Singular limits, saturation, and defects in block copolymer morphology

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RenDMS-0907777 A diblock copolymer molecule is a linear chain of anA-monomer block grafted covalently to a B-monomer block. Becauseof the repulsion between the unlike monomers, the different typesub-chains tend to segregate, but as they are chemically bondedin chain molecules, segregation of sub-chains cannot lead to amacroscopic phase separation. Only a local micro-phaseseparation occurs: micro-domains rich in A monomers andmicro-domains rich in B monomers emerge as a result. Anano-sized pattern formed from micro-domains is known as amorphology phase. The investigator develops singular limitmethods to study morphology phases that concentrate on points,curves, and surfaces in space, and that might not be found byexisting free energy methods. These methods reduce complicatednonlinear, nonlocal, variational, and partial differentialequation problems to simpler geometric problems. They analyzethe saturation phenomenon: a process of elongation, deformation,and breaking off of a small number of large objects to form alarge number of small objects. They also explain defects inmorphological phases caused by local-nonlocal competition ortopological constraints. In the case of block copolymer vesiclesthe investigator studies the bending rigidity in the free energy. The singular limit techniques are extended to problems withoutvariational structures, such as the Gierer-Meinhardt system forbiological morphogenesis in development. Block copolymers are soft condensed materials that incontrast to crystalline solids, are characterized by fluid-likedisorder on the molecular scale and a high degree of order on alonger length scale. An almost unlimited number of moleculararchitectures can be designed by modern nano-technologies toproduce materials with particular mechanical, electric, barrier,ionic and other physical properties. Common box tapes usetriblock copolymers to achieve pressure-sensitive adhesion. Block copolymers are blended with asphalt in road construction toreduce pavement cracking and rutting at low and high temperatureextremes. In this project the investigator studies patternformations within block copolymers that are related to changes inthe morphology phase of the material, and hence to larger-scalematerial properties. The project includes graduate students, whodevelop skills and knowledge in both mathematics and materialsscience.