Crystal structure and electrochemical properties of phosphosulphate NaFe2PO4(SO4)2

NASICON-type NaFe2(PO4)(SO4)2 (NFPS) electrode material is successfully synthesized via a rheological route. DSC/TG analysis shows that it is thermally stable up to 760 °C. A carbon composite NFPS/C was obtained using ball milling. NFPS and NFPS/C were investigated as cathode and anode materials for sodium-ion batteries. Analysis of migration paths was performed by the Voronoi-Dirichlet partition technique to determine all possible Na+ ion migration paths. The diffusion coefficient, estimated by GITT, is in the 10-12-10-13 cm2∙s-1 range, which corresponds to the fast Na+ ion migration in the structure. According to calculations, NFPS is a wide band gap material, which indicates its poor electrical conductivity.

Universality for Random Surfaces in Unconstrained Genus

Starting from an arbitrary sequence of polygons whose total perimeter is $2n$, we can build an (oriented) surface by pairing their sides in a uniform fashion. Chmutov & Pittel have shown that, regardless of the configuration of polygons we started with, the degree sequence of the graph obtained this way is remarkably constant in total variation distance and converges towards a Poisson–Dirichlet partition as $n \to \infty$. We actually show that several other geometric properties of the graph are universal. En route we provide an alternative proof of a weak version of the result of Chmutov & Pittel using probabilistic techniques and related to the circle of ideas around the peeling process of random planar maps. At this occasion we also fill a gap in the existing literature by surveying the properties of a uniform random map with $n$ edges. In particular we show that the diameter of a random map with $n$ edges converges in law towards a random variable taking only values in $\{2,3\}$.

Analysis of ion-migration paths in inorganic frameworks by means of tilings and Voronoi–Dirichlet partition: a comparison

Two methods using Voronoi–Dirichlet polyhedra (Voronoi–Dirichlet partition) or tiles (tiling) based on partitioning space are compared to investigate cavities and channels in crystal structures. The tiling method was applied for the first time to study ion conductivity in 105 ternary, lithium–oxygen-containing compounds, Li a X b O z , that were recently recognized as fast-ion conductors with the Voronoi–Dirichlet partition method. The two methods were found to be similar in predicting the occurrence of ionic conductivity, however, their conclusions on the dimensionality of conductivity were different in two cases. It is shown that such a contradiction can indicate a high anisotropy of conductivity. Both advantages and restrictions of the methods are discussed with respect to fast-ion conductors and zeolites.

Analysis of migration paths in fast-ion conductors with Voronoi–Dirichlet partition

In terms of the Voronoi–Dirichlet partition of the crystal space, definitions are given for such concepts as `void', `channel' and `migration path' for inorganic structures with three-dimensional networks of chemical bonds. A number of criteria are proposed for selecting significant voids and migration channels for alkali cations Li+–Cs+ based on the average characteristics of the Voronoi–Dirichlet polyhedra for alkali metals in oxygen-containing compounds. A general algorithm to analyze the voids in crystal structures has been developed and implemented in the computer package TOPOS. This approach was used to predict the positions of Li+ and Na+ cations and to analyze their possible migration paths in the solid superionic materials Li3 M 2P3O12 (M = Sc, Fe; LIPHOS) and Na1 + x Zr2Si x P3 − x O12 (NASICON), whose framework structures consist of connected M octahedra and T tetrahedra. Using this approach we determine the most probable places for charge carriers (coordinates of alkali cations) and the dimensionality of their conducting sublattice with high accuracy. The theoretically calculated coordinates of the alkali cations in MT frameworks are found to correlate to within 0.33 Å with experimental data for various phases of NASICON and LIPHOS. The proposed method of computer analysis is universal and suitable for investigating fast-ion conductors with other conducting components.

SIZE-BIASED PERMUTATION OF DIRICHLET PARTITIONS AND SEARCH-COST DISTRIBUTION

Consider the random Dirichlet partition of the interval into n fragments at temperature θ > 0. Explicit results on the law of its size-biased permutation are first supplied. Using these, new results on the comparative search cost distributions from Dirichlet partition and from its size-biased permutation are obtained.

Methods of crystallochemical analysis of supramolecular complexes by means of Voronoi–Dirichlet polyhedra: a study of cucurbituril host–guest compounds

Crystallochemical analysis methods based on the Voronoi–Dirichlet partition of crystal space are extended to supramolecular complexes of any complexity. The sizes and shapes of receptor cavities and substrate molecules are shown to be successfully estimated as volumes and the second moments of inertia of the corresponding molecular Voronoi–Dirichlet polyhedra. To predict which organic substrates can occupy the receptor cavity a mini-expert system known as MOLVOL was created, comprising a database on completely determined crystal structures of almost 60 000 organic molecular compounds. Using the developed methods, volumes and shapes are assessed for cucurbit[n]uril receptors (n = 5–10) and their cavities. A number of organic and inorganic molecules are found which can optimally fit the cavities inside the cucurbit[5]uril and cucurbit[6]uril molecules.