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We show existence of solution for a supercritical nonlinear Schrodinger equation on the whole RN by means of an approximation scheme. We prove a Sobolev embedding corresponding to the variable expo...

This paper is concerned with the existence of solutions for a class of elliptic equations on the unit ball with zero Dirichlet boundary condition. The nonlinearity is supercritical in the sense of Trudinger–Moser. Using a suitable approximating scheme we obtain the existence of at least one positive solution.

We establish the existence of positive solutions for a nonlinear elliptic Dirichlet problem in dimension N involving the N-Laplacian. The nonlinearity considered depends on the gradient of the unknown function and an exponential term. In such case, variational methods cannot be applied. Our approach is based on approximation scheme, where we consider a new class of normed spaces of finite dimension. As a particular case, we extended the result achieved by De Araujo and Montenegro [2016] for any N > 2 .

The purpose of the present paper is to study a class of semilinear elliptic Dirichlet boundary value problems in the ball, where the nonlinearities involve the sum of a sublinear variable exponent and a superlinear (may be supercritical) variable exponents of the form $$0\le f(r,u)\le a_1|u|^{p(r)-1}$$ , if $$u\ge 0$$ , where $$r = |x|$$ , $$p(r ) = 2^* + r^{\alpha }$$ , with $$\alpha > 0$$ , and $$2^* = 2N/(N - 2)$$ is the critical Sobolev embedding exponent. We do not impose the Ambrosetti–Rabinowitz condition on the nonlinearity (or some additional conditions) to obtain Palais–Smale or Cerami compactness condition. We employ techniques based on the Galerkin approximations scheme, combining with a Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces (due to do O et al. in Calc Var Partial Differ Equ 55:83, 2016), to establish the existence result.

Poly(ionic liquid)s are an interesting class of compounds because of their unique chemical and physical properties gathering the characteristics of ionic liquids and polymers. Pressure and temperature have been demonstrated to be alternative parameters to obtain polymers from monomeric species using only physical tools. In this work, we investigate the reaction under high pressure and room temperature of the ionic liquid 1-allyl-3-methylimidazolium iodide by using the diamond anvil cell technique in combination with synchrotron X-ray diffraction and electronic and vibrational spectroscopies. The results indicate a chemical reaction happening through the terminal double bond of the allyl group both in crystalline and glassy phases with the onset of the reaction around ∼7 GPa. Vibrational spectra present evidence for an oligomerization reaction in both the phases. The reaction occurring both in glassy and crystal phases indicates a mechanism not driven by collective motions and likely related to local topological arrangements. The results presented herein extend our understanding of ionic liquid instability boundaries under high pressure and contribute to the development of alternative synthetic routes to achieve poly(ionic liquids).

We consider the Hardy–Schrödinger operator L γ := - Δ 𝔹 n - γ ⁢ V 2 {L_{\gamma}:=-\Delta_{\mathbb{B}^{n}}-\gamma{V_{2}}} on the Poincaré ball model of the hyperbolic space 𝔹 n {\mathbb{B}^{n}} ( n ≥ 3 {n\geq 3} ). Here V 2 {V_{2}} is a radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e., V 2 ⁢ ( r ) ∼ 1 r 2 {V_{2}(r)\sim\frac{1}{r^{2}}} . As in the Euclidean setting, L γ {L_{\gamma}} is positive definite whenever γ < ( n - 2 ) 2 4 {\gamma , in which case we exhibit explicit solutions for the critical equation L γ ⁢ u = V 2 * ⁢ ( s ) ⁢ u 2 * ⁢ ( s ) - 1 {L_{\gamma}u=V_{2^{*}(s)}u^{2^{*}(s)-1}} in 𝔹 n , {\mathbb{B}^{n},} where 0 ≤ s < 2 {0\leq s , 2 * ⁢ ( s ) = 2 ⁢ ( n - s ) n - 2 {2^{*}(s)=\frac{2(n-s)}{n-2}} , and V 2 * ⁢ ( s ) {V_{2^{*}(s)}} is a weight that behaves like 1 r s {\frac{1}{r^{s}}} around 0. In dimensions n ≥ 5 {n\geq 5} , the equation L γ ⁢ u - λ ⁢ u = V 2 * ⁢ ( s ) ⁢ u 2 * ⁢ ( s ) - 1 {L_{\gamma}u-\lambda u=V_{2^{*}(s)}u^{2^{*}(s)-1}} in a domain Ω of 𝔹 n {\mathbb{B}^{n}} away from the boundary but containing 0 has a ground state solution, whenever 0 < γ ≤ n ⁢ ( n - 4 ) 4 {0 , and λ > n - 2 n - 4 ⁢ ( n ⁢ ( n - 4 ) 4 - γ ) {\lambda>\frac{n-2}{n-4}(\frac{n(n-4)}{4}-\gamma)} . On the other hand, in dimensions 3 and 4, the existence of solutions depends on whether the domain has a positive “hyperbolic mass” a notion that we introduce and analyze therein.

A Raman spectroscopy study of 1-ethyl-3-methylimidazolium hydrogen sulfate, [C2C1im][HSO4], as a function of temperature, has been performed to reveal the role played by anion–anion hydrogen bond on the phase transitions of this ionic liquid. Anion–anion hydrogen bonding implies high viscosity, good glass-forming ability, and also moderate fragility of [C2C1im][HSO4] in comparison with other ionic liquids. Heating [C2C1im][HSO4] from the glassy phase results in cold crystallization at ∼245 K. A solid–solid transition (crystal I → crystal II) is barely discernible in calorimetric measurements at typical heating rates, but it is clearly revealed by Raman spectroscopy and X-ray diffraction. Raman spectroscopy indicates that crystal I has extended ([HSO4]−)n chains of hydrogen-bonded anions but crystal II has not. Raman spectra recorded at isothermal condition show the ultraslow dynamics of cold crystallization, solid–solid transition, and continuous melting of [C2C1im][HSO4]. A brief comparison is also provi...

A local order-disorder transition has been disclosed in the thermophysical behavior of the ionic liquid 1-benzyl-3-methylimidazolium dicyanamide, [Bzmim][N(CN)2], and its microscopic nature revealed by spectroscopic techniques. Differential scanning calorimetry and specific heat measurements show a thermal event of small enthalpy variation taking place in the range 250-260 K, which is not due to crystallization or melting. Molecular dynamic simulations and X-ray diffraction measurements have been used to discuss the segregation of domains in the liquid structure of [Bzmim][N(CN)2]. Raman and NMR spectroscopy measurements as a function of temperature indicate that the microscopic origin of the event observed in the calorimetric measurements comes from structural rearrangement involving the benzyl group. The results indicate that the characteristic structural heterogeneity allow for rearrangements within local domains implying the good glass-forming ability for the low viscosity ionic liquid [Bzmim][N(CN)2]. This work sheds light on our understanding of the microscopic origin behind complex thermal behavior of ionic liquids.

The intermolecular dynamics in the THz frequency range of the ionic liquids n-butyl-trimethylammonium bis(trifluoromethanesulfonyl)imide, [N1114][NTf2], and methyl-tributylammonium bis(trifluoromethanesulfonyl)imide, [N1444][NTf2], were investigated by a combined usage of inelastic neutron scattering (INS), Raman, and far-infrared (FIR) spectroscopies and the power spectrum calculated by molecular dynamics (MD) simulations. The collective dynamics of the simulated systems is also discussed by the calculation of time correlation functions of charge and mass currents that are projected onto acoustic- and optic-like motions. The INS and Raman measurements have been performed as a function of temperature in the glassy, crystalline, and liquid phases. The excess in the vibrational density of states over the expectation of the Debye theory, the so-called boson peak, is found in the INS and Raman spectra as a peak at ∼2 meV (∼16 cm-1) and also in the direct measurement of heat capacity at very low temperatures (4-20 K). This low-frequency vibration is incorporated into the curve fits of Raman, FIR, and MD data at room temperature. Fits of spectra from these different sources in the range below 100 cm-1 are consistently achieved with three components at ca. 25, 50, and 80 cm-1, but with distinct relative intensities among the different techniques. It is proposed as the collective nature of the lowest-frequency component and the anion-cation intermolecular vibration nature of the highest-frequency component. The MD results indicate that there is no clear distinction between acoustic and optic vibrations in the spectral range investigated in this work for the ionic liquids [N1114][NTf2] and [N1444][NTf2]. The analysis carried out here agrees in part, but not entirely, with other propositions in the literature, mainly from optical Kerr effect (OKE) and FIR spectroscopies, concerning the intermolecular dynamics of ionic liquids.

In this work a series of metal complexes containing the rhodizonate ion have been synthesized and characterized by means of structural and spectroscopic measurements. Compounds of general formula MC6O6.nH2O, where M = Mn2+, Co2+, Ni2+, Cu2+ and Zn2+) were carefully obtained in an inert atmosphere, and the resulting complexes have been analyzed by powder X-ray technique, showing that manganese, cobalt, nickel and zinc compounds all present the same similar arrangement; the qualitative phase analysis indicate the Pmmm space group for manganese, cobalt, nickel and zinc salts, and P2/m for the copper salt. The unit cell volumes were determined, and they grow approximately with the number of water molecules in the structure measured by elemental and thermogravimetric analysis. The vibrational spectra are in concordance with the X-ray diffraction results since the salts of manganese, cobalt, nickel and zinc show a remarkably similar spectral profile, whereas the copper salt, shows a different spectral profile. It was observed that the number of bands in the Raman spectra of the compounds is greatly increased compared to the free rhodizonate ion. This fact is justified by the lower symmetry due to the complexation of the metal site. The vibrational modes related with the C=O stretching in the 1700–1500 cm−1 region are the most influenced with the metal coordination showing large displacements and band splitting. This last fact strongly supports the loss of symmetry experienced by the oxocarbon ion when coordinated to the metal centers in the complexes.

Vibrational spectroscopy has continued use as a powerful tool to characterize ionic liquids since the literature on room temperature molten salts experienced the rapid increase in number of publications in the 1990's. In the past years, infrared (IR) and Raman spectroscopies have provided insights on ionic interactions and the resulting liquid structure in ionic liquids. A large body of information is now available concerning vibrational spectra of ionic liquids made of many different combinations of anions and cations, but reviews on this literature are scarce. This review is an attempt at filling this gap. Some basic care needed while recording IR or Raman spectra of ionic liquids is explained. We have reviewed the conceptual basis of theoretical frameworks which have been used to interpret vibrational spectra of ionic liquids, helping the reader to distinguish the scope of application of different methods of calculation. Vibrational frequencies observed in IR and Raman spectra of ionic liquids based on different anions and cations are discussed and eventual disagreements between different sources are critically reviewed. The aim is that the reader can use this information while assigning vibrational spectra of an ionic liquid containing another particular combination of anions and cations. Different applications of IR and Raman spectroscopies are given for both pure ionic liquids and solutions. Further issues addressed in this review are the intermolecular vibrations that are more directly probed by the low-frequency range of IR and Raman spectra and the applications of vibrational spectroscopy in studying phase transitions of ionic liquids.

Solubility parameters such as Hildebrand's solubility parameter or its square, the cohesive energy density ced are often not able to predict solubilities of organic compounds in ionic liquids accurately. It has been suggested that this is an effect of the structural heterogeneity that many ionic liquids exhibit due to the non-polar alkyl chains forming separate domains from the charged parts of the cation and the anion. We here calculated the partial volumes of the polar and non-polar domains in Molecular Dynamics simulations of several 1-alkyl-3-methylimidazolium nitrates and 1-octyl-3-methylimidazolium hexafluorophosphate. We used these partial volumes to calculate the ced of the polar and non-polar domains. For 1-octyl-3-methylimidazolium nitrate the internal pressures Pi of the domains were also calculated. Our results show that ced and Pi are indeed very different in the domains, and that the domains containing the hydrophobic alkyl chains have solvent parameters similar to alkanes. For the domains that contain the charged part of the ions, ced and Pi are much higher, and Pi is comparable to what is found for high temperature molten salts. The results provide a possible explanation why solubility parameters seem incorrect for many ionic liquids, and may be useful in estimating more useful values using computer simulations.

X-ray diffraction and Raman scattering measurements of the ionic liquid 1-butyl-1-methylpyrrolidinium bis(trifluoromethanesufonyl)imide, [Pyrr1,4][NTf2], were performed in a wide range of temperature and pressure. Pressure dependence of glass transition Tg(p), cold-crystallization Tcc(p), and melting Tm(p) temperatures were obtained from atmospheric pressure up to ca. 2.0 GPa. The temperature dependence of Raman spectra within the Tcc

The phase diagram of the ionic liquid 1-butyl-1-methylpyrrolidinium bis(trifluoromethanesufonyl)imide, [Pyrr1,4][NTf2], was explored by synchroton X-ray diffraction and Raman scattering measurements as a function of temperature and pressure. Glass transition Tg(p) and melting Tm(p) temperatures were obtained from atmospheric pressure up to ca. 2.0 GPa. We found that both the Tg(p) and Tm(p) curves follow essentially the same pressure dependence. The similarity of pressure coefficients, dTg/dp ≈ dTm/dp, is explained within the non-equilibrium thermodynamics approach for the glass transition by assuming that one of the Ehrenfest equations is appropriated for Tg(p), whereas Tm(p) follows the Clausius-Clapeyron equation valid for the first-order transitions. The results highlight that ionic liquids are excellent model systems to address fundamental questions related to the glass transition.

By means of variational methods we investigate existence, nonexistence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

We study the existence of the homoclinic solution of a class of nonlinear fourth order ordinary differential equations, which involve particular cases of the extended Fisher–Kolmogorov and Swift–Hohenberg equations. We address the case when the source term possesses a nonlinear dependence on the first order derivative of the solution. The main result follows from an iterative method using the sequence of solutions obtained through the mountain-pass technique. In order to illustrate this method some numerical examples are shown.

Equation of state according to the Carnahan-Starling-van der Waals model has been proposed for ionic liquids based on the anions tetrafluoborate, [BF4]−, hexafluorophosphate, [PF6]−, and bis(trifluoromethanesufonyl)imide, [NTf2]−, from available volumetric data within the MPa range of pressure. Then, the equation of state is used to estimate density in the GPa range of pressure. The hard sphere part of the equation considered a binary mixture of cation and anion spheres whose diameters exhibit a satisfactory degree of transferability among different systems with a common ion. It is shown that pressure induced vibrational frequency shift, ∆ν, of the characteristic Raman band of the stretching mode of [BF4]−, [PF6]−, and [NTf2]− collapses into a master curve when ∆ν is plotted as a function of reduced density for each group of ionic liquids. Thus, Raman spectroscopy provides a validation test of the high-pressure extrapolation of the equation of state for ionic liquids.

Differential scanning calorimetry, X-ray diffraction, and Raman spectroscopy were used to reveal the premelting events precursors of melting of the ionic liquid triethylsulfonium bis(trifluoromethanesufonyl)imide, [S222][NTf2]. On heating the crystalline phase of [S222][NTf2], melting occurs along a sequence of at least three steps. First, the crystalline long-range order breaks down, but local order is retained. The second step is characterized by conformational freedom of the ethyl chains of cations related to premelting of nonpolar domains, and the complete melting finally occurs when anions acquire conformational freedom. This work provides a microscopic view on the mechanism of melting of [S222][NTf2] in line with the picture of melting taking place as a sequence of structural changes. The results of this work shed light on the understanding of the complex melting process of ionic liquids.

The aim of this paper is to prove the existence of a positive solution for a quasi-linear elliptic problem involving the (p, q)-Laplacian and a convection term, which means an expression that is not in the principal part and depends on the solution and its gradient. The solution is constructed through an approximating process based on gradient bounds and regularity up to the boundary. The positivity of the solution is shown by applying a new comparison principle, which is established here.

In this work we study the multiplicity results for a class of critical elliptic systems related to the Brézis–Nirenberg problem with the Neumann boundary condition on a ball. Our approach relies on a minimization argument for an auxiliary problem with a mixed boundary condition and on suitable estimates of the critical level for the system case.

In this paper, we deal with a quasilinear elliptic system in exterior domains with dependence on the gradient and coupling of the equations not only inside of the domain, but also on the boundary. We prove the existence of positive, negative or sign changing weak solutions. Our approach relies on an aproximation argument and an adequate elliptic “a priori” estimate.

We deal with the existence of solutions to a critical elliptic system involving the fractional Laplacian operator. We consider the primitive of the nonlinearity interacting with the spectrum of the operator. The one side resonant case is also considered. Variational methods are used to obtain the existence, and our result improves earlier results of the authors.

The interplay between crystallization and glass transition in the archetypal ionic liquid 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide, [C2C1im][NTf2], has been studied as a function of pressure up to ca. 12 GPa. Besides heterogeneous crystal nucleation, homogeneous nucleation in the sample inside the diamond anvil cell was also observed depending on compression/decompression rate. Amorphization of the crystal and glass formation under pressure has been followed by synchrotron X-ray diffraction. The characteristic Raman bands of the [NTf2]− anion provide a microscopic probe of the different phases. The crystalline phase is composed of the [NTf2]− cisoid conformer, but moisture implies formation of crystal with the transoid conformer. Raman spectra show that crystalline phases might become microscopically heterogeneous because of [NTf2]− conformational disorder. Raman mapping reveals the order–disorder evolution from crystal to glass. Crystals of [C2C1im][NTf2] formed under high pressure a...

Ionic liquids are an interesting class of materials due to their distinguished properties, allowing their use in an impressive range of applications, from catalysis to hypergolic fuels. However, the reactivity triggered by the application of high pressure can give rise to a new class of materials, which is not achieved under normal conditions. Here, we report on the high-pressure chemical instability of the ionic liquid 1-allyl-3-methylimidazolium dicyanamide, [allylC1im][N(CN)2], probed by both Raman and IR techniques and supported by quantum chemical calculations. Our results show a reaction occurring above 8 GPa, involving the terminal double bond of the allyl group, giving rise to an oligomeric product. The results presented herein contribute to our understanding of the stability of ionic liquids, which is of paramount interest for engineering applications. Moreover, gaining insight into this peculiar kind of reactivity could lead to the development of new or alternative synthetic routes to achieve, for example, poly(ionic liquids).

Assembling a diamond anvil cell for high-pressure measurements involves placing in a gasket hole the sample of interest, a pressure transmitting fluid, and a material for pressure calibration. In this communication, we propose the use of ionic liquids containing the bis(trifluoromethylsulfonyl)imide anion ([Tf2N]-), [(CF3SO2)2 N]-, as a simultaneous pressure transmitting and calibrant material for high-pressure Raman spectroscopy measurements of solid samples that are not soluble in ionic liquids. The position of the characteristic Raman band of the [Tf2N]- anion at 740 cm−1 exhibits linear frequency shift for pressures up to 2.5 GPa. High-pressure Raman spectra of different ionic liquids containing the same anion indicate that the actual magnitude of the pressure-induced frequency shift of the [Tf2N]- normal mode depends on the counterion, the typical shift being 4.2 cm−1/GPa. Ionic liquids based on the [Tf2N]- anion are also good pressure transmitting mediums because hydrostatic condition is kept at high pressure, and no crystallization is observed up to 4.0 GPa. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, we study the existence of homoclinic solutions for the following class of second order ordinary differential equations (ODEs) (P) { − ( A ( u ( t ) ) u ′ ( t ) ) ′ + u ( t ) = h ( t , u ( t ) ) + g ( t , u ′ ( t ) ) , t ∈ R u ( ± ∞ ) = u ′ ( ± ∞ ) = 0 , where A , h and g are nonnegative continuous functions. Using Galerkin’s Method and assuming additional hypotheses on A , h and g , we show the existence of a homoclinic solution to problem (P) .

The vibrational spectra and crystal structures of four lanthanide and potassium salts of 3,5-bis(dicyanomethylene)cyclopentane-1,2,4-trionate ( C 1 1 N 4 O 3 2 - ), known as croconate violet (CV), are described in this work. All LnKC 22 N 8 O 6 (Ln = La +3 , Nd +3 , Gd +3 and Ho +3 ) compounds are isostructural, crystallizing in the triclinic P 1 ¯ space group. In each compound the lanthanide ion is acting as both monodentate and chelate metal sites, whereas the potassium presents only monodentate coordination. The crystal structure shows the formation of a periodic 2D structure extended by K–N bonds parallel to the crystallographic [0 0 1] direction; these 2D sheets form hydrogen bonds with water molecules giving rise to a 3D extended arrangement. It is not possible to observe any type of π-interaction and the main forces responsible to stabilize the structures are the hydrogen bonds. The vibrational spectra of all the compounds are very similar, and the most important vibrational markers for the croconate violet ion, namely the ν (C N) and ν (C O) modes, behave differently: the ν (C N) modes are not shifted by the presence of the lanthanide ion species, only showing small band intensity differences, whereas the ν (C O) bands are shifted to higher wavenumbers, due to their coordination to the metal sites.

Ionic liquids with cyano anions have long been used because of their unique combination of low-melting temperatures, reduced viscosities, and increased conductivities. Recently we have shown that cyano anions in ionic liquids are particularly interesting for their potential use as electron donors to excited state photo-acceptors [B. Wu et al., J. Phys. Chem. B 119, 14790-14799 (2015)]. Here we report on bulk structural and quantum mechanical results for a series of ionic liquids based on the 1-ethyl-3-methylimidazolium cation, paired with the following five cyano anions: SeCN(-), SCN(-), N(CN)2 (-), C(CN)3 (-), and B(CN)4 (-). By combining molecular dynamics simulations, high-energy X-ray scattering measurements, and periodic boundary condition DFT calculations, we are able to obtain a comprehensive description of the liquid landscape as well as the nature of the HOMO-LUMO states for these ionic liquids in the condensed phase. Features in the structure functions for these ionic liquids are somewhat different than the commonly observed adjacency, charge-charge, and polarity peaks, especially for the bulkiest B(CN)4 (-) anion. While the other four cyano-anion ionic liquids present an anionic HOMO, the one for Im2,1 (+)/B(CN)4 (-) is cationic.

Combining truncature techniques with a variational approach we establish an existence result for nontrivial periodic solutions for a class of fourth-order ordinary differential equations involving extended Fisher–Kolmogorov and Swift–Hohenberg equations.

In this paper we prove existence and uniqueness of positive solutions of nonlinear singular biharmonic elliptic system in smooth bounded domains, with coupling of the equations, under Navier boundary condition. The solution is constructed through an approximating process based on a priori estimates, regularity up to the boundary and Hardy-Sobolev inequality.

Raman spectroscopy has been used to study phase transitions of ionic liquids based on the triflate anion, [TfO](-), as a function of pressure or temperature. Raman spectra of ionic liquids containing the cations 1-butyl-3-methylimidazolium, [C4C1Im](+), 1-octyl-3-methylimidazolium, [C8C1Im](+), 1-butyl-2,3-dimethylimidazolium, [C4C1C1Im](+), and 1-butyl-1-methylpyrrolidinium, [C4C1Pyr](+), were compared. Vibrational frequencies and binding energy of ionic pairs were calculated by quantum chemistry methods. The ionic liquids [C4C1Im][TfO] and [C4C1Pyr][TfO] crystallize at 1.0 GPa when the pressure is increased in steps of ∼ 0.2 GPa from the atmospheric pressure, whereas [C8C1Im][TfO] and [C4C1C1Im][TfO] do not crystallize up to 2.3 GPa of applied pressure. The low-frequency range of the Raman spectrum of [C4C1Im][TfO] indicates that the system undergoes glass transition, rather than crystallization, when the pressure applied on the liquid has been increased above 2.0 GPa in a single step. Strong hysteresis of spectral features (frequency shift and bandwidth) of the high-pressure crystalline phase when the pressure was released stepwise back to the atmospheric pressure has been found .

Raman band shape analysis of the C ≡ N stretching mode was carried out for ionic liquids based on [SCN] − , [N(CN) 2 ] − , [C(CN) 3 ] − , and [B(CN) 4 ] − , with a common cation, 1-ethyl-3-methylimidazolium, [C 2 C 1 im] + . Vibrational and reorientational time correlation functions were obtained by Fourier transforming isotropic and anisotropic Raman spectra. Comparison is provided between results for [C 2 C 1 im][SCN] and literature data available for molten alkali thiocyanates. Vibrational dephasing of the ω CN mode in [C 2 C 1 im][SCN] belongs to the regime of inhomogeneous broadening as it is determined by the distribution of vibrational frequencies, ω 2 >. Vibrational dephasing in [N(CN) 2 ] − , [C(CN) 3 ] − , and [B(CN) 4 ] − , depends on both the ω 2 > and the correlation time of frequency fluctuation. Decay rates of reorientational time correlation functions within the range of 1.0 ps do not correlate with the viscosity of these ionic liquids. Molecular dynamics simulations of these ionic liquids provide support to the experimental finding that the reorientational time correlation function of [SCN] − decays faster than the other anions. The Raman band shape analysis suggests that the short-time dynamics of these ionic liquids is structure-limited because of steric hindrance.

Raman spectroscopy has been used to decipher structural rearrangements in the protic ionic liquid propylammonium nitrate, [C3H7NH3][NO3], as a function of temperature (180-420 K) at atmospheric pressure and as a function of pressure (0.1 MPa-2.0 GPa) at room temperature. Spectral modifications of the Raman bands belonging to the anion and cation normal modes indicate structural changes occurring in both the polar and nonpolar nanoscale domains of [C3H7NH3][NO3]. The crystalline phase of [C3H7NH3][NO3] at low temperature has cations in the anti conformation and undertakes a transition with increasing temperature to a phase with cations mostly in the gauche conformation. The distorted network of hydrogen bonds gives a distribution of local environments around the anions that remains in the normal liquid phase at high temperature. The sample under high pressure might become microscopically heterogeneous, allowing for micro-Raman imaging of different ordered phases of [C3H7NH3][NO3] in a diamond anvil cell.

X-ray scattering experiments at room temperature were performed for the ionic liquids n-butyl-trimethylammonium bis(trifluoromethanesulfonyl)imide, [N1114][NTf2], and methyl-tributylammonium bis(trifluoromethanesulfonyl)imide, [N1444][NTf2]. The peak in the diffraction data characteristic of charge ordering in [N1444][NTf2] is shifted to longer distances in comparison to [N1114][NTf2], but the peak characteristic of short-range correlations is shifted in [N1444][NTf2] to shorter distances. Molecular dynamics (MD) simulations were performed for these ionic liquids using force fields available from the literature, although with new sets of partial charges for [N1114](+) and [N1444](+) proposed in this work. The shifting of charge and adjacency peaks to opposite directions in these ionic liquids was found in the static structure factor, S(k), calculated by MD simulations. Despite differences in cation sizes, the MD simulations unravel that anions are allowed as close to [N1444](+) as to [N1114](+) because anions are located in between the angle formed by the butyl chains. The more asymmetric molecular structure of the [N1114](+) cation implies differences in partial structure factors calculated for atoms belonging to polar or non-polar parts of [N1114][NTf2], whereas polar and non-polar structure factors are essentially the same in [N1444][NTf2]. Results of this work shed light on controversies in the literature on the liquid structure of tetraalkylammonium based ionic liquids.

Phase transitions of the ionic liquids n-butyl-trimethylammonium bis(trifluoromethanesulfonyl)imide, [N1114][NTf2], and methyl-tributylammonium bis(trifluoromethanesulfonyl)imide, [N1444][NTf2], were investigated by differential scanning calorimetry (DSC), X-ray diffraction (XRD) measurements, and Raman spectroscopy. XRD and Raman spectra were obtained as a function of temperature at atmospheric pressure, and also under high pressure at room temperature using a diamond anvil cell (DAC). [N1444][NTf2] experiences glass transition at low temperature, whereas [N1114][NTf2] crystallizes or not depending on the cooling rate. Both the ionic liquids exhibit glass transition under high pressure. XRD and low-frequency Raman spectra provide a consistent physical picture of structural ordering-disordering accompanying the thermal events of crystallization, glass transition, cold crystallization, pre-melting, and melting. Raman spectra in the high-frequency range of some specific cation and anion normal modes reveal conformational changes of the molecular structures along phase transitions.

This paper deals with a class of the semilinear elliptic equations of the Hénon-type in hyperbolic space. The problem involves a logarithm weight in the Poincaré ball model, bringing singularities on the boundary. Considering radial functions, a compact Sobolev embedding result is proved, which extends a former Ni result made for a unit ball in [Formula: see text] Combining this compactness embedding with the Mountain Pass Theorem, a result of the existence of positive solution is established.

Low frequency Raman spectra of ionic liquids have been obtained as a function of pressure up to ca. 4.0 GPa at room temperature and as a function of temperature along the supercooled liquid and glassy state at atmospheric pressure. Intermolecular vibrations are observed at ~20, ~70, and ~100 cm(-1) at room temperature in ionic liquids based on 1-alkyl-3-methylimidazolium cations. The component at ~100 cm(-1) is assigned to librational motion of the imidazolium ring because it is absent in non-aromatic ionic liquids. There is a correspondence between the position of intermolecular vibrational modes in the normal liquid state and the spectral features that the Raman spectra exhibit after partial crystallization of samples at low temperatures or high pressures. The pressure-induced frequency shift of the librational mode is larger than the other two components that exhibit similar frequency shifts. The lowest frequency vibration observed in a glassy state corresponds to the boson peak observed in light and neutron scattering spectra of glass-formers. The frequency of the boson peak is not dependent on the length scale of polar∕non-polar heterogeneity of ionic liquids, it depends instead on the strength of anion-cation interaction. As long as the boson peak is assigned to a mixing between localized modes and transverse acoustic excitations of high wavevectors, it is proposed that the other component observed in Raman spectra of ionic liquids has a partial character of longitudinal acoustic excitations.