Self-assembly of rigid magnetic rods consisting of single dipolar beads in two dimensions

Molecular dynamics simulations are used to investigate the structural properties of a two-dimensional ensemble of magnetic rods, which are modeled as aligned single dipolar beads. The obtained self-assembled configurations can be characterized as (1) clusters, (2) percolated, and (3) ordered structures, and their structural properties are investigated in detail. By increasing the aspect ratio of the magnetic rods, we show that the percolation transition is suppressed due to the reduced mobility of the rods in two dimensions. Such a behavior is opposite to the one observed in three dimensions. A magnetic bulk phase is found with local ferromagnetic order and an unusual nonmonotonic behavior of the nematic order is observed.

Figure 1

Schematic illustration of the interaction between two magnetic rods with indication of the important parameters of the pair interaction potential.

Figure 2

The pair interaction energy (a) as a function of the angle $\varphi$ for different inter-rod separation $\delta$, (b) as a function of the interparticle distance for $\varphi =\pi$, and (c) for $\varphi =0$. In (b) and (c) the different colors represent different values of the aspect ratio, indicated in (b).

Figure 3

The polymerization as a function of the packing fraction $\eta$ for different aspect ratios.

Figure 4

Some representative equilibrium configurations for $T^{*}=0.1$. Each color represents a different size of cluster. (d) and (e) Percolated systems. (a) $\eta =0.2$, $l=1$; (b) $\eta =0.2$, $l=3$; (c) $\eta =0.2$, $l=5$; (d) $\eta =0.4$, $l=1$; (e) $\eta =0.4$, $l=3$; (f) $\eta =0.4$, $l=5$.

Figure 5

The pair correlation function for different aspect ratio and for (a) $\eta =0.1$ and (b) $\eta =0.4$.

Figure 6

The angle correlation between the nearest neighboring particles for (a) $l=5$, (b) $l=3$, (c) $l=1$; and different values of $\eta$. Subsequent curves are shifted by 0.01 along the $y$ axis in order to accentuate the small differences. The angle correlation for $\theta =\pi$ as a function of the packing fraction $\eta$ is presented in the inset of each figure.

Figure 7

The average fraction of monomers in the largest cluster that is present in the ground state configuration as a function of packing fraction for a different aspect ratio. The horizontal dashed line at 0.5 represents the percolation threshold. The results of the simulation for twice the number of particles in the computational unit cell is represented by symbols.

Figure 8

The angle correlation between the nearest neighboring rods excluding the parallel head-to-tail alignment for $\eta =0.1$. Results for $\eta =0.4$ are shown in the inset. In both cases the $y$ axis is in log scale.

Figure 9

The cluster size distribution for $\eta =0.4$, and different aspect ratios. The dashed line represents the function $n\left(s\right)\approx s^{-2.05}$. In the inset, the $l=1$ case for ${\mu }^{*2}=250$. Both axes are in log scale.

Figure 10

The pair connectedness function for different aspect ratio values for packing fraction $\eta =0.4$. The $y$ axis is in log scale.

Figure 11

(a) The polarization and (b) the nematic order parameter for different aspect ratios as a function of the packing fraction. The orientation of some representative configurations are depicted in the right figures for $T^{*}=0.1$ and for $l=1$ with (c) $\eta =0.1$, (d) $\eta =0.4$, (e) $\eta =0.45$ and for $l=2$ with (f) $\eta =0.4$, (h) $\eta =0.45$ and for $l=5$ with (i) $\eta =0.1$, (j) $\eta =0.4$; (k) $\eta =0.45$. (g) The colors indicate the orientation of the dipoles in plane.