Hidden diversity of vacancy networks in Prussian blue analogues.

Prussian blue analogues (PBAs) are a diverse family of microporous inorganic solids, known for their gas storage ability1, metal-ion immobilization2, proton conduction3, and stimuli-dependent magnetic4,5, electronic6 and optical7 properties. This family of materials includes the double-metal cyanide catalysts8,9 and the hexacyanoferrate/hexacyanomanganate battery materials10,11. Central to the various physical properties of PBAs is their ability to reversibly transport mass, a process enabled by structural vacancies. Conventionally presumed to be random12,13, vacancy arrangements are crucial because they control micropore-network characteristics, and hence the diffusivity and adsorption profiles14,15. The long-standing obstacle to characterizing the vacancy networks of PBAs is the inaccessibility of single crystals16. Here we report the growth of single crystals of various PBAs and the measurement and interpretation of their X-ray diffuse scattering patterns. We identify a diversity of non-random vacancy arrangements that is hidden from conventional crystallographic powder analysis. Moreover, we explain this unexpected phase complexity in terms of a simple microscopic model that is based on local rules of electroneutrality and centrosymmetry. The hidden phase boundaries that emerge demarcate vacancy-network polymorphs with very different micropore characteristics. Our results establish a foundation for correlated defect engineering in PBAs as a means of controlling storage capacity, anisotropy and transport efficiency.

Percolation-Induced Ferrimagnetism from Vacancy Order in [Gua]Mn1-xFe2x/3(HCOO)3 Hybrid Perovskites.

We report the magnetic behavior of the hybrid perovskites [Gua]Mn1-xFe2x/3□x/3(HCOO)3 (0 ≤ x ≤ 0.88), showing that vacancy ordering drives bulk ferrimagnetism for x > 0.6. The behavior is rationalized in terms of a simple microscopic model of percolation-induced ferrimagnetism. Monte Carlo simulations driven by this model reproduce the experimental dependence of magnetic susceptibility on x and show that, at intermediate compositions, domains of short-range vacancy order lead to the emergence of local magnetization. Our results open up a new avenue for the design of multiferroic hybrid perovskites.

Torsional flexibility in zinc-benzenedicarboxylate metal-organic frameworks.

We explore the role and nature of torsional flexibility of carboxylate-benzene links in the structural chemistry of metal-organic frameworks (MOFs) based on Zn and benzenedicarboxlyate (bdc) linkers. A particular motivation is to understand the extent to which such flexibility is important in stabilising the unusual topologically aperiodic phase known as TRUMOF-1. We compare the torsion angle distributions of TRUMOF-1 models with those for crystalline Zn/1,3-bdc MOFs, including a number of new materials whose structures we report here. We find that both periodic and aperiodic Zn/1,3-bdc MOFs sample a similar range of torsion angles, and hence the formation of TRUMOF-1 does not require any additional flexibility beyond that already evident in chemically-related crystalline phases. Comparison with Zn/1,4-bdc MOFs does show, however, that the lower symmetry of the 1,3-bdc linker allows access to a broader range of torsion angles, reflecting a greater flexibility of this linker.

Understanding Defects in Amorphous Silicon with Million-Atom Simulations and Machine Learning.

The structure of amorphous silicon (a-Si) is widely thought of as a fourfold-connected random network, and yet it is defective atoms, with fewer or more than four bonds, that make it particularly interesting. Despite many attempts to explain such "dangling-bond" and "floating-bond" defects, respectively, a unified understanding is still missing. Here, we use advanced computational chemistry methods to reveal the complex structural and energetic landscape of defects in a-Si. We study an ultra-large-scale, quantum-accurate structural model containing a million atoms, and thousands of individual defects, allowing reliable defect-related statistics to be obtained. We combine structural descriptors and machine-learned atomic energies to develop a classification of the different types of defects in a-Si. The results suggest a revision of the established floating-bond model by showing that fivefold-bonded atoms in a-Si exhibit a wide range of local environments-analogous to fivefold centers in coordination chemistry. Furthermore, it is shown that fivefold (but not threefold) coordination defects tend to cluster together. Our study provides new insights into one of the most widely studied amorphous solids, and has general implications for understanding defects in disordered materials beyond silicon alone.

Predicting Distortion Magnitudes in Prussian Blue Analogues.

Based on simple electrostatic and harmonic potential considerations, we derive a straightforward expression linking the composition of a Prussian blue analogue (PBA) to its propensity to undergo collective structural distortions. We demonstrate the existence of a threshold value, below which PBAs are undistorted and above which PBAs distort by a degree that is controlled by a geometric tolerance factor. Our analysis rationalizes the presence, absence, and magnitude of distortions in a wide range of PBAs and distinguishes their structural chemistry from that of other hybrid perovskites.

K-Ion Slides in Prussian Blue Analogues.

We study the phenomenology of cooperative off-centering of K+ ions in potassiated Prussian blue analogues (PBAs). The principal distortion mechanism by which this off-centering occurs is termed a "K-ion slide", and its origin is shown to lie in the interaction between local electrostatic dipoles that couple through a combination of electrostatics and elastic strain. Using synchrotron powder X-ray diffraction measurements, we determine the crystal structures of a range of low-vacancy K2M[Fe(CN)6] PBAs (M = Ni, Co, Fe, Mn, Cd) and establish an empirical link between composition, temperature, and slide-distortion magnitude. Our results reflect the common underlying physics responsible for K-ion slides and their evolution with temperature and composition. Monte Carlo simulations driven by a simple model of dipolar interactions and strain coupling reproduce the general features of the experimental phase behavior. We discuss the implications of our study for optimizing the performance of PBA K-ion battery cathode materials and also its relevance to distortions in other, conceptually related, hybrid perovskites.

On the Origin of the Non-Arrhenius Na-ion Conductivity in Na3 OBr.

The sodium-rich antiperovskites (NaRAPs) with composition Na3 OB (B=Br, Cl, I, BH4 , etc.) are a family of materials that has recently attracted great interest for application as solid electrolytes in sodium metal batteries. Non-Arrhenius ionic conductivities have been reported for these materials, the origin of which is poorly understood. In this work, we combined temperature-resolved bulk and local characterisation methods to gain an insight into the origin of this unusual behaviour using Na3 OBr as a model system. We first excluded crystallographic disorder on the anion sites as the cause of the change in activation energy; then identified the presence of a poorly crystalline impurities, not detectable by XRD, and elucidated their effect on ionic conductivity. These findings improve understanding of the processing-structure-properties relationships pertaining to NaRAPs and highlight the need to determine these relationships in other materials systems, which will accelerate the development of high-performance solid electrolytes.

Revisiting metal fluorides as lithium-ion battery cathodes.

Metal fluorides, promising lithium-ion battery cathode materials, have been classified as conversion materials due to the reconstructive phase transitions widely presumed to occur upon lithiation. We challenge this view by studying FeF3 using X-ray total scattering and electron diffraction techniques that measure structure over multiple length scales coupled with density functional theory calculations, and by revisiting prior experimental studies of FeF2 and CuF2. Metal fluoride lithiation is instead dominated by diffusion-controlled displacement mechanisms, and a clear topological relationship between the metal fluoride F- sublattices and that of LiF is established. Initial lithiation of FeF3 forms FeF2 on the particle's surface, along with a cation-ordered and stacking-disordered phase, A-LixFeyF3, which is structurally related to α-/ß-LiMn2+Fe3+F6 and which topotactically transforms to B- and then C-LixFeyF3, before forming LiF and Fe. Lithiation of FeF2 and CuF2 results in a buffer phase between FeF2/CuF2 and LiF. The resulting principles will aid future developments of a wider range of isomorphic metal fluorides.

Hybrid Perovskites, Metal-Organic Frameworks, and Beyond: Unconventional Degrees of Freedom in Molecular Frameworks.

ConspectusThe structural degrees of freedom of a solid material are the various distortions most straightforwardly activated by external stimuli such as temperature, pressure, or adsorption. One of the most successful design strategies in materials chemistry involves controlling these individual distortions to produce useful collective functional responses. In a ferroelectric such as lead titanate, for example, the key degree of freedom involves asymmetric displacements of Pb2+ and Ti4+ cations; it is by coupling these together that the system as a whole interacts with external electric fields. Collective rotations of the polyhedral units in oxide ceramics are another commonly exploited distortion, driving anomalous behavior such as negative thermal expansion-the counterintuitive phenomenon of volume contraction on heating. An exciting development in the field has been to take advantage of the interplay between different distortion types: generating polarization by combining two different polyhedral rotations, for example. In this way, degrees of freedom act as geometric "elements" that can themselves be combined to engineer materials with new and interesting properties. Just as the discovery of new chemical elements quite obviously diversified chemical space, we might expect that identifying new and different types of structural degrees of freedom to be an important strategy for developing new kinds of functional materials. In this context, the broad family of molecular frameworks is emerging as an extraordinarily fertile source of new and unanticipated distortion types, the vast majority of which have no parallel in the established families of conventional solid-state chemistry.Framework materials are solids whose structures are assembled from two fundamental components: nodes and linkers. Quite simply, linkers join the nodes together to form scaffolding-like networks that extend from the atomic to the macroscopic scale. These structures usually contain cavities, which can also accommodate additional ions for charge balance. In the well-established systems-such as lead titanate-node, linker, and extra-framework ions are all individual atoms (Ti, O, and Pb, respectively). But in molecular frameworks, at least one of these components is a molecule.In this Account, we survey the unconventional degrees of freedom introduced through the simple act of replacing atoms by molecules. Our motivation is to understand the role these new distortions play (or might be expected to play) in different materials properties. The various degrees of freedom themselves-unconventional rotational, translational, orientational, and conformational states-are summarized and described in the context of relevant experimental examples. The much-improved prospect for generating emergent functionalities by combining these new distortion types is then discussed. We highlight a number of directions for future research-including the design and application of hierarchically structured phases of matter intermediate to solids and liquid crystals-which serve to highlight the extraordinary possibilities for this nascent field.

Geometric Frustration on the Trillium Lattice in a Magnetic Metal-Organic Framework.

In the dense metal-organic framework Na[Mn(HCOO)_{3}], Mn^{2+} ions (S=5/2) occupy the nodes of a "trillium" net. We show that the system is strongly magnetically frustrated: the Néel transition is suppressed well below the characteristic magnetic interaction strength; short-range magnetic order persists far above the Néel temperature; and the magnetic susceptibility exhibits a pseudo-plateau at 1/3-saturation magnetization. A simple model of nearest-neighbor Heisenberg antiferromagnetic and dipolar interactions accounts quantitatively for all observations, including an unusual 2-k magnetic ground state. We show that the relative strength of dipolar interactions is crucial to selecting this particular ground state. Geometric frustration within the classical spin liquid regime gives rise to a large magnetocaloric response at low applied fields that is degraded in powder samples as a consequence of the anisotropy of dipolar interactions.

Function from configurational degeneracy in disordered framework materials.

The existence of correlated disorder in molecular frameworks is an obvious mechanism by which unusual cooperative phenomena might be realised. We show that the use of local-symmetry lowering approaches can allow ostensibly high-symmetry framework structures to harbour exotic disordered states often studied in the context of spin lattice models. These states exhibit strongly cooperative behaviour that might be exploited in anomalous mechanical, host/guest, and information storage behaviour. Our contribution focuses in particular on the concepts of (i) combinatorial mechanics, (ii) adaptive flexibility, and (iii) error-correcting data storage in framework materials.

The crystallography of correlated disorder.

Classical crystallography can determine structures as complicated as multi-component ribosomal assemblies with atomic resolution, but is inadequate for disordered systems--even those as simple as water ice--that occupy the complex middle ground between liquid-like randomness and crystalline periodic order. Correlated disorder nevertheless has clear crystallographic signatures that map to the type of disorder, irrespective of the underlying physical or chemical interactions and material involved. This mapping hints at a common language for disordered states that will help us to understand, control and exploit the disorder responsible for many interesting physical properties.

Ordered B-Site Vacancies in an ABX3 Formate Perovskite.

We report the synthesis and structural characterization of the ABX3 perovskite frameworks [C(NH2)3]Mn1-x2+(Fe2x/33+,□x/3)(HCOO)3 (□ = B-site vacancy). For large x, the vacancies order, lowering the crystal symmetry. This system establishes B-site vacancies as a new type of defect in formate perovskites, with important chemical, structural, and functional implications. Monte Carlo simulations driven by nearest-neighbor vacancy repulsions show checkerboard vacancy order to emerge for x > 0.6, in accord with experiment.

Control of Metal-Organic Framework Crystallization by Metastable Intermediate Pre-equilibrium Species.

There is an increasing amount of interest in metal-organic frameworks (MOFs) for a variety of applications, from gas sensing and separations to electronics and catalysis. However, the mechanisms by which they crystallize remain poorly understood. Herein, an important new insight into MOF formation is reported. It is shown that, prior to network assembly, crystallization intermediates in the canonical ZIF-8 system exist in a dynamic pre-equilibrium, which depends on the reactant concentrations and the progress of reaction. Concentration can, therefore, be used as a synthetic handle to directly control particle size, with potential implications for industrial scale-up and gas sorption applications. These findings enable the rationalization of apparent contradictions between previous studies of ZIF-8 and opens up new opportunities for the control of crystallization in network solids more generally.

Negative Hydration Expansion in ZrW_{2}O_{8}: Microscopic Mechanism, Spaghetti Dynamics, and Negative Thermal Expansion.

We use a combination of x-ray diffraction, total scattering, and quantum mechanical calculations to determine the mechanism responsible for hydration-driven contraction in ZrW_{2}O_{8}. The inclusion of H_{2}O molecules within the ZrW_{2}O_{8} network drives the concerted formation of new W─O bonds to give one-dimensional (─W─O─)_{n} strings. The topology of the ZrW_{2}O_{8} network is such that there is no unique choice for the string trajectories: the same local changes in coordination can propagate with a large number of different periodicities. Consequently, ZrW_{2}O_{8}·H_{2}O is heavily disordered, with each configuration of strings forming a dense aperiodic "spaghetti." This new connectivity contracts the unit cell via large shifts in the Zr and W atom positions. Fluctuations of the undistorted parent structure towards this spaghetti phase emerge as the key negative thermal expansion (NTE) phonon modes in ZrW_{2}O_{8} itself. The large relative density of NTE phonon modes in ZrW_{2}O_{8} actually reflects the degeneracy of volume-contracting spaghetti excitations, itself a function of the particular topology of this remarkable material.

Metal-Organic Frameworks as Catalyst Supports: Influence of Lattice Disorder on Metal Nanoparticle Formation.

Because of their high tunability and surface area, metal-organic frameworks (MOFs) show great promise as supports for metal nanoparticles. Depending on the synthesis route, MOFs may contain defects. Here, we show that highly crystalline MIL-100(Fe) and disordered Basolite® F300, with identical iron 1,3,5-benzenetricarboxylate composition, exhibit very divergent properties when used as a support for Pd nanoparticle deposition. While MIL-100(Fe) shows a regular MTN-zeotype crystal structure with two types of cages, Basolite® F300 lacks long-range order beyond 8 Šand has a single-pore system. The medium-range configurational linker-node disorder in Basolite® F300 results in a reduced number of Lewis acid sites, yielding more hydrophobic surface properties compared to hydrophilic MIL-100(Fe). The hydrophilic/hydrophobic nature of MIL-100(Fe) and Basolite® F300 impacts the amount of Pd and particle size distribution of Pd nanoparticles deposited during colloidal synthesis and dry impregnation methods, respectively. It is suggested that polar (apolar) solvents/precursors attractively interact with hydrophilic (hydrophobic) MOF surfaces, allowing tools at hand to increase the level of control over, for example, the nanoparticle size distribution.

Metal-Organic Nanosheets Formed via Defect-Mediated Transformation of a Hafnium Metal-Organic Framework.

We report a hafnium-containing MOF, hcp UiO-67(Hf), which is a ligand-deficient layered analogue of the face-centered cubic fcu UiO-67(Hf). hcp UiO-67 accommodates its lower ligand:metal ratio compared to fcu UiO-67 through a new structural mechanism: the formation of a condensed "double cluster" (Hf12O8(OH)14), analogous to the condensation of coordination polyhedra in oxide frameworks. In oxide frameworks, variable stoichiometry can lead to more complex defect structures, e.g., crystallographic shear planes or modules with differing compositions, which can be the source of further chemical reactivity; likewise, the layered hcp UiO-67 can react further to reversibly form a two-dimensional metal-organic framework, hxl UiO-67. Both three-dimensional hcp UiO-67 and two-dimensional hxl UiO-67 can be delaminated to form metal-organic nanosheets. Delamination of hcp UiO-67 occurs through the cleavage of strong hafnium-carboxylate bonds and is effected under mild conditions, suggesting that defect-ordered MOFs could be a productive route to porous two-dimensional materials.

Incommensurate Chirality Density Wave Transition in a Hybrid Molecular Framework.

Using single-crystal x-ray diffraction we characterize the 235 K incommensurate phase transition in the hybrid molecular framework tetraethylammonium silver(I) dicyanoargentate, [NEt_{4}]Ag_{3}(CN)_{4}. We demonstrate the transition to involve spontaneous resolution of chiral [NEt_{4}]^{+} conformations, giving rise to a state in which molecular chirality is incommensurately modulated throughout the crystal lattice. We refer to this state as an incommensurate chirality density wave (XDW) phase, which represents a fundamentally new type of chiral symmetry breaking in the solid state. Drawing on parallels to the incommensurate ferroelectric transition of NaNO_{2}, we suggest the XDW state arises through coupling between acoustic (shear) and molecular rotoinversion modes. Such coupling is symmetry forbidden at the Brillouin zone center but symmetry allowed for small but finite modulation vectors q=[0,0,q_{z}]^{*}. The importance of long-wavelength chirality modulations in the physics of this hybrid framework may have implications for the generation of mesoscale chiral textures, as required for advanced photonic materials.

Orbital Dimer Model for the Spin-Glass State in Y_{2}Mo_{2}O_{7}.

The formation of a spin glass generally requires that magnetic exchange interactions are both frustrated and disordered. Consequently, the origin of spin-glass behavior in Y_{2}Mo_{2}O_{7}-in which magnetic Mo^{4+} ions occupy a frustrated pyrochlore lattice with minimal compositional disorder-has been a longstanding question. Here, we use neutron and x-ray pair-distribution function (PDF) analysis to develop a disorder model that resolves apparent incompatibilities between previously reported PDF, extended x-ray-absorption fine structure spectroscopy, and NMR studies, and provides a new and physical explanation of the exchange disorder responsible for spin-glass formation. We show that Mo^{4+} ions displace according to a local "two-in-two-out" rule on each Mo_{4} tetrahedron, driven by orbital dimerization of Jahn-Teller active Mo^{4+} ions. Long-range orbital order is prevented by the macroscopic degeneracy of dimer coverings permitted by the pyrochlore lattice. Cooperative O^{2-} displacements yield a distribution of Mo-O-Mo angles, which in turn introduces disorder into magnetic interactions. Our study demonstrates experimentally how frustration of atomic displacements can assume the role of compositional disorder in driving a spin-glass transition.