Figure 1

Schematic drawing of $\pm 1∕2$ disclination line segments. The disclination cores are indicated by a “⊗,” and the disclination line is locally perpendicular to the plane of the page. (a) $+1∕2$ wedge defect, (b) $-1∕2$ wedge defect, and (c) $+1∕2$ twist defect. The rotation of the director in (c) is about an axis lying in the plane of the page perpendicular to the disclination line, and the nail head representation indicates a projection of the director from below the plane of the page. Note that all three defects shown here are topologically equivalent and can be transformed into each other via suitable continuous rotation of the director field.Reuse & Permissions

Figure 2

(Color online) A barycentric coordinate representation of the Westin metrics based on condition, Eq. (12). The dashed line corresponds to $c_l=0.12$ and separates ordered uniaxial regions outside disclination cores (to the left of the line) from the core regions (to the right of the line). This choice of the numerical value for $c_l$ on the dashed line is explained in the text. The spheroids shown are drawn so that the principal axis directions and lengths are given by the eigenvectors and corresponding eigenvalues of the tensor $\mathsf{D}$ [31].Reuse & Permissions

Figure 3

(Color online) Illustration of a streamtube generated from the trajectory of the major eigenvector corresponding to the largest eigenvalue ${\lambda }_1$. The remaining two eigenvalues ${\lambda }_2$ and ${\lambda }_3$ determine the relative shape of the cross section of the streamtube. The orientation of the cross section is determined by the corresponding eigenvectors. The semiminor axis $r$ along the direction of the medium eigenvector is chosen for visual clarity.Reuse & Permissions

Figure 4

Illustration of the cubic $b$-spline weighting function $w\left(\mid {\mathbf{r}}_i-\mathbf{r}\mid \right)$ with kernel radius $r_K=7.3$. The function is constructed in a piecewise continuous fashion such that it is zero outside the kernel radius. Within the kernel $w\left(x\right)=\frac{1}{6}(3{\mid x\mid }^3-6x^2+4),0⩽\mid x\mid ⩽1;w\left(x\right)=\frac{1}{6}{(2-\mid x\mid )}^3,1⩽\mid x\mid ⩽2$, where $x=2\mid {\mathbf{r}}_i-\mathbf{r}\mid ∕r_K$.Reuse & Permissions

Figure 5

(Color online) (a) Visualization of the disclinations in the simulation data of Ref. 21 35 000 molecular dynamics time steps after an instantaneous thermal quench. The isosurfaces correspond to $c_l=0.12$. The color variation on the $c_l$ isosurfaces indicates variation of $c_p$, ranging from blue $(c_p\approx 0)$ to red $(c_p\approx 0.7)$. The inset shows for comparison the disclinations found at the same time step using the method of Ref. 21 which was based on standard lattice techniques. The general locations of the disclinations in the two figures are the same aside from the data padding added to all three directions in the main figure which was done to emphasize the spatial continuity of the disclinations due to periodic boundary conditions. (b) Isosurfaces corresponding to ${\lambda }_2=1∕3$, where ${\lambda }_2$ is the intermediate eigenvalue of the tensor $\mathsf{D}$, Eq. (5). This value of ${\lambda }_2$ corresponds to maximal biaxiality of the nematic order parameter tensor.Reuse & Permissions

Figure 6

(Color online) Wedge-twist loop visualized from the data of Ref. 21 70 000 time steps after the quench into the nematic phase. The green tubelike surface is an isosurface corresponding to $c_l=0.12$ and the red streamtubes (the thin lines) are along the director field. The variation in color along the streamtube indicates the relative value of $c_l$, with redder corresponding to larger values of this metric (streamtubes were selected so that $c_l⩾0.2$). If one encircles any segment of the loop, the director $\widehat{\mathbf{n}}$ rotates about the axis $\widehat{\Omega }$.Reuse & Permissions

Figure 7

(Color online) Disclination lines (which form closed loops given periodic boundary conditions) remaining near the end of the simulation of Ref. 21, 70 000 time steps after the quench. In (a) the two lines which are roughly parallel to the $z$ axis are viewed from above and appear to be purely wedge like, with charge $+1∕2$ on top and $-1∕2$ on the bottom. However, (b) (where the lines are viewed along the $x$ axis) and (c) (where the lines are viewed along the line $x=y$) indicate that the lines have a twistlike character as well. The red streamtubes (the thin lines) show the director field, while the blue streamsurfaces (the flat surfaces) show the planarity of the director field lines near the central portions of the defects indicated by the yellow-green tubelike isosurfaces corresponding to $c_l=0.12$. The color variation on the $c_l$ isosurfaces indicates variation of $c_p$, ranging from blue $(c_p\approx 0)$ to red $(c_p\approx 0.7)$. The streamsurfaces, corresponding to $c_p⩾0.2$, intersect the defect lines at regions of relatively high values of $c_p$. [For conciseness, parts (b) and (c) display only the $-1∕2$ line; the $1∕2$ line, apart from the sign difference in the topological charge, has a similar twistlike structure.]Reuse & Permissions

Figure 8

(Color online) The Westin metrics $c_l$ (red), $c_p$ (green), and $c_s$ (blue) as functions of the coordinate $y$ (in units of ${\sigma }_0$) for the two disclination lines [$-1∕2$ in (a), $+1∕2$ in (b)] shown in Fig. 7a. The coordinate system is shown in Figs. 5, 7a. The $y$ axis is parallel to one edge of the simulation cell, and the two disclination lines are roughly parallel to the $z$ axis. The value of the $x$ coordinate in the present figure has been chosen to coincide with the center of each line. The fixed value of $z$ is the same in (a) and (b) and is approximately one-third of the $z$ dimension of the simulation box.Reuse & Permissions

Figure 9

(Color online) Averages of $c_p$ at center of the disclinations of Figs. 7, 8 over the entire lengths of the disclination lines, plotted over 26 500 time steps starting at 70 000 time steps after the quench. The value of $c_p$ along the $1∕2$ and $-1∕2$ disclination lines is indicated by the solid and dashed curves, respectively. The time averages are indicated by the corresponding horizontal lines; the numerical values of these averages are given in Eqs. (19, 20).Reuse & Permissions