Configuration-dependent stretchable all-solid-state supercapacitors and hybrid supercapacitors

Given the rise in the popularity of wearable electronics that are able to deform into desirable configurations while maintaining electrochemical functionality,stretchable and flexible(hybrid)supercapacitors(SCs)have become increasingly of interest as innovative energy storage devices.Their outstanding power density,long lifetime with low capacitance loss,and appropriate energy density,in particular in hybrid cases make them ideal candidates for flexible electronics.The aim of this review paper is to provide an in-depth discussion of these stretchable and flexible SCs ranging from fabrication to electro-mechanical properties.This review paper begins with a short overview of the fundamentals of charge storage mechanisms and different types of multivalent metal-ion hybrid SCs.The research methods leading up to the current state of these stretchable and flexible SCs are then presented.This is followed by an in-depth presentation of the challenges associated with the fabrication methods for different configurations.Proposed novel strategies to maximize the elastic and electrochemical properties of stretchable/flexible or quasi-solid-state SCs are classified and the pros and cons associated with each are shown.The advances in mechanical properties and the expected advancements for the future of these SCs are discussed in the last section.

Peripheral octamethyl-substituted nickel(Ⅱ)-phthalocyanine-decorated carbon-nanotube electrodes for high-performance all-solid-state flexible symmetric supercapacitors

Construction of advanced electrode materials with unique performance for supercapacitors(SCs)is essential to achieving high implementation in the commercial market.Here,we report a novel peripheral octamethyl-substituted nickel(Ⅱ)phthalocyanine(Ni Me_(2)Pc)-based nanocomposite as the electrode material of all-solid-state SCs.The highly redox-active NiMe_(2)Pc/carboxylated carbon nanotube(CNTCOOH)dendritic nanocomposite provides rapid electron/electrolyte ion-transport pathways and exhibits excellent structural stability,resulting in high-capacity activity and impressive cycling stability.The composite prepared with the optimized weight ratio of Ni Me_(2)Pc:CNT-COOH(6:10)showed the highest specific capacitance of 330.5 F g^(-1)at 0.25 A g^(-1).The constructed NiMe_(2)Pc/CNT-COOH-based all-solid-state symmetric SC device showed excellent performance with a maximum energy density of 22.8 Wh kg^(-1)and outstanding cycling stability(111.6%retained after 35,000 cycles).Moreover,flexible carbon cloth significantly enhanced the energy density of the NiMe_(2)Pc/CNT-COOH all-solid-state symmetric device to 52.1 Wh kg^(-1)with 95.4%capacitance retention after 35,000 cycles,and it could be applied to highperformance flexible electronics applications.These findings provide a novel strategy to design phthalocyanine-based electrode materials for next-generation flexible SC devices.

Model reduction of fractional impedance spectra for time–frequency analysis of batteries, fuel cells, and supercapacitors

Joint time–frequency analysis is an emerging method for interpreting the underlying physics in fuel cells,batteries,and supercapacitors.To increase the reliability of time–frequency analysis,a theoretical correlation between frequency-domain stationary analysis and time-domain transient analysis is urgently required.The present work formularizes a thorough model reduction of fractional impedance spectra for electrochemical energy devices involving not only the model reduction from fractional-order models to integer-order models and from high-to low-order RC circuits but also insight into the evolution of the characteristic time constants during the whole reduction process.The following work has been carried out:(i)the model-reduction theory is addressed for typical Warburg elements and RC circuits based on the continued fraction expansion theory and the response error minimization technique,respectively;(ii)the order effect on the model reduction of typical Warburg elements is quantitatively evaluated by time–frequency analysis;(iii)the results of time–frequency analysis are confirmed to be useful to determine the reduction order in terms of the kinetic information needed to be captured;and(iv)the results of time–frequency analysis are validated for the model reduction of fractional impedance spectra for lithium-ion batteries,supercapacitors,and solid oxide fuel cells.In turn,the numerical validation has demonstrated the powerful function of the joint time–frequency analysis.The thorough model reduction of fractional impedance spectra addressed in the present work not only clarifies the relationship between time-domain transient analysis and frequency-domain stationary analysis but also enhances the reliability of the joint time–frequency analysis for electrochemical energy devices.

Recent advances in electrochemical performance of Mg-based electrochemical energy storage materials in supercapacitors:Enhancement and mechanism

The application of Mg-based electrochemical energy storage materials in high performance supercapacitors is an essential step to promote the exploitation and utilization of magnesium resources in the field of energy storage.Unfortunately,the inherent chemical properties of magnesium lead to poor cycling stability and electrochemical reactivity,which seriously limit the application of Mg-based materials in supercapacitors.Herein,in this review,more than 70 research papers published in recent 10 years were collected and analyzed.Some representative research works were selected,and the results of various regulative strategies to improve the electrochemical performance of Mg-based materials were discussed.The effects of various regulative strategies(such as constructing nanostructures,synthesizing composites,defect engineering,and binder-free synthesis,etc.)on the electrochemical performance and their mechanism are demonstrated using spinelstructured MgX_(2)O_(4) and layered structured Mg-X-LDHs as examples.In addition,the application of magnesium oxide and magnesium hydroxide in electrode materials,MXene's solid spacers and hard templates are introduced.Finally,the challenges and outlooks of Mg-based electrochemical energy storage materials in high performance supercapacitors are also discussed.

Enhancing MXene-based supercapacitors:Role of synthesis and 3D architectures

MXene has been the limelight for studies on electrode active materials,aiming at developing supercapacitors with boosted energy density to meet the emerging influx of wearable and portable electronic devices.Despite its various desirable properties including intrinsic flexibility,high specific surface area,excellent metallic conductivity and unique abundance of surface functionalities,its full potential for electrochemical performance is hindered by the notorious restacking phenomenon of MXene nanosheets.Ascribed to its two-dimensional(2D)nature and surface functional groups,inevitable Van der Waals interactions drive the agglomeration of nanosheets,ultimately reducing the exposure of electrochemically active sites to the electrolyte,as well as severely lengthening electrolyte ion transport pathways.As a result,energy and power density deteriorate,limiting the application versatility of MXene-based supercapacitors.Constructing 3D architectures using 2D nanosheets presents as a straightforward yet ingenious approach to mitigate the fatal flaws of MXene.However,the sheer number of distinct methodologies reported,thus far,calls for a systematic review that unravels the rationale behind such 3D MXene structural designs.Herein,this review aims to serve this purpose while also scrutinizing the structure–property relationship to correlate such structural modifications to their ensuing electrochemical performance enhancements.Besides,the physicochemical properties of MXene play fundamental roles in determining the effective charge storage capabilities of 3D MXene-based electrodes.This largely depends on different MXene synthesis techniques and synthesis condition variations,hence,elucidated in this review as well.Lastly,the challenges and perspectives for achieving viable commercialization of MXene-based supercapacitor electrodes are highlighted.

Emerging perovskite materials for supercapacitors:Structure,synthesis,modification,advanced characterization,theoretical calculation and electrochemical performance

As a new generation electrode materials for energy storage,perovskites have attracted wide attention because of their unique crystal structure,reversible active sites,rich oxygen vacancies,and good stability.In this review,the design and engineering progress of perovskite materials for supercapacitors(SCs)in recent years is summarized.Specifically,the review will focus on four types of perovskites,perovskite oxides,halide perovskites,fluoride perovskites,and multi-perovskites,within the context of their intrinsic structure and corresponding electrochemical performance.A series of experimental variables,such as synthesis,crystal structure,and electrochemical reaction mechanism,will be carefully analyzed by combining various advanced characterization techniques and theoretical calculations.The applications of these materials as electrodes are then featured for various SCs.Finally,we look forward to the prospects and challenges of perovskite-type SCs electrodes,as well as the future research direction.

Ti_(3)C_(2)T_(x) MXene/carbon composites for advanced supercapacitors:Synthesis,progress,and perspectives

MXenes are a family of two-dimensional(2D)layered transition metal carbides/nitrides that show promising potential for energy storage applications due to their high-specific surface areas,excellent electron conductivity,good hydrophilicity,and tunable terminations.Among various types of MXenes,Ti_(3)C_(2)T_(x) is the most widely studied for use in capacitive energy storage applications,especially in supercapacitors(SCs).However,the stacking and oxidation of MXene sheets inevitably lead to a significant loss of electrochemically active sites.To overcome such challenges,carbon materials are frequently incorporated into MXenes to enhance their electrochemical properties.This review introduces the common strategies used for synthesizing Ti_(3)C_(2)T_(x),followed by a comprehensive overview of recent developments in Ti_(3)C_(2)T_(x)/carbon composites as electrode materials for SCs.Ti_(3)C_(2)T_(x)/carbon composites are categorized based on the dimensions of carbons,including 0D carbon dots,1D carbon nanotubes and fibers,2D graphene,and 3D carbon materials(activated carbon,polymer-derived carbon,etc.).Finally,this review also provides a perspective on developing novel MXenes/carbon composites as electrodes for application in SCs.

A dynamic database of solid-state electrolyte(DDSE)picturing all-solid-state batteries

All-solid-state batteries(ASSBs)are a class of safer and higher-energy-density materials compared to conventional devices,from which solid-state electrolytes(SSEs)are their essential components.To date,investigations to search for high ion-conducting solid-state electrolytes have attracted broad concern.However,obtaining SSEs with high ionic conductivity is challenging due to the complex structural information and the less-explored structure-performance relationship.To provide a solution to these challenges,developing a database containing typical SSEs from available experimental reports would be a new avenue to understand the structureperformance relationships and find out new design guidelines for reasonable SSEs.Herein,a dynamic experimental database containing>600 materials was developed in a wide range of temperatures(132.40–1261.60 K),including mono-and divalent cations(e.g.,Li^(+),Na^(+),K^(+),Ag^(+),Ca^(2+),Mg^(2+),and Zn^(2+))and various types of anions(e.g.,halide,hydride,sulfide,and oxide).Data-mining was conducted to explore the relationships among different variates(e.g.,transport ion,composition,activation energy,and conductivity).Overall,we expect that this database can provide essential guidelines for the design and development of high-performance SSEs in ASSB applications.This database is dynamically updated,which can be accessed via our open-source online system.

Ag-integrated mixed metallic Co-Fe-Ni-Mn hydroxide composite as advanced electrode for high-performance hybrid supercapacitors

Direct growth of redox-active noble metals and rational design of multifunctional electrochemical active materials play crucial roles in developing novel electrode materials for energy storage devices.In this regard,silver(Ag)has attracted great attention in the design of efficient electrodes.Inspired by the house/building process,which means electing the right land,it lays a strong foundation and building essential columns for a complex structure.Herein,we report the construction of multifaceted heterostructure cobalt-iron hydroxide(CFOH)nanowires(NWs)@nickel cobalt manganese hydroxides and/or hydrate(NCMOH)nanosheets(NSs)on the Ag-deposited nickel foam and carbon cloth(i.e.,Ag/NF and Ag/CC)substrates.Moreover,the formation and charge storage mechanism of Ag are described,and these contribute to good conductive and redox chemistry features.The switching architectural integrity of metal and redox materials on metallic frames may significantly boost charge storage and rate performance with noticeable drop in resistance.The as-fabricated Ag@CFOH@NCMOH/NF electrode delivered superior areal capacity value of 2081.9μA h cm^(-2)at 5 mA cm^(-2).Moreover,as-assembled hybrid cell based on NF(HC/NF)device exhibited remarkable areal capacity value of 1.82 mA h cm^(-2)at 5 mA cm^(-2)with excellent rate capability of 74.77%even at 70 mA cm^(-2)Furthermore,HC/NF device achieved maximum energy and power densities of 1.39 mW h cm^(-2)and 42.35 mW cm^(-2),respectively.To verify practical applicability,both devices were also tested to serve as a self-charging station for various portable electronic devices.

Recent advances in transition metal phosphide materials:Synthesis and applications in supercapacitors

Supercapacitors(SCs)are considered promising energy storge systems because of their outstanding power density,fast charge and discharge rate and long-term cycling stability.The exploitation of cheap and efficient electrode materials is the key to improve the performance of supercapacitors.As the battery-type materials,transition metal phosphides(TMPs)possess high theoretical specific capacity,good electrical conductivity and superior structural stability,which have been extensively studied to be electrode materials for supercapacitors.In this review,we summarize the up-to-date progress on TMPs materials from diversified synthetic methods,diverse nanostructures and several prominent TMPs and their composites in application of supercapacitors.In the end,we also propose the remaining challenges toward the rational discovery and synthesis of high-performance TMP electrodes materials for energy storage.

Mechanical reliable,NIR light-induced rapid self-healing hydrogel electrolyte towards flexible zinc-ion hybrid supercapacitors with low-temperature adaptability and long service life

Hydrogel electrolytes hold great potential in flexible zinc ion supercapacitors(ZICs)due to their high conductivity,good safety,and flexibility.However,freezing of electrolytes at low temperature(subzero)leads to drastic reduction in ionic conductivity and mechanical properties that deteriorates the performance of flexible ZICs.Besides,the mechanical fracture during arbitrary deformations significantly prunes out the lifespan of the flexible device.Herein,a Zn^(2+)and Li^(+)co-doped,polypyrrole-dopamine decorated Sb_(2)S_(3)incorporated,and polyvinyl alcohol/poly(N-(2-hydroxyethyl)acrylamide)double-network hydrogel electrolyte is constructed with favorable mechanical reliability,anti-freezing,and self-healing ability.In addition,it delivers ultra-high ionic conductivity of 8.6 and 3.7 S m^(-1)at 20 and−30°C,respectively,and displays excellent mechanical properties to withstand tensile stress of 1.85 MPa with tensile elongation of 760%,together with fracture energy of 5.14 MJ m^(-3).Notably,the fractured hydrogel electrolyte can recover itself after only 90 s of infrared illumination,while regaining 83%of its tensile strain and almost 100%of its ionic conductivity during−30–60°C.Moreover,ZICs coupled with this hydrogel electrolyte not only show a wide voltage window(up to 2 V),but also provide high energy density of 230 Wh kg^(-1)at power density of 500 W kg^(-1)with a capacity retention of 86.7%after 20,000 cycles under 20°C.Furthermore,the ZICs are able to retain excellent capacity even under various mechanical deformation at−30°C.This contribution will open up new insights into design of advanced wearable flexible electronics with environmental adaptability and long-life span.

MXene Enhanced 3D Needled Waste Denim Felt for High‑Performance Flexible Supercapacitors

MXene,a transition metal carbide/nitride,has been prominent as an ideal electrochemical active material for supercapacitors.However,the low MXene load limits its practical applications.As environmental concerns and sustainable development become more widely recognized,it is necessary to explore a greener and cleaner technology to recycle textile by-products such as cotton.The present study proposes an effective 3D fabrication method that uses MXene to fabricate waste denim felt into ultralight and flexible supercapacitors through needling and carbonization.The 3D structure provided more sites for loading MXene onto Z-directional fiber bundles,resulting in more efficient ion exchange between the electrolyte and electrodes.Furthermore,the carbonization process removed the specific adverse groups in MXenes,further improving the specific capacitance,energy density,power density and electrical conductivity of supercapacitors.The electrodes achieve a maximum specific capacitance of 1748.5 mF cm-2 and demonstrate remarkable cycling stability maintaining more than 94%after 15,000 galvanostatic charge/discharge cycles.Besides,the obtained supercapacitors present a maximum specific capacitance of 577.5 mF cm^(-2),energy density of 80.2μWh cm^(-2)and power density of 3 mW cm^(-2),respectively.The resulting supercapacitors can be used to develop smart wearable power devices such as smartwatches,laying the foundation for a novel strategy of utilizing waste cotton in a high-quality manner.

Thin polymer electrolyte with MXene functional layer for uniform Li^(+) deposition in all-solid-state lithium battery

Solid polymer electrolyte(SPE) shows great potential for all-solid-state batteries because of the inherent safety and flexibility;however, the unfavourable Li+deposition and large thickness hamper its development and application. Herein, a laminar MXene functional layer-thin SPE layer-cathode integration(MXene-PEO-LFP) is designed and fabricated. The MXene functional layer formed by stacking rigid MXene nanosheets imparts higher compressive strength relative to PEO electrolyte layer. And the abundant negatively-charged groups on MXene functional layer effectively repel anions and attract cations to adjust the charge distribution behavior at electrolyte–anode interface. Furthermore,the functional layer with rich lithiophilic groups and outstanding electronic conductivity results in low Li nucleation overpotential and nucleation energy barrier. In consequence, the cell assembled with MXene-PEO-LFP, where the PEO electrolyte layer is only 12 μm, much thinner than most solid electrolytes, exhibits uniform, dendrite-free Li+deposition and excellent cycling stability. High capacity(142.8 mAh g-1), stable operation of 140 cycles(capacity decay per cycle, 0.065%), and low polarization potential(0.5 C) are obtained in this Li|MXene-PEO-LFP cell,which is superior to most PEO-based electrolytes under identical condition. This integrated design may provide a strategy for the large-scale application of thin polymer electrolytes in all-solid-state battery.

Viability of all-solid-state lithium metal battery coupled with oxide solid-state electrolyte and high-capacity cathode

Owing to the utilization of lithium metal as anode with the ultrahigh theoretical capacity density of 3860 mA h g^(-1)and oxide-based ceramic solid-state electrolytes(SE),e.g.,garnet-type Li7La_(3)Zr_(2)O_(12)(LLZO),all-state-state lithium metal batteries(ASLMBs)have been widely accepted as the promising alternatives for providing the satisfactory energy density and safety.However,its applications are still challenged by plenty of technical and scientific issues.In this contribution,the co-sintering temperature at 500℃is proved as a compromise method to fabricate the composite cathode with structural integrity and declined capacity fading of LiNi_(0.5)Co_(0.2)Mn_(0.3)O_(2)(NCM).On the other hand,it tends to form weaker grain boundary(GB)inside polycrystalline LLZO at inadequate sintering temperature for LLZO,which can induce the intergranular failure of SE during the growth of Li filament inside the unavoidable defect on the interface of SE.Therefore,increasing the strength of GB,refining the grain to 0.4μm,and precluding the interfacial defect are suggested to postpone the electro-chemo-mechanical failure of SE with weak GB.Moreover,the advanced sintering techniques to lower the co-sintering temperature for both NCM-LLZO composite cathode and LLZO SE can be posted out to realize the viability of state-of-the-art ASLMBs with higher energy density as well as the guaranteed safety.

All-solid-state flexible asymmetric supercapacitors with high energy and power densities based on NiCo_2S_4@MnS and active carbon

Electrode material based on a novel core–shell structure consisting of NiCoS（NCS） solid fiber core and Mn S（MS） sheet shell（NCS@MS） in situ grown on carbon cloth（CC） has been successfully prepared by a simple sulfurization-assisted hydrothermal method for high performance supercapacitor. The synthesized NiCoS@Mn S/CC electrode shows high capacitance of 1908.3 F gat a current density of 0.5 A gwhich is higher than those of NiCoSand Mn S at the same current density. A flexible all-solid-state asymmetric supercapacitor（ASC） is constructed by using NiCoS@Mn S/CC as positive electrode, active carbon/CC as negative electrode and KOH/poly（vinyl alcohol）(PVA) as electrolyte. The optimized ASC shows a maximum energy density of 23.3 Wh kgat 1 A g, a maximum power density of about7.5 kw kgat 10 A gand remarkable cycling stability. After 9000 cycles, the ASC still exhibited67.8% retention rate and largely unchanged charge/discharge curves. The excellent electrochemical properties are resulted from the novel core–shell structure of the NiCoS@Mn S/CC electrode, which possesses both high surface area for Faraday redox reaction and superior kinetics of charge transport. The NiCoS@Mn S/CC electrode shows a promising potential for energy storage applications in the future.

Wet-spun poly(ionic liquid)-graphene hybrid fibers for high performance all-solid-state flexible supercapacitors

It is crucial to develop flexible and wearable electronic devices that have attracted tremendous interest due to their merits on compactness,flexibility and high capacitive properties.Herein we report the continuously ordered macroscopic poly(ionic liquid)-graphene fibers by wet spinning method via liquid crystal assembly for supercapacitor application.The fabricated all-solid-state supercapacitors exhibited a high areal capacitance(268.2 mF cm 2)and volumetric capacitance(204.6 F cm 3)with an outstanding areal energy density(9.31μWh cm-2)and volumetric energy density(8.28 mWh cm-3).The fiber supercapacitors demonstrated exceptional cycle life for straight electrodes of about 10,000 cycles(94.2%capacitance retention)and flexibility at different angles(0°,45°,90°,180°)along with a good flexible cycling stability after 6000 cycles(92.7%capacitance retention).To date,such a novel poly(ionic liquid)-graphene fiber supercapacitors would be a new platform in real-time flexible electronics.

Flexible ultrathin all-solid-state supercapacitors

The flexible ultrathin all-solid-state supercapacitors with good electrochemical and mechanical performance were fabricated by the facile methods. The singlewall carbon nanotubes （SWCNTs）-polyaniline （PANI） film electrodes and poly（vinyl alcohol） （PVA）/H3PO4 electrolyte film were prepared by spray-printing and spincoating strategies, respectively. Thus, the thickness of a supercapacitor is only 8.4 μm. When the mass ratio of SWCNT to PANI is 1：1, the tensile strength and Young＇s modulus of the electrode are 10.9 and 655 MPa, respectively. The interior contact resistance of the supercapacitors based on this electrode is only 15-30 Ω. Furthermore, the specific capacitance of this electrode can reach about 355.5 F·g^-1, and the supercapacitor based on this electrode maintains 87.2% of its initial specific capacitance over 5000 charging/discharging cycles. Moreover, the supercapacitor shows an excellent electrochemical stability at different bending states. Therefore, the all-solid-state supercapacitors prepared by our strategies would meet the demands of wearable, lightweight, and compact energy storage devices.

Improving the performance of all-solid-state supercapacitors by modifying ionic liquid gel electrolytes with graphene nanosheets prepared by arc-discharge

Ionic liquid gel polymers have widely been used as the electrolytes in all-solid-state supercapacitors, but they suffer from low ionic conductivity and poor electrochemical performance. Arc discharge is a fast, low-cost and scalable method to prepare multi-layered graphene nanosheets, and as-made graphene nanosheets （denoted as ad-GNSs） with few defects, high electrical conductivity and high thermal stability should be favorable conductive additive materials. Here, a novel ionic liquid gel polymer electrolyte based on an ionic liquid （EM1MNTF2） and an copolymer （P（VDF-HFP）） was modified by the addition of ad-GNSs as an ionic conducting promoter. This modified gel electrolyte shows excellent thermal stability up to 400 ℃ and a wide electrochemical window of 3 V. An all-solid-state supercapacitor based on commercial activated carbon was fabricated using this modified ionic liquid gel polymer electrolyte, which shows obviously improved electrochemical behaviors compared with those of the corresponding all-solid-state supercapacitor using pure ionic liquid gel polymer electrolyte. Specially, smaller internal resistance, higher specific capacitance, better rate performance and cycling stability are achieved. These results indicate that the ionic liquid gel polymers modified by ad-GNSs would be promising and suitable gel electrolytes for high performance all-solid-state electrochemical devices.

Electrospun polyporous VN nanofibers for symmetric all-solid-state supercapacitors

To promote the energy density of symmetric all-solid-state supercapacitors(SCs),efforts have been dedicated to searching for high-performance electrode materials recently. In this paper,vanadium nitride(VN) nanofibers with mesoporous structure have been fabricated by a facile electrospinning method. Their crystal structures and morphology features were characterized by X-ray diffraction,scanning electron microscopy,and transmission electron microscopy. The mesoporous structure of VN nanofibers,which can provide short electrolyte diffusion routes and conducting electron transport pathways,is beneficial to their performance as a supercapacitor electrode. Under a stable electrochemical window of 1.0 V,VN nanofibers possess an excellent mass specific capacitance of 110.8 F/g at a scan rate of 5 mV/s. Moreover,the VN nanofibers were further assembled into symmetric all-solid-state SCs,achieving a high energy density of 0.89 mW·h/cm^3 and a high power density of 0.016 W/cm^3 over an operating potential range from 0 to 1.0 V. These results demonstrate that VN nanofibers could be potentially used for energy storage devices.

Activated carbon felts with exfoliated graphene nanosheets for flexible all-solid-state supercapacitors

The recent development of portable electronics promotes the growing demand for flexible energy storage devices. Supercapacitors are promising candidates due to their high power density. Therefore, flexible supercapacitors are desired. Here, the porous activated carbon felts(ACFs) with exfoliated graphene nanosheets and rich oxygen-containing groups were fabricated by a facile thermal treatment strategy.Such ACFs can act as the flexible electrodes of all-solid-state supercapacitors directly without the use of binder and conductive materials. They exhibit excellent electrochemical properties, such as high specific areal capacitance, superior rate ability and long-term cycling stability. Moreover, the fabricated flexible all-solid-state supercapacitors based on ACFs deliver stable electrochemical performance under different bending states.