We report on the modeling of the dynamics of confined lipid membranes. We derive a thin film model in the lubrication limit which describes an inextensible liquid membrane with bending rigidity confined between two adhesive walls. The resulting equations share similarities with the Swift–Hohenberg model. However, inextensibility is enforced by a time-dependent nonlocal tension. Depending on the excess membrane area available in the system, three different dynamical regimes, denoted as A, B and C, are found from the numerical solution of the model. In regime A, membranes with small excess area form flat adhesion domains and freeze. Such freezing is interpreted by means of an effective model for curvature-driven domain wall motion. The nonlocal membrane tension tends to a negative value corresponding to the linear stability threshold of flat domain walls in the Swift–Hohenberg equation. In regime B, membranes with intermediate excess areas exhibit endless coarsening with coexistence of flat adhesion domains and wrinkle domains. The tension tends to the nonlinear stability threshold of flat domain walls in the Swift–Hohenberg equation. The fraction of the system covered by the wrinkle phase increases linearly with the excess area in regime B. In regime C, membranes with large excess area are completely covered by a frozen labyrinthine pattern of wrinkles. As the excess area is increased, the tension increases and the wavelength of the wrinkles decreases. For large membrane area, there is a crossover to a regime where the extrema of the wrinkles are in contact with the walls. In all regimes after an initial transient, robust localised structures form, leading to an exact conservation of the number of adhesion domains.

中文翻译：

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密闭膜的粘附动力学

我们报告了密闭脂质膜动力学的建模。我们在润滑极限中导出了一个薄膜模型，该模型描述了一种不可伸展的液膜，其弯曲刚度限制在两个胶粘剂壁之间。所得方程与Swift–Hohenberg模型具有相似之处。但是，不可扩展性是由与时间有关的非局部张力引起​​的。根据系统中可用的多余膜面积，从模型的数值解中可以找到三种不同的动力学模式，分别表示为A，B和C。在方案A中，具有较小过量面积的膜形成平坦的粘附域并冻结。这种冻结是通过有效的曲率驱动的畴壁运动模型来解释的。在Swift–Hohenberg方程中，非局部膜张力趋向于为一个负值，该值对应于扁平畴壁的线性稳定性阈值。在方案B中，具有中间过量区域的膜表现出无休止的粗糙化，同时存在平坦的粘附域和皱纹域。在Swift–Hohenberg方程中，张力趋向于扁平畴壁的非线性稳定性阈值。在方案B中，被皱纹阶段覆盖的系统分数随过量区域线性增加。在方案C中，具有较大多余面积的膜完全被冷冻的迷宫式皱纹覆盖。随着多余区域的增加，张力增加，皱纹的波长减小。对于较大的膜面积，存在一种折皱方案，其中皱纹的末端与壁接触。