Theory of block copolymers. I. Domain formation in A-B block copolymers

Abstract

Microphase separation occurs in many block copolymers to give domain structures. In this first paper in a series dealing with domain formation and the consequences there of, a theory is presented for the formation of spherical domains in A-B block copolymers. The theory establishes criteria for the formation of domains and their size in terms of molecular and thermodynamic variables. It is shown that the considerable loss in configurational entropy due to the constraints on the spacial placement of chains in a domain structure requires that the critical block molecular weights required for domain formation are many-fold greater than required for phase separation of a simple mixture of the component blocks. The relation between domain radius R and molecular dimensions is obtained from the requirement that space in the domain must be filled with a constant density of segments. Segment densities are evaluated from solutions of the diffusion equation, treating the constraints on chain placement as boundary value problems. This gives the relationship R = 4/3 <L2>1/2, where <L2>1/2 is the root-mean-square end-to-end chain length. Because of chain perturbations in a domain system, <L2>1/2 is larger than the unperturbed value \documentclass{article}\pagestyle{empty}\begin{document}$ < {\rm L}^2 > _0^{1/2}$\end{document} normally expected for bulk polymers. A means to evaluate the perturbations is shown. The agreement between the predictions of the present theory and the limited published experimental information appears quite satisfactory.