# Simulation of Magnetically-Actuated Functional Gradient Nanocomposites
## 2 Theoretical Methods  
Magnetic particle transport can be treated as a drift diffusion process, supposing that a magnetic particle is extremely small with a diameter less than 100 nm in simulations and experiments. According to [PERSON]'s second law of diffusion, magnetic particle volume concentration \(n\) can be described by:  
\[\frac{\partial n}{\partial t}+\
abla\cdot J=S \tag{1}\]  
where \(J\) is the total flux vector of the particles; \(t\) is time; and \(S\) is the source term, which includes exchanges that occur at material interfaces or other chemical reactions. The total flux is the sum of chemical term \(J_{D}\) and drift term \(J_{F}\). \(J_{D}=-D\
abla n\) is the result of chemical concentration diffusion, and \(J_{F}=vn\) is the result of the action of magnetic forces on the particles, where \(D=\mu k_{B}T\) is the diffusion coefficient and \(v\) is particle drift velocity. Here, \(k_{B}\) is the Boltzmann's constant; \(\eta\) is the fluid viscosity (Stokes' approximation); and \(\mu\) is the mobility of a particle, which can be calculated by \(\mu=1/(6\pi\eta R_{p})\), where \(R_{P}\) is the effective hydrodynamic radius. Drift velocity \(v\) can be written as \(v=v_{f}+\mu\big{(}F_{m}+F_{g}\big{)}\), where \(v_{f}\) is the fluid velocity. \(F_{m}\) and \(F_{g}\) are the external magnetic and fluidic forces on particles, respectively. It should be noted that the gravitational force is usually negligible compared with the magnetic force. Therefore, the conservation equation (Equation (1)) can be rewritten as [28; 29]:  
\[\frac{\partial n(r,t)}{\partial t}=\frac{kT}{\big{(}6\pi\eta R_{p}\big{)}} \
abla(\
abla n(r,t))-\frac{1}{6\pi\eta R_{p}}\
abla\cdot(F(r)n(r,t)) \tag{2}\]When the particle concentration is high, it needs to be taken into account in the drift-diffusion process that particle motion actually takes place on spatial sites occupied by the atoms of materials. Hence, the governing equation should be described as:  
\[\frac{\partial n(r,t)}{\partial t}\ =\ \frac{kT}{\left(6\pi\eta R_{p}\right)} \
abla(f(n)\
abla n(r,t))\ -\ \frac{1}{6\pi\eta R_{p}}\
abla\cdot(f(n)F(r)n(r,t)) \tag{3}\]  
where \(f(n)\) is the factor that can be assumed to be \((1\ -\ n)\). The magnetic force is calculated using the \"effective\" dipole moment method, in which the particle is modeled as an \"equivalent\" point dipole with an effective moment \(m_{p,\mathit{eff}}\). The magnetic force \(F_{m}\) on the dipole is given by:  
\[F_{m}=\mu_{f}\Big{(}m_{p,eff}\cdot\
abla\Big{)}H_{a} \tag{4}\]  
where \(\mu_{f}\) is the permeability of the fluid and \(H_{a}\) is the externally applied magnetic field at the center of the particle. Moment \(m_{p,\mathit{eff}}\) can be determined using a magnetization model that takes into account self-demagnetization and the magnetic saturation of the particles. The dipole moment is \(m_{p,\mathit{eff}}=V_{p}M_{p}\), where \(V_{p}\) is the volume of the particle, and:  
\[M_{p}=\chi_{p}H_{in}=\chi_{p}\Big{(}H_{a}-H_{demag}\Big{)} \tag{5}\]  
where \(M_{p}\) is the magnetization of the particle; \(\chi_{p}=\mu_{p}/\mu_{0}-1\) is the susceptibility of the particle; and \(\mu_{p}\) is permeability. \(H_{in}\) is the internal magnetic field, which can be calculated by applied magnetic field \(H_{a}\) minus demagnetization field \(H_{demag}\). \(H_{demag}\) is the self-demagnetization field that accounts for the magnetization of the particle. Here, \(H_{demag}=M/3\) is used for a uniformly magnetized spherical particle with magnetization \(M\) in free space. Thus:  
\[m_{p,eff}=V_{p}f(H_{a})H_{a} \tag{6}\]  
where \(f(H_{a})\) is a function of magnetic susceptibilities, and can be expressed by [24]:  
\[f(H_{a})=\left\{ \begin{array}{ll}\frac{3(\chi_{p}-\chi_{f})}{( \chi_{p}-\chi_{f})+3},&\quad H_{a}<\bigg{(}\frac{(\chi_{p}-\chi_{f})+3}{3\chi _{p}}\bigg{)}M_{sp}\\ M_{sp}/H_{a},&\quad H_{a}\geq\bigg{(}\frac{(\chi_{p}-\chi_{f})+3}{3\chi_{p}} \bigg{)}M_{sp}\end{array} \right.