The packing of soft materials: Molecular asymmetry, geometric frustration and optimal lattices in block copolymer melts

Block copolymer systems are well known for their ability to self-assemble into a wide array of periodic structures. Due to the abundance and adaptability of physical theories describing polymers, this system is ideal for the development of robust and testible predictions about amphiphilic self-assembly phenomena at large. We review the results of field-theoretic treatments of block copolymer melts, with the aim of understanding how self-assembly in this system can be understood in terms of optimal lattice geometry. The self-consistent (mean) field theory of block copolymer melts as well as its low temperature limit, strong-segregation theory, are presented in detail, highlighting the special role played by asymmetry in the copolymer architecture. Special attention is paid to micellar configurations, where a well-defined and simple notion of optimal lattice geometry emerges from a particular asymptotic limit of the full self-consistent field theory. In this limit, the stability of competing arrangements of copolymer micelles can be assessed in terms of two discrete measures of the lattice geometry, emphasizing the non-trivial coupling between the internal configurations of the fundamentally soft micelles and the periodic symmetry of the lattice.