A diblock copolymer molecule is a linear chain of an A-monomer block grafted covalently to a B-monomer block. Because of the repulsion between the unlike monomers, the different type sub-chains tend to segregate, but as they are chemically bonded in chain molecules, segregation of sub-chains cannot lead to a macroscopic phase separation. Only a local micro-phase separation occurs: micro-domains rich in A monomers and micro-domains rich in B monomers emerge as a result. A nano-sized pattern formed from micro-domains is known as a morphology phase. The investigator develops singular limit methods to study morphology phases that concentrate on points, curves, and surfaces in space, and that might not be found by existing free energy methods. These methods reduce complicated nonlinear, nonlocal, variational, and partial differential equation problems to simpler geometric problems. They analyze the saturation phenomenon: a process of elongation, deformation, and breaking off of a small number of large objects to form a large number of small objects. They also explain defects in morphological phases caused by local-nonlocal competition or topological constraints. In the case of block copolymer vesicles the investigator studies the bending rigidity in the free energy. The singular limit techniques are extended to problems without variational structures, such as the Gierer-Meinhardt system for biological morphogenesis in development.
Block copolymers are soft condensed materials that in contrast to crystalline solids, are characterized by fluid-like disorder on the molecular scale and a high degree of order on a longer length scale. An almost unlimited number of molecular architectures can be designed by modern nano-technologies to produce materials with particular mechanical, electric, barrier, ionic and other physical properties. Common box tapes use triblock copolymers to achieve pressure-sensitive adhesion. Block copolymers are blended with asphalt in road construction to reduce pavement cracking and rutting at low and high temperature extremes. In this project the investigator studies pattern formations within block copolymers that are related to changes in the morphology phase of the material, and hence to larger-scale material properties. The project includes graduate students, who develop skills and knowledge in both mathematics and materials science.
