The ionic model: Extension to spatial charge distributions,derivation of an interaction potential for silica polymorphs

An interatomic interaction potential for silica polymorphs is derived based on the SCD model (cfr. Tijskens et al. 1994). This interaction potential incorporates all classical electrostatic interactions arising from the spherical part of the spatial extent of the atoms including many body interactions. The potential is derived from Hartree-Fock energies and electron densities for a set 72 [SiO₄]^4-- and [Si₂O₇]^6--clusters with variable configuration. The long range impact of the surroundings on these clusters in the infinite system has been successfully mimicked by embedding the clusters in a finite three-dimensional array of point charges. This three-dimensional array of point charges is optimized as to reproduce the average site potential and its gradient occurring in II–IV-coordinated silica polymorphs at the central atoms of the clusters. The resulting interaction potential consists of two functions of the configurational coordinates, ?, describing spherical “atomic” electron densities, σ_a(x, ?) for A=Si, O. All classical electrostatic interactions are derived from these densities. A Born-Mayer type correction term ΔE _qm(?) models the quantum mechanical interactions and the electrostatic interactions arising from the non-spherosymmetrical component of the electron density. The new interaction potential model shows a slightly improved reproduction of the potential surface with respect to the classical Born-Mayer ionic model and demonstrates the importance of many body interactions as charge transfer and expansion/contraction of the atomic electron densities in these systems. Also the dependence of the quantum mechanical correction term ΔE _qm(?) on the Si-O-Si-bond angle proves covalent effects to be larger than suggested by the classical Born-Mayer ionic model thereby clarifying the controversy in literature on the importance of covalent effects in silica polymorphs and polymerised silicates in general.     

Nonparametric estimation of spatial distributions

The indicator approach, whereby the data are used through their rank order, allows a nonparametric approach to the data bivariate distribution. Such rich structural information allows a nonparametric risk-qualified, estimation of local and global spatial distributions.     

The solubility of olivine in basaltic liquids: an ionic model

A simple ionic model which describes the solution of the forsterite component of olivine in silicate liquids is reported. The melting relation is represented: (Mg₂SiO₄)_ol = 2(Mg²⁺)_L + (SiO₄^4?_L and is extended to all silicate liquids by normalizing their compositions to 4 oxygens. At 1 bar, the temperature at which olivine is in equilibrium with any alkali-depleted basaltic composition can be calculated to within ±30°C. This error is increased considerably when applied to terrestrial basalts which contain several weight percent alkalis. Alkalis interfere with the equilibrium by generating strongly repulsive interionic forces which can be crudely modelled in a manner consistent with constraints imposed by regular solution theory.The model quantifies the reduced activity of SiO₄^4? monomers due to increasing SiO₂ concentrations in the melt. This is a consequence of polymerization which does not appear to operate gradually over the entire spectrum of mafic and ultramafic compositions. The coordination of alumina in melts which precipitate olivine only appears to be dominantly octahedral. Titanium acts as a polymerizing agent by interconnecting previously isolated SiO₄^4? monomers. Calcium associated with normative diopside tends to exhibit small but perceptible repulsive forces involving Mg²⁺.     

Atomic and ionic radii: a comparison with radii derived from electron density distributions

 The bonded radii of anions obtained in topological analyses of theoretical and experimental electron density distributions differ from atomic, ionic and crystal radii in that oxide-, fluoride-, nitride- and sulfide-anion radii are not constant for a given coordination number. They vary in a regular way with bond length and the electronegativity of the cation to which they are bonded, exhibiting radii close to atomic radii when bonded to a highly electronegative cation and radii close to ionic radii when bonded to a highly electropositive cation. The electron density distributions show that anions are not spherical but exhibit several different radii in different bonded directions. The bonded radii of cations correlate with ionic and atomic radii. But unlike ionic radii, the bonded radius of a cation shows a relatively small increase in value with an increase in coordination number. In contrast to atomic and ionic radii, the bonded radius of an ion in a crystal or molecule can be used as a reliable and well-defined estimate of its radius in the direction of its bonds. Received April 16, 1996 / Revised, accepted August 6, 1996     

Variational stabilization of the ionic charge densities in the electron-gas theory of crystals: Applications to MgO and CaO

The electron-gas theory of crystals is extended to include the effects of many-body forces that arise from both electrostatic and overlap interactions. These effects are incorporated through a self-consistent spherical relaxation of the ionic charge distributions such that the crystal binding energy is minimized. This variational model is used to compute the elastic constants and equations of state of MgO and CaO, and we compare its results to those derived from earlier electron-gas models. In the variational model, the anion charge distributions are markedly more sensitive to the local crystal environment than they are in the PIB or other electron-gas models. We find that for these oxides the variational model gives the best overall agreement with experiment for lattice constants, equations of state, dissociation energies, and elastic moduli.     

Nonparametric estimation of multivariate joint and conditional spatial distributions

To estimate the recoverable reserves of deposits containing more than one metal, linear combinations of sample grades based on the economic importance of each metal have been employed. These linear combinations, called equivalent grades, have inherent problems at least one of which is that the contribution of an individual metal is confounded with all others. As an alternative to equivalent grades, multivariable joint and conditional spatial distributions may be estimated using indicator variable methods. These spatial distributions may then be used to determine the joint or conditional amounts of the metals in the estimated recoverable reserve.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

Modelling Nd-isotopes with a coarse resolution ocean circulation model: Sensitivities to model parameters and source/sink distributions

The neodymium (Nd) isotopic composition (?_Nd) of seawater is a quasi-conservative tracer of water mass mixing and is assumed to hold great potential for paleoceanographic studies. Here we present a comprehensive approach for the simulation of the two neodymium isotopes ¹⁴³Nd, and ¹⁴⁴Nd using the Bern3D model, a low resolution ocean model. The high computational efficiency of the Bern3D model in conjunction with our comprehensive approach allows us to systematically and extensively explore the sensitivity of Nd concentrations and ?_Nd to the parametrisation of sources and sinks. Previous studies have been restricted in doing so either by the chosen approach or by computational costs. Our study thus presents the most comprehensive survey of the marine Nd cycle to date.Our model simulates both Nd concentrations as well as ?_Nd in good agreement with observations. ?_Nd covaries with salinity, thus underlining its potential as a water mass proxy. Results confirm that the continental margins are required as a Nd source to simulate Nd concentrations and ?_Nd consistent with observations. We estimate this source to be slightly smaller than reported in previous studies and find that above a certain magnitude its magnitude affects ?_Nd only to a small extent. On the other hand, the parametrisation of the reversible scavenging considerably affects the ability of the model to simulate both, Nd concentrations and ?_Nd. Furthermore, despite their small contribution, we find dust and rivers to be important components of the Nd cycle. In additional experiments, we systematically varied the diapycnal diffusivity as well as the Atlantic-to-Pacific freshwater flux to explore the sensitivity of Nd concentrations and its isotopic signature to the strength and geometry of the overturning circulation. These experiments reveal that Nd concentrations and ?_Nd are comparatively little affected by variations in diapycnal diffusivity and the Atlantic-to-Pacific freshwater flux. In contrast, an adequate representation of Nd sources and sinks is crucial to simulate Nd concentrations and ?_Nd consistent with observations. The good agreement of our results with observations paves the way for the evaluation of the paleoceanographic potential of ?_Nd in further model studies.     

Fuzzy-based spatial modeling approach to predict island karst distribution: a conceptual model

Previous studies have shown that island karst could successfully indicate paleoclimate change in the Quaternary Period. However, because of the relative inaccessibility of carbonate islands and their rural settings, the exploration of island karst features has been limited. To enhance future research, remote sensing and geospatial modeling were used in this study to improve the island karst exploration record. The results showed that fuzzy-based spatial modeling could successfully predict the island karst distributions on a simple carbonate island. The accuracy of the model was above 90 %. This method could apply to other coastal karst regions and carbonate islands in the future.     

The application of ionic equilibria to metamorphic differentiation: An example

Mineral segregations formed by metamorphic differentiation are an important source of information on diffusion processes in metamorphism. Segregations consisting of andalusite-biotite-quartz cores surrounded by a quartz-feldspar mantle in sillimanite-biotitefeldspar-quartz gneiss near Västervik, Sweden (Loberg, 1963) formed by core-to-mantle migration of K, and mantle-to-core migration of Fe, Mg and Ca. These migrations can be represented by a set of interconnected ionic equilibria involving reaction of microcline and Fe(OH)⁺ in the core to form andalusite plus biotite, and reaction of K⁺, sillimanite and biotite in the mantle to form microcline. Equilibrium constants for these reactions, calculated for conditions inferred from the mineral assemblage and biotite composition, indicate gradients of K⁺ activity (higher in core) and Fe(OH)⁺ activity (higher in mantle). These gradients result simply from the free energy difference between andalusite and sillimanite, without invoking pre-existing megascopic inhomogeneities in the rock or surface energy effects. Although small, these gradients appear to be capable of driving the segregation process.     

Adsorption of chromate on variable charge soils as influenced by ionic strength

The adsorption behavior of chromate on two variable charge soils (Oxisol and Ultisol) was investigated through batch experiments at different ionic strengths and pH values. The adsorption of chromate on the variable charge soils was found to be strongly dependent on the pH of the soil solutions. A characteristic pH was observed, which corresponds to the intersection of the chromate adsorption—pH curves at different ionic strengths. The characteristic pH values are 5.50 for Oxisol and 5.04 for Ultisol, close to the point of zero salt effect (PZSE) of these soils. The zeta potentials measured for these soils provide the evidences to support the interpretation of the effect of ionic strength on the adsorption of chromate on these variable charge soils. The adsorption behavior of chromate was interpreted by a schematic representation of chromate distribution at increasing ionic strength. The chromate desorption–pH curves were also found to intersect at pH of 5.15 and 4.89 for the Oxisol and Ultisol, respectively. It is considered that chromate adsorption by the variable charge soils was mainly determined by the electrostatic potential on the adsorption plane, which was controlled by the ionic strength of the soil solutions.     

A dynamic spatial diffusion model: an application to the kibbutz industry in Israel

A simple dynamic spatial diffusion model is suggested using the hitherto unreferred to concept ofinnovation adoption disparity between adoption units. Such disparity is an outcome of spatial variations in information flow caused by differential distances. The model is structured around the relationship between adoption disparity and distance in a temporal context. An application of the model to the diffusion of industrialization among Kibbutzim in Israel demonstrates its usefulness as a basic conceptual framework for understanding spatial difcesses. A possible extension of the model to inncorporate elements of the logistic curve is also proposed. *** DIRECT SUPPORT *** ABI02060 00004     

农田土壤水分和盐分空间分布的指示克立格分析评价

运用非参数地质统计学的多元指示克立格法,结合单元指示克立格法,对黄河河套灌区55hm2的盐渍化土地上两个比较关键时期的土壤水分和盐分进行空间分布的分析,给出了同一时期土壤水盐和不同时期土壤盐分满足一定标准的综合概率图,并借助该案例详细介绍了多元指示克立格的原理、步骤及应用范围。指示克立格方法为不同尺度(从农田到区域)上水土资源质量的评价提供了新的方法,能为水土资源的现代决策管理提供指导。     

Extension to barodesy to model void ratio and stress dependency of the K 0 value

Barodesy is a new framework for constitutive modelling of soils. The actual version uses a constant value of the coefficient of earth pressure at rest K ₀, which contradicts experimental findings. A straightforward modification is presented here to remove this shortcoming. The calibration of the material parameters remains as simple as in the original model.     

Unified scaling theory for distributions of temporal and spatial characteristics in seismology

The paper describes the unified scaling theory for distribution functions of temporal and spatial characteristics in seismology. It is based on the scaling of seismological characteristics calculated for various energy–spatial–temporal intervals. The common mathematical methods for the scaling of distribution functions are developed. The means to test possibility of such scaling are found as well. The relationship between the unified scaling theory and other present scaling approaches is determined. The theory is applied to two characteristics of different seismoactive regions. The first characteristic is the waiting time between earthquakes ΔT, the second one is a new space parameter ΔD_min, which is the minimum distance of a current seismic event to the nearest (in space) neighbor in an energy–spatial–temporal interval. The distribution of the characteristics ΔT and ΔD_min allows estimating the time interval to the next earthquake and the distance of the following earthquake from previous earthquakes. Thus, these characteristics are very important for seismic hazard estimations. Scaling of distributions functions is proven to be successful for ΔD_min in all energy–spatial–temporal intervals and for ΔT with variations of energy/magnitude range. The distribution function of ΔT for various time domains was stable in only 60% of the cases, and near to unstable for spatial variations.     

Estimating spatial distributions of heterogeneous subsurface characteristics by regionalized classification of electrofacies

Regionalized classification of electrofacies utilizes the statistical relationships between laboratory determined hydrologic properties and field-measured geophysical properties to estimate spatial distributions of porosity, permeability, and diagenetic characteristics. The method, illustrated with an application to the St. Peter Sandstone in the Michigan basin, combines techniques for multivariate analysis and spatial estimation. Core plug and borehole geophysical data are clustered into electrofacies that reflect the hydrologic properties and diagenetic characteristics of the formation. Electrofacies characteristics then are used to assign a class membership probability at locations where only geophysical data are available. Three-dimensional estimation of electrofacies occurrence is done by kriging datasets containing the probability of electrofacies membership at borehole locations. The discretization and kriging geometry allow three-dimensional estimation of hydrologic parameters for a large region that incorporates meter-scale heterogeneity. Finally, permeability and porosity are estimated at each grid location by probability-weighting. Because the electrofacies carry information about both the hydrologic and lithologic properties, the resulting spatial distributions provide an understanding of both the present-day flow characteristics and the extent of processes that control them.Managed by Martin Marietta Energy Systems. Inc., under contract DE-AC05-84OR21400 with the U.S. Department of Energy); Publication No. 4371, Environmental Sciences Division, ORNL.     

Extension of the specific interaction model to include gas solubilities in high temperature brines

A semi-empirical model for gas solubilities in high temperature brines was developed by modification of the Pitzer interaction model (Pitzer, 1973) and evaluated by least-squares fitting to available P-T-X data for carbon dioxide, hydrogen sulfide, and methane in pure water and in aqueous sodium chloride solutions. Over the range of experimental data used (25°–350°C, NaCl concentration 0–6 m, P_CO2 1–150 atm. P_CH4 1–30 atm, P_CH4 1–1700 atm), standard deviations of fit in the natural logarithm of the Henry's law ratio were 0.096 for CO₂, 0.093 for H₂S and 0.116 for CH₄.The model has several advantages. First, it has a theoretical basis which can easily be extended for the inclusion of more than one salt or gas. Second, the equations simplify to the empirically derived equation of Setschenow, in the limit of infinite dilution. Finally, the model is easily integrated into a framework of equations which can yield all of the thermodynamic properties of the system.An example in which values of the model parameters for interaction of gas with salt may be used to estimate the effect of dissolved gas or salt solubility is given.     

Thickness and spatial distributions of clastic dykes, northwest Sacramento Valley, California

The geometry and distribution of the clastic dykes of the Ono district, North Sacramento Valley are examined within stream sections. Five traverses along dry stream beds provide good exposure allowing the spacing, thickness and geometry of the dykes to be recorded. The spatial and thickness distribution of the dykes are considered using cumulative frequency plots, allowing a visual estimation of a best fit distribution. Dyke thickness conforms best to a log-normal distribution. There is also a characteristic minimum dyke thickness in a traverse and this is attributed to the minimum aperture that a fluid with sand clasts is able to exploit. Dyke spacing, however, shows a good correlation with a power-law distribution for four traverses, suggesting that there is a mechanistic control on the spatial distribution. Plotting dyke thickness against minimum dyke spacing reveals that thin dykes do not generally intrude in isolation. Unlike veins and igneous dykes, clastic dykes continue to provide preferential pathways for fluid flow, subsequent to their intrusion, thus inhibiting intrusion in the area surrounding a pre-existing dyke. A combination of this process and dyke branching provides the best model for the observed spatial and thickness distribution of clastic dykes seen in the Ono district, California.     

Computation of surface energies in an electrostatic point charge model: I. Theory

The attachment energies, the slice energies and the specific surface energies can be calculated in an electrostatic point charge model using the formula derived by Madelung for the potential introduced by an infinite row of equally spaced point charges. Power series are given for the Hankel function iH ₍₀₎ ⁽¹⁾ (iy) and Ψ(x)=d ln x!/dx. The logarithmic expression in the Madelung formula converges rapidly when applying a power series, which combines equally charged cations and anions. Besides the specific surface energy (γ ^hkl), the slice energy (E _s ^hkl ) and the attachment energy (E _a ^hkl ) can be considered as special categories of surface energies as they depend on surface configurations as well. The specific surface energy γ is the energy per unit area of surface needed to split the crystal parallel to a face (hkl). The attachment energy (E _a) is the energy released per mole, when a new slice of thickness d _hkl crystallizes on an already existing crystal face (hkl). The growth rate of the crystal face (hkl) is a function of its attachment energy. The slice energy (E _s) is the energy released per mole, when a new slice d _hkl is formed from the vapour neglecting the influence of edge energies. The lattice energy (E _c) which is the energy released per mole of a crystal crystallizing from the vapour, is given by the following relation: E _c=E _a+E _s.     

Application of empirical ionic models to SiO2 liquid: Potential model approximations and integration of SiO2 polymorph data

Structural and thermodynamic properties of crystalline SiO₂ and SiO₂ liquid have been examined with Monte Carlo (MC), molecular dynamics (MD), and energy minimization (EM) calculations using several ionic potential models obtained from the literature. The MC and MD methods calculate the same structural and thermodynamic properties for liquids when the same potential model is used. The Ewald (1921) method of calculating coulomb interactions reproduced most successfully the structure of liquid silica. Approximating the coulomb interaction by eliminating the inverse lattice sum results in predicted bond distances that are too short and an average 〈Si-O-Si〉 angle of approximately 180°. Introduction of a cut-off in the potential energy function produces irregular tetrahedra and inconsistencies in predicted Si-O coordination in silica liquid. The system internal energies show that liquid structures derived from random starting configurations can be metastable relative to structures calculated from crystalline starting configurations.The static lattice properties of the polymorphs alpha-quartz, coesite, and stishovite were used to evaluate further the accuracy of different sets of repulsive parameters for the full Ewald ionic model. Most of the models studied reproduced poorly the measured structures and elastic constants of the polymorphs. The major weakness of the ionic model is the unreasonably large Si-O bond strength (120 × 10⁻¹² ergs/bond) when formal ionic charges are used. Fractional charge models with a small Si-O bond strength (30 × 10⁻¹² ergs/bond) improve the agreement with experimental data. However, further improvement of the ionic model should include reducing the Si-O bond strength to values in better agreement with published estimates (7 × 10^{− 12} to 13 × 10⁻¹² ergs/bond). By using additional information to constrain the parameterization of the ionic model, such as estimated bond strengths and static properties of the silica polymorphs, a model more representative of the interparticle interactions may be obtained.     

Computation of surface energies in an electrostatic point charge model: II. Application to zircon (ZrSiO4)

The attachment energies, the slice energies and the specific surface energies can be calculated in an electrostatic point charge model using the formula derived by Madelung for the potential introduced by an infinite row of equally spaced point charges. Power series are given for the Hankel function iH ₍₀₎ ⁽¹⁾ (iy) and (x)=d ln x!/dx. The logarithmic expression in the Madelung formula converges rapidly when applying a power series, which combines equally charged cations and anions. Besides the specific surface energy ( ^hkl), the slice energy (E _s ^hkl ) and the attachment energy (E _a ^hkl ) can be considered as special categories of surface energies as they depend on surface configurations as well. The specific surface energy is the energy per unit area of surface needed to split the crystal parallel to a face (hkl). The attachment energy (E _a) is the energy released per mole, when a new slice of thickness d _hkl crystallizes on an already existing crystal face (hkl). The growth rate of the crystal face (hkl) is a function of its attachment energy. The slice energy (E _s) is the energy released per mole, when a new slice d _hkl is formed from the vapour neglecting the influence of edge energies. The lattice energy (E _c) which is the energy released per mole of a crystal crystallizing from the vapour, is given by the following relation: E _c=E _a+E _s.