Coupled Optimal Results with an Application to Nonlinear Integral Equations

In the present work, we consider the best proximal problem related to a coupled mapping, which we define using control functions and weak inequalities. As a consequence, we obtain some results on coupled fixed points. Our results generalize some recent results in the literature. Also, as an application of the results obtained, we present the solution to a system of a coupled Fredholm nonlinear integral equation. Our work is supported by several illustrations.The authors are grateful to the Basque Government by the support of this work through Grant IT1207-19

Revisiting Solid-solid Phase Transitions in Sodium and Potassium Tetrafluoroborate for Thermal Energy Storage

In situ synchrotron powder x-ray diffraction (PXRD) study was conducted on sodium and potassium tetrafluoroborate (NaBF4 and KBF4) to elucidate structural changes across solid-solid phase transitions over multiple heating-cooling cycles. The phase transition temperatures from diffraction measurements are consistent with the differential scanning calorimetry data (~240 °C for NaBF4 and ~290 °C for KBF4). The crystal structure of the high-temperature (HT) NaBF4 phase has been determined from synchrotron PXRD data. The HT disordered phase of NaBF4 crystallizes in the hexagonal, space group P63/mmc (No. 194) with a = 4.98936(2) Å, c = 7.73464(4) Å, V = 166.748(2) Å3, and Z = 2 at 250 °C. Density functional theory molecular dynamics (MD) calculations imply that the P63/mmc is indeed a stable structure for rotational NaBF4. MD simulations reproduce experimental phase sequence upon heating and indicates that F atoms are markedly more mobile than K and B atoms in the disordered state. Thermal expansion coefficients for both phases were determined from high precision lattice parameters at elevated temperatures, as obtained from Rietveld refinement of PXRD data. Interestingly for the HT-phase of NaBF4, the structure (upon heating) contracts slightly in the a-b plane but expands in the c direction such that overall thermal expansion is positive. Thermal conductivity at room temperature were measured and the values are 0.8-1.0 W.m-1K-1 for NaBF4 and 0.55-0.65 W.m-1K-1 for KBF4. The thermal conductivity and diffusivity showed a gradual decrease up to the transition temperature and then rose slightly. Both materials show good thermal and structural stabilities over multiple heating/cooling cycles.<br/

A Prolific Solvate Former, Galunisertib, under the Pressure of Crystal Structure Prediction, Produces Ten Diverse Polymorphs

The solid form screening of galunisertib produced many solvates, prompting an extensive investigation into possible risks to the development of the favored monohydrate form. Inspired by crystal structure prediction, the search for neat polymorphs was expanded to an unusual range of experiments, including melt crystallization under pressure, to work around solvate formation and the thermal instability of the molecule. Ten polymorphs of galunisertib were found; however, the structure predicted to be the most stable has yet to be obtained. We present the crystal structures of all ten unsolvated polymorphs of galunisertib, showing how state-of-the-art characterization methods can be combined with emerging computational modeling techniques to produce a complete structure landscape and assess the risk of late-appearing, more stable polymorphs. The exceptional conformational polymorphism of this prolific solvate former invites further development of methods, computational and experimental, that are applicable to larger, flexible molecules with complex solid form landscapes

Structural transformations of Li2C2 at high pressures

Structural changes of Li2C2 under pressure were studied by synchrotron x-ray diffraction in a diamond anvil cell under hydrostatic conditions and by using evolutionary search methodology for crystal structure prediction. We show that the high-pressure polymorph of Li2C2, which forms from the Immm ground-state structure (Z = 2) at around 15 GPa, adopts an orthorhombic Pnma structure with Z = 4. Acetylide C2 dumbbells characteristic of Immm Li2C2 are retained in Pnma Li2C2. The structure of Pnma Li2C2 relates closely to the anticotunnite-type structure. C2 dumbbell units are coordinated by nine Li atoms, as compared to eight in the antifluorite structure of Immm Li2C2. First-principles calculations predict a transition of Pnma Li2C2 at 32 GPa to a topologically identical phase with a higher Cmcm symmetry. The coordination of C2 dumbbell units by Li atoms is increased to 11. The structure of Cmcm Li2C2 relates closely to the Ni2 In-type structure. It is calculated that Cmcm Li2C2 becomes metallic at pressures above 40 GPa. In experiments, however, Pnma Li2C2 is susceptible to irreversible amorphization

Structure and Phase Stability of CaC2 Polymorphs, Li2C2 and Lithium Intercalated Graphite : A Revisit with High Pressure Experiments and Metal Hydride–Graphite Reactions

Alkali (A) and alkaline earth (AE) metals can form carbides and intercalated graphites with carbon. The carbides mostly represent acetylides which are salt-like compounds composed of C22− dumbbell anions and metal cations. Both the acetylide carbides and intercalated graphites are technologically important. Superconductivity has been observed in several intercalated graphites such as KC8 and CaC6. Li intercalated graphites are a major ingredient in Li ion batteries. CaC2 is an important commodity for producing acetylene and the fertilizer CaCN2. In spite of the extensive research on A–C and AE–C compounds, phase diagrams are largely unknown. The thermodynamic and kinetic properties of both carbides and intercalalated graphites are discussed controversially. Recent computational studies indicated that well-known carbides, like CaC2 and BaC2, are thermodynamically unstable. Additionally, computational studies predicted that acetylide carbides will generally form novel polymeric carbides (polycarbides) at high pressures. This thesis is intended to check the validity of theoretical predictions and to shed light on the complicated phase diagrams of the Li–C and the Ca–C systems. The Li–C and the Ca–C systems were investigated using well-controllable metal hydride–graphite reactions. Concerning the Li–C system, relative stabilities of the metastable lithium graphite intercalation compounds (Li-GICs) of stages I, IIa, IIb, III, IV and Id were studied close to the competing formation of the thermodynamically stable Li2C2. The stage IIa showed distinguished thermal stability. The phase Id showed thermodynamic stability and hence, was included in the Li–C phase diagram. In the Ca–C system, results from CaH2–graphite reactions indicate compositional variations between polymorphs I, II and III. The formation of CaC2  I was favored  only  at  1100  ◦C or  higher  temperature  and  with  excess calcium, which speculates phase I as carbon deficient CaC2−δ . To explore the potential existence of polycarbides, the acetylide carbides Li2C2 and CaC2 were investigated under various pressure and temperature conditions, employing diamond anvil cells for in situ studies and multi anvil techniques for large volume synthesis. The products were characterized by a combination of diffraction and spectroscopy techniques. For both Li2C2 and CaC2, a pressure induced structural transformation was observed at relatively low pressures (10–15 GPa), which was followed by an irreversible amorphization at higher pressures (25–30 GPa). For Li2C2 the structure of the high pressure phase prior to amorphization could be elucidated. The ground state with an antifluorite Immm structure (coordination number (CN) for C22− dumbbells = 8) transforms to a phase with an anticotunnite Pnma structure (CN for C22− dumbbells = 9). Polycarbides, as predicted from theory, could not be obtained.At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 2: Manuscript.</p

Intercalation compounds from lih and graphite: Relative stability of metastable stages and thermodynamic stability of dilute stage Id

The intercalation of lithium into graphite was studiedat temperatures between 400 and 550°C by heating mixtures ofLiH and graphite powders with molar ratios 4:1, 1:1, and 1:6 underdynamic vacuum for periods between 1 and 72 h. These conditionsprobe the formation and thermal stability of metastable staged Li−graphite intercalation compounds (Li-GICs) close to the competingformation of the thermodynamically stable carbide Li2C2. Li-GICsof stages I (LiC6,Aα), IIa(Li0.5C6,AαA), IIb(Li∼0.33C6,AαABβB),III (Li∼0.22C6,AαAB), IV (Li∼0.167C6), and dilute stage lithium Idhave been identified and characterized by powder X-ray diffractionand Raman spectroscopy. The rate and extent of intercalation (i.e.,the achieved stage of Li-GIC) depends on LiH activity andtemperature. Stage I was only observed for temperatures above 500°C. At 400°C, the highest intercalation corresponded to stage IIb, which was obtained after 2 and 24 h for 4:1 and 1:1 reactionmixtures, respectively. Lower-staged Li-GICs attained at temperatures below 500°C deintercalate upon prolonged dwelling withthe exception of stage IIa, which can be maintained for very long periods (several days) in the presence of LiH. At temperaturesabove 500°C, the kinetically controlled formation of Li-GICs is followed by Li2C2carbide formation. It is shown that the Li-GICIdcoexists with Li2C2at temperatures up to 800°C and that the Li content of Id(solubility of Li in graphite) increases between550 and 800°C. Consequently, Idwith a temperature-dependent homogeneity range should be added as a stable phase in theLi−C phase diagram. A sketch of a revised Li−C phase diagram is provided.</p