Aging of orientation fluctuations in stripe phases

Stripe patterns, observed in a large variety of physical systems, often exhibit a slow nonequilibrium dynamics because ordering is impeded by the presence of topological defects. Using computer simulations based on a well-established model for stripe formation, we show that a slow dynamics and aging occur also in stripe patterns free of topological defects. For a wide range of noise strengths, the two-time orientation correlation function follows a scaling form that is typical for systems exhibiting a growing length scale. In our case, the underlying mechanism is the coarsening of orientation fluctuations, ultimately leading to power-law spatial correlations perpendicular to the stripes. Our results show that even for the smallest amount of noise, stripe phases without topological defects do not reach equilibrium. This constitutes an important aspect of the dynamics of modulated phases.

Figure 1

Two types of slow dynamics in stripe patterns starting from different initial conditions. (a)–(c) Homogeneous initial state. (d)–(f) Striped initial state. The noise strength $\eta /{\eta }_{\mathrm{c}}=\frac{1}{30}$ in both cases. (a) and (d) show binarized snapshots representing the corresponding dynamics of the concentration field $\psi (\mathbf{r},t)$. The free-energy density ${\rho }_{\mathcal{F}}\left(t\right)$ decays slowly for homogeneous initial conditions (b), whereas it becomes stationary in the ordered system (e). During defect annihilation, the orientational order parameter $S\left(t\right)$ remains close to zero for a long time, but approaches unity at later times (c). In contrast, the ordered system exhibits a very slow but ongoing decay of $S\left(t\right)$ (f).

Figure 2

Orientation fluctuations within the ordered stripe pattern at late times. (a) Concentration profile $\psi (\mathbf{r},t=5.2\times {10}^5)$ for $\eta /{\eta }_{\mathrm{c}}=\frac{1}{3}$. (b) The corresponding orientation field $\theta (\mathbf{r},t=5.2\times {10}^5)$. The centers of stripes forming an orientational domain extending perpendicularly to the stripes are marked in (a) and (b). The snapshots are sections $200\times 200$ in size of a system with $L=517$. The further temporal evolutions of these patterns are shown in the videos S1 and S2, respectively (see Supplemental Material [56]).

Figure 3

Dependence of relaxations on waiting time $t_{\mathrm{w}}$ and noise strength $\eta$. The (a),(b) concentration autocorrelation function $C_{\psi }(t,t_{\mathrm{w}})$ is compared to the (c),(d) orientation autocorrelation function $C_{\theta }(t,t_{\mathrm{w}})$. Both functions are plotted as a function of the time difference $t-t_{\mathrm{w}}$ for different waiting times $t_{\mathrm{w}}$, with $t_{\mathrm{w}}=5\times {10}^2,{10}^3,2.5\times {10}^3,5\times {10}^3$ and ${10}^4$ (from dark to bright). Data for the two different noise strengths below and above ${\eta }_{\mathrm{c}}$ are shown as indicated. $C_{\psi }(t,t_{\mathrm{w}})$ exhibits a very slow decay and no dependence on the waiting time (a). In contrast, $C_{\theta }(t,t_{\mathrm{w}})$ decreases more slowly for longer waiting times (c). Both functions decay exponentially for $\eta >{\eta }_{\mathrm{c}}$ (b), (d). The insets show cropped snapshots (size $70\times 70$) of the concentration field $\psi (\mathbf{r},t)$ and the orientation field $\theta (\mathbf{r},t)$ for $t={10}^4$, respectively, demonstrating the absence of long-range order for $\eta >{\eta }_{\mathrm{c}}$.

Figure 4

Scaling form of the orientation autocorrelation function. (a) $C_{\theta }(t,t_{\mathrm{w}})$ as a function of $t_{\mathrm{w}}$ for a constant ratio $t/t_{\mathrm{w}}=3/2$ and different noise strengths. The straight red line represents a power law. (b) The rescaled orientation correlation function $C_{\theta }(t,t_{\mathrm{w}})t_{\mathrm{w}}^b$ plotted vs $(t-t_{\mathrm{w}})/t_{\mathrm{w}}$. The red line is the fit function $f_2\left(x\right)$ (see text). The waiting time $t_{\mathrm{w}}\in [{10}^3,2\times {10}^4]$.

Figure 5

Growth of spatial orientation correlations at early and late times. (a) Spatial orientation correlation function perpendicular to the stripes at early times $t=5\times {10}^1,5\times {10}^2$, and $5\times {10}^3$ as well as for $t=5\times {10}^5$ (bottom to top). An exponential function is plotted as a guide to the eye. (b) Orientational correlation length ${\xi }_{\theta }\left(t\right)$. The red line represents a power law with an exponent $\frac{1}{2}$. (c) $C_{\theta }(r_{\perp },r_{\parallel }=0,t)$ at late times $t=5\times {10}^5,{10}^5,2\times {10}^4$, and ${10}^4$ ($\eta /{\eta }_{\mathrm{c}}=\frac{1}{3}$, top to bottom). For $\eta /{\eta }_{\mathrm{c}}=\frac{1}{30}$, data for $t=5\times {10}^5,{10}^5,2\times {10}^4,{10}^4,5\times {10}^3,5\times {10}^2$, and $t=5\times {10}^1$ are shown. The straight red lines correspond to power laws with the indicated exponents. (d) Spatial orientation correlation function parallel to the stripes at $t=5\times {10}^5$. The arrows indicate the corresponding ordinate.

Figure 6

Orientation correlations after a quench to $\eta =0$. Stripe patterns which had previously evolved at a finite noise strength $\eta$ for a time $t_{\mathrm{w}}=5\times {10}^5$ were quenched to $\eta =0$. (a) The orientation autocorrelation function $C_{\theta }^{\mathrm{Q}}(t,t_{\mathrm{w}})$ is plotted on a double logarithmic scale as a function of the time $t-t_{\mathrm{w}}$ after the quench for different noise strengths $\eta$. The red lines are power laws fitted for ${10}^2\le t-t_{\mathrm{w}}\le {10}^3$, with the resulting exponents growing from $0.18$ for $\eta /{\eta }_{\mathrm{c}}=\frac{1}{150}$ to $0.33$ for $\eta /{\eta }_{\mathrm{c}}=\frac{2}{3}$. (b) Dependence of the exponents $b (\times ),b_{\mathrm{Q}} (\circ )$, and $c (+)$, obtained from temporal and spatial correlation functions, on the noise strength. For comparison, the red line represents a linear dependence on $\eta$. For $b_{\mathrm{Q}}$ and $c$, the error bars represent the standard deviation over 40 realizations of the system. The exponent $c$ has been obtained from a fit to the sample average [Fig. 4].