Photo-assisted hydrosilylation of silicon nanoparticles: Dependence of particle size on grafting chemistry

Silicon nanoparticles between 2.5 nm and 30 nm in diameter were functionalized by means of photoassisted hydrosilylation reactions in the aerosol phase with terminal alkenes of varying chain length. Using infrared spectroscopy and nuclear magnetic resonance, the chemical composition of the alkyl layer was determined for each combination of particle size and alkyl chain length. The spectroscopic techniques were used to determine that smaller particles functionalized with short chains in the aerosol phase tend to attach to the interior (β) alkenyl carbon atom, whereas particles >10 nm in diameter exhibit attachment primarily with the exterior (α) alkenyl carbon atom, regardless of chain length. © 2011 American Chemical Society.

The post-processed Galerkin method applied to non-linear shell vibrations

We present the results of a computational study of the post-processed Galerkin methods put forward by Garcia-Archilla et al. applied to the non-linear von Karman equations governing the dynamic response of a thin cylindrical panel periodically forced by a transverse point load. We spatially discretize the shell using finite differences to produce a large system of ordinary differential equations (ODEs). By analogy with spectral non-linear Galerkin methods we split this large system into a 'slowly' contracting subsystem and a 'quickly' contracting subsystem. We then compare the accuracy and efficiency of (i) ignoring the dynamics of the 'quick' system (analogous to a traditional spectral Galerkin truncation and sometimes referred to as 'subspace dynamics' in the finite element community when applied to numerical eigenvectors), (ii) slaving the dynamics of the quick system to the slow system during numerical integration (analogous to a non-linear Galerkin method), and (iii) ignoring the influence of the dynamics of the quick system on the evolution of the slow system until we require some output, when we 'lift' the variables from the slow system to the quick using the same slaving rule as in (ii). This corresponds to the post-processing of Garcia-Archilla et al. We find that method (iii) produces essentially the same accuracy as method (ii) but requires only the computational power of method (i) and is thus more efficient than either. In contrast with spectral methods, this type of finite-difference technique can be applied to irregularly shaped domains. We feel that post-processing of this form is a valuable method that can be implemented in computational schemes for a wide variety of partial differential equations (PDEs) of practical importance.