Self-consistent solution of magnetic and friction energy losses of a magnetic nanoparticle

We present a simple simulation model for analyzing magnetic and frictional losses of magnetic nanoparticles in viscous fluids subject to alternating magnetic fields. Assuming a particle size below the single-domain limit, we use a macrospin approach and solve the Landau-Lifshitz-Gilbert equation coupled to the mechanical torque equation. Despite its simplicity the presented model exhibits surprisingly rich physics and enables a detailed analysis of the different loss processes depending on field parameters and initial arrangement of the particle and the field. Depending on those parameters regions of different steady states emerge: a region with dominating magnetic relaxation and high magnetic losses and another region region with high frictional losses at low fields or low frequencies. The energy increases continuously even across regime boundaries up to frequencies above the viscous relaxation limit. At those higher frequencies the steady state can also depend on the initial orientation of the particle in the external field. The general behavior and special cases and their specific absorption rates are compared and discussed.

Figure 1

The Stoner-Wohlfarth astroid. The orange horizontal line indicates the orientation of the easy axis. The black arrow $H$ indicates the applied field, with an angle $\varphi$ relative to the easy axis, while the long blue arrow $M$ draws a tangent on the astroid to the tip of the field vector $H$, which gives the orientation of the magnetic moment of the particle. It is indicated by the short blue arrow, with angle $\theta$ relative to the field, drawn parallel to the long blue arrow at the center.

Figure 2

Plot of the oscillation of $\theta$ in time. Taken for different initial configurations for a particle with $10\mathrm{nm}$ radius, a field strength of $5\mathrm{kA}/m$ and a frequency of $100\mathrm{kHz}$, the trajectories are compared. Here shown: with decoupled magnetization and a surfactant layer (hybrid), when completely magnetized and no surfactant layer $\left({\mathit{hybrid}}_{\text{bulk}}\right)$, for the magnetic system (immobilized easy axis), for the mechanical system (rigid) and the analytic solution. The dashed light-gray line indicates the amplitude of the field. The initial angle for all systems was ${\varphi }_0={90}^{\circ }$.

Figure 3

Showing the region of magnetic reversal in the parameter space of field strength and frequency. In the dark region (switching) the particle always settles into a switching steady state. This means the magnetization will continuously switch in the steady state according to the oscillation of the AMF. In the transient region the steady state depends on the initial angle between easy axis and field. The particle rotates without the switching of the magnetization in the light region (nonswitching). The horizontal dashed lines indicate half ($\approx 60\mathrm{kA}/m$) and the full anisotropy field ($\approx 120\mathrm{kA}/m$).

Figure 4

The hysteresis curve in the global frame of reference of an MNP during a steady state cycle of the AMF for four different field strengths ($50\mathrm{kA}/m\mathrm{to}80\mathrm{kA}/m$ from the inside to the outside) at $500\mathrm{kHz}$. The particle is simulated with the default parameters. The simulation includes mechanical rotation and therefore the curve captures both mechanical and magnetic processes. $M/M_{\text{s}}$ corresponds to the alignment of the magnetization with the field. Full saturation is reached when the magnetization is completely aligned with the field.

Figure 5

The energy loss contributions of magnetic and friction processes for some sample of frequencies (increasing frequencies from top to bottom). The losses for calculations without inertia are also included in a paler shade and with dashed lines.

Figure 6

The energy losses as generated by magnetic processes (a), friction (b), and the total losses (c) in the space of the field parameters. The data points are logarithmically scaled according to the frequency. The values are taken for an initial angle ${\varphi }_0={45}^{\circ }$. The indicator lines in the color bars show the minimum and maximum energy value. The color gradients follow the same logarithmically scaling. Thin gray lines at $60\mathrm{kA}/m$ and $120\mathrm{kA}/m$ indicate half and full anisotropy field strength respectively.

Figure 7

The difference of calculating the total energy loss per cycle directly via the hysteresis or as the sum of magnetic losses and frictional losses from the angular velocity [see Eq. (19)] shown as a color gradient. These results are taken from simulations with an initial angle ${\varphi }_0={45}^{\circ }$.

Figure 8

The specific absorption rate (SAR) in a phase space diagram with logarithmic color scaling. The values are taken for an initial angle ${\varphi }_0={45}^{\circ }$.

Figure 9

The amplitude of the rotation of the easy axis as a gradient in degrees and the average angle of that oscillation written in numbers during one cycle of the AMF both given in degrees. The dashed curve indicates the viscous relaxation limit. The initial angle in the simulations is ${\varphi }_0={45}^{\circ }$.