Stripe patterns: Role of initial state and boundary conditions

This paper presents results on stripe patterns by numerical solution of the Swift-Hohenberg equation. The focus is on the role of initial state and boundary conditions. We choose initial states which generate simple defect configurations and study their evolution. Various classes of defects are identified and their motion and relaxation is studied numerically. We first study the dynamics of a straight front and present a comparison of numerical results with some analytical results. We then study the domain-wall dynamics in configurations containing two and three domains and identify some mechanisms of their relaxation. Rates of domain-wall relaxation depend on several features like incommensuration, dislocations and orientations in neighboring domains, in addition to the curvature of the walls. For a generic class of domain walls the relaxation process has an intrinsic frustration which leads to generation of dislocations. This process also generates stripe curvature thereby making relaxation nonmonotonic. We have also generated some other topological defects and studied their evolution and the effect of boundary conditions on their stability.

Figure 1

Comparison of numerical simulation to singular perturbation theory for the initial condition Eq. (12) is shown at different times: (a) $t=20$, (b) $t=40$, and (c) $t=120$. Panel (d) shows positions of the front $x\left(t\right)$ with time as dots. The continuous line is the fit according to Eq. (13).

Figure 2

Comparison of numerical simulation to singular perturbation theory for the initial condition given by Eq. (15) with $\alpha =0.5$ is shown at different times: (a) $t=20$, (b) $t=40$, and (c) $t=120$.

Figure 3

Four panels show patterns evolving from the initial condition in Eq. (22) at different times: (a) $t=0$, (b) $t=300$, (c) $t=3000$, and (d) $t=12000$. Parameters of the initial state are $\xi =1000$, $\vec{n}=(1/2,\sqrt{3}/2)$, and ${\vec{r}}_0$ is the center of the square cell.

Figure 4

(a) Evolution of the total number of defect points $N_D\left(t\right)$ and (b) the excess free energy $\Delta F\left(t\right)$ with time. The time $t$ is measured from the time the domain wall is formed. The inset shows a plot of $\Delta F$ with $N_D$. Continuous lines are a guide to the eyes.

Figure 5

(a) Plot of $d\left(\Delta F\right)/dt$ with $\Delta F$. The dotted line is the power-law fit at the late stage. (b) Motion of end points $X_{\max }$ and $Y_{\max }$ of the lower left of the domain wall in Fig. 3.

Figure 6

Patterns evolving from the initial condition of Fig. 3 for incommensurate $513\times 513$ lattice with $\Delta x=0.8$ at different times: (a) $t=300$, (b) $t=12000$, (c) $t=21000$, and (d) $t=30000$.

Figure 7

Evolution of (a) total number of defect points $N_D\left(t\right)$ and (b) excess free energy $\Delta F\left(t\right)$. (c) Plot of $d\left(\Delta F\right)/dt$ with $\Delta F$. Continuous lines are a guide to the eyes.

Figure 8

(a) Motion of end points $X_{\max }$ and $Y_{\max }$ of the left domain wall. (b) Motion of intermediate points $X_{\mathrm{in}}$ and $Y_{\mathrm{in}}$ of the left domain wall. Continuous lines are a guide to the eyes. (c) Power spectrum $P\left(\vec{k}\right)$ vs $k$.

Figure 9

Six panels show patterns evolving from the initial condition in Eq. (22) on a $513\times 513$ lattice at different times: (a) $t=0$, (b) $t=300$, (c) $t=10800$ and (d) $t=30000$, (e) $t=150000$ (f) $t=210000$. Parameters of the initial state are $\xi =0.01$, $\vec{n}=(\sqrt{3}/2,1/2)$, and ${\vec{r}}_0$ is the center of the square cell.

Figure 10

(a) Evolution of total number of defect sites $N_D\left(t\right)$ and number of defect sites $N_{GB1}\left(t\right)$ in the domain wall of horizontal stripes. (b) Early evolution of excess free energy $\Delta F\left(t\right)$. Inset show a plot of $\Delta F\left(t\right)$ with $N_D\left(t\right)$. (c) Plot of $d\left(\Delta F\right)/dt$ with $\Delta F$ at early times. (d) Evolution of total defect sites $N_D\left(t\right)$ over a longer range of time. Inset show a plot of $\Delta F\left(t\right)$ with $t$ for the same range.

Figure 11

(a) Evolution of average stripe curvature with time. (b) The distribution of stripe curvature at various times.

Figure 12

Patterns evolving from the initial condition in Eq. (24) with ${\xi }_x={\xi }_y=28.6$ are shown at different times: (a) $t=0$, (b) $t=600$, (c) $t=3000$, and (d) $t=12000$. The last two panels show the late stage configurations for an anisotropic seed with ${\xi }_x=19.6,{\xi }_y=28.6$ at times (e) $t=90000$ and (f) $t=180000$.

Figure 13

Generation and evolution of a convex disclination using the initial state of Eqs. (25) and (26). Panels show the patterns at different times: (a) $t=6$, (b) $t=30$, (c) $t=300$, and (d) $t=15000$.

Figure 14

Generation and evolution of a concave disclination using the initial state of Eqs. (25) and (27). Panels show the patterns at different times: (a) $t=6$, (b) $t=30$, (c) $t=300$, and (d) $t=90000$.

Figure 15

Generation and evolution of a saddle on a $1025\times 1025$ lattice with $\Delta x=\pi /8$ using the initial state of Eq. (28). Panels show the patterns at different times: (a) $t=0$, (b) $t=300$, (c) $t=12000$, and (d) $t=27000$.