# Simulation of Magnetically-Actuated Functional Gradient Nanocomposites
## 1 Introduction  
Biological materials with functional gradient designs within interfacial and surface regions are finding increasing use in bioapplications [1; 2; 3; 4; 5; 6; 7; 8; 9]. These can be inspired by many biomaterials in nature that are reported to be mechanically gradient, such as tree frog toe pads, squid beaks, sandworm jaws [10; 11; 12; 13], etc. Meanwhile, interface/surface failures for synthetic materials are ubiquitous in various fields, including the coating industry [14; 15], biomedical implants and restorations [16; 17], flexible and wearable electronics [13; 18], additive manufacturing, bioinspired adhesion [19; 20], etc. Most of these failures come from mechanical/thermal mismatches in integrated systems containing dissimilar materials or structures. However, it remains a challenging scientific and technological task to transplant these concepts of functional gradients into the design of synthetic interfaces/surfaces for extreme performance. It was found that a variety of biomedical applications can be achieved in systems with magnetic particles [5]. Recently, [PERSON] et al. developed a new manufacturing technique for controllable functional gradient nanocomposites (FGNCs) via use of combined magnetic and mechanical actuations; the results show that a near one order of magnitude smooth change of the local elastic moduli (\(\sim\)2.2 GPa to \(\sim\)9 GPa) can be enabled within regions as narrow as 10 microns [21]. These results indicate that precise control over the spatial distribution of magnetic particles in a monomeric fluid is important, and further understanding is required.  
The evolution of magnetic particles in an applied field can be affected by several factors, including magnetic field, viscous drag, gravity, Brownian dynamics, particle/fluid interactions,and particle-particle interactions (dipole-dipole interactions) [22; 23; 24]. Two different approaches (Eulerian approach and Lagrangian approach) are used to model magnetic-particle transport in fluid, based on whether the Brownian motion has a significant impact on particle dynamics. In the Eulerian approach, particle evolution is characterized by particle concentration, and it is governed by a diffusion equation that accounts for both magnetic force-induced drift and Brownian diffusion [25; 26; 27]. In the Lagrangian approach, particles are treated as discrete entities, as in molecular dynamics, and the Newtonian equation of each particle is solved independently at every time step [28]. Generally, the effects of Brownian motion are negligible in the Lagrangian approach. Based on the Eulerian approach, the transport and accumulation of magnetic nanoparticles in a passive magnetophoretic system are studied [29], which is commonly used to discover the accumulation based on the assumption of ultra-low concentrations.  
However, precise control of time-dependent spatially varying particle concentrations is needed in an experiment to acquire good, functional gradient materials, especially when the concentration is not low enough. To acquire tunable mechanical gradients in FCNCs, a permanent magnet was used to drive the magnetic-responsive nanofillers in a monomeric fluid until a steady-state concentration gradient was formed. In order to understand and design experiments, a more precise theoretical prediction is needed. Here, the drift-diffusion model, with a high concentration, was proposed to simulate functional gradient nanocomposites in polymers, in order to control the accumulation of magnetic particles in space at different times. Different magnetic forces will influence not only the accumulation of particles, as proposed in previous research, but also particle distribution in different times or spaces. At the same time, particle motion will be affected by a demagnetization field, which is induced by magnetic particle accumulation. In this paper, we completed a series of simulations in order to obtain the gradient concentration of nanoparticles at different scales and with different force distributions.