Self-Sorting Supramolecular Polymerization: Helical and Lamellar Aggregates of Tetra-Bay-Acyloxy Perylene Bisimide

A new perylene bisimide (PBI), with a fluorescence quantum yield up to unity, self-assembles into two polymorphic supramolecular polymers. This PBI bears four solubilizing acyloxy substituents at the bay positions and is unsubstituted at the imide position, thereby allowing hydrogen-bond-directed self-assembly in nonpolar solvents. The formation of the polymorphs is controlled by the cooling rate of hot monomer solutions. They show distinctive absorption profiles and morphologies and can be isolated in different polymorphic liquid-crystalline states. The interchromophoric arrangement causing the spectral features was elucidated, revealing the formation of columnar and lamellar phases, which are formed by either homo- or heterochiral self-assembly, respectively, of the atropoenantiomeric PBIs. Kinetic studies reveal a narcissistic self-sorting process upon fast cooling, and that the transformation into the heterochiral (racemic) sheetlike self-assemblies proceeds by dissociation via the monomeric state.     

Lower bounds for the blowup rate of solutions of the Zakharov equation in dimension two

We consider the blowup solution ( u,n,v )( t ) of the Zakharov equations where u : R ² → C, n : R ² → R, v: R² → R² in the energy space H₁ = {(u,n,v) η H¹ × L² × L²}. We show that there is a constant c depending on the L²-norm of u₀ such that where T is the blowup time. We check that this estimate is optimal and give further applications. © 1996 John Wiley & Sons, Inc.