Dynamics of magnetic particles near a surface: Model and experiments on field-induced disaggregation

Magnetic particles are widely used in biological research and bioanalytical applications. As the corresponding tools are progressively being miniaturized and integrated, the understanding of particle dynamics and the control of particles down to the level of single particles become important. Here, we describe a numerical model to simulate the dynamic behavior of ensembles of magnetic particles, taking account of magnetic interparticle interactions, interactions with the liquid medium and solid surfaces, as well as thermal diffusive motion of the particles. The model is verified using experimental data of magnetic field-induced disaggregation of magnetic particle clusters near a physical surface, wherein the magnetic field properties, particle size, cluster size, and cluster geometry were varied. Furthermore, the model clarifies how the cluster configuration, cluster alignment, magnitude of the field gradient, and the field repetition rate play a role in the particle disaggregation process. The simulation model will be very useful for further in silico studies on magnetic particle dynamics in biotechnological tools.

Figure 1

Overview of interactions of the magnetic particles. (a) Due to magnetic fields and magnetic field gradients, magnetic particles show dynamic behavior, interacting with the liquid, physical boundaries, and other particles. (b) Magnetic particles undergo thermal collision forces (i.e., exhibit Brownian motion), gravity, hydrodynamic interactions, and magnetic forces due to magnetic gradients and magnetic dipole-dipole interactions, which can be attractive or repulsive. The latter interaction has been exploited to disaggregate magnetic particle clusters [8].

Figure 2

Measured and simulated disaggregations of a two-particle cluster at a physical surface in an aqueous medium. Schematic (a) side view and (b) top view are shown indicating the vectors corresponding to the magnetic field, magnetic moments, and the magnetic dipole-dipole forces. (c) Measured top-view images at different times ($\mathbf{B}=15\mathrm{mT}{\widehat{\mathbf{e}}}_z$; $\nabla \mathbf{B}=-3.2T/\mathrm{m}{\widehat{\mathbf{e}}}_z$). (d) Simulated top-view images at different times. (e) In-plane center-to-center distance between the particles as obtained from the top-view experimental (black open circles) and simulated (red lines) data. Measurements were repeated on the same particle pair. The inset shows a log-log plot of the same data.

Figure 3

Computed particle trajectories at different initial tilt angles with respect to the surface. ($\mathbf{B}=15\mathrm{mT}{\widehat{\mathbf{e}}}_z$; $\nabla \mathbf{B}=-2.8\mathrm{T}/\mathrm{m}{\widehat{\mathbf{e}}}_z$) (a1) Side view and (a2) top view of simulations of 2.8 μm particles (M-270). (b1) Side view and (b2) top view of simulations of 1 μm particles (MyOne). The sketched spheres represent initial positions of the particles.

Figure 4

Computed probability of successful separation as a function of the initial tilt angle with respect to the surface. (a) Data for different particles using a downward force of $|{\mathbf{F}}_{\mathrm{down}}|$ = 0.26 pN. (b) Data for 2.8 μm sized particles (M-270) at different downward forces; i.e., combined magnetic gradient forces and buoyancy. (c) Comparison of data for 2.8 μm sized particles (M-270) in case the hydrodynamic interactions (H.I.s) between the particles and/or with the surface were excluded from the simulations. Each data point corresponds to the mean of 200 simulations. Particles that did not reform into a cluster within 10 s are counted as successful separations. The (small) standard error of the mean is represented by the shaded area around the data.

Figure 5

Disaggregation of magnetic particle chains at a physical surface. (a) Snapshots of the simulated disaggregation of a ten-particle chain. The disaggregation protocol leads to single particles and short chains oriented along the surface normal. (b) Probability to obtain chains oriented along the surface normal with $N$ particles after 10 s of application of a magnetic field along the surface normal ($\mathbf{B}=15\mathrm{mT}{\widehat{\mathbf{e}}}_z$; $|{\mathbf{F}}_{\mathrm{down}}|$ = 0.26 pN). For each initial chain size, 50 simulations were carried out. The standard error of the mean is represented by the shaded area. The black squares correspond to the probability to completely disaggregate a chain into separate single particles (i.e., after 10 s, all remaining clusters have a chain length of 1). The percentages do not add up to 100% because, in a single simulation, chains of different lengths may be obtained, e.g., $N$ = 1–3 [see panel (a)]. (c) Comparison of the disaggregation efficiency as found in simulation and in experiment. Experimental data points correspond to at least four measured chains, i.e., for smaller chains, more measurements (e.g., 20 two-particle clusters) were performed.

Figure 6

Disaggregation of sheetlike magnetic particle clusters at a physical surface. (a) Snapshots of the simulated disaggregation of a 19-particle sheetlike cluster. Besides obtaining single particles, smaller chains oriented along the surface normal are also obtained. (b) Probability to obtain chains with $N$ particles oriented along the surface normal after 10 s of application of a magnetic field along the surface normal ($\mathbf{B}=15\mathrm{mT}{\widehat{\mathbf{e}}}_z$; |F${}_{\mathrm{down}}$| = 0.26 pN). For each initial sheet size, 50 simulations were carried out. The standard error of the mean is represented by the shaded area. The black squares indicate the probability to completely disaggregate a sheetlike cluster into single particles. (c) Disaggregation efficiency computed from the simulation data.

Figure 7

Simulation of the disaggregation of a particle chain consisting of 23 particles by alternatingly applying an in-plane field (5 mT for 0.5 s) and an out-of-plane field (15 mT for 2 s) with respect to the surface, to align the (fragmented) chains to the surface and to disaggregate the chains into smaller chain fragments or separated particles, respectively. To keep the particles near the surface, a downward force of 0.26 pN was applied on each particle, corresponding to the sum of a field gradient force ($-2.8\mathrm{T}/\mathrm{m}{\widehat{\mathbf{e}}}_z$ and 0.22 pN) and a buoyancy force (0.04 pN).