Bismuth Redox Catalysis: An Emerging Main-Group Platform for Organic Synthesis

Bismuth has recently been shown to be able to maneuver between different oxidation states, enabling access to unique redox cycles that can be harnessed in the context of organic synthesis. Indeed, various catalytic Bi redox platforms have been discovered and revealed emerging opportunities in the field of main group redox catalysis. The goal of this perspective is to provide an overview of the synthetic methodologies that have been developed to date, which capitalize on the Bi redox cycling. Recent catalytic methods via low-valent Bi(II)/Bi(III), Bi(I)/Bi(III), and high-valent Bi(III)/Bi(V) redox couples are covered as well as their underlying mechanisms and key intermediates. In addition, we illustrate different design strategies stabilizing low-valent and high-valent bismuth species, and highlight the characteristic reactivity of bismuth complexes, compared to the lighter p-block and d-block elements. Although it is not redox catalysis in nature, we also discuss a recent example of non-Lewis acid, redox-neutral Bi(III) catalysis proceeding through catalytic organometallic steps. We close by discussing opportunities and future directions in this emerging field of catalysis. We hope that this Perspective will provide synthetic chemists with guiding principles for the future development of catalytic transformations employing bismuth

Compaction and dilation rate dependence of stresses in gas-fluidized beds

A particle dynamics-based hybrid model, consisting of monodisperse spherical solid particles and volume-averaged gas hydrodynamics, is used to study traveling planar waves (one-dimensional traveling waves) of voids formed in gas-fluidized beds of narrow cross sectional areas. Through ensemble-averaging in a co-traveling frame, we compute solid phase continuum variables (local volume fraction, average velocity, stress tensor, and granular temperature) across the waves, and examine the relations among them. We probe the consistency between such computationally obtained relations and constitutive models in the kinetic theory for granular materials which are widely used in the two-fluid modeling approach to fluidized beds. We demonstrate that solid phase continuum variables exhibit appreciable ``path dependence'', which is not captured by the commonly used kinetic theory-based models. We show that this path dependence is associated with the large rates of dilation and compaction that occur in the wave. We also examine the relations among solid phase continuum variables in beds of cohesive particles, which yield the same path dependence. Our results both for beds of cohesive and non-cohesive particles suggest that path-dependent constitutive models need to be developed.Comment: accepted for publication in Physics of Fluids (Burnett-order effect analysis added

Adjacency labeling schemes and induced-universal graphs

We describe a way of assigning labels to the vertices of any undirected graph on up to $n$ vertices, each composed of $n/2+O(1)$ bits, such that given the labels of two vertices, and no other information regarding the graph, it is possible to decide whether or not the vertices are adjacent in the graph. This is optimal, up to an additive constant, and constitutes the first improvement in almost 50 years of an $n/2+O(\log n)$ bound of Moon. As a consequence, we obtain an induced-universal graph for $n$-vertex graphs containing only $O(2^{n/2})$ vertices, which is optimal up to a multiplicative constant, solving an open problem of Vizing from 1968. We obtain similar tight results for directed graphs, tournaments and bipartite graphs