Elastic constants and the formation of topological defects in hybrid nematic cells: A Monte Carlo study

We present a Monte Carlo study of the effects of elastic anisotropy on the topological defects which can be formed in nematic films with hybrid boundary conditions. We simulate the polarized microscopy images and analyze their evolution in uniaxial systems for different values of the Frank elastic constants.

Figure 1

A schematic representation of the homeotropic and homogeneous aligned boundary conditions at the top and bottom surfaces. We employ periodic boundary conditions at the other four faces of the lattice.

Figure 2

Simulated optical patterns for a hybrid nematic film as obtained from a Monte Carlo simulation of a GHRL potential for different values of the three main elastic constants $K_i^{*}=K_i/{10}^{-12}\mathrm{N}, i=1,2,3$. Here, the images are from $(K_1^{*}=1,K_3^{*}=1)$ (top left) to $(K_1^{*}=1,K_3^{*}=9)$ (top right) and from $(K_1^{*}=9,K_3^{*}=1)$ (bottom left) to $(K_1^{*}=9,K_3^{*}=9)$ (bottom right) while $K_2^{*}=1$ is kept fixed.

Figure 3

Simulated optical patterns for a hybrid nematic film as obtained from a Monte Carlo simulation of a GHRL potential for different values of the three main elastic constants $K_i^{*}=K_i/{10}^{-12}\mathrm{N}, i=1,2,3$. Here, the images are from $(K_2^{*}=1,K_3^{*}=1)$ (top left) to $(K_2^{*}=1,K_3^{*}=9)$ (top right) and from $(K_2^{*}=9,K_3^{*}=1)$ (bottom left) to $(K_2^{*}=9,K_3^{*}=9)$ (bottom right) while $K_1^{*}=1$ is kept fixed.

Figure 4

Simulated optical patterns for a hybrid nematic film as obtained from a Monte Carlo simulation of a GHRL potential for different values of the three main elastic constants $K_i^{*}=K_i/{10}^{-12}\mathrm{N}, i=1,2,3$. Here, the images are from $(K_2^{*}=1,K_1^{*}=1)$ (top left) to $(K_2^{*}=1,K_1^{*}=9)$ (top right) and from $(K_2^{*}=9,K_1^{*}=1)$ (bottom left) to $(K_2^{*}=9,K_1^{*}=9)$ (bottom right) while $K_3^{*}=1$ is kept fixed.

Figure 5

Simulated optical pattern for a $200\times 200\times (10+2)$ nematic film for the elastic constants $K_1=1, K_2=1, K_3=1$. To better visualize the elliptic ring defect we have replicated the image obtained with the periodic boundary conditions. Bottom: Director field and order parameter isosurface obtained following the methodology presented in Ref. [45]. The white color encodes the director alignment along the $z$ axis, while red and blue correspond to alignment along the positive and negative $x$ axis, respectively. (Recall that the initial director orientation was along the positive $z$ axis.) The order parameter isosurfaces were built by setting the $c_l$ Westin metric [45] to 0.82.

Figure 6

As in Fig. 5 for $K_1=1, K_2=1, K_3=3$.

Figure 7

As in Fig. 5 for $K_1=1, K_2=1, K_3=4$.

Figure 8

As in Fig. 5 for $K_1=1, K_2=1, K_3=5$.

Figure 9

Snapshots of a portion of the first bottom nematic layer (left) and a vertical cut of the system crossing the ring defect (right) for $K_1=1, K_2=1, K_3=3$.

Figure 10

Snapshots of a portion of the first bottom nematic layer (left) and a vertical cut of the system close to a point defect (right) for $K_1=6, K_2=1, K_3=5$.