Adhesion of vesicles in two dimensions

The adhesion of vesicles in two dimensions is studied by solving the shape equations that determine the state of lowest energy. Two ensembles are considered where for a fixed circumference of the vesicle either a pressure difference between the exterior and the interior is applied or the enclosed area is prescribed. First, a short discussion of the shape of free vesicles is given. Then, vesicles confined to a wall by an attractive potential are considered for two cases: (i) For a contact potential, a universal boundary condition determines the contact curvature as a function of the potential strength and the bending rigidity. Bound shapes are calculated, and an adhesion transition between bound and free states is found, which arises from the competition between bending and adhesion energy. (ii) For adhesion in a potential with finite range, the crossover from the long-ranged to the short-ranged case is studied. For a short-ranged potential, a decrease in the strength of the potential can lead to a shape transition between a bound state and a ``pinned'' state, where the vesicle acquires its free shape but remains pinned by the potential. In such a potential, fluctuations lead to unbinding for which two different cases are found. Small vesicles unbind via fluctuations of their position, while large vesicles unbind via shape fluctuations.