Morphology patterns that appear in complex materials are intimately related to their physical properties. Examples include lamellar, cylindrical, and spherical patterns in block copolymers, neutral and charged Langmuir monolayers, and smectic liquid crystal films. The investigator and his collaborators analyze and predict morphology patterns based on mathematical models derived from first principles of statistical physics. They develop techniques to deal with nonlocal constitutive relations, singular perturbations, and critical eigenvalues. These mathematical phenomena are proved to be the reasons behind self-organization and pattern formation. The investigator finds effective reduction methods that accurately simplify the original infinite-dimensional problems to manageable finite-dimensional ones. The investigator also studies the dynamic phenomenon of pattern nucleation in materials by finding and characterizing some unstable solutions to the Euler-Lagrange equations of the free energy. The project expands our knowledge of singularly perturbed variational problems. It enhances classical theories such as Gamma convergence.
Complex materials, such as block copolymers, are used everywhere. The polyurethane foams used in upholstery and bedding are composed of multi-block copolymers known as thermoplastic elastomers that combine high temperature resilience and low temperature flexibility. 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. This project seeks to deepen our understanding of the mathematical theories underlying these systems. It produces effective methods that characterize and predict the mechanical, optical, electrical, ionic, barrier, and other physical properties of these materials. They help today's synthetic chemistry technologies to produce exquisitely structured materials to meet an ever rising demand from civil infrastructure and manufacturing.
