Coordination Chemistry of Gold Catalysts in Solution: A Detailed NMR Study

Coordination chemistry of gold catalysts bearing eight different ligands [L=PPh₃, JohnPhos (L2), Xphos (L3), DTBP, IMes, IPr, dppf, S‐tolBINAP (L8)] has been studied by NMR spectroscopy in solution at room temperature. Cationic or neutral mononuclear complexes LAuX (L=L2, L3, IMes, IPr; X=charged or neutral ligand) underwent simple ligand exchange without giving any higher coordinate complexes. For L2AuX the following ligand strength series was determined: MeOH?hex‐3‐yne <MeCN≈OTf^??Me₂S<2,6‐lutidine<4‐picoline<CF₃CO₂^?≈DMAP<TMTU<PPh₃<OH^?≈Cl^?. Some heteroligand complexes DTBPAuX exist in solution in equilibrium with the corresponding symmetrical species. Binuclear complexes dppf(AuOTf)₂ and S‐tolBINAP(AuOTf)₂ showed different behavior in exchange reactions with ligands depending on the ligand strength. Thus, PPh₃ causes abstraction of one gold atom to give mononuclear complexes LLAuPPh₃⁺ and (Ph₃P)_nAu⁺, but other N and S ligands give ordinary dicationic species LL(AuNu)₂²⁺. In reactions with different bases, LAu⁺ provided new oxonium ions whose chemistry was also studied: (DTBPAu)₃O⁺, (L2Au)₂OH⁺, (L2Au)₃O⁺, (L3Au)₂OH⁺, and (IMesAu)₂OH⁺. Ultimately, formation of gold hydroxide LAuOH (L=L2, L3, IMes) was studied. Ligand‐ or base‐assisted interconversions between (L2Au)₂OH⁺, (L2Au)₃O⁺, and L2AuOH are described. Reactions of dppf(AuOTf)₂ and S‐tolBINAP(AuOTf)₂ with bases provided more interesting oxonium ions, whose molecular composition was found to be [dppf(Au)₂]₃O₂²⁺, L8(Au)₂OH⁺, and [L8(Au)₂]₃O₂²⁺, but their exact structure was not established. Several reactions between different oxonium species were conducted to observe mixed heteroligand oxonium species. Reaction of L2AuNCMe⁺ with S^2? was studied; several new complexes with sulfide are described. For many reversible reactions the corresponding equilibrium constants were determined.     

Application of the Discrete Element Method to crack propagation and crack branching in a vitreous dense biopolymer material

Crack propagation in a vitreous biopolymer material is simulated using the Discrete Element Method (DEM), which models the brittle material as an assembly of particles bonded together. The simulations are compared to experiments combining a high-speed camera monitoring of crack branching together with a micromechancial testing of samples where local mixture mode is generated by introducing a stress concentrator. Our experimental results show unstable crack propagation and branching occurring upon crack deviation by the action of the stress concentrator. The validity of the DEM simulations is checked by comparing its result to the Finite Element Method (FEM) and to an analytical expression under similar conditions. DEM results show a higher sensitivity to mixed mode compared to FEM and a better match with the analytical formulation. Finally, crack branching is correctly predicted using DEM without any specific criterion for the initiation of secondary cracks.     

Inequalities between successive minima and intrinsic volumes of a convex body

The second theorem of Minkowski establishes a relation between the successive minima and the volume of a 0-symmetric convex body. Based on this theorem we will prove a series of inequalities connecting the product of certain successive minima with certain intrinsic volumes.