Transition to coarsening for confined one-dimensional interfaces with bending rigidity

We discuss the nonlinear dynamics and fluctuations of interfaces with bending rigidity under the competing attractions of two walls with arbitrary permeabilities. This system mimics the dynamics of confined membranes. We use a two-dimensional hydrodynamic model, where membranes are effectively one-dimensional objects. In a previous work [T. Le Goff et al., Phys. Rev. E 90, 032114 (2014)], we have shown that this model predicts frozen states caused by bending rigidity-induced oscillatory interactions between kinks (or domain walls). We here demonstrate that in the presence of tension, potential asymmetry, or thermal noise, there is a finite threshold above which frozen states disappear, and perpetual coarsening is restored. Depending on the driving force, the transition to coarsening exhibits different scenarios. First, for membranes under tension, small tensions can only lead to transient coarsening or partial disordering, while above a finite threshold, membrane oscillations disappear and perpetual coarsening is found. Second, potential asymmetry is relevant in the nonconserved case only, i.e., for permeable walls, where it induces a drift force on the kinks, leading to a fast coarsening process via kink-antikink annihilation. However, below some threshold, the drift force can be balanced by the oscillatory interactions between kinks, and frozen adhesion patches can still be observed. Finally, at long times, noise restores coarsening with standard exponents depending on the permeability of the walls. However, the typical time for the appearance of coarsening exhibits an Arrhenius form. As a consequence, a finite noise amplitude is needed in order to observe coarsening in observable time.

Figure 1

Schematic of a confined membrane. The wall on the top is at $z=h_0$, and the bottom wall is at $z=-h_0$. The blue solid line at $h(x,t)$ is the height of the membrane. Quantities above the membrane are written with $+$ and those below with $-$. We discuss the influence of three physical ingredients on the dynamics of confined membranes (a) a tension $\sigma$, (b) an asymmetry of the adhesion potential $\mathcal{U}\left(h\right)$, and (c) thermal fluctuations.

Figure 2

Nonconserved dynamics. (a) Average wavelength as a function of time. Black solid line corresponds to $\gamma =2$, and the red dashed line to $\gamma =3$. (b) Zeros of the membrane profile for $\gamma =2$ and (c) for $\gamma =3$. Black points correspond to the condition $h=0$ and ${\partial }_{x}h>0$, and red points correspond to $h=0$ and ${\partial }_{x}h<0$.

Figure 3

Conserved dynamics. (a) Membrane profile as a function of time for $\gamma =2$. (b) Trajectories of the zeros of the membrane profile. Bottom: Full simulation; top: subsequent dynamics obtained from the kink model. (c) Same plots as (b) for $\gamma =3$. (d) Average wavelength as a function of time. Interrupted coarsening is found in the lower curve with $\gamma =2$: The black solid line corresponds to the full dynamics, and the green dashed line is obtained from the kink model for $\gamma =2$. Endless coarsening is found in the upper curve with $\gamma =3$: The red solid line denotes full dynamics and the blue dotted line the kink model.

Figure 4

In black with circles, ${\mathcal{L}}_{\lambda }$ obtained by simulations as a function of $\lambda$ for $\gamma =2$ and mean height $\overline{H}=0$. The red vertical line gives the position of the most unstable wavelength ${\lambda }_m$ in the linear analysis. The solid green histogram is the distribution of wavelengths for the early dynamics of the membrane with $\gamma =2$, and the dotted blue histogram is the long-time distribution.

Figure 5

Membrane dynamics in an asymmetric double-well potential. (a) Asymmetric potential $U\left(H\right)$ for $H_m=0.9$ and $\beta ={\beta }_0\simeq 0.281$ or $\beta ={\beta }_c\approx 0.041$. (b) Velocity $V$ of an isolated kink as a function of $\beta$. Circles: Full simulations. The red solid line is the linear prediction for small $\beta$, see text. [(c) and (d)] Nonconserved dynamics in an asymmetric potential for (c) $\beta =0.1$, which is intermediate between ${\beta }_c$ and ${\beta }_0$, and (d) $\beta =0.04$, which is smaller than ${\beta }_c$. (e) Steady-state distance length $l$ of an isolated adhesion patch on the upper wall as a function of $\beta$. The circles are found from full simulation, and the the line is the prediction of the kink model. The solid line represents the stable steady states and the dashed line the unstable ones. The insets show the dynamics of the membrane in specific cases.

Figure 6

Activated coarsening. (a) Normalized average distance $\lambda U_m^{1/4}{2}^{-1/2}$ between kinks as a function of normalized time. Noisy-TDGL kink dynamics: black solid lines, with, from bottom to top, $\widehat{D}=0.002,0.01,0.1,0.2,0.3$. Noisy-CH kink dynamics: red dashed lines, with, from bottom to top, $\widehat{D}=0.002,0.1,0.3$. Thin lines are expected power laws at long times, and the inset in semilog coordinates shows logarithmic coarsening in the case of low noise amplitude $\widehat{D}=0.002$ for TDGL (solid line) and CH (dashed line). (b) Normalized average distance $\lambda U_m^{1/4}{2}^{-1/2}$ between kinks as a function of normalized time. Noisy-TDGL4 kink dynamics: black solid lines, with, from bottom to top, $\widehat{D}=0.09,0.1,0.11,0.12$. Noisy-CH4 kink dynamics: red dashed lines, with, from bottom to top, $\widehat{D}=0.07,0.070.075,0.08,0.10.15,0.20.3$. (c) Time $t_c$ for the initiation of coarsening: Disks for TDGL4 and diamonds for CH4. (d) Energy of a periodic steady state ${\mathcal{E}}_{\lambda }^s$ as a function of its period $\lambda$ for the potential defined in Eq. (24), with $H_m=0.9$. The symbols (dashed line) indicate the full numerical solution of the membrane profile. The red solid line is the large $\lambda$ approximation from Eq. (44).