Anticlinic order of long-range repulsive rodlike magnetic particles in two dimensions

In the field of liquid crystals, it is well known that rodlike molecules interacting via long-range attractive interactions or short-range repulsive potentials can exhibit orientational order. In this work, we are interested in what would happen to systems of rodlike particles interacting via a long-range repulsive potential. In our model, each particle consists of a number of point dipoles uniformly distributed along the particle length, with all dipoles pointing along the $z$ axis so that the rodlike particles repel each other when they lie in the $x\text{−}y$ plane. Dipoles from different particles interact via an $r^{-3}$ potential, where $r$ is the distance between the dipoles. We have considered two model systems, each with $N$ particles in a unit cell with periodic boundary conditions. In the first, particle centers are fixed on a square or triangular lattice but they are free to rotate. In the second, particles are free to translate as well as rotate in cells with variable shapes. Here they self-assemble to form configurations where the stress tensors are isotropic. Our numerical results show that, at low temperatures, the particles tend to form stripes with alternating orientations, resembling herringbone patterns or the anticlinic Sm-${\mathrm{C}}_A$ liquid crystal phase.

Figure 1

(a) Representative energy contours of two particles separated on the $x$ axis by a distance of $d=1.6$ as a function of angles ${\theta }_1$ and ${\theta }_2$. The number of dipoles on each particle is $n=21$. The minimum-energy state is attained when both angles are $\pi /2$, corresponding to the configuration in panel (b1). The maximum energy state is attained when both angles are zero, corresponding to the configuration in panel (b2). If ${\theta }_1=0$, then the minimum-energy state is attained when ${\theta }_2=\pi /2$, corresponding to the configuration in panel (b3). In the schematic of panel (b), only three dipoles are shown on each particle to illustrate the directions of dipoles.

Figure 2

(a) Equilibrium configurations of particles on a square lattice at six representative number densities. Each particle has 41 uniformly distributed dipoles along its length. (b) The angles of particles in two adjacent stripes on the square lattices as function of the number density $\rho$.

Figure 3

(a) Equilibrium configurations of particles on a triangular lattice at six representative number densities. Each particle has 41 uniformly distributed dipoles. (b) The angles of the particles in two adjacent stripes on the triangular lattices as function of the number density $\rho$.

Figure 4

One of the equilibrium configurations of rare-earth magnets in a triangular lattice.

Figure 5

(a) Equilibrium configurations of 100 free particles in a unit cell at four representative number densities. The unit cell is bounded by thin solid lines, and the centered rectangular lattice is indicated by dashed lines. Each particle has 41 dipoles uniformly distributed along its length. The red particles are on the vertices of the centered rectangular lattice, and the blue particles are at the center of the lattice. (b) The angle of the particles on the vertices of the centered rectangular lattice as function of the number density $\rho$. (c) The aspect ratio of the centered rectangular lattice as function of the number density $\rho$.