Disentangling local heat contributions in interacting magnetic nanoparticles

Recent experiments on magnetic nanoparticle hyperthermia show that the heat dissipated by particles must be considered locally instead of characterizing it as a global quantity. Here we show theoretically that the complex energy transfer between nanoparticles interacting via magnetic dipolar fields can lead to negative local hysteresis loops and does not allow the use of these local hysteresis loops as a temperature measure. Our model shows that interacting nanoparticles release heat not only when the nanoparticle magnetization switches between different energy wells, but also in the intrawell motion, when the effective magnetic field is changed because the magnetization of another particle has switched. The temperature dynamics has a highly nontrivial dependence on the amount of precession, which is controlled by the magnetic damping. Our results constitute a step forward in modeling magnetic nanoparticles for hyperthermia and other heating applications.

Figure 1

Illustration of global vs local heat in a nanoparticle ensemble: do individual (single-particle) loops represent local heating?

Figure 2

Global (dashed area) and local (red for large particle A and blue for small particle B) hysteresis cycles for two interacting nanoparticles, separated by different distances $x$ along the perpendicular to the field direction. Cases are (a) noninteracting, (b) $x=7.1$-nm separation, and (c) 1.5-nm separation. Schematics of the two particles are drawn below each set of cycles, with their anisotropy axes drawn by dashed lines.

Figure 3

Dynamics of (a) normalized magnetization, (b) total energy (normalized by the maximum possible stored energy $2M_{s}2H_K{V}_{tot}=8K{V}_{tot}$, where $V_{tot}$ is the system volume and $H_K=2K/M_s$ is the anisotropy field), and (c) temperature change $\mathrm{\Delta }T$ as computed by Eq. (1). The blue line corresponds to nanoparticle B and the red line to nanoparticle A. The nanoparticles are separated by $x=7.1$ nm.

Figure 4

A comparison of energy change (work plus heat, solid bars) and heat released (hashed bars) during the two irreversible magnetization jumps for the two particles A (red) and B (blue). The energy changes $\mathrm{\Delta }{E}_{\min }$ are evaluated from individual hysteresis loops, while the heat released $Q$ is calculated using Eq. (1) with large damping ${\alpha }_{\perp }=1.0$. The results are normalized to the total energy “stored” in the global hysteresis loop. The nanoparticles are separated by 7.1 nm, as in Fig. 3. The sketches illustrate whether the magnetization of the particle switches, or wobbles, within the same energy well.

Figure 5

(a) Temperature dynamics of both nanoparticles A (red) and B (blue) for four values of the damping parameter. The temperature changes of particles A and B as a function of ${\alpha }_{\perp }$ at jumps 1 and 2 are shown in (b) and (c), respectively. Again, the nanoparticles are separated by $x=7.1$ nm.