Magnetic Particle Dynamics and Self-Assembly

The interest in magnetic nanoparticles and ferrofluids has grown substantially in recent years as their applications continue to proliferate. Current applications include field directed transport of biomolecules and therapeutic drugs, enhanced gene transfection, bioseparation and sorting, high-density magnetic data storage, ferrofluidic seals and pumps, microfluidic mixers, and highly sensitive magnetoresistive-based sensors, among many others. However, despite the widespread and growing use of magnetic nanoparticles, there are many fundamental aspects of their behavior that remain unknown. Areas of particular interest are use of field-directed assembly to form micro- and nano-structured magnetic media and the controlled manipulation of assembled structures using time-varying fields.

We are developing computational models for predicting the assembly of magnetic particles in high-gradient fields and the dynamics of particle-based microstructures in time-dependent fields. The model involves the numerical integration of a Langevin equation that accounts for interparticle dipole-dipole effects, (that drive particle assembly), viscous drag, fluid-mediated (hydrodynamic) particle-particle interactions and a stochastic force to account for Brownian motion. We use dynamic time-stepping and analytical expressions for the magnetic force to greatly accelerate the computations. We discuss the model in detail and demonstrate its use in the analysis of both high-gradient and time-varying magnetic particle systems.

The self-assembly of colloidal magnetic nanoparticles into extended spatial patterns can be controlled with nanoscale precision using magnetic template structures. In this approach, magnetic field-directed self-assembly is enhanced using soft-magnetic template elements that are embedded in a nonmagnetic substrate and magnetized using a uniform bias field. The combination of a uniform field with localized gradient fields produced by the template elements provides a unique force field that enables extraordinary control of particle placement during assembly. We have developed computational models that simulates the assembly process taking into account magnetic and hydrodynamic forces including interparticle interactions, Brownian diffusion, Van der Waals force and effects of surfactants. Our analysis shows that multilayered hexagonal packed particle superstructures can be assembled within milliseconds. It enables the assembly of complex structures, which opens up opportunities for the scalable fabrication of novel functional materials for a broad range of applications.

Figures 1, 2, 3, 4 and 5 illustrate the directed assembly of core-shell Fe₃O₄-SiO₂ nanoparticles using a soft-magnetic cylinder template element embedded in a nonmagnetic substrate. When a uniform field is applied, the cylinder becomes magnetized and produces a localized high gradient field that focuses the particles onto the top of the cylinder where they assemble.  Fig. 1 shows the template geometry and force profiles. Fig. 4 is an animation of the assembly process, which in this case results in a tapered four layer structure.  Fig. 5 shows details of the progression of the formation of four layers of hexagonal packed particles.  Experimental data showing the assembly of magnetic particles on a cylindrical template is shown in Fig. 6.

Figure 8 shows the directed assembly of core-shell Fe₃O₄-SiO₂ nanoparticles using a soft-magnetic ring structurtemplate embedded in a nonmagnetic substrate.* When a uniform field is applied, the ring becomes magnetized and produces a localized high gradient field that attracts the particles to the ring annulus where they assemble.  The combined use of a uniform field with an induced gradient-field provides localized regions of attractive and repulsive magnetic force (Fig 8. a,b,c,d) that enable nanoscale precision of particle placement. An interesting feature of the assembly process is that the particles assemble into a ring-like pattern and are evenly spaced due to a repulsive dipole-dipole force between neighboring particles that exists once the particles reach the substrate as shown in Figs. 9 and 10.

*Yellen, Benjamin B., Ondrej Hovorka, and Gary Friedman."Arranging matter by magnetic nanoparticle assemblers." Proceedings of the National Academy of Sciences of the United States of America 102, no. 25 (2005): 8860-8864.

*Xiaozheng Xue and Edward P. Furlani. "Template-assisted Nano-patterning of Magnetic Core-shell Particles in Gradient Fields." Physical Chemistry Chemical Physics (2014).

In a uniform magnetic field, magnetic particles become magnetized and assemble into chain-like microstructures due to dipole-dipole interactions. The assembled chains tend to align with the direction of the external field as shown in the simulation of Fig. 11.* In this analysis, a uniform field is applied upward in the z-direction through a micro-channel  that contains an initial random distribution of superparamagnetic beads and an array of spherical (gold colored) magnetic elements embedded in its base. In the presence of an applied field, the beads become magnetized and assemble into discrete chain-like structures.  These structures in turn, are attracted to the anchored magnetic elements. The analysis shows the self-assembly of particle chains and the subsequent attachment of the chains onto the embedded magnetic dipolar elements. The computational model takes into account fully-coupled particle fluid interaction wherein the fluid provides a viscous drag on particle motion and the moving particles, in turn, alter the fluid flow. The Flow-3D CFD program (www.flow3d.com  )was used for this analysis.

*Chenxu Liu, Xiaozheng Xue and Edward P. Furlani. "Numerical Analysis of Fully-Coupled Particle-Fluid Transport and Free-Flow Magnetophoretic Sorting in Microfluidic Systems." Proc. Int. NSTI Nanotech Conf., Washington DC, June 2014.

Time-varying magnetic fields can be used to manipulate magnetic particles for applications such as micromixing. Figure 12 shows a simulation of the behavior of a magnetic particle chain in the presence of a rotating magnetic field. The stability of the chain-like structure depends on the strength and frequency of the external magnetic field, viscosity of the surrounding fluid, and the properties of particles, etc. When the frequency is low, the chain is stable and there is a slight delay between the orientation of the chain and the external magnetic field, which depends on the length of the chain.

However, in a high frequency magnetic field, the chain breaks into two parts that rotate independently with the external field without delay. These chain segments temporarily reassemble into a longer chain and then break apart again in a time-wise periodic fashion as shown in the simulation.

• Yellen, Benjamin B., Ondrej Hovorka, and Gary Friedman. "Arranging matter by magnetic nanoparticle assemblers." Proceedings of the National Academy of Sciences of the United States of America 102, no. 25 (2005): 8860-8864.
• Xiaozheng Xue and Edward P. Furlani. "Template-assisted Nano-patterning of Magnetic Core-shell Particles in Gradient Fields." Physical Chemistry Chemical Physics (2014).
• Chenxu Liu, Xiaozheng Xue and Edward P. Furlani. "Numerical Analysis of Fully-Coupled Particle-Fluid Transport and Free-Flow Magnetophoretic Sorting in Microfluidic Systems."  Proc. Int. NSTI Nanotech Conf., Washington DC, June 2014.
• Xue, X., et al. (2015). "Self-Assembly of Crystalline Structures of Magnetic Core–Shell Nanoparticles for Fabrication of Nanostructured Materials." ACS Applied Materials & Interfaces 7(40): 22515-22524.
• Xue, X. Z. and E. P. Furlani (2015). "Analysis of the Dynamics of Magnetic Core-Shell Nanoparticles and Self-Assembly of Crystalline Superstructures in Gradient Fields." Journal of Physical Chemistry C 119(10): 5714-5726.

• Xiaozheng Xue (PhD)
• Jianchao Wang (MS)