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Weak cross-linkers can improve the ultimate strength and tear resistance of polymer networks. 

We present a modified Lake–Thomas theory that accounts for the molecular details of network connectivity upon crack propagation in polymer networks. This theory includes not only the energy stored in the breaking network strands (bridging strands) but also the energy stored in the tree-like structure of the strands connecting the bridging strands to the network continuum, which remains intact as the crack propagates. The energy stored in each of the generations of this tree depends nonmonotonically on the generation index due to the nonlinear elasticity of the stretched network strands. Further, the energy required to break a single bridging strand is not necessarily dominated by the energy stored in the bridging strand itself but in the higher generations of the tree. We describe the effect of mechanophores with stored length on the energy stored in the tree-like structure. In comparison with the “strong” mechanophores that can only be activated in the bridging strand, “weak” mechanophores that can be activated both in the bridging strand and in other generations could provide more energy dissipation due to their larger contribution to higher generations of the tree. 

Many transition-metal-oxide-based catalysts have been investigated to chemically bind soluble lithium polysulfides and accelerate their redox kinetics in lithium-sulfur (Li-S) battery chemistry. However, the intrinsic poor electrical conductivities of these oxides restrict their catalytic performance, consequently limiting the sulfur utilization and the rate performance of Li-S batteries. Herein, we report a freestanding electrocatalytic sulfur host consisting of hydrogen-treated VO2 nanoparticles (H-VO2) anchored on nitrogen-doped carbonized bacterial cellulose aerogels (N-CBC). The hydrogen treatment enables the formation and stabilization of the rutile VO2(R) phase with metallic conductivity at room temperature, significantly enhancing its catalytic capability compared to the as-synthesized insulative VO2(M) phase. Several measurements characterize the electrocatalytic performance of this unique H-VO2@N-CBC structure. In particular, the two kinetic barriers between S8, polysulfides, and Li2S are largely reduced by 28.2 and 43.3 kJ/mol, respectively. Accordingly, the Li-S battery performance, in terms of sulfur utilization and charge/discharge rate, is greatly improved. This work suggests an effective strategy to develop conductive catalysts based on a typical transition metal oxide (VO2) for Li-S batteries. 

Abstract We perform a systematic survey of active galactic nuclei (AGNs) continuum lags using ∼3 days cadence gri -band light curves from the Zwicky Transient Facility. We select a sample of 94 type 1 AGNs at z < 0.8 with significant and consistent inter-band lags based on the interpolated cross-correlation function method and the Bayesian method JAVELIN . Within the framework of the “lamp-post” reprocessing model, our findings are: (1) The continuum emission (CE) sizes inferred from the data are larger than the disk sizes predicted by the standard thin-disk model. (2) For a subset of the sample, the CE size exceeds the theoretical limit of the self-gravity radius (12 lt-days) for geometrically thin disks. (3) The CE size scales with continuum luminosity as R CE ∝ L 0.48±0.04 with a scatter of 0.2 dex, analogous to the well-known radius–luminosity relation of broad H β . These findings suggest a significant contribution of diffuse continuum emission from the broad-line region (BLR) to AGN continuum lags. We find that the R CE – L relation can be explained by a photoionization model that assumes ∼23% of the total flux comes from the diffuse BLR emission. In addition, the ratio of the CE size and model-predicted disk size anticorrelates with the continuum luminosity, which is indicative of a potential nondisk BLR lag contribution evolving with the luminosity. Finally, a robust positive correlation between the CE size and black hole mass is detected.