Influence of field amplitude and dipolar interactions on the dynamic response of immobilized magnetic nanoparticles: Perpendicular mutual alignment of an alternating magnetic field and the easy axes

In this paper, the dynamic magnetic properties of an ensemble of interacting immobilized magnetic nanoparticles with aligned easy axes in an applied ac magnetic field directed perpendicular to the easy axes are considered. The system models soft, magnetically sensitive composites synthesized from liquid dispersions of the magnetic nanoparticles in a strong static magnetic field, followed by the carrier liquid's polymerization. After polymerization, the nanoparticles lose translational degrees of freedom; they react to an ac magnetic field via Néel rotation, when the particle's magnetic moment deviates from the easy axis inside the particle body. Based on a numerical solution of the Fokker-Planck equation for the probability density of the magnetic moment orientation, the dynamic magnetization, frequency-dependent susceptibility, and relaxation times of the particle's magnetic moments are determined. It is shown that the system's magnetic response is formed under the influence of competing interactions, such as dipole-dipole, field-dipole, and dipole–easy-axis interactions. The contribution of each interaction to the magnetic nanoparticle's dynamic response is analyzed. The obtained results provide a theoretical basis for predicting the properties of soft, magnetically sensitive composites, which are increasingly used in high-tech industrial and biomedical technologies.

Figure 1

Sketch of the sample considered here: a frozen configuration of magnetic nanoparticle positions and easy-axis orientations. The black dashed arrows show the easy-axis direction $\widehat{\mathbf{n}}$, while the red solid arrows represent the magnetic moment orientation $\widehat{\mathbf{m}}$; the ac magnetic field $\mathbf{H}$ is directed perpendicular to the easy magnetization axes.

Figure 2

Laboratory coordinate system. An immobilized particle's position is defined by radius vector $\mathbf{r}=r(sin\xi cos\psi ,sin\xi sin\psi ,cos\xi )$. The particle's orientation is given by the magnetic easy-axis vector $\widehat{\mathbf{n}}=(0,0,1)$ frozen in the particle's body. The particle magnetic moment's orientation is determined by the vector $\widehat{\mathbf{m}}=(sin\theta cos\varphi ,sin\theta sin\varphi ,cos\theta )$, which can differ from the easy-axis vector.

Figure 3

Real and imaginary parts of the susceptibility for noninteracting (dashed lines and open symbols) and interacting (solid lines and closed symbols) immobilized MNPs with ${\chi }_L=1$ at $\alpha =0.01$. Lines correspond to analytical expressions (16) and (17) from [40] obtained for low-field amplitudes. The symbols are from numerical solution of the FPE. The anisotropy constants are $\sigma =0$ (blue squares), $\sigma =3$ (red triangles), and $\sigma =5$ (black circles).

Figure 4

Numerical results of the real and imaginary parts of the susceptibility for noninteracting (open symbols) and interacting (closed symbols) immobilized MNPs with ${\chi }_L=0.5$ and $\sigma =0$. The numerical data are compared with the theory developed for mobile MNPs: Dashed lines correspond to the analytical expressions (18) and (19) from [42] obtained for noninteracting MNPs; solid lines show the theory for interacting MNPs (20) and (21) from [39]. The Langevin parameters are $\alpha =0.01$ (blue), $\alpha =1$ (red), and $\alpha =5$ (black).

Figure 5

Real and imaginary parts of the susceptibility for noninteracting (dashed lines) and interacting (solid lines) particles at $\alpha =1, \sigma =1$, and different values of the Langevin susceptibility ${\chi }_L=0.1$, 1, and 1.5.

Figure 6

Contour plots of the ratio $\chi (\omega \to 0)/{\chi }_{id}(\omega \to 0)$ depending on field amplitude $\alpha$ and magnetic anisotropy constant $\sigma$ at (a) ${\chi }_L=0.5$ and (b) ${\chi }_L=1.5$.

Figure 7

Contour plots $1/{\omega }_{\max }{\tau }_D={\tau }_{\mathrm{relax}}{\tau }_D$ depending on field amplitude $\alpha$ and magnetic anisotropy constant $\sigma$ for immobilized interacting MNPs at ${\chi }_L=1.5$.

Figure 8

Ratio of the effective relaxation times of magnetic moments ${\tau }_{\mathrm{relax}}/{\tau }_{\mathrm{relax}}^{id}$ depending on $\sigma$ for immobilized MNPs in an ac magnetic field with amplitude $\alpha =1$. The blue line corresponds to a system with a Langevin susceptibility ${\chi }_L=0.5$, the red line to ${\chi }_L=1.0$, and the black line to ${\chi }_L=1.5$.

Figure 9

Dynamic hysteresis loop $M^{*}\left(H^{*}\right)=M\left(H^{*}\right)/\rho m$, where $H^{*}=\alpha cos\omega t$ for noninteracting MNPs (dotted line) and interacting MNPs (solid line) with ${\chi }_L=1.5, \sigma =1$, and $\alpha =1$ at $\omega {\tau }_D=0.1$, 1, and 10.

Figure 10

Real and imaginary parts of the susceptibility with ${\chi }_L=1$ at field amplitude $\alpha =0.1$, 1, and 5 with (a) and (b) $\sigma =1$ and (c) and (d) $\sigma =5$.

Figure 11

Dynamic hysteresis loops for the immobilized interacting MNPs with ${\chi }_L=1$ and $\sigma =1$ in an ac magnetic field with frequency (a) $\omega {\tau }_D=0.1$ and (b) $\omega {\tau }_D=1$. Blue corresponds to the amplitude of the field $\alpha =1$, red to $\alpha =2$, and black to $\alpha =5$.

Figure 12

Ratio ${\tau }_{\mathrm{relax}}/{\tau }_D$ for immobilized interacting MNPs with ${\chi }_L=1$ as a function of field amplitude $\alpha$ at $\sigma =0.1$, 1, and 5.