Dynamics of disk pairs in a nematic liquid crystal

We use a hybrid lattice Boltzmann method to study the behavior of sets of ferromagnetic colloidal disks in a nematic liquid crystal. When a weak rotating magnetic field acts on the system, the disks rotate following the magnetic field. This leads to a distortion in the liquid crystal that drives translational motion of the disks. If the concentration of disks is high, disks get locked together: a stable chain configuration is created, where each disk lays on the nearest neighbor. For intermediate concentrations of disks, a different behavior is observed. When disks are rotated by the magnetic field by more than ${90}^{\circ }$ from their initial orientation, the distortion in the liquid crystal leads to a simultaneous flip of both disks. The final disk positions depends only weakly on the initial configuration. Consecutive rotations of magnetic field push disks towards an equidistant configuration. Periodicity of the systems studied and analysis of the flipping motion of a single disk imply that one can use weak rotating magnetic fields to create stable crystal structures of disks.

Figure 1

Schematic representation of the disks in the simulation cell. Boundary conditions at the top and bottom walls impose a preferred “far-field” direction for the director field, whereas the director is normal to the face of the disk. The magnetic moment is parallel to the face of the disk. Note that the system periodically repeats in the $x$ and $y$ directions. The director field is only shown in the bottom half of the box.

Figure 2

Coupling of the liquid crystal to an immersed object. Direction of each arrow indicates the direction of the coupling it relates to.

Figure 3

Motion of disks pairs. Disks are represented by sets of (orange) spheres, and their magnetic moments are shown as semitransparent (red and purple) lines. The arrow corresponds to the direction of a magnetic field. The center part on each picture is an actual simulation domain, while pictures on the left and right are the same domain repeated periodically. The color scale on the right shows the correspondence between the largest eigenvalue and colors from black to light gray. (a) Initial configuration. (b) Configuration at $t=30082\mu \mathrm{s}$. (c) Configuration at $t=115215\mu \mathrm{s}$ after which the disks do not move.

Figure 4

(a) Distance between the centers of mass of the disks (in the actual simulation cell, not the periodic images) in Fig. 3 with time (continuous blue line, corresponds to the scale on the left) together with the free energy of the system (purple dots, right scale, normalized by its maximum value). (b) Changes in $x$ (continuous lines), $y$ (dashed lines), and $z$ (dotted lines) coordinates of the centers of mass of the disks with time. Disk 1 is the left disk in the center domain in Fig. 3, disk 2 is the right one.

Figure 5

Motion of the disks pair. Disks are represented by set of (orange) spheres, their magnetic moments are shown as semitransparent (red and purple) lines. The arrow corresponds to the direction of a magnetic field. Color scale is the same as in Fig. 3. The time since the beginining of simulation is (a) 0, (b) $13380 \mu \mathrm{s}$, (c) $68272 \mu \mathrm{s}$, (d) $75627 \mu \mathrm{s}$, (e) $82575 \mu \mathrm{s}$, (f) $97405 \mu \mathrm{s}$, (g) $117179$ $\mu \mathrm{s}$, (h) $139981 \mu \mathrm{s}$, (i) $147720 \mu \mathrm{s}$, (j) $152510 \mu \mathrm{s}$, (k) $163107 \mu \mathrm{s}$, and (l) $244588 \mu \mathrm{s}$.

Figure 6

Motion of the centers of mass of the disks started at distance $1.72R$. (a) Changes of the distance between the centers of mass of the disks from the middle domain in Fig. 3 with time (continuous blue line, corresponds to the scale on the left) together with normalized by the largest value of the free energy of the system (purple dots, right scale). (b) Changes in $x$ (continuous lines), $y$ (dashed lines), and $z$ (dotted lines) coordinates of the centers of mass of the disks with time. Here disk 1 corresponds to the left disk in Fig. 5, and disk 2 corresponds to the right one.

Figure 7

Motion of the centers of mass of the disks started at distance $3.4R$. (a) Changes of the distance between the centers of mass of the disks from the middle domain in Fig. 3 with time (continuous blue line, corresponds to the scale on the left) together with normalized by the largest value of the free energy of the system (purple dots, right scale). (b) Changes in $x$ (continuous lines), $y$ (dashed lines), and $z$ (dotted lines) coordinates of the centers of mass of the disks with time. Here disk 1 corresponds to the disk started on the left, and disk 2 corresponds to the one started on the right.

Figure 8

Changes in $x$ (continuous lines), $y$ (dashed lines), and $z$ (dotted lines) coordinates of the centers of mass of the disks with time. Here disk 1 corresponds to the disk started on the right, and disk 2 corresponds to the one started on the left.

Figure 9

Motion of the disks pair for a second half turn rotation. Disks are represented by set of (orange) spheres, and their magnetic moments are shown as semitransparent (red and purple) lines. The arrow corresponds to the direction of a magnetic field. Color scale is the same as in Fig. 3. The time since the beginning of simulation is (a) $244588 \mu \mathrm{s}$, (b) $666415 \mu \mathrm{s}$, (c) $685132 \mu \mathrm{s}$, (d) $703736 \mu \mathrm{s}$, (e) $725998 \mu \mathrm{s}$, (f) $732004\mu \mathrm{s}$, (g) $745000 \mu \mathrm{s}$, (h) $757500 \mu s$, (i) $782500 \mu \mathrm{s}$, (j) $832500\mu \mathrm{s}$, (k) $807500 \mu \mathrm{s}$, and (l) $307500 \mu \mathrm{s}$.

Figure 10

Changes in $x$ (continuous lines), $y$ (dashed lines), and $z$ (dotted lines) coordinates of the centers of mass of the disks with time during the simulation with counterclockwise rotation of the magnetic field (reversed motion of the magnetic field from the simulation described in Sec. 3b1). Here disk 1 corresponds to the disk started on the right, and disk 2 corresponds to the one started on the left.

Figure 11

Motion of the disks pair during backward rotation. Disks are represented in orange, and their magnetic moments are shown as semitransparent red and purple lines. The arrow corresponds to the direction of a magnetic field. The color scale is the same as in Fig. 3. Time since the beginning of the simulation is (a) $244588$ $\mu \mathrm{s}$, (b) $654637\mu \mathrm{s}$, (c) $663755 \mu \mathrm{s}$, (d) $671435 \mu \mathrm{s}$, (e) $686000 \mu \mathrm{s}$, (f) $757750 \mu \mathrm{s}$, (g) $775250 \mu \mathrm{s}$, (h) $854000 \mu \mathrm{s}$, (i) $889000 \mu \mathrm{s}$, (j) $932750 \mu \mathrm{s}$, (k) $1055250 \mu \mathrm{s}$, and (l) $1589000 \mu \mathrm{s}$.

Figure 12

Periodic structure of disks resulting from two half circle rotations of the magnetic field (top view). Disks are represented as a set of nodes colored in orange.

Figure 13

(a) Schematic representation of the distribution of the nodes of the object to fluid mesh simplified to two dimensions. (b) Edgeless representation of the disk: Red dots corresponds to the nodes comprising the disk.