Mesophases of Regularly Branched Copolymers
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Mesophases of Regularly Branched Copolymers Galen T. Pickett Department of Physics and Astronomy California State University Long Beach 1250 Bellflower Blvd. Long Beach, CA 90840 ABSTRACT The phase diagram of A/B block copolymers is determined when the A block is a flexible, linear homopolymer, and the B block has a regularly branched tree structure with G generations. The predictions of a classical-path analysis and a lattice self-consistent mean field calculation are in general agreement. The phase diagram is considerably skewed toward keeping the A blocks inside cylindrical or spherical cores with the branched block forming a corona. There is little, if any, tendency for the branched arms to fold back to facilitate packing. INTRODUCTION As the architecture of hyperbranched molecules [1] has come under more and more spectacular control [2, 3, 4, 5], these molecules could well hold the key to the designed self-assembly of interfacial layers with bio-specific adhesion and surface protection [6] as well as a host of other applications. The application of interest here is gaining extra control over the morphology of linear-dendritic block copolymer melts. Ordinary multiblock copolymers are composed of long runs of same-type monomers (blocks) placed on a linear chain. The strong-segregation-limit phase diagram can be strongly affected by the branching of one of the blocks [7, 8, 9]. Roughly speaking, branchings of the B block favor interfacial curvature forcing the B chains to splay outward, relieving some packing constraints. The diagrams can be shifted towards keeping an extreme minority branched species on the exterior of the domains [8, 9], with a consequently severe distortion of the packed domains. An initial attempt to determine the properties of this system [8] employs an Alexander-deGennes ansatz [10, 11], in which all dendrimer free ends are localized on the same surface. In relaxing this requirement, I have turned to the “classical path” approximation [12, 13] and to Scheutjens and Fleer numerical lattice calculations [14]. While the classical path, the Alexander and the lattice calculations all give similarly distorted phase diagrams, the Alexander approximation is found to be particularly unsuited to estimating free energies for branched chains. CLASSICAL PATH MODEL The polymers in question have an A block composed of NA monomers grafted to a dendrimer composed of B monomers and G generations where each of the 2G 1 branched arms is composed of NB monomers. Thus, the total number of monomers on the chain is N NA 2G 1 NB . The thermodynamic incompatibility of the A and B monomers results in the surface tension γ between A and B rich domains. In the lamellar phase, the A blocks stretch a distance HA from the common AB interface, and the B blocks stretch a distance HB . Letting
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H HA HB be the overall stripe width, and letting φ stand for the overall volume fraction of the B species per chain, with σ copolymers per unit AB interface, the lamellar phase free energy is Flam
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