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At least 1,441 records · Page 80
All sourcesNASA NTRSDOE OSTI
DOE OSTI2017

Low- and high-order accurate boundary conditions: From Stokes to Darcy porous flow modeled with standard and improved Brinkman lattice Boltzmann schemes

The present contribution focuses on the accuracy of reflection-type boundary conditions in the Stokes–Brinkman–Darcy modeling of porous flows solved with the lattice Boltzmann method (LBM), which we operate with the two-relaxation-time (TRT) collision and the Brinkman-force based scheme (BF), called BF-TRT scheme. In parallel, we compare it with the Stokes–Brinkman–Darcy linear finite element method (FEM) where the Dirichlet boundary conditions are enforced on grid vertices. In bulk, both BF-TRT and FEM share the same defect: in their discretization a correction to the modeled Brinkman equation appears, given by the discrete Laplacian of the velocity-proportional resistance force. This correction modifies the effective Brinkman viscosity, playing a crucial role in the triggering of spurious oscillations in the bulk solution. While the exact form of this defect is available in lattice-aligned, straight or diagonal, flows; in arbitrary flow/lattice orientations its approximation is constructed. At boundaries, we verify that such a Brinkman viscosity correction has an even more harmful impact. Already at the first order, it shifts the location of the no-slip wall condition supported by traditional LBM boundary schemes, such as the bounce-back rule. For that reason, this work develops a new class of boundary schemes to prescribe the Dirichlet velocity condition at an arbitrary wall/boundary-node distance and that supports a higher order accuracy in the accommodation of the TRT-Brinkman solutions. For their modeling, we consider the standard BF scheme and its improved version, called IBF; this latter is generalized in this work to suppress or to reduce the viscosity correction in arbitrarily oriented flows. Our framework extends the one- and two-point families of linear and parabolic link-wise boundary schemes, respectively called B-LI and B-MLI, which avoid the interference of the Brinkman viscosity correction in their closure relations. The performance of LBM and FEM is thoroughly evaluated in three benchmark tests, which are run throughout three distinctive permeability regimes. The first configuration is a horizontal porous channel, studied with a symbolic approach, where we construct the exact solutions of FEM and BF/IBF with different boundary schemes. The second problem refers to an inclined porous channel flow, which brings in as new challenge the formation of spurious boundary layers in LBM; that is, numerical artefacts that arise due to a deficient accommodation of the bulk solution by the low-accurate boundary scheme. The third problem considers a porous flow past a periodic square array of solid cylinders, which intensifies the previous two tests with the simulation of a more complex flow pattern. The ensemble of numerical tests provides guidelines on the effect of grid resolution and the TRT free collision parameter over the accuracy and the quality of the velocity field, spanning from Stokes to Darcy permeability regimes. It is shown that, with the use of the high-order accurate boundary schemes, the simple, uniform-mesh-based TRT-LBM formulation can even surpass the accuracy of FEM employing hardworking body-fitted meshes.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

Implementing contact angle boundary conditions for second-order Phase-Field models of wall-bounded multiphase flows

In the present work, a general formulation is proposed to implement the contact angle boundary conditions for the second-order Phase-Field models, which is applicable to N-phase (N ≥ 2) moving contact line problems. To remedy the issue of mass change due to the contact angle boundary condition, a source term or Lagrange multiplier is added to the original second-order Phase-Field models, which is determined by the consistent and conservative volume distribution algorithm so that the summation of the order parameters and the consistency of reduction are not influenced. To physically couple the proposed formulation to the hydrodynamics, especially for large-density-ratio problems, the consistent formulation is employed. The reduction-consistent conservative Allen-Cahn models are chosen as examples to illustrate the application of the proposed formulation. The numerical scheme that preserves the consistency and conservation of the proposed formulation is employed to demonstrate its effectiveness. Results produced by the proposed formulation are in good agreement with the exact and/or asymptotic solutions. The proposed method captures complex dynamics of moving contact line problems having large density ratios.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements

Within finite element models of fluids, vector-valued fields such as velocity or momentum variables are commonly discretised using the Raviart-Thomas elements. However, when using the lowest-order quadrilateral Raviart-Thomas elements, standard finite element discretisations of the vector transport equation typically have a low order of spatial accuracy. This paper describes two schemes that improve the accuracy of transporting such vector-valued fields on two-dimensional curved manifolds. The first scheme that is presented reconstructs the transported field in a higher-order function space, where the transport equation is then solved. The second scheme applies a mixed finite element formulation to the vector transport equation, simultaneously solving for the transported field and its vorticity. In this work, an approach to stabilising this mixed vector-vorticity formulation is presented that uses a Streamline Upwind Petrov-Galerkin (SUPG) method. These schemes are then demonstrated, along with their accuracy properties, through some numerical tests. Two new test cases are used to assess the transport of vector-valued fields on curved manifolds, solving the vector transport equation in isolation. The improvement of the schemes is also shown through two standard test cases for rotating shallow-water models.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Stabilized bases for high-order, interpolation semi-Lagrangian, element-based tracer transport

In a computational fluid model of the atmosphere, the advective transport of trace species, or tracers, can be computationally expensive. For efficiency, models often use semi-Lagrangian advection methods. High-order interpolation semi-Lagrangian (ISL) methods, in particular, can be extremely efficient, if the problem of property preservation specific to them can be addressed. Atmosphere models often use geometrically and logically nonuniform grids for efficiency and, as a result, element-based discretizations. Such grids and discretizations make stability a particular problem for ISL methods. Generally, high-order, element-based ISL methods that use the natural polynomial interpolant associated with a nodal finite-element discretization are unstable. Here, we derive new bases having order of accuracy up to nine, with positive nodal weights, that stabilize the element-based ISL method. We use these bases to construct the linear advection operator in the property-preserving Interpolation Semi-Lagrangian Element-based Transport (Islet) method. Then we discuss key software implementation details. Finally, we show performance results for the Energy Exascale Earth System Model's atmosphere dynamical core, comparing the original and new transport methods. These simulations used up to 27,600 Graphical Processing Units (GPU) on the Oak Ridge Leadership Computing Facility's Summit supercomputer.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A reduced-order model for nonlinear radiative transfer problems based on moment equations and POD-Petrov-Galerkin projection of the normalized Boltzmann transport equation

A data-driven projection-based reduced-order model (ROM) for nonlinear thermal radiative transfer (TRT) problems is presented. The TRT ROM is formulated by (i) a hierarchy of low-order quasidiffusion (aka variable Eddington factor) equations for moments of the radiation intensity and (ii) the normalized Boltzmann transport equation (BTE). The multilevel system of moment equations is derived by projection of the BTE onto a sequence of subspaces which represent elements of the phase space of the problem. Exact closure for the moment equations is provided by the Eddington tensor. A Petrov-Galerkin (PG) projection of the normalized BTE is formulated using a proper orthogonal decomposition (POD) basis representing the normalized radiation intensity over the whole phase space and time. The Eddington tensor linearly depends on the solution of the normalized BTE. By linear superposition of the POD basis functions, a low-rank expansion of the Eddington tensor is constructed with coefficients defined by the PG projected normalized BTE. The material energy balance (MEB) equation is coupled with the effective gray low-order equations which exist on the same dimensional scale as the MEB equation. The resulting TRT ROM is structure and asymptotic preserving. A detailed analysis of the ROM is performed on the classical Fleck-Cummings (F-C) TRT multigroup test problem in 2D geometry. Numerical results are presented to demonstrate the ROM's effectiveness in the simulation of radiation wave phenomena. Importantly, the ROM is shown to produce solutions with sufficiently high accuracy while using low-rank approximation of the normalized BTE solution. Essential physical characteristics of supersonic radiation wave are preserved in the ROM solutions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2026

Efficient data-driven regression for reduced-order modeling of spatial pattern formation

We present an efficient data-driven regression approach for constructing reduced-order models (ROMs) of reaction-diffusion systems exhibiting pattern formation. The ROMs are learned non-intrusively from available training data of physically accurate numerical simulations. The method can be applied to general nonlinear systems through the use of polynomial model form, while not requiring knowledge of the underlying physical model, governing equations, or numerical solvers. The process of learning ROMs is posed as a low-cost least-squares problem in a reduced-order subspace identified via Proper Orthogonal Decomposition (POD). Numerical experiments on classical pattern-forming systems–including the Schnakenberg and Mimura–Tsujikawa models–demonstrate that higher-order surrogate models significantly improve prediction accuracy while maintaining low computational cost. The proposed method provides a flexible, non-intrusive model reduction framework, well suited for the analysis of complex spatio-temporal pattern formation phenomena.

Data-driven modeling↗
DOE OSTI2016

Dependence of short and intermediate-range order on preparation in experimental and modeled pure a-Si

Variability in the short-to-intermediate range order of pure amorphous silicon prepared by different experimental and computational techniques is probed by measuring mass density, atomic coordination, bond-angle deviation, and dihedral angle deviation. It is found that there is significant variability in order parameters at these length scales in this archetypal covalently bonded, monoatomic system. This diversity strongly reflects preparation technique and thermal history in both experimental and simulated systems. Experiment and simulation do not fully quantitatively agree, partly due to differences in the way parameters are accessed. However, qualitative agreement in the trends is identified. Relaxed forms of amorphous silicon closely resemble continuous random networks generated by a hybrid method of bond-switching Monte Carlo and molecular dynamics simulation. As-prepared ion implanted amorphous silicon can be adequately modeled using a structure generated from amorphization via ion bombardement using energetic recoils. Preparation methods which narrowly avoid crystallization such as experimental pressure-induced amorphization or simulated melt-quenching result in inhomogeneous structures that contain regions with significant variations in atomic ordering. Ad hoc simulated structures containing small (1 nm) diamond cubic crystal inclusions were found to possess relatively high bond-angle deviations and low dihedral angle deviations, a trend that could not be reconciled with any experimental material.

36 MATERIALS SCIENCE↗
DOE OSTI2016

Highly conducting divalent Mg{sup 2+} cation solid electrolytes with well-ordered three-dimensional network structure

A three-dimensionally well-ordered NASICON-type Mg{sup 2+} cation conductor, (Mg{sub x}Hf{sub 1−x}){sub 4/(4−2x)}Nb(PO{sub 4}){sub 3}, was firstly developed by partial substitution of lower valent Mg{sup 2+} cation onto the Hf{sup 4+} sites in a HfNb(PO{sub 4}){sub 3} solid to realize high Mg{sup 2+} cation conductivity even at moderate temperatures. Due to the formation of well-ordered NASICON-type structure, both the high Mg{sup 2+} cation conductivity below 450 °C and the low activation energy for Mg{sup 2+} cation migration was successfully realized for the (Mg{sub 0.1}Hf{sub 0.9}){sub 4/3.8}Nb(PO{sub 4}){sub 3} solid. Pure Mg{sup 2+} cation conduction in the NASICON-type (Mg{sub 0.1}Hf{sub 0.9}){sub 4/3.8}Nb(PO{sub 4}){sub 3} solid was directly and quantitatively demonstrated by means of two kinds of dc electrolysis. - Graphical abstract: Image of the Mg{sup 2+} cation conduction in NASICON-type (Mg{sub 0.1}Hf{sub 0.9}){sub 4/3.8}Nb(PO{sub 4}){sub 3} and its Mg{sup 2+} conductivity. - Highlights: • We develop a three-dimensionally well-ordered NASICON-type Mg{sup 2+} cation conductor. • A high magnesium cation conductivity is realized even at moderate temperatures. • Divalent magnesium cation conduction is demonstrated directly and quantitatively.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2017

Structural, optical and femtosecond third-order nonlinear optical properties of LiNbO{sub 3} thin films

Highlights: • Fs third order NLO properties of LiNbO{sub 3} thin films were investigated. • Al:ZnO was used as a buffer layer to control the growth of LiNbO{sub 3}. • The magnitude of the χ{sup (3)} of all films was ∼10{sup −11} e.s.u. • χ{sup (3)} obtained here is much higher that of single crystal Z-cut LiNbO{sub 3}. - Abstract: LiNbO{sub 3} thin films were deposited by means of Al:ZnO as intermediate layer using pulsed laser deposition and their structural, linear and nonlinear optical (NLO) properties were investigated. The films were found to be polycrystalline, (110) orientation for Al:ZnO/fused silica and (300) orientation for Al:ZnO/quartz substrates. The films demonstrated a transparency of >80% in the visible and near-infrared region. The NLO studies were performed with the Z-scan technique using ∼150 fs, 80 MHz pulses at 800 nm. The studies revealed that the polycrystalline LiNbO{sub 3}films and (300) LiNbO{sub 3}films demonstrated reverse saturable absorption while a transition to saturable absorption was observed in the case of (110) LiNbO{sub 3}. The presence of two-photon absorption is attributed to the indirect transition via intermediate defect levels which is related to the structural asymmetry. The large nonlinearity in refractive index could be a result of the strain induced ferroelectric polarization. The magnitude of the third order optical nonlinearity of all films was ∼10{sup −11} e.s.u. which is orders of magnitude higher than the value obtained for single crystal Z-cut LiNbO{sub 3}.

36 MATERIALS SCIENCE↗
DOE OSTI2018

Room temperature magnetic ordering and analysis by bound magnetic polaron model of Yb3+ doped nanocrystalline zinc oxide (Zn0.98Yb0.02O)

Highlights: • ZnO and Yb doped ZnO are synthesized by co-precipitation method. • Rietvelt anlysis of XRD patterns confirms the complete substitution of Yb{sup 3+} in ZnO. • Vacancy and defect mediated RTFM is introduced in doped ZnO. • BMP model successfully analyze the magnetic property of Zn{sub 0.98}Yb{sub 0.02}O. - Abstract: Nanocrystalline ZnO and Yb-doped ZnO (Zn{sub 0.98}Yb{sub 0.02}O) are prepared by simple co-precipitation method. The pure crystallographic phases of ZnO and Zn{sub 0.98}Yb{sub 0.02}O are confirmed by Rietveld analyses of XRD patterns. Raman and photoluminescence spectra recorded at room temperature (RT) indicates that lattice defects and oxygen vacancy in Zn{sub 0.98}Yb{sub 0.02}O is more than that of ZnO. Magnetic measurements of ZnO and Zn{sub 0.98}Yb{sub 0.02}O are carried out from 300 to 5 K. Ferromagnetic ordering is observed in Zn{sub 0.98}Yb{sub 0.02}O at RT and the strength of the ordering increases with the lowering of temperature. Ferromagnetism of Zn{sub 0.98}Yb{sub 0.02}O is attributed to the oxygen vacancy introduced by annealing in vacuum. Magnetization vs. temperature curve is well fitted by the combination of 3D spin wave and Curie-Weiss law, which suggest simultaneous presence of para- and ferro-magnetic phases in Zn{sub 0.98}Yb{sub 0.02}O. Ferromagnetic ordering is successfully explained by vacancy mediated bound-magnetic polaron model.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Towards the design of nature-inspired materials: Impact of complex pore morphologies via higher-order homogenization

Even though the development of novel materials that mimic nature is widely used in a variety of engineering and scientific fields, the relationship between effective material properties and underlying, often complex pore morphology is still not fully understood. To address this knowledge gap and accelerate the development of novel nature-inspired materials, this paper adopts a higher-order asymptotic homogenization method to numerically investigate the effect of complex micropore morphology on the effective mechanical properties of a porous system. Specifically, we create unique pore morphologies with varying levels of complexity that serve as a more realistic representation of natural materials. Here, we then use the second-order homogenization method to capture the role of pore size, shape, orientation, and distribution on effective properties. By creating different pore morphologies, we systematically studied the relationship between morphology and effective mechanical properties. The results highlight the necessity of higher-order parameters to fully capture the role of realistic pore morphologies on effective mechanical properties and provide a path forward in the design of nature-inspired materials.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Identifying point defects and ordering in the high-entropy layered oxide Li 1.5 MO 3-δ (M=Mn, Al, Fe, Co, Ni) for energy storage applications

High-entropy layered oxides (HELOs) represent a very promising class of next-generation battery cathodes, combining the well-studied properties of layered cathode materials such as LiCoO 2 with the chemical tunability and stability of high-entropy materials. HELO materials often form particles with complex defects and domain structures, complicating accurate characterization of structure and cation mixing. Understanding disorder and order in HELO materials is necessary for understanding their performance and utility as cathodes. Here we demonstrate the characterization of the HELO Li 1.5 MO 3-δ (M = Mn, Al, Fe, Co, Ni), wherein X-ray powder diffraction, transmission electron microscopy imaging, and electron diffraction patterns are analyzed to reveal the presence of ordering in the HELO, with imaging and diffraction simulation employed to compare experimental results to atomic modeling. Finally, without rigorous characterization at the atomic scale, important features such as defect ordering can be easily overlooked and therefore remain unconsidered when interpreting experimental results.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Closing in on critical net-baryon fluctuations at LHC energies: Cumulants up to third order in Pb–Pb collisions

Fluctuation measurements are important sources of information on the mechanism of particle production at LHC energies. This article reports the first experimental results on third-order cumulants of the net-proton distributions in Pb–Pb collisions at a center-of-mass energy $\sqrt{s_{NN}}$ = 5.02 TeV recorded by the ALICE detector. The results on the second-order cumulants of net-proton distributions at $\sqrt{s_{NN}}$ = 2.76 and 5.02 TeV are also discussed in view of effects due to the global and local baryon number conservation. The results demonstrate the presence of long-range rapidity correlations between protons and antiprotons. Such correlations originate from the early phase of the collision. The experimental results are compared with HIJING and EPOS model calculations, and the dependence of the fluctuation measurements on the phase-space coverage is examined in the context of lattice quantum chromodynamics (LQCD) and hadron resonance gas (HRG) model estimations. The measured third-order cumulants are consistent with zero within experimental uncertainties of about 4% and are described well by LQCD and HRG predictions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Ab initio computations of the fourth-order charge density moments of 48 Ca and 208 Pb

Neutron skins of neutron-rich nuclei connect nuclei with the matter in neutron stars. High-precision measurements of nuclear charge densities to extract higher-order moments are proposed to be sensitive to neutron radii and skin thicknesses. We investigate the charge density of 48 Ca and 208 Pb, leading candidates for such studies, with ab initio nuclear structure calculations. We find strong correlations between the fourth-order charge density moment R$^{4}_{ch}$ and the charge and neutron radii, allowing us to predict R$^{4}_{ch}$ for 48 Ca and 208 Pb. We find a substantially weaker correlation between the fourth-order charge density moment and the neutron skin, limiting the ability of high-precision electron scattering to determine the neutron skin in a model-independent manner.

Ab initio nuclear theory↗
DOE OSTI2026

Long-range magnetic order and relaxor ferroelectricity in a hexagonal high-entropy ferrite

Multiferroics that combine ferroelectricity and magnetic order are attractive for electronic and spintronic technologies, yet chemical disorder that promotes relaxor ferroelectricity usually suppresses long-range magnetic order. Here, we report entropy-stabilized relaxor multiferroicity in epitaxial hexagonal (Tb0.2Dy0.2Ho0.2Lu0.2Yb0.2)FeO3 thin films. Structural, magnetic, dielectric, and synchrotron spectroscopic measurements show the coexistence of relaxor ferroelectricity and long-range ferromagnetic order. We find that improper ferroelectricity remains robust against A-site configurational disorder, while the Fe sublattice preserves magnetic exchange. This separation of the microscopic origins of the polar and magnetic responses enables chemically disordered multiferroicity. Our results establish entropy engineering in hexagonal ferrites as a route toward multifunctional oxide thin films and provide a general design strategy for high-entropy multiferroics.

Miertschin, Duncan [Baylor University]↗
DOE OSTI2023

Genome-scale phylogeny and comparative genomics of the fungal order Sordariales

The order Sordariales is taxonomically diverse, and harbours many species with different lifestyles and large economic importance. Despite its importance, a robust genome-scale phylogeny, and associated comparative genomic analysis of the order is lacking. In this study, we examined whole-genome data from 99 Sordariales, including 52 newly sequenced genomes, and seven outgroup taxa. We inferred a comprehensive phylogeny that resolved several contentious relationships amongst families in the order, and cleared-up intrafamily relationships within the Podosporaceae. Extensive comparative genomics showed that genomes from the three largest families in the dataset (Chaetomiaceae, Podosporaceae and Sordariaceae) differ greatly in GC content, genome size, gene number, repeat percentage, evolutionary rate, and genome content affected by repeat-induced point mutations (RIP). All genomic traits showed phylogenetic signal, and ancestral state reconstruction revealed that the variation of the properties stems primarily from within-family evolution. Together, the results provide a thorough framework for understanding genome evolution in this important group of fungi.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2024

Role of Short-Range Order on Diffusion Coefficients in the Li–Mg Alloy

Li–Mg alloys are important because of their beneficial role in fostering uniform plating and stripping of lithium in all-solid-state batteries. The alloy Li x Mg 1–x forms a solid solution on the BCC crystal structure when the lithium content is greater than x ≈ 0.3. The activation barriers of lithium and magnesium exchanges with a vacancy, crucial for substitutional diffusion, are predicted to be exceptionally low and almost identical, with negligible dependence on the alloy composition. The equilibrium vacancy concentration at room temperature is predicted to be very low, and it also remains almost constant with no dependence on Mg content in the alloy (for x Li ≥ 0.5). Nevertheless, both experiments and kinetic Monte Carlo simulations indicate that the tracer diffusion coefficients decrease by almost an order of magnitude with the addition of Mg to the alloy. In this contribution, the crucial role that chemical short-range order plays in affecting the diffusion coefficients is studied. Chemical short-range order is found to increase the effective activation barrier for lithium and magnesium diffusion by making successive atomic hops with vacancies correlated.

36 MATERIALS SCIENCE↗
DOE OSTI2016

First 14-Layer Twinned Hexagonal Perovskite Ba 14 Mn 1.75 Ta 10.5 O 42 : Atomic-Scale Imaging of Cation Ordering

Formation of hexagonal perovskite with mixed cubic and hexagonal stacking of AO 3 layers becomes more and more difficult when the number of layers in the stacking repeating unit increases. So far, the highest number of layers reported for twinned hexagonal perovskite is 12, with alternative 5 consecutive cubic layers and one hexagonal layer in the (ccccch) 2 sequence. Here, we present the unexpected formation of a 14-layer twinned hexagonal perovskite with a stacking sequence (cccccch) 2 for the BaO 3 layers on the Ba 14 Mn 1.75 Ta 10.5 O 42 (Ba 8 MnTa 6 O 24 ) composition, the first example of twinned hexagonal perovskite with a periodicity exceeding 12-layers. The B-cation and vacancy distributions are characterized by multiple efficient and complementary techniques including neutron and synchrotron powder diffraction, scanning transmission electron microscopy-high angle annular dark field (STEM-HAADF) imaging, and electron energy loss spectroscopy (EELS) and X-ray energy dispersive spectroscopy (EDS) elemental mapping. Atomic-resolution STEM-HAADF imaging and EELS/EDS elemental mapping enables direct observation of high-spin d 5 Mn 2+ cation ordering in the d 0 Ta 5+ host, thus demonstrating the great potential of this technique for probing cation ordering and performing structure determination. Moreover, atomic mapping allows for the observation of local defect structure variants, which can be a powerful tool for future new material design. The large high-spin Mn 2+ cation and Ta-vacancy pair formation in face-sharing octahedral sites play key roles on both the stabilization of this 14-layer twinned hexagonal perovskite structure and the Mn 2+ ordering in the central corner-sharing octahedral (CSO) positions within the five-consecutive CSO layers. Here, compared with the 8-layer twinned Ba 8 ZnTa 6 O 24 material, the low quality factor in microwave frequency and enhanced ultraviolet and visible light absorption of Ba 14 Mn 1.75 Ta 10.5 O 42 as well as the photocatalytic activity on water splitting are discussed in terms of the presence of high-spin Mn 2+ cations in the structure.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗