Phase diagram of selectively cross-linked block copolymers shows chemically microstructured gel

Abstract

We study analytically the intricate phase behavior of cross-linked $AB$ diblock copolymer melts, which can undergo two main phase transitions due to quenched random constraints: Gelation, i.e., spatially random localization of polymers forming a system-spanning cluster, is driven by increasing the number parameter $\mu$ of irreversible, type-selective cross-links between random pairs of $A$ blocks. Self-assembly into a periodic pattern of $A$/$B$-rich microdomains (microphase separation) is controlled by the $AB$ incompatibility $\chi$ inversely proportional to temperature. Our model aims to capture the system's essential microscopic features, including an ensemble of random networks that reflects spatial correlations at the instant of cross-linking. We identify suitable order parameters and derive a free-energy functional in the spirit of Landau theory that allows us to trace a phase diagram in the plane of $\mu$ and $\chi$. Selective cross-links promote microphase separation at higher critical temperatures than in uncross-linked diblock copolymer melts. Microphase separation in the liquid state facilitates gelation, giving rise to a novel gel state whose chemical composition density mirrors the periodic $AB$ pattern.