Dynamics of anisotropic superparamagnetic particles in a precessing magnetic field

A bifurcation diagram for anisotropic magnetic particles in a precessing magnetic field is analyzed. It is found that a synchronous regime in the case of a prolate particle exists for all precession angles of the magnetic field if the frequency of field rotation is below some critical value. An oblate particle has a synchronous regime in a limited range of precession angle. To understand the flow of suspensions of these particles in precessing fields, it is essential to take into account the differing dynamics of prolate and oblate particles.

Figure 1

Orientation angles of the vector $\vec{n}$.

Figure 2

Nullclines for the case with one fixed point. Dashed lines, nullclines $f_1(\alpha ,\beta ,\omega ,\vartheta )=0$; solid lines, $f_2(\alpha ,\beta ,\omega ,\vartheta )=0$. $\omega =1.9,\vartheta ={80}^{\circ }$.

Figure 3

Nullclines for the case with three fixed points. Dashed lines, nullclines $f_1(\alpha ,\beta ,\omega ,\vartheta )=0$; solid lines, $f_2(\alpha ,\beta ,\omega ,\vartheta )=0$. $\omega =2.3,\vartheta ={80}^{\circ }$.

Figure 4

Nullclines near saddle-node bifurcation. Dashed lines, nullclines $f_1(\alpha ,\beta ,\omega ,\vartheta )=0$; solid lines, $f_2(\alpha ,\beta ,\omega ,\vartheta )=0$. $\omega =1.94,\vartheta ={80}^{\circ }$.

Figure 5

Real parts of growth increments for small perturbations near fixed points as a function of the parameter $\omega$. Solid line, stable node; dotted line, saddle point; and dashed line, unstable node. $\vartheta ={71}^{\circ }$.

Figure 6

Real parts of growth increments for small perturbations near fixed points as a function of the parameter $\omega$. Solid line, stable node; dotted line, saddle point; and dashed line, unstable node. $\vartheta ={79}^{\circ }$.

Figure 7

Phase portrait in region III. $\omega =2.0,\vartheta ={79}^{\circ }$.

Figure 8

Phase diagram. The prolate particle has a synchronous regime in region $\text{II}\cup \text{III}$, and the oblate particle in region $\text{I}\cup \text{III}$. The point shows a codimension-2 bifurcation point with coordinates $(70.{53}^{\circ },1.73)$.

Figure 9

Phase trajectory of the prolate particle. $\omega =1.8,\vartheta ={80}^{\circ }$.

Figure 10

Phase trajectory of the oblate particle. $\omega =-2.2,\vartheta ={75}^{\circ }$.