  We study the buckling of elastic spherical shells under osmotic pressure with
the osmolyte concentration of the exterior solution as control parameter. We
compare our results for the bifurcation behavior with results for buckling
under mechanical pressure control, that is, with an empty capsule interior. We
find striking differences for the buckling states between osmotic and
mechanical buckling. Mechanical pressure control always leads to fully
collapsed states with opposite sides in contact, whereas uncollapsed states
with a single finite dimple are generic for osmotic pressure control. For
sufficiently large interior osmolyte concentrations, osmotic pressure control
is qualititatively similar to buckling under volume control with the volume
prescribed by the osmolyte concentrations inside and outside the shell. We
present a quantitative theory which also captures the influence of shell
elasticity on the relation between osmotic pressure and volume. These findings
are relevant for the control of buckled shapes in applications. We show how the
osmolyte concentration can be used to control the volume of buckled shells. An
accurate analytic formula is derived for the relation between the osmotic
pressure, the elastic moduli and the volume of buckled capsules. This also
allows to use elastic capsules as osmotic pressure sensors or to deduce elastic
properties and the internal osmolyte concentration from shape changes in
response to osmotic pressure changes. We apply our findings to published
experimental data on polyelectrolyte capsules.
