Microphase Boundaries and Chain Conformations in Multiply Branched Diblock Copolymers Amalie Frischknecht* ,†,1 and G. H. Fredrickson ‡ Departments of Physics and Chemical Engineering, University of California, Santa Barbara, California 93106 Received March 12, 1999; Revised Manuscript Received July 16, 1999 ABSTRACT: We consider a melt of diblock copolymers consisting of a linear A block connected to a regularly branched B block. A simple “Alexander-de Gennes”-type calculation of the free energy for lamellar, cylindrical, and spherical microphases in strong segregation reveals large shifts in the phase boundaries as a function of the number of generations of the branched block and the functionality of the branch points. Modifying the calculation for the case of just two generations in the branched block, we find that a significant fraction of the second-generation branches fold back to lower the free energy. This is consistent with recent dendrimer theory, which also predicts that branches fold back toward the center. Nevertheless, controlled introduction of branching into copolymer blocks can evidently be quite effective in shifting phase boundaries and thereby influencing mesoscale morphology. I. Introduction Block copolymer systems are extremely versatile in that variations of composition, architecture, and choice of monomers can lead to dramatic changes in self- assembly at the mesoscale and, consequently, in proper- ties. 2 For example, commercial materials based on blocks of polystyrene and polybutadiene have properties that vary from high modulus, tough thermoplastics to soft, highly extensible thermoplastic elastomers. 3 To design block copolymers for specific applications it is necessary to have in hand two types of fundamental or empirical knowledge. The first is the relationship between the chemical composition and molecular archi- tecture of the copolymer and its self-assembly behavior. Second, one must establish the relationship between the various copolymer mesophases and the properties that are critical to the application. For example, toughness and high modulus are simultaneously achieved in certain grades of the styrenic block copolymers mar- keted by Phillips (K-Resins). 4 The high modulus derives from their high styrene content, while toughness can be attributed to their unique mesophase morphology. Phillips utilizes a complex radial architecture to achieve these key properties, which would be difficult or impos- sible to reproduce with a simple diblock or triblock architecture. The present paper is a contribution to the first body of knowledgesspecifically, we explore theo- retically the effect of introducing multiple branching into a copolymer block on the location of phase bound- aries between lamellar, cylindrical, and spherical mes- ophases. Theoretical advances over the past three decades have greatly improved our understanding of the connection between copolymer architecture (and composition) and self-assembly behavior in the melt state. Already in the work of Helfand in the late 1970s, 5 the relationship between block copolymer composition and the thermo- dynamic stability of the lamellar, cylindrical, and spherical mesophases was firmly established. In the 1980s, Leibler, 6 Semenov 7 and others further developed the analytical machinery to do such calculations. Mat- sen and Schick’s work of the 1990s was a significant advance on the numerical analysis side, which greatly facilitated the exploration of phase diagrams for a variety of new block copolymer architectures and com- plex mesophases. 8 Diblock copolymers with similarly shaped monomers and simple interactions (described by a composition- independent Flory  parameter) are well-known to have a phase diagram that is symmetric in the composition variable φ (the volume fraction of the B block). This symmetry can be weakly broken (i.e. phase boundaries shifted) by changing one statistical segment length relative to the other or by switching to a triblock architecture. 9 Calculations for (AB) n starblock copoly- mers have also demonstrated modest compositional shifts of phase boundaries for practical values of the number of starblock arms, n. 10 Much more substantial shifts of phase boundaries were predicted by Milner 11 for A n B m copolymers (n * m), where n A blocks and m B blocks are connected at a central junction point. This includes the important case of a simple A 2 B graft (or “Y”) copolymer. By forcing unequal numbers of A and B blocks to emerge from the junction points localized at A-B interfaces, the resulting imbalance of chain elasticity in such copolymer melts drives strong interfacial curvature and produces large phase boundary shifts from the symmetric diblock phase diagram. Milner’s predictions have been largely con- firmed by experimental investigations. 12 A disadvantage of the A n B m architecture is that it is more difficult to synthesize with precise control over arm number (particularly on commercial scales) than the more familiar (AB) n starblock architecture. An alternative architecture that might also lead to strongly asymmetric phase diagrams would be an AB diblock (or BAB triblock) copolymer with a linear A block and a multiply branched (but flexible) B block(s). Such materi- als might be prepared in styrenic copolymers by intro- ducing a small amount of a multifunctional monomer into a typical anionic polymerization recipe. Similar architectures involving a rigid dendrimer B block and a flexible A block have recently been synthesized, 13 but † Department of Physics. ‡ Department of Chemical Engineering. 6831 Macromolecules 1999, 32, 6831-6836 10.1021/ma990372w CCC: $18.00 © 1999 American Chemical Society Published on Web 09/11/1999