Exact solution of pulled, directed vesicles with sticky walls in two dimensions

We analyse a directed lattice vesicle model incorporating both the binding-unbinding transition and the vesicle inflation-deflation transition. From the exact solution we derive the phase diagram for this model and elucidate scaling properties around the binding-unbinding critical point in this larger parameter space. We also consider how the phase diagram changes when a perpendicular force is applied to the end of a directed vesicle.