The drainage of liquid in a foam may be described in terms of a nonlinear partial differential equation for the foam density as a function of time and vertical position. We review the history and recent development of this theory, analysing various exact and approximate solutions and relating them to each other.
Rechargeable solid-state sodium-ion batteries (SSSBs) hold great promise for safer and more energy-dense energy storage. However, the poor electrochemical stability between current sulfide-based solid electrolytes and high-voltage oxide cathodes has limited their long-term cycling performance and practicality. Here, we report the discovery of the ion conductor Na3-xY1-xZrxCl6 (NYZC) that is both electrochemically stable (up to 3.8 V vs. Na/Na+) and chemically compatible with oxide cathodes. Its high ionic conductivity of 6.6 × 10−5 S cm−1 at ambient temperature, several orders of magnitude higher than oxide coatings, is attributed to abundant Na vacancies and cooperative MCl6 rotation, resulting in an extremely low interfacial impedance. A SSSB comprising a NaCrO2 + NYZC composite cathode, Na3PS4 electrolyte, and Na-Sn anode exhibits an exceptional first-cycle Coulombic efficiency of 97.1% at room temperature and can cycle over 1000 cycles with 89.3% capacity retention at 40 °C. These findings highlight the immense potential of halides for SSSB applications.
In the limit of strong electron-phonon coupling, we use either a Pekar-type or an oscillator wave function for the center-of-mass coordinate and either a Coulomb or an oscillator wave function for the relative coordinate, and are able to reproduce all the results from the literature for the large-bipolaron binding energy. Lower bounds are constructed for the critical ratio g, of dielectric constants below which bipolarons can exist. It is found that, in the strong-coupling limit, the stability region for bipolaron formation is much larger in two dimensions (2D) than in 3D. We introduce a model that combines the averaging of the relative coordinate over the asymptotically best wave function with a path-integral treatment of the center-of-mass motion. The stability region for bipolaron formation is increased compared with the full path-integral treatment at large values of the coupling constant a. The critical values are a, =9.3 in 3D and a, =4.5 in 2D. Phase diagrams for the presented models are also obtained in both 2D and 3D.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.