Origins of Complex Self-Assembly in Block Copolymers

Amphiphilic molecules are renowned for their ability to partition chemically immiscible components into nanoscale domains. Often these domains exhibit intriguing complex periodic geometries with long-range order. Surprisingly, the diverse systems that selfassemble in this manner, surfactants, lipids, soaps, and block copolymers, exhibit topologically identical geometries, suggesting to researchers that a common set of principles govern amphiphilic phase selection. From this association has emerged the belief that constant mean curvature (CMC) interfaces are generally good models for block copolymer microdomain geometries. By taking advantage of new developments in polymer theory, we accurately examine this hypothesis for the first time, and find it to be wrong. Furthermore, our study reveals new explanations for complex phase selection that are relevant to numerous block copolymer systems. A block copolymer consists of chemically distinct polymer chains (i.e., blocks) joined together to form a single macromolecule. As a consequence of a general tendency for the blocks to separate, tempered by the restriction imposed by the covalent bonds that connect them, these molecules exhibit amphiphilic behavior. Even in the simplest case, AB diblock copolymers, a rich assortment of ordered phases has been documented.1-8 The composition of the AB diblock (i.e., the volume fraction f of block A) controls the geometry of the structure (see Figure 1). For nearly symmetric diblocks (f ∼ /2), a lamellar (L) phase occurs. For moderate asymmetries, a complex bicontinuous state, known as the gyroid (G) phase, has been observed in which the minority blocks form domains consisting of two interweaving threefold-coordinated lattices.1,2 (Prior to the discovery of the G phase, a double-diamond (D) structure formed from two fourfold-coordinated lattices3 was erroneously associated with the bicontinuous state in these materials.4) Another complex structure, the perforated lamellar (PL) phase, occurs when the minority-component layers of the L phase develop a hexagonal arrangement of passages.5 At yet higher asymmetries, the minority component forms hexagonally packed cylinders (C) and then spheres (S) arranged on a bodycentered cubic lattice. Eventually, as f f 0 or 1, a disordered phase results. The complete mean field or rather self-consistent field theory (SCFT) for block copolymers was developed by Helfand and co-workers.9 However, at the time of its development, it had to be supplemented with approximations limiting its effectiveness. Nevertheless, important advances were made by examining this theory in the limits of weak10 and strong11 segregation. (The degree to which the A and B blocks segregate is determined by the product oN, where o is the FloryHuggins A/B interaction parameter and N is the total degree of polymerization.) The combination of these works established that the underlying physics controlling block copolymer phase behavior involves a competition between interfacial tension and the entropic penalty for stretching polymer coils so as to fill space uniformly. The balance determines the equilibrium size of the microdomains and dictates the geometry of the structure. Although these earlier approaches correctly predicted the classical phases (i.e., L, C, and S),10,11 they failed to account for the more recently discovered complex phases (i.e., G and PL).12,13 With new advances,14 it is now possible to implement the full SCFT. The first calculations14,15 to do so evaluated the free energies of the structures described above and established the phase diagram. This demonstrated that complex phase behavior occurs in the intermediatesegregation regime as opposed to the weakand strongsegregation regimes treated by Leibler10 and Semenov,11 respectively. For intermediate segregation (e.g., oN ) 20), the new calculations predict the sequence L f G f C f S f disordered as f progresses from /2 to either 0 or 1. Although PL is absent from this sequence, it is nearly stable at the L/G phase boundary, consistent with where it is observed experimentally.6 This supports very recent experiments indicating that the PL structure is only a long-lived metastable state. The D phase is clearly unstable, in agreement with current experiments.4,6 Given this theoretical accomplishment, we now probe deeper into the theory14 to examine the physical factors responsible for complex phase behavior. As described below, the explanation lies in the detailed shape of the dividing interface between the A and B microdomains. Earlier works, such as that of Semenov,11 illustrate that the phase transitions are driven by a tendency to curve the interface as the diblocks become asymmetric in composition. The curvature allows the molecules to balance the degree of stretching between the A and B blocks. We demonstrate this quantitatively in Figure * Present address: Polymer Science Centre, University of Reading, Whiteknights, Reading RG6 6AF, UK. Figure 1. Area-averaged mean curvature 〈H〉 as a function of the A-block volume fraction f for each of the structures shown schematically calculated using self-consistent meanfield theory.14,15 The stable and metastable states are shown with solid and dashed lines, respectively, and transitions are denoted by dots. As the molecules become asymmetric, structures with more curvature are preferred. 7641 Macromolecules 1996, 29, 7641-7644