Brick Layer Model Research Articles

Influence of Microstructure on the Material Properties of LLZO Ceramics Derived by Impedance Spectroscopy and Brick Layer Model Analysis.

Variants of garnet-type Li7La3Zr2O12 are being intensively studied as separator materials in solid-state battery research. The material-specific transport properties, such as bulk and grain boundary conductivity, are of prime interest and are mostly investigated by impedance spectroscopy. Data evaluation is usually based on the one-dimensional (1D) brick layer model, which assumes a homogeneous microstructure of identical grains. Real samples show microstructural inhomogeneities in grain size and porosity due to the complex behavior of grain growth in garnets that is very sensitive to the sintering protocol. However, the true microstructure is often omitted in impedance data analysis, hindering the interlaboratory reproducibility and comparability of results reported in the literature. Here, we use a combinatorial approach of structural analysis and three-dimensional (3D) transport modeling to explore the effects of microstructure on the derived material-specific properties of garnet-type ceramics. For this purpose, Al-doped Li7La3Zr2O12 pellets with different microstructures are fabricated and electrochemically characterized. A machine learning-assisted image segmentation approach is used for statistical analysis and quantification of the microstructural changes during sintering. A detailed analysis of transport through statistically modeled twin microstructures demonstrates that the transport parameters derived from a 1D brick layer model approach show uncertainties up to 150%, only due to variations in grain size. These uncertainties can be even larger in the presence of porosity. This study helps to better understand the role of the microstructure of polycrystalline electroceramics and its influence on experimental results.

Thermochemical Expansion and Protonic and Electronic Hole Conductivity of Grain Interior and Grain Boundaries in 10 Mole% Y-Substituted SrZrO3.

Proton conducting acceptor-doped SrZrO3 has a history as long as that of BaZrO3 , but has attracted less interest. Inspired by its higher transport number of ionic conduction in wet oxygen revealed by our recent work, we here explore further aspects of doped SrZrO3 as electrolyte in proton ceramic electrochemical cells. In-situ high temperature XRD (HT-XRD) analysis of SrZr0.9 Y0.1 O3-δ (SZY10) indicated an anisotropic chemical expansion of hydration, stronger along the b than the a direction, and negative in the c direction. A systematic electromotive force (EMF) and impedance spectroscopy study as a function of and allowed determination of partial conductivities of electron holes and ions (mainly protons) in bulk (grain interior) and grain boundaries. Enthalpies and preexponentials were determined and interpreted for bulk and grain boundary partial conductivities based on defect chemistry and a brick layer model. The hole conductivity in bulk is modest and ensures high ionic transport numbers in oxidizing atmospheres, while grain boundaries exhibit lower ionic transport numbers from a relatively higher hole conductivity attributed primarily to tunnelling past the deepest part of the space charge region. Y-doped SrZrO3 (SZY) materials exhibit lower proton conductivities but excel over Y-doped BaZrO3 (BZY) in terms of thermal expansion compatibility with electrode materials and higher ionic transport numbers in oxidizing atmospheres and may hence be candidates for functional layers between BZY-based electrolytes and positrodes in proton ceramic electrochemical cells.

A brick layer model for surface conduction in porous ceramics

We provide herein a review and in-depth derivation and discussion of a surface brick layer model (SBLM) that scales the conductance of a surface layer – such as protonic conductance in adsorbed water – and the macroscopic conductivity for the porous material. The model is derived on basis of grains and pores of similar sizes and will be most realistic for porous compacts or loosely sintered ceramics of isotropic grains, although it still provides order-of-magnitude guidance also for textured or less porous materials. Brief comparison is made with other relevant models and literature. The model is illustrated and evaluated with data primarily from measurements of surface protonic conduction in porous nanoscopic ceramic oxides.

Charge transport and dielectric characteristics of Sm x Bi1−x FeO3 thin films from the perspective of grain and grain boundary properties

Samarium-substituted bismuth ferrite (Sm x Bi1−x FeO3) compositions comprise a system of important materials due to their combination of multiferroic properties. Several dielectric and charge transport reports in literature can be found in this system. However, as a typical polycrystalline electroceramic, their grains and grain boundaries (GBs) are expected to possess very different properties. To this date, these distinctions have not been determined for this system. In this work, through measurements via impedance spectroscopy on Sm x Bi1−x FeO3 thin films, we show that using a brick layer model allows the separation of the electrical properties of grains and GBs. Results indicate that grains have dielectric permittivity and electrical conductivity much higher than GBs. Their properties mostly control the characteristics observed in the studied thin films. The introduction of samarium reduces the electrical conductivity and increases the activation energies for charge transport in grains and GBs. In turn, dielectric permittivity is reduced in grains and subtly increased in GBs.

Investigating the grain boundary features of lithium titanium phosphate as an electrolyte for all-solid-state lithium-ion batteries and their optimization by boron doping

This work reports a detailed analysis of the grain boundary electrochemical properties of lithium titanium phosphate for application in all-solid-state lithium-ion batteries and their improvement by boron addition, Li1+xTi2−xBx(PO4)3 (x = 0, 0.2). The results demonstrate that highly resistive grain boundaries exist in the undoped material, which preclude its ability to attain a high overall Li-ion conductivity. Conversely, boron addition provides a significant improvement of both bulk and grain boundary conductivities, resulting from the Li-enrichment associated with charge compensation for the boron doping. A detailed analysis using the brick layer model and a space charge analysis reveals a lower depletion of Li+ species at the space charge layers of the grain boundaries of the boron-doped sample, which allow a higher intrinsic grain boundary conductivity to be offered by this material. This factor permits a much higher overall conductivity to be attained in the boron-containing sample, despite a finer microstructure. Overall, this work provides new insights regarding the electrochemical nature of the grain boundary of NASICON-based solid-state Li-electrolytes, underscoring the effectiveness of composition-driven grain boundary engineering for performance improvement; a factor that is currently understudied for this category of material.

Microstructural and high-temperature dielectric, piezoelectric and complex impedance spectroscopic properties of K0.5Bi0.5TiO3 modified NBT-BT lead-free ferroelectric ceramics

Solid solutions (1-x-y)(Na0.5Bi0.5)TiO3-xBaTiO3-y(K0.5 Bi0.5)TiO3 with (x (mol.%) = 0, 7 and 100); y(mol.%) = 0, 20 and 100) compositions have been prepared by a conventional solid-state reaction method, and their structure, dielectric properties and depolarization temperature have been examined. At room temperature, X-ray diffraction (XRD) patterns reveal that the crystalline structure of the ceramics was perovskite. The morphotropic phase boundary (MPB) of the ternary system lying between rhombohedral (R3c) and tetragonal (P4mm) phases is in the range of (x (mol.%) = 7 and y (mol.%) = 20). The Raman-active modes for 0.73NBT-0.07BT-0.20KBT were separated and identified under the framework of group theory. SEM micrographs illustrate the quasi-uniform distribution of the grains, which are compact. The dielectric properties of the ceramics were studied in the frequency range of 1 kHz–100 kHz from ambient temperature to 600 °C. Dielectric measurements indicate that all ceramics show a diffuse phase transition near the temperature (Tm) for diffusivity of the order of 1.4–1.7 and a shift of (Tm) towards high temperatures. The resistance and capacitance of the various contributors (grain and grain boundary) in our samples are also discussed using a brick-layer model. Excellent piezoelectric properties for d33 = 146 pCN−1 and electromechanical coupling factors kp = 29.4% were observed at morphotropic phase boundary (MPB), which was assumed to be associated with the coexistence of rhombohedral and tetragonal phases and accurate grain size. This work establishes a new approach for improving lead-free piezoelectric ceramics based on 0.73NBT-0.07BT-0.20KBT.

Surface protonic conductivity in chemisorbed water in porous nanoscopic CeO2

CeO2 surfaces play decisive roles in heterogeneous catalysis of important processes. Here, we investigate adsorption and dissociation of water and migration of protons on internal surfaces of nanoscopic porous CeO2. Sorption and thermogravimetry confirm literature suggestions that the surface is hydrogenated to Ce3+ ions and protons H+. The following chemisorption is dissociative, yet weak, and physisorption sets in only at the very highest relative humidities, reflecting hydrophobic behaviour. We link sample conductivities to surface protonic conductances via a brick layer model and show that behaviours at high, intermediate, and low temperatures with, respectively, positive, close to zero, and negative apparent activation energies and pH2O1/2, pH2O1, and pH2O3/2 dependences, can be attributed to different models of migration all within the chemisorbed layer, without contribution from physisorbed water. While CeO2 may special in this respect due to the effect of the hydrogenated surface, we believe the extended models of transport in the chemisorbed layer may apply also to other oxides. Unsaturated chemisorption may play an important role for CeO2 as catalyst in that the surface is left available for reactant molecules, still with availability of dissociated and mobile protons in the chemisorbed layer and electronic defects by Ce3+ in the surface.

A Physical Derivation of the Capacitive Brick Layer Model in Polycrystalline Ceramics from Fundamental Electrodynamic Equations

The brick layer model in electrochemical impedance spectroscopy has been successfully used for decades to describe the behavior of polycrystalline materials. Equivalent circuits were created to obtain information about grain and grain boundary properties from electrochemical impedance spectroscopy measurements. Previous publications have expanded on initial interpretations and described grains and grain boundaries as plate capacitors filled with the grain and grain boundary material, respectively. However, this approach poses a number of significant issues, since key assumptions for the equivalent circuits do not match with the actual experimental situation of the material when exposed to an outer electric field, thus calling into question the entire approach for the interpretation.

Investigating the Correlation between the Microstructure and Electrical Properties of FeSbO4 Ceramics.

FeSbO4 powder was prepared using the solid-state reaction method in this work. Afterward, the dense and porous ceramics were obtained by sintering the pressed powder calcined at temperatures of 900 and 1000 °C for 4 h. Rietveld profile analysis of the X-ray powder diffraction data showed that FeSbO4 adopts the trirutile-type structure (space group P42/mnm, with a ≅ 4.63 Å and c ≅ 9.23 Å). SEM images showed that the powder calcined at 900 °C after being sintered at 1200 °C resulted in ceramics of higher crystallinity, larger grains, and consequently, low porosity. The dielectric properties were measured in the frequency range of 10−1 Hz–1 MHz as a function of temperature (25–250 °C). The real (σ′) and imaginary (σ″) parts of the complex conductivity increase with rising annealing temperature for both samples. The real conductivity in the AC region for 𝑓 = 100 kHz was and for the ceramic samples obtained from the powder calcined at 900 (C-900) and 1000 °C (C-1000), respectively. Furthermore, the dielectric constants (k′) measured at room temperature and were (C-900) and (C-1000), while the activation energies of the grain region were Ea = 0.53 eV and Ea = 0.49 eV, respectively. Similar activation energy (Ea = 0.52 eV and 0.49 eV) was also obtained by the brick-layer model and confirmed by the adjustment of activation energy by DC measurements which indicated an absence of the porosity influence on the parameter. Additionally, loss factor values were obtained to be equal to 3.8 (C-900) and 5.99 (C-1000) for measurements performed at 100 Hz, suggesting a contribution of the conductivity originated from the combination or accommodation of the pores in the grain boundary region. Our results prove that the microstructural factors that play a critical role in the electrical and dielectric properties are the average grain size and the porosity interspersed with the grain boundary region.

High-safety composite solid electrolyte based on inorganic matrix for solid-state lithium-metal batteries

Composite solid electrolytes (CSEs) combine the advantages of polymer electrolytes and inorganic ceramic electrolytes, which have attracted increasing attention for solid-state lithium-metal batteries. However, the most studied polymer-based CSEs are easily ignited and can be penetrated by lithium (Li) dendrite, leading to safety issue. Herein, we report CSEs with very high inorganic ceramic loadings of >90% by a cold sintering method at a low temperature of 200°C. The ceramic-based CSEs show better safety and mechanical properties than polymer-based CSEs. The Li symmetric cell–symmetric Li-ion cells have the same material as the positive and negative electrode [when cells are assembled one would already contain lithium and the other would not]–using the ceramic-based CSE delivers a stable cycling performance of over 550 h at the current density of 0.3 mA/cm2. More importantly, according to the electrochemical impedance spectroscopy (EIS), Debye diagram, and power law equation, the Li-ion conduction mechanism in CSEs is systematically investigated. Using a modified brick layer model, the conductivities for Li-ions transport parallel with ceramic/polymer interfaces are calculated to be five orders of magnitudes higher than that perpendicular to the interfaces. This work provides a new analytical method for inorganic matrix based CSEs for high-safety batteries and conducts an in-depth research on the ion conduction mechanism.

Effect of sintering temperature on microstructure and electrical properties of Ce0.82Dy0.155Sr0.025O2-δ electrolyte

Ce0.82Dy0.155Sr0.025O2-? (SDDC) powder was synthesized by a sucrose-pectin modified sol-gel process. The powder was pressed and sintered at different temperatures, 1200, 1250 and 1300?C, to prepare dense ceramics with a relative density of 91, 93 and 95%TD, respectively. Crystal structure and phase composition were analysed by XRD using Rietveld refinement. The sample compositional analysis and element mapping were carried out by energy-dispersive spectroscopy. Microstructure images were recorded by scanning electron microscope. The electrical properties were measured by an impedance analyser. The grain resistance and the grain boundary resistance changed with the increased grain size. The SDDC electrolyte sintered at 1300?C exhibits an enhanced total electrical conductivity of 2.45?10?2 S/cm at 700?C and the activation energy of 0.76 eV. The porosity corrections are applied to the Bricklayer model to calculate the specific grain and grain boundary conductivities.

Correlation between structural, electrical and electrochemical performance of Zn doped high voltage spinel LiNi0.5-xZnxMn1.5O4 porous microspheres as a cathode material for Li-Ion batteries

This work reports on the structural, electrical and electrochemical properties of Zn doped LiNi0.5-xZnxMn1.5O4 (x = 0.0, 0.05, 0.1 and 0.2) porous microspheres cathode materials obtained by a hydrothermal route followed by low-temperature annealing. Structural properties are investigated by X-ray diffraction (XRD), nuclear magnetic resonance (NMR) spectroscopy; scanning electron microscopy (SEM), X-ray photoelectron spectroscopy (XPS), Raman, and Fourier transform infrared (FTIR) spectroscopy. The XRD patterns and NMR studies further demonstrate the successful incorporation of Zn into the spinel structure. SEM reveals morphology with porous spheres with a size range of 0.2 μm – 1 μm. Temperature dependent electrical conductivity was measured by impedance spectroscopy. The activation energies of total conductivity are determined as 0.41, 0.41, 0.47, and 0.48 eV for x = 0, 0.05, 0.1, and 0.2, respectively. Influence of Zn doping on the grain, grain boundary and total conductivities of LiNi0.5-xZnxMn1.5O4 were analyzed through brick-layer model. The grain conductivity increases with Zn content, whereas the total conductivity increases till x = 0.1. Overall, electrical properties of LiNi0.5-xZnxMn1.5O4 are governed by grain boundary conduction. Increasing the Zn content up to x = 0.1 in LiNi0.5-xZnxMn1.5O4 leads to an increment in specific capacity at a 1C rate. Among them, the LiNi0.4Zn0.1Mn1.5O4 exhibits a high discharge capacity of 114 mAh g−1 at 1C with capacity retention of 91.5% after 50 cycles. We also investigated the effect of Zn doping on Li+ ion diffusivity and observed improved Li+ ion diffusion (7.4⋅10−13 cm2 s−1) for LiNi0.4Zn0.1Mn1.5O4. Based on these electrochemical studies it is proposed that this material can be utilized as a cathode in Li-ion batteries for high-power applications.

Understanding the Impact of Microstructure on Charge Transport in Polycrystalline Materials Through Impedance Modelling

Charge transport in polycrystalline electronic or ionic conductors is usually analyzed by serial macroscopic equivalent circuits, e.g., the brick layer model, which assume a homogeneous electric potential distribution across the sample. In such analyses, the microstructure is highly idealized and usually not representative of the actual microstructure. Here, we use a network model approach to investigate the impact of the sample’s microstructure on the impedance. We find that this influence can be severe and should not be ignored. The interplay between microscopic transport paths affects the impedance response, which is reflected in both the frequency and the time domain. Especially in the distribution of relaxation times additional signals are identified and studied systematically. These additional contributions cannot be assigned to a microscopic transport process as usually done in a conventional analysis based on an equivalent circuit model fitted to the impedance data. The neglect of the peculiarities of the real microstructure in impedance analyses based on the brick layer model may yield deviations in the order of 100 % in terms of the derived microscopic transport parameters. The microstructures used as input for the modelling are digitalized electron microscope images of polycrystalline samples.

Calcination Temperature Dependent Dielectric Properties of Nanocrystalline BaSnO3

BaSnO3 nanocrystals were synthesized using a simple solid-state route varying the calcination temperature to investigate the effect of grain growth on dielectric properties of this popular perovskite oxide. Duration of calcination was optimized to obtain the pure phase. The samples were characterized by X-ray diffraction, field-emission scanning electron microscopy, transmission electron microscopy and energy dispersive X-ray analysis to investigate crystal structure, morphological information, and elemental composition respectively. The average crystallite size increased from 35 to 49 nm, whereas the porosity decreased from 20.41% to 11.03% with tuning of calcination temperature. The dielectric constant and conductivity increased with calcination temperature whereas the tangent loss maximum was decreased by ∼9%. A non-Debye type relaxation mechanism was confirmed for all samples using Cole-Davidson relations. The outcomes of the electrical properties were correlated with the grain-grain boundary effect, porosity, and brick layer model. It was clearly observed that the effect of high calcination temperature can enhance the room-temperature dielectric constant and conductivity of the nano-BaSnO3 perovskites.

Dielectric properties of A-site Mn-doped bismuth sodium titanate perovskite: (Bi0·5Na0.5)0.9Mn0·1TiO3

We synthesized A-site Mn-doped Bi0·5Na0·5TiO3 polycrystalline ceramic ((Bi0·5Na0.5)0.9Mn0·1TiO3 ceramic (BNMT10)) via solid-state reaction route and investigated the structural and temperature-dependent dielectric properties. The dielectric properties of the synthesized samples are explained using the brick-layer model. The relaxation time and corresponding activation energies were obtained through the Arrhenius mechanism by using the imaginary impedance and modulus results. Analysis of the activation energy and other dielectric results suggests that the electric conduction in our BNMT10 sample is due to single and doubly charged oxygen ion vacancy-defects at low and high temperatures, respectively. The grains and grain boundaries contribute to the inherent defect-based conduction mechanism and the dielectric behavior. The electric modulus spectroscopy results discerned the non-Debye type relaxation processes that these processes consist of the localized and long-range relaxation processes. Our results demonstrate that the dielectric behavior of Bi0·5Na0·5TiO3 ceramics can be enhanced through Mn-doping at the A-site.

Synthesis, structure, microstructure, and electrical properties of Ce0.81Nd0.095Sm0.095O2−δ

Maltose and pectin-assisted low-temperature combustion modified sol–gel process was employed to synthesize nanopowders of Ce0.8Sm0.2O2−δ (SDC 20), Ce0.81Nd0.095Sm0.095O2−δ (NSDC). The sample's powder was sintered at 1250oC for 6h to form dense ceramics with a relative density greater than 95%. The Rietveld refinement of powder XRD patterns indicates a single-phase material with a cubic fluorite structure. The Raman spectroscopy studies confirm solid solution with evidence of more oxygen vacancies in sample NSDC to sample SDC 20. The SEM images of the samples show a high-dense surface with few pores. The EDX results confirm the sample's chemical composition. Impedance measurements were carried out to study electrical properties. The NSDC sample showed improved conductivity (3.06×10-2S/cm at 600°C) over SDC 20 (2.16×10-2S/cm at 600°C) with lower activation energy (0.81eV). Specific grain (σ*g) and grain boundary (σ*gb) conductivity values were calculated using porosity corrections to the Bricklayer model.

A Simulation Study for Optimizing Grain Size in Poly-Crystalline Silicon Material Using Impedance Spectroscopy

This paper uses impedance spectroscopy as a simulation tool to analyze grain boundaries properties in poly-crystalline materials. The interfacial structure at grain boundaries in poly-crystalline materials plays a decisive role in their electrical behaviors and degrades the performance in integrated circuits, gas sensors and solar photovoltaics. A numerical simulation is carried out to optimize grain size of poly-crystalline material to behave as mono-like material using impedance spectroscopy technique. The grain boundary potential (Vb) obtained from thermionic emission diffusion theory based continuum and localized energy models are utilized for obtaining the resistance (R), capacitance (C) and angular frequency (??) of the bulk of the grain (Rb, Cb and ??b) and that of grain boundaries (Rgb, Cgb and ??gb) using the brick layer model. Two markedly different impedance spectra are obtained, one for the bulk of the grain and another for the grain boundaries. It is found that with an increase in the grain size (Lg), the signatures of grain boundaries spectra diminish and at a particular grain size, disappear completely which indicates that the poly-silicon material behaves comparable to mono-crystalline material. In this simulation work, silicon material is taken as the reference due to its abundant availability and huge market potential for photovoltaic and semiconductor applications but the simulation can be implemented on any poly-crystalline materials. The effect of varying silicon material resistivity (1.0???3.5 ??cm) and temperature (300-420 K) on grain size was also analysed using the impedance spectra. The findings of the paper will be helpful for the silicon ingot growth industries to decide the optimum grain size during the polysilicon materials growth.

Phase stability and properties of Bi6Te2O13+δ (δ = 0 or 2)

The crystalline Bi6Te2O15 and Bi6Te2O13 phases with orthorhombic and cubic symmetries, respectively, were synthesized by solid-state reaction synthesis. A study of the thermal stability of both phases was carried out, being verified that the high temperature cubic phase can be kept at room temperature in a metastable condition during long time. The Raman scattering spectrum, optical reflectivity and photoluminescence emission of these phases were measured for the first time. The optical band gaps Eg = 3.29eV, for the orthorhombic phase, and Eg = 3.08eV, for the cubic phase, were calculated from the reflectivity spectra. Using the brick-layer model the electrical conductivity of a dense ceramic of Bi6Te2O15 was measured, taking in account the grain and grain-boundary effects.

Proton conductivity and transport number of complex perovskite barium strontium tantalate

Previously, complex perovskites barium strontium tantalate, i.e., Ba3Sr1.4T1.6O8.4 (BST40), has exhibited excellent proton conductivity due to its high basicity and oxygen vacancy concentration. Herein, we report proton conduction mechanism of BST40 using brick-layer model and defect equilibria model. Briefly, BST40 is prepared by solid-state synthesis method, resulting in hexagonal perovskite structure with space group of P63/m. Interestingly, grain interior and grain boundary thickness of BST40 are almost equal and the grain boundary conduction mainly dominates the conductivity at T > 500 °C. Total standard molar hydration enthalpy of BST40 and grain interior is found to be −79.6 kJ/mol and −101.3 kJ/mol, respectively. Activation energy for BST40, protons, oxygen-ions, and holes in oxygen-containing humid atmosphere is found to be 0.63 eV, 0.51 eV, 0.92 eV, and 1.08 eV, respectively. These values are higher than corresponding values for the grain interior. At 700 °C, proton transport number of BST40 and grain interior is found to be 0.7 and 0.72, respectively, which is significantly higher than other congeneric complex perovskite-type proton conductors. Overall, these results confirm that BST40 possesses excellent proton conductivity.

Electrical Conductivity of Y<sup>3+</sup> Doped Ba(Ce,Zr)O<sub>3</sub> in Wet N<sub>2</sub> Atmosphere Prepared with the Addition of Brij-97

Y-doped barium cerate-zirconate ceramic oxide is proven to be a competent material as an electrolyte with high proton conductivity as well as chemical and mechanical stabilities in carbon dioxide and water vapour atmospheres. This ceramic oxide requires high processing temperature which will results in the increase of particle/grain size. Hence, modification on the synthesis route has been studied in reducing the particle/grain size of the ceramic by lowering the calcination temperature. In this work, BaCe0.54Zr0.36Y0.1O2.95 (BCZY) powder was synthesized with addition of surfactant (Brij-97) through an established modified sol-gel route. Single BCZY perovskite phase was successfully obtained at calcination temperature of 950°C which was lower than our previous study (T=1100°C). The prepared sample was made into pellet by a dry pressing technique with diameter, d=13 mm and thickness, t~2 mm and then subjected to a two-step sintering method prior to morphological and electrical measurements. Impedance measurement was carried out at intermediate temperatures (500-800°C) using an Electrochemical Impedance Spectroscopy (EIS) in wet nitrogen atmosphere. Impedance spectrum was analysed to obtain the behaviour of grain core and grain boundary responses by a fitting procedure using a brick-layer model. Scanning electron microscope (SEM) analysis of fractured pellet revealed that BCZY prepared with the assisted of Brij-97 exhibited dense, homogenous and less agglomerate grain with grain size around 88 nm, which may explain the enhancement in the total conductivity of the BCZY electrolyte.