Repdigits as sums of three Pell numbers

In this paper, all base 10 repdigits expressible as sums of three Pell numbers are found.     

Efficient synthesis of branched poly(acrylic acid) derivatives via postpolymerization modification

The utility of pentafluorophenyl esters for the selective introduction of functional units and branch points in well-defined poly(acrylic acid) (PAA) derivatives is demonstrated using a combination of controlled radical polymerization and postpolymerization modification. Reversible addition-fragmentation chain transfer enables the synthesis of well-defined copolymers—poly(pentafluorophenyl acrylate-co-tert-butyl acrylate)—with the active ester repeat units serving as attachment points for reaction with primary amines, specifically tris(2-(t-butoxycarbonyl)ethyl)methyl amine (Behera's amine). Deprotection using trifluoroacetic acid removes both the backbone and side chain t-butyl esters to give a series of branched PAA derivatives containing novel tricarboxylic acid side chains that are well suited to complexation and multidentate interactions. Surprisingly, the active ester homopolymer is shown to have the highest reactivity with Behera's amine when compared to copolymers with lower incorporation of pentafluorophenyl esters, suggesting an intriguing interplay of neighboring group effects and steric interactions. The ability to tune the efficiency of postpolymerization modification gives a library of PAA derivatives.     

Two-Dimensional Germanium Sulfide Nanosheets as an Ultra-Stable and High Capacity Anode for Lithium Ion Batteries

Lithium ion batteries (LIBs) at present still suffer from low rate capability and poor cycle life during fast ion insertion/extraction processes. Searching for high-capacity and stable anode materials is still an ongoing challenge. Herein, a facile strategy for the synthesis of ultrathin GeS₂ nanosheets with the thickness of 1.1 nm is reported. When used as anodes for LIBs, the two-dimensional (2D) structure can effectively increase the electrode/electrolyte interface area, facilitate the ion transport, and buffer the volume expansion. Benefiting from these merits, the as-synthesized GeS₂ nanosheets deliver high specific capacity (1335 mAh g⁻¹ at 0.15 A g⁻¹), extraordinary rate performance (337 mAh g⁻¹ at 15 A g⁻¹) and stable cycling performance (974 mAh g⁻¹ after 200 cycles at 0.5 A g⁻¹). Importantly, our fabricated Li-ion full cells manifest an impressive specific capacity of 577 mAh g⁻¹ after 50 cycles at 0.1 A g⁻¹ and a high energy density of 361 Wh kg⁻¹ at a power density of 346 W kg⁻¹. Furthermore, the electrochemical reaction mechanism is investigated by the means of ex-situ high-resolution transmission electron microscopy. These results suggest that GeS₂ can use to be an alternative anode material and encourage more efforts to develop other high-performance LIBs anodes.     

Atomic-Scale Dispersed Fe-Based Catalysts Confined on Nitrogen-Doped Graphene for Li-S Batteries: Polysulfides with Enhanced Conversion Efficiency

Lithium-sulfur batteries have been considered as potential electrochemical energy-storage devices owing to their satisfactory theoretical energy density. Nonetheless, the inferior conversion efficiency of polysulfides in essence leads to fast capacity decay during the discharge/charge cycle. In this work, it is successfully demonstrated that the conversion efficiency of lithium polysulfides is remarkably enhanced by employing a well-distributed atomic-scale Fe-based catalyst immobilized on nitrogen-doped graphene (Fe@NG) as a coating of separator in lithium-sulfur batteries. The quantitative electrocatalytic efficiency of the conversion of lithium polysulfides is determined through cyclic voltammetry. It is also proven that the Fe-N_X configuration with highly catalytic activity is quite beneficial for the conversion of lithium polysulfides. In addition, the adsorption and permeation experiments distinctly indicate that the strong anchoring effect, originated from the charge redistribution of N doping into the graphene matrix, inhibits the movement of lithium polysulfides. Thanks to these advantages, if the as-prepared Fe@NG catalyst is combined with polypropylene and applied as a separator (Fe@NG/PP) in Li-S batteries, a high initial capacity (1616 mA h g⁻¹ at 0.1 C), excellent capacity retention (93 % at 0.2 C, 70 % at 2 C), and superb rate performance (820 mA h g⁻¹ at 2 C) are achieved.     

Carbon-Based Materials as Lithium Hosts for Lithium Batteries

Lithium (Li) metal has attracted significant attention in areas that range from basic research to various commercial applications due to its high theoretical specific capacity (3860 mA h g⁻¹) and low electrochemical potential (−3.04 vs. standard hydrogen electrode). However, dendrites often form on the surfaces of Li metal anodes during cycling and thus lead to battery failure and, in some cases, raise safety concerns. To overcome this problem, a variety of approaches that vary the electrolyte, membrane, and/or anode have been proposed. Among these efforts, the use of three-dimensional frameworks as Li hosts, which can homogenize and minimize the current density at the anode surface, is an effective approach to suppress the formation of Li dendrites. Herein, we describe the development of using carbon-based materials as Li hosts. While these materials can be fabricated into a variety of porous structures, they have a number of intrinsic advantages including low costs, high specific surface areas, high electrical conductivities, and wide electrochemical stabilities. After briefly summarizing the formation mechanisms of Li dendrites, various methods for controlling structural and surface chemistry will be described for different types of carbon-based materials from the viewpoint of improving their performance as Li hosts. Finally, we provide perspective on the future development of Li host materials needed to meet the requirements for their use in flexible and wearable devices and other contemporary energy storage techniques.