Hierarchical One-Dimensional Ammonium Nickel Phosphate Microrods for High-Performance Pseudocapacitors

High-performance electrochemical capacitors will drive the next-generation portable, flexible and wearable electronics. Unlike the conventional all-carbon supercapacitors (electric double layer capacitors, EDLC) with high power but poor energy density, pseudocapacitors capitalize the high energy density inherent to reversible redox reactions and provide a facile means to enhancing the energy ratings of supercapacitors. The high length-to-diameter ratio and anisotropic character of 1-D architecture makes them suitable for use in energy storage. For the first time, we report 1-D microrod structures (~ 36 nm width) of ammonium nickel phosphate hydrate (ANPmr) as a pseudocapacitor with high energy rating and power handling. To confirm the data, the ANPmr-based pseudocapacitor was subjected to various configurations (i.e., half-cell, symmetric, asymmetric, and flexible all-solid-state) and in each case it gave excellent values compared to any accessible literature to date. We clearly demonstrate that a flexible all-solid-state ANPmr-based pseudocapacitor achieved high areal capacitance of 66 mF cm−2 with extra-ordinary energy (21.2 mWh cm−2) and power (12.7 mW cm−2) densities. This work opens doors for a facile, robust and scalable preparation strategy for low-cost, earth-abundant electrode materials for high-performance pseudocapacitors.


Results and Discussion
A schematic representation of the synthetic strategy adopted for the formation of three different nanostructures of NH 4   microplatelets (ANP mp ) and microrods (ANP mr ) were synthesized in EG, water, and EG/water mixture via facile solvo/hydrothermal processes, respectively. The morphology and size of as-synthesized NH 4 NiPO 4 . H 2 O were confirmed with the SEM and TEM analyses. Figure 2 (a-f) shows the SEM and TEM images of ANP md , ANP mp and ANP mr showing the unique formation of different morphologies of microplatelets, microdendrites and microrods. It is interesting to note from Fig. 2 that the obtained microplatelets ( Fig. 2a) have average dimensions of 400-600 nm in length and nearly 428 nm in diameter (Fig. 2d). Figure 2b,e illustrate the SEM and TEM images of ANP md with average dimension of 100-300 nm in length. As shown in Fig. 2c, the obtained microrods have average dimensions of 200-300 nm in length and 35-40 nm in diameter, with few of them as nanorods. As illustrated by the HRTEM image, (Fig. 2f and insets), the observed microrods (with nanometric width) are polycrystalline in nature with clear lattice fringes. The d-spacings of the lattice fringes is found to be 0.29 nm, corresponding to the (121) plane of ANP mr along with SAED pattern corresponding to (121) plane. Note that these microrods could only be formed at 48 h as shown from the SEM and TEM images ( Supplementary Information, Figure-S1).
From the results, it is evident that spatial localization of water molecules in solvent mixture is critical in mediating the shape growth; the highly viscous solvent EG (η = 21 mPa s, 20°C) restrains the mobility of reactants compared to water (η = 1.0087 × 10 −3 mPa s, 20 °C). The solubility and mobility of reactants considerably favours the homogeneous nucleation process when an appropriate amount of EG/H 2 O is used 17 . In general, the formation mechanism of the microstructures seems to occur via the hydrolysis of acetate to acetic acid and hydroxide ion in aqueous solution, followed by the reaction between the PO 4 3− , NH 4 + and Ni 2+ ions. The diffusion of the active sites of PO 4 3− , NH 4 + and Ni 2+ ions increases with a rise in temperature thus enhancing the nucleation process. The viscosity of EG decreased with increase in temperature thereby facilitating fast nucleation by reducing the interlayer spacing thus enhance the anisotropic growth of nanorods 18 . The NH 4 NiPO 4 .H 2 O layers are formed by sharing the highly distorted NiO 6 octahedra corners with cross-linked distorted PO 4 3− tetrahedra and NH 4 + ions inserted between the inorganic layers via hydrogen bonding 19 . There are other 1-D materials, such as LnPO 4 and CePO 4 , whose formation is driven by diffusion-controlled growth mechanism, i.e., attachment of infinite linear chains along the axis of its crystalline phase 20 . The structural arrangement of NH 4 NiPO 4 .H 2 O also contains open channels of octahedra along the parallel and perpendicular axis in (010) and (001) planes which may tend to form linear chain extending of octahedral along the axis. Additionally, a drastic change in morphology has also been reported when using different reactants. For instance, Zhao et al. 13   nickel acetate gives dendrite-like NH 4 NiPO 4 .H 2 O morphology. The nitrate anion, which is a weak base and a good-leaving group, can easily substitute with one another and proceeds to 3-D growth process, whereas strong base of acetate anions surrounding the cation impede the 3-D growth. However, acetate ions and equal mixture of water and EG play a crucial role in the formation of unique morphologies of 1-D NH 4  O and all the observed peaks can be readily indexed to a pure orthorhombic phase (space group: Pmn2) with the cell parameters of a = 5.425 Å, b = 8.77 Å and c = 4.31 Å in accordance with the JCPDS card no 86-0585. Interestingly, an increase in intensity of the (121) peak is observed compared to the (200) peak infers the preferential orientation or anisotropic growth along the c-axis (insert Fig. 3). The broad peak of the ANPmr is a clear indication that its particle size is smaller than those of the ANP mp and ANP md .
Electrochemical characterization of three-electrode system. Figure 4 compares the electrochemical performance of the three-electrode configurations of the three microstructures. The cyclic voltammetric evolutions (Fig. 3a)  The microrods showed well-defined electrochemistry with large current density and narrower peak-to-peak potential (Δ E p ≈ 150 mV at 20 mVs −1 ) compared to the > 200 mV shown by the microdendrites and microplatelets, meaning that ANP mr exhibits better electrochemical reversibility and faster electron transfer kinetics. Also, unlike the others, the microrods showed no additional oxidation peaks, indicating that the only oxidation process is that of Ni 2+ /Ni 3+ without any other phase changes. Figure 4b shows typical charge/discharge curves of the ANP mr at different current densities (1 to 50 A g −1 ). At all the current densities investigated (Fig. 4c), the ANP mr electrode showed the best performance compared to the ANP md and ANP mp , achieving remarkable a maximum reversible specific capacity of ~1400 F g −1 . Interestingly, the ANP mr showed an extra-ordinary high rate capability proven by the high capacitance of 545 F g −1 at a very high current density of 10 A g −1 which is extremely high in comparison with the capacitance value reported in the literature to date. The high value reflects the effective ion migration even at a higher speed which is influenced by high surface area of microrods with minimum diffusion length of ion accessibility. Upon continuous cycling, the electrode experienced capacitance loss at the initial cycles, stabilized at about 200 cycles and then retained nearly ~80% of its original specific capacitance after 5,000 cycles (Fig. 4d). Figure 4e highlights the Ragone plot of the calculated power and energy densities. The results are higher than most recently reported supercapacitor nanomaterials based on 3-electrode configurations (Supplementary Information, Table-S1). To provide further insights into the pseudocapacitive behaviour of the ANP materials, EIS experiments were conducted at open-circuit voltage at room temperature. Figure 4f compares the EIS (Nyquist plots) of the three ANP-based electrodes. The EIS data were satisfactorily fitted with the electrical equivalent circuit (EEC) comprising two Voigt RC elements, involving a series resistance (R s ), charge-transfer resistance (R ct ) and constant-phase elements (CPE or Q). As summarised in the Table-S2 (Supplementary  Information) the R s of the ANP mr (ca. 0.8 Ω ) is smaller than those of the ANP mp (1.48 Ω ) and ANP md (1.72 Ω ). Also, the total R ct values of the ANP mr (ca. 25.6 Ω ) is smaller than those of the ANP md (ca. 31.4 Ω ) and ANP mp (ca. 41.4 Ω ). These results clearly indicate that the rod-like morphology provides the least internal resistance of the electrode and permits faster charge transportation compared to other morphologies investigated in this study. In addition, from the Fig. 4f, the ANP mr showed near-vertical line as expected of a high-performing pseudocapacitance compared to others. The "knee" or "onset" frequency (f o ), which is a measure of the power capability of a supercapacitor, decreases as ANP mr (5 kHz) > ANP md (1.7 kHz) > ANP mp (1.18 kHz), confirming the higher energy-storage capability of the ANP mr over other electrodes. The experimentally observed impedance curve was best fitted with the equivalent circuit and the calculated value of ESR of ANP mr , ANP md and ANP mp were found to be 0.79, 1.72 and 1.48 Ω . It is noted that the impedance of CPE is defined as equation (1): where Q represents the frequency-independent constant relating to the surface electroactive properties, w is the radial frequency, the exponent n arises from the slope of log Z vs. log f (and has values − 1 ≤ n ≤ 1). If n = 0, the CPE behaves like a pure resistor; n = 1, CPE behaves as a pure capacitor, n = -1, CPE behaves as an inductor; while n = 0.5 corresponds to Warburg impedance (Z w ) which is related to the diffusion of the ions. The n values observed for the ANP mr (n 1 = 0.58 and n 2 = 0.53) while other electrodes were ca. 0.8, which suggests that the rod-like structure allows for improved ion diffusion than others. Further, the impedance curves were measured after 5,000 consecutive charge-discharge cycles ( Supplementary Information, Figure-S2).

Symmetric pseudocapacitors in alkaline electrolyte (3 M KOH).
Considering the high-performance of the ANP mr electrode at half-cell configuration, subsequent studies on full-cell pseudocapacitor devices were devoted to the ANP mr . Figure 5 summarises the performance of the ANP mr as a symmetric pseudocapacitor in an aqueous alkaline electrolyte (3 M KOH) using carbon cloth as substrate/current collector, showing typical CV curve at a scan rate of 10 mVs −1 (Fig. 5a), charge-discharge curves at current density 10 mA cm −2 (Fig. 5b), areal capacitance (Fig. 5c), cycle stability (Fig. 5d), Ragone plots (Fig. 5e) and Nyquist plot (Fig. 5f) of ANP mr symmetric pseudocapacitors. The symmetric pseudocapacitor showed satisfactory rate capability, evident at different scan rate and current densities (Supplementary Information, Figure-S3). Interestingly, the electrode gave high areal capacitance of 138 mF cm −2 at 20 mA cm −2 , 92% of which (126 mF cm −2 ) was retained even at a current density of 50 mA cm −2 . The high rate capability can be attributed to the microrods maintaining their excellent structural stability and charge propagation even at higher current densities. It was found that the ANP mr has the highest areal capacitance compared with recently reported symmetric supercapacitor materials (cf. Supplementary Information, Table-S3). More importantly, the areal capacitance retain more than 97% of its initial values after 5000 continuous charge-discharge cycles with 100% columbic efficiency (Fig. 5d). As shown in Fig. 5e, the ANP mr delivered the highest energy and power densities of 69 mWh cm −2 and 145 mW cm −2 at a current density of 20 mA cm −2 and was found that the ANP mr has energy values compared to that of literature values.
As shown in the EIS spectra (Fig. 5f), the observed high frequency intercept show that ANP mr (0.13 Ω ) have much smaller ESR with inclined vertical line after the semicircle with the response time of 8 ms was lower than the values reported in liquid electrolyte used supercapacitors, onion-like carbon (26 ms) 25 and biscrolled yarn (17 ms) 26 . As summarised in the Table-S2 (Supplementary Information), the ANP mr -based symmetric cell gave very small R s (ca. 0.13 Ω ) and total R ct (R ct1 + R ct2 ≈ 33 Ω ), with each n value (n 1 or n 2 ) greater than 0.8 but less than 1, clearly confirming the pseudocapacitive properties of the ANP mr when deployed in full-cell system.

Asymmetric pseudocapacitors in neutral aqueous electrolyte (1 M Na 2 SO 4 ).
We also prepared asymmetric pseudocapacitors in order to further increase the energy density of the device. Typically, ANP mr coated carbon cloth electrodes were used as positive and activated carbon (Norit ® supra) coated carbon cloth electrode as negative in 1M Na 2 SO 4 neutral aqueous electrolyte. The cyclic voltammograms of ANP mr (Fig. 6a) obtained at a scan rate of 25 mVs −1 shows rectangular shapes. Figure 6b show the charge-discharge at a current density of 10 mA cm −2 , the cell gave high areal capacitance of 221 mF cm −2 at 20 mA cm −2 , 90% of (201 mF cm −2 ) which was retained even at a current density of 50 mA cm −2 . It was found that this asymmetric capacitor has best areal capacitances compared to many nanostructured electrodes reported earlier in asymmetric capacitors, such as H-TiO 2 @ MnO 2 (0.9 F cm −3 ) 30 , TiO 2 /NiO nanotube array (2.9 mF cm −2 at 0.4 mA cm −2 ) 31 and Fe 3 O 4 -SnO 2 core-shell nanorod film (7 mF cm −2 ) 32 . As shown in Fig. 6d, 50 h voltage-floating tests show excellent capacity retention for the ANP mr cell with areal capacitance of 135 mF cm −2 , and long cycle stability with an almost 100% columbic efficiency. Ragone plot of ANP mr asymmetric capacitors (Fig. 6e) exhibits extraordinary energy (134.6 mWh cm −2 ) and power (325.6 mW cm −2 ) densities at a current density of 20 mA cm −2 . These values are much higher when compared with other asymmetric pseudocapacitors (Supplementary Table-S3). Figure 6f shows the Nyquist plot of ANP mr //AC asymmetric pseudocapacitor in 1M Na 2 SO 4 neutral aqueous electrolyte showed very small ESR (0.55 Ω ) with inclined vertical line with the response time of 18 s. From Table-S2 (Supplementary Information), the ANP mr -based asymmetric cell showed similar EIS behaviour as its symmetric cell counterpart; very small R s (ca. 0.55 Ω ) and total R ct (ca. 41 Ω ), with each n value (n 1 or n 2 ) greater than 0.8 but less than 1, further corroborating that the pseudocapacitive behaviour of the ANP mr when used in full-cell system. In addition, it may be observed that the CV curve (Fig. 5a) is higher in the positive direction than in the negative side. This behaviour is related to the series resistance of the asymmetric configuration; the R s value of the asymmetric configuration is much higher (ca. 0.55 Ω ) than the symmetric counterpart (ca. 0.13 Ω ), which explains why we did not observe this for the symmetric configuration.
All-solid-state flexible symmetric pseudocapacitors. Finally, we explored the performance of the as-prepared ANP materials (ANP mp , ANP md and ANP mr ) as all-solid-state flexible symmetric pseudocapacitors using PVA-KOH polymer electrolyte and carbon cloth as substrate/current collector (Fig. 7). Also, nickel foam was used as the substrate/current collector for the ANP mr -based all-solid-state pseudocapacitors (see Supplementary Information, Figure-S4) but we found carbon cloth to be easier to handle, more flexible, with more improved electrochemical properties than nickel foam. Thus, all further studies were devoted to the use of carbon cloth. From every analysis, the ANP mr showed better electrochemical performance than the other two ANP materials ( Supplementary Information,  Figure-S5). Figure 7a exemplifies typical galvanostatic charge-discharge experiments, while Fig. 6b compares the specific capacitance values obtained at various current densities (0.1-0.8 mAcm −2 ). The ANP mr gave an excellent specific areal capacitance of 66 mF cm −2 at 0.1 mA cm −2 (Fig. 7b), and even at a higher current density of 0.8 mA cm −2 the capacitance remained as high as 3 mF cm −2 . This value is much higher compared to the values reported for other all-solid-state symmetric and asymmetric supercapacitors ( Supplementary Information, Table-S4). For example, areal capacitance achieved with ANP mr was better than the electrochemical double layer microcapacitors which delivered 0.4-2 mF cm −2 at scan rates of 1-100 mVs −1 27 , and graphene or carbon nanotube based flexible supercapacitors that showed 3-50 mF cm −2 28 . From the prolonged cycle stability performed at the scan rate of 0.6 mA cm −2 , the ANP mr retained ~97% of its initial capacitance even after 5000 consecutive cycles (Fig. 7c). The Ragone plot (Fig. 7d) showed significantly higher energy (21.2 mWh cm −2 ) and power (12.7 mW cm −2 ) densities compared to the values reported in the literature to date for all-solid-state SCs (Fig. 6d, and Supplementary Information, Table-S4). From the Nyquist plots of ANP mr (Supplementary Information, Figure-S5(e)) the equivalent series resistance (ESR) values obtained before and after 5000 cycles were 4.5 and 23 Ω , respectively. The response time (60 ms) was lower than the reported values for solid electrolytes (80 ms), and activated carbon (700 ms) 22 . Indeed, detailed examination of the EIS data of the various ANP materials in Table-S2 (i.e., ANP mp , ANP md and the three ANP mr obtained at 24, 36 and 48 h) showed an interesting trend on the conductivity of the various morphologies. The three rod-like morphologies gave the least series resistance (i.e., R s ≈ 5 Ω ) compared to the platelet-like (R s ≈ 11 Ω ) and dendrite-like (R s ≈ 25 Ω ) morphologies. The total R ct values decrease as follows: ANP mr@36h (ca. 140 Ω ) > ANP md (ca. 115 Ω ) > ANP mp (ca. 93 Ω ) > ANP mr@24h (ca. 32 Ω ) > ANP mr@48h (ca. 24 Ω ). This is an interesting result as it clearly corroborates other electrochemical data that shows that our rod-like morphology, obtained at optimized 48 h and 200 °C, gave the best conditions for ion mobility in pseudocapacitor devices.
To further understand the reason for the high-performance of the ANP mr , we examined its specific surface area and porosity by performing the Brunauer-Emmett-Teller (BET) measurements ( Supplementary  Information, Fig. S6 showing the N 2 adsorption-desorption isotherm of the nanorods). The observed BET surface area was 214 m 2 g −1 with an average pore size distribution of 2-20 nm with a pore volume of 0.7 m 3 g −1 (BJH desorption), inferring the co-existence of mesoporous and microporous microrods.
As a proof of concept, Fig. 8 describes the bendability of the ANP mr -based all-solid-state symmetric pseudocapacitor (ASSSP) and its ability to light up a 1.67 V LED when connected in series. Interestingly, when the ASSSP was bent to nearly 120° and subjected to 1000 charge-discharge cycles, it was able to maintain its performance with ca. 100% coulombic efficiency. Conclusions Novel NH 4 NiPO 4 .H 2 O with unique morphologies (microdendrites, microplatelets and microrods) of different supercapacitive properties has been reported. The microrod morphology gave an extraordinarily high specific capacitance, power and energy densities in half-cell and full-cell configurations (i.e., symmetric and asymmetric cells, including all-solid-state flexible pseudocapacitors) in different electrolytes. The well-aligned microrods (ca. 35.8 nm diameter) with meso-and microporous surface enhance ion propagation and interfacial interactions compared to the long range plates (428 nm in diameter) with larger thickness or the microdendrites with smaller branched structure (ca. 100 nm) with uneven surface. The all-solid-state symmetric pseudocapacitor fabricated from the ANP microrods proved it can generate power even when bent to 120° and can drive an LED when connected in series. The study proves that rod-like morphology (with ~ 36 nm width) provides a significant and promising direction for novel 1-D and 2-D materials to obtain high-performance pseudocapacitors, especially for flexible and wearable electronics. The study has opened new doors of research opportunities for this type of materials. Such opportunities include examination of other redox-active metals other than nickel, and interrogation of the impact of tuning the reported synthesis protocols (e.g., in terms of changes in temperature, solvents, reaction times, work-up conditions, etc) or new synthesis procedure on the final structures and physicochemical properties. These constitute the directions of the on-going investigations in our laboratory.

Synthesis and characterisation of NH 4 NiPO 4 .H 2 O nanorods (with micrometric length).
Analytical grade chemicals, Nickel (II) acetate tetrahydrate (Ni(CH 3 COO) 2 .4H 2 O), ammonium dihydrophosphate (NH 4 H 2 PO 4 ) and ethylene glycol (EG) were procured from Sigm-Aldrich, and used as received without further purification. In a typical synthesis, 0.5 g of Ni(CH 3 COO) 2 .4H 2 O and NH 4 H 2 PO 4 were dissolved thoroughly in 40 ml deionized water. Subsequently, an equal amount of EG was added into the above solution (water and EG volume ratio is 1:1). After vigorous stirring for 1 h, the mixture was then transferred into an autoclave and heated at 200 °C for 48 h. The ANP mr products obtained at 200 °C for 24 and 36 h duration were also tested for comparison. The resulted greenish yellow precipitates were thoroughly washed with deionised water and ethanol to remove any unreacted materials. Finally, the powder was dried slowly at 50 °C in oven or left in air to dry at room temperature overnight. The microrod-like product (evident from SEM and TEM images) is abbreviated herein as ANP mr . The control experiments were also performed using ethylene glycol alone or water alone at the same time period of 48 h and constant temperature of 200 °C. From SEM and TEM, the product from water alone gave platelet-like morphology (abbreviated herein as ANPmp), while that from ethylene glycol alone gave dendrite-like morphology (abbreviated herein as ANP md ).
The formation of NH 4 NiPO 4 .H 2 O was investigated by PANalytical X'Pert PRO diffractometer equipped with Ni-filtered Cu K-alpha radiation (λ = 1.541841 A). The morphology of the as-synthesized powders was analysed using JEOL-JSM 7500F scanning electron microscope operated at 2.0 kV. TEM and HRTEM images were obtained from JEOL-Jem 2100 microscope operated at an acceleration voltage of 200 kV. BET measurements were performed to measure the specific surface area and pore size based on the N 2 adsorption-desorption method by using Micromeritics TriStar II instrument.
Materials, preparation and pseudocapacitor fabrication. Nickel foam (Celmet: thickness = 1.6 mm, surface area 7500 m 2 , cell size = 0.5 mm, 48-52 cells per inch) and carbon cloth (B-1/C, E-TEK) were used as substrates and current collectors in the fabrication of the half-cell (three-electrode system) and full-cell (two-electrode systems), respectively. Prior to use, the nickel foam was properly cleaned as we described before 39 by first sonicating in 1 M HCl solution for 30 min, washed several times with copious amount of de-ionized water, and then dried under vacuum. The electrolyte materials, polyvinyl alcohol (PVA), potassium hydroxide (KOH) and sodium sulphate (Na 2 SO 4 ) were procured from Sigma-Aldrich and used as received. For the three-electrode system, the electrode materials were prepared by coating a slurry mixture of NH 4 NiPO 4 .H 2 O (ANP), carbon black and polyvinylidene fluoride (PVDF) (80:15:5 weight ratio) on a piece of clean nickel foam and dried overnight in a vacuum oven at 80 °C. The mass of the active material on nickel foam was 0.32 mg for ANP mr , 0.35 mg for ANP md and 0.34 mg for ANP mp . For the symmetric and asymmetric systems, the electrode materials were prepared by coating a slurry mixture of ANP mr , activated carbon (AC, Norit ® supra 30) and PVDF (50:40:10 weight ratio) on carbon cloth (disc = 1.6 cm 2 ). The symmteric (ANP mr //ANP mr ) and asymmetric (ANP mr //AC) pseudocapacitors were performed in 3 M KOH and 1 M Na 2 SO 4 , respectively. The all-solid-state flexible symmetric pseudocapacitor (square = 1 cm 2 ) was fabricated in a similar manner as the symmetric cells but using a polymer gel electrolyte (PVA-KOH). In a typical polymer electrolyte preparation, 8 g of PVA and 4 g of KOH were dissolved in deionised water (40 ml) and the mixture was stirred at 90 °C for 1 h to form a gel. Various forms of the ANP systems (ANP mp , ANP md , ANP mr as well the ANP mr obtained at different 24 and 36 h at the same temperature) were investigated as possible candidates as all-solid-state flexible pseudocapacitors.
Electrochemical measurements and calculations. All electrochemical tests involving cyclic voltammetry (CV), electrochemical impedance spectroscopy (EIS) and galvanostatic charge-discharge analysis (including voltage-holding experiments) were performed at room temperature using computer-controlled Scientific RepoRts | 5:17629 | DOI: 10.1038/srep17629 Multi-channel Potentiostat/Galvanostat Bio-Logic VMP3 work station driven by EC-Lab ® v10.40 software with Z-fit tool for EIS data analysis. In a typical three-electrode system, ANP-coated nickel foam was used as the working electrode, platinum mesh as the counter electrode, and Ag/AgCl (3 M KCl) as the reference electrode, in 3 M KOH aqueous solution. EIS measurements were carried out in the frequency ranging from 10 kHz to 10 mHz at the open circuit voltage with AC voltage amplitude of 1.5 mV.
For the half-cells (3-electrode configurations), the specific capacitance (C sp ), maximum specific power density (P max ) and specific energy density (E sp ) was evaluated using the conventional equations (2)(3)(4) 40  where i (A) is the applied current, Δ V (V)/Δ t (s) the slope of the discharge curve and S (cm −2 ) the total geometric surface area of the two electrodes, C (F) the calculated capacitance, V (V) is the maximum voltage obtained during charge, and R ir is the internal resistance which is determined from the voltage drop at the beginning of each discharge, while the ΔV ir represents the voltage drop.