On the multiple Borsuk numbers of sets

The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.Comment: 16 pages, 3 figure

Mixed Electronic and Ionic Conduction Properties of Lithium Lanthanum Titanate

With the continued increase in Li‐metal anode rate capability, there is an equally important need to develop high‐rate cathode architectures for solid‐state batteries. A proposed method of improving charge transport in the cathode is introducing a mixed electronic and ionic conductor (MEIC) which can eliminate the need for conductive additives that occlude electrolyte–electrode interfaces and lower the net additive required in the cathode. This study takes advantage of a reduced perovskite electrolyte, Li0.33La0.57TiO3 (LLTO), to act as a model MEIC. It is found that the ionic conductivity of reduced LLTO is comparable to oxidized LLTO (σbulk = 10−3–10−4 S cm−1, σGB = 10−5–10−6 S cm−1) and the electronic conductivity is 1 mS cm−1. The ionic transference numbers are 0.9995 and 0.0095 in the oxidized and reduced state, respectively. Furthermore, two methods for controlling the transference numbers are evaluated. It is found that the electronic conductivity cannot easily be controlled by changing O2 overpressures, but increasing the ionic conductivity can be achieved by increasing grain size. This work identifies a possible class of MEIC materials that may improve rate capabilities of cathodes in solid‐state architectures and motivate a deeper understanding of MEICs in the context of solid‐state batteries.Lithium lanthanum titanate can exhibit both high ionic conductivity and high electronic conductivity depending on its oxidation state. While electronic conductivity in solid electrolytes is typically undesirable, the ability to conduct both Li‐ions and electrons may pose useful for cathode composites. This work investigates the conduction properties of lithium lanthanum titanate and potential methods to control transference number.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/154428/1/adfm201909140_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154428/2/adfm201909140-sup-0001-SuppMat.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154428/3/adfm201909140.pd

Converting ceria polyhedral nanoparticles into single-crystal nanospheres

Ceria nanopartides are one of the key abrasive materials for chemical-mechanical planarization of advanced integrated circuits. However, ceria nanoparticles synthesized by existing techniques are irregularly faceted, and they scratch the silicon wafers and increase defect concentrations. We developed an approach for large-scale synthesis of single-crystal ceria nanospheres that can reduce the polishing defects by 80% and increase the silica removal rate by 50%, facilitating precise and reliable mass-manufacturing of chips for nanoelectronics. We doped the ceria system with titanium, using flame temperatures that facilitate crystallization of the ceria yet retain the titania in a molten state. In conjunction with molecular dynamics simulation, we show that under these conditions, the inner ceria core evolves in a single-crystal spherical shape without faceting, because throughout the crystallization it is completely encapsulated by a molten 1- to 2-nanometer shell of titania that, in liquid state, minimizes the surface energy. The principle demonstrated here could be applied to other oxide systems