# Simulation of Magnetically-Actuated Functional Gradient Nanocomposites
## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c} \hline
**Temperature** & **Magnet Length** & **Magnet Radius** & **Time Step** & **Grid Size** \\ \(\mathbf{T}\)**(K)** & \(\mathbf{L_{m}}\)**(mm)** & \(\mathbf{R_{m}}\)**(mm)** & \(\mathbf{dt}\)**(s)** & \(\mathbf{dx}\)**(m)** \\ \hline
300.0 & 5.0 & 3.0 & \(6\times 10^{-4}\) & \(2\times 10^{-7}\)-\(2\times 10^{-5}\) \\ \hline
**Fluid Viscosity** & **Magnet Saturation Magnetization** & **Particle Saturation Magnetization** & **Particle Radius** \\ \(\mathbf{\eta}\)**(Ns/m\({}^{2}\))** & \(\mathbf{M_{s}}\)**(A/m)** & \(\mathbf{M_{sp}}\)**(A/m)** & \(\mathbf{R_{p}}\)**(nm)** \\ \hline
0.225 & \(9.55\times 10^{5}\) & \(4.78\times 10^{5}\) & 15.0 \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. can almost be assembled in the right half of the sample. If the magnetic force is smaller (0.4 fN), due to the diffusion effect, the stable particle concentration gradually grows, from left to right. If the magnetic force is smaller than 0.02 fN, there is almost no change in the particle concentration in the sample. To compare the results with the experiment, we changed the particle concentration to the numbers of particles existing in the domain (Figure 2). The distribution of particles in Figure 2a-c comes from the assumption that the magnetic force only exists in the \(z\) direction; thus, the distribution of the cross-section of the sample (the plane perpendicular to the \(x\) direction) can be simplified to a one-dimensional model, which corresponds with the distribution in Figure 1d. It is easy to see the difference among the three states, and the results of the three different concentration distributions were consistent with experiments. It is obvious that if the sample under the magnet is not at the center point, there will be a radial force, as seen in Figure 1c. Here, we consider the sample close to the magnet as having both the radial and axial force components. For simplicity, we assume that the magnetic force is constant in the \(y\) and \(z\) directions to study magnetic particle accumulation evolution. In Figure 3, the particle concentration shows high inhomogeneity when the force exists in both the \(z\) and \(y\) directions. If the force component in the \(y\) direction is zero, the accumulation will only exist in the \(z\) direction. If the force in the \(y\) direction is small (Figure 3b), accumulation in the \(y\) direction will exist. Due to the drift in the \(z\) and \(y\) directions, most magnetic particles will accumulate in the upper right corner. When the force in the \(y\) direction increases, the accumulation in the upper right corner will be larger, until it is sufficient to destroy the concentration gradient and mechanics gradient. This is the reason why in experiments, a larger cylindrical magnet is needed to make the magnetic force homogeneous in the sample in the \(y\) direction (radial direction) [21]. However, it is interesting that by adjusting the magnetic force in the \(z\) and \(y\) directions, one can precisely control the highly inhomogeneous or particular magnetic particle concentration distributions, and thus the high mechanics inhomogeneity in a more complex design. Figure 1: (**a**) Magnetic field and force distribution in the \(z\) direction of magnet; (**b**,**c**) magnetic force in the axial and radial directions in different locations; (**d**) concentration distribution after 1.2 \(\times\) 10\({}^{3}\) s with different applied forces. As the sample scale increases, the diffusion process will not be the dominant process, and the magnetic force will change at different positions of the sample. Therefore, the force will gradually decrease when the distance from the magnet increases. Here, we consider the distance between the sample and the bottom of magnet as 2 mm.