Brown University, Providence, RI/USA
This workshop will explore emergent phenomena in the context of small clusters, supramolecular self-assembly and the shape of self-assembled structures such as polymer vesicles. The emphasis will be on surprises which arise when common conditions are not satisfied, for instance when the number of components is small, or they are highly non-spherical, or there are several types of components. Interactions vary from hard sphere repulsion to competition between coarse-grained liquid-crystalline ordering competing with shape deformation. Examples of this behavior are common in materials such as bulk homopolymers (rubber), copolymers, liquid crystals and colloidal aggregates. A basic mathematical setting would be to consider small clusters of hard spheres with isotropic short-range attractions and study the shape of the clusters as a function of the number of components. One known surprise is that highly symmetric structures are suppressed by rotational entropy. This emphasizes the need to accurately count the number of particle configurations that lead to the same final state. Small clusters can also generate anisotropic building blocks which can in turn serve as nano- or meso-scale building blocks for supermolecules and bulk materials (supramolecular chemistry) freed from the limited scope of atoms and quantum-mechanical bonding. These structures frequently possess topological defects in their ground states because they lower the energy. The challenge is to determine the shape and equilibrium defect structure of such superatoms and the number and geometry of their arrangement. The number of defects determines the effective valence of the super atoms and the global geometry of their arrangement determines the types of directional bonding possible when defects are linked together. The phenomenon of the appearance of singularities/defects because they are minimizers not necessarily required by topology or boundary conditions is also encountered in the study of harmonic maps. Moving up to self-assembly of large numbers of units, block copolymers self-assemble into a wide variety of structures including vesicles, nano-fibers and tori. Many of the structures formed are essentially two-dimensional surfaces embedded in R3. The mathematical challenge is to find both the shape and the order of the assembled object. This requires minimizing of a functional that depends on both the local and global order of the relevant matter fields and the shape of the surface.