Osmotic buckling of spherical capsules

Abstract

We study the buckling of elastic spherical shells under osmotic pressure with the osmolyte concentration of the exterior solution as a 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 qualitatively 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 relationship 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 analytical formula is derived for the relationship between the osmotic pressure, the elastic moduli and the volume of buckled capsules. This also allows use of elastic capsules as osmotic pressure sensors or deduction of 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.

Shallow shell theory of the buckling energy barrier: From the Pogorelov state to softening and imperfection sensitivity close to the buckling pressure

“…confirming the parameter dependence p c1 ∼ p c γ −1/4 [17]. Using the Pogorelov theory p c1 0.901(1 − ν 2 ) −1/4 p c γ −1/4 has been obtained [47], which slightly deviates from our above result p c1 0.786 p c γ −1/4 from shallow shell theory.…”

“…The linear stiffness k with respect to a point force is experimentally accessible in mechanical compression tests such as plate compression [49,50] or compression by microscopy tips [50,51] because all of these compression devices effectively act as point forces in the initial regime. Therefore our result (21) for the softening of the shell can be directly tested if such compression tests are combined with external pressure p. For microcapsules in liquids external pressure can be generated as osmotic pressure [17], and for macroscopic capsules as mechanical air or liquid pressure. For compressive pressure p = p c /2 at half the buckling pressure, Eq.…”

“…The energy barrier represents also an important feature of an spherical shell from a general theoretical point of view as the barrier vanishes upon approaching the buckling bifurcation and how it vanishes characterizes the critical behavior of the buckling bifurcation. In the mechanics literature, the unstable barrier state is often referred to as post-buckling state [14] as the shell already contains a dimple; the catastrophic nature of the buckling instability is reflected in a decreasing pressure p < p c of the barrier state, which leads to a snapthrough buckling [3,17]. Many quantitative analytical results on buckling of spherical shells are based on the Pogorelov theory, where the dimple is approximated as a mirror-inverted spherical cap-shaped indentation [18].…”

“…Such a rounded spherical cap is an approximative isometry of the spherical rest shape. For a fixed mechanical pressure p ≥ p c , the dimple will actually snap through and grow until opposite sides are in contact, whereas for osmotic pressure control or even volume control, a stable dimple shape is reached before opposite sides come into contact [3,17]. A deep dimple can also assume a polygonal shape in a secondary buckling transition [43][44][45].…”

“…Most of the known results for the barrier state and the energy barrier height have been derived in the limit p p c either from numerical work starting from energy minimization [6,7,9] or based on the Pogorelov energy scaling of the buckled state, which is only valid for relatively deep mirror-buckled indentations at p p c . The scaling of the energy barrier height E B ∝ (p/p c ) −3 and the depth of the barrier indentation z B ∝ (p/p c ) −2 can be derived using this Pogorelov scaling [9,16,17]. In the Pogorelov approach, numerical prefactors in the scaling results can be obtained from only an approximative variational energy minimization for the rounding of the sharp edge of inverted spherical cap shapes.…”