# Simulation of Magnetically-Actuated Functional Gradient Nanocomposites
## 2 Theoretical Methods
\tag{7}\]  
where \(\chi_{p}\) and \(\chi_{f}\) are the magnetic susceptibilities of the particle and fluid, and \(M_{sp}\) is the saturation magnetization of the particle. Thus, the magnetic force on the magnetic particle can be written as:  
\[F_{m}=\mu_{f}V_{p}f(H_{a})(H_{a}\cdot\
abla)H_{a} \tag{8}\]  
For example, the magnetic field distribution (\(H_{az}\)) and magnetic force (\(F_{mz}\)) along the axial of a cylindrical rare-earth magnet can be calculated using [30]:  
\[H_{az}(z)\ =\ \frac{M_{s}}{2}\left(\frac{z+L_{m}}{\sqrt{\big{(}z+L_{m}\big{)}^ {2}+R_{m}^{2}}}\ -\ \frac{z}{\sqrt{z^{2}+R^{2}}}\right) \tag{9}\]  
\[ \begin{array}{l}F_{mz}(z)=\mu_{0}V_{p}f(H_{a})\frac{M_{s}^{2}R_{m}^{2}}{4} \big{(}\frac{z+L_{m}}{\big{[}(z+L_{m})^{2}+R_{m}^{2}\big{]}^{2}}+\\ \frac{z}{\big{(}z^{2}+R_{m}^{2}\big{)}^{2}}\ -\ \frac{(z+L_{m})\big{[}(z+L_{m})^{2}+R_{m}^{2} \big{]}+\big{[}z^{2}+R_{m}^{2}\big{]}}{\big{(}z^{2}+R_{m}^{2}\big{)}^{3/2}[(z+L _{m})^{2}+R_{m}^{2}\big{]}^{3/2}}\big{)}\end{array}  \tag{10}\]where \(z\) is the distance from the sample to the magnet, and \(L_{m}\) and \(R_{m}\) are the length and radius of the magnet, respectively. If we need to estimate the radial force, the method in Appendix A can be used to calculate both radial and axial force. Assuming that the scale of the sample is relatively small (-20 \(\upmu\)m), the strength of the magnetic force can be seen as a uniform distribution due to the fact that distance \(Z_{0}\) is constant. \[ \begin{split} F_{mz0}(z)&=\mu_{0}V_{p}f(H_{a}) \frac{M_{\mathrm{d}}^{2}R_{m}^{2}}{4}(\frac{z_{0}+L_{m}}{\left[\left(z_{0}+L_{ m}\right)^{2}+R_{m}^{2}\right]^{2}}+\frac{z_{0}}{\left(z_{0}^{2}+R_{m}^{2}\right)^{2}} \\ &\qquad-\frac{\left(z_{0}+L_{m}\right)\left[\left(z_{0}+L_{m} \right)^{2}+R_{m}^{2}\right]+z_{0}\left(z_{0}^{2}+R_{m}^{2}\right)}{\left(z_{0 }^{2}+R_{m}^{2}\right)^{3/2}\left[\left(z_{0}+L_{m}\right)^{2}+R_{m}^{2}\right] ^{3/2}})\end{split}  \tag{11}\]  
Substituting Equation (11) to Equation (3), the drift-diffusion equation can be simplified as:  
\[\frac{\partial n(r,t)}{\partial t}=\frac{kT}{\left(6\pi\eta R_{p}\right)} \
abla(f(n)\
abla n(r,t))-\frac{F_{mz0}}{6\pi\eta R_{p}}\
abla(f(n)n(r,t)) \tag{12}\]  
Since the particle concentration is high, the drift-diffusion transport equation should be modified, compared with the dilute case. First, the mobility coefficient should be modified due to particle accumulation. According to the literature [31], in a polymer liquid, the mobility of nanoparticles can be affected by several factors, including particle size, solvent viscosity, and solute concentration. The mobility of particles decreases with solution volume fraction with the power relation \(D(n)\sim\)\(n^{-\alpha}\), where \(\alpha\) is the parameter to be determined under different experimental conditions. During the accumulation of magnetic particles in space, there will be a long-range interaction coming from the magnetic charge interaction. Here, we consider the magnetic charge interaction as a demagnetization field (\(H_{md}\)) in the whole matrix. Then, the magnetic force on the dipole is:  
\[F_{m}=\mu_{f}\Big{(}m_{p,eff}\cdot\
abla\Big{)}(H_{a}+H_{md}) \tag{13}\]  
The dipole moment of the magnetic particle \(m_{p,eff}=V_{p}M_{p}\) should be modified, and:  
\[M_{p}=\chi_{p}H_{in}=\chi_{p}\Big{(}H_{a}+H_{md}-H_{demag}\Big{)} \tag{14}\]  
The magnetic force on a magnetic particle can be written as:  
\[F_{m}=\mu_{f}V_{p}f(H)(H\cdot\
abla)H \tag{15}\]  
where \(H\) is the total magnetic field and \(H=H_{a}+H_{md}\).