Super-efficiency measurement under variable return to scale: an approach based on a new directional distance function

A modified super-efficiency model based on the directional distance function (DDF) has recently been developed in order to tackle the infeasibility issue in the two exceptions of the Nerlove–Luenberger (N–L) super-efficiency model under variable return to scale (VRS). However, we find that model does not fully eliminate the infeasibility issue. This paper chooses an appropriate reference bundle in the DDF so that the resulting DDF-based VRS super-efficiency model is always feasible. The proposed new model successfully addresses the infeasibility issue of conventional VRS super-efficiency models and fully eliminates the infeasibility issue in the two exceptions of the VRS N–L super-efficiency model. Additional advantages of the new model include: it is unit-invariant and does not need to predetermine any parameter. Theoretical analyses and numerical examples support the practicality and superiority of our model when compared with other super-efficiency models.     

An Unprecedented Two‐Fold Nested Super‐Polyrotaxane: Sulfate‐Directed Hierarchical Polythreading Assembly of Uranyl Polyrotaxane Moieties

The hierarchical assembly of well‐organized submoieties could lead to more complicated superstructures with intriguing properties. We describe herein an unprecedented polyrotaxane polythreading framework containing a two‐fold nested super‐polyrotaxane substructure, which was synthesized through a uranyl‐directed hierarchical polythreading assembly of one‐dimensional polyrotaxane chains and two‐dimensional polyrotaxane networks. This special assembly mode actually affords a new way of supramolecular chemistry instead of covalently linked bulky stoppers to construct stable interlocked rotaxane moieties. An investigation of the synthesis condition shows that sulfate can assume a vital role in mediating the formation of different uranyl species, especially the unique trinuclear uranyl moiety [(UO₂)₃O(OH)₂]²⁺, involving a notable bent [O=U=O] bond with a bond angle of 172.0(9)°. Detailed analysis of the coordination features, the thermal stability as well as a fluorescence, and electrochemical characterization demonstrate that the uniqueness of this super‐polyrotaxane structure is mainly closely related to the trinuclear uranyl moiety, which is confirmed by quantum chemical calculations.     

Tailorable PC71BM Isomers: Using the Most Prevalent Electron Acceptor to Obtain High‐Performance Polymer Solar Cells

Despite being widely used as electron acceptor in polymer solar cells, commercially available PC₇₁BM (phenyl‐C₇₁‐butyric acid methyl ester) usually has a “random” composition of mixed regioisomers or stereoisomers. Here PC₇₁BM has been isolated into three typical isomers, α‐, β₁‐ and β₂‐PC₇₁BM, to establish the isomer‐dependent photovoltaic performance on changing the ternary composition of α‐, β₁‐ and β₂‐PC₇₁BM. Mixing the isomers in a ratio of α/β₁/β₂=8:1:1 resulted in the best power conversion efficiency (PCE) of 7.67 % for the polymer solar cells with PTB7:PC₇₁BM as photoactive layer (PTB7=poly[[4,8‐bis[(2‐ethylhexyl)oxy]benzo[1,2‐b:4,5‐b′]dithiophene‐2,6‐diyl][3‐fluoro‐2‐[(2‐ethylhexyl)carbonyl]thieno[3,4‐b]thiophenediyl]]). The three typical PC₇₁BM isomers, even though sharing similar LUMO energy levels and light absorption, render starkly different photovoltaic performances with average‐performing PCE of 1.28–7.44 % due to diverse self‐aggregation of individual or mixed PC₇₁BM isomers in the otherwise same polymer solar cells.