Liouville property for groups and conformal dimension

Conformal dimension is a fundamental invariant of metric spaces, particularly suited to the study of self-similar spaces, such as spaces with an expanding self-covering (e.g. Julia sets of complex rational functions). The dynamics of these systems are encoded by the associated iterated monodromy groups, which are examples of contracting self-similar groups. Their amenability is a well-known open question. We show that if $G$ is an iterated monodromy group, and if the (Alfhors-regular) conformal dimension of the underlying space is strictly less than 2, then every symmetric random walk with finite second moment on $G$ has the Liouville property. As a corollary, every such group is amenable. This criterion applies to all examples of contracting groups previously known to be amenable, and to many new ones. In particular, it implies that for every post-critically finite complex rational function $f$ whose Julia set is not the whole sphere, the iterated monodromy group of $f$ is amenable.Comment: 36 pages, 5 figures, v2: minor change

Tris(1,10-phenanthroline-κ2 N,N′)iron(II) bis­[(1,10-phenanthroline-κ2 N,N′)tetra­kis­(thio­cyanato-κN)chromate(III)] acetonitrile tris­olvate monohydrate

Single crystals of the title heterometallic compound, [Fe(C12H8N2)3][Cr(NCS)4(C12H8N2)]2·3CH3CN·H2O or [Fe(Cphen)3][Cr(NCS)4(phen)]2·3CH3CN·H2O, were pre­pared using the one-pot open-air reaction of iron powder, Reineckes salt and 1,10-phenanthroline (phen) in acetonitrile. The asymetric unit consists of an [Fe(phen)3]2+ cation, two [Cr(phen)(NCS)4]− anions, three acetonitrile solvent mol­ecules and a water mol­ecule. The Fe and Cr atoms both show a slightly distorted octa­hedral FeN6 and CrN6 coordination geometry with adjacent angles in the range 79.67 (12)–95.21 (12)°. No classical hydrogen bonding involving the water molecule is observed

Zr(IV) and Hf(IV) based metal–organic frameworks with reo-topology

Zr and Hf based MOFs with enhanced pore accessibility for large molecules and good hydrothermal stability were obtained using a bent dithienothiophene dicarboxylate and Zr4+ or Hf4+ source. A modulator (benzoic acid) facilitates formation of an eight-connecting cluster leading to a new framework which adopts reo topology.Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich

Flexible metal–organic frameworks

Advances in flexible and functional metal–organic frameworks (MOFs), also called soft porous crystals, are reviewed by covering the literature of the five years period 2009–2013 with reference to the early pertinent work since the late 1990s. Flexible MOFs combine the crystalline order of the underlying coordination network with cooperative structural transformability. These materials can respond to physical and chemical stimuli of various kinds in a tunable fashion by molecular design, which does not exist for other known solid-state materials. Among the fascinating properties are so-called breathing and swelling phenomena as a function of host–guest interactions. Phase transitions are triggered by guest adsorption/desorption, photochemical, thermal, and mechanical stimuli. Other important flexible properties of MOFs, such as linker rotation and sub-net sliding, which are not necessarily accompanied by crystallographic phase transitions, are briefly mentioned as well. Emphasis is given on reviewing the recent progress in application of in situ characterization techniques and the results of theoretical approaches to characterize and understand the breathing mechanisms and phase transitions. The flexible MOF systems, which are discussed, are categorized by the type of metal-nodes involved and how their coordination chemistry with the linker molecules controls the framework dynamics. Aspects of tailoring the flexible and responsive properties by the mixed component solid-solution concept are included, and as well examples of possible applications of flexible metal–organic frameworks for separation, catalysis, sensing, and biomedicine

Metal-Organic Frameworks in Germany: from Synthesis to Function

Metal-organic frameworks (MOFs) are constructed from a combination of inorganic and organic units to produce materials which display high porosity, among other unique and exciting properties. MOFs have shown promise in many wide-ranging applications, such as catalysis and gas separations. In this review, we highlight MOF research conducted by Germany-based research groups. Specifically, we feature approaches for the synthesis of new MOFs, high-throughput MOF production, advanced characterization methods and examples of advanced functions and properties

Impact of Defects and Crystal Size on Negative Gas Adsorption in DUT-49 Analyzed by in Situ ¹²⁹Xe NMR Spectroscopy

The origin of crystal-size-dependent adsorption behavior of flexible metal-organic frameworks is increasingly studied. In this contribution, we probe the solid-fluid interactions of DUT-49 crystals of different size by in situ 129Xe NMR spectroscopy at 200 K. With decreasing size of the crystals, the average solid-fluid interactions are found to decrease reflected by a decrease in chemical shift of adsorbed xenon from 230 to 200 ppm, explaining the lack of adsorption-induced transitions for smaller crystals. However, recent studies propose that these results can also originate from the presence of lattice defects. To investigate the influence of defects on the adsorption behavior of DUT-49, we synthesized a series of samples with tailored defect concentrations and characterized them by in situ 129Xe NMR. In comparison to the results obtained for crystals with different size, we find pronounced changes of the adsorption behavior and influence of the chemical shift only for very high concentrations of defects, which further emphasizes the important role of particle size phenomena

A new metal–organic framework with ultra-high surface area

A new mesoporous MOF, Zn4O(bpdc)(btctb)4/3 (DUT-32), containing linear ditopic (bpdc2−; 4,4′-biphenylenedicarboxylic acid) and tritopic (btctb3−; 4,4′,4′′-[benzene-1,3,5-triyltris(carbonylimino)]tris-benzoate) linkers, was synthesised. The highly porous solid has a total pore volume of 3.16 cm3 g−1 and a specific BET surface area of 6411 m2 g−1, adding this compound to the top ten porous materials with the highest BET surface area