A mapping of the electron localization function for the silica polymorphs: evidence for domains of electron pairs and sites of potential electrophilic attack

The electron localization function, η, evaluated for first-principles geometry optimized model structures generated for quartz and coesite, reveals that the oxide anions are coordinated by two hemispherically shaped η-isosurfaces located along each of the SiO bond vectors comprising the SiOSi angles. With one exception, they are also coordinated by larger banana-shaped isosurfaces oriented perpendicular to the plane centered in the vicinity of the apex of each angle. The hemispherical isosurfaces, ascribed to domains of localized bond-pair electrons, are centered ～0.70 Å along the bond vectors from the oxide anions and the banana-shaped isosurfaces, ascribed to domains of localized nonbonding lone-pair electrons, are centered ～0.60 Å from the apex of the angle. The oxide anion comprising the straight SiOSi angle in coesite is the one exception in that the banana-shaped isosurface is missing; however, it is coordinated by two hemispherically shaped isosurfaces that lie along the bond vectors. In the case of a first-principles model structure generated for stishovite, the oxide anion is coordinated by five hemispherically shaped η-isosurfaces, one located along each of the three SiO bond vectors (ascribed to domains of bonding-electron pairs) that are linked to the anion with the remaining two (ascribed to domains of nonbonding-electron pairs) located on opposite sides of the plane defined by three vectors, each isosurface at a distance of ～0.5 Å from the anion. The distribution of the five isosurfaces is in a one-to-one correspondence with the distribution of the maxima displayed by experimental Δρ and theoretical ??²ρ maps. Isosurface η maps calculated for quartz and the (HO) ₃ SiOSi(OH) ₃ molecule also exhibit maxima that correspond with the (3,?3) maxima displayed by distributions of ??²ρ. Deformation maps observed for the SiOSi bridges for the silica polymorphs and a number of silicates are similar to that calculated for the molecule but, for the majority, the maxima ascribed to lone-pair features are absent. The domains of localized nonbonding-electron pair coordinating the oxide anions of quartz and coesite provide a basis for explaining the flexibility and the wide range of the SiOSi angles exhibited by the silica polymorphs with four-coordinate Si. They also provide a basis for explaining why the SiO bond length in coesite decreases with increasing angle. As found in studies of the interactions of solute molecules with a solvent, a mapping of η-isosurfaces for geometry-optimized silicates is expected to become a powerful tool for deducing potential sites of electrophilic attack and reactivity for Earth materials. The positions of the features ascribed to the lone pairs in coesite correspond with the positions of the H atoms recently reported for an H-doped coesite crystal.     

ELF isosurface maps for the Al2SiO5 polymorphs

This study examines the electron localization function (ELF) isosurfaces of the Al₂SiO₅ polymorphs kyanite, sillimanite, and andalusite to see how differences in coordination and geometry of the cations and anions affect the ELF isosurfaces. Examination of the ELF isosurfaces indicates that their shapes are dependent on the coordination and geometry of the oxygen atoms and are not sensitive to coordination of the surrounding cations. Of the 18 crystallographically distinct oxygen atoms in the Al₂SiO₅ polymorphs, 13 are bonded to two aluminum atoms and one silicon atom (Al₂–O–Si) and are associated with two different ELF isosurface shapes. The shape of the ELF isosurface is dependent on the distance at which the oxygen atom lies from a plane defined by the three surrounding cations: at a distance greater than 0.2 Å the ELF can be defined as horseshoe-shaped and at a distance less then 0.2 Å it can be described as concave hemispherical. This feature is also seen in the ELF isosurfaces for the oxygens bonded to three aluminum atoms (Al₃–O) where the isosurfaces can be defined as trigonally toroidal and uniaxially trigonally toroidal. The changes in the ELF isosurfaces for the three coordinated oxygens are also indicative of changes in hybridization. The ELF isosurface for the two-fold coordinated oxygen (Al–O–Si) has a large mushroom-shaped isosurface along the Al–O bond and a concave hemispherical isosurface along the Si–O. The four-fold coordinated oxygen (Al₄–O) contains two concave hemispherical isosurfaces along the shorter Al–O bonds and a banana-shaped isosurface, which encompasses the longer Al–O bonds. In addition, this study shows the homeomorphic relationship between the ELF isosurfaces and electron density difference maps with respect to number and arrangement of domains.     

A modeling of the structure and compressibility of quartz with a molecular potential and its transferability to cristobalite and coesite

Several computer models of quartz were developed and tested. A simple model based on a potential energy function, derived in large part from quantum mechanical calculations on the molecule H₆Si₂O₇, was found to reproduce the compressibility curve for quartz up to pressures of 8 GPa. The potential includes quadratic expressions for the SiO bond lengths, the OSiO angles and a parameter spanning the SiOSi angle together with an exponential OO repulsion term for non co-dimer O atoms. The variations in the cell edges and in the SiOSi angle, as a function of pressure, parallel observed trends when the bond lengths and angles calculated for the molecule are used as rgressor values. Poisson ratios calculated using the model match those observed. Two configurations for quartz related by the Dauphiné twin law are generated as minimum energy structures of the model with about equal frequencies as observed in nature. It is shown that the model, devised for quartz, can also be applied to the silica polymorph cristobalite, giving reasonable estimates of its compressibility curve, structural parameters and its negative Poisson ratio. When the observed bond lengths and angles are used as regressor values, the model generates the absolute coordinates of the atoms and the cell dimensions for quartz to within 0.005 Å and those of cristobalite to within 0.001 Å, on average, both at zero pressure. When applied to coesite, the model yields a zero pressure structure that is close to that observed but which is significantly softer than observed. The resulting SiO bond lengths are linearly correlated with f _s (O), as observed for coesite, despite the use of a single bond length and a single SiOSi angle as regressor values in the calculation. When the structures are optimized assuming P1 space group symmetry and triclinic cell dimensions, the resulting frameworks of silicate tetrahedra exhibit the translational, rotational and reflection symmetries observed for quartz, cristobalite and coesite. The fact that the resulting frameworks exhibit observed space group symmetries is evidence that the symmetry adopted by the silica polymorphs can be explained by short ranged forces.     

Ab initio calculated geometries and charge distributions for H4SiO4 and H6Si2O7 compared with experimental values for silicates and siloxanes

Ab initio STO-3G molecular orbital theory has been used to calculate energy-optimized Si-O bond lengths and angles for molecular orthosilicic and pyrosilicic acids. The resulting bond length for orthosilicic acid and the nonbridging bonds for pyrosilicic acid compare well with Si-OH bonds observed for a number of hydrated silicate minerals. Minimum energy Si-O bond lengths to the bridging oxygen of the pyrosilicic molecule show a close correspondence with bridging bond length data observed for the silica polymorphs and for gas phase and molecular crystal siloxanes when plotted against the SiOSi angle. In addition, the calculations show that the mean Si-O bond length of a silicate tetrahedron increases slightly as the SiOSi angle narrows. The close correspondence between the Si-O bond length and angle variations calculated for pyrosilicic acid and those observed for the silica polymorphs and siloxanes substantiates the suggestion that local bonding forces in solids are not very different from those in molecules and clusters consisting of the same atoms with the same coordination numbers. An extended basis calculation for H₄SiO₄ implies that there are about 0.6 electrons in the 3d-orbitals on Si. An analysis of bond overlap populations obtained from STO-3G* calculations for H₆Si₂O₇ indicates that Si-O bond length and SiOSi angle correlations may be ascribed to changes in the hybridization state of the bridging oxygen and (d – p) π-bonding involving all five of the 3d AO's of Si and the lone-pair AO's of the oxygen. Theoretical density difference maps calculated for H₆Si₂O₇ show a build-up of charge density between Si and O, with the peak-height charge densities of the nonbridging bonds exceeding those of the bridging bonds by about 0.05 e Å^?3. In addition, atomic charges (+1.3 and ?0.65) calculated for Si and O in a SiO₂ moiety of the low quartz structure conform reasonably well with the electroneutrality postulate and with experimental charges obtained from monopole and radial refinements of diffraction data recorded for low quartz and coesite.     

Framework silica structures generated using simulated annealing with a potential energy function based on an H6Si2O7 molecule

A simulated annealing technique was used to search for global and local minimum energy structures of a potential energy model for silica. The model is based on ab initio SCF MO calculations on the disilicic acid molecule, H₆Si₂O₇. Starting with 4 SiO₂ units, with the atoms randomly distributed in the unit cell, 23 distinct silica tetrahedral framework structures were found, with a variety of space group symmetries and cell dimensions. Despite the assumption of P 1 space group symmetry for the starting structure, only 7 of the local minimum energy structures were found to possess triclinic symmetry with the remainder exhibiting symmetries ranging from P c to to within 0.001 Å. Although the interaction potential for the disilicic acid molecule has a single minimum energy SiO bond length and SiOSi angle, the local minimum energy structures exhibit angles that range between 105° and 180° and bond lengths that range between 1.55 and 1.68 Å. The correlation observed for coesite and the other silica polymorphs between SiO bond length and f_s(O) is reproduced. The generated structures show a wide variety of coordination sequences, ring sizes and framework densities, the later ranging from 19.8 to 35.5 Si/1000 ų. The energies of these structures correlate with their framework densities, particularly for higher energy structures.     

A CNDO/2 molecular orbital study of the silica polymorphs quartz,cristobalite, and coesite

CNDO/2 MO calculations on H₁₂Si₅O₁₆ clusters modeling silicate tetrahedral linkage in the silica polymorphs show total energy minima at bent SiOSi angles and a correlation between the Si-O bond lengths, d(Si-O), used in the calculation and the minimum energy value of the SiOSi angle. Calculations on hydrogen saturated Si₅O₁₆ clusters isolated from the structures of low quartz, low cristobalite and coesite which were adjusted by DLS methods so that all d(Si-O) equal 1.61 Å and all _L OSiO equal 109.47° yield Mulliken bond overlap populations, n(Si-O), and Si-O two-center energies, E(Si-O), which correlate with observed bond lengths; shorter bonds involve larger n(Si-O) values and more negative E(Si-O) values.

     

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     

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     

Periodic Hartree-Fock study of minerals: Tetracoordinated silica polymorphs

Periodic Hartree-Fock STO-3G calculations have been performed on several tetracoordinated silica polymorphs: low and high quartz, low and idealized high cristobalite and prototype tridymite. The optimized structural parameters are in overall good agreement with experimental data. In the particular case of -quartz, the SiO₄ tetrahedra are found to be irregular. The optimized values of the two different SiO bond lengths are respectively 1.608 Å and 1.613 Å. The potential energy versus tilt angle curves suggest a picture of the high temperature phases in terms of delocalized oxygen atoms which is consistent with a disordered structure. Finally, the bonding in silica polymorphs is discussed from electron density maps and Mulliken population analysis.     

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.     

A modeling of the structure and favorable H-docking sites and defects for the high-pressure silica polymorph stishovite

Employing first-principles methods, the docking sites for H were determined and H, Al, and vacancy defects were modeled with an infinite periodic array of super unit cells each consisting of 27 contiguous symmetry nonequivalent unit cells of the crystal structure of stishovite. A geometry optimization of the super-cell structure reproduces the observed bulk structure within the experimental error when P1 translational symmetry was assumed and an array of infinite extent was generated. A mapping of the valence electrons for the structure displays mushroom-shaped isosurfaces on the O atom, one on each side of the plane of the OSi₃ triangle in the nonbonded region. An H atom, placed in a cell near the center of the super cell, was found to dock upon geometry optimization at a distance of 1.69 Å from the O atom with the OH vector oriented nearly perpendicular to the plane of the triangle such that the OH vector makes a angle of 91° with respect to [001]. However, an optimization of a super cell with an Al atom replacing Si and an H atom placed nearby in a centrally located cell resulted in an OH distance of 1.02 Å with the OH vector oriented perpendicular to [001] as observed in infrared studies. The geometry-optimized position of the H atom was found to be in close agreement with that (0.44, 0.12, 0.0) determined in an earlier study of the theoretical electron density distribution. The docking of the H atom at this site was found to be 330 kJ mol^–1 more stable than a docking of the atom just off the shared OO edge of the octahedra as determined for rutile. A geometry optimization of a super cell with a missing Si generated a vacant octahedra that is 20% larger than that of the SiO₆ octahedra. The valence electron density distribution displayed by the two-coordinate O atoms that coordinate the vacant octahedral site is very similar to those displayed by the bent SiOSi angles in coesite. The internal distortions induced by the defect were found to diminish rather rapidly with distance, with the structure annealing to that observed in the bulk crystal to within about three coordination spheres.     

Calculated thermodynamic properties of silica polymorphs

Accurate interatomic potentials have been employed to compute the phonon density of states of αquartz, stishovite and coesite polymorphs of silica. The temperature variation of several thermodynamic properties is calculated by using the phonon density of states to describe the vibrational entropy contribution to the free energy. Results for these polymorphs are in surprisingly good agreement with available experimental data. Moreover, the microscopic origin of quantitative differences in the heat capacity behavior of low and high density polymorphs is established.     

Applications of quantum mechanical potential surfaces to mineral physics calculations

Ab-Initio quantum mechanical calculations on molecular clusters are used to obtain potential surfaces for the SiO bond in silicates. These potential surfaces form the basis for extracting the key parameters in various commonly employed potential functions. Applications to the usual ionic model demonstates a close relation between the ab-initio derived ionic potential and those empirically dervied. The ionic model is then used to predict structures and elastic properties of orthosilicates and of the silica polymorphs. The deficiencies in the ionic model lead to the application of the quantum results to covalent models. These latter models are then used in theoretical calculations of the properties of silica polymorphs.     

Deep-sea biogenic silica: new structural and analytical data from infrared analysis - geological implications

The main parameters of infrared spectra have been investigated for marine siliceous skeletons and for silica polymorphs. From infrared (IR)data, the mean values for the Si-O bond length (1.62 Å) and for the Si-O-Si angle (about 142°6) are inferred for biogenic silica. A model is proposed for the molecular structure of biogenic, amorphous silica, which consists of a discontinuous, three-dimensional framework of short chains of [SiO₄] tetrahedra, bounded with apical hydroxyls. According to the relationship between the structure of the silica polymorphs and their infrared spectra, it is shown that a quantitative analysis of biogenic silica seems to be possible in quartz-rich sediments. Some geological implications of these data are emphasized, including the processes of siliceous skeleton sedimentation and diagenetic evolution.     

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.     

Shock-induced growth and metastability of stishovite and coesite in lithic clasts from suevite of the Ries impact crater (Germany)

The microtextures of stishovite and coesite in shocked non-porous lithic clasts from suevite of the Ries impact structure were studied in transmitted light and under the scanning electron microscope. Both high-pressure silica phases were identified in situ by laser-Raman spectroscopy. They formed from silica melt as well as by solid-state transformation. In weakly shocked rocks (stage I), fine-grained stishovite (≤1.8 μm) occurs in thin pseudotachylite veins of quartz-rich rocks, where it obviously nucleated from high-pressure frictional melts. Generally no stishovite was found in planar deformation features (PDFs) within grains of rock-forming quartz. The single exception is a highly shocked quartz grain, trapped between a pseudotachylite vein and a large ilmenite grain, in which stishovite occurs within two sets of lamellae. It is assumed that in this case the small stishovite grains formed by the interplay of conductive heating and shock reverberation. In strongly shocked rocks (stages Ib–III, above ∼30 GPa), grains of former quartz typically contain abundant and variably sized stishovite (<6 μm) embedded within a dense amorphous silica phase in the interstices between PDFs. The formation of transparent diaplectic glass in adjacent domains results from the breakdown of stishovite and the transformation of the dense amorphous phase and PDFs to diaplectic glass in the solid state. Coesite formed during unloading occurs in two textural varieties. Granular micrometre-sized coesite occurs embedded in silica melt glass along former fractures and grain boundaries. These former high-pressure melt pockets are surrounded by diaplectic glass or by domains consisting of microcrystalline coesite and earlier formed stishovite. The latter is mostly replaced by amorphous silica.     

蛇绿岩地幔岩中自由SiO2的发现及其地质意义

自由SiO_2系指石英及其同质多型物(polymorphs)柯石英、斯石英等。石英广泛分布于地壳中的各种岩石中,柯石英和斯石英只存在于超高压岩石和陨石坑中。由于石英和非饱和SiO_2的橄榄石不能共生,因此在地幔橄榄岩和超镁铁岩中不存在原生石英。最近笔者在西藏罗布莎蛇绿岩的地幔岩(方辉橄榄岩)的豆荚状铬铁矿中发现了自由SiO_2和柯石英相。根据高温高压相平衡实验资料,橄榄石、辉石这样的硅酸盐矿物在地幔深部的压力条件下可以分解成简单氧化物,如FeO(方铁矿)、MgO(方镁石)以及SiO_2(斯石英)等。由此推测,西藏蛇绿岩地幔岩中自由SiO_2可能是来自于下地幔的矿物,是地幔柱作用将其搬运到上地幔浅部。     

Water in coesite: Incorporation mechanism and operation condition,solubility and P-T dependence,and contribution to water transport and coesite preservation

A series of coesite,coexisting with or without a liquid phase,was synthesized in the nominal system SiO2-H2O at800-1450℃and 5 GPa.Micro-Raman spectroscopy was used to identity the crystalline phase,electron microprobe and LA-ICP-MS were employed to quantity some major and trace elements,and unpolarized FTIR spectroscopy was applied to probe the different types of hydrogen defects,explore water-incorporation mechanisms and quantify water contents.Trace amounts of A1 and B were detected in the coesite.Combining our results with the results in the literatures,we have found no positive correlation between the Al contents and the"Al"-based hydrogen concentrations,suggesting that previously proposed hydrogen-incorporation mechanism H^++Al^3+■Si^4+does not function in coesite.In contrast,we have confirmed the positive correlation between the B contents and the B-based hydrogen concentrations.The hydrogen-incorporation mechanism H^++B3^+■Si^4+readily takes place in coesite at different P-T conditions,and significantly increases the water content at both liquid-saturated and liquid-undersaturated conditions.For the SiO2-H2O system,we have found that type-Ⅰhydrogarnet substitution plays a dictating role in incorporating water into coesite at liquid-saturated condition,type-II hydrogarnet substitution contributes significantly at nearly dry condition,and both operate at conditions in between.The water solubility of coesite,as dictated by the type-Ⅰhydrogarnet substitution,positively correlates with both P and T,cH2O=-105(30)+5.2(32)×P+0.112(26)×T,with cH2O in wt ppm,P in GPa and T in℃.Due to its low water solubility and small fraction in subducted slabs,coesite may contribute insignificantly to the vertical water transport in subduction zones.Furthermore,the water solubility of any coesite in exhuming ultra-high pressure metamorphic rocks should be virtually zero as coesite becomes metastable.With an adequately fast waterdiffusion rate,this metastable coesite should be completely dry,which may have been the key factor to the partial preservation of most natural Coe.As a byproduct,a new IR experimental protocol for accurate water determination in optically anisotropic nominally anhydrous minerals has been found.Aided with the empirical method of Paterson(1982)it employs multiple unpolarized IR spectra,collected from randomly-orientated mineral grains,to approximate both total integrated absorbance and total integrated molar absorption coefficient.Its success relies on a high-level orientation randomness in the IR analyses.