Structure of a two-dimensional superparamagnetic system in a quadratic trap

Ground-state structures of a two-dimensional (2D) system composed of superparamagnetic charged particles are investigated by means of molecular dynamics simulation. The charged particles trapped in a quadratic potential interact with each other via the repulsive, attractive, and magnetic dipole-dipole forces. Simulations are performed within two regimes: a one-component system and a two-component system where the charged particles have the identical charge-to-mass ratio. The effects of magnetic dipole-dipole interaction, mixing ratio of the two species and confinement frequency on the ground-state structures are discussed. It is found that as the strength of the magnetic dipole increases, the charged particles tend to self-organize into chainlike structures. The two species particles exhibit different structural features, depending on the competition of electrostatic repulsive interaction, magnetic dipole-dipole interaction and confinement force. The potential lanes are observed through analyzing the global potential of the magnetic particles, which guide the unmagnetic particles aligning themselves in the direction of the potential lanes.

Figure 1

Sketch of a pair of superparamagnetic particles $Q_1$ and $Q_2$. Magnetic moment $\mathbf{M}$ for both of the magnetic particles is aligned to the external magnetic field. The tilt angle of $\mathbf{M}$ with respect to the 2D (x-y) plane is $\alpha$. The project of $\mathbf{M}$ onto the 2D plane, i.e., $M_x$, is assumed to be along the $x$ direction without losing generality. The angle between $M_x$ and connection vector of the pair particles is $\theta$.

Figure 2

Configurations of the one-component system with $N=256$ and $\alpha ={60}^{\circ }$. In the columns, attraction strength $B$ is decreased from the top to the bottom. In the rows, magnetic dipole strength $\eta$ is increased from the left to the right. Three shells in the panel c(1) are indicated with the green lines.

Figure 3

Configurations of the one-component system with $N=256$ and $\eta =1$. In the rows, the chainlike structures are replaced by the circular shells with inner hexagon crystals with increasing $\alpha$. In the columns, the structures basically remain unchanged with decreasing $B$.

Figure 4

Contours of the interaction potential (combination of ${\mathrm{\Phi }}_R, {\mathrm{\Phi }}_A$ and ${\mathrm{\Phi }}_M$) for one charged particle. With increasing $\alpha$, the potential tends to be radically asymmetric. For each column, the potential is independent of $B$. $\kappa =0.9, \beta =4$, and $\eta =1$.

Figure 5

Configurations of the one-component system with $N=20$. In the columns, the cluster size is decreased with increasing $\omega$, which has no influence on the ground-state configurations. In the rows, the chainlike structures tend to appear with increasing $\eta$ and the 2D balls expand. $\kappa =1, B=0.1, \beta =3$, and $\alpha ={60}^{\circ }$.

Figure 6

Configurations of the two-component system with $N=64$ and $\alpha ={60}^{\circ }$. In the columns, proportion of the species 1 particles $x_1$ is increased from the top to the bottom. In the rows, magnetic dipole strength $\eta$ is increased from the left to the right.

Figure 7

Configurations of the two-component system with $N=400, x_1=0.5$, and $\mu =2$. The species 1 particles (green ones) are magnetic and the species 2 particles (red ones) are unmagnetic.

Figure 8

Global potential distribution of the species 1 particles. In panel (a), only ${\mathrm{\Phi }}_R$ of the species 1 particles exist in the real space. In panels (b) and (c), ${\mathrm{\Phi }}_M$ of species 1 particles is added in and the potential lanes appear, particularly for the bigger $\eta$.

Figure 9

Configurations of the two-component system with $N=16, x_1=0.5, \mu =2$, and $\eta =0$. The species 1 particles (green ones) nearly keep still, while the species 2 particles (red ones) move inward as $\omega$ increases.