A mean-field phase diagram for conformationally symmetric diblock melts using the standard Gaussian polymer model is presented. Our calculation, which traverses the weak- to strong-segregation regimes, is free of traditional approximations. Regions of stability are determined for disordered (DIS) melts and for ordered structures including lamellae (L), hexagonally packed cylinders (H), body-centered cubic spheres (Q Im3̄m ), close-packed spheres (CPS), and the bicontinuous cubic network with Ia3̄d symmetry (Q Ia3̄d ). The CPS phase exists in narrow regions along the order−disorder transition for χN ≥ 17.67. Results suggest that the Q Ia3̄d phase is not stable above χN ∼ 60. Along the L/Q Ia3̄d phase boundaries, a hexagonally perforated lamellar (HPL) phase is found to be nearly stable. Our results for the bicontinuous Pn3̄m cubic (Q Pn3̄m ) phase, known as the OBDD, indicate that it is an unstable structure in diblock melts. Earlier approximation schemes used to examine mean-field behavior are reviewed, and comparisons are made with our more accurate calculation.

No abstract

As a result of important advances over the last decade, block copolymer melts have become an excellent model system for studying fundamental phenomena associated with molecular self-assembly. During this time, good quantitative agreement has been achieved between theory and experiment in regards to equilibrium phase behaviour, and with it has emerged a thorough understanding in terms of simple intuitive explanations. The theoretical contributions to this effort are largely attributed to mean-field calculations on a standard Gaussian model. Here, we review this present understanding of block copolymer phase behaviour, the model and its underlying assumptions, the mean-field approximation and its limitations, and the attempts to incorporate fluctuation corrections. Rather than following the traditional rigorous derivations, we present slightly more intuitive and transparent ones in such a way to stress the close connection between the related calculations. In this way, we hope to provide a valuable introduction to block copolymer theory.

The interactions between mesophase-forming copolymers and nanoscopic particles can lead to highly organized hybrid materials. The morphology of such composites depends not only on the characteristics of the copolymers, but also on the features of the nanoparticles. To explore this vast parameter space and predict the mesophases of the hybrids, we have developed a mean field theory for mixtures of soft, flexible chains and hard spheres. Applied to diblock-nanoparticle mixtures, the theory predicts ordered phases where particles and diblocks self-assemble into spatially periodic structures. The method can be applied to other copolymer-particle mixtures and can be used to design novel composite architectures.

No abstract

We examine weakly segregated blends of AB diblock copolymer and A homopolymer with similar degrees of polymerization. The relative stability of numerous phases is examined and phase diagrams are constructed using self-consistent field theory. While the pure diblock system is only found to exhibit the body-centered cubic (spherical), hexagonal (cylindrical), bicontinuous Ia3d cubic (gyroid), and lamellar ordered phases, we find that the addition of homopolymer stabilizes close-packed spherical, bicontinuous Pn3m cubic (double-diamond), and hexagonally-perforated (catenoid) lamellar phases. We find that, in general, the minority-component region of a microstructure can only accommodate a limited amount of homopolymer before macrophase separation occurs. On the other hand, the majority-component regions can swell indefinitely with the addition of homopolymer, eventually resulting in an unbinding transition. We associate the region of highly-swollen microstructures with the micellar region observed in real systems. For the lamellar and hexagonal phases, we examine the distribution of homopolymer within tbe microstructure, and for the lamellar phase, we calculate the effect of homopolymer on the dimensions of the Aand B-rich microdomains.

We develop a numerical method for examining complex morphologies in thin films of block copolymer using self-consistent field theory. Applying the method to confined films of symmetric diblock copolymer, we evaluate the stability of parallel, perpendicular, and mixed lamellar phases. In general, lamellar domains formed by the diblocks are oriented parallel to the film by surface fields. However, their orientation can flip to perpendicular when the natural period of the lamellae is incommensurate with the film thickness. Experiments and Monte Carlo simulations have indicated that mixed lamellar phases may also occur, but for symmetric diblocks, we find these phases to be slightly unstable relative to perpendicular lamellae. Nevertheless, just a small asymmetry in the molecule stabilizes a mixed lamellar phase. Although our work focuses on confined films, we do discuss the behavior that results when films are unconfined.

Melts of ABA triblock copolymer molecules with identical end blocks are examined using self-consistent field theory ͑SCFT͒. Phase diagrams are calculated and compared with those of homologous AB diblock copolymers formed by snipping the triblocks in half. This creates additional end segments which decreases the degree of segregation. Consequently, triblock melts remain ordered to higher temperatures than their diblock counterparts. We also find that middle-block domains are easier to stretch than end-block domains. As a result, domain spacings are slightly larger, the complex phase regions are shifted towards smaller A-segment compositions, and the perforated-lamellar phase becomes more metastable in triblock melts as compared to diblock melts. Although triblock and diblock melts exhibit very similar phase behavior, their mechanical properties can differ substantially due to triblock copolymers that bridge between otherwise disconnected A domains. We evaluate the bridging fraction for lamellar, cylindrical, and spherical morphologies to be about 40%-45%, 60%-65%, and 75%-80%, respectively. These fractions only depend weakly on the degree of segregation and the copolymer composition.