Coarsening of stripe patterns: Variations with quench depth and scaling

The coarsening of stripe patterns when the system is evolved from random initial states is studied by varying the quench depth $\epsilon$, which is a measure of distance from the transition point of the stripe phase. The dynamics of the growth of stripe order, which is characterized by two length scales, depends on the quench depth. The growth exponents of the two length scales vary continuously with $\epsilon$. The decay exponents for free energy, stripe curvature, and densities of defects like grain boundaries and dislocations also show similar variation. This implies a breakdown of the standard picture of nonequilibrium dynamical scaling. In order to understand the variations with $\epsilon$ we propose an additional scaling with a length scale dependent on $\epsilon$. The main contribution to this length scale comes from the “pinning potential,” which is unique to systems where the order parameter is spatially periodic. The periodic order parameter gives rise to an $\epsilon$-dependent potential, which can pin defects like grain boundaries, dislocations, etc. This additional scaling provides a compact description of variations of growth exponents with quench depth in terms of just one exponent for each of the length scales. The relaxation of free energy, stripe curvature, and the defect densities have also been related to these length scales. The study is done at zero temperature using Swift-Hohenberg equation in two dimensions.

Figure 1

(a) $S_{\psi }(q,t)$ versus $q$ at different times for $\epsilon =0.1$. (b) $S_{\psi }(q,t)$ versus $q$ at different values of $\epsilon$ at a time $t=30,000$. (c) Data collapse of $S_{\psi }(q,t)$ according to Eq. (5) for $\epsilon =0.1$. (d) Variations of $\delta q$ with time on a log-log plot for four values of $\epsilon$.

Figure 2

Plots of $C_{nn}(r,t)$ with $r$, (a) for $\epsilon =0.1$ at different times, (b) at $t=30000$ for different values of $\epsilon$. (c) $R_{\alpha }\left(t\right)$ with $t$ on a log-log plot for different values of $\alpha$, (d) Data collapse for $C_{nn}(r,t)$ according to Eq. (6) for $\epsilon =0.1$.

Figure 3

(a) $P(\kappa ,t)$ versus $\kappa$ at different times for $\epsilon =0.1$. (b) $P(\kappa ,t)$ versus $\kappa$ at different values of $\epsilon$ at time $t=30000$. (c) Scaling of $P(\kappa ,t)$ with time at $\epsilon =0.1$. Best data collapse is obtained for ${\varphi }_c=0.33$. (d) Relaxation of the average stripe curvature $\overline{\kappa }$ with time for four values of $\epsilon$.

Figure 4

(a) Excess free energy with time at four different values of $\epsilon$. (b) $N_{\text{GB}}$ vs. $t$ on a log-log plot. (c) $N_{\text{Dis}}$ vs. $t$ on a log-log plot.

Figure 5

(a) Plot of $\Delta V_p\left(\tau \right)=V_p\left(t\right)-V_p\left(t_0\right)$ vs. $\tau =t-t_0$ for four different values of $\epsilon$. (b) Data collapse of $\Delta V_p\left(\tau \right)$ for $\epsilon$ with a scaling of the form $\Delta V_p\left(\tau \right)={\epsilon }^{1.55}f\left(\tau \right)$, for the time range $300<t<3000$. (c) Data collapse of $\Delta V_p\left(\tau \right)$ for $\epsilon$ for the time range $3000<t<30000$. Black line is the fit to the collapsed data using the fitting function $f\left(\tau \right)=a{e}^{-b/\sqrt{\tau }}$.

Figure 6

(a) The plot of $l_p/{\tilde{R}}_g$ against $x=t^{0.31}/{\tilde{R}}_g$ establishing scaling of $l_p$ with $\epsilon$ as in Eq. (21). (b) The plot of $l_n/{\tilde{R}}_g$ against $y=t^{0.45}/{\tilde{R}}_g$ establishing scaling of $l_n$ with $\epsilon$.

Figure 7

(a) $\Delta F\left(t\right)$ vs. ${\left(l_p\right)}^{-1}$. (b) Variation of K with $\epsilon$. (c) $N_{\text{GB}}$ vs. ${\left(l_n\right)}^{-1}$. (d) Average stripe curvature $\overline{\kappa }\left(t\right)$ as a function of ${\left(l_p\right)}^{-1}$.