Catalyst screening: Refinement of the origin of the volcano curve and its implication in heterogeneous catalysis

Understanding the overall catalytic activity trend for rational catalyst design is one of the core goals in heterogeneous catalysis. In the past two decades, the development of density functional theory (DFT) and surface kinetics make it feasible to theoretically evaluate and predict the catalytic activity variation of catalysts within a descriptor-based framework. Thereinto, the concept of the volcano curve, which reveals the general activity trend, usually constitutes the basic foundation of catalyst screening. However, although it is a widely accepted concept in heterogeneous catalysis, its origin lacks a clear physical picture and definite interpretation. Herein, starting with a brief review of the development of the catalyst screening framework, we use a two-step kinetic model to refine and clarify the origin of the volcano curve with a full analytical analysis by integrating the surface kinetics and the results of first-principles calculations. It is mathematically demonstrated that the volcano curve is an essential property in catalysis, which results from the self-poisoning effect accompanying the catalytic adsorption process. Specifically, when adsorption is strong, it is the rapid decrease of surface free sites rather than the augmentation of energy barriers that inhibits the overall reaction rate and results in the volcano curve. Some interesting points and implications in assisting catalyst screening are also discussed based on the kinetic derivation. Moreover, recent applications of the volcano curve for catalyst design in two important photoelectrocatalytic processes (the hydrogen evolution reaction and dye-sensitized solar cells) are also briefly discussed.

Optimal 5-step nilpotent quadratic algebras

By the Golod–Shafarevich theorem, an associative algebra $R$ given by $n$ generators and $<n^2/3$ homogeneous quadratic relations is not 5-step nilpotent. We prove that this estimate is optimal. Namely, we show that for every positive integer $n$, there is an algebra $R$ given by $n$ generators and $\lceil n^2/3\rceil$ homogeneous quadratic relations such that $R$ is 5-step nilpotent.

Two problems from the Polishchuk and Positselski book on Quadratic algebras

In the book ’Quadratic algebras’ by Polishchuk and Positselski [23] algebras with a small number of generators (n = 2, 3) are considered. For some number r of relations possible Hilbert series are listed, and those appearing as series of Koszul algebras are specified. The first case, where it was not possible to do, namely the case of three generators n = 3 and six relations r = 6 is formulated as an open problem. We give here a complete answer to this question, namely for quadratic algebras with dimA_1 = dimA_2 = 3, we list all possible Hilbert series, and find out which of them can come from Koszul algebras, and which can not. As a consequence of this classification, we found an algebra, which serves as a counterexample to another problem from the same book [23] (Chapter 7, Sec. 1, Conjecture 2), saying that Koszul algebra of finite global homological dimension d has dimA_1 > d. Namely, the 3-generated algebra A given by relations xx + yx = xz = zy = 0 is Koszul and its Koszul dual algebra A^! has Hilbert series of degree 4: HA! (t) = 1 + 3t + 3t^2 + 2t^3 + t^4, hence A has global homological dimension 4.