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668 records · Page 4

Tailored Solvent Treatment for Optimized Production of Upcycled Anodes from End-Of-Life Li-Ion Batteries

Recycling processes for lithium-ion batteries typically overlook graphite because of its lower market value relative to that of transition-metal-containing cathode materials. However, graphite recovered from cycled lithium-ion batteries holds additional engineered value associated with the solid-electrolyte interphase (SEI). The SEI contributes critical electronic passivation of the graphite surface but becomes highly resistive with extended cycling, yielding poor cell performance. In this work, we apply tailored solvent treatment to end-of-life (EOL) graphite anodes to selectively remove adverse SEI components while retaining beneficially passivating species. We evaluate a series of polar protic solvents to achieve targeted removal of SEI components and control selectivity through rational variation in solvent properties. The physiochemical properties of treatment solvents correlate with both the retained SEI composition and the corresponding electrochemical performance of solvent-treated “upcycled” graphite anodes. Within the initial set of solvents evaluated, top-performing candidates show capacity and Coulombic efficiency nearly equivalent to those of an analogous pristine anode, as well as promising electrochemical performance enhancement with regard to irreversible capacity-loss metrics. This study establishes critical design principles for an optimized anode upcycling method that enhances the value of recycled graphite by retaining and upgrading the SEI.

25 ENERGY STORAGE

Monitoring Depolymerization in Mesopores Using Dynamic Properties of Polymeric Melt Accessed via Dielectric Spectroscopy

Traditional design principles for heterogeneous catalysis guide the use of catalytic particles with mesosized (∼2–50 nm) pores to increase the number of surface-active sites by way of an increased surface area. However, the entry of long-chain polymers into such pores may be significantly limited by the size and entanglement of polymers in the melt state, thereby decreasing the number of accessible sites. Assessment of catalyst performance from traditional reactor-based studies averages over intrapore reaction events as well as reactions on the surface of a particle, resulting in an inability to distinguish between differences in site accessibility and activity. Techniques that assess the intrapore performance can inform the design of future heterogeneous catalysts for polymer upcycling. In this work, we demonstrate the use of broadband dielectric spectroscopy to monitor depolymerization of a polymer melt within mesopores via changes in the segmental relaxation time scale of amorphous polymer chains. In particular, we highlight the use of an anodic aluminum oxide (AAO) membrane as a readily available model for catalyst pores with a well-characterized pore morphology. The decrease in the segmental relaxation (α-relaxation) time of the melt with increasing chain scission emerges as a measure of the extent of polymer deconstruction inside mesopores. To demonstrate the utility of this technique, we demonstrate the decomposition of two commercial poly(propylene carbonate) polymers with different decomposition rates within mesopores. As the polymers depolymerize, their segmental relaxation time decreases as the molecular weight decreases (as predicted by the Fox–Flory equation). The BDS-measured change in segmental relaxation time mirrors the expected trend based on change in molecular weight measured by size exclusion chromatography.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH

Structural and spectroscopic characterization of a Tb(IV) polyoxometalate

There has been a renaissance in high valent lanthanide chemistry, which has resulted in the first examples of tetravalent praseodymium (Pr) and terbium (Tb) in molecular systems. These feats have been achieved with tailored ligands that facilitate tetravalent lanthanide stability in non-aqueous conditions. The next step in realizing the potential of high valent lanthanide chemistry is moving towards complexes that are stable in ambient and aqueous conditions. In this investigation, we advance this paradigm by obtaining definitive evidence of Tb(IV) in a molecular system under aqueous conditions utilizing the lacunary Wells-Dawson polyoxometalate, K 10 P 2 W 17 O 61 •20H 2 O. Herein we present the single crystal structure of K 16 Tb(IV)(P 2 W 17 O 61 ) 2 •39.85H 2 O (Tb(IV)W 34 ) as well as extensive spectroscopic evidence obtained via UV–Vis-NIR, X-ray absorption near edge spectroscopy, continuous wave X-band electron paramagnetic resonance spectroscopy and SQUID magnetometry measurements to confirm the oxidation state of Tb in W 34 complexes as +4.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH

Phonon-mediated ultrafast dynamics in self-assembled monolayers of 4-mercaptobenzoic acid on gold

Non-equilibrium interactions between plasmonic metals and adsorbed molecules lie at the heart of emerging applications such as plasmonic photocatalysis and sensing, though the ultrafast charge and energy transfer mechanisms arising from these interactions are not well understood. Herein, we investigate the ultrafast dynamics of Au nano-islands tethered with a self-assembled monolayer (SAM) of electron-withdrawing 4-mercaptobenzoic acid (4MBA) molecules. Ultrafast UV-visible transient absorption spectroscopy following excitation of the interband transition in Au reveals three well-known, characteristic time constants that quantify electron–electron (el–el), electron–phonon (el–ph) and phonon–phonon (ph–ph) scattering lifetimes. When comparing the dynamics of bare Au and 4MBA-Au, we find that the el–ph and ph–ph scattering lifetimes are notably longer in 4MBA-Au. Density functional perturbation theory calculations ascribe the elongation in el–ph lifetimes in 4MBA-Au to the significant coupling of acoustic phonon modes of Au with certain molecular vibrations of 4MBA, leading to decreased spatial overlap between carrier electronic states and the acoustic modes. We speculate that the elongation of ph–ph scattering lifetimes in 4MBA-Au arises due to poor thermal conductivity of the SAM which disrupts efficient energy dissipation from Au to the environment, thus slowing down the thermalization of phonons. This work provides a glimpse into how molecular adsorbates modify the charge carrier and phonon dynamics of Au and sets the stage for further systematic exploration of plasmonic metal–molecule interactions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH

Resonant Metasurfaces with Van Der Waals Hyperbolic Nanoantennas and Extreme Light Confinement

This work reports on a metasurface based on optical nanoantennas made of van der Waals material hexagonal boron nitride. The optical nanoantenna made of hyperbolic material was shown to support strong localized resonant modes stemming from the propagating high-k waves in the hyperbolic material. An analytical approach was used to determine the mode profile and type of cuboid nanoantenna resonances. An electric quadrupolar mode was demonstrated to be associated with a resonant magnetic response of the nanoantenna, which resembles the induction of resonant magnetic modes in high-refractive-index nanoantennas. The analytical model accurately predicts the modes of cuboid nanoantennas due to the strong boundary reflections of the high-k waves, a capability that does not extend to plasmonic or high-refractive-index nanoantennas, where the imperfect reflection and leakage of the mode from the cavity complicate the analysis. In the reported metasurface, excitations of the multipolar resonant modes are accompanied by directional scattering and a decrease in the metasurface reflectance to zero, which is manifested as the resonant Kerker effect. Van der Waals nanoantennas are envisioned to support localized resonances and can become an important functional element of metasurfaces and transdimensional photonic components. By designing efficient subwavelength scatterers with high-quality-factor resonances, this work demonstrates that this type of nanoantenna made of naturally occurring hyperbolic material is a viable substitute for plasmonic and all-dielectric nanoantennas in developing ultra-compact photonic components.

Chemistry

Hierarchical Epoxy Structures via Tunable Polymerization-Induced Phase Separation Combined with Additive Manufacturing

Polymerization-induced phase separation (PIPS) allows for the control of thermoset morphologies and properties, enabling the tuning of domain sizes and thermomechanical response. However, its use in generating substructural features in additively manufactured materials has been limited. In this work, we combine epoxy PIPS with UV curable acrylate and rheological modifiers to print nano- to macro-phase separating materials via a two-step, dual-cure approach. This method enables direct ink write printing of hierarchical structures with both controlled morphologies through phase separation and macroscale architecture through print design. We find that formulations for phase-separating materials require judicious incorporation of additives to enable printability and to provide sufficient green strength. Atomic force microscopy-nano infrared mapping reveals tunable, reticulated nano- to micron-scale domains of the resultant multiphase materials and their morphology changes due to additives, resulting in alterations to thermomechanical and tensile properties. Shape memory behavior is also demonstrated through multimaterial additive manufacturing of epoxies with functionally graded internal morphology using active mixing techniques, highlighting this method’s ability to fabricate complex architectures with controlled morphologies and thermomechanical response.

Van Meter, Kylie E [Organic Materials Science, San

Dimensional Evolution Guides Property Control in the A n Cu 4– n TiS 4 Semiconductor Series

Through progressive reduction of the three-dimensional (3D) covalent network of Cu 4 TiS 4 , we isolate seven new members of the A n Cu 4–n TiS 4 family (A = alkali metal; n = 0–4), spanning 3D, 2D, 1D, and 0D structural fragments. The dimensional reduction is rational, as it preserves the edge-sharing connectivity between [CuS 4 ] 7– and [TiS 4 ] 4– tetrahedra across the series. This structural evolution is driven by the stepwise substitution of Cu with alkali metals, guiding the formation of fragments with reduced dimensionality. The effects of “n” and “A” on the crystal structures, stabilities, electronic structures, and optoelectronic properties are profound, demonstrating that the manipulation of alkali metal size and A n Cu 4–n TiS 4 stoichiometry enables predictable variations in structure and properties. For example, the n = 0 and n = 4 end members of the A n Cu 4–n TiS 4 family set the range of achievable band gaps with 2.00 eV for Cu 4 TiS 4 , 2.60 eV for Na 4 TiS 4 , and intermediate values for the n = 1–3 members. Notably, CsCu 3 TiS 4 exhibits exceptional air stability and congruent melting, with density functional theory (DFT) calculating moderate hole and electron effective masses in specific crystallographic directions (mh = 1.24m 0 , me = 0.87m 0 ). Additionally, A 3 CuTiS 4 (A = Na, K, Rb) displays direct band gap behavior and long photoluminescence lifetimes of 2.3–8.6 μs, and K 3 CuTiS 4 has a PLQY of 5.19%. These findings underscore the potential of the A n Cu 4–n TiS 4 family for applications in optoelectronics and demonstrate widely applicable design concepts that unveil rational stoichiometries within a given composition space to generate a series of crystal structures related through an evolving covalent dimensionality that corresponds to a predictable electronic structure and property progression.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH

Direct Observation of Elusive (DTBM‐SEGPHOS)CuH Monomer Enables Mechanistic Insights Into Hydrocupration, Aggregation, and Dynamics of Alkene Functionalization Catalysis

The bulky diphosphine DTBM-SEGPHOS is widely employed in CuH-catalyzed transformations as it provides remarkably active catalyst systems. The transient (DTBM-SEGPHOS)CuH monomer (LCuH) is the often-invoked active species. However, its instability has prevented spectroscopic characterization and mechanistic elucidation, hindering mechanistic understanding. We report low-temperature NMR spectroscopic characterization of LCuH, enabling quantitative kinetic analysis of the stoichiometric hydrocupration and catalytic hydroboration of cyclopentene, as well as the structural identification of two CuH clusters. LCuH inserts cyclopentene at −43°C, reaffirming its high reactivity toward olefins. LCuH deactivates to form L 2 Cu 3 H 3 and L 2 Cu 4 H 4 clusters, in which LCuH dimerization initiates aggregation. Kinetic analysis of reactions of unactivated alkenes indicates that competing on-cycle alkene hydrocupration and LCuH dimerization impact performance, as catalyst deactivation and turnover occur on comparable timescales. Structure–activity analysis using atomistic simulations shows that the steric profile of DTBM-SEGPHOS increases the CuH dimerization barrier by ∼7.7 kcal mol−1 compared to that of SEGPHOS, rationalizing the unique ability of DTBM-SEGPHOS to stabilize a reactive monomer for hydrocupration of broader alkene substrates. These findings illustrate the fundamental design principle that steric control of aggregation governs CuH catalyst performance, explaining both the exceptional activity of (DTBM-SEGPHOS)CuH and the limitations imposed by competing deactivation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH

Understanding Discharge‐Driven Growth of Cathode Impedance in Ni‐Rich NMC Cathodes

Degradation of LiNi x Mn y Co 1-x-y O 2 (NMC)-based lithium-ion batteries depends strongly on cut-off voltage ranges. In addition to the high upper cut-off voltage, a high depth of discharge (i.e., lower cut-off voltage) significantly worsens cathode impedance growth and capacity fade during long-term cycling. However, there is currently no consensus on the mechanism behind the negative role of a deep discharge. Here, this phenomenon was investigated in graphite||NMC cells with single-crystal cathodes (LiNi 0.6 Co 0.2 Mn 0.2 O 2 (NMC622) or LiNi 0.76 Co 0.14 Mn 0.10 O 2 (NMC76)) using targeted aging protocols (constant high-voltage holds vs. charge–discharge cycling), while monitoring transition-metal (TM) dissolution, cathode-electrolyte interface (CEI) impedance, and NMC surface composition. We demonstrate a correlation between discharge-driven CEI impedance growth and increased TM dissolution. Furthermore, this degradation pathway is more pronounced in lower-Ni NMC622 than in higher-Ni (NMC76) under comparable delithiation states at charge, with both compositions undergoing the H2→H3 phase transition. X-ray photoelectron spectroscopy (XPS) reveals NMC composition-dependent evolution of surface lattice oxygen and restructured surface layer composition between charged and discharged states. These findings add mechanistic depth to the role of discharge as an active driver of interfacial degradation and provide new insights into its composition dependence.

25 ENERGY STORAGE

Data-Driven Discovery and Experimental Validation of Solvent Polarity Effects on Conjugated Polymer Solution-to-Film Assembly Pathways

Understanding how solvent properties influence the solution-to-film assembly of conjugated polymers remains a critical challenge due to the complex and intertwined nature of polymer–solvent interactions. In this study, we integrate a data-driven framework with experimental validation to identify key parameters influencing the assembly and performance of poly[2,5-(2-octyldodecyl)-3,6-diketopyrrolopyrrole-alt-5,5-(2,5-di(thien-2-yl)thieno[3,2-b]thiophene)] (DPP-DTT) in organic field-effect transistors (OFETs). A machine learning (ML) approach identified the normalized Reichardt polarity parameter (E T N ) as a significant descriptor correlated with DPP-DTT hole mobility (μ). Systematic DPP-DTT devices fabricated using solvents across a wide E T N range revealed that higher E T N solvents yield enhanced μ. To elucidate the structural origins of high μ, we conducted comprehensive analyses using UV–vis–NIR spectroscopy and grazing incidence wide angle X-ray scattering (GIWAXS) measurements. The results revealed that films processed from high E T N solvents exhibit reduced paracrystallinity. By analyzing the solution-state behavior using optical microscopy and solution WAXS, we revealed polymer solubility differences in the various solvents and associated distinct polymer assembly pathways, elucidating why the high E T N solvent produces long-range ordered films. Notably, the high E T N solvent shows a pronounced preference for liquid-crystal (LC)-mediated assembly, providing a mechanistic explanation for the enhanced structural order. Therefore, these results demonstrate that solvent polarity, as evaluated by E T N , serves as an important parameter that plays a significant role in the DPP-DTT assembly pathway and resultant solid-state morphology. This work provides a strategy for integrating data science with experiments to identify critical parameters associated with complex polymer systems and helps guide rational process design for high-performance organic electronics.

36 MATERIALS SCIENCE

Robust Spectral Anomaly Detection in EELS Spectral Images via 3D Convolutional Variational Autoencoders

Abstract A 3D Convolutional Variational Autoencoder (3D‐CVAE) is introduced for automated anomaly detection in electron energy‐loss spectroscopy spectrum imaging (EELS‐SI) data. This approach leverages the full 3D structure of EELS‐SI data to detect subtle spectral anomalies while preserving both spatial and spectral correlations across the datacube. By employing cross‐entropy loss and training on bulk spectra, the model learns to reconstruct bulk features characteristic of the defect‐free material. In exploring methods for anomaly detection, both the 3D‐CVAE approach and principal component analysis (PCA) are evaluated, testing their performance using FeL‐edge ΔEpeak shifts designed to simulate material defects. These results show that 3D‐CVAE achieves superior anomaly detection and maintains consistent performance across various shift magnitudes. The method demonstrates clear bimodal separation between bulk and anomalous spectra, enabling reliable classification. Further analysis verifies that lower‐dimensional representations are robust to anomalies in the data. While performance advantages over PCA diminish with decreasing anomaly concentration, our method maintains high reconstruction quality even in challenging, noise‐dominated spectral regions. This approach provides a robust framework for unsupervised automated detection of spectral anomalies in EELS‐SI data, particularly valuable for analyzing complex material systems.

Chemistry

Laser-induced graphene gas sensors for environmental monitoring

Artemesia tridentatais a foundational plant taxon in western North America and an important medicinal plant threatened by climate change. Low-cost fabrication of sensors is critical for developing large-area sensor networks for understanding and monitoring a range of environmental conditions. However, the availability of materials and manufacturing processes is still in the early stages, limiting the capacity to develop cost-effective sensors at a large scale. In this study, we demonstrate the fabrication of low-cost flexible sensors using laser-induced graphene (LIG); a graphitic material synthesized using a 450-nm wavelength bench top laser patterned onto polyimide substrates. We demonstrate the effect of the intensity and focus of the incident beam on the morphology and electrical properties of the synthesized material. Raman analyses of the synthesized LIG show a defect-rich graphene with a crystallite size in the tens of nanometers. This shows that the high level of disorder within the LIG structure, along with the porous nature of the material provide a good surface for gas adsorption. The initial characterization of the material has shown an analyte response represented by a change in resistance of up to 5% in the presence of volatile organic compounds (VOCs) that are emitted and detected byArtemisiaspecies. Bend testing up to 100 cycles provides evidence that these sensors will remain resilient when deployed across the landscapes to assess VOC signaling in plant communities. The versatile low-cost laser writing technique highlights the promise of low-cost and scalable fabrication of LIG sensors for gas sensor monitoring.

Chemistry

Crack‐Free Single‐Crystalline LiNiO 2 for High Energy Density All‐Solid‐State Batteries

Abstract Single‐crystalline layered oxide (LiNi 1‐ x ‐ y Mn x Co y O 2 ) cathodes have been found to exhibit exceptional electrochemical properties when coupled with various inorganic solid electrolytes (ISEs) in all‐solid‐state batteries (ASSBs). Their advantages stem from the robust morphological integrity with the absence of grain boundaries and the high electrochemical oxidative stability. Here, ASSBs featuring single‐crystalline LiNiO 2 (LNO) with the highest Ni content are reported, offering a high theoretical specific capacity of 275 mAh g ‐1 alongside a high average discharge voltage (3.7 V vs Li + /Li). Through a careful investigation, it is demonstrated that micron‐sized single‐crystalline LNO (µSC‐LNO) composite cathodes with a halide ISE exhibit a high initial discharge capacity of 205 mAh g ‐1 with an outstanding cycle performance over 200 cycles in room‐temperature ASSBs. The significance of engineering parameters is emphasized, such as particle size and specific density, in promoting a homogeneous and fast Li + transport within the composite cathodes. Furthermore, the formation of undesirable interphase between the halide ISE in the cathode and sulfide ISE separator is elucidated, which may be a critical factor impeding long‐term cyclability of ASSBs. This work provides insights into the design of composite cathodes for high‐energy‐density ASSBs.

Chemistry

Glancing Angle Deposition in Gas Sensing: Bridging Morphological Innovations and Sensor Performances

Glancing Angle Deposition (GLAD) has emerged as a versatile and powerful nanofabrication technique for developing next-generation gas sensors by enabling precise control over nanostructure geometry, porosity, and material composition. Through dynamic substrate tilting and rotation, GLAD facilitates the fabrication of highly porous, anisotropic nanostructures, such as aligned, tilted, zigzag, helical, and multilayered nanorods, with tunable surface area and diffusion pathways optimized for gas detection. This review provides a comprehensive synthesis of recent advances in GLAD-based gas sensor design, focusing on how structural engineering and material integration converge to enhance sensor performance. Key materials strategies include the construction of heterojunctions and core–shell architectures, controlled doping, and nanoparticle decoration using noble metals or metal oxides to amplify charge transfer, catalytic activity, and redox responsiveness. GLAD-fabricated nanostructures have been effectively deployed across multiple gas sensing modalities, including resistive, capacitive, piezoelectric, and optical platforms, where their high aspect ratios, tailored porosity, and defect-rich surfaces facilitate enhanced gas adsorption kinetics and efficient signal transduction. These devices exhibit high sensitivity and selectivity toward a range of analytes, including NO2, CO, H2S, and volatile organic compounds (VOCs), with detection limits often reaching the parts-per-billion level. Emerging innovations, such as photo-assisted sensing and integration with artificial intelligence for data analysis and pattern recognition, further extend the capabilities of GLAD-based systems for multifunctional, real-time, and adaptive sensing. Finally, current challenges and future research directions are discussed, emphasizing the promise of GLAD as a scalable platform for next-generation gas sensing technologies.

Chemistry

Utilization of a bipolar lead acid battery for the advanced launch system

The development of a battery comprised of bipolar lead acid modules is discussed. The battery is designed to satisfy the requirements of the Advanced Launch System (ALS). The battery will have the following design features: (1) conventional lead acid chemistry; (2) thin electrode/active materials; (3) a thin separator; (4) sealed construction (gas recombinant); and (5) welded plastic frames for the external seal.

Gentry, William O.

Super-resolution imaging reveals resistance to mass transfer in functionalized stationary phases

Chemical separations are costly in terms of energy, time, and money. Separation methods are optimized with inefficient trial-and-error approaches that lack insight into the molecular dynamics that lead to the success or failure of a separation and, hence, ways to improve the process. We perform super-resolution imaging of fluorescent analytes in five different commercial liquid chromatography materials. Unexpectedly, we observe that chemical functionalization can block more than 50% of the material’s porous interior, rendering it inaccessible to small-molecule analytes. Only in situ imaging unveils the inaccessibility when compared to the industry-accepted ex situ characterization methods. Selectively removing some of the functionalization with solvent restores pore access without substantially altering the single-molecule kinetics that underlie the separation and agree with bulk chromatography measurements. Our molecular results determine that commercial “fully porous” stationary phases are over-functionalized and provide an alternative avenue to characterize and direct separation material design from the bottom-up.

Science & Technology - Other Topics

The Role of Asparagine as a Gatekeeper Residue in the Selective Binding of Rare Earth Elements by Lanthanide‐Binding Peptides

Abstract Lanthanide‐binding tag (LBT) peptides selectively complex lanthanide cations (Ln 3+ ) in their binding pockets and are promising for lanthanide separation. However, designing LBTs that selectively target specific Ln 3+ cations remains a challenge due to limited molecular‐level understanding and control of interactions within the lanthanide‐binding pocket. In this study, we reveal that the N5 asparagine residue acts as a gatekeeper in the binding pocket, resulting in a 100‐fold selectivity for smaller Lu 3+ over larger La 3+ cations. Nuclear magnetic resonance spectroscopy and molecular dynamics simulations show that the N5 residue weakly binds to the larger La 3+ cation, permitting H 2 O molecules inside the pocket. For the smaller Lu 3+ cations, the N5 residue forms an inter‐arm hydrogen bond with the E14 glutamic acid residue, locking the Lu 3+ cation in the pocket and preventing H 2 O infiltration. Mutating the N5 asparagine to a D5 aspartic acid prevents such a hydrogen bond, eliminating the gatekeeping mechanism and precipitously reducing selectivity. The resulting binding affinity to Ln 3+ cations is non‐monotonic but generally increases with cation size. These results suggest a molecular design paradigm: the reduced affinity for larger lanthanides is due to open pocket conformations, while the selectivity of smaller Ln 3+ cations over larger ones is due to the gatekeeping hydrogen bond.

Chemistry

Validation of the SINDA/FLUINT Code Using Several Analytical Solutions

The Systems Improved Numerical Differencing Analyzer and Fluid Integrator (SINDA/FLUINT) code has often been used to determine the transient and steady-state response of various thermal and fluid flow networks. While this code is an often used design and analysis tool, the validation of this program has been limited to a few simple studies. For the current study, the SINDA/FLUINT code was compared to four different analytical solutions. The thermal analyzer portion of the code (conduction and radiative heat transfer, SINDA portion) was first compared to two separate solutions. The first comparison examined a semi-infinite slab with a periodic surface temperature boundary condition. Next, a small, uniform temperature object (lumped capacitance) was allowed to radiate to a fixed temperature sink. The fluid portion of the code (FLUINT) was also compared to two different analytical solutions. The first study examined a tank filling process by an ideal gas in which there is both control volume work and heat transfer. The final comparison considered the flow in a pipe joining two infinite reservoirs of pressure. The results of all these studies showed that for the situations examined here, the SINDA/FLUINT code was able to match the results of the analytical solutions.

John R Keller