Elasticity of polymer vesicles by osmotic pressure: An intermediate theory between fluid membranes and solid shells

The entropy of a polymer confined in a curved surface and the elastic free energy of a membrane consisting of polymers are obtained by scaling analysis. It is found that the elastic free energy of the membrane has the form of the in-plane strain energy plus Helfrich's curvature energy [Z. Naturforsch. C 28, 693 (1973)]. The elastic constants in the free energy are obtained by discussing two simplified models: one is the polymer membrane without in-plane strains and asymmetry between its two sides, which is the counterpart of quantum mechanics in a curved surface [H. Jensen and H. Koppe, Ann. Phys. (N.Y) 63, 586 (1971)]; the other is the planar rubber membrane with homogeneous in-plane strains. The equations to describe equilibrium shape and in-plane strains of the polymer vesicles by osmotic pressure are derived by taking the first-order variation of the total free energy containing the elastic free energy, the surface tension energy, and the term induced by osmotic pressure. The critical pressure above which a spherical polymer vesicle will lose its stability is obtained by taking the second-order variation of the total free energy. It is found that the in-plane mode also plays an important role in the critical pressure because it couples with the out-of-plane mode. Theoretical results reveal that polymer vesicles possess mechanical properties intermediate between those of fluid membranes and solid shells.