An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy

Using the approach of Rulla (1996 SIAM J. Numer. Anal. 33, 68-87)for analysing the time discretization error and assuming moreregularity on the initial data, we improve on the error boundderived by Barrett and Blowey (1996 IMA J. Numer. Anal. 16,257-287) for a fully practical piecewise linear finite elementapproximation with a backward Euler time discretization of amodel for phase separation of a multi-component alloy.     

Reaction mechanism governing formation of 1,3-bis(diphenylphosphino)propane-protected gold nanoclusters

This report outlines the determination of a reaction mechanism that can be manipulated to develop directed syntheses of gold monolayer-protected clusters (MPCs) prepared by reduction of solutions containing 1,3-bis(diphenylphosphino)propane (L(3)) ligand and Au(PPh(3))Cl. Nanocluster synthesis was initiated by reduction of two-coordinate phosphine-ligated [Au(I)LL'](+) complexes (L, L' = PPh(3), L(3)), resulting in free radical complexes. The [Au(0)LL'](?) free radicals nucleated, forming a broad size distribution of ligated clusters. Timed UV-vis spectroscopy and electrospray ionization mass spectrometry monitored the ligated Au(x), 6 ≤ x ≤ 13, clusters, which comprise reaction intermediates and final products. By employing different solvents and reducing agents, reaction conditions were varied to highlight the largest portion of the reaction mechanism. We identified several solution-phase reaction classes, including dissolution of the gold precursor, reduction, continuous nucleation/core growth, ligand exchange, ion-molecule reactions, and etching of colloids and larger clusters. Simple theories can account for the reaction intermediates and final products. The initial distribution of the nucleation products contains mainly neutral clusters. However, the rate of reduction controls the amount of reaction overlap occurring in the system, allowing a clear distinction between reduction/nucleation and subsequent solution-phase processing. During solution-phase processing, the complexes undergo core etching and core growth reactions, including reactions that convert neutral clusters to cations, in a cyclic process that promotes formation of stable clusters of specific metal nuclearity. These processes comprise "size-selective" processing that can narrow a broad distribution into specific nuclearities, enabling development of tunable syntheses.     

Finite element approximation of a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix

We consider a model for phase separation of a multi-componentalloy with a concentration-dependent mobility matrix and logarithmicfree energy. In particular we prove that there exists a uniquesolution for sufficiently smooth initial data. Further, we provean error bound for a fully practical piecewise linear finiteelement approximation in one and two space dimensions. Finallynumerical experiments with three components in one space dimensionare presented.