Study on factors governing the conductivity performance of acylated chitosan-NaI electrolyte system

Hexanoyl chitosan and lauroyl chitosan were prepared by acyl modification of chitosan. Films of hexanoyl chitosan- and lauroyl chitosan-based polymer electrolytes incorporated with different weight concentrations of sodium iodide (NaI) were prepared using the solution casting technique. FTIR and differential scanning calorimetry (DSC) results suggested that NaI interacted with both hexanoyl chitosan and lauroyl chitosan. Maximum conductivities of 1.3 × 10^?6 and 1.1 × 10^?8 S cm^?1 are achieved for hexanoyl chitosan and lauroyl chitosan, respectively. Higher conductivity in hexanoyl chitosan is attributed to higher ion mobility as supported by DSC results. The dielectric constants of neat hexanoyl chitosan and lauroyl chitosan are 2.7 and 1.9, respectively, estimated from impedance spectroscopy. Higher dielectric constant of hexanoyl chitosan resulted in greater NaI dissociation and hence higher conductivity. Deconvolution of O═C-NHR and OCOR bands of polymer has been carried out to estimate the amount of dissociated Na⁺ ions from NaI. The findings were in good agreement with conductivity results. In order to assess quantitatively, the conductivity, parameter number, n, and mobility, μ, of ions were calculated using impedance spectroscopy. XRD results showed the influence of NaI on the crystalline content of the electrolyte system. Sample with lower crystalline content exhibited higher conductivity.     

Thermal and electrochemical properties of poly(butylene sulfite)-based polymer electrolyte

Poly(butylene sulfite) (poly-1) was synthesized by cationic ring-opening polymerization of butylene sulfite (1), which was prepared by the reaction of 1,4-butanediol and thionyl chloride, with trifluoromethanesulfonic acid (TfOH) in bulk. The polymer electrolytes composed of poly-1 with lithium salts such as bis(trifluoromethanesulfonyl)imide (LiN(SO₂CF₃)₂, LiTFSI) and bis(fluorosulfonyl)imide (LiN(SO₂F)₂, LiFSI) were prepared, and their ionic conductivities, thermal, and electrochemical properties were investigated. Ionic conductivities of the polymer electrolytes for the poly-1/LiTFSI system increased with lithium salt concentrations, reached maximum values at the [LiTFSI]/[repeating unit] ratio of 1/10, and then decreased in further more salt concentrations. The highest ionic conductivity values at the [LiTFSI]/[repeating unit] ratio of 1/10 were 2.36?×?10^?4 S/cm at 80 °C and 1.01?×?10^?5 S/cm at 20 °C. On the other hand, ionic conductivities of the polymer electrolytes for the poly-1/LiFSI system increased with an increase in lithium salt concentrations, and ionic conductivity values at the [LiFSI]/[repeating unit] ratio of 1/1 were 1.25?×?10^?3 S/cm at 80 °C and 5.93?×?10^?5 S/cm at 20 °C. Glass transition temperature (T _g) increased with lithium salt concentrations for the poly-1/LiTFSI system, but T _g for the poly-1/LiFSI system was almost constant regardless of lithium salt concentrations. Both polymer electrolytes showed high transference number of lithium ion: 0.57 for the poly-1/LiTFSI system and 0.56 for the poly-1/LiFSI system, respectively. The polymer electrolytes for the poly-1/LiTFSI system were thermally more stable than those for the poly-1/LiFSI system.     

Enhancement of the electrochemical properties with the effect of alkali metal systems on PEO/PVdF-HFP complex polymer electrolytes

Polymer blend electrolytes based on poly(ethylene oxide) (PEO) and poly(vinylidene fluoride-hexafluoropropylene) (PVdF-HFP) were prepared by using different lithium salts LiX (X = ClO₄, BF₄, CF₃SO₃, and N [CF₃SO₂]₂) using solution casting technique. To confirm the structure and complexation of the electrolyte films, the prepared electrolytes were subjected to X-ray diffraction (XRD) and Fourier transform infrared spectroscopy (FTIR) analysis. Alternating current (AC) impedance analysis was performed for all the electrolyte samples at various temperatures from 303 to 343 K. The result suggests that among the various lithium salts, LiN[CF₃SO₂]₂-based electrolytes exhibited the highest ionic conductivity at 8.20 × 10^?4 S/cm. The linear variation of the ionic conductivity of the polymer electrolytes with increasing temperature suggests the Arrhenius-type thermally activated process. Activation energies were found to decrease when doping with lithium imide salt. The dielectric behavior has been analyzed using dielectric permittivity (ε*), electric modulus (M*), and dissipation factor (tanδ) of the samples. Cyclic voltammetry has been performed for the electrolyte films to study their cyclability and reversibility. Thermogravimetric and differential thermal analysis (TG/DTA) was used to ascertain the thermal stability of the electrolytes, and the porous nature of the electrolytes was identified using scanning electron microscopy via ion hopping conduction. Surface morphology of the sample having maximum conductivity was studied by an atomic force microscope (AFM).     

Preparation,characterization, thermal,and electrical properties of lithium salt-magnetite/epoxidized natural rubber composite polymer electrolytes

The various solid lithium salt-magnetite/epoxidized natural rubber (LiX-Fe₃O₄/ENR) composite polymer electrolytes (CPEs) were obtained via solvent casting method. The CPEs were characterized using SEM/X-mapping, TEM, FTIR, DSC, TG analysis, and impedance spectroscopy. The CPEs demonstrate similar thermal behavior as their respective LiX-ENR polymer electrolytes (PEs) where X?=?COOCF₃ ^?, I^?, CF₃SO₃ ^?, and ClO₄ ^?. The presence of Fe₃O₄ particles in the CPEs enhanced the conductivity where an improvement of 1–2 orders of magnitude in CPEs’ conductivity is observed as compared to the PE counterparts. The CPEs showed an ion transference number (t _ion) of >0.92 suggesting that ionic conduction remain dominant. In these CPEs, the Fe₃O₄ particles facilitated the movement of charge carrier via space-charge creation at the particle/polymer interface as well as increasing the amorphocity of the ENR matrix. The LiX (where X?=?COOCF₃ ^?, I^?, and CF₃SO₃ ^?), however, gave no significant effect to the thermal stability of ENR in the CPE while LiClO₄ destabilized the ENR in the CPE. In contrast, the LiBF₄-Fe₃O₄/ENR was thermally less stable (<20 °C) as compared to the respective LiBF₄-ENR PE. Nevertheless, the activation energy for the degradation (E _d ) of ENR in the CPEs is higher than the Fe₃O₄/ENR composite.     

Green electrolytes based on dextran-chitosan blend and the effect of NH<Subscript>4</Subscript>SCN as proton provider on the electrical response studies

Dextran-chitosan blend added with ammonium thiocyanate (NH₄SCN)-based solid polymer electrolytes are prepared by solution cast method. The interaction between the components of the electrolyte is verified by Fourier transform infrared (FTIR) analysis. The blend of 40 wt% dextran-60 wt% chitosan is found to be the most amorphous ratio. The room temperature conductivity of undoped 40 wt% dextran-60 wt% chitosan blend film is identified to be (3.84?±?0.97)?×?10^?10 S cm^?1. The inclusion of 40 wt.% NH₄SCN to the polymer blend has optimized the room temperature conductivity up (1.28?±?0.43)?×?10^?4 S cm^?1. Result from X-ray diffraction (XRD) and differential scanning calorimetry (DSC) analysis shows that the electrolyte with the highest conductivity value has the lowest degree of crystallinity (χ _c) and the glass transition temperature (T _g), respectively. Temperature-dependence of conductivity follows Arrhenius theory. From transport analysis, the conductivity is noticed to be influenced by the mobility (μ) and number density (n) of ions. Conductivity trend is further verified by field emission scanning electron microscopy (FESEM) and dielectric results.     

Dielectric relaxation and ac conductivity of sodium tungsten phosphate glasses

Studies of dielectric relaxation and ac conductivity have been made on three samples of sodium tungsten phosphate glasses over a temperature range of 77–420 K. Complex relative permitivity data have been analyzed using dielectric modulus approach. Conductivity relaxation frequency increases with the increase of temperature. Activation energy for conductivity relaxation has also been evaluated. Measured ac conductivity (σ_m(ω)) has been found to be higher than σ_dc at low temperatures whereas at high temperature σ_m(ω) becomes equal to σ_dc at all frequencies. The ac conductivity obeys the relation σ_ac(ω)=Aω ^S over a considerable range of low temperatures. Values of exponent S are nearly equal to unity at about 78 K and the values decrease non-linearly with the increase of temperature. Values of the number density of states at Fermi level (N(E _F)) have been evaluated at 80 K assuming values of electron wave function decay constant α to be 0.5 (Å)^?1. Values of N(E _F) have the order 10²⁰ which are well within the range suggested for localized states. Present values of N(E _F) are smaller than those for tungsten phosphate glasses.     

Ion dynamics in methylcellulose–LiBOB solid polymer electrolytes

A polymer electrolyte system comprising methylcellulose (MC) as the host polymer and lithium bis(oxalato) borate (LiBOB) as the lithium ion source has been prepared via the solution cast technique. The electrolyte with the highest conductivity of 2.79 μS cm^?1 has a composition of 75 wt% MC–25 wt% LiBOB. The mobile ion concentration (n) in this sample was estimated to be 5.70?×?10²⁰ cm^?3. A good correlation between ionic conductivity, dielectric constant, and free ion concentration has been observed. The ratio of mobile ion number density (n) at a particular temperature to the concentration n ₀ of free ions at T?=?∞ (n/n ₀) and the power law exponents (s) exhibit opposite trends when varied with salt concentration.     

Studies on structural and electrical properties of Mg<Subscript>0. 5+y</Subscript> (Zr<Subscript>2-y</Subscript>Fe<Subscript>y</Subscript>) <Subscript>2</Subscript> (PO<Subscript>4</Subscript>) <Subscript>3</Subscript> ceramic electrolytes

The sample of Mg_{0. 5+y} (Zr_1-y Fe_y) ₂ (PO₄) ₃ (0.0 ≤y ≤0.5) was synthesized using the sol-gel method. The structures of the samples were investigated using X-ray diffraction and Fourier transform infrared spectroscopy measurement. XRD studies showed that samples had a monoclinic structure which was iso-structured with the parent compound, Mg_0.5Zr (PO₄) ₃. The complex impedance spectroscopy was carried out in the frequency range 1–6 MHz and temperature range 303 to 773 K to study the electrical properties of the electrolytes. The substitutions of Fe³⁺ with Zr⁴⁺ in the Mg_0.5Zr (PO₄) ₃ structure was introduced as an extrainterstitial Mg²⁺ ion in the modified structured. The compound of Mg_0.5+y (Zr_1-y Fe_y)₂(PO₄)₃ with y?=?0.4 gives a maximum conductivity value of 1.25?×?10^?5 S cm^?1 at room temperature and 7.18?×?10^?5 S cm^?1 at 773 K. Charge carrier concentration, mobile ion concentration, and ion hopping rate are calculated by fitting the conductance spectra to power law variation, σ _ac (ω)?=?σ _o ? +?Aω ^α . The charge carrier concentration and mobile ion concentration increases with increase of Fe³⁺ inclusion. This implies the increase in conductivity of the compounds was due to extra interstitial Mg²⁺ ions.     

Derivation of Hamaker Dispersion Energy of Amorphous Carbon Surfaces in Contact with Liquids Using Photoelectron Energy-Loss Spectra

Hamaker interaction energies and cutoff distances have been calculated for disordered carbon films, in contact with purely dispersive (diiodomethane) or polar (water) liquids, using their experimental dielectric functions ε (q, ω) obtained over a broad energy range. In contrast with previous works, a q-averaged <ε (q, ω) >  _q is derived from photoelectron energy-loss spectroscopy (XPS-PEELS) where the energy loss function (ELF) < Im[?1/ε (q, ω)] >  _q is a weighted average over allowed transferred wave vector values, q, given by the physics of bulk plasmon excitation. For microcrystalline diamond and amorphous carbon films with a wide range of (sp³/sp² + sp³) hybridization, non-retarded Hamaker energies, A ₁₃₂ (L < 1 nm), were calculated in several configurations, and distance and wavenumber cutoff values were then calculated based on A ₁₃₂ and the dispersive work of adhesion obtained from contact angles. A geometric average approximation, H _0?CVL ?=?(H _0?CVC H _0?LVL )^1/2, holds for the cutoff separation distances obtained for carbon-vacuum-liquid (CVL), carbon-vacuum-carbon (CVC) and liquid-vacuum-liquid (LVL) equilibrium configurations. The linear dependence found for A _CVL, A _CLC and A _CLV values as a function of A _CVC, for each liquid, allows predictive relationships for Hamaker energies (in any configuration) using experimental determination of the dispersive component of the surface tension, \( {\gamma}_{CV}^d \), and a guess value of the cutoff distance H _0?CVC of the solid.

Open image in new window

Graphical Abstract

     

Design of amphiphilic poly(vinylidene fluoride-<Emphasis Type="Italic">co</Emphasis>-hexafluoropropylene)-based gel electrolytes for high-performance lithium-ion batteries

Gel polymer electrolytes (GPE) based on electrospun polymer membranes, poly(vinylidene fluoride-co-hexafluoropropylene), grafted poly(poly(ethylene glycol) methyl ether methacrylate) (PVDF-HFP-g-PPEGMA), and poly(vinylidene difluoride-co-hexafluoropropylene) (PVDF-HFP) are prepared for lithium ion batteries by incorporating with 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide (BMITFSI). The uniform porosity and the compatibility of blend electrospun membranes avoiding the pore blocking are beneficial to enhance the electrolyte uptakes. The GPE based on the fibrous PVDF-HFP-g-PPEGMA/PVDF-HFP activated with 1 M LiTFSI (BMITFSI) show a maximum ionic conductivity of 2.3 × 10^?3 S cm^?1 at room temperature and electrochemical stability of up to 5.2 V. The Li/GPE/LiFePO₄ cells with GPE based on PVDF-HFP-g-PPEGMA/PVDF-HFP blend electrospun membrane deliver specific capacities of 163, 141, and 125 mAh g^?1 at 0.1, 0.5, and 1C rates, respectively, and remains well after 50 cycles for each rate. Therefore, the novel GPE have been demonstrated to be suitable for lithium-ion battery applications.     

A novel and shortcut method to prepare ionic liquid gel polymer electrolyte membranes for lithium-ion battery

The ionic liquid polymer electrolyte (IL-PE) membrane is prepared by ultraviolet (UV) cross-linking technology with polyurethane acrylate (PUA), methyl methacrylate (MMA), ionic liquid (Py₁₃TFSI), lithium salt (LiTFSI), ethylene glycol dimethacrylate (EGDMA), and benzoyl peroxide (BPO). N-methyl-N-propyl pyrrolidinium bis(trifluoromethanesulfonyl)imide (Py₁₃TFSI) ionic liquid is synthesized by mixing N-methyl-N-propyl pyrrolidinium bromide (Py₁₃Br) and lithium bis(trifluoromethanesulfonyl)imide (LiTFSI). The addition of Py₁₃TFSI to polymer electrolyte membranes leads to network structures by the chain cross-linking. The resultant electrolyte membranes display the room temperature ionic conductivity of 1.37 × 10^?3 S cm^?1 and the lithium ions transference number of 0.22. The electrochemical stability window of IL-PE is about 4.8 V (vs. Li⁺/Li), indicating sufficient electrochemical stability. The interfacial resistances between the IL-PE and the electrodes have the less change after 10 cycles than before 10 cycles. IL-PE has better compatibility with the LiFePO₄ electrode and the Li electrode after 10 cycles. The first discharge performance of Li/IL-PE/LiFePO₄ half-cell shows a capacity of 151.9 mAh g^?1 and coulombic efficiency of 87.9%. The discharge capacity is 131.9 mAh g^?1 with 95.5% coulombic efficiency after 80 cycles. Therefore, the battery using the IL-PE exhibits a good cycle and rate performance.     

A first-principle study on the phase transition,electronic structure,and mechanical properties of three-phase ZrTi<Subscript>2</Subscript> alloy under high pressure*

We employed density-functional theory (DFT) within the generalized gradient approximation(GGA) to investigate the ZrTi₂ alloy, and obtained its structural phase transition,mechanical behavior, Gibbs free energy as a function of pressure, P-V equation of state,electronic and Mulliken population analysis results. The lattice parameters andP-V EOS for α, β and ω phases revealed by ourcalculations are consistent with other experimental and computational values. The elasticconstants obtained suggest that ω-ZrTi₂ and α-ZrTi₂ are mechanically stable, and that β-ZrTi₂ is mechanically unstableat 0 GPa, but becomes more stable with increasing pressure. Our calculated resultsindicate a phase transition sequence of α → ω → β forZrTi₂. Both thebulk modulus B and shear modulus G increase linearly withincreasing pressure for three phases. The G/B values illustrated goodductility of ZrTi₂alloy for three phases, with ω<α<β at0 GPa. The Mulliken population analysis showed that the increment of d electron occupancystabilized the β phase. A low value for B '₀ is the feature of EOS for ZrTi₂ and this softness in the EOS isrepresentative of pressure induced s-d electron transfer.     

Physical properties of FeSe<Subscript>0.5</Subscript>Te<Subscript>0.5</Subscript> single crystals grown under different conditions

We report on structural, magnetic, conductivity, and thermodynamic studies of FeSe_0.5Te_0.5 single crystals grown by self-flux and Bridgman methods. The lowest values of the susceptibility in thenormal state, the highest transition temperature T _c of 14.4 K, and the largest heat-capacity anomaly at T _c were obtained for pure (oxygen-free) samples. The criticalcurrent density j _c of 8.6 × 10⁴A/cm² (at 2 K) achieved in pure samples is attributed to intrinsic inhomogeneity due to disorder at the anion sites. The samples containing an impurity phase of Fe₃O₄ show increased j _c up to2.3 × 10⁵A/cm² due to additional pinning centers. The upper critical field\(H_{c2}\)of ~500 kOe is estimated from the resistivity studyin magnetic fields parallel to the c-axis using a criterion of a 50%drop of the normal state resistivity R _n . The anisotropy ofthe upper critical fieldγ _{H c2} =H ^ab _c2/H _c2 ^c reaches a value ~6 at\(T\longrightarrow T_c\). Extremely low values of the residualSommerfeld coefficient \(\gamma_r\) of about 1 mJ/mol K²,compared to the normal state Sommerfeld coefficient γ _n = 25mJ/mol K² for pure samples indicate a high volume fraction of thesuperconducting phase (up to 97%). The electronic contribution to the specific heat in thesuperconducting state is well described within a single-band BCS model with a temperature dependent gapΔ(0 K) = 27(1) K. A broad cusp-like anomaly in the electronic specific heat observed at low temperatures in samples with suppressed bulk superconductivity is ascribed to a splitting of the ground state of the Fe²⁺ ions at the 2c sites. This contribution is fully suppressed in the ordered state in samples with bulk superconductivity.     

High-temperature complex impedance and modulus spectroscopic studies of doped Na<Subscript>0.5</Subscript>Bi<Subscript>0.5</Subscript>TiO<Subscript>3</Subscript>-BaTiO<Subscript>3</Subscript> ferroelectric ceramics

Lead-free Na_0.5Bi_0.5TiO₃ (NBT) and (1 ? x)Na_0.5Bi_0.5TiO₃ + xBaTiO₃ with x = 0.1 and 0.2 (where x = 0.1 and 0.2 are named as NBT1 and NBT2, respectively), (1 ? y)Na_0.5Bi_0.5TiO₃ + yBa_0.925Nd_0.05TiO₃ with y = 0.1 and 0.2 (where y = 0.1 and 0.2 are named as NBT3 and NBT4, respectively)-based relaxor ferroelectric ceramics were prepared using the sol-gel method. The crystal structure was investigated by X-ray diffraction (XRD) at room temperature (RT). The XRD patterns confirmed the presence of the rhombohedral phase in all the samples. The electrical properties of the present NBT-based samples were investigated by complex impedance and the modulus spectroscopy technique in the temperature range of RT–600 °C. The AC conductivity was found to increase with the substitution of Ba²⁺ ions to the NBT sample whereas it significantly decreased with the addition of Nd³⁺ ions. The more anion vacancies in Ba-added samples and the lower anion vacancies in Nd-added samples were found to be responsible for higher and lower conductivities, respectively.     

Synthesis and characterization of Ce-doped Sm<Subscript>2</Subscript>CuO<Subscript>4 + <Emphasis Type="Italic">δ</Emphasis></Subscript> cathode for IT-SOFC applications

Single-phase solid solutions of Sm_2 ? x Ce _x CuO_4 + δ (0.05 ≤ x ≤ 0.20) with tetragonal structure are synthesized using acetate combustion followed by sintering at 1373 K for 10 h. X-ray powder diffraction and transmission electron microscopy studies confirmed the formation of solid solution in a single phase. Maximum conductivity (σ = 96.0 ± 0.5 S cm^?1 at 973 K) giving composition Sm_1.90Ce_0.10CuO₄ offers the minimum activation energy (E _a = 0.32 ± 0.004 eV) among all prepared compositions. Lowest cathode polarization resistance (R _p = 3.92 ± 0.07 Ω cm² at 973 K) and activation energy (E _a = 1.12 ± 0.03 eV) values across the measured temperature range are obtained for Sm_1.90Ce_0.10CuO₄. The impedance data fitted well to the Gerischer model indicates that a chemical-electrochemical-chemical-type reaction occurred at the mixed electronic-ionic conducting cathode.     

New Bounds on the Maximum Ionization of Atoms

We prove that the maximum number N _c of non-relativistic electrons that a nucleus of charge Z can bind is less than 1.22Z + 3Z ^1/3. This improves Lieb’s upper bound N _c  < 2Z + 1 Lieb (Phys Rev A 29:3018–3028, 1984) when Z ≥ 6. Our method also applies to non-relativistic atoms in magnetic field and to pseudo-relativistic atoms. We show that in these cases, under appropriate conditions, \({\limsup_{Z \to \infty}N_c/Z \le 1.22}\).     

The Boundary Control Approach to the Titchmarsh-Weyl <Emphasis Type="Italic">m</Emphasis>–Function. I. The Response Operator and the <Emphasis Type="Italic">A</Emphasis>–Amplitude

We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the A?amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl m?function associated with the Schrödinger operator H = ?? _x ²  + q(x) on L ₂(0, ∞) with Dirichlet boundary condition at x = 0.     

Reversibler Gleichfeld- und Mikrowellen-Durchschlag in n-Typ-Germanium bei 4,2 °K

The microwave induced breakdown characteristic inn-type germanium at 4.2 °K has been observed and compared with the d.c. induced breakdown characteristic obtained from the same sample. The effective microwave breakdown field intensity is nearly equal to the field intensity observed in d.c. induced breakdown. However, in the breakdown region with conductivities greater than 0.1 ohm^?1 cm^?1 a relaxation effect was found and interpreted qualitatively as momentun relaxation. In the initial breakdown the relaxation time τ _m is small,ω ²τ _m ² being ?1 whereω/2π=9·10⁹s^?1 is the microwave frequency. The relaxation time is determined by predominant neutral impurity scattering with at last 5·10¹⁴ impurities per cm³. This scattering mechanism becomes ineffective when the impurities are ionized by hot carriers. Ionized impurity scattering or acoustic phonon scattering will then be predominant with increased and energy dependent values of τ _m . The increased phase shift between carriers and field causes a decreased energy transfer from the field to the carriers, an accordingly smaller ionization rate, and finally results in a nearly constant a.c. conductivity. The observed anisotropy of the breakdown field intensity is in qualitative agreement with the assumption that only carriers in “hot” valleys of the conduction band initiate the breakdown by impact ionization. The unsufficient quantitative agreement may be due to an inhomogeneity of doping which is suggested by comparing the values of the ohmic low temperature conductivity of the samples.     

A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity

We revisit and prove some convexity inequalities for trace functions conjectured in this paper’s antecedent. The main functional considered is

$ \Phi_{p,q} (A_1,\, A_2, \ldots, A_m) = \left({\rm Tr}\left[\left( \, {\sum\limits_{j=1}^m A_j^p } \, \right) ^{q/p} \right] \right)^{1/q} $

for m positive definite operators A _j . In our earlier paper, we only considered the case q = 1 and proved the concavity of Φ _p,1 for 0 < p ≤ 1 and the convexity for p = 2. We conjectured the convexity of Φ _p,1 for 1 < p < 2. Here we not only settle the unresolved case of joint convexity for 1 ≤ p ≤ 2, we are also able to include the parameter q ≥ 1 and still retain the convexity. Among other things this leads to a definition of an L ^q (L ^p ) norm for operators when 1 ≤ p ≤ 2 and a Minkowski inequality for operators on a tensor product of three Hilbert spaces – which leads to another proof of strong subadditivity of entropy. We also prove convexity/concavity properties of some other, related functionals.

     

Temperature Dependence of the Thermal Conductivity of a Trapped Dipolar Bose-Condensed Gas

The thermal conductivity of a trapped dipolar Bose condensed gas is calculated as a function of temperature in the framework of linear response theory. The contributions of the interactions between condensed and noncondensed atoms and between noncondensed atoms in the presence of both contact and dipole-dipole interactions are taken into account to the thermal relaxation time, by evaluating the self-energies of the system in the Beliaev approximation. We will show that above the Bose-Einstein condensation temperature (T?>?T _BEC ) in the absence of dipole-dipole interaction, the temperature dependence of the thermal conductivity reduces to that of an ideal Bose gas. In a trapped Bose-condensed gas for temperature interval k _B T?<<?n ₀ g _B , E _p ?<<?k _B T (n ₀ is the condensed density and g _B is the strength of the contact interaction), the relaxation rates due to dipolar and contact interactions between condensed and noncondensed atoms change as \( {\tau}_{dd12}^{-1}\propto {e}^{-E/{k}_BT} \) and τ _c12?∝?T ^?5, respectively, and the contact interaction plays the dominant role in the temperature dependence of the thermal conductivity, which leads to the T ^?3 behavior of the thermal conductivity. In the low-temperature limit, k _B T?<<?n ₀ g _B , E _p ?>>?k _B T, since the relaxation rate \( {\tau}_{c12}^{-1} \) is independent of temperature and the relaxation rate due to dipolar interaction goes to zero exponentially, the T ² temperature behavior for the thermal conductivity comes from the thermal mean velocity of the particles. We will also show that in the high-temperature limit (k _B T?>?n ₀ g _B ) and low momenta, the relaxation rates \( {\tau}_{c12}^{-1} \) and \( {\tau}_{dd12}^{-1} \) change linearly with temperature for both dipolar and contact interactions and the thermal conductivity scales linearly with temperature.