Hysteresis effects in magnetic nanoparticles: A simplified rate-equation approach

Fine tuning and optimization of magnetic hyperthermia treatments using versatile trapezoidal driving-field waveforms

“…14,15 2.2 Rate equations: advantages and limits of applications Rate equations are being applied since about one decade to study the response of uniaxial magnetic NPs described as double-well systems (DWSs). 41,59 The method has been successfully applied to systems of non-interacting particles, under both equilibrium 43,59 and off-equilibrium conditions. In particular, rate equations are used to describe important offequilibrium properties, such as the behaviour of the low-eld magnetization below the blocking temperature in eld cooling/zero eld cooling experiments, 43 and isothermal magnetic hysteresis loops.…”

“…In particular, rate equations are used to describe important offequilibrium properties, such as the behaviour of the low-eld magnetization below the blocking temperature in eld cooling/zero eld cooling experiments, 43 and isothermal magnetic hysteresis loops. 28,41 The method provides an accurate picture of the process of magnetization of an assembly of DWSs, under both linear and non-linear, static and dynamic conditions. Rate equations work well over an extended range of driving-eld amplitudes and frequencies; 41 they describe with precision the transition between superparamagnetic and blocked regimes in a nanoparticle, 43 and are appropriate for treating the off-equilibrium response of nanoparticles subjected to a cyclic eld of frequency up to hundreds of kHz (ref.…”

“…Finally, the rate-equation method is appropriate for studying both monodisperse and polydisperse systems of nanoparticles with easy axes randomly pointing in all directions. 41 Of course, more exact approaches to the dynamics of magnetization in nanostructures exist. They involve solving either the Landau-Lifshitz 60 /Landau-Lifshitz-Gilbert 61 equations, or the Fokker-Planck-Brown equation for the magnetization dynamics.…”

“…9-20 kA m À1 ) the limits of validity of the linear theory are overcome, and the magnetic response of the system is no longer linear, as clearly shown by the rate equation approach. 28,41 The present heating model takes in due account the true magnetic losses of typical nanoparticles under standard operating conditions. In particular, the rate equations governing the evolution of the populations in each potential-energy well of the DWS are solved for both monodisperse and polydisperse NP systems, so that the hysteresis loops and the heating power P in are obtained as functions of temperature T. When a trapezoidal waveform is applied, the heating power is a function of the taper parameter too, as shown in the right panel of Fig.…”

“…The loops are obtained using a rate-equation approach, which is able to suitably describe the dynamics of magnetization in NPs subjected to an alternating eld of high frequency while maintaining intrinsic simplicity and effectiveness. 41 The heating efficacy of a set of magnetite nanoparticles subjected to a trapezoidal driving eld waveform is nally evaluated by making use of a simple heating model to picture a small portion of living tissue. The proposed technique of nanoparticle activation by using trapezoidal waveforms is shown to give a substantial contribution to the optimization of magnetic hyperthermia.…”