Pathways to equilibrium orientation fluctuations in finite stripe-forming systems

Small-angle orientation fluctuations in ordered stripe-forming systems free of topological defects can exhibit aging and anisotropic growth of two length scales. In infinitely extended systems, the stripe orientation field develops a dominant modulation length ${\lambda }_{\parallel }^{*}\left(t\right)$ in the direction parallel to the stripes, which increases with time $t$ as ${\lambda }_{\parallel }^{*}\left(t\right)\sim t^{1/4}$. Simultaneously, the orientation correlation length ${\xi }_{\perp }\left(t\right)$ in the direction perpendicular to the stripes increases as ${\xi }_{\perp }\left(t\right)\sim t^{1/2}$ [Riesch et al., Interface Focus 7, 20160146 (2017)]. Here we show that finite systems of size $L_{\perp }\times L_{\parallel }$ with periodic boundary conditions reach equilibrium when the dominant modulation length ${\lambda }_{\parallel }^{*}\left(t\right)$ reaches the system size $L_{\parallel }$ in the stripe direction. The equilibration time ${\tau }_{\mathrm{eq}}^{\parallel }$ is solely determined by $L_{\parallel }$, with ${\tau }_{\mathrm{eq}}^{\parallel }\sim L_{\parallel }^4$. In systems with $L_{\perp }<L_{\parallel }^2/2\pi {\lambda }_{\mathrm{p}}$, where ${\lambda }_{\mathrm{p}}$ is the undulation penetration length, the initial aging and coarsening dynamics changes at the crossover time ${\tau }_{\mathrm{C}}^{\perp }\sim L_{\perp }^2$ to an aging and coarsening dynamics described by the one-dimensional Mullins-Herring equation, before reaching equilibrium at ${\tau }_{\parallel }^{\mathrm{eq}}$. Our work reveals the two pathways to equilibrium in stripe phases with periodic boundary conditions, the finite-size scaling behavior of equilibrium orientation fluctuations, and the characteristic exponents associated with the influence of a finite system size.

Figure 1

(a) Snapshot of an ordered stripe-forming system displaying orientation fluctuations. The concentration field $\psi (\mathbf{r},t=5\times {10}^5)$ is shown for a noise strength $\eta /{\eta }_{\mathrm{c}}=\frac{1}{3}$. The image shows a portion of the $55{\lambda }_0\times 55{\lambda }_0$ large simulation area. (b) Illustration of the stripe displacement $u$ and the stripe orientation $\theta$. The thick black lines represent the center lines of individual stripes, while the dashed line indicates $u=0$. ${\lambda }_0$ indicates the wavelength of the stripe pattern. Adapted from Ref. [17].

Figure 2

(a) Examples of orientation fields $\theta (\mathbf{r},t)$ for stripe-forming systems with different extensions $L_{\perp }$ and $L_{\parallel }$. The dynamics was advanced until equilibrium was attained. Top left: $L_{\perp }=3{\lambda }_0$, $L_{\parallel }=16{\lambda }_0$. Top right: $L_{\perp }=L_{\parallel }=16{\lambda }_0$. Bottom left: $L_{\perp }=L_{\parallel }=3{\lambda }_0$. Bottom right: $L_{\perp }=16{\lambda }_0$, $L_{\parallel }=3{\lambda }_0$. (b) Orientation fields from the same systems as in (a) where fluctuations smaller than ${\lambda }_0$ were removed. The orientation angle in the two bottom panels was multiplied by a factor of three.

Figure 3

Pathway to equilibrium in systems comprised of short stripes. (a) Temporal evolution of the spatial correlation function $C_{\theta }(r_{\perp },r_{\parallel }=0,t)$ for $L_{\perp }=384{\lambda }_0$ and $L_{\parallel }=8{\lambda }_0$. The solid red lines represent the analytical result of Eq. (25) at the same times. The curves for $t=5\times {10}^4$ and $5\times {10}^5$ overlap each other, indicating that the coarsening has stopped. (b) Growing length scales ${\xi }_{\perp }\left(t\right)$ and ${\lambda }_{\parallel }^{*}\left(t\right)$ extracted from the same system as shown in (a). In the log-log plot, the crossover from an algebraic increase to a constant value at the equilibration time ${\tau }_{\mathrm{eq}}^{\parallel }$ [Eq. (19)] is clearly visible. The lower horizontal line represents the system size $L_{\parallel }$, which limits ${\lambda }_{\parallel }^{*}\left(t\right)$. ${\xi }_{\perp }\left(t\right)$ approaches the equilibrium correlation length ${\tilde{\xi }}_{\perp }^{\mathrm{eq}}$ [derived from Eq. (27)], indicated by the upper horizontal line. (c) Equilibrium dynamics of the two-time correlation function in a system with $L_{\perp }=384{\lambda }_0$ and $L_{\parallel }=3{\lambda }_0$. For the range of times shown here, $C_{\theta }(t,t_{\mathrm{w}})$ only depends on the difference $t-t_{\mathrm{w}}$. In the inset, the same data are shown in a semilogarithmic plot. The solid red line is the theoretical result given in Eq. (28).

Figure 4

Scaling behavior of the equilibrium spatial orientation correlation function $C_{\theta }^{\mathrm{eq}}(r_{\perp },r_{\parallel }=0)$ in rectangular systems containing short stripes ($L_{\perp }\gg L_{\parallel }$). (a) The equilibrium correlation function is plotted as a function of the distance $r_{\perp }$ for different system sizes $L_{\parallel }$. The equilibrium correlation length increases with $L_{\parallel }$. (b) The rescaled equilibrium correlation function $C_{\theta }^{\mathrm{eq}}(r_{\perp },r_{\parallel }=0)L_{\parallel }$ is plotted as a function of the scaling variable $r_{\perp }/L_{\parallel }^2$ for different system sizes. The black line represents the analytical result given by Eq. (27).

Figure 5

Equilibrium behavior of the two-time correlation function $C_{\theta }(t,t_{\mathrm{w}})$ in systems containing short stripes. (a) $C_{\theta }(t,t_{\mathrm{w}})$ is plotted as a function of $t-t_{\mathrm{w}}$ for different system sizes $L_{\parallel }$. The waiting time $t_{\mathrm{w}}=5\times {10}^5$. (b) The same data, rescaled by $L_{\parallel }$, are plotted as a function of $(t-t_{\mathrm{w}})/L_{\parallel }^4$. The black line represents Eq. (29).

Figure 6

Coarsening and aging in systems comprised of long stripes. (a) Growth of the length scales ${\lambda }_{\parallel }^{*}\left(t\right)$ and ${\xi }_{\perp }\left(t\right)$ for $L_{\perp }=8{\lambda }_0$. The solid red line is a power law $\sim t^{1/4}$, indicating the ongoing growth of the dominant modulation length ${\lambda }_{\parallel }^{*}\left(t\right)$. The correlation length ${\xi }_{\perp }\left(t\right)$ approaches the system size $L_{\perp }$, indicated by the dashed horizontal line. The dashed vertical line represents the characteristic time ${\tau }_{\mathrm{C}}^{\perp }$ [Eq. (18)]. (b) Aging dynamics in the same system. $C_{\theta }(t-t_{\mathrm{w}})$ is plotted as a function of the difference $t-t_{\mathrm{w}}$. The solid red lines represent Eq. (31). (c) Scaling behavior of $C_{\theta }(t,t_{\mathrm{w}})$. The data for different system sizes and waiting times collapse onto a single curve upon rescaling. The solid red line represents the rescaled $C_{\theta }(t,t_{\mathrm{w}})$ according to Eq. (31). The dashed black line is a power law indicating the asymptotic behavior of the scaling function.

Figure 7

Pathway to equilibrium in square systems. (a) Growth of the length scales ${\lambda }_{\parallel }^{*}\left(t\right)$ and ${\xi }_{\perp }\left(t\right)$ in a system with $L_{\perp }=L_{\parallel }=16{\lambda }_0$. The corresponding characteristic times ${\tau }_{\mathrm{C}}^{\perp }$ and ${\tau }_{\mathrm{eq}}^{\parallel }$ are marked with vertical dashed lines. The dashed horizontal line indicates the system size, which limits the growth of both length scales. Note that ${\tau }_{\mathrm{C}}^{\perp }\ll {\tau }_{\mathrm{eq}}^{\parallel }$. (b) Equilibrium dynamics of the two-time correlation function for $L_{\perp }=L_{\parallel }=3{\lambda }_0$, where $C_{\theta }^{\mathrm{eq}}(t-t_{\mathrm{w}})$ depends only on the difference $t-t_{\mathrm{w}}$. The inset shows the same data on a semilogarithmic scale. The solid red line is an exponential according to Eq. (33).

Figure 8

Growth and saturation of the peak intensity $S_{\theta }^{*}\left(t\right)$ in small square systems. The continuous black lines represent the theoretical result [Eq. (34)], while the dashed lines indicate the equilibration time ${\tau }_{\mathrm{eq}}^{\parallel }$. The inset shows the evolution of the structure factor $S_{\theta }(q_m^{\perp }=0,q_n^{\parallel },t)$ for the system with $L_{\perp }=L_{\parallel }=8{\lambda }_0$.

Figure 9

Common scaling behavior of the structure factor $S_{\theta }^{\mathrm{eq}}(q_m^{\perp }=0,q_n^{\parallel })$ in small square systems ($+$) and systems containing many short stripes ($\circ$). The black line represents a power law drawn to illustrate the scaling behavior at small wave numbers $q_n^{\parallel }$.

Figure 10

Finite-size scaling of the peak intensity $S_{\theta }^{*}$ in (a), (b) square systems and (c), (d) rectangular systems consisting of short stripes. In (a) and (c), the equilibrium value $S_{\theta }^{*,\mathrm{eq}}$ is plotted as a function of the system sizes $L_{\parallel }$. The dashed red lines represent a power law. In (b) and (d), the peak intensity $S_{\theta }^{*}\left(t\right)$ has been rescaled, demonstrating a scaling collapse for different system sizes. The black lines represent Eq. (34).

Figure 11

Map of the pathways to equilibrium. Closed (open) circles indicate that equilibrium was reached (not reached) in numerical simulations of Eq. (3). The dashed vertical lines represent the equilibration time ${\tau }_{\mathrm{eq}}^{\parallel }$ for a given value of $L_{\parallel }$. The solid red line represents the condition ${\tau }_{\mathrm{C}}^{\perp }<{\tau }_{\mathrm{eq}}^{\parallel }$. Systems located below this line approach equilibrium via the 1D MHc behavior. Systems above the solid red line display 2D equilibration behavior.