Mixed Brownian alignment and Néel rotations in superparamagnetic iron oxide nanoparticle suspensions driven by an ac field

Superparamagnetic iron oxide nanoparticles with highly nonlinear magnetic behavior are attractive for biomedical applications like magnetic particle imaging and magnetic fluid hyperthermia. Such particles display interesting magnetic properties in alternating magnetic fields and here we document experiments that show differences between the magnetization dynamics of certain particles in frozen and melted states. This effect goes beyond the small temperature difference ($\mathrm{\Delta }T\sim 20{}^{\circ }\text{C}$) and we show the dynamics to be a mixture of Brownian alignment of the particles and Néel rotation of their moments occurring in liquid particle suspensions. These phenomena can be modeled in a stochastic differential equation approach by postulating log-normal distributions and partial Brownian alignment of an effective anisotropy axis. We emphasize that precise particle-specific characterization through experiments and nonlinear simulations is necessary to predict dynamics in solution and optimize their behavior for emerging biomedical applications including magnetic particle imaging.

Figure 1

The magnetocrystalline energy surface for cubic magnetite (negative $K_1$), which has easy directions in the [111] family. After (a) applying an arbitrary magnetic field vector $H_{\mathrm{app}}$ in a fixed direction, particles in solution undergo Brownian rotation through a small angle until (b) the nearest easy axis is aligned with the applied field.

Figure 2

SPIOs were characterized by complementary methods to obtain magnetic, size, and structural information: (a) VSM curve, (b) TEM image of nanoparticles showing monodisperse size distribution, (c) DLS plot, and (d) x-ray diffraction plot.

Figure 3

(a) The simulated magnetic field of the permanent magnet (lines of equal field strength are shown in the figure). The test tube containing the SPIOs was frozen in this orientation at various distances to achieve the varying magnetic field amplitudes. In the example illustrated in (a), the aligning field was applied perpendicular to the direction of the oscillating (drive) field, which was always applied parallel to the long axis of the sample tube. The surface field for these magnets is 0.12 T, so we see that at 5 cm the field is in the single millitesla range as shown in experiment. Notably at this distance we can see the variations in the field are minimal within the test tube. (b) Measured variation of field strength with distance from the magnet. Error bars represent the variation in field within the sample tube for the stated field orientation, and they differ because the tube geometry is anisotropic.

Figure 4

Distribution of the effective easy axes of 500 particles. This figure should aid the visualization of the asymmetry that was included in the simulations in order to mimic partial alignment of an effective easy axis with the static field.

Figure 5

MPS results of the same sample measured in fluid state (at ${20}^{\circ }\mathrm{C}$) and frozen (in zero magnetic field, 0 mT/${\mu }_0$). The excitation field was 50 mT/${\mu }_0$ amplitude at 26 kHz: (a) ${\chi }_{\mathrm{diff}}$ and (b) $m\left(H\right)$ curves calculated by integrating data in (a).

Figure 6

MPS (26 kHz, 50 mT/${\mu }_0$) signal parameters of MNPs, field frozen (10 mT/${\mu }_0$) along different directions with respect to the MPS applied field. (a) ${\chi }_{\mathrm{diff}}$ and (b) integrated $m\left(H\right)$ curves. MPS (26 kHz, 50 mT/${\mu }_0$) signal parameters of MNPs, frozen in different magnetic fields along parallel orientation: (c) ${\chi }_{\mathrm{diff}}$ and (d) $m\left(H\right)$ curves. MPS (50 mT/${\mu }_0$, different frequencies) signal parameters of MNPs, frozen in 5 mT/${\mu }_0$ along parallel orientation: (e) ${\chi }_{\mathrm{diff}}$ and (f) $m\left(H\right)$ curves.

Figure 7

Comparison plot of MNPs response for various combinations of freezing fields, orientations, and drive frequencies: (a) ${\chi }_{\mathrm{diff}}^{\max }$ and (b) coercive field.

Figure 8

MPS (26 kHz, 20 mT/${\mu }_0$) response of SPIOs in water at room temperature and $0^{\circ }\mathrm{C}$ (liquid state). Error bars in (a) represent the standard deviation of three measurements.

Figure 9

${\chi }_{\mathrm{diff}}^{\max }$ data (at 26 kHz and 20 mT/${\mu }_0$) of different concentrations: (a) at different freezing fields and (b) at 10 mT/${\mu }_0$ parallel freezing.

Figure 10

Comparison between simulation and experiment for nanoparticles that were frozen while exposed to a 10 mT/${\mu }_0$ static field aligned parallel or perpendicular to the oscillating field. Then spectroscopy was performed using a 40-kHz, 20 mT/${\mu }_0$ field. The log-normal distributions of particle diameter and anisotropy constant are shown on the right by number in a representative 1000-particle subsample.

Figure 11

Comparison between simulation and experiment for nanoparticles either in either a frozen or melted state. Spectroscopy was performed with a 26-kHz, 31 mT/${\mu }_0$ field. The log-normal distributions of particle diameter and anisotropy constant are shown on the right by number in a representative 1000-particle subsample.

Figure 12

Magnetizations were simulated using a 40-kHz, 20 mT/${\mu }_0$ field at room temperature to show how the (a) differential magnetic susceptibility in arbitrary units ${\chi }_{\mathrm{diff}}$ (arb. units) and (b) hysteresis loops are affected by increasing the percentage of particles aligned in the direction of the applied field. The size and anisotropy distribution are identical to that in other simulations.