Synthesis and negative thermal expansion property of Y2−xLaxW3O12 (0≤x≤2)

Y_2−xLa_xW₃O₁₂ solid solutions were successfully synthesized by the solid state reaction method. The microstructure, hygroscopicity and thermal expansion property of the resulting samples were investigated by X-ray diffraction (XRD), thermogravimetric analysis (TGA), field emission scanning electron microscopy (FESEM) and thermal mechanical analysis (TMA). Results indicate that the structural phase transition of the Y_2−xLa_xW₃O₁₂ changes from orthorhombic to monoclinic with increasing substituted content of lanthanum. The pure phase can form for 0≤x≤0.4 with orthorhombic structure and for 1.5≤x≤2 with monoclinic one. High lanthanum content leads to a low relative density of Y_2−xLa_xW₃O₁₂ ceramic. Thermal expansion coefficients of the Y_2−xLa_xW₃O₁₂ (0≤x≤2) ceramics also vary from −9.59×10⁻⁶ K⁻¹ to 2.06×10⁻⁶ K⁻¹ with increasing substituted content of lanthanum. The obtained Y_0.25La_1.75W₃O₁₂ ceramic shows almost zero thermal expansion and its average linear thermal expansion coefficient is −0.66×10⁻⁶ K⁻¹ from 103 °C to 700 °C.     

Synthesis and tunable thermal expansion properties of Sc2−xYxW3O12 solid solutions

A new series of rare earth solid solutions Sc_2−xY_xW₃O₁₂ was successfully synthesized by the conventional solid-state method. Effects of doping ion yttrium on the crystal structure, morphology and thermal expansion property of as-prepared Sc_2−xY_xW₃O₁₂ ceramics were investigated by X-ray diffraction (XRD), thermogravimetric analysis (TG), field emission scanning electron microscope (FE-SEM) and thermal mechanical analyzer (TMA). Results indicate that the obtained Sc_2−xY_xW₃O₁₂ samples with Y doping of 0≤x≤0.5 are in the form of orthorhombic Sc₂W₃O₁₂-structure and show negative thermal expansion (NTE) from room temperature to 600 °C; while as-synthesized materials with Y doping of 1.5≤x≤2 take hygroscopic Y₂W₃O₁₂·nH₂O-structure at room temperature and exhibit NTE only after losing water molecules. It is suggested that the obvious difference in crystal structure leads to different thermal expansion behaviors in Sc_2−xY_xW₃O₁₂. Thus it is proposed that thermal expansion properties of Sc_2−xY_xW₃O₁₂ can be adjusted by the employment of Y dopant; the obtained Sc_1.5Y_0.5W₃O₁₂ ceramic shows almost zero thermal expansion and its average linear thermal expansion coefficient is −0.00683×10⁻⁶ °C⁻¹ in the 25–250 °C range.     

Synthesis and thermal expansion properties of Y2−xLaxMo3O12 (x=0, 0.5, 2)

Y_2−xLa_xMo₃O₁₂ (x=0, 0.5, 2) ceramics were successfully synthesized by the solid state reaction method. The microstructure, composition and thermal expansion property of the resulting samples were investigated by X-ray diffraction (XRD), thermogravimetric analysis (TGA), field emission scanning electron microscopy (FESEM), energy dispersive spectroscopy (EDS) and dilatometry. Results indicate that the Y_1.5La_0.5Mo₃O₁₂ crystallizes in monoclinic Tb₂Mo₃O₁₂-type structure and it is non-hygroscopic. The Y_1.5La_0.5Mo₃O₁₂ ceramic is denser than the Y₂Mo₃O₁₂ and La₂Mo₃O₁₂ ceramics, and its relative density can reach 94.12% of the theoretical value. Most importantly, it shows almost zero thermal expansion and its thermal coefficient is 0.87×10⁻⁶ K⁻¹ from 178 °C to 600 °C. Y₂Mo₃O₁₂ ceramic shows negative thermal expansion whereas La₂Mo₃O₁₂ ceramic shows positive thermal expansion, their thermal expansion coefficients being−12.06×10⁻⁶ K⁻¹ and 8.88×10⁻⁶×10⁻⁶ K⁻¹, respectively.     

Synthesis and transport properties in La2−xAxMo2O9−δ (A = Ca, Sr, Ba, K) series

The La_2−xA_xMo₂O_9−δ (A = Ca²⁺, Sr²⁺, Ba²⁺ and K⁺) series has been synthesised as nanocrystalline materials via a modification of the freeze-drying method. The resulting materials have been characterised by X-ray diffraction (XRD), thermal analysis (TG/DTA, DSC), scanning electron microscopy (SEM) and high-resolution transmission electron microscopy (HRTEM). The high-temperature β-polymorph is stabilised for dopant content x > 0.01. The nanocrystalline powders were used to obtain dense ceramic materials with optimised microstructure and relative density >95%. The overall conductivity determined by impedance spectroscopy depends on both the ionic radius and dopant content. The conductivity decreases slightly as the dopant content increases in addition a maximum conductivity value was found for Sr²⁺ substitution, which show an ionic radii slightly higher than La³⁺ (e.g. 0.08 S cm⁻¹ for La₂Mo₂O₉ and 0.06 S cm⁻¹ for La_1.9Sr_0.1Mo₂O_9−δ at 973 K). The creation of extrinsic vacancies upon substitution results in a wider stability range under reducing conditions and prevents amorphisation, although the stability is not enhanced significantly when compared to samples with higher tungsten content. These materials present high thermal expansion coefficients in the range of (13-16) × 10⁻⁶ K⁻¹ between room temperature and 753 K and (18-20) × 10⁻⁶ K⁻¹ above 823 K. The ionic transport numbers determined by a modified emf method remain above 0.98 under an oxygen partial pressure gradient of O₂/air and decreases substantially under wet 5% H₂-Ar/air when approaching to the degradation temperature above 973 K due to an increase of the electronic contribution to the overall conductivity.     

Synthesis and NTC properties of YCr1−xMnxO3 ceramics sintered under nitrogen atmosphere

YCr_1−xMn_xO₃ (0 ≤ x ≤ 0.8) negative temperature coefficient (NTC) compositions were synthesized by classical solid state reaction at 1200 °C, and sintered under nitrogen atmosphere at 1500 °C and 1600 °C. XRD patterns analysis has revealed that for x ≤ 0.6, the structure consists of a solid solution of an orthorhombic perovskite YCrO₃ phase with Mn substitute for Cr. For x ≥ 0.8, a second phase with a structure similar to the hexagonal YMnO₃ phase appears. SEM images and calculated open porosity have shown that the substitution of Mn for Cr results in a decrease in porosity. Whatever the sintering temperature, the electrical characterizations (between 25 and 900 °C) have shown that the increase in the manganese content involves the decrease in both resistivity and material constant B (parameter which characterizes the thermal sensitivity of material) when x ≤ 0.6. The magnitude order of the resistivity at 25 °C is of 10⁴-10⁸ Ω cm and activation energies vary from 0.28 to 0.99 eV at low and high temperatures, respectively.     

Preparation and characterization of negative thermal expansion Sc2W3O12/Cu core-shell composite

A composite of Sc₂W₃O₁₂/Cu where Sc₂W₃O₁₂, the core, is coated by the Cu shell was synthesized using simple electroless plating method. As-prepared Sc₂W₃O₁₂/Cu composites were characterized by X-ray diffraction (XRD), scanning electron microscopy (SEM), energy dispersive spectrometry (EDS) and thermomechanical analyzer (TMA) techniques. The study results show that the Pd-Sn activator was successfully formed on the surface of Sc₂W₃O₁₂ after the sensitization and activation. In the electroless plating process, Cu nanocrystals formed firstly, and then grew together to form a continuous coating. Sc₂W₃O₁₂/Cu core-shell composites exhibit a negative linear coefficient of thermal expansion CTE = −4.47 × 10⁻⁶ °C⁻¹ from room temperature to 200 °C.     

Electrical conductivity and redox stability of La2Mo2−xWxO9 materials

A series of compounds La₂Mo_2−xW_xO₉ (x = 0-2) were synthesized using a freeze-dried precursor method at relatively low temperatures (673-823 K). These materials were characterised by thermogravimetric and differential thermal analysis (TG/DTA), differential scanning calorimetric (DSC), X-ray diffraction (XRD), and transmission electron microscopy (TEM) and dilatometric measurements. Oxygen stoichiometry was evaluated by coulometric titration and thermogravimetric analysis at 873-1273 K. The ionic and electronic conductivities of these materials were analysed by impedance spectroscopy and a Hebb-Wagner ion-blocking method under moderately reducing conditions. The presence of W⁶⁺ leads to an increase of the stability range (about 10⁻¹⁶ Pa for La₂Mo_0.5W_1.5O₉ at 1073 K) and prevents oxygen loss and amorphisation. Within the stability range, the electronic conductivity increases gradually as the temperature increases and as the oxygen partial pressure reduces. This indicates that the electronic transport is mainly n-type as a result of the oxygen-content decreasing in the molybdate lattice. Further reduction of the oxygen partial pressure gave rise to the decomposition of La₂Mo_2−xW_xO₉, leading to the formation of new phases with molybdenum in lower oxidation states, which further enhances the electronic conductivity. The results of the coulometric titration and the thermogravimetric studies under a dry 5% H₂/Ar flow suggest that tungsten doped lanthanum molybdate materials can be used as electrolyte only at low temperature and under moderate reducing conditions.     

Kinetics and PEMFC performance of RuxMoySez nanoparticles as a cathode catalyst

Kinetics of Ru_xMo_ySe_z nanoparticles dispersed on carbon powder was studied in 0.5 M H₂SO₄ electrolyte towards the oxygen reduction reaction (ORR) and as cathode catalysts for a proton exchange membrane fuel cell (PEMFC). Ru_xMo_ySe_z catalyst was synthesized by decarbonylation of transition-metal carbonyl compounds for 3 h in organic solvent. The powder was characterized by X-ray diffraction (XRD), and transmission electron microscopy (TEM) techniques. Catalyst is composed of uniform agglomerates of nanocrystalline particles with an estimated composition of Ru₆Mo₁Se₃, embedded in an amorphous phase. The electrochemical activity was studied by rotating disk electrode (RDE) and rotating ring-disk electrode (RRDE) techniques. Tafel slopes for the ORR remain invariant with temperature at −0.116 V dec⁻¹ with an increase of the charge transfer coefficient in dα/dT = 1.6 × 10⁻³, attributed to an entropy turnover contribution to the electrocatalytic reaction. The effect of temperature on the ORR kinetics was analyzed resulting in an apparent activation energy of 45.6 ± 0.5 kJ mol⁻¹. The catalyst generates less than 2.5% hydrogen peroxide during oxygen reduction. The Ru_xMo_ySe_z nanoparticles dispersed on a carbon powder were tested as cathode electrocatalyst in a single fuel cell. The membrane-electrode assembly (MEA), included Nafion^® 112 as polymer electrolyte membrane and commercial carbon supported Pt (10 wt%Pt/C-Etek) as anode catalyst. It was found that the maximum performance achieved for the electro-reduction of oxygen was with a loading of 1.0 mg cm⁻² Ru_xMo_ySe_z 20 wt%/C, arriving to a power density of 240 mW cm⁻² at 0.3 V and 80 °C.     

Tuning the phase transition temperature of Cr2(MoO4)3 by A-site substitution of scandium

A series of Cr_2-xSc_x(MoO₄)₃ solid solutions with tunable monoclinic-to-orthorhombic phase transition temperature have been synthesized via solid-state reaction. X-ray diffraction (XRD), transmission electron microscopy (TEM) and X-ray photoelectron spectroscopy (XPS) results show that all synthesized Cr_2-xSc_x(MoO₄)₃ (0?< x?≤?1.4) solid solutions are single phased with no impurities identified, which reveals that Cr³⁺ has been substituted by Sc³⁺ in Cr₂(MoO₄)₃. Monoclinic to orthorhombic phase transition temperature of Cr_2-xSc_x(MoO₄)₃ can be effectively tuned from 372?°C to room temperature as the substituted Sc³⁺-content (x) varies from 0 to 1.4. The synthesized Cr_0.6Sc_1.4(MoO₄)₃ crystalizes in an orthorhombic structure at room temperature, exhibiting anisotropic negative thermal expansion throughout the testing temperature range. The coefficient of thermal expansion measured by thermal mechanical analyzer (TMA) for Cr_0.6Sc_1.4(MoO₄)₃ is ?11.17?×?10^?6 °C^?1 in the testing temperature range of 30–600?°C. Moreover, the crystallization, micromorphology, density and coefficient of thermal expansion of Cr_0.6Sc_1.4(MoO₄)₃ are obviously sensitive to the twice sintering temperature, whereas none of such sensitivity is found for the phase transition temperature.     

Synthesis of negative thermal expansion materials ZrW2−xMoxO8 (0 ≤ x ≤ 2) using hydrothermal method

Negative thermal expansion materials ZrW_2−xMo_xO₈ (0 ≤ x ≤ 2) have been successfully synthesized by the reaction of a mixture of ammonium tungstate and ammonium molybdate with zirconium oxynitrate using a hydrothermal method. Effect of substituted ion Mo on the microstructure, α-to-β and cubic to trigonal phase transition in resulting ZrW_2−xMo_xO₈ powders was examined by the XRD experiments. It was found that the structural phase transition temperature decreased slightly with increasing substituted content. The cubic to trigonal phase transition was also influenced by substituted content. The resulting products decomposed to WO₃/MoO₃ and ZrO₂ as temperature increasing when x ≤ 0.5 and while x > 0.5, the cubic phase transited to trigonal phase. The effect of substituted Mo on the morphology of resulting products was also investigated by SEM experiments.     

Synthesis and electrochemical properties of layered Li[Li(1/3−x/3)CrxMn(2/3−2x/3)]O2 prepared by sol-gel method

The Li[Li_(1/3−x/3)Cr_xMn_(2/3−2x/3)]O₂ (0.15 ≤ x ≤ 0.3) cathode materials were synthesized by sol-gel process using aqueous solutions of metal acetates and citric acid as the chelating agent. The precipitate of metal citrate was dried in a vacuum oven for 10 h at 100 °C. After drying, the gel precursor was calcined at 300 °C for about 10 h. The resulted powder was ground and heated at 900 °C. The structural characterization was carried out by fitting the XRD data with Rietveld program. The samples exhibited a well defined layered structure and the unit cell parameters linearly increased with increasing chromium contents in Li[Li_(1/3−x/3)Cr_xMn_(2/3−2x/3)]O₂ Surface morphology was determined by SEM and HRTEM and it is found that the cathode material consisted of highly ordered single crystalline particles with layered-hexagonal structure. Test cells were assembled and cycled in the voltage range of 2.0-4.9 V with a current density of 7.947 mA/g. Electrode with (x = 0.2) delivered a high reversible capacity of around 280 mA h/g in cycling.     

Magnetic and thermal expansion properties of chromium-substituted lithium ferrite

The structure, magnetic, and thermal expansion properties of chromium-substituted lithium ferrite have been investigated. The lattice constant (Å) decreases linearly as a (x) = 8.32366 − 0.04338x for Li_0.5Fe_2.5−xCr_xO₄ (x = 0.0–1.0). When increasing Cr content, the initial permeability decreased gradually. The average thermal expansion coefficient of Li_0.5Fe_2.5−xCr_xO₄ (x = 0.0–1.0) varied from 15.34 to 17.77 ppm/°C, with increasing Cr content, the average thermal expansion coefficient decreased. The average thermal expansion coefficient (ppm/°C) in the range of 25–850 °C give the polynomial correlation as follows, TEC (x) = 1 7.775 − 0.216x − 0.723x² − 1.493x³ for Li_0.5Fe_2.5−xCr_xO₄ (x = 0.0–1.0).     

Preparation, characterization and electrical conductivity studies of NdYb1−xGdxZr2O7 (0 ≤ x ≤ 1.0) ceramics

Several compositions of NdYb_1−xGd_xZr₂O₇ (0 ≤ x ≤ 1.0) ceramics were prepared by pressureless-sintering method at 1973 K for 10 h in air. The relative density, microstructure and electrical conductivity of NdYb_1−xGd_xZr₂O₇ ceramics were analyzed by the Archimedes method, X-ray diffraction, scanning electron microscopy and impedance plots measurements. NdYb_1−xGd_xZr₂O₇ (0 ≤ x ≤ 0.3) ceramics have a single phase of defect fluorite-type structure, and NdYb_1−xGd_xZr₂O₇ (0.7 ≤ x ≤ 1.0) ceramics exhibit a single phase of pyrochlore-type structure; however, the NdYb_0.5Gd_0.5Zr₂O₇ composition shows mixed phases of both defect fluorite-type and pyrochlore-type structures. The measured values of the grain conductivity obey the Arrhenius relation. The grain conductivity of each composition in NdYb_1−xGd_xZr₂O₇ ceramics gradually increases with increasing temperature from 673 to 1173 K. NdYb_1−xGd_xZr₂O₇ ceramics are oxide-ion conductor in the oxygen partial pressure range of 1.0 × 10⁻⁴ to 1.0 atm at all test temperature levels. The highest grain conductivity value obtained in this work is 1.79 × 10⁻² S cm⁻¹ at 1173 K for NdYb_0.3Gd_0.7Zr₂O₇ composition.     

Preparation and characterization of novel spinel Li4Ti5O12−xBrx anode materials

Br-doped Li₄Ti₅O₁₂ in the form of Li₄Ti₅O_12−xBr_x (0 ≤ x ≤ 0.3) compounds were successfully synthesized via solid state reaction. The structure and electrochemical properties of the spinel Li₄Ti₅O_12−xBr_x (0 ≤ x ≤ 0.3) materials were investigated. The Li₄Ti₅O_12−xBr_x (x = 0.2) presents the best discharge capacity among all the samples, and shows better reversibility and higher cyclic stability compared with pristine Li₄Ti₅O₁₂, especially at high current rates. When the discharge rate was 0.5 C, the Li₄Ti₅O_12−xBr_x (x = 0.2) sample presented the excellent discharge capacity of 172 mAh g⁻¹, which was very close to its theoretical capacity (175 mAh g⁻¹), while that of the pristine Li₄Ti₅O₁₂ was 123.2 mAh g⁻¹ only.     

Nanosized cordierite–sapphirine–spinel glass-ceramics from natural raw materials

Inexpensive nanosized sintered cordierite glass-ceramic was prepared from quartz sand, kaolin, and magnesite. The addition of nucleation catalysts, such as TiO_2, Cr₂O_3, and admixed TiO₂–Cr₂O₃, was tested in the cordierite base glass. Cordierite, sapphirine, spinel, magnesium aluminium silicate, and cristobalite were developed using the crystallisation process. These glass-ceramics have ultra-fine grain sizes with nanorounded crystals measuring less than 200 nm, particularly in the Cr₂O₃-containing samples. Due to its different crystalline phases, the new glass-ceramics varied in hardness from 6374 to 8139 MPa and had coefficients of thermal expansion (CTE) from 0.83 to 6.89×10⁻⁶ °C⁻¹. In glass-ceramic samples, the spinel and sapphirine imparted high CTE (from 6.89 to 5.31×10⁻⁶ °C⁻¹) and hardness values (from 8139 to 7894 MPa), whereas cordierite provided lower CTE (from 0.83 to 2.66×10⁻⁶ °C⁻¹) and hardness values (from 7453 to 6374 MPa).     

New NiFe2−xCrxO4 spinel films for O2 evolution in alkaline solutions

Spinel-type ternary ferrites with composition NiFe_2−xCr_xO₄ (0 ≤ x ≤ 1) were synthesized by a precipitation method and their physicochemical and electrocatalytic properties have been investigated using IR, XRD, BET surface area, XPS, impedance and Tafel polarization techniques. The study indicated that substitution of Cr from 0.2 to 1.0 mol in the spinel matrix increased the apparent electrocatalytic activity of the base oxide towards the O₂ evolution reaction in 1 M KOH at 25 °C. The apparent electrocatalytic activity of the oxide with 0.8-1.0 mol Cr was found to be the greatest among the present series of oxides investigated. It is noteworthy that the electrocatalytic activity of the oxide with x = 0.8-1.0 was also greater than those of other spinel/perovskite O₂ evolving electrocatalysts reported in literature.     

Structural and electrochemical studies of (SnxCo1−x)60C40 alloys prepared by mechanical attriting

Using a vertical-axis attritor, samples of (Sn_xCo_1−x)₆₀C₄₀ for 0 ≤ x ≤ 0.7 have been prepared in increments of Δx = 0.1. The effect of Sn content on the structure and performance of the Sn-Co-C nanocomposites was examined by X-ray diffraction (XRD), ¹¹⁹Sn Mössbauer effect spectroscopy and electrochemical methods. XRD shows a diffraction pattern characteristic of a nanostructured material having amorphous CoSn grains in a carbon matrix for x = 0.5. Broad Bragg peaks of Co₃C, SnCo₃C_0.7 and CoSn₂ were observed for samples with 0 ≤ x ≤ 0.1, 0.1 ≤ x ≤ 0.4 and 0.5 < x ≤ 0.7, respectively. ¹¹⁹Sn Mössbauer effect spectroscopy shows the presence of an amorphous CoSn component in samples with 0.3 ≤ x ≤ 0.6. Samples with a large amount of the SnCo₃C_0.7 phase showed small capacity for lithium. Therefore, SnCo₃C_0.7 is best described as an inactive phase. The sample with x = 0.7 shows the highest specific capacity (about 600 mAh/g) of all samples prepared but exhibited poor capacity retention after cycle 45. Excellent charge-discharge capacity retention, reasonable specific capacity (>450 mAh/g) along with stable differential capacities were observed for samples near x = 0.5.     

Perovskite-type La2−xSrxNiO4 (0 ≤ x ≤ 1) as active anode materials for methanol oxidation in alkaline solutions

Perovskite-type ternary oxides with molecular formulae, La_2−xSr_xNiO₄ (0 ≤ x ≤ 1), were prepared by a modified citric acid sol-gel route at 600 °C for their possible use in a direct methanol fuel cell (DMFC). The study was conducted by cyclic voltammetry, chronoamperometry, impedance and anodic Tafel polarization techniques. The results showed that the electrocatalytic activity of the base oxide (x = 0) in 1 M KOH plus 1 M CH₃OH at 25 °C increases with x, the observed current densities being 23.6, 47.3, 43.2 and 50.9 mA cm⁻² at a scan rate of 10 mV s⁻¹ and E = 0.6 V versus Hg/HgO for oxides with x = 0, 0.25, 0.5 and 1.0, respectively. All the four perovskite anodes used in this study did not indicate any poisoning by the methanol oxidation intermediates/products. The methanol electro-oxidation reaction followed a Tafel slope of ∼2 × 2.303RT/3F (=40 mV decade⁻¹) on each oxide catalyst, regardless of Sr content.     

Structure,dielectric and piezoelectric properties of Ba0.90Ca0.10Ti1−xSnxO3 lead-free ceramics

Lead-free piezoelectric ceramics Ba_0.90Ca_0.10Ti_1−xSn_xO₃ have been prepared by a conventional ceramic fabrication technique and the effects of Sn⁴⁺ on the structure, dielectric and piezoelectric properties of the ceramics have been investigated. All the ceramics exhibit a pure perovskite structure. After the substitution of Sn⁴⁺, the crystal structure of ceramics is transformed gradually from a tetragonal to an orthorhombic phase, and becomes a pseudo-cubic phase at x≥0.14. The substitution also decreases the Curie temperature greatly from 138 °C at x=0 to 33 °C at x=0.12, and shifts the orthorhombic–tetragonal phase transition to higher temperatures. Coexistence of the orthorhombic and tetragonal phases is formed in the ceramic at x=0.10, leading to significant improvements in the piezoelectric properties: d₃₃=521 pC/N and k_p=45.5%. Our results also reveal that the ceramics sintered at higher temperatures contain larger grains, and thus exhibit more noticeable tetragonal–orthorhombic phase transition and enhanced ferroelectric and piezoelectric properties.     

Structure,magnetic and electrical transport properties of the perovskites Sm1−xCaxFe1−xMnxO3

The structure of series Sm_1−xCa_xFe_1−xMn_xO₃ (0.0 ≤ x ≤ 1.0) compounds was investigated. The lattice parameters increase with coupled substitution Sm³⁺ by Ca²⁺ and Mn⁴⁺ for Fe³⁺. The variation of parameter, c, is larger than that of a and b, respectivly. The detailed analysis of magnetic properties of series Sm_1−xCa_xFe_1−xMn_xO₃ (0.1 ≤ x ≤ 0.9) shows that local magnetic interaction between Fe³⁺ and Fe³⁺ and Mn⁴⁺ and Mn⁴⁺ at below magnetic transition temperature is antiferromagnetic. Above magnetic transition temperature the presence of large magnetic cluster is proposed and the sizes of magnetic clusters decrease with Mn⁴⁺. The electrical transport behaviors related with small polaron hopping and variable range hopping models.