Catalyst design relies heavily on electronic metal-support interactions, but the metal-support interface with an uncontrollable electronic or coordination environment makes it challenging. Herein, we outline a promising approach for the rational design of catalysts involving heteroatoms as anchors for Pd nanoparticles for ethanol oxidation reaction (EOR) catalysis. The doped B and N atoms from dimethylamine borane (DB) occupy the position of the Ti₃C₂ lattice to anchor the supported Pd nanoparticles. The electrons transfer from the support to B atoms, and then to the metal Pd to form a stable electronic center. A strong electronic interaction can be produced and the d-band center can be shifted down, driving Pd into the dominant metallic state and making Pd nanoparticles deposit uniformly on the support. As-obtained Pd/DB–Ti₃C₂ exhibits superior durability to its counterpart (˜14.6% retention) with 91.1% retention after 2000 cycles, placing it among the top single metal anodic catalysts. Further, in situ Raman and density functional theory computations confirm that Pd/DB–Ti₃C₂ is capable of dehydrogenating ethanol at low reaction energies.

For the porous-membrane-based osmotic energy generator, the potential synergistic enhancement mechanism of various key parameters is still controversial, especially because optimizing the trade-off between permeability and selectivity is still a challenge. Here, to construct a permeability and selectivity synergistically enhanced osmotic energy generator, the two-dimensional porous membranes with tunable charge density are prepared by inserting sulfonated polyether sulfone into graphene oxide. Influences of charge density and pore size on the ion transport are explored, and the ionic behaviors in the channel are calculated by numerical simulations. The mechanism of ion transport in the process is studied in depth, and the fundamental principles of energy conversion are revealed. The results demonstrate that charge density and pore size should be matched to construct the optimal ion channel. This collaborative enhancement strategy of permeability and selectivity has significantly improved the output power in osmotic energy generation; compared to the pure graphene oxide membrane, the composite membrane presents almost 20 times improvement.

Electrochemical reduction of CO₂ to syngas (CO and H₂) offers an efficient way to mitigate carbon emissions and store intermittent renewable energy in chemicals. Herein, the hierarchical one-dimensional/three-dimensional nitrogen-doped porous carbon (1D/3D NPC) is prepared by carbonizing the composite of Zn-MOF-74 crystals in situ grown on a commercial melamine sponge (MS), for electrochemical CO₂ reduction reaction (CO₂RR). The 1D/3D NPC exhibits a high CO/H₂ ratio (5.06) and CO yield (31 mmol g^–1 h^–1) at –0.55 V, which are 13.7 times and 21.4 times those of 1D porous carbon (derived from Zn-MOF-74) and N-doped carbon (carbonized by MS), respectively. This is attributed to the unique spatial environment of 1D/3D NPC, which increases the adsorption capacity of CO₂ and promotes electron transfer from the 3D N-doped carbon framework to 1D carbon, improving the reaction kinetics of CO₂RR. Experimental results and charge density difference plots indicate that the active site of CO₂RR is the positively charged carbon atom adjacent to graphitic N on 1D carbon and the active site of HER is the pyridinic N on 1D carbon. The presence of pyridinic N and pyrrolic N reduces the number of electron transfer, decreasing the reaction kinetics and the activity of CO₂RR. The CO/H₂ ratio is related to the distribution of N species and the specific surface area, which are determined by the degree of spatial confinement effect. The CO/H₂ ratios can be regulated by adjusting the carbonization temperature to adjust the degree of spatial confinement effect. Given the low cost of feedstock and easy strategy, 1D/3D NPC catalysts have great potential for industrial application.

The rapid advancement in the miniaturization, integration, and intelligence of electronic devices has escalated the demand for customizable micro-supercapacitors (MSCs) with high energy density. However, efficient microfabrication of safe and high-energy MXene MSCs for integrating microelectronics remains a significant challenge due to the low voltage window in aqueous electrolytes (typically ≤0.6 V) and limited areal mass loading of MXene microelectrodes. Here, we tackle these challenges by developing a high-concentration (18 mol kg^–1) “water-in-LiBr” (WiB) gel electrolyte for MXene symmetric MSCs (M-SMSCs), demonstrating a record high voltage window of 1.8 V. Subsequently, additive-free aqueous MXene ink with excellent rheological behavior is developed for three-dimensional (3D) printing customizable all-MXene microelectrodes on various substrates. Leveraging the synergy of a high-voltage WiB gel electrolyte and 3D-printed microelectrodes, quasi-solid-state M-SMSCs operating stably at 1.8 V are constructed, and achieve an ultrahigh areal energy density of 1772 µWh cm^–2 and excellent low-temperature tolerance, with a long-term operation at –40°C. Finally, by extending the 3D printing protocol, M-SMSCs are integrated with humidity sensors on a single planar substrate, demonstrating their reliability in miniaturized integrated microsystems.

By employing metal oxides as oxygen carriers, chemical looping demonstrates its effectiveness in transferring oxygen between reduction and oxidation environments to partially oxidize fuels into syngas and convert CO₂ into CO. Generally, NiFe₂O₄ oxygen carriers have demonstrated remarkable efficiency in chemical looping CO₂ conversion. Nevertheless, the intricate process of atomic migration and evolution within the internal structure of bimetallic oxygen carriers during continuous high-temperature redox cycling remains unclear. Consequently, the lack of a fundamental understanding of the complex ionic migration and oxygen transfer associated with energy conversion processes hampers the design of high-performance oxygen carriers. Thus, in this study, we employed in situ characterization techniques and theoretical calculations to investigate the ion migration behavior and structural evolution in the bulk of NiFe₂O₄ oxygen carriers during H₂ reduction and CO₂/lab air oxidation cycles. We discovered that during the H₂ reduction step, lattice oxygen rapidly migrates to vacancy layers to replenish consumed active oxygen species, while Ni leaches from the material and migrates to the surface. During the CO₂ splitting step, Ni migrates toward the core of the bimetallic oxygen carrier, forming Fe–Ni alloys. During the air oxidation step, Fe–Ni migrates outward, creating a hollow structure owing to the Kirkendall effect triggered by the swift transfer of lattice oxygen. The metal atom migration paths depend on the oxygen transfer rates. These discoveries highlight the significance of regulating the release–recovery rate of lattice oxygen to uphold the structures and reactivity of oxygen carriers. This work offers a comprehensive understanding of the oxidation/reduction-driven atomic interdiffusion behavior of bimetallic oxygen carriers.

Material composition and structural design are important factors influencing the electromagnetic wave (EMW) absorption performance of materials. To alleviate the impedance mismatch attributed to the high dielectric constant of Ti₃C₂T_x MXene, we have successfully synthesized core-shell structured SiO₂@MXene@MoS₂ nanospheres. This architecture, comprising SiO₂ as the core, MXene as the intermediate layer, and MoS₂ as the outer shell, is achieved through an electrostatic self-assembly method combined with a hydrothermal process. This complex core-shell structure not only provides a variety of loss mechanisms that effectively dissipate electromagnetic energy but also prevents self-aggregation of MXene and MoS₂ nanosheets. Notably, the synergistic combination of SiO₂ and MoS₂ with highly conductive MXene enables the suitable dielectric constant of the composites, ensuring optimal impedance matching. Therefore, the core-shell structured SiO₂@MXene@MoS₂ nanospheres exhibit excellent EMW absorption performance, featuring a remarkable minimum reflection loss (RL_min) of –52.11 dB (2.4 mm). It is noteworthy that these nanospheres achieve an ultra-wide effective absorption bandwidth (EAB) of 6.72 GHz. This work provides a novel approach for designing and synthesizing high-performance EMW absorbers characterized by “wide bandwidth and strong reflection loss.”

Metal–N–C single-atom catalysts, mostly prepared from pyrolysis of metal-organic precursors, are widely used in heterogeneous electrocatalysis. Since metal sites with diverse local structures coexist in this type of material and it is challenging to characterize the local structure, a reliable structure–property relationship is difficult to establish. Conjugated macrocyclic complexes adsorbed on carbon support are well-defined models to mimic the single-atom catalysts. Metal–N₄ site with four electroneutral pyridine-type ligands embedded in a graphene layer is the most commonly proposed structure of the active site of single-atom catalysts, but its molecular counterpart has not been reported. In this work, we synthesized the conjugated macrocyclic complexes with a metal center (Co, Fe, or Ni) coordinated with four electroneutral pyridinic ligands as model catalysts for CO₂ electroreduction. For comparison, the complexes with anionic quadri-pyridine macrocyclic ligand were also prepared. The Co complex with the electroneutral ligand expressed a turnover frequency of CO formation more than an order of magnitude higher than that of the Co complex with the anionic ligand. Constrained ab initio molecular dynamics simulations based on the well-defined structures of the model catalysts indicate that the Co complex with the electroneutral ligand possesses a stronger ability to mediate electron transfer from carbon to CO₂.

Nanofluidic channels inspired by electric eels open a new era of efficient harvesting of clean blue osmotic energy from salinity gradients. Limited by less charge and weak ion selectivity of the raw material itself, energy conversion through nanofluidic channels is still facing considerable challenges. Here, a facile and efficient strategy to enhance osmotic energy harvesting based on drastically increasing surface charge density of MXenes subnanochannels via oxygen plasma is proposed. This plasma could break Ti–C bonds in the MXenes subnanochannels and effectively facilitate the formation of more Ti–O, C═O, O–OH, and rutile with a stronger negative charge and work function, which leads the surface potential of MXenes membrane to increase from 205 to 430 mV. This significant rise of surface charge endows the MXenes membrane with high cation selectivity, which could make the output power density of the MXenes membrane increase by 248.2%, reaching a high value of 5.92 W m^–2 in the artificial sea-river water system. Furthermore, with the assistance of low-quality heat at 50°C, the osmotic power is enhanced to an ultrahigh value of 9.68 W m^–2, which outperforms those of the state-of-the-art two-dimensional (2D) nanochannel membranes. This exciting breakthrough demonstrates the enormous potential of the facile plasma-treated 2D membranes for osmotic energy harvesting.

To attain a circular carbon economy and resolve CO₂ electroreduction technology obstacles, single-atom catalysts (SACs) have emerged as a logical option for electrocatalysis because of their extraordinary catalytic activity. Among SACs, metal–organic frameworks (MOFs) have been recognized as promising support materials because of their exceptional ability to prevent metal aggregation. This study shows that atomically dispersed Ni single atoms on a precisely engineered MOF nanosheet display a high Faradaic efficiency of approximately 100% for CO formation in H-cell and three-compartment microfluidic flow-cell reactors and an excellent turnover frequency of 23,699 h^–1, validating their intrinsic catalytic potential. These results suggest that crystallographic variations affect the abundant vacancy sites on the MOF nanosheets, which are linked to the evaporation of Zn-containing organic linkers during pyrolysis. Furthermore, using X-ray absorption spectroscopy and density functional theory calculations, a comprehensive investigation of the unsaturated atomic coordination environments and the underlying mechanism involving CO* preadsorbed sites as initial states was possible and provided valuable insights.

Biomass-derived carbon is a promising electrode material in energy storage devices. However, how to improve its low capacity and stability, and slow diffusion kinetics during lithium storage remains a challenge. In this research, we propose a “self-assembly-template” method to prepare B, N codoped porous carbon (BN-C) with a nanosandwich structure and abundant pyridinic N-B species. The nanosandwich structure can increase powder density and cycle stability by constructing a stable solid electrolyte interphase film, shortening the Li⁺ diffusion pathway, and accommodating volume expansion during repeated charging/discharging. The abundant pyridinic N-B species can simultaneously promote the adsorption/desorption of Li⁺/PF₆^– and reduce the diffusion barrier. The BN-C electrode showed a high lithium-ion storage capacity of above 1140 mAh g^–1 at 0.05 A g^–1 and superior stability (96.5% retained after 2000 cycles). Moreover, owing to the synergistic effect of the nanosandwich structure and pyridinic N-B species, the assembled symmetrical BN-C//BN-C full cell shows a high energy density of 234.7 W h kg^–1, high power density of 39.38 kW kg^–1, and excellent cycling stability, superior to most of the other cells reported in the literature. As the density functional theory simulation demonstrated, pyridinic N-B shows enhanced adsorption activity for Li⁺ and PF₆^–, which promotes an increase in the capacity of the anode and cathode, respectively. Meanwhile, the relatively lower diffusion barrier of pyridinic N-B promotes Li⁺ migration, resulting in good rate performance. Therefore, this study provides a new approach for the synergistic modulation of a nanostructure and an active site simultaneously to fabricate the carbon electrode material in energy storage devices.

Vanadium oxide cathode materials with stable crystal structure and fast Zn²⁺ storage capabilities are extremely important to achieving outstanding electrochemical performance in aqueous zinc-ion batteries. In this work, a one-step hydrothermal method was used to manipulate the bimetallic ion intercalation into the interlayer of vanadium oxide. The pre-intercalated Cu ions act as pillars to pin the vanadium oxide (V-O) layers, establishing stabilized two-dimensional channels for fast Zn²⁺ diffusion. The occupation of Mn ions between V-O interlayer further expands the layer spacing and increases the concentration of oxygen defects (O_d), which boosts the Zn²⁺ diffusion kinetics. As a result, as-prepared Cu_0.17Mn_0.03V₂O_5–□·2.16H₂O cathode shows outstanding Zn-storage capabilities under room- and low-temperature environments (e.g., 440.3 mAh g^–1 at room temperature and 294.3 mAh g^–1 at –60°C). Importantly, it shows a long cycling life and high capacity retention of 93.4% over 2500 cycles at 2 A g^–1 at –60°C. Furthermore, the reversible intercalation chemistry mechanisms during discharging/charging processes were revealed via operando X-ray powder diffraction and ex situ Raman characterizations. The strategy of a couple of 3d transition metal doping provides a solution for the development of superior room-/low-temperature vanadium-based cathode materials.

Compared with the extensively used ester-based electrolyte, the hard carbon (HC) electrode is more compatible with the ether-based counterpart in sodium-ion batteries, which can lead to improved cycling stability and robust rate capability. However, the impact of salt anion on the electrochemical performance of HC electrodes has yet to be fully understood. In this study, the anionic chemistry in regulating the stability of electrolytes and the performance of sodium-ion batteries have been systematically investigated. This work shows discrepancies in the reductive stability of the anionic group, redox kinetics, and component/structure of solid electrolyte interface (SEI) with different salts (NaBF₄, NaPF₆, and NaSO₃CF₃) in the typical ether solvent (diglyme). Particularly, the density functional theory calculation manifests the preferred decomposition of PF₆^– due to the reduced reductive stability of anions in the solvation structure, thus leading to the formation of NaF-rich SEI. Further investigation on redox kinetics reveals that the NaPF₆/diglyme can induce the fast ionic diffusion dynamic and low charge transfer barrier for HC electrode, thus resulting in superior sodium storage performance in terms of rate capability and cycling life, which outperforms those of NaBF₄/diglyme and NaSO₃CF₃/diglyme. Importantly, this work offers valuable insights for optimizing the electrochemical behaviors of electrode materials by regulating the anionic group in the electrolyte.

The emergence of Li–Mg hybrid batteries has been receiving attention, owing to their enhanced electrochemical kinetics and reduced overpotential. Nevertheless, the persistent challenge of uneven Mg electrodeposition remains a significant impediment to their practical integration. Herein, we developed an ingenious approach that centered around epitaxial electrocrystallization and meticulously controlled growth of magnesium crystals on a specialized MgMOF substrate. The chosen MgMOF substrate demonstrated a robust affinity for magnesium and showed minimal lattice misfit with Mg, establishing the crucial prerequisites for successful heteroepitaxial electrocrystallization. Moreover, the incorporation of periodic electric fields and successive nanochannels within the MgMOF structure created a spatially confined environment that considerably promoted uniform magnesium nucleation at the molecular scale. Taking inspiration from the “blockchain” concept prevalent in the realm of big data, we seamlessly integrated a conductive polypyrrole framework, acting as a connecting “chain,” to interlink the “blocks” comprising the MgMOF cavities. This innovative design significantly amplified charge-transfer efficiency, thereby increasing overall electrochemical kinetics. The resulting architecture (MgMOF@PPy@CC) served as an exceptional host for heteroepitaxial Mg electrodeposition, showcasing remarkable electrostripping/plating kinetics and excellent cycling performance. Surprisingly, a symmetrical cell incorporating the MgMOF@PPy@CC electrode demonstrated impressive stability even under ultrahigh current density conditions (10 mA cm^–2), maintaining operation for an extended 1200 h, surpassing previously reported benchmarks. Significantly, on coupling the MgMOF@PPy@CC anode with a Mo₆S₈ cathode, the assembled battery showed an extended lifespan of 10,000 cycles at 70 C, with an outstanding capacity retention of 96.23%. This study provides a fresh perspective on the rational design of epitaxial electrocrystallization driven by metal–organic framework (MOF) substrates, paving the way toward the advancement of cutting-edge batteries.

Engineering high-performance and low-cost bifunctional catalysts for H₂ (hydrogen evolution reaction [HER]) and O₂ (oxygen evolution reaction [OER]) evolution under industrial electrocatalytic conditions remains challenging. Here, for the first time, we use the stronger electronegativity of a rare-Earth yttrium ion (Y³⁺) to induce in situ NiCo-layered double-hydroxide nanosheets from NiCo foam (NCF) treated by a dielectric barrier discharge plasma NCF (PNCF), and then obtain nitrogen-doped YNiCo phosphide (N-YNiCoP/PNCF) after the phosphating process using radiofrequency plasma in nitrogen. The obtained N-YNiCoP/PNCF has a large specific surface area, rich heterointerfaces, and an optimized electronic structure, inducing high electrocatalytic activity in HER (331 mV vs. 2000 mA cm^–2) and OER (464 mV vs. 2000 mA cm^–2) reactions in 1 M KOH electrolyte. X-ray absorption spectroscopy and density functional theory quantum chemistry calculations reveal that the coordination number of CoNi decreased with the incorporation of Y atoms, which induce much shorter bonds of Ni and Co ions and promote long-term stability of N-YNiCoP in HER and OER under the simulated industrial conditions. Meanwhile, the CoN-YP₅ heterointerface formed by plasma N-doping is the active center for overall water splitting. This work expands the applications of rare-Earth elements in engineering bifunctional electrocatalysts and provides a new avenue for designing high-performance transition-metal-based catalysts in the renewable energy field.

The application of Li metal anodes in rechargeable batteries is impeded by safety issues arising from the severe volume changes and formation of dendritic Li deposits. Three-dimensional hollow carbon is receiving increasing attention as a host material capable of accommodating Li metal inside its cavity; however, uncontrollable and nonuniform deposition of Li remains a challenge. In this study, we synthesize metal–organic framework-derived carbon microcapsules with heteroatom clusters (Zn and Ag) on the capsule walls and it is demonstrated that Ag-assisted nucleation of Li metal alters the outward-to-inward growth in the microcapsule host. Zn-incorporated microcapsules are prepared via chemical etching of zeolitic imidazole framework-8 polyhedra and are subsequently decorated with Ag by a galvanic displacement reaction between Ag⁺ and metallic Zn. Galvanically introduced Ag significantly reduces the energy barrier and increases the reaction rate for Li nucleation in the microcapsule host upon Li plating. Through combined electrochemical, microstructural, and computational studies, we verify the beneficial role of Ag-assisted Li nucleation in facilitating inward growth inside the cavity of the microcapsule host and, in turn, enhancing electrochemical performance. This study provides new insights into the design of reversible host materials for practical Li metal batteries.

A stable and highly active core-shell heterostructure electrocatalyst is essential for catalyzing oxygen evolution reaction (OER). Here, a dual-trimetallic core-shell heterostructure OER electrocatalyst that consists of a NiFeWS₂ inner core and an amorphous NiFeW(OH)_z outer shell is designed and synthesized using in situ electrochemical tuning. The electrochemical measurements of different as-synthesized catalysts with a similar mass loading suggest that the core-shell Ni_0.66Fe_0.17W_0.17S₂@amorphous NiFeW(OH)_z nanosheets exhibit the highest overall performance compared with that of other bimetallic reference catalysts for the OER. Additionally, the nanosheet arrays were in situ grown on hydrophilic-treated carbon paper to fabricate an integrated three-dimensional electrode that affords a current density of 10 mA cm^–2 at a small overpotential of 182 mV and a low Tafel slope of 35 mV decade^–1 in basic media. The Faradaic efficiency of core-shell Ni_0.66Fe_0.17W_0.17S₂@amorphous NiFeW(OH)_z is as high as 99.5% for OER. The scanning electron microscope, transmission electron microscope, and X-ray photoelectron spectroscopy analyses confirm that this electrode has excellent stability in morphology and elementary composition after long-term electrochemical measurements. Importantly, density functional theory calculations further indicate that the core-shell heterojunction increased the conductivity of the catalyst, optimized the adsorption energy of the OER intermediates, and improved the OER activity. This study provides a universal strategy for designing more active core-shell structure electrocatalysts based on the rule of coordinated regulation between electronic transport and active sites.

The key to designing photocatalysts is to orient the migration of photogenerated electrons to the target active sites rather than dissipate at inert sites. Herein, we demonstrate that the doping of phosphorus (P) significantly enriches photogenerated electrons at Ni active sites and enhances the performance for CO₂ reduction into syngas. During photocatalytic CO₂ reduction, Ni single-atom-anchored P-modulated carbon nitride showed an impressive syngas yield rate of 85 µmol g_cat^–1 h^–1 and continuously adjustable CO/H₂ ratios ranging from 5:1 to 1:2, which exceeded those of most of the reported carbon nitride-based single-atom catalysts. Mechanistic studies reveal that P doping improves the conductivity of catalysts, which promotes photogenerated electron transfer to the Ni active sites rather than dissipate randomly at low-activity nonmetallic sites, facilitating the CO₂-to-syngas photoreduction process.

Use of a flexible thermoelectric source is a feasible approach to realizing self-powered wearable electronics and the Internet of Things. Inorganic thin films are promising candidates for fabricating flexible power supply, but obtaining high-thermoelectric-performance thin films remains a big challenge. In the present work, a p-type Bi_xSb_2–xTe₃ thin film is designed with a high figure of merit of 1.11 at 393 K and exceptional flexibility (less than 5% increase in resistance after 1000 cycles of bending at a radius of ˜5 mm). The favorable comprehensive performance of the Bi_xSb_2–xTe₃ flexible thin film is due to its excellent crystallinity, optimized carrier concentration, and low elastic modulus, which have been verified by experiments and theoretical calculations. Further, a flexible device is fabricated using the prepared p-type Bi_xSb_2–xTe₃ and n-type Ag₂Se thin films. Consequently, an outstanding power density of ˜1028 µW cm^–2 is achieved at a temperature difference of 25 K. This work extends a novel concept to the fabrication of high-performance flexible thin films and devices for wearable energy harvesting.

Remarkable progress has characterized the field of electrocatalysis in recent decades, driven in part by an enhanced comprehension of catalyst structures and mechanisms at the nanoscale. Atomically precise metal nanoclusters, serving as exemplary models, significantly expand the range of accessible structures through diverse cores and ligands, creating an exceptional platform for the investigation of catalytic reactions. Notably, ligand-protected Au nanoclusters (NCs) with precisely defined core numbers offer a distinct advantage in elucidating the correlation between their specific structures and the reaction mechanisms in electrocatalysis. The strategic modulation of the fine microstructures of Au NCs presents crucial opportunities for tailoring their electrocatalytic performance across various reactions. This review delves into the profound structural effects of Au NC cores and ligands in electrocatalysis, elucidating their underlying mechanisms. A detailed exploration of the fundamentals of Au NCs, considering core and ligand structures, follows. Subsequently, the interaction between the core and ligand structures of Au NCs and their impact on electrocatalytic performance in diverse reactions are examined. Concluding the discourse, challenges and personal prospects are presented to guide the rational design of efficient electrocatalysts and advance electrocatalytic reactions.

Electrocatalytic water splitting has been identified as a potential candidate for producing clean hydrogen energy with zero carbon emission. However, the sluggish kinetics of oxygen evolution reaction on the anode side of the water-splitting device significantly hinders its practical applications. Generally, the efficiency of oxygen evolution processes depends greatly on the availability of cost-effective catalysts with high activity and selectivity. In recent years, extensive theoretical and experimental studies have demonstrated that cobalt (Co)-based nanomaterials, especially low-dimensional Co-based nanomaterials with a huge specific surface area and abundant unsaturated active sites, have emerged as versatile electrocatalysts for oxygen evolution reactions, and thus, great progress has been made in the rational design and synthesis of Co-based nanomaterials for electrocatalytic oxygen evolution reactions. Considering the remarkable progress in this area, in this timely review, we highlight the most recent developments in Co-based nanomaterials relating to their dimensional control, defect regulation (conductivity), electronic structure regulation, and so forth. Furthermore, a brief conclusion about recent progress achieved in oxygen evolution on Co-based nanomaterials, as well as an outlook on future research challenges, is given.