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All sources NASA NTRS DOE OSTI

DOE OSTI2016

Carbon Nanotube Dispersion in Solvents and Polymer Solutions: Mechanisms, Assembly, and Preferences

Debundling and dispersion of carbon nanotubes (CNTs) in polymer solutions play a major role in the preparation of carbon nanofibers due to early effects on interfacial ordering and mechanical properties. A roadblock toward ultrastrong fibers is the difficulty to achieve homogeneous dispersions of CNTs in polyacrylonitrile (PAN) and poly(methyl methacrylate) (PMMA) precursor solutions in solvents such as dimethyl sulfoxide (DMSO), N,N-dimethylacetamide (DMAc), and N,N-dimethylformamide (DMF). In this contribution, molecular dynamics simulations with accurate interatomic potentials for graphitic materials that include virtual π electrons are reported to analyze the interaction of pristine single wall CNTs with the solvents and polymer solutions at 25 °C. The results explain the barriers toward dispersion of SWCNTs and quantify CNT-solvent, polymer-solvent, as well as CNT-polymer interactions in atomic detail. Debundling of CNTs is overall endothermic and unfavorable with dispersion energies of +20 to +30 mJ/m 2 in the pure solvents, + 20 to +40 mJ/m 2 in PAN solutions, and +20 to +60 mJ/m 2 in PMMA solutions. Differences arise due to molecular geometry, polar, van der Waals, and CH-π interactions. Among the pure solvents, DMF restricts CNT dispersion less due to the planar geometry and stronger van der Waals interactions. PAN and PMMA interact favorably with the pure solvents with dissolution energies of -0.7 to -1.1 kcal per mole monomer and -1.5 to -2.2 kcal per mole monomer, respectively. Adsorption of PMMA onto CNTs is stronger than that of PAN in all solvents as the molecular geometry enables more van der Waals contacts between alkyl groups and the CNT surface. Polar side groups in both polymers prefer interactions with the polar solvents. Higher polymer concentrations in solution lead to polymer aggregation via alkyl groups and reduce adsorption onto CNTs. PAN and PMMA solutions in DMSO and dilute solutions in DMF support CNT dispersion more than other combinations whereby the polymers significantly adsorb onto CNTs in DMSO solution. The observations by molecular simulations are consistent with available experimental data and solubility parameters and aid in the design of carbon nanofibers. The methods can be applied to other multiphase graphitic materials.

Chemistry↗

DOE OSTI2016

The thermodynamic stability of intermediate solid solutions in LiFePO 4 nanoparticles

Theoretical predictions from first principles and recent advances in in situ electrochemical characterization techniques have confirmed the presence of solid-solution states during electrochemical (de)lithiation of LiFePO 4 nanoparticles. Surprisingly, however, such thermodynamically unfavorable solid solution states have been observed at rates as low as 0.1C. Given the high diffusivity of Li in LiFePO 4 and the thermodynamic instability of homogeneous solid solution states, spinodal decomposition to a thermodynamically favorable two-phase state is expected to occur on time scales as rapid as 1–100 ms. Here, we resolve this apparent paradox by demonstrating that, given the symmetry of the low-energy solid-solution Li/Va orderings and the 1D character of Li diffusion, spinodal decomposition from a solid solution preferentially leads to the formation of a diffuse ac interface with a large intermediate solid-solution region, as opposed to the commonly assumed bc interface. Our first principles predictions not only rationalize the persistence of solid-solution states at low-to-moderate C-rates in high-rate LiFePO 4 electrodes, but also explain the observations of large intermediate solid-solution regions at an ac interface in single Li x FePO 4 particles quenched from a high-temperature solid solution.

36 MATERIALS SCIENCE↗

DOE OSTI2019

Group-invariant solutions for one dimensional, inviscid hydrodynamics

In this paper, the results of the Lie group method carried out by Ovsiannikov are utilized to study the one-dimensional hydrodynamic equations governing compressible, inviscid fluid flow in the absence of heat conduction. One-parameter subgroups of the admissible R-parameter Lie group of point transformations of the system are applied to reduce the first-order, non-linear system of partial differential equations (PDE)s to a first-order system of ordinary differential equations (ODE)s. Closed-form solutions to the reduced ODE systems are subsequently determined using a linear velocity profile ansatz. These solutions are valid in one-dimensional (1D) planar, cylindrical and spherical geometries and are connected to solutions of the governing system of PDEs through an inverse map. The linear velocity type solutions were first considered by Sedov in 1953 and constitute a subclass of all possible similarity solutions of the compressible hydrodynamics equations. They further serve as illustrative examples of using solutions obtained for the reduced ODEs to find solutions of the associated system of PDEs. Consequently, the reduced systems of ODEs are provided in their entirety inviting additional solutions to be determined via alternative explicit or numerical solution techniques.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2016

Fast computation of synthetic seismograms within a medium containing remote localized perturbations: a numerical solution to the scattering problem

We derive a fast discrete solution to the scattering problem. This solution allows us to compute accurate synthetic seismograms or waveforms for arbitrary locations of sources and receivers within a medium containing localized perturbations. The key to efficiency is that wave propagation modelling does not need to be carried out in the entire volume that encompasses the sources and the receivers but only within the subvolume containing the perturbations or scatterers. The proposed solution has important applications, for example, it permits the imaging of remote targets located in regions where no sources or receivers are present. Our solution relies on domain decomposition: within a small volume that contains the scatterers, wave propagation is modelled numerically, while in the surrounding volume, where the medium isn't perturbed, the response is obtained through wavefield extrapolation. The originality of this work is the derivation of discrete formulas for representation theorems and Kirchhof–Helmholtz integrals that naturally adapt to the numerical scheme employed for modelling wave propagation. Our solution applies, for example, to finite difference methods or finite/spectral elements methods. The synthetic seismograms obtained with our solution can be considered ‘exact’ as the total numerical error is comparable to that of the method employed for modelling wave propagation. We detail a basic implementation of our solution in the acoustic case using the finite difference method and present numerical examples that demonstrate the accuracy of the method. We show that ignoring some terms accounting for higher order scattering effects in our solution has a limited effect on the computed seismograms and significantly reduces the computational effort. Finally, we show that our solution can be used to compute localized sensitivity kernels and we discuss applications to target oriented imaging. Extension to the elastic case is straightforward and summarized in a dedicated section.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Approximate Mixed-Integer Programming Solution with Machine Learning Technique and Linear Programming Relaxation

Security constrained unit commitment (SCUC) is solved in the day-ahead electricity market to determine electricity generation schedule for the following date. The associated mixed integer programming (MIP) problem is known computationally hard. Heuristics like rounding can be computationally efficient for solving large MIP problems however often produces lower quality solutions due to discrepancy between MIP solution and linear programming relaxation (LPR) solution, i.e., variables taking different discrete values between MIP and LPR solutions. In this work, we applied the machine learning-based algorithms—Classification and Regression Tree (CART) model and random forest (RF) model—to obtain an LPR-based approximation that close to an MIP solution without actually solving the MIP problem. By computing feature importance derived from the CART and RF models, we assessed the link between coefficients in the MIP model, the LPR solution, and the optimal solution of the MIP problem. We applied the CART and RF approaches to solve SCUC problems using day-ahead energy market cases from Midcontinent Independent System Operator. The results show that our method is computationally efficient and could achieve an approximation sufficiently close to the SCUC solution with <0.15% error and ~78% reduction in the discrepancies between LPR and MIP solutions.

Lin, Xinming↗

DOE OSTI2024

On the Existence of Steady-State Solutions to the Equations Governing Fluid Flow in Networks

The steady-state solution of fluid flow in pipeline infrastructure networks driven by junction/node potentials is a crucial ingredient in various decision-support tools for system design and operation. While the nonlinear system is known to have a unique solution (when one exists), the absence of a definite result on the existence of solutions hobbles the development of computational algorithms, for it is not possible to distinguish between algorithm failure and non-existence of a solution. In this letter, we show that for any fluid whose equation of state is a scaled monomial, a unique solution exists for such nonlinear systems if the term solution is interpreted in terms of potentials and flows rather than pressures and flows. However, for gases following the CNGA equation of state, while the question of existence remains open, we construct an alternative system that always has a unique solution and show that the solution to this system is a good approximant of the true solution. Further, the existence result for flow of natural gas in networks also applies to other fluid flow networks such as water distribution networks or networks that transport carbon dioxide in carbon capture and sequestration. Most importantly, our result enables correct diagnosis of algorithmic failure, problem stiffness, and non-convergence in computational algorithms.

42 ENGINEERING↗

DOE OSTI2017

Thermodynamics of copper‐manganese and copper‐iron spinel solid solutions

Abstract High‐temperature oxide melt solution calorimetry has been performed to investigate the energetics of spinel solid solutions in the Mn 3 O 4 ‐CuMn 2 O 4 and Fe 3 O 4 ‐CuFe 2 O 4 systems. The spinel solid solutions were synthesized by a ceramic route and calcined at appropriate temperatures to obtain single phase samples. The drop solution enthalpies of the solid solutions in the Mn 3 O 4 ‐CuMn 2 O 4 system are the same within experimental error as the enthalpies of drop solution of mechanical mixtures of the end‐members, indicating a zero heat of mixing, i.e., ideal mixing in terms of enthalpy. In Fe 3 O 4 ‐CuFe 2 O 4 , the drop solution enthalpy of the solid solutions shows a positive deviation from those of the mechanical mixture of the end‐members, suggesting negative mixing enthalpy. The formation enthalpies of the spinel solid solutions from their constituent oxides plus oxygen and from the elements were also determined.

Sahu, Sulata K.↗

DOE OSTI2023

Burst Pressure Solutions of Thin and Thick-Walled Cylindrical Vessels

Pressure vessels (PVs) are widely used in the energy industry. Accurate burst pressure is critical to structural design and safe operation for both thin and thick-walled PVs. The traditional strength theories utilized a single-parameter material property, such as the yield stress or the ultimate tensile stress (UTS) to develop failure models for determining the yield or ultimate pressure carrying capacity in the PV design. The UTS-based Barlow formula is a typical burst pressure model developed from the Tresca strength theory that provides the basis for developing regulation rules and failure models for different industry design codes, such as ASME BPVC, ASME B31.3, and ASME B31G, among others. In order to reduce the conservatism of the Tresca strength model, ASME BPVC recently adapted failure models developed from the von Mises strength theory for the PV design and analysis. It has been commonly accepted that the burst pressure of pipelines depends on the UTS and strain hardening exponent, n, of the pipeline steel. An average shear stress yield theory was thus developed, and the Zhu-Leis solution of burst pressure was obtained as a function of UTS and n for thin-walled line pipes. Experiments showed that the Zhu-Leis solution provides an accurate, reliable prediction of burst pressure for defect-free thin-walled pipes. In order to extend the Zhu–Leis solution to thick-walled cylindrical PVs, this paper defined three new flow stresses, modified the traditional strength theories, and obtained three new burst pressure solutions that are valid for both thin and thick-walled cylindrical vessels. The proposed flow stresses are able to describe the tensile strength and the plastic flow response of PVs for a strain hardening steel. The associated strength theories were then developed in terms of the Tresca, von Mises, and Zhu-Leis yield criteria. From these new strength theories, three burst pressure solutions were obtained for thick-walled cylinders, where the von Mises solution is an upper bound prediction, the Tresca solution is a lower bound prediction, and the Zhu-Leis solution is an intermediate prediction of burst pressure for thick-walled cylinders. Lastly, the proposed burst pressure solutions were evaluated and validated by two large datasets of full-scale burst tests for thick-walled tubes and for thin-walled pipes.

42 ENGINEERING↗

DOE OSTI2019

Application of symmetries to differential equations: symmetry reduction and solution transformation examples

In the following lecture notes, we provide an outline of what we mean by symmetries of differential equations and illustrate their utility in simplification of differential equations which may lead to solutions. Loosely, a symmetry of a differential equation is a local group of transformations acting on the space of independent and dependent variables that transform solutions of a differential equation to new solutions, i.e., if Δ(x,u,p)=0 denotes a differential equation with a solution prescribed by [x,u,p], then knowlege of a symmetry facilitates the attainment of a new solution, [$\tilde{x}$, $\tilde{u}$, $\tilde{p}$], such that Δ(x,u,p)=Δ($\tilde{x}$, $\tilde{u}$, $\tilde{p}$)=0. In Equations (1) and (2), x, u and p denote the independent variables, dependent variables and partial derivatives, respectively and the tildes denote the transformations. Equation (2) will subsequently be referred to as the symmetry criterion. We can apply the knowlede of the existence of a symmetry in two ways: 1) If not solution is known apriori, the symmetry can be applied to simplify the differential equation through the method of symmetry reduction. Often, the resultant simplification may enable a solution to be determined; 2) If we already possess a solution to a different equation, knowledge of an admissible symmetry enables us to generate new solutions.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Corrosion of 347 stainless steel in the presence of uranyl sulfate solution and radiation

The U.S. molybdenum 99 (Mo-99) industry is pursuing production of fission-made Mo-99 using a uranium solution such as uranyl sulfate. In this process, uranyl sulfate solution containing low-enriched uranium will be bombarded by neutrons creating Mo-99 and other fission products. During the production, the uranyl sulfate solution will be irradiated until an acceptable activity level of Mo-99 is produced. The uranyl sulfate solution containing Mo-99 and other fission products will then undergo a series of separation steps. First, uranyl sulfate can be separated from Mo-99 using a primary titania column to recover Mo-99, with the uranyl sulfate solution to be used for another irradiation cycle. Then, raffinate from a primary titania column containing Mo-99 can be concentrated and purified using a LEU modified Cintichem process developed by Argonne National Laboratory. During irradiation, the temperature of the uranyl sulfate solution can reach near boiling (up to ~80° C assumed), causing radiolysis of water and the resultant formation of hydrogen peroxide. Because high-radiation fields will be present during each irradiation cycle, it is important to determine the corrosion rates of SS-347 under such conditions to estimate the life cycle of the target solution vessel. The buildup of corrosion products from the SS components in the uranyl sulfate solution also needs to be well understood because potential accumulation of iron, nickel, and other corrosion products may affect the Mo-99 recovery and purification process. To study the corrosion rates of SS-347 material under conditions relevant to future Mo-99 production facility, SS-347 coupons in uranyl sulfate solution at ~80° C were irradiated using Argonne’s Van de Graaff generator, which can generate high-radiation fields without fissioning of uranium or production of activation products.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2022

Effects of Chlorine Capture and a Proposed Density Law on the Reactivity of Plutonium Solution Systems

During fissionable material processing, all normal and credible abnormal conditions must remain safely subcritical. Nuclear Criticality Safety (NCS) uses a number of methods to determine subcriticality, one of which is the use of neutron transport codes such as MCNP6. In order to create models for use with MCNP6, both the geometry and materials in fissionable material processes must be known, or assumptions must be made and quantified for the impact to bias. One of the systems with a significant amount of bias due to material modeling assumptions is in the area of aqueous plutonium processing. These solutions are typically plutonium nitrate solutions or plutonium chloride solutions, which are modeled as fictitious plutonium metal-water mixtures because little is known about the actual density of the solution and there is no current predictive capability approved for use at Los Alamos National Laboratory (LANL) for modeling them. This research is currently underway to fill the gap and develop an algorithm for use with MCNP6 to model the density of plutonium chloride solutions. The method is to be validated with experimental data for density, and also validated with critical experiments using MCNP6. Note that the Chlorine Worth Study (CWS) was performed in December 2021 to help bridge the gap in chlorine data for critical experiments, and is currently awaiting International Criticality Safety Benchmark Evaluation Project (ICSBEP) review. This study was performed by LANL at the National Criticality Experiments Research Center (NCERC) at the Nevada National Security Site (NNSS). Additional information regarding this experiment may currently be found in LA-UR- 22-29180. Additionally, the Chemistry-Actinide Analytical Chemistry (C-AAC) at LANL has performed a number of solution density measurements for PuCl 3 -HC 1 -H 2 O, allowing for such data be used to create a semi-empirical density via the Pitzer method. The published dataset for the measurements is documented in LA-UR-22-25454. This method has already been tested successfully for aqueous plutonium nitrate solutions in SCALE. Current solution density measurements exist of plutonium concentrations of 0-~142g/L, all at 2M HC1, at temperatures 20-40°C. Additional data was taken for HC1-corrected density values, which essentially mimics the data for a pure PuCl x -water solution. The calculations in this report aim to support the current research by demonstrating the difference in system reactivity for the current modeling method when compared to the new proposed modeling with a density law implementation, which is being written as a Python tool to be used with MCNP6.

38 RADIATION CHEMISTRY, RADIOCHEMISTRY, AND NUCLEA↗

DOE OSTI2016

THE WIND OF ROTATING B SUPERGIANTS. I. DOMAINS OF SLOW AND FAST SOLUTION REGIMES

In the scenario of rotating radiation-driven wind theory for massive stars, three types of stationary hydrodynamic solutions are currently known: the classical ( fast ) m-CAK solution, the Ω- slow solution that arises for fast rotators, and the so-called δ - slow solution if high values of the δ line-force parameter are allowed independently of the rotation speed. Compared to the fast solution, both “ slow solutions” have lower terminal velocities. As the study of the parameter domain for the slow solution is still incomplete, we perform a comprehensive analysis of the distinctive flow regimes for B supergiants that emerge from a fine grid of rotation values, Ω, and various ionization conditions in the wind ( δ ) parameter. The wind ionization defines two domains: one for fast outflowing winds and the other for slow expanding flows. Both domains are clear-cut by a gap, where a kink/plateau structure of the velocity law could exist for a finite interval of δ . The location and width of the gap depend on T {sub eff} and Ω. There is a smooth and continuous transition between the Ω- slow and δ - slow regimes, a single Ω δ - slow regime. We discuss different situations where the slow solutions can be found and the possibility of a switch between fast and slow solutions in B supergiant winds. We compare the theoretical terminal velocity with observations of B and A supergiants and find that the fast regime prevails mostly for early B supergiants while the slow wind regime matches better for A and B mid- and late-type supergiants.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1973

Application of matched asymptotic expansions to lunar and interplanetary trajectories. Volume 2: Derivations of second-order asymptotic boundary value solutions

Previously published asymptotic solutions for lunar and interplanetery trajectories have been modified and combined to formulate a general analytical solution to the problem of N-bodies. The earlier first-order solutions, derived by the method of matched asymptotic expansions, have been extended to second order for the purpose of obtaining increased accuracy. The complete derivation of the second-order solution, including the application of a regorous matching principle, is given. It is shown that the outer and inner expansions can be matched in a region of order mu to the alpha power, where 2/5 alpha 1/2, and mu (the moon/earth or planet/sun mass ratio) is much less than one. The second-order asymptotic solution has been used as a basis for formulating a number of analytical two-point boundary value solutions. These include earth-to-moon, one- and two-impulse moon-to-Earth, and interplanetary solutions. Each is presented as an explicit analytical solution which does not require iterative steps to satisfy the boundary conditions. The complete derivation of each solution is shown, as well as instructions for numerical evaluation. For Vol. 1, see N73-27738.

Lancaster, J. E.↗

NASA NTRS1975

A similarity solution of the Navier-Stokes equations with wall catalysis and slip for hypersonic, low Reynolds number flow over spheres

The slip conditions for a multicomponent mixture with diffusion, wall-catalyzed atom recombination and thermal radiation are derived. The more realistic multicomponent species slip conditions are shown to be necessary for accurate merged shock layer solutions on a sphere. These slip conditions are used in a first-order similarity solution of the Navier-Stokes equations with nonequilibrium chemistry for the merged shock layer. Results of this quick numerical solution are compared with a time dependent solution around the sphere and with measured arc jet results at low Reynolds numbers. The similarity solution, unlike the time dependent solution, shows smooth radial profiles of the pressure and smooth variations of velocity slip, skin friction, temperature slip and heat transfer around the body. The present first-order similarity solution is valid up to 25 deg from the stagnation point and takes less than 1% of the computer time to run a time dependent scheme. The smaller stand-off distance obtained from the similarity solution is supported by experimental data. The measured heat flux is closer to the similarity solution than the time dependent method at the stagnation point and shows the proper variation with circumferential angle up to at least 40 deg.

Hendricks, W. L.↗

NASA NTRS1976

Solution of the minimum time-to-climb problem by matched asymptotic expansions

Application of singular perturbation techniques to trajectory optimization problems of flight mechanics is discussed. The method of matched asymptotic expansions is used to obtain an approximate solution to the aircraft minimum time-to-climb problem. Outer, boundary-layer, and composite solutions are obtained to zeroth and first orders. A stability criterion is derived for the zeroth-order boundary-layer solutions (the theory requires a form of boundary-layer stability). A numerical example is considered for which it is shown that the stability criterion is satisfied and a useful numerical solution is obtained. The zeroth-order solution proves to be a poor approximation, but the first-order solution gives a good approximation for both the trajectory and the minimum time-to-climb. The computational cost of the singular perturbation solution is considerably less than that of a steepest descent solution. Thus singular perturbation methods appear to be promising for the solution of optimal control problems.

Ardema, M. D.↗

NASA NTRS1986

Numerical solution of random singular integral equation appearing in crack problems

The solution of several elasticity problems, and particularly crack problems, can be reduced to the solution of one-dimensional singular integral equations with a Cauchy-type kernel or to a system of uncoupled singular integral equations. Here a method for the numerical solution of random singular integral equations of Cauchy type is presented. The solution technique involves a Chebyshev series approximation, the coefficients of which are the solutions of a system of random linear equations. This method is applied to the problem of periodic array of straight cracks inside an infinite isotropic elastic medium and subjected to a nonuniform pressure distribution along the crack edges. The statistical properties of the random solution are evaluated numerically, and the random solution is used to determine the values of the stress-intensity factors at the crack tips. The error, expressed as the difference between the mean of the random solution and the deterministic solution, is established. Values of stress-intensity factors at the crack tip for different random input functions are presented.

Sambandham, M.↗

NASA NTRS1994

Radiation boundary condition and anisotropy correction for finite difference solutions of the Helmholtz equation

In this paper finite-difference solutions of the Helmholtz equation in an open domain are considered. By using a second-order central difference scheme and the Bayliss-Turkel radiation boundary condition, reasonably accurate solutions can be obtained when the number of grid points per acoustic wavelength used is large. However, when a smaller number of grid points per wavelength is used excessive reflections occur which tend to overwhelm the computed solutions. Excessive reflections are due to the incompability between the governing finite difference equation and the Bayliss-Turkel radiation boundary condition. The Bayliss-Turkel radiation boundary condition was developed from the asymptotic solution of the partial differential equation. To obtain compatibility, the radiation boundary condition should be constructed from the asymptotic solution of the finite difference equation instead. Examples are provided using the improved radiation boundary condition based on the asymptotic solution of the governing finite difference equation. The computed results are free of reflections even when only five grid points per wavelength are used. The improved radiation boundary condition has also been tested for problems with complex acoustic sources and sources embedded in a uniform mean flow. The present method of developing a radiation boundary condition is also applicable to higher order finite difference schemes. In all these cases no reflected waves could be detected. The use of finite difference approximation inevita bly introduces anisotropy into the governing field equation. The effect of anisotropy is to distort the directional distribution of the amplitude and phase of the computed solution. It can be quite large when the number of grid points per wavelength used in the computation is small. A way to correct this effect is proposed. The correction factor developed from the asymptotic solutions is source independent and, hence, can be determined once and for all. The effectiveness of the correction factor in providing improvements to the computed solution is demonstrated in this paper.

Tam, Christopher K. W.↗

NASA NTRS1995

Ca-Rich Carbonate Melts: A Regular-Solution Model, with Applications to Carbonatite Magma + Vapor Equilibria and Carbonate Lavas on Venus

A thermochemical model of the activities of species in carbonate-rich melts would be useful in quantifying chemical equilibria between carbonatite magmas and vapors and in extrapolating liquidus equilibria to unexplored PTX. A regular-solution model of Ca-rich carbonate melts is developed here, using the fact that they are ionic liquids, and can be treated (to a first approximation) as interpenetrating regular solutions of cations and of anions. Thermochemical data on systems of alkali metal cations with carbonate and other anions are drawn from the literature; data on systems with alkaline earth (and other) cations and carbonate (and other) anions are derived here from liquidus phase equilibria. The model is validated in that all available data (at 1 kbar) are consistent with single values for the melting temperature and heat of fusion for calcite, and all liquidi are consistent with the liquids acting as regular solutions. At 1 kbar, the metastable congruent melting temperature of calcite (CaCO3) is inferred to be 1596 K, with (Delta)bar-H(sub fus)(calcite) = 31.5 +/- 1 kJ/mol. Regular solution interaction parameters (W) for Ca(2+) and alkali metal cations are in the range -3 to -12 kJ/sq mol; W for Ca(2+)-Ba(2+) is approximately -11 kJ/sq mol; W for Ca(2+)-Mg(2+) is approximately -40 kJ/sq mol, and W for Ca(2+)-La(3+) is approximately +85 kJ/sq mol. Solutions of carbonate and most anions (including OH(-), F(-), and SO4(2-)) are nearly ideal, with W between 0(ideal) and -2.5 kJ/sq mol. The interaction of carbonate and phosphate ions is strongly nonideal, which is consistent with the suggestion of carbonate-phosphate liquid immiscibility. Interaction of carbonate and sulfide ions is also nonideal and suggestive of carbonate-sulfide liquid immiscibility. Solution of H2O, for all but the most H2O-rich compositions, can be modeled as a disproportionation to hydronium (H3O(+)) and hydroxyl (OH(-)) ions with W for Ca(2+)-H3O(+) (approximately) equals 33 kJ/sq mol. The regular-solution model of carbonate melts can be applied to problems of carbonatite magma + vapor equilibria and of extrapolating liquidus equilibria to unstudied systems. Calculations on one carbonatite (the Husereau dike, Oka complex, Quebec, Canada) show that the anion solution of its magma contained an OH mole fraction of (approximately) 0.07, although the vapor in equilibrium with the magma had P(H2O) = 8.5 x P(CO2). F in carbonatite systems is calculated to be strongly partitioned into the magma (as F(-)) relative to coexisting vapor. In the Husereau carbonatite magma, the anion solution contained an F(-) mole fraction of (approximately) 6 x 10(exp -5).

Treiman, Allan H.↗