Influence of a bias dc field and an ac field amplitude on the dynamic susceptibility of a moderately concentrated ferrofluid

In this paper, we study the effect of a bias dc field on the dynamic response of a moderately concentrated ferrofluid to an ac magnetic field of arbitrary amplitude. The ferrofluid is modeled by an ensemble of interacting moving magnetic particles; the reaction of particle magnetic moments to ac and dc magnetic fields occurs according to the Brownian mechanism; and the ac and dc magnetic fields are parallel. Based on a numerical solution of the Fokker-Planck equation for the probability density of the orientation of the magnetic moment of a random magnetic particle, dynamic magnetization and susceptibility are determined and analyzed for various values of the ac field amplitude, the dc field strength, and the intensity of dipole-dipole interactions. It is shown that the system's magnetic response is formed under the influence of competing interactions, such as dipole-dipole, dipole-ac field, and dipole-dc field interactions. When the energies of these interactions are comparable, unexpected effects are observed: the system's susceptibility can either increase or decrease with increasing ac field amplitude. This behavior is associated with the formation of nose-to-tail dipolar structures under the action of the dc field, which can hinder or promote the system's dynamic response to the ac field. The obtained results provide a theoretical basis for predicting the dynamic properties of ferrofluids to improve their use in biomedical applications, such as, in magnetic induction hyperthermia.

Figure 1

Laboratory coordinate system.

Figure 2

Dynamic hysteresis loop of an ensemble of moving interacting magnetic particles with ${\chi }_L=1$; the results are presented for four sets of frequencies $\omega {\tau }_B$ and ac field amplitudes ${\alpha }_{ac}$ (a) $\omega {\tau }_B=1, {\alpha }_{ac}=1$, (b) $\omega {\tau }_B=1, {\alpha }_{ac}=5$, (c) $\omega {\tau }_B=0.1, {\alpha }_{ac}=1$, (d) $\omega {\tau }_B=0.1, {\alpha }_{ac}=5$ for different values of the bias dc field ${\alpha }_{dc}=0$ (red), ${\alpha }_{dc}=1$ (blue), ${\alpha }_{dc}=5$ (black).

Figure 3

The effect of the dipole-dipole interaction on the dynamic hysteresis loop of an ensemble of moving magnetic particles with ${\chi }_L=1.5$ in an ac field with frequency $\omega {\tau }_B=0.1$, amplitude ${\alpha }_{ac}=1$ and a bias dc field ${\alpha }_{dc}=0$ (red), ${\alpha }_{dc}=1$ (blue), ${\alpha }_{dc}=5$ (black). Solid lines are used for the system with interactions; dashed lines correspond to the system without interactions.

Figure 4

Dependence of dynamic susceptibility at $\omega \to 0$ on the Langevin susceptibility ${\chi }_L$. The points correspond to numerical results for ${\alpha }_{ac}=0.01$ and ${\alpha }_{dc}=0$ (red dots); ${\alpha }_{ac}=1$ and ${\alpha }_{dc}=3$ (blue dots); ${\alpha }_{ac}=2$ and ${\alpha }_{dc}=2$ (black dots); ${\alpha }_{ac}=3$ and ${\alpha }_{dc}=1$ (green dots). The black dashed line corresponds to MMF1 [51]; the black solid line corresponds to the Langevin theory.

Figure 5

The effect of the intensity of the bias dc field on the susceptibility of interacting particles with ${\chi }_L=0.5$ at low frequencies of the ac field with the amplitude ${\alpha }_{ac}=0.01$ (red solid line), ${\alpha }_{ac}=1$ (blue solid line), ${\alpha }_{ac}=3$ (green solid line), ${\alpha }_{ac}=5$ (black solid line), ${\alpha }_{ac}=10$ (purple solid line). Solid lines denote numerical results, dashed lines are used for theory [43].

Figure 6

The effect of the amplitude of the ac field at low frequencies ($\omega \to 0$) on the susceptibility of an ensemble of interacting (solid lines) and noninteracting (dot-and-dash lines) magnetic particles with ${\chi }_L=0.5$, under the influence of the dc magnetic field with intensity: ${\alpha }_{dc}=0$ (red), ${\alpha }_{dc}=1$ (blue), ${\alpha }_{dc}=3$ (green), ${\alpha }_{dc}=5$ (black), ${\alpha }_{dc}=10$ (purple). Solid and dot-and-dash lines denote numerical results, dashed line is used for the approximation formula from [40].

Figure 7

Numerical results of dynamic susceptibility of magnetic particles to an ac field with amplitude ${\alpha }_{ac}=1$ for different values of the Langevin susceptibility (a) ${\chi }_L=1$, (b) ${\chi }_L=2$. The intensities of the bias dc field are ${\alpha }_{dc}=0$ (red), ${\alpha }_{dc}=1$ (blue), ${\alpha }_{dc}=5$ (black). The solid line corresponds to a system with interactions, the dashed line indicates a system without interactions.

Figure 8

Numerical results of dynamic susceptibility of an ensemble of interacting magnetic particles with ${\chi }_L=1$ to an ac field with amplitude ${\alpha }_{ac}=0.01$ (red), ${\alpha }_{ac}=2$ (blue), ${\alpha }_{ac}=4$ (black), and ${\alpha }_{ac}=6$ (green). The intensities of the bias dc field are (a) ${\alpha }_{dc}=1$, (b) ${\alpha }_{dc}=3$, (c) ${\alpha }_{dc}=5$.

Figure 9

Numerical results of dynamic susceptibility for a suspension of the magnetite particles with diameter $d=10$ nm and volume concentration $\phi =0.08$ in applied ac and dc magnetic fields with $H_{ac}=H_{dc}=12{\mathrm{kAm}}^{-1}$ at $T=270$ K (green), $T=297$ K (blue), and $T=330$ K (red).