A physical basis for Pauling's definition of bond strength

 The average strength, s, of the bonded interactions comprising a cation containing oxide anion coordination polyhedron and the value of the electron density, ρ(r _c ), at the bond-critical points are inversely correlated with bond length. In each case, the observed bond lengths, R, were modeled with power-law expressions defined in terms of s/r and ρ(r _c )/r, respectively, where r is the Periodic Table row number of the cation involved in the bonded interaction. On the basis of the close connection between bond strength and the value of the electron density at the bond-critical point, we conclude that bond strength is a direct measure of bond type; the greater its value, the greater the localization of electron density in the binding region and the greater the shared–electron covalent character of the bonded interaction. Received: 15 October 2002 / Accepted: 17 February 2003 Present address:G. V. Gibbs in care of M. Spackman Department of Chemistry, University of New England, Armidale 2351, Australia Acknowledgements The NSF is thanked for supporting this study with grant EAR–9627458. The paper was written while GVG was a Visiting NSF Scholar at The University of Arizona. The faculty and graduate students of the Department of Geosciences and Bob Downs and Marelina Stimpf in particular are thanked for making the visit great fun.     

A study of the bonded interactions in nitride molecules in terms of bond critical point properties and relative electronegativities

Bond critical point properties calculated for the MN bonds in a number of geometry optimized nitride molecules containing first- and second-row M cations are compared with those calculated for a number of oxide molecules. As reported for the oxides, the value of the electron density, ρ(r _c ), at the bond critical points, r _c , increases with decreasing bond length while for the more electronegative cations, the local energy density, H(r _c ) decreases nonlinearly in value as the relative electronegativities of the M-cations, χ _M , tend to increase. In the majority of cases, χ_M, |λ₁|/λ₃ and ∇²ρ(r _c ) increase with decreasing minimum energy bond lengths. The bond lengths adopted by the molecules are indicated to be an important determinant of the critical point properties of the electron density distributions. The relative electronegativities derived from the electron density distributions of the nitrides agree with those derived for the oxides and Pauling’s electronegativities to within ∼5%, on average. Received: 3 February 1997 / Revised, accepted: 11 July 1997     

Critical point properties of electron density distributions for oxide molecules containing first and second row cations

 Minimum energy geometries and electron density distributions, ϱ(r), for ∼40 polyatomic oxide molecules containing first and second row M-cations have been calculated at the Hartree-Fock level with a 6-311++G** basis set. The nature of the bonded interactions in these molecules is examined in terms of the relative electronegativities, χ _M , of the M-cations and the properties of the electron density distribution, ϱ(r _c ), evaluated at the bond critical points, r _c , along each MO bond. As ϱ(r _c ) and the Laplacian of ϱ(r _c ) increase, χ _M increases indicating an increase in the covalent character of the bonded interactions between M and O. The ratios of the curvatures of ϱ(r _c ) indicate that the NO bond is predominantly covalent, that the CO and SO bonds are of intermediate type and that the remaining MO bonds are indicated to be predominantly ionic in character. A comparison of the critical point properties of ϱ(r _c ) and χ _M indicates that the minimum energy MO bond length is an important determinate of the properties of ϱ(r _c ) and the character of the MO bonds. On the other hand, values of the local energy density, H(r _c ), indicate that the LiO, BeO, NaO, MgO and AlO bonds are predominantly ionic and that the BO, CO, NO, SiO, PO and SO bonds are predominantly covalent in character. The χ _M -values provided by the properties of ϱ(r _c ) indicate that the covalent component of a bond increases with decreasing bond length, coordination number and increasing bond strength. Each MO bond seems to represent a unique entity and to possess a distinct set of ϱ(r _c ) properties, the distinction being greater for the more electronegative cations. The bonded radius of the oxide ion, r _b (O), and the χ _M -values determined from ϱ(r _c ) correlate with values determined from promolecule electron density distributions. In addition, r _b (O) and χ _M -values determined from experimental electron density distributions for crystals correlate with values determined from procrystal electron density distributions. The number of critical points and bond paths are modeled rather faithfully by procrystal and promolecule electron density distributions, despite the neglect of the binding forces in their constructions. Received: October 15, 1996/Revised, accepted: February 10, 1997     

SiO bonded interactions in coesite: a comparison of crystalline, molecular and experimental electron density distributions

Bond critical point properties of electron density distributions calculated for representative Si₅O₁₆ moieties of the structure of coesite are compared with those observed and calculated for the bulk crystal. The values calculated for the moieties agree with those observed to within ∼5%, on average, whereas those calculated for the crystal agree to within ∼10%. As the SiOSi angles increase and the SiO bonds shorten, there is a progressive build-up in the calculated electron density along the bonds. This is accompanied by an increase in both the curvatures of the electron density, both perpendicular and parallel to each bond, and the Laplacian of the electron density distribution at the bond critical points. The cross sections of the bonds at the critical points become more circular as the angle approaches 180o. Also, the bonded radius of the oxide anion decreases about twice as much as that of the Si cation as the SiO bond length decreases and the fraction of s-character of the bond is indicated to increase. A knowledge of electron density distributions is central to our understanding of the forces that govern the structure, properties, solid state reactions, surface reactions and phase transformations of minerals. The software (CRYSTAL95 and TOPOND) used in this study to calculate the bond critical properties of the electron density and Laplacian distributions is bound to promote a deeper understanding of crystal chemistry and properties. Received: 23 February 1998 / Revised, accepted: 16 July 1998     

Pauling bond strength, bond length and electron density distribution

The power law regression equation, <R(M–O)> = 1.46(<ρ(r _c)>/r)^?0.19, relating the average experimental bond lengths, <R(M–O)>, to the average accumulation of the electron density at the bond critical point, <ρ(r _c)>, between bonded pairs of metal and oxygen atoms (r is the row number of the M atom), determined at ambient conditions for oxide crystals, is similar to the regression equation R(M–O) = 1.41(ρ(r _c)/r)^?0.21 determined for three perovskite crystals at pressures as high as 80 GPa. The pair are also comparable with the equation <R(M–O)> = 1.43(<s>/r)^?0.21 determined for oxide crystals at ambient conditions and <R(M–O)> = 1.39(<s>/r)^?0.22 determined for geometry-optimized hydroxyacid molecules that relate the geometry-optimized bond lengths to the average Pauling bond strength, <s>, for the M–O bonded interactions. On the basis of the correspondence between the equations relating <ρ(r _c)> and <s> with bond length, it seems plausible that the Pauling bond strength might serve a rough estimate of the accumulation of the electron density between M–O bonded pairs of atoms. Similar expressions, relating bond length and bond strength hold for fluoride, nitride and sulfide molecules and crystals. The similarity of the expressions for the crystals and molecules is compelling evidence that molecular and crystalline M–O bonded interactions are intrinsically related. The value of <ρ(r _c)> = r[(1.41)/<R(M–O)>]^4.76 determined for the average bond length for a given coordination polyhedron closely matches the Pauling’s electrostatic bond strength reaching each the coordinating anions of the coordinated polyhedron. Despite the relative simplicity of the expression, it appears to be more general in its application in that it holds for the bulk of the M–O bonded pairs of atoms of the periodic table.     

Bond length and radii variations in fluoride and oxide molecules and crystals

Molecular orbital calculations completed on fluoride molecules containing first and second row cations have generated bond lengths, R, that match those observed for coordinated polyhedra in crystals to within ～0.04 Å, on average. The calculated bond lengths and those observed for fluoride crystals can be ranked with the expression R=Kp ^?0.22, where p=s/r, s is the Pauling strength of the bond, r is the row number of the cation and K=1.34. The exponent -0.22 (≈ -2/9) is the same as that observed for oxide, nitride and sulfide molecules and crystals. Bonded radii for the fluoride anion, obtained from theoretical electron density maps, increase linearly with bond length. Those calculated for the cations as well as for the fluoride anion match calculated promolecule radii to within ～0.03 Å, on average, suggesting that the electron density distributions in the vicinity of the minima along the bond paths possess a significant atomic component despite bond type. Bonded radii for Si and O ions provided by experimental electron density maps measured for the oxides coesite, danburite and stishovite match those calculated for a series of monosilicic acid molecules. The resulting radii increase with bond length and coordination number with the radius of the oxide ion increasing at a faster rate than that of the Si cation. The oxide ion within danburite exhibits several distinct radii, ranging between 0.9 and 1.2 Å, rather than a single radius with each exhibiting a different radius along each of the nonequivalent bonds with B, Si and Ca. Promolecule radii calculated for the coordinated polyhedra in danburite match procrystal radii obtained in a structure analysis to within 0.002 Å. The close agreement between these two sets of radii and experimentally determined bonded radii lends credence to Slater's statement that the difference between the electron density distribution observed for a crystal and that calculated for a procrystal (IAM) model of the crystal “would be small and subtle, and very hard to determine by examination of the total charge density.”     

Resonance bond numbers: A graph-theoretic study of bond length variations in silicate crystals

The resonance bond number n, as defined in this paper, is designed to describe the strength of an XO bond as a function of the kinds of atoms present and which atoms are bonded. The calculation of n is made on a fragment extracted from the crystal encompassing the XO bond. If this fragment consists of only the X atom and its coordinating O atoms, then n is numerically equal to the Pauling bond strength, s. In this study a graph-theoretic algorithm is developed permitting the calculation of n using fragments including up to 50 atoms. This algorithm was used to calculate n for all of the bonds in ten silicate crystals. Since bond strength is be inversely related to bond length, we examined the relationship between these two variables and found that n can be used to explain over 70 percent of the variation of XO bond lengths from their average values in the crystals. A fit of the parameter n/r, where r is the row number in the periodic table of the metal atom X, to the observed bond lengths in these crystals yielded the equation R(XO)=1.39(n/r)^?0.22 which explains over 95.5 percent of the variation of bond lengths in the crystals. The fact that the same formula with s replacing n was found in an earlier study to be a good estimator of average bond lengths in crystals shows that n relates to individual variations in bond lengths in crystals in the same way that s relates to average bond lengths in crystals. Using minimum energy SiO, AlO and MgO bond lengths and harmonic force constant data calculated for these bonds in hydroxyacid molecules, theoretical equations similar to those used by Pauling to explain bond length variations in hydrocarbons are derived. Bond lengths calculated with these equations for the 10 crystals shows that 95 percent of the variation of the observed bond lengths in these crystals can be explained in terms of n by this purely theoretical model.     

SiO and GeO bonded interactions as inferred from the bond critical point properties of electron density distributions

The topological properties of the electron density distributions for more than 20 hydroxyacid, geometry optimized molecules with SiO and GeO bonds with 3-, 4-, 6- and 8-coordinate Si and Ge cations were calculated. Electronegativities calculated with the bond critical point (bcp) properties of the distributions indicate, for a given coordination number, that the electronegativity of Ge (∼1.85) is slightly larger than that of Si (∼1.80) with the electronegativities of both atoms increasing with decreasing bond length. With an increase in the electron density, the curvatures and the Laplacian of the electron density at the critical point of each bond increase with decreasing bond length. The covalent character of the bonds are assessed, using bond critical point properties and electronegativity values calculated from the electron density distributions. A mapping of the (3, −3) critical points of the valence shell concentrations of the oxide anions for bridging SiOSi and GeOGe dimers reveals a location and disposition of localized nonbonding electron pairs that is consistent with the bridging angles observed for silicates and germanates. The bcp properties of electron density distributions of the SiO bonds calculated for representative molecular models of the coesite structure agree with average values obtained in X-ray diffraction studies of coesite and danburite to within ∼5%. Received: 18 August 1997 / Revised, accepted: 19 February 1998     

Molecular mimicry of the bond length-bond strength variations in oxide crystals

Minimum energy theoretical bond lengths R _t obtained with robust split-basis molecular orbital calculations for 27 hydroxyacid molecules containing first- and second-row cations X ⁿ⁺ reproduce XO bond lengths in crystals. Plots of ln(R _t ) vs. ln(s), where s is the Pauling bond strength, define two different but essentially parallel trends (for first- and second-row cations, respectively) as observed for crystals. A new bond strength parameter p=s/r is defined where r=1 for first- and r=2 for second-row main-group cations. When a ln(R _t ) vs. ln(p) plot is prepared with these theoretical bond lengths, a single trend is obtained. A regression analysis of this data set shows that more than 99 percent of the variation of ln(R _t can be explained in terms of a linear dependence on ln(p), yielding R=1.39 p ^?0.22 as an estimator of the bond lengths. A comparison of 153 mean XO bond lengths compiled by Shannon (1976) for main-group closed-shell X-cations from all 6 rows of the periodic table with those estimated with this formula for r=1, 2, ..., 6, respectively, shows that these bond lengths are estimated within 0.05 Å on average with nearly 85 percent estimated within 0.10 Å of the observed value. More than 97 percent of the variation of these observed bond lengths can be ranked in terms of a linear dependence on the estimated bond lengths. The success of these calculations is further evidence that the forces that govern bond length variations in oxide crystals behave as if they are short-ranged.     

Electron density distribution and bond critical point properties for forsterite, Mg2 SiO4, determined with synchrotron single crystal X-ray diffraction data

A generalized X-ray scattering factor model experimental electron density distribution has been generated for the orthosilicate forsterite, using an essentially extinction and absorption free set of single crystal diffraction data recorded with intense, high energy synchrotron X-ray radiation (E=100.6 keV). A refinement of the model converged with an R(F)=0.0061. An evaluation of the bond critical point, bcp, properties of the distribution at the (3, –1) stationary points for the SiO and MgO bonded interactions, yielded values that agree typically within ~5%, on average, with theoretical values generated with quantum chemical computational strategies, using relatively robust basis sets. On the basis of this result, the modeling of the experimental distribution is considered to be adequate. As the bcp properties increase in magnitude, the MgO and SiO bonds decrease in length as calculated for a number of rock forming silicates. As asserted by Coppens (X-ray charge densities and chemical bonding. Oxford University Press, Oxford, 1997), large negative ²(r_c) values, characteristic of shared interactions involving first row atoms, may not be characteristic of closed shell covalent bonded interactions involving second row Si, P and S atoms bonded to O. This study adds new evidence to the overall relatively good agreement between theoretical bcp properties generated with computational quantum strategies, on the one hand, and experimental properties generated with single crystal high energy synchrotron diffraction data on the other. The similarity of results not only provides a basis for using computational strategies for studying and modeling structures, defects and the reactivity of representative structures, but it also provides a basis for improving our understanding of the crystal chemistry of earth materials and the character of the SiO bonded interaction.     

Electron density distributions and bonded interactions for the fibrous zeolites natrolite, mesolite and scolecite and related materials

 For the fibrous zeolites natrolite, Na₂[Al₂Si₃O₁₀]·2H₂O, mesolite, Na₂Ca₂[Al₂Si₃O₁₀]₃·8H₂O, and scolecite, Ca[Al₂Si₃O₁₀]·3H₂O, with topologically identical aluminosilicate framework structures, accurate single-crystal X-ray diffraction data have been analyzed by least-squares refinements using generalized scattering factor (GSF) models. The final agreement indices were R(F ) = 0.0061, 0.0165, and 0.0073, respectively. Ensuing calculations of static deformation [Δρ(r)], and total, [ρ(r)], model electron density distributions served to study chemical bonding, in particular by topological electron density analyses yielding bond critical point (bcp) properties and in situ cation electronegativities. The results for 32 SiO, 24 AlO, 14 CaO, and 12 NaO unique bonds are compiled and analyzed in terms of both mean values and correlations between bond lengths, bonded oxygen radii, bcp densities, curvatures at the bcps, and electronegativities. Comparison with recent literature data obtained from both experimental electron density studies on minerals and model calculations for geometry-optimized molecules shows that the majority of the present findings conforms well with chemical expectation and with the trends observed from molecular modeling. For the SiO bond, the shared interaction is indicated to increase with decreasing bond length, whereas the AlO bond is of distinctly more polar nature, as is the NaO bond compared to CaO. Also, the observed ranges of the Si and Al in situ electronegativities and their mean electronegativities agree well with both Pauling's values and model calculation results, and statistically significant correlations are obtained which are consistent with trends described for oxide and nitride molecules. Received: 10 May 1999 / Revised, accepted: 14 September 1999     

A molecular orbital study of bond length and angle variations in framework structures

Molecular orbital calculations on a variety of silicate and aluminosilicate molecules have been used to explore the bonding forces that govern tetrahedral bond length variations, r(TO), in framework silicates and aluminosilicates. Not only do the calculations provide insight into the variety of structural types and the substitution limits of one tetrahedral atom for another, but they also provide an understanding of the interrelationships among r(TO) and linkage factors, bond strength sum, coordination number, and angles within and between tetrahedra. A study of these interrelationships for a theoretical data set shows that r(SiO) and r(AlO) are linearly correlated with (1) p _o, the bond strength sum to a bridging oxygen, (2) f _s(O), the fractional s-character of a bridging oxygen, and (3) f _s (T), the fractional s-character of the T atom. In a multiple linear regression analysis of the data, 92% of the variation of r(SiO) and 99% of the variation of r(AlO) can be explained in terms of a linear dependence on p _o, f _s (O), and f _s (T). Analogous regression analyses completed for observed r(Al, SiO) bond length data from a number of silica polymorphs and ordered aluminosilicates account for more than 75% of the bond length variation. The lower percentage of bond length variation explained is ascribed in part to the random and systematic errors in the experimental data which have a negligible effect on the theoretical data. The modeling of more than 75% of the variation of r(Al, SiO) in the framework silicates using the same model used for silicate and aluminosilicate molecules strengthens the viewpoint that the bonding forces that govern the shapes of such molecules are quite similar to the forces that govern the shapes of chemically similar groups in solids. The different regression coefficients calculated for f _s (T) indicate that SiO and AlO bond length variations in framework structures should not be treated as a single population in estimating the average Al, Si content of a tetrahedral site.     

Experimental and theoretical bond critical point properties for model electron density distributions for earth materials

Generalized X-ray scattering factor model experimental electron density distributions and bond critical point, bcp, properties generated in recent studies for the earth materials stishovite, forsterite, fayalite and cuprite with high energy single crystal synchrotron X-ray diffraction data and those generated with high resolution diffraction data for coesite and senarmonite were found to be adequate and in relatively good agreement, ~5% on average, with those calculated with quantum chemical methods with relatively robust basis sets. High resolution low energy single crystal diffraction data, recorded for the molecular sieve AlPO₄-15, were also found to yield model electron density distribution values at the bcp points for the AlO and PO bonded interactions that are in relatively good to moderately good agreement with the theoretical values, but the Laplacian values of the distribution at the points for the two bonded interactions were found to be in relatively poor agreement. In several cases, experimental bcp properties, generated with conventional low energy X-ray diffraction data for several rock forming minerals, were found overall to be in poorer agreement with the theoretical properties. The overall agreement between theoretical bcp properties generated with computational quantum methods and experimental properties generated with synchrotron high energy radiation not only provides a basis for using computational strategies for studying and modeling structures and their electron density distributions, but it also provides a basis for improving our understanding of the crystal chemistry and bonded interactions for earth materials. Theoretical bond critical point properties generated with computational quantum methods are believed to rival the accuracy of those determined experimentally. As such the calculations provide a powerful and efficient method for evaluating electron density distributions and the bonded interactions for a wide range of earth materials.Dedicated to Professor Robert F. Stewart of Carnegie Mellon University on his retirement for his brilliant and original work modeling electron density distributions.     

A comparison of procrystal and ab initio model representations of the electron-density distributions of minerals

 The procrystal calculation of the electron density is a very rapid procedure that offers a quick way to analyze various bonding properties of a crystal. This study explores the extent to which the positions, number, and properties of bond-critical points determined from the procrystal representations of the electron density for minerals are similar to those of first-principles ab initio model distributions. The purpose of the study is to determine the limits imposed upon interpretation of the procrystal electron density. Procrystal calculations of the electron density for more than 300 MO bonds in crystals were compared with those previously calculated using CRYSTAL98 and TOPOND software. For every bond-critical point found in the ab initio calculations, an equivalent one was also found in the procrystal model, with similar magnitudes of electron density, and at similar positions along the bonds. The curvatures of the electron densities obtained from the ab initio and the procrystal distributions are highly correlated. It is concluded that the procrystal distributions are capable of providing good estimates of the bonded radii of the atoms and the properties of the electron-density distributions at the bond-critical points. Because the procrystal model is so fast to compute, it is especially useful in addressing the question as to whether a pair of atoms is bonded or not. If the Bader criteria for bonding are accepted, then the successful generation of the bond-critical points by the procrystal model demonstrates that bonding is an atomic feature. The main difference between the critical-point properties of the procrystal and the ab initio model is that the curvature in the electron density perpendicular to the bond path of the ab initio model is sharper than for the procrystal model. This is interpreted as indicating that the electrons that migrate into a bond originate from its sides, and not from the regions closer to the nuclei. This observation also suggests that ab initio optimization routines could see an improvement in speed if the parameters relating to the angular components of atomic wave functions were to vary before the radial components. Received: 6 August 2001 / Accepted: 21 November 2001     

Experimental multipole-refined and theoretical charge density study of LiGaSi<Subscript>2</Subscript>O<Subscript>6</Subscript> clinopyroxene at ambient conditions

The synthetic LiGaSi₂O₆ clinopyroxene is monoclinic C2/c at room-T. Its experimental electron density, ρ(r), has been derived starting from accurate room-T single-crystal diffraction data. Topological analysis confirms an intermediate ionic-covalent character for Si–O bonding, as found by previous electron-density studies on other silicates such as diopside, coesite and stishovite. The non-bridging Si–O bonds have more covalent character than the bridging ones. The Ga–O bonds have different bonding characters, the Ga–O2 bond being more covalent than the two Ga–O1 bonds. Li–O bonds are classified as pure closed-shell ionic interactions. Similar to spodumene (LiAlSi₂O₆), Li has sixfold coordination, but the bond critical points associated to the two longest bonds are characterized by very low electron density values. Similar to what previously found in spodumene and diopside, O···O interactions were detected from the topological analysis of ρ(r), and indicate a cooperative interaction among the lone pairs of neighbouring oxygen atoms. In particular, this kind of interaction has been obtained for the O1···O1 edge shared between two Ga octahedra. Integration over the atomic basins gives net charges of −1.39(10), 2.82(10), 1.91(10) and 0.82(8) e for O (averaged), Si, Ga and Li atoms, respectively. Periodic Hartree–Fock and DFT calculations confirm the results obtained by multipole refinement of the experimental data. Moreover, the theoretical topological properties of the electron density distribution on the Si₂O₆ group are very similar to those calculated for spodumene. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

A crystal chemical study of protoanthophyllite: orthoamphiboles with the protoamphibole structure

Two new protoamphibole-type amphiboles with space group type Pnmn, have been found in nature: protoferro-anthophyllite (Fe_0.80Mn_0.20)₂ (Fe_0.98Mg_0.02)₅ (Si₄O₁₁)₂(OH)₂, and protomangano-ferro-anthophyllite, (Mn_0.70Fe_0.30)₂ (Fe_0.82Mg_0.18)₅ (Si₄O₁₁)₂(OH)₂. Protoferro-anthophyllite (PFA) occurs in pegmatites at both Gifu Prefecture, Japan and at Cheyenne Mountain, El Paso County, Colorado, USA. Protomangano-ferro-anthophyllite, (PMFA) occurs in pegmatites at Fukushima Prefecture and in a Mn mine at Tochigi Prefecture, Japan. Structure determinations of the two amphiboles show that both are isostructural with the synthetic fluorian-amphibole, protoamphibole (= protofluorian-lithian-anthophyllite). A calculation of the procrystal electron density distributions, the bond paths and the bond critical point properties of PFA, PMFA, grunerite and protoamphibole indicates that the M4 cation in these amphiboles is 4-coordinated. A calculation of the electron density distributions at the Becke3LYP/6-311G(2d,p) level for model silicate tetrahedra for these amphiboles and anthophyllite reveals that the value of the electron density at the bond critical points, ρ(r _c ), for the SiO(nbr) bonds is larger, on average (0.93 e/ų), than that for the SiO(br) bonds (0.90 e/ų). The observed SiO bond lengths decrease linearly with increasing ρ(r _c ) while the magnitudes of the curvatures of ρ(r _c ) both perpendicular and parallel to the bonds and the Laplacian of ρ(r _c ) each increases. These trends are associated with an increase in the electronegativity of the Si cation, a possible increase in the covalent character of the SiO bond and a tendency for SiO(nbr) bonds to be involved in wider OSiO angles than SiO(br) bonds. It is possible, if not likely, that protoanthophyllite has often been misidentified as anthophyllite.     

Oxide solution mechanisms in silicate melts: Systematic variations in the activity coefficient of SiO2

Phase equilibria and spectroscopic data are used to develop a simple model for the interaction of various oxide components and molten SiO₂. Network modifying oxides, M_xO_y produce nonbridging oxygens thereby depolymerizing the SiO₂ network. The energetics of nonbridging oxygen formation are least favorable when the field strength of the metal cation is high. This produces relatively strong M-O and Si-O-Si bridging bonds at the expense of weaker Si-O-M bonds (De Jonget al., 1980). This relationship is manifested by an increase in positive deviations from ideality with increasing cation field strength in M_xO_y-SiO₂ systems; the activity coefficient of SiO₂ is inversely correlated with Si-O-M bond strength. Network forming oxides (aluminates, phosphates, titanates. zirconates, etc.) may copolymerize with the SiO₂ network. Mixing on the same quasi-lattice produces solutions which approach ideality. Deviations from ideality in such solutions can be linked to distortions in the SiO₂ network. Discrete anion formers (phosphates, titanates, chromates, zirconates) complex with metal oxides other than SiO₂ to form discrete structural units which do not copolymerize with SiO₂. The SiO₂ network is essentially shielded from the high charge density cations in such systems and unmixing is common. As a result, the relative deviations from ideality in such melts are high. It is important to recognize that oxides such as P₂O₅, TiO₂ and ZrO₂ may act as either network-formers or discrete anion formers depending upon melt composition, and are probably distributed between these two “sites” in most geologically important liquids. The latter structural role is favored in more basic compositions.     

Granular contact dynamics with elastic bond model

This paper proposes an elastic bond model in the framework of contact dynamics based on mathematic programming. The bond model developed in this paper can be used to model cemented materials. The formulation can be reduced to model pure static problems without introducing any artificial damping. In addition, omitting the elastic terms in the objective function turns the formulation into rigid bond model, which can be used for the modeling of rigid or stiffly bonded materials. The developed bond model has the advantage over the explicit DEM that large time step or displacement increment can be used. The tensile and shear strength criteria of the bond model are formulated based on the modified Mohr–Coulomb failure criterion. The torque transmission of bonds is introduced based on rolling resistance model. The loss of shear or tensile strength, or torque transmission will lead to the breakage of bonds, and turn the bond into purely frictional contact. Three simple examples are first used to validate the bond model. Numerical examples of uniaxial and biaxial compression tests are used to show its potential in modeling cemented geomaterials. Numerical results show that elastic bonds are indeed necessary for the modeling of cemented granular material under static conditions.     

Maximum entropy method: an unconventional approach to explore observables related to the electron density in phengites

The maximum entropy method (MEM) is used here to get an insight into the electron density [ρ(r)] of phengites 2M ₁ and 3T, paying special attention to the M1-formally empty site and charge distribution. Room temperature single crystal X-ray diffraction data have been used as experimental input for MEM. The results obtained by MEM have been compared with those from conventional structure refinement which, in turn, has provided the prior-electron density to start the entropy maximization process. MEM reveals a comparatively non-committal approach, able to produce information related to the M1-site fractional occupancy, and yields results consistent with those from the difference Fourier synthesis, but free of the uncertainties due to the abrupt truncation of the series. The charge distribution is investigated by means of the notion of ‘‘site basin’’, i.e., those site-centered volumes delimited by a surface such as ∇ρ·n = 0. In particular, we observe: (1) the overall partitioning of the basin total charge between cation and anion sites, and the interlayer site charge seems to depend on sample composition, and (2) the apical-oxygen plane total basin charge and hydroxyl basin charge are presumably related to the polytype. The MEM-determined electron density does not allow full exploration of the critical points for very complex structures as micas, insofar as conventional room temperature experimental diffraction data are used.