Response and instabilities of the lamellar phase of diblock copolymers under uniaxial stress

Abstract

We study the quasistatic behavior of the lamellar phase of diblock copolymers under uniaxial compression and tension along the normal direction of the layers, in both the weak segregation limit (WSL) and the strong segregation limit (SSL). In the SSL, we derive a (nonlinear) continuum free energy description of the system in terms of local displacement of the lamellar layers, and use this free energy to study the mechanical behaviors. While compression induces the usual Hookian elastic response (for strains or stresses that are not too large), tension leads to square-lattice wave undulations in the transverse directions when the strain exceeds a critical value. In the WSL close to the order–disorder transition temperature, compression can ``melt'' the lamellar phase to the isotropic phase; such a melting can take the form of three types of instabilities, a quasithermodynamic instability, a spinodal at controlled strain, and a mechanical instability at controlled stress. It is shown that the third instability always precedes the second one under controlled-stress conditions. For a weakly first-order transition, the quasithermodynamic instability precedes the mechanical instability; but for a (hypothetical) second-order transition, the mechanical instability appears first as the stress is increased. In the case of tension, a transverse square-lattice wave deformation again develops at a critical strain. This deformation can be followed by a subsequent melting of types similar to the compressional case, upon further increase of the stress or strain. In both the SSL and WSL, the modulus undergoes an abrupt decrease when layer undulation develops, to a value 7/15 of that before the onset of undulation. Because the critical strain for the onset of undulation is usually very small, the modulus for tension will appear different from the modulus for compression, the former being 7/15 of the latter. As a result of this decrease in the modulus, melting of the lamellar phase in the WSL will occur at larger strains under tension than under compression.

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