Third harmonic of the dynamic magnetic susceptibility of a concentrated ferrofluid: Numerical calculation and simple approximation formula

Information about the nonlinear magnetic response of dispersions of magnetic particles is the basis for biomedical applications. In this paper, using analytical and numerical methods, the third harmonic of the dynamic susceptibility of an ensemble of moving magnetic particles in an ac magnetic field with an arbitrary amplitude is studied, taking into account interparticle interactions. A simple approximation formula is proposed to predict the third harmonic as a function of two parameters: the Langevin susceptibility ${\chi }_L$, which is used to estimate the particle dipole-dipole interactions, and the Langevin parameter $\xi$, which represents the ratio of the energy of the magnetic moment interacting with the magnetic field to the thermal energy. The derived approximation formula corresponds with the known single-particle theories in the limit case of a small particle's concentration and is valid for concentrated dispersions of magnetic particles (with the Langevin susceptibility up to ${\chi }_L\le 3$) in high-amplitude ac fields (with the Langevin parameter up to $\xi \le 10$).