Numerical calculation of interaction forces between paramagnetic colloids in two-dimensional systems

Typically the force between paramagnetic particles in a uniform magnetic field is described using the dipolar model, which is inaccurate when particles are in close proximity to each other. Instead, the exact force between paramagnetic particles can be determined by solving a three-dimensional Laplace's equation for magnetostatics under specified boundary conditions and calculating the Maxwell stress tensor. The analytical solution to this multi-boundary-condition Laplace's equation can be obtained by using a solid harmonics expansion in conjunction with the Hobson formula. However, for a multibody system, finite truncation of the Hobson formula does not lead to convergence of the expansion at all points, which makes the approximation physically unrealistic. Here we present a numerical method for solving this Laplace's equation for magnetostatics. This method uses a smoothed representation to replace all the boundary conditions. A two-step propagation is used to dramatically accelerate the calculation without losing accuracy. Using this method, we calculate the force between two paramagnetic particles in a uniform and a rotational external field and compare our results with other models. Furthermore, the many-body effects for three-particle, ten-particle, and 24-particle systems are examined using the same method. We also calculate the interaction between particles with different magnetic susceptibilities and particle diameters. The Laplace's equation solver method described in this article that is used to determine the force between paramagnetic particles is shown to be very useful for dynamic simulations for both two-particle systems and a large cluster of particles.

Figure 1

Plot showing the calculation of the magnetic field strength around (a) one paramagnetic particle and (b) two paramagnetic particles in a uniform external magnetic field of 6 G in the $x$ direction. The scales correspond to the value of the magnetic field in units of A/m. (c) Schematic of the two grids and parameters used in the numerical calculation of the magnetic field for a two-particle system.

Figure 2

(a), (b) The contour plots of the $\Delta H_x$ and $\Delta H_y$ middle $x$-$y$ plane for $r$ = 1.4$D$ and $\alpha =0$. (c), (d) The contour plots of the $\Delta H_x$ and $\Delta H_y$ middle $x$-$y$ plane for $r$ = 1.1$D$ and $\alpha =0$. The scales correspond to the value of the magnetic field in units of A/m.

Figure 3

Magnetic force and total force on a particle in a particle pair system in a uniform field for (a) $B_0$ = 6 G and α = 0°, (b) $B_0$ = 6 G and α = 90°, (c) $B_0$ = 2 G and α = 0°, and (d) $B_0$ = 2 G and α = 90°. A dashed line (cyan) is used for DM, a solid line (blue) for MDM, a large dotted line (magenta) for LES, and a small dotted line (black) for the polynomial fit. All insets show the ratio of the LES results over the DM results for a large $r/D$ range.

Figure 4

The pair magnetic force in a rotational magnetic field (a) at $r/D$ = 1.4 for different angles for a particle pair, (b) at $r/D$ = 1.1 for different angles for a particle pair, (c) angular average for a particle pair, and (d) angular average for a particle pair and three-particle system. In (c), a dashed line (cyan) is used for the DM results, a solid line (blue) for the MDM results, a line with circular markers (magenta) for the LES results, and a small dotted line (black) for the polynomial fit. The inset shows the ratio of results from LES over those from DM. In (d), the difference between a pair and a three-particle system is compared by a solid line (blue) for the MDM pair results and a small dotted line (green) for MDM three-particle results, and a line with circular markers (magenta) for the LES pair results line with square markers (yellow) for the LES three-particle results. The inset shows the ratio of effective pair force results from LES on the three-particle system over pair force results from LES.

Figure 5

Schematics of a (a) ten-particle system and (b) 24-particle system used to calculate force. The many-body effect on different particles with $r/D$ = 1.1 in a rotational magnetic field for a (c) ten-particle system and (d) 24-particle system. (e) Magnetic field strength on particles in a three-particle chain. The scales correspond to the value of the magnetic field in units of A/m. Magnetic force in the 24-particle system in a rotational magnetic field for different angles with $r/D$ = 1.1 for (f) exterior particle 1 and (g) interior particle 8. In (f) and (g), dashed lines (cyan) are used for the DM results, solid lines (blue) for the MDM results, and circular markers (magenta) for the LES results. The insets in (f) and (g) show the angular averaged force calculated using a different method, where $D$ represents DM, $M$ represents MDM, $S$ represents superposition of LES, and $L$ represents LES.

Figure 6

(a) Angular average of the pair magnetic force in a rotational magnetic field for $B_0$ = 6 G and $\chi =5$ from DM (cyan dashed line), MDM (blue line), and MDM on a three-particle system (green dotted line), LES (magenta line with circular markers), and LES on a three-particle system (yellow line with square markers). (b) Angular average of the pair magnetic force over the particle diameter in a rotational field for $B_0$ = 6 G, $\chi =0.96$, and different $D$: DM for both $D$ = 2.8 and 6 μm (cyan dashed line), MDM for both $D$ = 2.8 and 6 μm (blue line), LES for $D$ = 2.8 μm (red line with circular markers), LES for $D$ = 6 μm (green line with square markers), and Eq. (16) (black small dotted line).