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Results for “Thermodynamic limit”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 595 records · Page 33
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Competitive effects of glucan’s main hydrolysates on biochar formation: A combined experiment and density functional theory analysis

We report the complexity of polysaccharide hydrothermal products increases the difficulty of exploring the formation of biochar, limiting the development of biochar. This work clarifies the completive effects of glucan’s main hydrolysates on biochar formation from three aspects: experimental, thermodynamic, and kinetic. The products distribution illustrates that 5-HMF, FA, and LA are mainly involved in the formation of biochar. Biochar mainly includes furan ring, ether group, and ester group by the analysis of magic-angle-spinning nuclear magnetic resonance, X-ray photoelectron spectroscopy, and elemental analysis. Combined experiments and density functional theory analysis, the etherification reaction of 5-HMF itself is most likely to occur and is key to form biochar, followed by the esterification of FA with 5-HMF, and then the etherification of 5-HMF and LA. The further verified experiments also manifest these results. This work will develop a foundation for exploring the complex formation mechanism of cellulose-based biochar.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2025

Crystallographic stability of the BCC phase during quenching of metastable beta titanium alloy Ti-5553 and comparison of structural criteria for the predictive capability of α” martensite

This work evaluates the compositional standard currently developed for Ti-5553 powder and explores the metastable β region for sensitivity to martensitic formation during rapid quenching from the melt. Ti-5553 is a beta-stabilized titanium alloy that is increasingly being used in additive manufacturing applications. A series of alloys within the Ti-5553 compositional space was processed using two-piston spat quenching to perform rapid solidification and quenching for each of these alloy compositions. Typical empirical models such as molybdenum equivalency are not able to fully separate the retained BCC and martensitic compositions. Further, the use of thermodynamic data to estimate the transformation energy needed to form martensite can differentiate the alloys and provide a metric to further develop compositional limits for metastable beta titanium alloy development.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Influence of External Conditions on the Black-to-Yellow Phase Transition of CsPbI 3 Based on First-Principles Calculations: Pressure and Moisture

All-inorganic CsPbI 3 perovskite has emerged as a promising candidate for next-generation solar cells. However, owing to the poor structural stability of black phase CsPbI 3 perovskite at room temperature, it spontaneously transforms to the yellow, photoinactive, nonperovskite phase at room temperature, limiting further development of CsPbI 3 perovskite solar cells. Here, to better understand the mechanism driving such undesirable phase transformation, we examine the thermodynamics and kinetics associated with γ-to-δ phase transformation using density functional theory calculation. A solid-state nudged elastic band method was employed to find minimum energy pathways assuming a concerted solid-state phase transformation between these two phases. The gas molecules representing atmospheric (H 2 O, O 2 , and N 2 ) and inert (Ar) ambient environments, as well as applied external pressure, were considered to examine the influence of external conditions on the black-to-yellow phase transition. Our calculation reveals that, with increasing pressure, the thermodynamic driving force for converting the γ-phase to the δ-phase increases, whereas a larger activation energy must be overcome for the phase transition to occur. We also investigate the moisture-induced phase transformation with an H 2 O molecule and its dissociated species (H + /OH – ) and demonstrate that the reaction energy barriers can be significantly lowered in the presence of H 2 O or OH – . On the other hand, other nonpolar species (O 2 , N 2 , and Ar) have a negligible effect on the phase transformation kinetics, while they might reduce the thermodynamic driving force for the phase transformation and suppress the undesirable phase transformation. Our theoretical prediction could support recent observations that the γ-to-δ phase transformation occurs rapidly and catalytically in the presence of moisture, whereas in dry argon and dry oxygen atmospheres, the γ-CsPbI 3 remains stable.

14 SOLAR ENERGY↗
DOE OSTI2026

High-Voltage Sodium–Metal Batteries with Asymmetric Fluoroalkoxylated Organoborate Anion Chemistry

The high reactivity of sodium (Na) metal restricts its compatibility to ether-based electrolytes, while the poor oxidative stability of ethers precludes their coupling to high-voltage cathodes, fundamentally limiting the operating voltage and energy density of sodium–metal batteries (SMBs). We present here a coordination-asymmetry strategy to reconcile this thermodynamic mismatch by generating in situ an asymmetric fluoroalkoxylated organoborate anion, [FB(OCH(CF 3 ) 2 ) 3 ] − (BOF – ), via a Lewis acid–base adduct reaction in ether electrolyte. The asymmetric ligand architecture differentiates oxidative and fluorination pathways: oxidizable B–O moieties mediate controlled interfacial reconstruction, whereas the terminal B–F units supply fluorine for chemical passivation. This self-adaptive chemistry yields nanoscale, conformal, compositionally graded interphases: a boron-oxide/boron-oxycarbide-rich cathode-electrolyte interphase (CEI) that mitigates ether oxidation and a bilayered inorganic–organic solid–electrolyte interphase (SEI) that regulates Na deposition. The nanostructured interphases enable highly reversible Na plating/stripping with an average Coulombic efficiency (CE) of 99.98% and sustain stable 4.3 V operation of anode-free SMBs in oxidation-prone ether electrolytes. Furthermore, this work establishes asymmetric boron coordination as a molecular-level design principle for creating chemically adaptive interphases that overcome the redox asymmetry in energy-dense electrochemical systems.

Anions↗
DOE OSTI2024

The persistence of memory in ionic conduction probed by nonlinear optics

Predicting practical rates of transport in condensed phases enables the rational design of materials, devices and processes. This is especially critical to developing low-carbon energy technologies such as rechargeable batteries. For ionic conduction, the collective mechanisms, variation of conductivity with timescales and confinement, and ambiguity in the phononic origin of translation, call for a direct probe of the fundamental steps of ionic diffusion: ion hops. However, such hops are rare-event large-amplitude translations, and are challenging to excite and detect. Here we use single-cycle terahertz pumps to impulsively trigger ionic hopping in battery solid electrolytes. This is visualized by an induced transient birefringence, enabling direct probing of anisotropy in ionic hopping on the picosecond timescale. The relaxation of the transient signal measures the decay of orientational memory, and the production of entropy in diffusion. We extend experimental results using in silico transient birefringence to identify vibrational attempt frequencies for ion hopping. Using nonlinear optical methods, we probe ion transport at its fastest limit, distinguish correlated conduction mechanisms from a true random walk at the atomic scale, and demonstrate the connection between activated transport and the thermodynamics of information.

25 ENERGY STORAGE↗
NASA NTRS1992

High pressure turbomachinery ground test facility

Turbomachinery test facilities are at present scarce to non-existent world-wide. The turbomachinery test facility at Stennis Space Center will provide for advanced development and research and development capabilities for liquid hydrogen/liquid oxygen propellant rocket engine components. The facility will provide ultra-high pressure via gas generators to deliver the needed turbine drive on various turbomachinery. State of the art process control systems will provide the vital pressure, temperature and flow requirements during tests. These systems will better control adverse transient conditions during start-up and shutdown, and by using advanced control theory, as well as incorporate test article health monitoring. Also, digital data acquisition systems will obtain high frequency (up to 20 KHz) and low frequency (up to 1 KHz) data during the test. Pressures of up to 15,000 psi will be generated to pressurize high pressure tanks supplying cryogens to various test article inlets thus pushing turbopump materials and manufacturing processes to their limits. By planning for future projects the test facility will be easily adaptable to multi-program test configurations over a range of thermodynamic positions.

Scheuermann, Patrick E.↗
NASA NTRS2002

Energetics of muscle contraction: the whole is less than the sum of its parts

Understanding muscle energetics is a problem in optimizing supply of ATP to the demands of ATPases. The complexity of reactions and their fluxes to achieve this balance is greatly reduced by recognizing constraints imposed by the integration of common metabolites at fixed stoichiometry among modular units. ATPase is driven externally. Oxidative phosphorylation and glycogenolysis are the suppliers. We focus on their regulation which involves different controls, but reduces to two principles that enable facile experimental analysis of the supply and demand fluxes. The ratio of concentration of phosphocreatine (PCr) to ATP, not their individual values, sets the range of achievable concentrations of ADP in resting and active muscle (at fixed pH) in different cell types. This principle defines the fraction of available flux of oxidative phosphorylation utilized (at fixed enzyme activities). Then the kinetics of PCr recovery defines the kinetics of oxygen supply and substrate utilization. The second principle is the constancy of PCr and H(+) (lactate) production by glycogenolysis due to the coupling of ATPase and glycolysis. This principle enables glycogenolytic flux to be measured from intracellular proton loads. Further simplification occurs because the magnitude of the interacting fluxes and metabolite concentrations are specified within narrow limits when both the resting and active fluxes are quantified. Thus there is a small set of rules for assessing and understanding the thermodynamics and kinetics of muscle energetics.

NASA Program Biomedical Research and Countermeasur↗
NASA NTRS2006

Vertical Moist Thermodynamic Structure and Spatial-Temporal Evolution of the MJO in AIRS Observations

The atmospheric moisture and temperature profiles from the Atmospheric Infrared Sounder (AIRS)/Advanced Microwave Sounding Unit on the NASA Aqua mission, in combination with the precipitation from the Tropical Rainfall Measuring Mission (TRMM), are employed to study the vertical moist thermodynamic structure and spatial-temporal evolution of the Madden-Julian oscillation (MJO). The AIRS data indicate that, in the Indian Ocean and western Pacific, the temperature anomaly exhibits a trimodal vertical structure: a warm (cold) anomaly in the free troposphere (800-250 hPa) and a cold (warm) anomaly near the tropopause (above 250 hPa) and in the lower troposphere (below 800 hPa) associated with enhanced (suppressed) convection. The AIRS moisture anomaly also shows markedly different vertical structures as a function of longitude and the strength of convection anomaly. Most significantly, the AIRS data demonstrate that, over the Indian Ocean and western Pacific, the enhanced (suppressed) convection is generally preceded in both time and space by a low-level warm and moist (cold and dry) anomaly and followed by a low-level cold and dry (warm and moist) anomaly. The MJO vertical moist thermodynamic structure from the AIRS data is in general agreement, particularly in the free troposphere, with previous studies based on global reanalysis and limited radiosonde data. However, major differences in the lower-troposphere moisture and temperature structure between the AIRS observations and the NCEP reanalysis are found over the Indian and Pacific Oceans, where there are very few conventional data to constrain the reanalysis. Specifically, the anomalous lower-troposphere temperature structure is much less well defined in NCEP than in AIRS for the western Pacific, and even has the opposite sign anomalies compared to AIRS relative to the wet/dry phase of the MJO in the Indian Ocean. Moreover, there are well-defined eastward-tilting variations of moisture with height in AIRS over the central and eastern Pacific that are less well defined, and in some cases absent, in NCEP. In addition, the correlation between MJO-related mid-tropospheric water vapor anomalies and TRMM precipitation anomalies is considerably more robust in AIRS than in NCEP, especially over the Indian Ocean. Overall, the AIRS results are quite consistent with those predicted by the frictional Kelvin-Rossby wave/conditional instability of the second kind (CISK) theory for the MJO.

troposhere↗
NASA NTRS2012

Characterization and Computational Modeling of Minor Phases in Alloy LSHR

The minor phases of powder metallurgy disk superalloy LSHR were studied. Samples were consistently heat treated at three different temperatures for long times to approach equilibrium. Additional heat treatments were also performed for shorter times, to assess minor phase kinetics in non-equilibrium conditions. Minor phases including MC carbides, M23C6 carbides, M3B2 borides, and sigma were identified. Their average sizes and total area fractions were determined. CALPHAD thermodynamics databases and PrecipiCalc(TradeMark), a computational precipitation modeling tool, were employed with Ni-base thermodynamics and diffusion databases to model and simulate the phase microstructural evolution observed in the experiments with an objective to identify the model limitations and the directions of model enhancement.

Jou, Herng-Jeng↗
NASA NTRS2018

Thermodynamic and Dynamic Aspects of Ice Nucleation

It is known that ice nucleating particles (INP) immersed within supercooled droplets promote the formation of ice. Common theoretical models used to represent this process assume that the immersed particle lowers the work of ice nucleation without significantly affecting the dynamics of water in the vicinity of the particle. This is contrary to evidence showing that immersed surfaces significantly affect the viscosity and diffusivity of vicinal water. To study how this may affect ice formation this work introduces a model linking the ice nucleation rate to the modification of the dynamics and thermodynamics of vicinal water by immersed particles. It is shown that INP that significantly reduce the work of ice nucleation also pose strong limitations to the growth of the nascent ice germs. This leads to the onset of a new ice nucleation regime, called spinodal ice nucleation, where the dynamics of ice germ growth instead of the ice germ size determines the nucleation rate. Nucleation in this regime is characterized by an enhanced sensitivity to particle area and cooling rate. Comparison of the predicted ice nucleation rate against experimental measurements for a diverse set of species relevant to cloud formation suggests that spinodal ice nucleation may be common in nature.

Barahona, Donifan↗
NASA NTRS2018

High-resolution SOFIA/EXES Spectroscopy of SO2 Gas in the Massive Young Stellar Object MonR2 IRS3: Implications for the Sulfur Budget

Sulfur has been observed to be severely depleted in dense clouds leading to uncertainty in the molecules that contain it and the chemistry behind their evolution. Here, we aim to shed light on the sulfur chemistry in young stellar objects (YSOs) by using high-resolution infrared spectroscopy of absorption by the ν3 rovibrational band of SO2 obtained with the Echelon-Cross-Echelle Spectrograph on the Stratospheric Observatory for Infrared Astronomy. Using local thermodynamic equilibrium models we derive physical parameters for the SO2 gas in the massive YSO MonR2 IRS3. This yields a SO2/H abundance lower limit of 5.6 ± 0.5 × 10(exp -7), or >4% of the cosmic sulfur budget, and an intrinsic line width (Doppler parameter) of b < 3.20 km s(exp -1). The small line widths and high temperature (Tex = 234 ± 15 K) locate the gas in a relatively quiescent region near the YSO, presumably in the hot core where ices have evaporated. This sublimation unlocks a volatile sulfur reservoir (e.g., sulfur allotropes as detected abundantly in comet 67P/Churyumov–Gerasimenko), which is followed by SO2 formation by warm, dense gas-phase chemistry. The narrowness of the lines makes formation of SO2 from sulfur sputtered off grains in shocks less likely toward MonR2 IRS3.

Infrared ISM↗
DOE OSTI2023

Deuterated water as a substrate-agnostic isotope tracer for investigating reversibility and thermodynamics of reactions in central carbon metabolism

Stable isotope tracers are a powerful tool for the quantitative analysis of microbial metabolism, enabling pathway elucidation, metabolic flux quantification, and assessment of reaction and pathway thermodynamics. 13 C and 2 H metabolic flux analysis commonly relies on isotopically labeled carbon substrates, such as glucose. However, the use of 2 H-labeled nutrient substrates faces limitations due to their high cost and limited availability in comparison to 13 C-tracers. Furthermore, isotope tracer studies in industrially relevant bacteria that metabolize complex substrates such as cellulose, hemicellulose, or lignocellulosic biomass, are challenging given the difficulty in obtaining these as isotopically labeled substrates. In this study, we examine the potential of deuterated water ( 2 H 2 O) as an affordable, substrate-neutral isotope tracer for studying central carbon metabolism. We apply 2 H 2 O labeling to investigate the reversibility of glycolytic reactions across three industrially relevant bacterial species -C. thermocellum, Z. mobilis, and E. coli-harboring distinct glycolytic pathways with unique thermodynamics. We demonstrate that 2 H 2 O labeling recapitulates previous reversibility and thermodynamic findings obtained with established 13 C and 2 H labeled nutrient substrates. Furthermore, we exemplify the utility of this 2 H 2 O labeling approach by applying it to high-substrate C. thermocellum fermentations -a setting in which the use of conventional tracers is impractical-thereby identifying the glycolytic enzyme phosphofructokinase as a major bottleneck during high-substrate fermentations and unveiling critical insights that will steer future engineering efforts to enhance ethanol production in this cellulolytic organism. This study demonstrates the utility of deuterated water as a substrate-agnostic isotope tracer for examining flux and reversibility of central carbon metabolic reactions, which yields biological insights comparable to those obtained using costly 2 H-labeled nutrient substrates.

09 BIOMASS FUELS↗
NASA NTRS1978

Chemical weathering on Mars - Thermodynamic stabilities of primary minerals /and their alteration products/ from mafic igneous rocks

Chemical weathering on Mars is examined theoretically from the standpoint of thermodynamic equilibrium between primary rock-forming minerals and the atmospheric gases O2, H2O, and CO2. The primary minerals considered are those common to mafic igneous rocks and include olivine, pyroxene, plagioclase, magnetite, troilite, pyrrhotite, and apatite. The importance of kinetics and reaction mechanisms in controlling possible weathering processes on Mars is discussed within the limits of currently available data, and the possible influence of liquid water on Martian weathering processes is evaluated where appropriate. For gas-solid weathering of mafic igneous rocks at the Martian surface, it is concluded that upon attainment of thermodynamic equilibrium: (1) oxides and carbonates should dominate the mineral assemblage of weathering products; (2) hematite rather than goethite should be the stable mineral form of Fe (III); (3) FeSO4 or FeSO4.H2O could be the stable weathering product of iron sulfides in the absence of liquid water; and (4) kaolinite is apparently the only clay mineral that should be thermodynamically stable over all ranges of temperature and water-vapor abundance at the Martian surface.

Gooding, J. L.↗
DOE OSTI2026

The critical role of intrinsic defects and many-body interactions on the stability of MnBi2Te4

Intrinsic antisite defects pose a major challenge to understanding and predicting the exotic properties of the layered topological magnetic insulator MnBi2Te4 (MBT). In this work, we study the origin of the abundance of intrinsic defects in MBT, including many-body defect–defect interactions and many-body electronic correlations. Until now, ab initio methods have struggled to explain thermodynamic stability and properties influenced by defect behavior in MBT. We model native Mn–Bi antisite defects in MBT at finite temperatures using a cluster expansion that includes defect–defect interactions. To overcome the limitations of conventional density functional theory (DFT), we introduce a hybrid approach that incorporates high-accuracy quantum Monte Carlo (QMC) calculations, introducing missing correlations. This strategy allows for accurate estimation of defect energetics and finite-temperature properties. We compute the configurational free energy, defect concentration, and configurational heat capacity, revealing a second-order order–disorder phase transition near the experimental synthesis temperature. Our study provides the first theoretical insight into the thermodynamics of intrinsic defects in MBT. The negative free energy relative to pristine MBT at synthesis temperatures indicates that Mn–Bi antisite formation is thermodynamically spontaneous. We also present a broadly applicable general framework for correcting low-level theoretical theories using highly accurate many-body corrections from QMC.

Ghaffar, Abdul [ORNL] (ORCID:0000000241190168)↗
DOE OSTI2022

Investigations into EGR dilution tolerance in a pre-chamber ignited GDI engine

EGR is a useful means to improve fuel economy in spark-ignited gasoline engines, especially under part-load conditions by means of reducing throttling losses. There is also a significant reduction in NOx emissions due to the reduced peak cylinder temperatures. However, the ability of conventional spark ignition systems to reliably ignite dilute mixtures limits the dilution tolerance due to the onset of combustion instability. In this study, an active pre-chamber system was evaluated at part load condition with various EGR rates using a regular grade full boiling range gasoline. A premixed air-fuel mixture was supplied to the pre-chamber, which assists in scavenging the residual mass fractions to enable reliable ignition of EGR dilute mixtures in the main combustion chamber. A range of pre-chamber injection, air-fuel mixture formulation, and ignition timing strategies were evaluated during the EGR sweeps. In addition, 0-D and 1-D simulations were conducted in GT-Power to assess the thermodynamic state and composition in the pre-chamber, as well as estimate the jet momentum at the various dilution rates. The active pre-chamber extended the EGR dilution limit from 20% for conventional SI to above 30%, with a higher pre-chamber flow rate (more scavenging) resulting in increased combustion stability. A tradeoff between the jet momentum and spark timing was observed when analyzing the GT-Power simulations and the pre-chamber jet momentum was found to decrease with increasing EGR rate. At unstable high EGR operation, no relation could be found between the pre-chamber indicated data and main chamber combustion stability.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2020

Thermodynamic Consistent Neural Networks for Learning Material Interfacial Mechanics

For multilayer materials in thin substrate systems, interfacial failure is one of the most challenges. The traction-separation relations (TSR) quantitatively describe the mechanical behavior of a material interface undergoing openings, which is critical to understand and predict interfacial failures under complex loadings. However, existing theoretical models have limitations on enough complexity and flexibility to well learn the real-world TSR from experimental observations. A neural network can fit well along with the loading paths but often fails to obey the laws of physics, due to a lack of experimental data and understanding of the hidden physical mechanism. In this paper, we propose a thermodynamic consistent neural network (TCNN) approach to build a data-driven model of the TSR with sparse experimental data. The TCNN leverages recent advances in physics-informed neural networks (PINN) that encode prior physical information into the loss function and efficiently train the neural networks using automatic differentiation. We investigate three thermodynamic consistent principles, i.e., positive energy dissipation, steepest energy dissipation gradient, and energy conservative loading path. All of them are mathematically formulated and embedded into a neural network model with a novel defined loss function. A real-world experiment demonstrates the superior performance of TCNN, and we find that TCNN provides an accurate prediction of the whole TSR surface and significantly reduces the violated prediction against the laws of physics.

Zhang, Jiaxin↗
DOE OSTI2023

FWP FEAA149: “Next Generation Environmental Barrier Coatings”

Environmental barrier coatings (EBCs) are required coatings for utilization of SiC/SiC ceramic matrix composite (CMC) components in gas turbines, where the EBC represents the life-limiting factor for such components. EBC/CMC systems have shown success in aero-engine applications with increased turbine inlet temperatures and improved efficiencies, which are achieved through higher temperature stability, lower density, and decreased reliance on cooling air compared to traditional superalloys. While industrial gas turbines (IGTs) do not currently utilize SiC/SiC CMCs, the current shift towards low-carbon or carbon-free fuel sources for power generation could result in a need for EBC/CMC components with higher temperature capabilities. In this work, three tasks were outlined to improve understanding of EBC lifetimes to encourage use in IGTs with carbon-free fuel such as hydrogen: 1. Define the bond coating oxidation kinetics and EBC failure criteria, 2. Measure thermal expansion coefficients of each layered material, and 3. Perform advanced characterization and modeling to assess EBC lifetimes. Cyclic steam oxidation tests were conducted on various EBC/Si/SiC chemistries and EBC/SiC architectures to define substrate oxidation kinetics and EBC failure modes. An open-source code was developed to quantify the undulating thermally grown oxide thickness with thousands of measurements from specimen cross-section images. Bond coating oxidation kinetics were determined and used to develop a kinetic and thermodynamic model for predicting EBC lifetimes. High-temperature Raman spectroscopy was utilized for determining the SiO 2 thermally grown oxide phase transformation as the life-limiting feature for EBCs. Model efforts supported the claim that the SiO 2 phase transformation causes elevated stress during thermal cycling with associated cracking that decreases the adhesion strength of the EBC, eventually resulting in coating spallation. The finite element model subroutine will be made publicly available upon internal review. Further development of an EBC lifetime model for IGTs involves definition of a critical SiO 2 thickness for EBC spallation and must also consider both environmental (gas velocity, pressure, etc.) and specimen (EBC dopants, layer architectures, etc.) effects into predicted bond coating oxidation kinetics for long-lifetime components.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Thermodynamic assessment of the quaternary WTaCrV refractory high entropy alloy as a means to guide experimental approaches

The deployment of fusion energy poses challenges for materials in plasma facing components to withstand high temperatures and thermal gradients, particle implantation and neutron damage. The current material of choice is tungsten, although property degradation limits its consideration in future fusion reactors. Hence, materials with better resistance to harsh environments need to be developed for fusion energy to become a reality. High entropy alloys are being explored as potential candidates with some compositions showing good radiation resistance to defect cluster formation. One of these materials is the WTaCrV system, although only one composition has been tested under ion irradiation. In this work, we study the thermodynamic properties of the entire quaternary alloy composition range. Coupling first principles calculations, cluster expansion approaches, and Monte Carlo methods, we access the free energy functionals, short-range ordering as a function of temperature, and atomic configurations that can be compared to experimental observations. We use this data to inform experiments into compositions with higher propensity to form solid solutions, instead of phase separating. With this formalism we have developed thermodynamic database (TDB) files that can be used to plot quaternary phase diagrams.

Cluster Expansion↗