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Modern syntheses of colloidal nanocrystals yield extraordinarily narrow size distributions that are believed to result from a rapid “burst of nucleation” (La Mer, JACS , 1950, 72 (11), 4847–4854) followed by diffusion limited growth and size distribution focusing (Reiss, J. Chem. Phys. , 1951, 19 , 482). Using a combination of in situ X-ray scattering, optical absorption, and 13 C nuclear magnetic resonance (NMR) spectroscopy, we monitor the kinetics of PbS solute generation, nucleation, and crystal growth from three thiourea precursors whose conversion reactivity spans a 2-fold range. In all three cases, nucleation is found to be slow and continues during >50% of the precipitation. A population balance model based on a size dependent growth law (1/ r ) fits the data with a single growth rate constant ( k G ) across all three precursors. However, the magnitude of the k G and the lack of solvent viscosity dependence indicates that the rate limiting step is not diffusion from solution to the nanoparticle surface. Several surface reaction limited mechanisms and a ligand penetration model that fits data our experiments using a single fit parameter are proposed to explain the results. 

Abstract Carbon–carbon bond cleavage reactions, adapted to deconstruct aliphatic hydrocarbon polymers and recover the intrinsic energy and carbon value in plastic waste, have typically been catalysed by metal nanoparticles or air-sensitive organometallics. Metal oxides that serve as supports for these catalysts are typically considered to be inert. Here we show that Earth-abundant, non-reducible zirconia catalyses the hydrogenolysis of polyolefins with activity rivalling that of precious metal nanoparticles. To harness this unusual reactivity, our catalytic architecture localizes ultrasmall amorphous zirconia nanoparticles between two fused platelets of mesoporous silica. Macromolecules translocate from bulk through radial mesopores to the highly active zirconia particles, where the chains undergo selective hydrogenolytic cleavage into a narrow, C 18 -centred distribution. Calculations indicated that C–H bond heterolysis across a Zr–O bond of a Zr(O) 2 adatom model for unsaturated surface sites gives a zirconium hydrocarbyl, which cleaves a C–C bond via β-alkyl elimination. 

Ab initio computational studies have made tremendous progress in describing the behavior of molecular (homogeneous) catalysts and crystalline versions of heterogeneous catalysts, but not for amorphous heterogeneous catalysts. Even widely used industrial amorphous catalysts like atomically dispersed Cr on silica remain poorly understood and largely intractable to computational investigation. The central problems are that (i) the amorphous support presents an unknown quenched disordered structure, (ii) metal atoms attach to various surface grafting sites with different rates, and (iii) the resulting grafted sites have different activation and catalytic reaction kinetics. This study combines kernel regression and importance sampling techniques to efficiently model grafting of metal ions onto a non-uniform ensemble of support environments. Our analysis uses a simple model of the quenched disordered support environment, grafting chemistry, and catalytic activity of the resulting grafted sites. 

Ab initio calculations have greatly advanced our understanding of homogeneous catalysts and crystalline heterogeneous catalysts. In contrast, amorphous heterogeneous catalysts remain poorly understood. The principal difficulties include (i) the nature of the disorder is quenched and unknown; (ii) each active site has a different local environment and activity; (iii) active sites are rare, often less than ∼20% of potential sites, depending on the catalyst and its preparation method. Few (if any) studies of amorphous heterogeneous catalysts have ever attempted to compute site-averaged kinetics, because the exponential dependence on variable activation energy requires an intractable number of ab initio calculations to converge. We present a new algorithm using machine learning techniques (metric learning kernel regression) and importance sampling to efficiently learn the distribution of activation energies. We demonstrate the algorithm by computing the site-averaged activity for a model amorphous catalyst with quenched disorder.