Rigid Unit Modes Research Articles

The ISOTILT program for discovering cooperative rigid-unit modes (RUMs) in crystalline solids

The <i>ISOTILT</i> program for discovering cooperative rigid-unit modes (RUMs) in crystalline solids

Anharmonicity and scissoring modes in the negative thermal expansion materials ScF3 and CaZrF6

We use a symmetry-motivated approach to analysing X-ray pair distribution functions to study the mechanism of negative thermal expansion in two ReO$_3$-like compounds; ScF$_3$ and CaZrF$_6$. Both average and local structure suggest that it is the flexibility of M-F-M linkages (M = Ca, Zr, Sc) due to dynamic rigid and semi-rigid "scissoring" modes that facilitates the observed NTE behaviour. The amplitudes of these dynamic distortions are greater for CaZrF$_6$ than for ScF$_3$, which corresponds well with the larger magnitude of the thermal expansion reported in the literature for the former. We show that this flexbility is enhanced in CaZrF$_6$ due to the rock-salt ordering mixing the characters of two of these scissoring modes. Additionally, we show that in ScF$_3$ anharmonic coupling between the modes responsible for the structural flexibility and the rigid unit modes contributes to the unusually high NTE behaviour in this material.

Tilts and shifts in molecular perovskites

A survey of the rigid unit modes in molecular perovskites is presented, showing how the prevalence of conventional tilts, unconventional tilts and columnar shifts vary across the different classes of molecular perovskites.

Structural evolution of fused silica below the glass-transition temperature revealed by in-situ neutron total scattering

The common belief that glass structure is completely frozen-in at room temperature is challenged at both macroscopic and atomic scales. Here, we demonstrate an analytical method to elucidate the fine details of a continuous structural change of fused silica (FS) at temperatures below the glass-transition temperature using in-situ neutron total scattering. We find that the SiO4 tetrahedron expands through the entire temperature range with a local coefficient of thermal expansion of 9.1 × 10−6 K−1, while the average medium-range distance, derived from the first sharp diffraction peak of the structure factor, expands at a rate of 21 × 10−6 K−1. Such an expansion difference reflects glass-structure changes within the “rigid-unit mode” model, where each tetrahedron behaves as a rigid unit and the flexible rotations between rigid units lead to more than two times higher medium-range thermal expansion. We further demonstrate that such rotations change the shape of individual rings, leading to a measurable change in the first sharp diffraction peak (FSDP). This study paves the way to measure the structural changes of other silicate glasses, especially through the glass transition.

Phonon dynamics in the layered negative thermal expansion compounds CuxNi2−x(CN)4

This study explores the relationship between phonon dynamics and negative thermal expansion (NTE) in Cu$_x$Ni$_{2-x}$(CN)$_4$. The partial replacement of nickel (II) by copper (II) in Ni(CN)$_2$ leads to a line phase, CuNi(CN)$_4$ (x = 1), and a solid solution, Cu$_x$Ni$_{2-x}$(CN)$_4$ (0 $\leq$ x $\leq$ 0.5). CuNi(CN)$_4$ adopts a layered structure related to that of Ni(CN)$_2$ (x = 0), and interestingly exhibits 2D NTE which is about 1.5 times larger. Inelastic neutron scattering (INS) measurements combined with first principles lattice dynamical calculations provide insights into the effect of Cu$^{2+}$ on the underlying mechanisms behind the anomalous thermal behavior in all the Cu$_x$Ni$_{2-x}$(CN)$_4$ compounds. The solid solutions are presently reported to also show 2D NTE. The INS results highlight that as the Cu$^{2+}$ content increases in Cu$_x$Ni$_{2-x}$(CN)$_4$, large shifts to lower energies are observed in modes consisting of localized in- and out-of-plane librational motions of the CN ligand, which contribute to the NTE in CuNi(CN)$_4$. Mode Gr\"uneisen parameters calculated for CuNi(CN)$_4$ show that acoustic and low-energy optic modes contribute the most to the NTE, as previously shown in Ni(CN)$_2$. However, mode eigenvectors reveal a large deformation of the [CuN4] units compared to the [NiC4] units, resulting in phonon modes not found in Ni(CN)$_2$, whose NTE-driving phonons consist predominately of rigid-unit modes. The deformations in CuNi(CN)$_4$ arise because the d$^9$ square-planar center is easier to deform than the d$^8$ one, resulting in a greater range of out-of-plane motions for the adjoining ligands.

The incommensurately modulated structures of volcanic plagioclase: displacement, ordering and phase transition.

Four basaltic phenocryst samples of plagioclase, with compositions ranging from An48 (andesine) to An64 (labradorite), have been studied with single-crystal X-ray and neutron diffraction techniques. The samples were also subjected to a heating experiment at 1100°C for two weeks in an effort to minimize the Al-Si ordering in their structures. The average and the modulated structures of the samples (before and after the heating experiment) were compared, in order to understand the mechanism of the phase transition from the disordered C\bar 1 structure to the e-plagioclase structure. A comparison between the structures from neutron and X-ray diffraction data shows that the 〈T-O〉 distance does not solely depend on the Al occupancy as previously thought. A dramatic decrease of the Al-Si ordering is observed after heating at 1100°C for two weeks for all four samples, with an obvious change in the intensities of the satellite reflections (e-reflections) in the diffraction pattern. Evident changes in the modulation period were also observed for the more calcic samples. No obvious change in the Ca-Na ordering was observed after the heating experiment. An in situ heating X-ray diffraction experiment was carried out on the andesine sample (An48) to study the change in the satellite intensity at high temperature. A dramatic weakening of the satellite peaks was observed between 477°C and 537°C, which strongly supports the displacive nature of the initiation of e2 ordering. Rigid-Unit Mode (RUM) analysis of the plagioclase structure suggests the initial position of the e-reflections is determined by the anti-RUMs in the framework.

Flexibility of network materials and the Rigid Unit Mode model: a personal perspective.

The Rigid Unit Mode model was initially developed to understand the origin of displacive phase transitions in network silicate structures. Here, we review the successes of the model, and consider how it might apply to a wider range of network structures. This article is part of the theme issue 'Mineralomimesis: natural and synthetic frameworks in science and technology'.

Lattice dynamics and negative thermal expansion in the framework compound ZnNi(CN)4 with two-dimensional and three-dimensional local environments

ZnNi(CN)$_4$ is a 3D framework material consisting of two interpenetrating PtS-type networks in which tetrahedral [ZnN$_4$] units are linked by square-planar [NiC$_4$] units. Both the parent compounds, cubic Zn(CN)$_2$ and layered Ni(CN)$_2$, are known to exhibit 3D and 2D negative thermal expansion (NTE), respectively. Temperature-dependent inelastic neutron scattering measurements were performed on a powdered sample of ZnNi(CN)$_4$ to probe phonon dynamics. The measurements were underpinned by ab initio lattice dynamical calculations. Good agreement was found between the measured and calculated generalized phonon density-of-states, validating our theoretical model and indicating that it is a good representation of the dynamics of the structural units. The calculated linear thermal expansion coefficients are $\alpha_a$=-21.2 $\times$ 10$^{-6}$ K$^{-1}$ and $\alpha_c$=+14.6$\times$10$^{-6}$K$^{-1}$, leading to an overall volume expansion coefficient $\alpha_V$ of -26.95$\times$10$^{-6}$K$^{-1}$, pointing towards pronounced NTE behavior. Analysis of the derived mode-Gr\"uneisen parameters shows that optic modes around 12 and 40 meV make a significant contribution to NTE. These modes involve localized rotational motions of the [NiC$_4$] and/or [ZnN$_4$] rigid units, echoing what has previously been observed in Zn(CN)$_2$ and Ni(CN)$_2$. However, in ZnNi(CN)$_4$, modes below 10 meV have the most negative Gr\"uneisen parameters. Analysis of their eigenvectors reveals that a large transverse motion of the Ni atom in the direction perpendicular to its square-planar environment induces a distortion of the units. This mode is a consequence of the Ni atom being constrained only in 2D within a 3D framework. Hence, although rigid-unit modes account for some of the NTE-driving phonons, the added d-o-f compared with Zn(CN)$_2$ results in modes with twisting motions, capable of inducing greater NTE.

An algebraic approach to cooperative rotations in networks of interconnected rigid units.

Crystalline solids consisting of three-dimensional networks of interconnected rigid units are ubiquitous amongst functional materials. In many cases, application-critical properties are sensitive to rigid-unit rotations at low temperature, high pressure or specific stoichiometry. The shared atoms that connect rigid units impose severe constraints on any rotational degrees of freedom, which must then be cooperative throughout the entire network. Successful efforts to identify cooperative-rotational rigid-unit modes (RUMs) in crystals have employed split-atom harmonic potentials, exhaustive testing of the rotational symmetry modes allowed by group representation theory, and even simple geometric considerations. This article presents a purely algebraic approach to RUM identification wherein the conditions of connectedness are used to construct a linear system of equations in the rotational symmetry-mode amplitudes.

Algebraic search for cooperative rotational rigid-unit modes

Algebraic search for cooperative rotational rigid-unit modes

Origin of negative thermal expansion in Zn2GeO4 revealed by high pressure study

Zn2GeO4, as an open-framework structure compound, exhibits negative thermal expansion (NTE) below room temperature. In this work, we investigated the structural stability and phonon modes employing the x-ray diffraction and Raman spectroscopy under high pressure up to 23.0 GPa within a diamond anvil cell, and we observed that a pressure-induced irreversible amorphization took place around 10.1 GPa. Bulk modulus, pressure coefficients, and Grüneisen parameters were measured for the initial rhombohedral structure. Several low-frequency rigid-unit modes are found to have negative Grüneisen parameter, which accounts for the primary part of NTE in Zn2GeO4. These results further confirm the hypothesis that the pressure-induced amorphization and the negative thermal expansion are correlated phenomena.

3D negative thermal expansion in orthorhombic MIL-68(In).

Negative thermal expansion (NTE), lattice contraction upon heating, has been found in some metal organic frameworks (MOFs). To date, MOF compounds have been found to present either isotropic NTE in the cubic phase or 1D or 2D NTE in a low symmetry structure. Herein, we report anisotropic NTE in orthorhombic MIL-68(In) with the 3D NTE of αa = -5.6 × 10-6 K-1, αb = -2.7 × 10-6 K-1 and αc = -4.0 × 10-6 K-1. A synchrotron X-ray diffraction study shows that the transverse vibration of the phenyl ring leads to the NTE in the ab plane. Further high-energy synchrotron X-ray total scattering investigation reveals that the rotation of rigid octahedrons gives rise to the NTE along the c-axis. Such abnormal 3D NTE in orthorhombic MIL-68(In) is accounted for by the combination of transverse vibration and rigid unit modes. This is the first case to show 3D NTE in a low symmetry MOF compound and it will be beneficial to tailor the thermal expansion of functional MOF materials.

Algebraic search for cooperative rotational rigid-unit modes

Algebraic search for cooperative rotational rigid-unit modes

Tunable Negative Thermal Expansion in Layered Perovskites from Quasi-Two-Dimensional Vibrations.

We identify a quasi-two-dimensional (quasi-2D) phonon mode in the layered-perovskite Ca_{3}Ti_{2}O_{7}, which exhibits an acoustic branch with quadratic dispersion. Using first-principles methods, we show this mode exhibits atomic displacements perpendicular to the layered [CaTiO_{3}]_{2} blocks comprising the structure and a negative Grüneisen parameter. Owing to these quasi-2D structural and dynamical features, we find that the mode can be utilized to realize unusual membrane effects, including a tunable negative thermal expansion (NTE) and a rare pressure-independent thermal softening of the bulk modulus. Detailed microscopic analysis shows that the NTE relies on strong intralayer Ti-O covalent bonding and weaker interlayer interactions, which is in contrast to conventional NTE mechanisms for perovskites, such as rigid-unit modes, structural transitions, and electronic or magnetic ordering. The general application of the quasi-2D lattice dynamics opens exciting avenues for the control of lattice dynamical and thermodynamic responses of other complex layered compounds through rational chemical substitution, as we show in A_{3}Zr_{2}O_{7} (A=Ca, Sr), and by heterostructuring.

Negative thermal expansion and electrical properties of α-Cu2V2O7

The orthorhombic copper(II) vanadate (α-Cu2V2O7) was synthesized using a solid state method, and its negative thermal expansion (NTE) and electrical properties were systematically studied. The results show that the α-Cu2V2O7 exhibits NTE property linearly from room temperature (RT) to 470K with the coefficient of thermal expansion (CTE) −5.8×10−6K−1. The coupling rotation of rigid unit modes (RUMs) caused by transverse vibrations of bridge oxygen atoms is attributed to the NTE behavior. Moreover, NTE property performs nonlinearity in 470–610K, and the average CTE is about −2.4×10−6K−1 which is caused by the synthetic action of d9 Jahn-Teller effect of Cu2+ and small amount of micro-pores. The material exhibits low thermal expansion between 610 and 720K with CTE 0.43×10−6K−1. The distortion effect was confirmed by UV–vis absorption spectroscopy. The impedance spectroscopy analysis shows that the α-Cu2V2O7 has a certain semiconductor property, which is consistent with the results of ab-initio calculation. The NTE property and pyroconductivity bring a good potential application in composite conductive ceramic and high temperature conductive materials.

Unusual Polarization Ordering in Lanthanum Modified Lead Zirconate Titanate (Pb0.97La0.03)(Zr0.90Ti0.10)0.9925O3

Lanthanum incorporation in lead zirconate titanate (PLZT) increases the stability range of the antiferroelectric phase in the Zr‐rich side of the phase diagram. The paper presents the analysis, which has been carried out on (Pb0.97La0.03)(Zr0.90Ti0.10)0.9925O3 ceramic system since it exhibits interesting results with ferroelectric (FE) and antiferroelectric (AFE) phases, and coexistence of both properties in some temperature range. Transmission electron microscopy evidences a complex domain structure, where nanodomains appear within larger ones. The data are consistent with the possibility to have in this material a ferroelastic phase of novel type in which the coexistence mechanism between FE and AFE properties is associated with the oscillation in space of the polarization. The possibility for a determining role of a rigid unit mode in structural instabilities, dependence on composition and doping, and pretransitional effects in the perovskite family is underlined by the obtained results for the studied system.

Molecular dynamics simulation study of various zeolitic imidazolate framework structures.

We report the results of a series of molecular dynamics simulations on a number of zinc zeolitic imidazolate framework (ZIF) structures together with some lattice dynamics calculations on ZIF-4, providing information about the flexibilities of these structures. The simulations have used a force field we developed based on ab initio calculations of clusters of ligands and metal cations. We have shown that there are instabilities of the structures of some ZIF structures at low temperatures and high pressures. A rigidity analysis based on the Rigid Unit Mode model shows considerable degree of network flexibility, including a significant elastic flexibility.

Rigid unit modes insp−sp2hybridized carbon systems: Origin of negative thermal expansion

Using density functional theory combined with quasi-harmonic approximation, we investigate the thermal expansion behaviors of three different types ({\alpha}, {\beta}, and {\gamma}) of graphyne which is a two-dimensional carbon allotrope composed of sp and $sp^2$ bonds. For each type of graphyne, we obtain the temperature dependent area variation by minimizing its free energy as a function of temperature, which is calculated by considering all the phonon modes in the whole Brillouin zone. We find that all three types of graphyne exhibit negative in-plane thermal expansion up to $T\lesssim1000$ K. The observed in-plane thermal contraction can be attributed partially to the ripple effect, similarly in graphene. The ripple effect itself, however, is not sufficient to explain anomalously larger thermal contraction found in graphyne than in graphene. Our deliberate analysis on the phonon modes observed in graphyne enables us to discover another source causing such thermal expansion anomaly. We find that there are particular phonon modes with frequencies around a few hundreds of cm$^{-1}$ existing exclusively in graphyne that may fill empty spaces resulting in area reduction. These modes are identified as "rigid unit modes" corresponding to the libration of each rigid unit composed of $sp^2$ bonds.

Intrinsic flexibility of porous materials; theory, modelling and the flexibility window of the EMT zeolite framework.

Framework materials have structures containing strongly bonded polyhedral groups of atoms connected through their vertices. Typically the energy cost for variations of the inter-polyhedral geometry is much less than the cost of distortions of the polyhedra themselves - as in the case of silicates, where the geometry of the SiO4 tetrahedral group is much more strongly constrained than the Si-O-Si bridging angle. As a result, framework materials frequently display intrinsic flexibility, and their dynamic and static properties are strongly influenced by low-energy collective motions of the polyhedra. Insight into these motions can be obtained in reciprocal space through the `rigid unit mode' (RUM) model, and in real-space through template-based geometric simulations. We briefly review the framework flexibility phenomena in energy-relevant materials, including ionic conductors, perovskites and zeolites. In particular we examine the `flexibility window' phenomenon in zeolites and present novel results on the flexibility window of the EMT framework, which shed light on the role of structure-directing agents. Our key finding is that the crown ether, despite its steric bulk, does not limit the geometric flexibility of the framework.

X-ray absorption spectroscopy study of local dynamics and thermal expansion inReO3

The thermal expansion of polycrystalline ${\mathrm{ReO}}_{3}$ is studied in vacuum by x-ray diffraction from 300 to 600 K and by extended x-ray absorption fine structure (EXAFS) at the Re ${L}_{3}$ edge from 30 to 600 K. A detailed EXAFS analysis is presented up to the sixth coordination shell around Re. The crystal thermal expansion is weak throughout the explored range, negative below 100 K, positive from 150 to 500 K, and negative again above 500 K. The analysis of mean square relative displacements obtained by EXAFS and of mean square displacements available from x-ray and neutron diffraction measurements confirms the presence of significant static disorder in the investigated samples. EXAFS results suggest that below 500 K in ${\mathrm{ReO}}_{3}$ at least some of the octahedra are slightly and disorderly rotated by an average angle $\ensuremath{\langle}\ensuremath{\theta}\ensuremath{\rangle}$, whose value decreases when the temperature increases. This explains why ${\mathrm{ReO}}_{3}$, in spite of the possibility of supporting rigid unit modes, shows a weak positive expansion at increasing temperatures, from 150 up to 500 K. Results are discussed and compared with recent experimental and theoretical work on materials characterized by negative thermal expansion.