Deformation of injected vesicles adhering onto flat rigid substrates

This study is concerned with the determination of the mechanical behaviour of closed fluid lipid bilayer membranes (vesicles) under a uniform hydrostatic pressure, pressed against and adhering onto a flat homogeneous rigid substrate. Assuming that the initial and deformed shapes of the vesicle are axisymmetric, a variational statement of the problem is developed on the ground of the so-called spontaneous curvature model. In this setting, the vesicle is regarded as a closed surface in the three-dimensional Euclidean space and its equilibrium shapes are supposed to provide stationary values of the bending energy functional under the constraint of fixed total area. The corresponding Euler–Lagrange equations and natural boundary conditions are derived, the work done by the pressure being taken into account, and used to determine the forces and moments in the membrane. Several examples of surfaces representing possible equilibrium shapes of so loaded membranes are determined numerically.