Diblock copolymer/homopolymer blends: Derivation of a density functional theory

Melts of diblock copolymer/homopolymer blends exhibit multiscale phase separation: (i) macrophase separation into homopolymer- and copolymer-rich macrodomains followed by (ii) microphase separation into A- and B-rich microdomains within the copolymer-rich macrodomains (cf. [S. Koizumi, H. Hasegawa, T. Hashimoto, Macromolecules 27 (1994) 6532; S. Koizumi, H. Hasegawa, T. Hashimoto, Macromolecules 27 (1994) 7893; H. Tanaka, H. Hasegawa, T. Hashimoto, Macromolecules 24 (1991) 240]). Following our previous derivation in [R. Choksi, X. Ren, On a derivation of a density functional theory for microphase separation of diblock copolymers, J. Stat. Phys. 113 (2003) 151-176], we derive a density functional theory for blends. This theory has been shown numerically to capture the multiscale separation (cf. [T. Ohta, A. Ito, Dynamics of phase separation in copolymer-homopolymer mixtures, Phys. Rev. E 52-5 (1995) 5250-5260; A. Ito, Domain patterns in copolymer-homopolymer mixtures, Phys. Rev. E 58-5 (1998) 6158-6165]). We also prove a result on local minimizers in one space dimension, confirming a lamellar multiscale phase separation.