Two-dimensional liquid crystalline growth within a phase-field-crystal model

By using a two-dimensional phase-field-crystal (PFC) model, the liquid crystalline growth of the plastic triangular phase is simulated with emphasis on crystal shape and topological defect formation. The equilibrium shape of a plastic triangular crystal (PTC) grown from an isotropic phase is compared with that grown from a columnar or smectic-$A$ (CSA) phase. While the shape of a PTC nucleus in the isotropic phase is almost identical to that of the classical PFC model, the shape of a PTC nucleus in CSA is affected by the orientation of stripes in the CSA phase, and irregular hexagonal, elliptical, octagonal, and rectangular shapes are obtained. Concerning the dynamics of the growth process, we analyze the topological structure of the nematic order, which starts from nucleation of $+\frac{1}{2}$ and $-\frac{1}{2}$ disclination pairs at the PTC growth front and evolves into hexagonal cells consisting of $+1$ vortices surrounded by six satellite $-\frac{1}{2}$ disclinations. It is found that the orientational and the positional order do not evolve simultaneously; the orientational order evolves behind the positional order, leading to a large transition zone, which can span over several lattice spacings.

Figure 1

LC-PFC phase diagrams as a function of ${\overline{\psi }}_0$ and $A_1$ for $B_3=0$, (a) $-0.4$, and (b) $-1.6$, which correspond to zero, weak, and strong coupling strength between the density field and the nematic-order field, respectively. The small letters correspond to parameters used in our simulations.

Figure 2

The equilibrium shape of the PTC nucleus in the isotropic phase for (a) $B_3=0$ and (b) $B_3=-1.6$ and (c) the corresponding contour line and aspect ratio $\alpha$ as a function of $A_1$. The considered examples a-d and e-h correspond to the depicted points in Fig. 1. The insets show the obtained equilibrium shapes for two selected points. (d) and (e) The morphology of a PTC nucleus in a CSA phase for $A_1=1.5$, where the close-packed direction of the PTC phase is parallel to the stripe in the CSA phase. (d) shows the equilibrium PTC shapes for $B_3=0$ for the considered examples ${\mathrm{a}}^{\prime }\text{−}{\mathrm{d}}^{\prime }$, and (e) shows the shapes of growing PTC for $B_3=-1.6$ for ${\mathrm{h}}^{\prime }\text{−}{\mathrm{e}}^{\prime }$, again corresponding to the depicted points in Fig. 1. The insets again show selected growth shapes. (f) The aspect ratio $\alpha$ as a function of $A_1$ for the equilibrium PTC shape for $B_3=0$ and the growing shape for $B_3=-1.6$. (g) The definition of crystal unit vectors $a_1$ and $a_2$.

Figure 3

The PTC shape in the CSA phase for $A_1=1.5$ and (a) and (c) $B_3=0$ and (b) and (d) $B_3=-1.6$, where the close-packed crystal direction is perpendicular to the stripe in the CSA phase. (c) and (d) show a sketch of the PTC shape in the CSA phase. The red dashed lines visualize the extrapolation to a hexagonal shape.

Figure 4

The topological structure of the PTC phase grown from (a) the isotropic phase and (b) the CSA phase. A magnified view of the area denoted by black boxes in (a) and (b) is given in (c) and (d), respectively, where the short dashes represent the director field of the topological structure.

Figure 5

Snapshots of the topological defect structure formation in the region enclosed by the square box in Fig. 4 during the PTC growth in the CSA phase. Here d1 and d2 are a pair of $+\frac{1}{2}$ disclinations. The PTC-CSA interface moves from bottom to top from (a) to (f), and for (d) and (e) the region around d1 and d2 is located inside the PTC-CSA interface.

Figure 6

The growth kinetics of the PTC-CSA interface. The displacement of PTC-CSA interface vs (a) time $t$ and (b) growth velocity for various $B_3$. $X$ is the position of the PTC-CSA interface at different times, and $X_0$ is the initial position at time $t_0$.

Figure 7

The moving trajectory of two $+\frac{1}{2}$ disclinations, d1 and d2, corresponding to Fig. 5, as represented by (a) the $S(\vec{r},t)$ profile and (b) the profile of the density field corresponding to time $t=130$ (i.e., $\tilde{t}=9.95$), and (c) the position and (d) the velocity of d1 and d2. The region between the two vertical dotted lines in (b) indicates the PTC-CSA interface (the left and the right boundaries of the interface are determined by the criterion that the height of the peaks is lower than 0.95 of that in the bulk PTC phase and the CSA phase, respectively). The inset in (b) is a zoom of the $X$ interval as in (a), with the arrows indicating the position of the d1 and d2 defects at time $\tilde{t}=9.95$.

Figure 8

The influences of coupling strength $B_3$ on topological defects formation kinetics: (a) the velocity of two $+\frac{1}{2}$ disclinations during the formation of $+1$ vortex for various $B_3$ and (b) the peak value of the velocity curves as shown in (a) as a function of $B_3$.