Periodic Area Minimizing Surfaces in Microstructural Science

Abstract

An A/B block copolymer consists of two macromolecules bonded together. In forming an equilibrium structure, such a material may separate into distinct phases, creating domains of component A and component B A dominant factor in the determination of the domain morphology is area-minimization of the intermaterial surface, subject to fixed volume fraction. Surfaces that satisfy this mathematical condition are said to have constant mean curvature. The geometry of such surfaces strongly influences physical properties of the material, and they have been proposed as structural models in a variety of physical and biological systems. We have discovered domain structures in phase-separated diblock copolymers that closely approximate periodic constant mean curvature surfaces. Transmission electron microscopy and computer simulation are used to deduce the three-dimensional microstructure by comparison of tilt series with two-dimensional image projection simulations of 3-D mathematical models. Three structures are discussed: the first of which is the double diamond microdomain morphology associated to a newly discovered family of triply periodic constant mean curvature surfaces. Second, a doubly periodic boundary between lamellar microdomains, corresponding to a classically known surface (called Scherk’s First Surface), is described. Finally, we show a lamellar morphology in thin films that is apparently related to a new family of periodic surfaces.