  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
\textbf{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 curved surface [Jensen and Koppe, Ann. Phys.
\textbf{63}, 586 (1971)]; another 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 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 important role
in the critical pressure because it couples with the out-of-plane mode.
Theoretical results reveal that polymer vesicles possess the mechanical
properties intermediate between fluid membranes and solid shells.

Soft Condensed Matter