The entropy of a polymer confined in a curved surface and the elastic freeenergy 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 thein-plane strain energy plus Helfrich's curvature energy [Z. Naturforsch. C\textbf{28}, 693 (1973)]. The elastic constants in the free energy are obtainedby discussing two simplified models: one is the polymer membrane withoutin-plane strains and asymmetry between its two sides, which is the counterpartof quantum mechanics in curved surface [Jensen and Koppe, Ann. Phys.\textbf{63}, 586 (1971)]; another is the planar rubber membrane withhomogeneous in-plane strains. The equations to describe equilibrium shape andin-plane strains of the polymer vesicles by osmotic pressure are derived bytaking the first order variation of the total free energy containing theelastic free energy, the surface tension energy and the term induced by osmoticpressure. The critical pressure, above which spherical polymer vesicle willlose its stability, is obtained by taking the second order variation of thetotal free energy. It is found that the in-plane mode also plays important rolein the critical pressure because it couples with the out-of-plane mode.Theoretical results reveal that polymer vesicles possess the mechanicalproperties intermediate between fluid membranes and solid shells.
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机译：通过比例分析获得了限制在曲面中的聚合物的熵和由聚合物组成的膜的弹性自由能，发现膜的弹性自由能具有面内应变能加Helfrich曲率能的形式[Z. Naturforsch。 C \ textbf {28}，693（1973）]。通过讨论两个简化的模型获得自由能中的弹性常数：一个是聚合物膜，其两侧之间没有面内应变且不对称，这是曲面中量子力学的对应物[Jensen and Koppe，Ann。 Phys。\ textbf {63}，586（1971）]；另一个是具有均匀面内应变的平面橡胶膜。通过对包含弹性自由能，表面张力能和渗透压诱导的项的总自由能的一阶变化，得出通过渗透压描述聚合物囊泡的平衡形状和面内应变的方程式。通过取总自由能的二阶变化来获得临界压力，高于该临界压力球形聚合物囊泡将失去其稳定性。发现平面内模式在临界压力中也起着重要作用，因为它与平面外模式耦合。理论结果表明，聚合物囊泡具有介于流体膜和固体壳之间的机械性能。
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