Periodic Area Minimization Surfaces in Microstructural Science

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 of component B. A dominant factor in the determination of the domain morphology is area-minimization of the intermaterial surface, subject to fixed volume fractions. Surfaces that satisfy this mathematical condition are said to have constant mean curvature. The geometry of such surfaces strongly influences material physical properties. We have discovered domain structures in microphase-separated diblock copolymers that closely approximate periodic surfaces of constant mean curvature. 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 three-dimensional mathematical models. Two structures are discussed: First is the double diamond microdomain morphology, associated with a newly discovered family of triply periodic constant mean curvature surfaces. Second, a doubly periodic boundary between lamellar microdomains, corresponding to a classically known minimal surface (Scherk’s First Surface), is described.