Determination of the critical interspacing for the noninteracting magnetic nanoparticle system

The dipole–dipole interactions of monodisperse Fe3O4 nanoparticles (NPs) can be directly controlled by a uniform SiO2 shell with different thickness, i.e., different interspacings. Thus, the interacting strength of a serial of Fe3O4–SiO2 NPs system can be revealed by fitting the blocking temperature TB measured at ac fields to the Vogel–Fulcher law. The interspacing over five times of diameter for less than 8.0 nm Fe3O4 NPs is the critical value to achieve a noninteracting system. Furthermore, a general equation to evaluate critical interspacing for noninteracting magnetic NPs systems with different sizes and saturation magnetizations was calculated by Monte Carlo method.