A study of morphologies in block copolymers and Langmuir films

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Morphology patterns that appear in complex materials areintimately related to their physical properties. Examples includelamellar, cylindrical, and spherical patterns in block copolymers,neutral and charged Langmuir monolayers, and smectic liquidcrystal films. The investigator and his collaborators analyze andpredict morphology patterns based on mathematical models derivedfrom first principles of statistical physics. They developtechniques to deal with nonlocal constitutive relations, singularperturbations, and critical eigenvalues. These mathematicalphenomena are proved to be the reasons behind self-organizationand pattern formation. The investigator finds effective reductionmethods that accurately simplify the original infinite-dimensionalproblems to manageable finite-dimensional ones. The investigatoralso studies the dynamic phenomenon of pattern nucleation inmaterials by finding and characterizing some unstable solutions tothe Euler-Lagrange equations of the free energy. The projectexpands our knowledge of singularly perturbed variationalproblems. It enhances classical theories such as Gammaconvergence. Complex materials, such as block copolymers, are usedeverywhere. The polyurethane foams used in upholstery and beddingare composed of multi-block copolymers known as thermoplasticelastomers that combine high temperature resilience and lowtemperature flexibility. Common box tapes use triblock copolymersto achieve pressure-sensitive adhesion. Block copolymers areblended with asphalt in road construction to reduce pavementcracking and rutting at low and high temperature extremes. Thisproject seeks to deepen our understanding of the mathematicaltheories underlying these systems. It produces effective methodsthat characterize and predict the mechanical, optical, electrical,ionic, barrier, and other physical properties of these materials. They help today's synthetic chemistry technologies to produceexquisitely structured materials to meet an ever rising demandfrom civil infrastructure and manufacturing.