Mark W. Matsen

The standard Gaussian model for block copolymer melts

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.

Block copolymer microstructures in the intermediate-segregation regime

A detailed examination of the intermediate-segregation regime of diblock copolymer melts is presented using the incompressible Gaussian chain model and self-consistent field theory (SCFT). We find that the competition between interfacial tension and chain stretching used to describe behavior in the strong-segregation regime also explains behavior in this regime. Phase transitions from lamellae (L) to cylinders (C) to spheres (S) occur due to the spontaneous curvature produced as the asymmetry in the diblock composition increases. Complex phases, gyroid (G), perforated lamellar (PL), and double diamond (D), have curvatures between those of L and C, and therefore they compete for stability along the L/C boundary. Nevertheless, only G exhibits a region of stability. To explain why, we recognize that interfacial tension prefers the formation of constant mean curvature (CMC) surfaces to reduce interfacial area, and chain stretching favors domains of uniform thickness so as to avoid packing frustration. While t...

Thin films of block copolymer

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.

Equilibrium behavior of symmetric ABA triblock copolymer melts

Melts of ABA triblock copolymer molecules with identical end blocks are examined using self-consistent field theory (SCFT). 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.

Conformationally asymmetric block copolymers

The standard parameters controlling AB diblock copolymer phase behavior are xN and fA, where x is an A-B segment interaction parameter, N is the overall degree of polymerization, and fA is the volume fraction of the A block. Recently, it has been recognized that the ratio of the A and B statistical segment lengths aA/aB also represents another important parameter. Here, we theoretically examine the effects of this latter parameter on the phase behavior using the standard Gaussian chain model. Calculations are performed using both self-consistent field theory (SCFT) and strong segregation theory (SST). The ratio aA/aB is shown to have strong effects on order- order phase boundaries. Furthermore, it significantly affects the relative stability of the complex phases. In particular, it enhances the metastability of the perforated lamel- lar phase and may actually cause it to become an equilibrium structure. We also illustrate that varying aA/aB produces large changes in the relative domain spacings at order-order phase boundaries, which could strongly affect the kinetics of these transitions. q 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 945-952, 1997

Gyroid versus double-diamond in ABC triblock copolymer melts

Monodisperse melts of ABC linear triblock copolymer are examined using self-consistent field theory (SCFT). Our study is restricted to symmetric triblocks, where the A and C blocks are equal in size and the A/B and B/C interactions are identical. Furthermore, we focus on the regime where B forms the majority domain. This system has been studied earlier using density functional theory (DFT), strong-segregation theory (SST), and Monte Carlo (MC), and it corresponds closely to a series of isoprene–styrene–vinylpyridine triblocks examined by Mogi and co-workers. In agreement with these previous studies, we find stable lamellar, complex, cylindrical, and spherical phases. In the spherical phase, the minority A and C domains alternate on a body-centered cubic lattice. In order to produce alternating A and C domains in the cylinder phase, the melt chooses a tetragonal packing rather than the usual hexagonal one. This amplifies the packing frustration in the cylinder phase, which results in a large complex phase ...

Liquid-crystalline behavior of rod-coil diblock copolymers

The exact mean-field phase behavior of the Semenov–Vasilenko model for rod-coil diblock copolymers is studied by applying self-consistent field techniques. The behavior depends on three quantities: the rod/coil immiscibility χN, the coil volume fraction f, and the ratio ν of the characteristic coil to rod dimensions. When χN≲5, the rods and coils mix producing a nematic phase, and at larger χN, they microphase separate forming a lamellar phase. The nonlamellar phases expected at f≳0.7 are not treated here. The lamellar phase is typically a smectic-C structure with monolayers of tilted rods. A thorough understanding of the model is achieved by closely examining segment distributions and various contributions to the free energy. The tilt angle θ is generally controlled by a competition between rod/coil interfacial tension and stretching of the coils. Lowering f reduces the latter contribution, causing θ→0 and producing a continuous transition to a smectic-A structure. Beyond that, there is a tendency to form structures with the rods arranged in bilayers, but this is strongly suppressed by a large steric penalty. We suggest that small amounts of solvent can greatly alleviate this penalty, and therefore could significantly affect some aspects of the phase behavior.

Autophobic dewetting of homopolymer on a brush and entropic attraction between opposing brushes in a homopolymer matrix

The wetting behavior of homopolymer on a chemically identical polymer brush is mathematically equivalent to the interaction between a pair of opposing brushes in a matrix of parent homopolymer. We examine both systems using self-consistent field theory (SCFT) with a new highly efficient and accurate algorithm. Our calculations provide compelling evidence that the global minimum in the free energy curve remains at a finite film thickness, implying that the brush/homopolymer interfacial tension, γb/h, is always positive favoring dewetting, or equivalently that an attraction always exists between opposing brushes. Nevertheless, we identify a region at low homopolymer molecular weights where γb/h is negligible, in which case complete wetting is highly metastable and the attraction between opposing brushes is extraordinarily weak. Furthermore, we demonstrate that SCFT is in good quantitative agreement with experiment. In contrast, we find that earlier predictions based on strong-stretching theory are terribly ...

Self-assembly of block copolymers

Recent advances in the application of self-consistent field theory permit the calculation of the free energy of ordered phases with any symmetry. Hence it is now possible to study, within this theory, the complex phases of self-assembling polymer systems starting from standard model polymer Hamiltonians.

Fast and accurate SCFT calculations for periodic block-copolymer morphologies using the spectral method with Anderson mixing.

We study the numerical efficiency of solving the self-consistent field theory (SCFT) for periodic block-copolymer morphologies by combining the spectral method with Anderson mixing. Using AB diblock-copolymer melts as an example, we demonstrate that this approach can be orders of magnitude faster than competing methods, permitting precise calculations with relatively little computational cost. Moreover, our results raise significant doubts that the gyroid (G) phase extends to infinite