Rotational properties of ferromagnetic nanoparticles driven by a precessing magnetic field in a viscous fluid

We study the deterministic and stochastic rotational dynamics of ferromagnetic nanoparticles in a precessing magnetic field. Our approach is based on the system of effective Langevin equations and on the corresponding Fokker-Planck equation. Two key characteristics of the rotational dynamics, namely the average angular frequency of precession of nanoparticles and their average magnetization, are of interest. Using the Langevin and Fokker-Planck equations, we calculate both analytically and numerically these characteristics in the deterministic and stochastic cases, determine their dependence on the model parameters, and analyze in detail the role of thermal fluctuations.

Figure 1

Schematic representation of the model. A nanoparticle of radius $a$ with frozen magnetization $\mathbf{M}$ rotates about the origin of the Cartesian coordinate system in a viscous fluid under the action of a precessing magnetic field $\mathbf{H}$.

Figure 2

Nanoparticle magnetization $\mu$ and nanoparticle angular frequency $\chi$ as functions of the reduced magnetic field frequency $\mathrm{\Omega }/h$. The solid curves represent the analytical results (4.5) and (4.7), and the symbols indicate the numerical results obtained at $t={10}^7{\tau }_1$ by solving Eqs. (4.1) for maghemite nanoparticles in water. The numerical values of $\mu$ and $\chi$, which are calculated at $h_z={10}^{-3}$ and $h_z=0$, respectively, do not depend on the initial values $\theta \left(0\right)$ and $\psi \left(0\right)$ of the polar and lag angles.

Figure 3

Examples of steady-state trajectories of the rotational motion of nanoparticles illustrating their dependence on the initial polar angle $\theta \left(0\right)$. The trajectories are obtained from the numerical solution of Eqs. (4.1) for $\mathrm{\Omega }=2.5,h=1,h_z=0,\psi \left(0\right)=0$, and $\theta \left(0\right)=0.4$ (a); $\theta \left(0\right)=\pi /2$ (b); and $\theta \left(0\right)=2.7$ (c). In this figure, the angles are measured in radians, the lag angle $\psi \left(t\right)$ is reduced to the interval $[0,\pi ]$, the arrows indicate the direction of time evolution, and the degenerate trajectory $b$ corresponds to the nanoparticle rotation in the $xy$ plane [i.e., $\theta \left(t\right)=\pi /2]$.

Figure 4

Dependence of $\sigma$ on $\mathrm{\Theta }$ for $\mathrm{\Omega }/h=0.5$ (1) and $\mathrm{\Omega }/h=1.5$ (2). The solid curves correspond to the theoretical result $\sigma =(\mathrm{\Omega }/h){sin}^2\mathrm{\Theta }$ and the symbols represent the numerical values of $\sigma$. The latter are obtained by solving Eqs. (5.11) for $h=2.5\times {10}^{-2},\mathrm{\Omega }=1.25\times {10}^{-2}$ (1) and $\mathrm{\Omega }=3.75\times {10}^{-2}$ (2); the other parameters are given in the text.

Figure 5

Average value of the reduced angular frequency of nanoparticles, $\langle \chi \rangle$, as a function of $\mathrm{\Omega }$ for $h_z=0,h=1$ and different values of the parameter $\kappa$. The solid curves show the results obtained by numerical solution of Eqs. (5.11), and the dashed curve represents Eq. (4.7), which corresponds to the noiseless case $(\langle \chi \rangle {|}_{\kappa =\infty }=\chi )$.

Figure 6

Dependence of $\varepsilon$ on $\mathrm{\Omega }$ for different values of $\kappa$. The simulation parameters are chosen as in Fig. 5.

Figure 7

Average value of the reduced angular frequency of precession of nanoparticles as a function of $\mathrm{\Omega }$ for $h=1,\kappa =5$, and different values of the parameter $h_z$.

Figure 8

Average value of the reduced angular frequency of precession of nanoparticles as a function of $h_z$ for $h=1,\kappa =5$ and two values of $\mathrm{\Omega }$. In this case, the characteristic frequency ${\mathrm{\Omega }}_0$ that separates the regimes of monotonic $(\mathrm{\Omega }<{\mathrm{\Omega }}_0)$ and nonmonotonic $(\mathrm{\Omega }>{\mathrm{\Omega }}_0)$ dependence of $\langle \chi \rangle$ on $h_z$ is equal to 0.78.

Figure 9

Frequency dependence of the average reduced magnetization for $h_z=0.3,h=1$, and different values of the parameter $\kappa$. The dashed curve represents the theoretical result (4.4) for the noiseless case.