N. Priezjev and R. A. Pelcovits

The coarsening dynamics of  biaxial nematic liquid crystals is very interesting and unusual because of the nature of the topological line defects. There are four topologically distinct classes of line defects in biaxial nematics, which are disclination lines distinguished by the rotation of the long and short axes of the rectangular building particles about the core of the line. Three of these classes (which we denote as C_u, C_v, C_w) correspond to 180 degree rotations of two of the three molecular axes, while the fourth class corresponds to a 360 degree rotation. The fundamental homotopy group of biaxial nematics is non-Abelian leading to a number of interesting consequences, e.g., the merging of two defects will depend on the path they follow, and two 180 degrees disclinations of different types will be connected by a 360 degrees "umbilical" cord after crossing each other. The latter fact is known to result in obstruction to crossing of the lines and might lead to slow kinetics of biaxial nematics. We have studied biaxial nematics using Langevin molecular dynamics on a lattice model.

As discussed in greater detail in our paper, we carried out simulations of quenches from a completely disordered state (where many defects are present) to the zero temperature ordered state (no defects). Three distinct coarsening sequences are possible depending on the parameters of the model, which in turn determine the energies of the different types of defects.  In this animation  we consider a parameterization which favors the classes  C_u (the blue lines) and  C_w  (green lines) over C_v (red lines). Note that red lines decay very rapidly and blue and green loops coarsen independently at later times.  The system size is 40³ and periodic boundary conditions are used so all the segments are connected to form loops.

The most interesting and novel coarsening sequence occurs when all three elastic constants associated with the axis rotations are equal. In this animation  we show the equilibration process for a system of size 40³. Note the formation of a network of nearly equal populations of all three types of defects, which meet at a "junction'' points. Neighboring junction points pinch together as the system coarsens, leading to the creation of nonintersecting loops which then shrink. 

It is also possible to have a coarsening sequence where only one class of defects survives until late times  In this animation  one can see a rather quick equilibration of  C_v (red) and  C_w (green)  defects, while the C_u  (blue) lines remain for a long time.

Here is some related earlier work we carried out on the coarsening of uniaxial nematics:

J. L. Billeter, A. M. Smondyrev, G. B. Loriot and R. A. Pelcovits

We simulated using molecular dynamics the Gay-Berne model of liquid crystals after a quench from the isotropic to nematic phase in a system of 65536 particles. Disclination lines as well as type 1 lines and monopoles were observed.  The full details of our work can be found in our paper.  Here we show several animated gifs that show the coarsening sequence.

While our simulation was done with the fluid Gay-Berne model, we broke the system into a lattice of cubic bins in order to implement defect finding algorithms. The animations are for a 16 x 16 x 8 lattice with approximately 30 Gay-Berne particles in each bin. The defect finding algorithms are described in our preprint.

In the animations the red lines are disclination lines (half-integer), the blue lines are type 1 lines (which can escape), and the green points are monopoles. With periodic boundary conditions all of the disclination lines form closed loops. Note that the type 1 lines fluctuate more rapidly than the disclination lines. The type 1 lines also tend to appear as precursors to the formation of bends and kinks in the disclination lines. The monopoles fluctuate even more rapidly than the type 1 lines. Monopole formation upon disclination loop collapse was never observed, a result consistent with the presence of only twist disclination loops which possess zero monopole charge.

Animation #1 This animation shows the entire coarsening sequence. The time intervals between the frames vary between 1 and 3 thousand time steps.

Animation #2 This is a large file (1.3 MB), consisting of 220 consecutive time steps starting at 29,000 steps after the quench.

Animation #3  This is a coarser version of animation #2. It consists of every tenth step of animation #2.