Kenton D. Hammonds

Rigid-unit phonon modes and structural phase transitions in framework silicates

Abstract The rigid-unit mode model provides many new insights into the stability and physical properties of framework silicates. In this model the SiO4 and AlO4 tetrahedra are treated as very stiff, to a first approximation as completely rigid, in comparison with intertetrahedral forces. In this paper we apply the model to several important examples. The model is reviewed by a detailed study of quartz, and it is shown that the α-β phase transition involves a rigid-unit mode that preserves the Si-O-Si bond angle. The model is used to explain the phase transitions in cristobalite and the different feldspar, sodalite, and leucite structures. We also use the model to explain the nature of the high-temperature disordered phases of cristobalite and tridymite, to interpret the observations of streaks of diffuse scattering in electron diffraction patterns, to interpret the structures in the kalsilite-nepheline solid solution, to explain volume anomalies in the cubic leucite structures, and to explain qualitatively the negative linear thermal expansion in cordierite. The results for the highest symmetry sodalite structure show that there is a rigid-unit mode at every wave vector, a finding with significant implications for the understanding of the sorption and catalytic behavior of zeolites.

Rigid unit modes in framework silicates

Abstract We describe a model for framework silicates in which the SiO4 (and AlO4) tetrahedra are treated as perfectly rigid and freely jointed. From this model we are able to identify low-energy modes of distortion of the structure, which we call Rigid unit modes. These modes can act as soft modes to allow easy distortions at phase transition. We discuss three forces that will operate at a phase transition in conjunction with the candidate soft modes to determine which of the rigid unit modes will actually precipitate a phase transition, and illustrate these ideas by detailed discussions of the phase transitions in quartz, leucite and cristobalite. The model can also be used to estimate the transition temperature, and the theory highlights an important role for the stiffness of the tetrahedra.

The phenomenon of low Al-Si ordering temperatures in aluminosilicate framework structures

Abstract The enthalpy of ordering of Al and Si between tetrahedral sites in framework structures is now known from several studies to be in the range 0.3-1.4 eV per two Al-O-Si linkages, depending on the structure and type of linkage between tetrahedra. On the basis of this value a simple Bragg-Williams model predicts an ordering phase transition at a temperature, Tc, on the order of a few thousand degrees after adjusting the estimate to account for fluctuations in short-range order. However, there are several materials with Tc below 1800 °C (a typical melting point), with some having Tc so low that it cannot be observed because of the slowness of the ordering kinetics, e.g., gehlenite and leucite. We discuss two mechanisms that can reduce Tc substantially: low effective dimensionality and low Al concentration. The cases of sillimanite, gehlenite, cordierite, and leucite, and some simple demonstration systems, are treated quantitatively with the aid of Monte Carlo computer simulations. Other materials (kalsilite, anorthite, albite, and nepheline) are discussed qualitatively in terms of the same principles.

On the convergence of the SHAKE algorithm

Abstract Several aspects of the convergence speed for the SHAKE iterative technique are reviewed. Two examples of molecular geometry which are known to have slow SHAKE convergence are used to show that simple changes to the procedure can result in a five-fold improvement in convergence speed. Fortran code, relevant to fully atomic models of alkane chains, is presented which is suitable for use in MD programs. Aspects of convergence criteria and constraint tolerance are also discussed.

Rigid unit modes in crystal structures with octahedrally coordinated atoms

Abstract Within the framework of the Adam-Gibbs (configurational entropy) theory of viscosity, it is shown that for a given composition, the ratio of parameters Be (a temperature independent constant) to Sc(Tg) (the configurational entropy at the glass transition) is proportional to the height of the average potential energy barrier to viscous flow (Δμ) and the size of the rearranging domains at the glass transition [z*(Tg)]. The ratio Be/Sc(Tg) is evaluated for several silicate and aluminosilicate compositions of variable polymerization. It is found that the ratio Be/Sc(Tg) shows simple compositional variations that correspond closely to those that may be expected qualitatively for the height of the potential energy barrier to viscous flow. Assuming that z(Tg) is a constant for all compositions, the available data for Be/Sc(Tg) are parameterized as a function of Δμ. The physical basis of this parameterization will therefore allow extension to more complex systems as additional data become available. The Ae term in the Adam-Gibbs equation (another temperature independent constant) shows only minor compositional variation (-2.6 ± 1). but the variation that does exist is found to be a linear function of Be/tetrahedron. The proposed parameterizations of Be/Sc(Tg) and Ae are shown to be sufficient for estimating the glass transition temperature to within 15-20 K. Calculated glass transition temperatures may be combined with existing models for viscosities in the range 1O-1O5 Pa·s. Interpolation provides the whole viscosity curve and thus also an estimate of the departure from Arrhenian behavior. Although further work is necessary to verify) and extend the parameterizations to compositions of direct geological relevance, this work represents a step toward a fully generalizable predictive model of silicate melt viscosity based within a physical framework.

A COMPUTATIONAL STUDY OF AL/SI ORDERING IN CORDIERITE

The ordering of Al and Si in Mg cordierite Mg2Al4Si5O18 is considered using computer simulation. First the enthalpy of interaction Jij between sites is derived by computer modelling 101 different Al/Si configurations and analysing their energies. They are compared with similar results for three other minerals and with ab initio calculations to assess the whole approach. Secondly the ordering process is studied using Monte Carlo simulation applied to the Jij. The ordering phase transition temperature Tc is found as 1800°C in reasonable agreement with the experimental estimate of 1450° C. These are much lower than the estimate Tc(ABW)≈7600°C obtained from Bragg-Williams theory. Strong short-range order sets in below Tc(ABW), and the reasons for much lower temperature Tc of long-range ordering are discussed. Strong short-range also sets in very rapidly in a simulated anneal, in agreement with experiment. Thirdly an attempt is made to compare our calculated enthalpies directly with the results of NMR and calorimetry experiments, not completely successfully. A free energy ΔG≈4.6 eV for the activation barrier for ordering is suggested.

Rigid unit modes and the negative thermal expansion in ZrW2O8

Abstract ZrW2O8 has a negative coefficient of thermal expansion from 0.3 K to its decomposition temperature around 1050 K. The negative thermal expansion is isotropic and is not disrupted by the structural phase transition at 430 K. A complicated distribution in reciprocal space of Rigid Unit Mode (RUM) phonons has been found for ZrW2O8 using Lattice Dynamics. The RUMS are very low energy phonons which cause collective rotations of ZrO6 octahedra and WO4 tetrahedra within the crystal structure. These distortions enable ZrW2O8 to contract upon heating.

Rigid unit modes in the high-temperature phase of SiO2 tridymite: calculations and electron diffraction

Calculations of the rigid unit mode (RUM) spectrum of the high-temperature phase of SiO2 tridymite are used to explain the patterns of diffuse scattering seen in transmission electron microscopy experiments. These results show that RUMs can occur with wave vectors on curved surfaces in reciprocal space rather than being confined to symmetry points, lines or planes. The fact that the calculations reproduce the detail seen in the diffuse scattering provides a striking nontrivial confirmation of the validity of the rigid unit mode model.

Computational studies of the structure of carbon dioxide monolayers physisorbed on the basal plane of graphite

The molecular-dynamics method has been used to study carbon dioxide physisorbed on the basal plane of graphite at temperatures between 100 and 130K at monolayer and submonolayer coverages. Additionally, energy-minimization calculations have been used to explore the relative stability of a number of solid structures of the adsorbate. Three models of carbon dioxide, which have been successful in describing the properties of the bulk phase, where tested in these simulations of the adsorbate. The results at submonolayer coverage suggest that the adsorbate forms a two-sublattice incommensurate herringbone structure. These solid patches have approximately the correct melting point. At monolayer coverage the existence of a four-sublattice pinwheel structure was only observed for a model with an artificially enhanced quadrupole moment. Further refinement of the potential model will require additional calorimetric or diffraction experiments.

Rigid unit modes and the phase transition and structural distortions of zeolite rho

Abstract The rigid unit mode model (RUM) was applied to the framework structure of zeolite rho. In conjunction with the ideas of soft-mode phase transition theory much of the complex behaviour that have been observed experimentally for this zeolite can be explained. In particular, all possible distortions of the S6R, S8R and D8R are determined and how these distortions influence the positioning of the cations and evoke the observed negative thermal expansion of this material. The large volume reduction observed through the low to high temperature phase transition of zeolite Sr–rho, the large distance relocation of Cd2+-ions that accompanies the I 4 3m to Im 3 m phase transition of zeolite Cd–rho, and the dehydration behaviour of zeolite rho, beryllophosphate and -arsenate zeolite rho could be explained. Also the possibility of an Im 3 phase of zeolite rho was predicted and its framework structure was described.