Spin morphologies and heat dissipation in spherical assemblies of magnetic nanoparticles

Aggregates of magnetic nanoparticles (MNPs) exhibit unusual properties due to the interplay of small system size and long-range dipole-dipole interactions. Using the micromagnetic simulation software oommf, we study the spin morphologies and heat dissipation in micron-size spherical assemblies of MNPs. In particular, we examine the sensitivity of these properties to the dipolar strength, manipulated by the interparticle separation. As oommf is not designed for such a study, we have incorporated a novel scaling protocol for this purpose. We believe that it is essential for all studies where volume fractions are varied. Our main results are as follows: (i) Dense assemblies exhibit strong dipolar effects which yield local magnetic order in the core but not on the surface, where moments are randomly oriented. (ii) The probability distribution of ground-state energy exhibits a long high-energy tail for surface spins in contrast to small tails for the core spins. Consequently, there is a wide variation in the energy of surface spins but not the core spins. (iii) There is strong correlation between ground-state energy and heating properties on application of an oscillating magnetic field $h\left(t\right)=h_{o}cos2\pi ft$: the particles in the core heat uniformly, while those on the surface exhibit a wide range from cold to intensely hot. (iv) Specific choices of $h_o$ and $f$ yield characteristic spatial heat distributions, e.g., hot surface and cold core, or vice versa. (iv) For all values of $h_o$ and $f$ that we consider, heating was maximum at a specific volume fraction. These results are especially relevant in the context of contemporary applications such as hyperthermia and chemotherapy, and also for novel materials such as smart polymer beads and superspin glasses.

Figure 1

(a) Schematic of the sphere of radius $R$ containing cubic MNPs with lateral dimension $l$ arranged on a lattice with spacing $a$. (b) Enlarged portion of the basic unit cell in (a).

Figure 2

Scaled hysteresis loops (refer to text) corresponding to assemblies of ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticles in blue (black) symbols and Co nanoparticles in red (gray) symbols for indicated values of the lattice spacing $a$ and field strength $H_o$. The frequency of the applied field $f={10}^7$ Hz. The solid black curve is for $a=\infty$ corresponding to a noninteracting assembly of MNPs. This data has been obtained using kMCS as detailed in the text.

Figure 3

Typical ground-state morphologies of ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticles ($l=10$ nm) assembled in a sphere of radius $R=200$ nm on a cubic lattice with spacing (a) $a=10$ nm corresponding to $\mathrm{\Theta }\simeq 1.75$ and (b) $a=20$ nm corresponding to $\mathrm{\Theta }\simeq 0.21$. The morphology in (a) contains 33 496 MNPs, while that in (b) contains 4186 MNPs. The slices in (c) and (d) are taken at $z=0$ in the $xy$ plane. The average value of the component of the magnetic moment ${\overline{\mu }}_i$ vs $r$ for (e) $a=10\mathrm{nm}$ and (f) $a=20\mathrm{nm}$ is also indicated. The data have been averaged over $\sim {10}^3$ values for $r=20$ nm and $\sim {10}^5$ values for $r=200$ nm.

Figure 4

Variation of $\mathrm{\Theta }=D/KV$ for cubic ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticles with $l=10$ nm as a function of $r/a$, where $r$ is the center-to-center distance between two particles and $a$ is the spacing of the underlying lattice. The data sets are for $a=10$ and 20 nm. The inset shows a magnified view for $r/a\ge 5$, i.e., for the fifth neighbor and beyond. For $\mathrm{\Theta }>1$, the dipolar effects continue to be significant even beyond $r/a=10$ or the tenth neighbor. Note that $\mathrm{\Theta }$ is the largest when $a=l$, which corresponds to 10 nm in the present case.

Figure 5

Probability distributions $P\left(E\right)$ vs $E$ of the ground-state energy for ${\mathrm{Fe}}_3{\mathrm{O}}_4$ MNPs packed in a sphere of radius $R=200\mathrm{nm}$ for lattice spacing (a) $a=10\mathrm{nm}$ and (b) $a=20\mathrm{nm}$. These values of $a$ correspond to $\mathrm{\Theta }\simeq 1.7$ and 0.2, respectively. (c),(d) The corresponding probability distribution of energy $P_s\left(E\right)$ vs $E$ for MNPs in concentric spherical shells of specified radii. (e),(f) The variation of the average energy of a nanoparticle in the shell ${\overline{E}}_s$ vs $r$. The $y$ axis has been scaled by the maximum value $E_m$ to enable comparisons. Clearly, there are strong surface effects for the $\mathrm{\Theta }\simeq 1.7$ case.

Figure 6

Heat dissipated in MNP spheres on application of the oscillating field $H=H_{o}cos2\pi ft$ indicated by corresponding color charts in J ${\mathrm{m}}^{-3}$. The interparticle separation in the left panel is 10 nm, while that in the right panel is 20 nm. The distinct parameters are (a) $a=10\mathrm{nm}, H_o=0.4\mathrm{T}, f={10}^7\mathrm{Hz}$; (b) $a=10\mathrm{nm}, H_o=0.4\mathrm{T}, f={10}^9\mathrm{Hz}$; (c) $a=20\mathrm{nm}, H_o=0.4\mathrm{T}, f={10}^7\mathrm{Hz}$; and (d) $a=20$ nm, $H_o=0.4\mathrm{T}, f={10}^9\mathrm{Hz}$. The corresponding 2-$d$ slices are indicated, respectively, in (e)–(h).

Figure 7

Probability distribution $P\left(E_H\right)$ vs $E_H$ of heat dissipated by MNPs for $a=10$ and 20 nm from oommf in blue (black) symbols and kMCS in red (gray) symbols. The field strength $H_o=0.2$ T and the frequency $f={10}^7$ Hz.

Figure 8

Probability distribution of heat dissipation $P_s\left(E_H\right)$ vs $E_H$ for MNPs in concentric spherical shells of specified radii for (a) $a=10\mathrm{nm}, H_o=0.4\mathrm{T}, f={10}^7\mathrm{Hz}$; (b) $a=20\mathrm{nm}, H_o=0.4\mathrm{T}, f={10}^7\mathrm{Hz}$; (c) $a=10\mathrm{nm}, H_o=0.4\mathrm{T}, f={10}^9\mathrm{Hz}$; and (d) $a=20\mathrm{nm}, H_o=0.4\mathrm{T}, f={10}^9\mathrm{Hz}$. (e)–(h) Corresponding variation of ${\overline{E}}_s$ vs $r$. The $y$ axis has been scaled by $E_m$ for comparison. Note that the data points at the periphery are averaged over ${10}^5$ values and are very accurate.

Figure 9

Pictorial representation of Figs. 8–8 showing the average heat dissipation in concentric spherical shells for (a) $a=10\mathrm{nm}, H_o=0.4\mathrm{T}, f={10}^7\mathrm{Hz}$; (b) $a=20\mathrm{nm}, H_o=0.4\mathrm{T}, f={10}^7\mathrm{Hz}$; (c) $a=10\mathrm{nm}, H_o=0.4\mathrm{T}, f={10}^9\mathrm{Hz}$; and (d) $a=20\mathrm{nm}, H_o=0.4\mathrm{T}, f={10}^9\mathrm{Hz}$. The innermost shell with averaging $<{10}^3$ has been left blank in each of the pictographs.

Figure 10

Variation of the hysteresis loop area as a function of center-to-center interparticle separation $a$ for cubic particles ($l=10\mathrm{nm}$) of ${\mathrm{Fe}}_3{\mathrm{O}}_4$ assembled in a sphere ($R=200\mathrm{nm}$). The rescaled amplitude $H_o$ of the applied field is (a) 0.2, (b) 0.4, and (c) 0.6 T. Each set contains evaluations for three values of the frequency $f={10}^7, {10}^8$, and ${10}^9\mathrm{Hz}$. All data have been averaged over 50 initial conditions. Maximum heat dissipation is observed at a critical separation $a^{*}$ and not at $a=l$ which maximizes the dipolar strength.