976 resultados para AMORPHOUS MOLECULAR MATERIALS


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A new solid state organometallic route to embedded nanoparticle-containing inorganic materials is shown, through pyrolysis of metal-containing derivatives of cyclotriphosphazenes. Pyrolysis in air and at 800 °C of new molecular precursors gives individual single-crystal nanoparticles of SiP2O7, TiO2, P4O7, WP2O7 and SiO2, depending on the precursor used. High resolution transmission electron microscopy investigations reveal, in most cases, perfect single crystals of metal oxides and the first nanostructures of negative thermal expansion metal phosphates with diameters in the range 2–6 nm for all products. While all nanoparticles are new by this method, WP2O7 and SiP2O7 nanoparticles are reported for the first time. In situ recrystallization formation of nanocrystals of SiP2O7 was also observed due to electron beam induced reactions during measurements of the nanoparticulate pyrolytic products SiO2 and P4O7. The possible mechanism for the formation of the nanoparticles at much lower temperatures than their bulk counterparts in both cases is discussed. Degrees of stabilization from the formation of P4O7 affects the nanocrystalline products: nanoparticles are observed for WP2O7, with coalescing crystallization occurring for the amorphous host in which SiP2O7 crystals form as a solid within a solid. The approach allows the simple formation of multimetallic, monometallic, metal-oxide and metal phosphate nanocrystals embedded in an amorphous dielectric. The method and can be extended to nearly any metal capable of successful coordination as an organometallic to allow embedded nanoparticle layers and features to be deposited or written on surfaces for application as high mobility pyrophosphate lithium–ion cathode materials, catalysis and nanocrystal embedded dielectric layers.

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The synthesis and characterization of new organosilicon derivatives of N3P3Cl6, N3P3[NH(CH2)3Si(OEt)3]6 (1), N3P3[NH(CH2)3Si(OEt)3]3[NCH3(CH2)3CN]3 (2), and N3P3[NH(CH2)3Si(OEt)3]3[HOC6H4(CH2)CN]3 (3) are reported. Pyrolysis of 1, 2, and 3 in air and at several temperatures results in nanostructured materials whose composition and morphology depend on the temperature of pyrolysis and the substituents of the phosphazenes ring. The products stem from the reaction of SiO2 with P2O5, leading to either crystalline Si5(PO4)6O, SiP2O7 or an amorphous phase as the glass Si5(PO4)6O/3SiO2·2P2O5, depending on the temperature and nature of the trimer precursors. From 1 at 800 °C, core−shell microspheres of SiO2 coated with Si5(PO4)6O are obtained, while in other cases, mesoporous or dense structures are observed. Atomic force microscopy examination after deposition of the materials on monocrystalline silicon wafers evidences morphology strongly dependent on the precursors. Isolated islands of size ∼9 nm are observed from 1, whereas dense nanostructures with a mean height of 13 nm are formed from 3. Brunauer−Emmett−Teller measurements show mesoporous materials with low surface areas. The proposed growth mechanism involves the formation of cross-linking structures and of vacancies by carbonization of the organic matter, where the silicon compounds nucleate. Thus, for the first time, unique silicon nanostructured materials are obtained from cyclic phosphazenes containing silicon.

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The thesis primarily reports the synthesis, characterization and application of novel mixed mode stationary phases for Hydrophilic Interaction Liquid Chromatography (HILIC). HILIC is a rapidly emerging chromatographic mode that is finding great applicability in the analysis of polar organic molecules. In addition, there is a chapter on the analysis of Bisphenol A and related species using capillary electrophoresis (CE) coupled with boron-doped diamond electrodes for electrochemical detection. The synthesis and characterization of the novel mixed mode stationary phases prepared in this work is an important contribution to the field as the materials prepared exhibited better performance than similar materials obtained commercially. In addition a more thorough characterization of the materials (e.g.,thermogravimetric analysis, various NMR modes, elemental analysis, etc.) and resulting columns (e.g., H) than is typically encountered. The application of these new materials to the analysis of sugars using evaporative light scattering is also novel. In CE studies, electrochemical detection is sufficiently rare that the work is also novel.

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In this paper, we use a model of hydrogenated amorphous silicon generated from molecular dynamics with density functional theory calculations to examine how the atomic geometry and the optical and mobility gaps are influenced by mild hydrogen oversaturation. The optical and mobility gaps show a volcano curve as the hydrogen content varies from undersaturation to mild oversaturation, with largest gaps obtained at the saturation hydrogen concentration. At the same time, mid-gap states associated with dangling bonds and strained Si-Si bonds disappear at saturation but reappear at mild oversaturation, which is consistent with the evolution of optical gap. The distribution of Si-Si bond distances provides the key to the change in electronic properties. In the undersaturation regime, the new electronic states in the gap arise from the presence of dangling bonds and strained Si-Si bonds, which are longer than the equilibrium Si-Si distance. Increasing hydrogen concentration up to saturation reduces the strained bonds and removes dangling bonds. In the case of mild oversaturation, the mid-gap states arise exclusively from an increase in the density of strained Si-Si bonds. Analysis of our structure shows that the extra hydrogen atoms form a bridge between neighbouring silicon atoms, thus increasing the Si-Si distance and increasing disorder in the sample.

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The coupling of mechanical stress fields in polymers to covalent chemistry (polymer mechanochemistry) has provided access to previously unattainable chemical reactions and polymer transformations. In the bulk, mechanochemical activation has been used as the basis for new classes of stress-responsive polymers that demonstrate stress/strain sensing, shear-induced intermolecular reactivity for molecular level remodeling and self-strengthening, and the release of acids and other small molecules that are potentially capable of triggering further chemical response. The potential utility of polymer mechanochemistry in functional materials is limited, however, by the fact that to date, all reported covalent activation in the bulk occurs in concert with plastic yield and deformation, so that the structure of the activated object is vastly different from its nascent form. Mechanochemically activated materials have thus been limited to “single use” demonstrations, rather than as multi-functional materials for structural and/or device applications. Here, we report that filled polydimethylsiloxane (PDMS) elastomers provide a robust elastic substrate into which mechanophores can be embedded and activated under conditions from which the sample regains its original shape and properties. Fabrication is straightforward and easily accessible, providing access for the first time to objects and devices that either release or reversibly activate chemical functionality over hundreds of loading cycles.

While the mechanically accelerated ring-opening reaction of spiropyran to merocyanine and associated color change provides a useful method by which to image the molecular scale stress/strain distribution within a polymer, the magnitude of the forces necessary for activation had yet to be quantified. Here, we report single molecule force spectroscopy studies of two spiropyran isomers. Ring opening on the timescale of tens of milliseconds is found to require forces of ~240 pN, well below that of previously characterized covalent mechanophores. The lower threshold force is a combination of a low force-free activation energy and the fact that the change in rate with force (activation length) of each isomer is greater than that inferred in other systems. Importantly, quantifying the magnitude of forces required to activate individual spiropyran-based force-probes enables the probe behave as a “scout” of molecular forces in materials; the observed behavior of which can be extrapolated to predict the reactivity of potential mechanophores within a given material and deformation.

We subsequently translated the design platform to existing dynamic soft technologies to fabricate the first mechanochemically responsive devices; first, by remotely inducing dielectric patterning of an elastic substrate to produce assorted fluorescent patterns in concert with topological changes; and second, by adopting a soft robotic platform to produce a color change from the strains inherent to pneumatically actuated robotic motion. Shown herein, covalent polymer mechanochemistry provides a viable mechanism to convert the same mechanical potential energy used for actuation into value-added, constructive covalent chemical responses. The color change associated with actuation suggests opportunities for not only new color changing or camouflaging strategies, but also the possibility for simultaneous activation of latent chemistry (e.g., release of small molecules, change in mechanical properties, activation of catalysts, etc.) in soft robots. In addition, mechanochromic stress mapping in a functional actuating device might provide a useful design and optimization tool, revealing spatial and temporal force evolution within the actuator in a way that might also be coupled to feedback loops that allow autonomous, self-regulation of activity.

In the future, both the specific material and the general approach should be useful in enriching the responsive functionality of soft elastomeric materials and devices. We anticipate the development of new mechanophores that, like the materials, are reversibly and repeatedly activated, expanding the capabilities of soft, active devices and further permitting dynamic control over chemical reactivity that is otherwise inaccessible, each in response to a single remote signal.

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The use of DNA as a polymeric building material transcends its function in biology and is exciting in bionanotechnology for applications ranging from biosensing, to diagnostics, and to targeted drug delivery. These applications are enabled by DNA’s unique structural and chemical properties, embodied as a directional polyanion that exhibits molecular recognition capabilities. Hence, the efficient and precise synthesis of high molecular weight DNA materials has become key to advance DNA bionanotechnology. Current synthesis methods largely rely on either solid phase chemical synthesis or template-dependent polymerase amplification. The inherent step-by-step fashion of solid phase synthesis limits the length of the resulting DNA to typically less than 150 nucleotides. In contrast, polymerase based enzymatic synthesis methods (e.g., polymerase chain reaction) are not limited by product length, but require a DNA template to guide the synthesis. Furthermore, advanced DNA bionanotechnology requires tailorable structural and self-assembly properties. Current synthesis methods, however, often involve multiple conjugating reactions and extensive purification steps.

The research described in this dissertation aims to develop a facile method to synthesize high molecular weight, single stranded DNA (or polynucleotide) with versatile functionalities. We exploit the ability of a template-independent DNA polymerase−terminal deoxynucleotidyl transferase (TdT) to catalyze the polymerization of 2’-deoxyribonucleoside 5’-triphosphates (dNTP, monomer) from the 3’-hydroxyl group of an oligodeoxyribonucleotide (initiator). We termed this enzymatic synthesis method: TdT catalyzed enzymatic polymerization, or TcEP.

Specifically, this dissertation is structured to address three specific research aims. With the objective to generate high molecular weight polynucleotides, Specific Aim 1 studies the reaction kinetics of TcEP by investigating the polymerization of 2’-deoxythymidine 5’-triphosphates (monomer) from the 3’-hydroxyl group of oligodeoxyribothymidine (initiator) using in situ 1H NMR and fluorescent gel electrophoresis. We found that TcEP kinetics follows the “living” chain-growth polycondensation mechanism, and like in “living” polymerizations, the molecular weight of the final product is determined by the starting molar ratio of monomer to initiator. The distribution of the molecular weight is crucially influenced by the molar ratio of initiator to TdT. We developed a reaction kinetics model that allows us to quantitatively describe the reaction and predict the molecular weight of the reaction products.

Specific Aim 2 further explores TcEP’s ability to transcend homo-polynucleotide synthesis by varying the choices of initiators and monomers. We investigated the effects of initiator length and sequence on TcEP, and found that the minimum length of an effective initiator should be 10 nucleotides and that the formation of secondary structures close to the 3’-hydroxyl group can impede the polymerization reaction. We also demonstrated TcEP’s capacity to incorporate a wide range of unnatural dNTPs into the growing chain, such as, hydrophobic fluorescent dNTP and fluoro modified dNTP. By harnessing the encoded nucleotide sequence of an initiator and the chemical diversity of monomers, TcEP enables us to introduce molecular recognition capabilities and chemical functionalities on the 5’-terminus and 3’-terminus, respectively.

Building on TcEP’s synthesis capacities, in Specific Aim 3 we invented a two-step strategy to synthesize diblock amphiphilic polynucleotides, in which the first, hydrophilic block serves as a macro-initiator for the growth of the second block, comprised of natural and/or unnatural nucleotides. By tuning the hydrophilic length, we synthesized the amphiphilic diblock polynucleotides that can self-assemble into micellar structures ranging from star-like to crew-cut morphologies. The observed self-assembly behaviors agree with predictions from dissipative particle dynamics simulations as well as scaling law for polyelectrolyte block copolymers.

In summary, we developed an enzymatic synthesis method (i.e., TcEP) that enables the facile synthesis of high molecular weight polynucleotides with low polydispersity. Although we can control the nucleotide sequence only to a limited extent, TcEP offers a method to integrate an oligodeoxyribonucleotide with specific sequence at the 5’-terminus and to incorporate functional groups along the growing chains simultaneously. Additionally, we used TcEP to synthesize amphiphilic polynucleotides that display self-assemble ability. We anticipate that our facile synthesis method will not only advance molecular biology, but also invigorate materials science and bionanotechnology.

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Surface-enhanced Raman spectroscopy (SERS) is now widely used as a rapid and inexpensive tool for chemical/biochemical analysis. The method can give enormous increases in the intensities of the Raman signals of low-concentration molecular targets if they are adsorbed on suitable enhancing substrates, which are typically composed of nanostructured Ag or Au. However, the features of SERS that allow it to be used as a chemical sensor also mean that it can be used as a powerful probe of the surface chemistry of any nanostructured material that can provide SERS enhancement. This is important because it is the surface chemistry that controls how these materials interact with their local environment and, in real applications, this interaction can be more important than more commonly measured properties such as morphology or plasmonic absorption. Here, the opportunity that this approach to SERS provides is illustrated with examples where the surface chemistry is both characterized and controlled in order to create functional nanomaterials.

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Silica additives in bone substitute materials are topical, clinically interesting and have significant support in the Orthopaedic field. Biosilica, e.g isolated from diatoms, has many advantages over its synthetic counterparts, e.g. it is amorphous, thus will be absorbed by the body, however, issues such as purity, presence of endotoxins and cytotoxicity need to be addressed before it can be further exploited. Biosilica isolated from Cyclotella Meneghiniana was then tested in a mouse model, to test the immunological response, organ toxicity (kidney, spleen, liver) and route of metabolism/excretion of silica. Five-week-old Balb-c mice were injected subcutaneously with a single high dose (50mg/ml) of Si-frustules, Si-frustules + organic linker and vehicle only control. Animals were sacrificed at 1d and 28d. The animal studies were conducted under an ethically approved protocol at Queen’s University, Belfast. The animals showed no adverse stress during the experiment and remained healthy until sacrifice. Blood results using ICP-OES analysis suggest the frustules were metabolized between comparator groups at different rates, and clearly showed elevated levels of silicon in groups injected with frustules relative to control. The histology of organs showed no variation in morphology of mice injected frustules relative compared to the control group.
Acknowledgements: The authors would like to thank Marie Curie International Outgoing Fellowships from the EU and Beaufort Marine Biodiscovery Award as part of the Marine Biotechnology Ireland Programme for providing financial support to this project.

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Epitaxial van der Waals (vdW) heterostructures of organic and layered materials are demonstrated to create high-performance organic electronic devices. High-quality rubrene films with large single-crystalline domains are grown on h-BN dielectric layers via vdW epitaxy. In addition, high carrier mobility comparable to free-standing single-crystal counterparts is achieved by forming interfacial electrical contacts with graphene electrodes.

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Understanding the effect of electric fields on the physical and chemical properties of two-dimensional (2D) nanostructures is instrumental in the design of novel electronic and optoelectronic devices. Several of those properties are characterized in terms of the dielectric constant which play an important role on capacitance, conductivity, screening, dielectric losses and refractive index. Here we review our recent theoretical studies using density functional calculations including van der Waals interactions on two types of layered materials of similar two-dimensional molecular geometry but remarkably different electronic structures, that is, graphene and molybdenum disulphide (MoS2). We focus on such two-dimensional crystals because of they complementary physical and chemical properties, and the appealing interest to incorporate them in the next generation of electronic and optoelectronic devices. We predict that the effective dielectric constant (ε) of few-layer graphene and MoS2 is tunable by external electric fields (E ext). We show that at low fields (E ext < 0.01 V/Å) ε assumes a nearly constant value ∼4 for both materials, but increases at higher fields to values that depend on the layer thickness. The thicker the structure the stronger is the modulation of ε with the electric field. Increasing of the external field perpendicular to the layer surface above a critical value can drive the systems to an unstable state where the layers are weakly coupled and can be easily separated. The observed dependence of ε on the external field is due to charge polarization driven by the bias, which show several similar characteristics despite of the layer considered. All these results provide key information about control and understanding of the screening properties in two-dimensional crystals beyond graphene and MoS2

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Though controversial the question of applying data protection laws to biological materials has only gotten a little attention in data privacy discourse. This article aims to contribute to this dearth by arguing that despite absence of positive intention from the architects to apply the EU Data privacy law to biological materials, a range of developments in Molecular Biology and nano-technology—usually mediated by advances in ICT—may provide persuasive grounds to do so. In addition, paucity of sufficient explication of key terms like ‘data/information’ in these legislations may fuel such tendency whereby laws originally intended for the informational world may end up applying to the biological world. The article also analyzes various predicaments that may arise from applying data privacy laws to biological materials. A focus is made on legislative sources at the EU level though national laws are relied on when pertinent.

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Thesis (Ph.D.)--University of Washington, 2016-08

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Le béton conventionnel (BC) a de nombreux problèmes tels que la corrosion de l’acier d'armature et les faibles résistances des constructions en béton. Par conséquent, la plupart des structures fabriquées avec du BC exigent une maintenance fréquent. Le béton fibré à ultra-hautes performances (BFUP) peut être conçu pour éliminer certaines des faiblesses caractéristiques du BC. Le BFUP est défini à travers le monde comme un béton ayant des propriétés mécaniques, de ductilité et de durabilité supérieures. Le BFUP classique comprend entre 800 kg/m³ et 1000 kg/m³ de ciment, de 25 à 35% massique (%m) de fumée de silice (FS), de 0 à 40%m de poudre de quartz (PQ) et 110-140%m de sable de quartz (SQ) (les pourcentages massiques sont basés sur la masse totale en ciment des mélanges). Le BFUP contient des fibres d'acier pour améliorer sa ductilité et sa résistance aux efforts de traction. Les quantités importantes de ciment utilisées pour produire un BFUP affectent non seulement les coûts de production et la consommation de ressources naturelles comme le calcaire, l'argile, le charbon et l'énergie électrique, mais affectent également négativement les dommages sur l'environnement en raison de la production substantielle de gaz à effet de serre dont le gas carbonique (CO[indice inférieur 2]). Par ailleurs, la distribution granulométrique du ciment présente des vides microscopiques qui peuvent être remplis avec des matières plus fines telles que la FS. Par contre, une grande quantité de FS est nécessaire pour combler ces vides uniquement avec de la FS (25 à 30%m du ciment) ce qui engendre des coûts élevés puisqu’il s’agit d’une ressource limitée. Aussi, la FS diminue de manière significative l’ouvrabilité des BFUP en raison de sa surface spécifique Blaine élevée. L’utilisation du PQ et du SQ est également coûteuse et consomme des ressources naturelles importantes. D’ailleurs, les PQ et SQ sont considérés comme des obstacles pour l’utilisation des BFUP à grande échelle dans le marché du béton, car ils ne parviennent pas à satisfaire les exigences environnementales. D’ailleurs, un rapport d'Environnement Canada stipule que le quartz provoque des dommages environnementaux immédiats et à long terme en raison de son effet biologique. Le BFUP est généralement vendu sur le marché comme un produit préemballé, ce qui limite les modifications de conception par l'utilisateur. Il est normalement transporté sur de longues distances, contrairement aux composantes des BC. Ceci contribue également à la génération de gaz à effet de serre et conduit à un coût plus élevé du produit final. Par conséquent, il existe le besoin de développer d’autres matériaux disponibles localement ayant des fonctions similaires pour remplacer partiellement ou totalement la fumée de silice, le sable de quartz ou la poudre de quartz, et donc de réduire la teneur en ciment dans BFUP, tout en ayant des propriétés comparables ou meilleures. De grandes quantités de déchets verre ne peuvent pas être recyclées en raison de leur fragilité, de leur couleur, ou des coûts élevés de recyclage. La plupart des déchets de verre vont dans les sites d'enfouissement, ce qui est indésirable puisqu’il s’agit d’un matériau non biodégradable et donc moins respectueux de l'environnement. Au cours des dernières années, des études ont été réalisées afin d’utiliser des déchets de verre comme ajout cimentaire alternatif (ACA) ou comme granulats ultrafins dans le béton, en fonction de la distribution granulométrique et de la composition chimique de ceux-ci. Cette thèse présente un nouveau type de béton écologique à base de déchets de verre à ultra-hautes performances (BEVUP) développé à l'Université de Sherbrooke. Les bétons ont été conçus à l’aide de déchets verre de particules de tailles variées et de l’optimisation granulaire de la des matrices granulaires et cimentaires. Les BEVUP peuvent être conçus avec une quantité réduite de ciment (400 à 800 kg/m³), de FS (50 à 220 kg/m³), de PQ (0 à 400 kg/m³), et de SQ (0-1200 kg/m³), tout en intégrant divers produits de déchets de verre: du sable de verre (SV) (0-1200 kg/m³) ayant un diamètre moyen (d[indice inférieur 50]) de 275 µm, une grande quantité de poudre de verre (PV) (200-700 kg/m³) ayant un d50 de 11 µm, une teneur modérée de poudre de verre fine (PVF) (50-200 kg/m³) avec d[indice inférieur] 50 de 3,8 µm. Le BEVUP contient également des fibres d'acier (pour augmenter la résistance à la traction et améliorer la ductilité), du superplastifiants (10-60 kg/m³) ainsi qu’un rapport eau-liant (E/L) aussi bas que celui de BFUP. Le remplacement du ciment et des particules de FS avec des particules de verre non-absorbantes et lisse améliore la rhéologie des BEVUP. De plus, l’utilisation de la PVF en remplacement de la FS réduit la surface spécifique totale nette d’un mélange de FS et de PVF. Puisque la surface spécifique nette des particules diminue, la quantité d’eau nécessaire pour lubrifier les surfaces des particules est moindre, ce qui permet d’obtenir un affaissement supérieur pour un même E/L. Aussi, l'utilisation de déchets de verre dans le béton abaisse la chaleur cumulative d'hydratation, ce qui contribue à minimiser le retrait de fissuration potentiel. En fonction de la composition des BEVUP et de la température de cure, ce type de béton peut atteindre des résistances à la compression allant de 130 à 230 MPa, des résistances à la flexion supérieures à 20 MPa, des résistances à la traction supérieure à 10 MPa et un module d'élasticité supérieur à 40 GPa. Les performances mécaniques de BEVUP sont améliorées grâce à la réactivité du verre amorphe, à l'optimisation granulométrique et la densification des mélanges. Les produits de déchets de verre dans les BEVUP ont un comportement pouzzolanique et réagissent avec la portlandite générée par l'hydratation du ciment. Cependant, ceci n’est pas le cas avec le sable de quartz ni la poudre de quartz dans le BFUP classique, qui réagissent à la température élevée de 400 °C. L'addition des déchets de verre améliore la densification de l'interface entre les particules. Les particules de déchets de verre ont une grande rigidité, ce qui augmente le module d'élasticité du béton. Le BEVUP a également une très bonne durabilité. Sa porosité capillaire est très faible, et le matériau est extrêmement résistant à la pénétration d’ions chlorure (≈ 8 coulombs). Sa résistance à l'abrasion (indice de pertes volumiques) est inférieure à 1,3. Le BEVUP ne subit pratiquement aucune détérioration aux cycles de gel-dégel, même après 1000 cycles. Après une évaluation des BEVUP en laboratoire, une mise à l'échelle a été réalisée avec un malaxeur de béton industriel et une validation en chantier avec de la construction de deux passerelles. Les propriétés mécaniques supérieures des BEVUP a permis de concevoir les passerelles avec des sections réduites d’environ de 60% par rapport aux sections faites de BC. Le BEVUP offre plusieurs avantages économiques et environnementaux. Il réduit le coût de production et l’empreinte carbone des structures construites de béton fibré à ultra-hautes performances (BFUP) classique, en utilisant des matériaux disponibles localement. Il réduit les émissions de CO[indice inférieur 2] associées à la production de clinkers de ciment (50% de remplacement du ciment) et utilise efficacement les ressources naturelles. De plus, la production de BEVUP permet de réduire les quantités de déchets de verre stockés ou mis en décharge qui causent des problèmes environnementaux et pourrait permettre de sauver des millions de dollars qui pourraient être dépensés dans le traitement de ces déchets. Enfin, il offre une solution alternative aux entreprises de construction dans la production de BFUP à moindre coût.

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Nanocrystalline samples of Ba1-xCaxF2 prepared by high-energy milling show an unusually high F-ion conductivity, which exhibit a maximum in the magnitude and a minimum in the activation energy at x = 0.5. Here, we report an X-ray absorption spectroscopy (XAS) at the Ca and Sr K edges and the Ba L-3 edge and a molecular dynamics (MD) simulation study of the pure and mixed fluorides. The XAS measurements on the pure binary fluorides, CaF2, SrF2 and BaF2 show that high-energy ball-milling produces very little amorphous material, in contrast to the results for ball milled oxides. XAS measurements of Ba1-xCaxF2 reveal that for 0 < x < 1 there is considerable disorder in the local environments of the cations which is highest for x = 0.5. Hence the maximum in the conductivity corresponds to the composition with the maximum level of local disorder. The MD calculations also show a highly disordered structure consistent with the XAS results and similarly showing maximum disorder at x = 0.5.