Frozen states and order-disorder transition in the dynamics of confined membranes

The adhesion dynamics of a membrane confined between two permeable walls is studied using a two-dimensional hydrodynamic model. The membrane morphology decomposes into adhesion patches on the upper and the lower walls and obeys a nonlinear evolution equation that resembles that of phase-separation dynamics, which is known to lead to coarsening, i.e., to the endless growth of the adhesion patches. However, due to the membrane bending rigidity, the system evolves toward a frozen state without coarsening. This frozen state exhibits an order-disorder transition when increasing the permeability of the walls.

Figure 1

Schematics of a membrane confined between two permeable walls.

Figure 2

Arrested dynamics and order-disorder transition. (a) Nonpermeable case. Frozen ordered patterns obtained from the numerical solution of the CH4 equation, Eq. (14). (b) Permeable case. Frozen disordered patterns obtained from the numerical solution of the TDGL4 equation, Eq. (13). In (a) and (b), the vertical scale is increased by a factor $\sim 10$ for a better visibility of the membrane morphology. (c) Saturation of the spatially averaged wavelength $\langle \lambda \rangle$ as a function of time. The black solid line and the red dotted line correspond to TDGL4 Eq. (13) and CH4 Eq. (14), respectively.

Figure 3

Snapshots of hydrodynamics flows and membrane profile during the dynamics in the conserved case [Eq. (14)]. Horizontal arrows represent the hydrodynamic flow. (a) The initial membrane profile is a single period of a sinusoid. (b) Intermediate times. (c) The final membrane profile exhibits plateaus separated by kinks.

Figure 4

(a) Linear dispersion relation. (b) Histogram of the distances between kinks. The solid line represents the value of minus the total curvature energy in one steady-state period ${\mathcal{L}}_{\lambda }=-{\int }_0^{\lambda }{\left[{\partial }_{XX}{H}_{\lambda }\left(X\right)\right]}^2$, obtained numerically from the periodic double-kink solution shown in Fig. 5. As discussed in Sec. 6, ${\mathcal{L}}_{\lambda }$ controls the stability of the steady states. The dashed line corresponds to the approximate expression of Eq. (26) with ${\mathcal{L}}_0=-0.43225$.

Figure 5

Periodic double-kink steady-state profile. The insets show a zoom on an oscillatory kink tail, and the oscillations of ${(H-H_m)}^2$ in log scale away from a kink. The transient dynamics leading to this periodic steady state is shown in Fig. 3.