  As solute molecules permeate into a vesicle due to a concentration difference
across its membrane, the vesicle swells through osmosis. The swelling can be
divided into two stages: (a) an "ironing" stage, where the volume-to-area ratio
of the vesicle increases without a significant change in its area; (b) a
stretching stage, where the vesicle grows while remaining essentially
spherical, until it ruptures. We show that the crossover between these two
stages can be represented as a broadened continuous phase transition.
Consequently, the swelling curves for different vesicles and different
permeating solutes can be rescaled into a single, theoretically predicted,
universal curve. Such a data collapse is demonstrated for giant unilamellar
POPC vesicles, osmotically swollen due to the permeation of urea, glycerol, or
ethylene glycol. We thereby gain a sensitive measurement of the solutes'
membrane permeability coefficients, finding a concentration-independent
coefficient for urea, while those of glycerol and ethylene glycol are found to
increase with solute concentration. In addition, we use the width of the
transition, as extracted from the data collapse, to infer the number of
independent bending modes that affect the thermodynamics of the vesicle in the
transition region.
