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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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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Determination of the Structures of Lignin Subunits and Nanoparticles in Solution by Small‐Angle Neutron Scattering: Towards Improving Lignin Valorization

Abstract Lignin nanoparticles (LNPs) are usually produced from lignin solution through supersaturation. The structure of the lignin in solution is still poorly understood due to structural variability of isolated lignins. Here, lignins were extracted from different plants to establish a general pattern of their structure in several lignin solvents. Lignin molecules (lignin subunits) and larger aggregates were observed in dimethyl sulfoxide (DMSO), ethylene glycol (EG) and 0.1 N NaOD solutions by small‐angle neutron scattering (SANS). It was proposed that the aggregates were composed of lignin subunits with a higher molecular weight and a higher ratio of the aliphatic to phenolic hydroxyl groups. The size, shape, and compactness are important factors that affect the uses of the LNPs, which were obtained from the SANS data for the first time. A discrepancy in the radius between SANS and DLS was discovered, pointing to a large hydration shell around the LNPs in aqueous solutions. The cytotoxicity of the corncob lignin, kraft lignin, and their LNPs were measured and compared.

Zhang, Xin↗
DOE OSTI2023

Aqueous processing of organic semiconductors enabled by stable nanoparticles with built-in surfactants

The introduction of oligoether side chains onto a polymer backbone can help to stabilise polymeric dispersions in water without the necessity of surfactants or additives when conjugated polymer nanoparticles are prepared. A series of poly(3-hexylthiophene) (P3HT) derivatives with different content of a polar thiophene derivative 3-((2-methoxyethoxy)methyl)thiophene was interrogated to find the effect of the polar chains on the stability of the formed nanoparticles, as well as their structural, optical, electrochemical, and electrical properties. Findings indicated that incorporation of 10–20 percent of the polar side chain led to particles that are stable over a period of 42 days, with constant particle size and polydispersity, however the particles from the polymer with 30 percent polar side chain showed aggregation effects. The polymer dispersions showed a stronger solid-like behaviour in water with decreasing polar side chain content, while thin film deposition from water was found to afford globular morphologies and crystallites with more isotropic orientation compared to conventional solution-processed films. As a proof-of-principle, field-effect transistors were fabricated directly from the aqueous dispersions demonstrating that polymers with hydrophilic moieties can be processed in water without the requirement of surfactants.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Self-similar mesocrystals form via interface-driven nucleation and assembly

Crystallization by particle attachment (CPA) is a frequently occurring mechanism of colloidal crystallization that results in hierarchical morphologies. CPA has been exploited to create nanomaterials with unusual properties and is implicated in the development of complex mineral textures. Oriented attachment–a form of CPA in which particles align along specific crystallographic directions–produces mesocrystals that diffract as single crystals do, although the constituent particles are still discernible. The conventional view of CPA is that nucleation provides a supply of particles that aggregate via Brownian motion biased by attractive interparticle potentials. However, mesocrystals often exhibit regular morphologies and uniform sizes. Although many crystal systems form mesocrystals and individual attachment events have been directly visualized10, how random attachment events lead to well defined, self-similar morphologies remains unknown, as does the role of surface-bound ligands, which are ubiquitous in nanoparticle systems. Attempts to understand mesocrystal formation are further complicated in many systems by the presence of precursor nanoparticles with a phase distinct from that of the bulk. Some studies propose that such particles convert before attachment, whereas others attribute conversion to the attachment process itself and yet others conclude that transformation occurs after the mesocrystals exceed a characteristic size. Here we investigate mesocrystal formation by iron oxides, which are important colloidal phases in natural environments and classic examples of systems forming ubiquitous precursor phases and undergoing CPA accompanied by phase transformations. Combining in situ transmission electron microscopy (TEM) at 80 degrees Celsius with ‘freeze-and-look’ TEM, we tracked the formation of haematite (Hm) mesocrystals in the presence of oxalate (Ox), which is abundant in soils, where iron oxides are common. In this work, we find that isolated Hm particles rarely appear, but once formed, interfacial gradients at the Ox-covered surfaces drive Hm particles to nucleate repeatedly about two nanometres from the surfaces, to which they then attach, thereby generating mesocrystals. Comparison to natural and synthetic systems suggests that interface-driven pathways are widespread.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Surface structure of mass-selected niobium oxide nanoclusters on Au(111)

The structures formed by the deposition of mass-selected niobium oxide clusters, Nb 3 O y ( y = 5, 6, 7), onto Au(111) were studied by scanning tunneling microscopy. The as-deposited Nb 3 O 7 clusters assemble into large dendritic structures that grow on the terraces as well as extend from the top and bottom of step edges. The Nb 3 O 6 cluster also forms dendritic assemblies but they are generally much smaller in size. The assemblies are composed of smaller discrete structures (<1 nm) which are likely to be single clusters. Furthermore, the dendritic assemblies for both the Nb 3 O 7 and Nb 3 O 6 clusters have fractal dimensions of about 1.7 which is very close to that expected for simple diffusion limited aggregation. Annealing the Nb 3 O 7,6 /Au(111) surfaces up to 550 K results in changes in assembly sizes and increases in heights, while heating to 700 results in the disruption of the assemblies into smaller structures. By contrast, the as-deposited Nb 3 O 5 /Au(111) surface at RT exhibits compact cluster structures which become 3D nanoparticles when annealed above 550 K. Differences in the observed surface structures and thermal stability are attributed to differences in metal-oxygen stoichiometry which can influence cluster binding energies, mobility and inter-cluster interactions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

The influence of temperature history and flow mixing on the vapor-phase speciation of uranium oxide nanoparticles

The transport of radioactive material following a nuclear event depends heavily on fallout characteristics (e.g., composition, shape, and size), which in turn are controlled by the conditions under which the fallout forms. To this end, we use a novel plasma flow reactor (PFR) to explore how conditions including temperature history and gas mixing influence the chemical speciation and physical characteristics of uranium oxide nanoparticles. The PFR consists of a glass tube attached to an inductively coupled plasma (ICP) torch via an adaptor piece (ring flow injector) located downstream of the plasma-generating RF coil. We increase the gas flow rates from normal ICP operating conditions to higher values both upstream at the RF coil and downstream at the ring flow injector to create four distinct temperature profiles and two distinct gas flow mixing conditions (normal and higher Ar atomic ratios). Ex situ transmission electron microscopy (TEM) analysis is used to determine the crystal structure, morphology, and size distribution of the synthesized particles as a function of temperature and mixing. The particles display a striking dependence on these conditions in terms of the resulting speciation of uranium oxide: lower temperature (higher upstream) and slower cooling (normal downstream with normal Ar atomic ratio mixing) favor the formation of α-UO 3 particles, while faster cooling (higher downstream with higher Ar atomic ratio mixing) and higher temperature (normal upstream) favor the formation of UO 2 particles. The extent of gas mixing dictates particle aggregation: better mixing encourages the particles to aggregate more extensively. Here these results demonstrate the sensitivity of uranium to initial local conditions and can be used to improve our understanding of the kinetics that drive uranium speciation and ultimately inform fallout transport models.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2020

Collaborative Research: Understanding Nanoparticle-Plasma Interactions in Dusty Non-Thermal Plasmas by Nanoparticle Probes and Online Aerosol Characterization (Final Technical Report)

This report describes results from the University of Minnesota’s contribution to studies performed under the award DE-SC0018202 (principal Investigator: Christopher J. Hogan Jr.. This award is a collaborative research grant with support equally provided to Case Western Reserve University and to the University of Illinois (with principal investigator R. Mohan Sankaran). To date ten peer-reviewed published papers have resulted from the studies performed with University of Minnesota personnel contributing; this report describes the results of these studies as well as several unpublished studies, grouped into seven separate study types. The overarching goal of the carried out studies was to understand nanoparticle behavior and dynamics in non-thermal plasma synthesis reactors, specifically adapting aerosol measurement technology to provide online measurements of particle size, charge, and structure in plasma reactor systems.

42 ENGINEERING↗
DOE OSTI2020

Sub‐5 nm Intermetallic Nanoparticles Confined in Mesoporous Silica Wells for Selective Hydrogenation of Acetylene to Ethylene

Abstract We report the synthesis and catalytic application of intermetallic PtSn nanoparticles confined in MSWs (Mesoporous Silica Wells) in the semi‐hydrogenation of acetylene. Platinum nanoparticles with a size of 2.9 nm supported on amine‐functionalized ∼180 nm silica spheres were enmeshed in a mesoporous silica shell. After the addition of tin via a polyol synthesis method, we obtained the final PtSn x @MSW (x=0.30, 0.50, 0.70, 1.0 and 2.0). Owing to the mesoporous encapsulation the sub‐5 nm bimetallic particles are shown to be monodisperse and well protected against aggregation on the silica sphere even under high‐temperature treatment conditions. In the gas phase semi‐hydrogenation of acetylene, the PtSn@MSW with Sn/Pt molar ratios equal to or greater than one displayed higher selectivity to ethylene. Meanwhile, increasing Sn content decreases acetylene conversion. The intermetallic PtSn@MSW catalysts also displayed long‐term stability and regenerability due to the confined environment provided by MSW.

Maligal‐Ganesh, Raghu V.↗
DOE OSTI2020

Capture of Iodine from Nuclear-Fuel-Reprocessing Off-Gas: Influence of Aging on a Reduced Silver Mordenite Adsorbent after Exposure to NO/NO 2

Iodine radioisotopes released during nuclear fuel reprocessing must be removed from the off-gas stream before discharge. One promising material for iodine capture is reduced silver mordenite (Ag 0 Z). Nevertheless, the adsorbent’s capacity will degrade, or age, over time when the material is exposed to other off-gas constituents. Though the overall impact of aging is known, the underlying physical and chemical processes are not. To examine these processes, Ag 0 Z samples were prepared and aged in 2% NO 2 in dry air and in 1% NO in N 2 gas streams at 150 °C for up to six months. Aged samples were then characterized using scanning electron microscopy, X-ray diffraction, Raman spectroscopy, and X-ray absorption spectroscopy. These techniques show that aging involves two overarching processes: (i) oxidation of the silver nanoparticles present in Ag 0 Z and (ii) migration of oxidized silver into the mordenite’s inner network. Silver on the nanoparticle’s surface is oxidized through adsorption of O 2 , NO, and NO 2 . Additionally, Raman spectroscopy and X-ray absorption spectroscopy indicate that nitrates are the primary products of this adsorption. Most of these nitrates migrate into the interior of the mordenite and exchange at framework binding sites, returning silver to its unreduced state (AgZ). The remaining nitrates exist at a persistent concentration without aggregating into bulk-phase AgNO 3 . X-ray absorption spectroscopy results further indicate that iodine adsorption occurs on not just Ag 0 Z but also on AgZ and a portion of the nitrates in the system. AgZ adsorbs a sizable quantity of iodine early in the aging process, but its capacity drops rapidly over time. For well-aged samples, nitrates are responsible for up to 95% of mordenite’s iodine capacity. Overall, these results have enhanced our understanding of the aging process in silver mordenite and are expected to guide the development of superior adsorbents for the capture of radioactive iodine from reprocessing off-gas.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2024

Random forest prediction of crystal structure from electron diffraction patterns incorporating multiple scattering

Diffraction is the most common method to solve for unknown or partially known crystal structures. However, it remains a challenge to determine the crystal structure of a new material that may have nanoscale size or heterogeneities. Here, in this study, we train an architecture of hierarchical random forest models capable of predicting the crystal system, space group, and lattice parameters from one or more unknown two-dimensional electron diffraction patterns. Our initial model correctly identifies the crystal system of a simulated electron diffraction pattern from a 20-nm-thick specimen of arbitrary orientation 67% of the time. We achieve a topline accuracy of 79% when aggregating predictions from ten patterns of the same material but different zone axes. The space group and lattice predictions range from 70% to 90% accuracy and median errors of 0.01-0.5Å, respectively, for cubic, hexagonal, trigonal, and tetragonal crystal systems while being less reliable on orthorhombic and monoclinic systems. We apply this architecture to a four-dimensional scanning transmission electron microscopy scan of gold nanoparticles, where it accurately predicts the crystal structure and lattice constants. These random forest models can be used to significantly accelerate the analysis of electron diffraction patterns, particularly in the case of unknown crystal structures. Additionally, due to the speed of inference, these models could be integrated into live transmission electron microscopy experiments, allowing real-Time labeling of a specimen.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Energetic decomposition yields efficient bimetallic Cu MOF-derived catalysts

Metal–organic frameworks (MOFs) have recently emerged as efficient self-sacrificial templates to fabricate porous carbon-supported metal nanoparticles (NPs). Due to observed increased activity, catalysts containing bimetallic NPs represent an active frontier for heterogeneous catalyst development. A strategy to synthesize active catalysts with highly dispersed bimetallic metal/metal oxides within a porous carbon matrix via rapid MOF decomposition using dopants is presented in this paper. A 2,4,6-trinitrotoluene (TNT) additive enhances the rapid thermolysis of the metal-doped Cu-based MOFs to minimize particle aggregation. Catalyst characterization reveals that a secondary metal increases dispersion of both metals over the carbon composite support. The catalyst preparation method influences both the metal particle size and oxidation state. Catalytic performance shows increased rates for 4-nitrophenol reduction even with <1 wt% of added secondary metal. Among the synthesized catalysts, the Ni–CuO@C bimetallic catalyst exhibits outstanding activity. This synthetic strategy is useful for creating highly efficient, robust, non-noble metal catalysts for development of sustainable chemical processes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Environmental controls on the kinetics of iron-sulfur cluster nucleation and nanoparticle formation

Anoxic, sulfidic conditions have been prevalent since the early Proterozoic and favor aqueous iron-sulfur (FeS aq ) clusters as a major fraction of the soluble, reduced iron and sulfur pool. FeS aq cluster formation and nucleation is driven by the high affinity between ferrous iron (Fe(II)) and sulfide (HS − ), ultimately yielding particles that precipitate as iron sulfide minerals. FeS aq clusters were recently shown to be bioavailable sources of iron and sulfur for a variety of anaerobes, yet little is known of the factors that influence the kinetics of their formation and nucleation. Here we apply computational and spectroscopic approaches to investigate the dynamics of FeS aq nucleation, cluster growth, precipitation, and redissolution as a function of Fe(II)/HS − concentration, temperature, and pH. Experiments were conducted under excess HS − to mimic euxinic conditions common to contemporary anaerobic aquatic ecosystems and those of the Proterozoic. Density functional theory calculations reveal the key role of water oxygen-iron interactions in stabilizing small FeS aq clusters and promoting solubility. Dynamic light scattering revealed a concentration-dependent increase in the kinetics of FeS aq nucleation and cluster aggregation. Increasing temperature promoted FeS aq cluster nucleation and aggregation while also enhancing dissolution. Alkaline pH also promoted FeS aq nucleation and cluster aggregation. At 25 °C, pH 7.0, and at reactant concentrations of 30 µM, FeS aq clusters < 10 nm in diameter remained in solution for > 2 h. These results underscore the importance of temperature, pH, and reactant concentration in the kinetics of FeS aq nucleation and cluster growth that, in turn, influence their bioavailability in anaerobic ecosystems.

Aquatic ecosystems↗
DOE OSTI2021

Semi-grand canonical Monte Carlo simulation of the acrolein induced surface segregation and aggregation of AgPd with machine learning surrogate models

The single atom alloy of AgPd has been found to be a promising catalyst for the selective hydrogenation of acrolein. It is also known that the formation of Pd islands on the surface will greatly reduce the selectivity of the reaction. As a result, the surface segregation and aggregation of Pd on the AgPd surface under reaction conditions of selective hydrogenation of acrolein are of great interest. In this work, we lay out a workflow that can predict the surface segregation and aggregation of Pd on a FCC(111) AgPd surface with and without the presence of acrolein. We use machine learning surrogate models to predict the AgPd bulk energy, AgPd slab energy, and acrolein adsorption energy on AgPd slabs. Then, we use the semi-grand canonical Monte Carlo simulation to predict the surface segregation and aggregation under different bulk Pd concentrations. Under vacuum conditions, our method predicts that only trace amount of Pd will exist on the surface at Pd bulk concentrations less than 20%. However, with the presence of acrolein, Pd will start to aggregate as dimers on the surface at Pd bulk concentrations as low as 6.5%.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

From Powder to Power: Tailored Pre-Milling Strategy that Optimizes Microstructure for Efficient Hydrogen Evolution

The hydrogen evolution reaction (HER) plays a critical role in enabling large-scale electrolytic hydrogen production and advancing future technologies and fuel production. Among non-precious metals, NiMo-based catalysts are particularly attractive due to their capability to promote both water dissociation and hydrogen adsorption in alkaline media. However, conventional NiMo catalysts often suffer from incomplete alloying, particle aggregation, weak metal–support interactions, and surface oxidation, which significantly limit active site utilization, electronic conductivity, and long-term stability. Herein, we report a scalable, solid-state, two-step ball milling strategy for constructing an efficient NiMo/C catalyst for the HER. Pre-milling the metal precursors prior to carbon incorporation promotes efficient solid-state activation, leading to the formation of uniformly alloyed, ultrafine, and defect-rich Ni–Mo nanoparticles with robust metal–carbon interfacial anchoring. Benefiting from this integrated structural design, the optimized NiMo/C catalyst exhibits low overpotentials and Tafel slopes, reduced charge-transfer resistance, and excellent durability under alkaline conditions. Comprehensive structural and surface analysis reveal that the enhanced HER performance can be attributed to the synergistic interplay of homogeneous Ni–Mo alloying, uniform nanoparticle dispersion on a defect-rich carbon scaffold, enhanced accessibility of catalytically active sites, and optimized electronic coupling that stabilizes the catalyst surface. This work highlights the two-step solid-state ball milling strategy as a simple, robust, and scalable route for the preparation of effective and durable non-precious metal HER electrocatalysts, offering practical insights toward large-scale and inexpensive hydrogen production.

Yang, Xiaoxuan [Oak Ridge National Laboratory (ORN↗
DOE OSTI2020

Growth and structural studies of In/Au(111) alloys and InO x /Au(111) inverse oxide/metal model catalysts

Indium oxide has received attention as an exciting candidate for catalyzing the CO 2 hydrogenation to methanol due to its high selectivity (>80%). Compared to the extent of research on the activity of indium oxide-based powder catalysts, very little is known about the phenomena associated with the formation of surface alloys involving indium or the growth mechanism for indium oxide nanoparticles. In this report, scanning tunneling microscopy (STM) and X-ray photoelectron spectroscopy (XPS) were employed to elucidate the growth mode, structure, and chemical state of In/Au(111) alloys and InO x /Au(111) inverse model catalysts. Here, our study reveals distinct morphological differences between In/Au(111) and InO x /Au(111), and the InO x structure also depends strongly on the preparation conditions. In/Au surface alloy systems with extremely low coverage (0.02 ML) form islands preferentially on the elbow sites of reconstructed Au(111) herringbone, regardless of hcp and fcc stacking. At higher coverage (0.1 ML), the In islands expand over the herringbone in the <110> direction and create two dimensional domain structures over the entire surfaces. Moreover, this 2D domain structure is disturbed by temperature with high dispersion of indium atoms observed during the annealing process. Oxidation of the In/Au(111) surface alloys with O 2 at 550 K produces InO x /Au(111) systems which contain various sizes of InO x aggregates (from 0.7 to 10 nm). On the other hand, InO x /Au(111) surfaces prepared by vapor deposition of In at 550 K in an O 2 background exhibit highly dispersed and uniformly small InO x particles (~1 nm). Both InO x systems were confirmed to be partially oxidized by XPS.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Combustion synthesis of Eu 2 O 3 nanomaterials with tunable phase composition and morphology

Combustion reactions in europium nitrate – acetylacetone – 2-methoxyethanol solutions and gels were investigated to produce europium (III) oxide (Eu 2 O 3 ) nanocrystalline materials and thin films. Thermal analyses of solutions indicated that the 2-methoxyethanol solvent also acts as a fuel in the absence of acetylacetone. Adding acetylacetone increases the overall heat of the reaction. Thermal analysis results revealed that the slow oxidation of unburned hydrocarbon residues follows the primary combustion reaction. Time-temperature profile measurements of the bulk combustion synthesis process in air and nitrogen atmospheres enabled the extraction of the maximum reaction temperature and the heating and cooling rates in the combustion zone. Several direct correlations exist between measured combustion parameters and the phase composition of the products. Combustion in air results in mixed-phase cubic and monoclinic nanocrystalline Eu 2 O 3 . Increasing the acetylacetone concentration in solutions increases the synthesis temperature and decreases the quantity of cubic Eu 2 O 3 . Here, the reaction of solutions in a nitrogen atmosphere or diluted with Eu 2 O 3 provides control of the product phase composition and reduces the quantity of the monoclinic phase. Transmission electron microscopy imaging shows that the Eu 2 O 3 end products are highly porous aggregates of nanocrystalline particles. Electrospraying of reactive solutions onto different substrates followed by short annealing makes the preparation of Eu 2 O 3 materials with diverse morphologies possible.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS2000

Metal Nanoparticle Aerogel Composites

We have fabricated sol-gels containing gold and silver nanoparticles. Formation of an aerogel produces a blue shift in the surface plasmon resonance as a result of the decrease in the dielectric constant of the matrix upon supercritical extraction of the solvent. However, as a result of chemical interface damping this blue shift does not obey effective medium theories. Annealing the samples in a reducing atmosphere at 400 C eliminates this discrepancy and results in narrowing and further blue shifting of the plasmon resonance. Metal particle aggregation also results in a deviation from the predictions of effective medium theories, but can be controlled through careful handling and by avoiding the use of alcohol. By applying effective medium theories to the heterogeneous interlayer surrounding each metal particle, we extend the technique of immersion spectroscopy to inhomogeneous materials characterized by spatially dependent dielectric constants, such as aerogels. We demonstrate that the shift in the surface plasmon wavelength provides the average fractional composition of each component (air and silica) in this inhomogeneous layer, i.e. the porosity of the aerogel or equivalently, for these materials, the catalytic dispersion. Additionally, the kinetics suggest that collective particle interactions in coagulated metal clusters are perturbed during silica gelation resulting in a change in the aggregate geometry.

Smith, David D.↗
DOE OSTI2024

Reexpansion of charged nanoparticle assemblies in concentrated electrolytes

Electrostatic forces in solutions are highly relevant to a variety of fields, ranging from electrochemical energy storage to biology. However, their manifestation in concentrated electrolytes is not fully understood, as exemplified by counterintuitive observations of colloidal stability and long-ranged repulsions in molten salts. Highly charged biomolecules, such as DNA, respond sensitively to ions in dilute solutions. Here, we use non-base-pairing DNA-coated nanoparticles (DNA-NP) to analyze electrostatic interactions in concentrated salt solutions. Despite their negative charge, these conjugates form colloidal crystals in solutions of sufficient divalent cation concentration. We utilize small-angle X-ray scattering (SAXS) to study such DNA-NP assemblies across the full accessible concentration ranges of aqueous CaCl 2 , MgCl 2 , and SrCl 2 solutions. SAXS shows that the crystallinity and phases of the assembled structures vary with cation type. For all tested salts, the aggregates contract with added ions at low salinities and then begin expanding above a cation-dependent threshold salt concentration. Wide-angle X-ray scattering (WAXS) reveals enhanced positional correlations between ions in the solution at high salt concentrations. Complementary molecular dynamics simulations show that these ion–ion interactions reduce the favorability of dense ion configurations within the DNA brushes below that of the bulk solution. Measurements in solutions with lowered permittivity demonstrate a simultaneous increase in ion coupling and decrease in the concentration at which aggregate expansion begins, thus confirming the connection between these phenomena. Our work demonstrates that interactions between charged objects continue to evolve considerably into the high-concentration regime, where classical theories project electrostatics to be of negligible consequence.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗