Micro-mutual-dipolar model for rapid calculation of forces between paramagnetic colloids

Typically, the force between paramagnetic particles in a uniform magnetic field is calculated using either dipole-based models or the Maxwell stress tensor combined with Laplace's equation for magnetostatics. Dipole-based models are fast but involve many assumptions, leading to inaccuracies in determining forces for clusters of particles. The Maxwell stress tensor yields an exact force calculation, but solving Laplace's equation is very time consuming. Here, we present a more elaborate dipole-based model: the micro-mutual-dipolar model. Our model has a time complexity that is similar to that of other dipole-based models but is much more accurate especially when used to calculate the force of small aggregates. Using this model, we calculate the force between two paramagnetic spheres in a uniform magnetic field and a circular rotational magnetic field and compare our results with those of other models. The forces for three-particle and ten-particle systems dispersed in two-dimensional (2D) space are examined using the same model. We also apply this model to calculate the force between two paramagnetic disks dispersed in 2D space. The micro-mutual-dipolar model is demonstrated to be useful for force calculations in dynamic simulations of small clusters of particles for which both accuracy and efficiency are desirable.

Figure 1

(a) The magnetic field strength distribution of a two-sphere system at the midplane $z=0$ with $\alpha ={30}^{\circ }$ determined by directly solving Laplace's equation for magnetostatics. For clarity, the logarithm of the magnetic field strength is plotted. The color scale represents the magnetic field strength in units of ln(A/m). (b) The difference between the dipole moments calculated using different dipole-based models at $\alpha ={30}^{\circ }$. The solid arrow (cyan) denotes the DM, the dashed arrow (blue) denotes the MDM, and the dotted arrow (red) denotes the MMDM. Note that the positions of the MMDM dipole moments differ from those of the DM or MDM dipole moments as shown in the zoomed-in inset. (c) The magnitude of dipole moments calculated using different models at $r=1.1\mathrm{D}$. (d) The trajectories of the location of the shifted dipole moments on the left particle for different center-to-center distances calculated using the MMDM. The curves from inside to outside correspond the positions of shifted dipole for $r/\mathrm{D}=1.1,1.05$, and 1.1, respectively. The red dot on the curve corresponds to the condition in (b).

Figure 2

The magnetic force between two paramagnetic spheres calculated using different models (a) for different $r$ 's at $\alpha =0^{\circ }$, (b) for different $r$'s at $\alpha ={90}^{\circ }$, and (c) for different $\alpha$'s at $r=1.1\mathrm{D}$. Note that positive values indicate repulsive forces, whereas negative values indicate attractive forces.

Figure 3

(a) The pair magnetic force and effective pair magnetic force at $r=1.1\mathrm{D}$ for a three-sphere system in a CRM field calculated using different methods. The inset shows the pair magnetic force calculated using different methods with an increased sphere susceptibility ${\chi }_{\mathrm{eff}}$. (b) The magnetic force at $r=1.1\mathrm{D}$ on sphere 1 of a ten-sphere system for different $\alpha$'s calculated using different methods.

Figure 4

The magnetic force between two paramagnetic disks calculated using different models (a) for different $r$'s at $\alpha =0^{\circ }$ and (b) for different $r$'s at $\alpha ={90}^{\circ }$. (c) The time complexity for different sphere numbers and different methods. The inset shows the same complexity with a shorter $x$-axis interval and logarithm scales for both axes.