Convergence of phase-field approximations to the Gibbs–Thomson law

We prove the convergence of phase-field approximations of the Gibbs–Thomson law. This establishes a relation between the first variation of the Van der Waals–Cahn–Hilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs–Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn–Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta–Kawasaki as a model for micro-phase separation in block-copolymers.     

Synthese von enantiomerenreinen ‘Grasshopper’-Ketonen und verwandten Verbindungen

Synthesis of Optically Pure Grasshopper Ketone and of Its Diastereoisomers and Related Compounds Starting from our previously described synthon 1 , the synthesis of four enantiomerically pure grasshopper ketons (diastereoisomeric 4-(2′,4′-dihydroxy-2′,6′,6′-trimethylcyclohexylidene)but-3-en-2-ons) and of their oxo derivatives was performed. Spectral and chiroptical data are presented.     

Eine Methode zur Sichtbarmachung der Benetzung

Zusammenfassung Es wird eine besondere optische Anordnung beschrieben, mit deren Hilfe die Benetzung in Glasr?hren sehr deutlich sichtbar gemacht werden kann. Die Methode gestattet auch, sehr dünne, sonst unsichtbare Benetzungsfilme nachzuweisen.