanced magnetization characteristics and better performance in medical applications. Additionally, it was found that stronger magnetization can be expected from a CMNP connecting 10 or more magnetic cores with a size of approximately 10 to 12 nm.

Dipolar interaction between magnetic nanoparticles (MNPs) affects their magnetization properties, and the effect is particularly significant for multi-core MNPs and high-concentration MNP samples. In this work, we performed numerical simulations using the Landau–Lifshitz–Gilbert equation to investigate the effect of dipolar interaction on magnetization properties. In these simulations, we introduced a cubic model with 10 00 MNPs whose core size d c ranges from 10 to 20 nm and whose easy axes are randomly oriented. The distance d g between two adjacent MNPs in the cubic model was varied from 10–70 nm. We quantified the critical interparticle distance d crit — defined as the minimum surface-to-surface distance between magnetic nanoparticles (MNPs) at which dipolar interaction becomes negligible. Assuming a uniform spherical coating layer, this corresponds to d crit = 2 t m , where t m is the minimum coating thickness required to eliminate dipolar effects for a given d c . Our results reveal that d crit varies with harmonic order: the values of d crit for the third M 3 and fifth M 5 harmonic components of magnetization are larger than that for the fundamental component M 1 , indicating that higher-order harmonics commonly used in MPI are more sensitive to dipolar interaction. Moreover, we found d crit is also affected by AC field frequency. A similar frequency dependence is observed for the hysteresis loop area A , which is relevant to magnetic hyperthermia, where higher frequencies require a larger d crit (and hence a thicker coating) to suppress dipolar interactions. These results may have important implications for the design of core–shell MNPs tailored for biomedical applications.

Brownian relaxation is one of the primary mechanisms that allows magnetic nanoparticles (MNPs) to convert magnetic energy into thermal energy under an excitation magnetic field. Accurately characterizing the MNP’s magnetization dynamics dominated by Brownian relaxation is crucial for achieving high-precision temperature estimation. However, the lack of a readily applicable analytical expression remains a major obstacle to the advancement of magnetic nanoparticle hyperthermia (MNPH). In this paper, the perturbation method was applied to derive analytical expressions from the Fokker-Planck equation, which characterized MNP’s magnetization behaviors under the AC and DC magnetic fields. Numerical simulations were conducted to validate the accuracy of the analytical expressions and to explore the correlation between temperature and the magnetization. Then, a temperature analysis model based on magnetization harmonics was constructed. The first to second harmonic ratios and first harmonic phases were used to calculate MNP’s temperature, respectively. The experimental results demonstrate that within 310 K to 320 K, the estimation error of the temperature using the amplitude ratio of the first to second harmonics is below 0.02 K, while the error using the first harmonic phase is below 0.03 K. The derived analytical expressions are expected to enhance the accuracy of MNP temperature measurements and facilitate their broader applications in MNPH and MNP imaging.