## 2 Materials and Methods
### Magnetic Field Forces and Magnetization Models  
In general, the magnetic force on a magnetic particle was calculated by the following Equation (6) [23]  
\[\mathbf{F}_{M}=V_{p}\mu_{0}\mathbf{M}\
abla\mathbf{H}, \tag{6}\]  
where \(V_{p}\) is the volume of the particle, \(M\) is the magnetic moment per unit volume of the magnetic particle in the magnetic field, i.e., the magnetization strength, and \(\mathbf{H}\) is the magnetic field strength. According to the Equation (6), the magnetic field force on a magnetic particle is not only influenced by the external magnetic field strength \(\mathbf{H}\), but the magnetization \(\mathbf{M}\) of the magnetic particle itself also affects the magnitude of the magnetic field force. Regarding the magnetization model of magnetic particles in a magnetic field, many researchers simply described it using Equation (7) [14; 24; 25; 26]. This equation considered that the magnetization strength of a magnetic particle varies linearly with the external magnetic field strength, and the slope \(\chi\) is the magnetic susceptibility of the magnetic particle. \[\mathbf{M}=\chi\mathbf{H}, \tag{7}\]  
In fact, the magnetization of the magnetic particles can reach saturation when the external magnetic field is sufficiently strong, i.e., the magnetization strength will no longer vary linearly with the magnetic field strength; at that time, Equation (7) is no longer applicable either. To take into account the effect of the magnetic saturation phenomenon, we proposed a magnetization model based on the actual magnetic characteristic curves of magnetic particles. Taking a common superparamagnetic Fe\({}_{3}\)O\({}_{4}\) nanoparticle as an example, its diameter is about 300 nm. The magnetic hysteresis loop was measured by a magnetic property measurement system (VSM, 7404, LakeShore, Columbus, OH, USA), as shown in Figure 2. The Fe\({}_{3}\)O\({}_{4}\) nanoparticles exhibited typical superparamagnetism, and the remanence was almost zero. The magnetization process of magnetic particles can be described in two stages: the first stage was defined as the initial magnetization stage, and the second stage was defined as the magnetization saturation stage. In the initial magnetization stage, the magnetization intensity of the magnetic particle was approximately linear with increasing external magnetic field strength. In the magnetization saturation stage, the magnetization intensity of the magnetic particle no longer changed with increasing external magnetic field strength. Here, two straight lines were plotted tangent to the magnetization curves of the two stages, as shown in Figure 2, and the coordinates of the intersection of the tangent lines were defined as (\(H_{0}\), \(M_{0}\)). When \(\mid\!\!\!\!\!\!\!\!\!\!\!\!\  
susceptibility \(\chi\) was equal to 7.74. These values will be used for further discussion and analysis in the following section. \[\mathbf{M}=\left\{ \begin{array}{ll}\chi\mathbf{H}_{\mathbf{\mathrm{r}}},&\mathbf{|H|<H_{0}}\\ \frac{M_{\mathbf{\mathrm{r}}}\mathbf{H}}{|\mathbf{H}|},&\mathbf{|H|\geq H_{0}}^{\prime}\\ \end{array} \right. \tag{8}\]  
Then, the magnetic field force equation becomes  
\[\mathbf{F}_{\mathbf{M}}=\left\{ \begin{array}{ll}V_{p}u_{0}\chi(H\
abla\mathbf{H}),&\mathbf{|H| <H_{0}}\\ V_{p}u_{0}\frac{M_{\mathbf{\mathrm{r}}}}{|\mathbf{H}|}(H\
abla\mathbf{H}),&\mathbf{|H|\geq H_{0 }}^{\prime},\\ \end{array} \right. \tag{9}\]  
The magnetic volume force vector in a two-dimensional coordinate system can be described by the following Equations (10) and (11)  
\[\mathbf{f}_{\mathbf{M}\mathbf{x}}=\left\{