MODELING THE ELECTROCHEMICAL CHARACTERISTICS OF GRAPHITE AND TRANSITION METAL OXIDE THIN FILM ELECTRODES: A QUASI-METALLIC APPROACH
M.D.Levi and D.Aurbach
Department of Chemistry, Bar-Ilan University, 52900 Ramat-Gan, ISRAEL
ABSTRACT
This paper reviews our last findings related to application of a Frumkin-type intercalation isotherm as a tool for a quantitative description of electrochemical insertion of Li-ions into various Li-insertion anodes and cathodes. Four major electroanalytical techniques, namely, slow-scan rate cyclic voltammetry (SSCV), potentiostatic intermittent titration (PITT), galvanostatic intermittent titration (GITT) and electrochemical impedance spectroscopy (EIS), are frequently used for the determination of chemical diffusion coefficient of Li-ions, D. We have defined their characteristic time-invariant functions (E), specific of each technique (I -1/2, I t1/2, 1/2 dE /dt and Aw, respectively) in such a way that the diffusion time constant can be expressed as a combination of the related function with the differential intercalation capacitance, Cint. Such form of presentation allows (E) to be inter-related, thus demonstrating the equivalence in application of three differential techniques -PITT, GITT and EIS- for obtaining kinetic data. A common feature observed in the experimental plots, D vs. E , for a variety of Li-insertion compounds, appears in the form of deep minima corresponding to the peak values of the differential intercalation capacitance. This was elucidated on the basis of an analysis which combines the concept of Frumkin intercalation isotherm with a simple mechanistic diffusion model.
INTRODUCTION
Li-insertion cathodes and anodes, like lithiated transition metal oxides and graphites, respectively, have recently attracted much attention as being the most promising materials for high-energy Li-ion batteries. The first Li-ion battery appeared on the market in 1991. The principle on which the Li-ion battery works can be described as follows. The cathode material is usually one of the transition metal oxides of the general formulae LixMO2 where M stands for Mn, Ni or Co, whereas the anode is graphite or disordered hard carbon, LixC6. Here X=1 for the cathode materials and graphite, and X>1 for the disordered carbon. During charging, Li+ is extracted from the cathode and then transported through liquid or polymeric electrolyte solution and finely inserted into the carbonaceous anode. During discharge, the process is reverted: Li-ion is transferred from the anode to the cathode. It is not surprising that this kind of battery is often called the “Rocking Chair Battery” or the “Swing Battery”. Numerous attempts have been made to prepare materials with high specific capacity, good rechargeability and fast Li-ion transfer kinetics. Although theoretical and methodological basis for quantitative thermodynamic and kinetic characterization of intercalation phenomena has been already reviewed (see an excellent treatise on solid-state electrochemistry (B. G. Bruce, Ed. [1]) there still remains the question related to the link between the lattice models, more precisely the quantity of d/dX ( denotes the chemical potential of ions, whereas X is the
232 intercalation level), and the solid-state chemical diffusion coefficient, D [2]. In our recent paper [3], we tried to take into account some kinetics effects, such as slow interfacial ion transfer and its connection to intercalation isotherms. Practical Li- insertion electrodes, as evidenced by their electrochemical impedance spectra measured in aprotic solvents, almost always reveal this interfacial kinetic contribution in the medium frequency domain of their spectra [4-6]. Moreover, the picture of intercalation processes becomes even more complicated if there is a preferential orientation of powder particles in the composite electrode coatings, or in the case of a broad particle size distribution [3]. The goal of this paper is to review Li insertion processes into thin-film electrodes, both with respect to their kinetics and thermodynamics. We show that a first-order phase transition during charge and discharge of Li-insertion compounds, complicated by slow Li-ion transfer kinetics, can be described by a simple Frumkin-type isotherm, taking into account the short-range interactions between the intercalation sites. The proposed description of insertion processes in terms of intercalation isotherm easily enables the comparison between the expressions for the solid-state diffusion time constant obtained for different electroanalytical techniques, and for finding the relationship between their characteristic time-invariant (but potential-dependent) functions.
RESULTS AND DISCUSSION
1. The Frumkin-Type Intercalation Isotherm
The approach used for the description of Li-intercalation into inorganic hosts will be illustrated here, using the insertion of Li into CoO2 as an example. A clear voltammetric peak (and hence, the corresponding plateau on the charge and discharge curves) is well documented [3,5] and relates to X ranging from 1 to 0.75 (referring to LixCoO2). The number of Li-ions per unit intercalation site is assumed to be one corresponding to the following stoichiometry:
- + 4 LiCoO2 = 4 L 0.75CoO2 + e + Li (1) Definition of the stoichiometry of the intercalation reaction is an important step in the kinetic description. This is because the intercalation site plays the role of a molecule, if one formally compares reaction (1) with the conventional localized redox-species reactions [7]. The stoichiometry following Eq. 1 relates exactly to a two-phase co- existence region (3.6 – 4.0 V vs. Li/Li+), as is evident from the in-situ XRD characterizations [8]. Thus, advancement of the above insertion/deinsertion reaction can be realized in terms of occupation of intercalation sites with Li+. Concerning the graphite anode, a single intercalation site includes 6 carbon atoms, and thus, the occupation of sites is defined by X in LixC6. A schematic view of a thin inorganic matrix, capable of incorporating of Li+ cations (anions are completely excluded) which thus exhibits permselective behavior is shown in Fig.1. One of the important principles of thermodynamic equilibrium applied to such an intercalation electrode demands that the total charge due to cations inserted into the matrix bulk from the solution be compensated (locally) by an equal amount of electronic species transferred from the substrate metal (see Fig. 1). The presence of two kinds of species during intercalation results in a rather complicated picture of potential distribution in the host bulk and across the both
233 current collector / intercalation electrode and intercalation electrode / solution interfaces, as compared with the distribution in the classical case of metal/solution interface. Two kinds of mobile species in the insertion compounds evidently create two possible kinetic limitations. These limitations are due to the transfer of the electronic species across the current collector / intercalation electrode interface and ionic species across the intercalation electrode / solution interface. Moreover, two additional mass-transport steps may appear due to the movement of both species from the electrode host’s boundaries to its interior. Significantly, the equilibrium potential distribution may change during Li insertion processes, as intercalation proceeds. Fig. 2 contains a schematic view of a typical case of a dielectric film possessing different partition constants for the electronic equilibrium across the Me/host interface [9]. A dielectric film possessing a very low concentration of mobile electronic species reveals a potential distributed within the whole film. As the electronic partition constant becomes higher, the potential in the host starts to drop in close proximity to the electrode host/solution interface. This activity reflects behavior typical of inorganic semiconductors in contact with electrolyte solutions. Further advancement of the intercalation process with a variation of electrode potential results in equal potential drops across both interfaces (symmetrical “electron-counterion’ case [9]). This result appears as a consequence of the local electroneutrality in the host bulk, assuming a 1:1 ratio between the elementary charges of the electronic and ionic species [9,10], and an equilibrium for both the electronic species across the current collector / electrode host interface and the ionic species across the electrode host/solution interface. Finally, when the electronic partition constant is high, typical of a metal, the distribution of interfacial potential becomes similar to the classical distribution, with a single potential drop in the part of solution close to its boundary with the host material. We designate this case as a quasi-metallic approximation to intercalation electrodes. Mathematical expressions describing equilibrium and kinetic characteristics of these electrodes are then reduced to the simplest form (see below). Once reaction products, stoichiometry and the type of the potential distribution across the interfaces are defined, one can proceed with formal kinetic analysis of intercalation reactions. The combination of the Frumkin-type isotherm with the Butler-Volmer equation for slow charge/discharge transfer (here we suggest Li+ transfer across the electrode host/solution interface) results in an equation which is valid for description of a quasi-equilibrium intercalation/deintercalation reaction [5,11,12]:
Idim=(ko/f){(1-X) exp[-(1-)gX) exp [(1-)f(E-Eo)]-X exp(gX) exp[-f(E-Eo)] (2)
Here Idim is the dimensionless current; k = (ko/f) is the dimensionless rate constant, with ko and representing the standard heterogeneous rate constant (cm/s) and the potential scan rate (V/s), respectively, is the thickness of the host matrix (cm); g is the dimensionless interaction parameter, and f is defined as f = F/RT (39.8 V-1 at room temperature). The charge-transfer coefficient in Eq. (2) is taken symmetrically for the anodic and cathodic reactions, which is a reasonable initial approximation for many electrochemical reactions: = 0.5. X in Eq. 2 denotes the Li content in Li1-X CoO2. The rate of anodic (deintercalation) reaction is proportional to X in LiXCoO2 (or (1-X) in Li1-X CoO2).
234 At equilibrium, the net current passed through the intercalation electrode should be zero. By equalizing both terms of Eq. 2, one immediately obtains the Frumkin isotherm:
X/(1-X) = exp [f (E - Eo)] exp (-gX) (3)
Frumkin isotherm (Eq. 3), in contrast to Langmuir one, includes the interaction term exp (-gX). This makes Frumkin isotherm steeper than the Langmuir one for attractive interactions (g < 0), or, on the contrary, flatter for repulsive interactions between the intercalation sites (g >0). There appears a critical value of gcrit = -4 such that as g < gcrit attractive interactions become extremely intensive, resulting in first-order phase transition. Thus gcrit is appropriate for distinguishing between monotonous and non- monotonous character of intercalation reactions. The extent to which the intercalation reaction may deviate from equilibrium is is expressed by the dimensionless constant K = ko/fThe larger is K, the less is the deviation of the charge and discharge process from equilibrium conditions. For a given intercalation reaction (characterized by ko), the system is closer to equilibrium at lower scan rates, and for thinner coatings. We have recently successfully applied kinetic equation (2) for the simulation of cyclic voltammetric response for Li intercalation reactions [3,5,12]. One important result is worth to be reviewed here. Fig. 3 shows a best fit of a SSCV curve of LixCoO2 electrode with Eq. 2, in the range of potentials from 3.5 to 4.4V. The scan rate was 10 -7 V/s, the fitted parameters were g = - 4.2 and ko = 8.0 10 cm/s. In this case, as is seen from the above figure, a satisfactory agreement between the experimental and theoretical curves is observed only for the anodic peak (see Fig. 3). The height of the theoretical cathodic peak is somewhat larger than that of the experimental peak, whereas the peak-potential separation agrees with the theoretical one. Simulation of a (capacitive-like) plateau on the SSCV curve located at higher anodic potentials is beyond the scope of Eq. 2. Fig.4a presents as an example a family of theoretical SSCV curves simulated according to Eq.2 with g = - 4.2 and different effective scan rates (K between 0.4 and 400). Experimental SSCV curves for the same range of are shown in Fig.4 b. Both families of curves are in broad agreement with each other in the range of between 10 and 50 V/s. We noted two limiting ranges of covering > 50 V/s and possibly <10V/s in which a considerable difference between both sets of curves had been observed. At higher scan rates, experimental SSCV curves revealed a deviation towards a diffusion-controlled behavior. The deviation of experimental curves from the theoretical ones in the limit of low scan rates may be of principal importance: at g < - 4 , as discussed above, Li intercalation proceeds via first-order phase transition, thus the differential capacity peak starts to increase enormously approaching a delta- function behavior (see Fig. 4a). Deviation of the experimental curves from this limiting behavior can be explained in terms of large Ohmic potential drops developed as the current increases steeply (flattens the actual voltammetric peak).
235 2. Simultaneous Application of SSCV, PITT, GITT and EIS for Studying Li-Ion Solid-State Diffusion Kinetics
As mentioned above (see Introduction section), thin composite electrodes are not well-defined systems compared to conventional metal and semiconductor electrodes, thus in order to obtain reliable results one has to utilize for their characterization a variety of electroanalytical techniques. We recently demonstrated the advantage of simultaneous application of SSCV, PITT, EIS and GITT for characterization of lithiated graphite and several transition metal oxides with subsequent modeling of their electroanalytical responses [3,4,5,11,12]. Here we present a short review of the results obtained. All four electroanalytical techniques under consideration are related to the same finite-space solid-state diffusion appearing in the host material after application of the input signal specific of the technique used: linear potential scan in SSCV, small potential steps in PITT, current pulse in GITT and small ac voltage in EIS. The fundamentals of one-dimensional finite-space diffusion problem and the routes for quantitative treatments of the output responses were developed by K Aoki et al. [13] (SSCV), W.Weppner and R.A.Huggins [14,15] (PITT and GITT), and C.Ho et al [16] (EIS). A primary diffusion parameter obtained by these techniques is the characteristic diffusion time which is defined for one-dimensional case as = l2/D
(4) where l is the characteristic diffusion length, D denotes the chemical diffusion coefficient connected through thermodynamics to the Frumkin intercalation isotherm. Note that correct calculations of D through Eq. 4 depend on the adopted values of l which for the powdered composite electrode is expected to correlate with the average particle size rather than with the electrode’s thickness. This should be taken into account when comparing D for the same materials obtained by different authors. Table 1 summarizes expressions for the characteristic time-invariant (but potential- dependent) functions (E) and the differential intercalation capacity Cint that were derived for four basic electroanalytical techniques. In this Table, GITT 1 corresponds to a limiting case of short current pulses, whereas GITT 2 refers to small current pulses of longer duration (for further details see below). The first line in Table 1 presents (E) characteristic of each technique. For SSCV, (E) is defined for the entire range of intercalation electrode potentials. Note that (E) in Table 1 corresponds to a short-time domain of the responses except for SSCV, which is a large-amplitude technique. The second line specifies the expression for the differential intercalation capacity Cint. In a recent paper [3] we presented a rigorous proof that for each specific technique used is expressed through combinations (within the accuracy of a constant) of the terms listed in the two lines of
Table 1, (e.g. (E) and Cint(E)). (E) is a specific form of presentation of experimental data for each technique. For example, for SSCV this function is defined as a hight of the voltammetric peak normalized with respect to square-root of the scan rate . For PITT, (E) is the so-
236 called Cottrell slope I t1/2, i.e. the product of gradually decaying current and square- root of time elapsing after application of a small potential step to intercalation electrode. In the case of GITT 1, a current pulse applied to the system during a period
of time t perturbes the system from equilibrium; t and s are the changes in the electrode potential during the pulse (dynamic characteristic) and after its switching- off (equilibrium or steady-state characteristic), respectively.The slope of the plot E vs. t1/2 characterizes time dependence of the potential measured after application of a small amplitude constant current (GITT 2). Finally, Aw is known in the theory of -1/2 -1/2 electrochemical impedance as Warburg slope equal to Aw = Re/ = Im/ (Re and Im are the differences in the real and imaginary components of the impedance, respectively, corresponding to a finite variation in the angular frequency of the alternative current, ). Table 1. Characteristic time-invariant function (E) and differential intercalation capacity, Cint for four basic electroanalytical techniques used for determination of solid-state diffusion time constant. Technique Time-invariant SSCV PITT GITT 1 GITT 2 EIS
-1/2 1/2 1/2 function, Ip I t t/s dE/dt Aw (E)
Cint=Qm dX/dE I(E)/ QmX It/s ItmX(E)/E -1/Z
The second line in Table 1 lists expressions for the differential intercalation capacity Cint=Qm dX/dE with Qm standing for the maximum intercalation charge. The form of dX/dE can be easily specified for each involved technique [3]. The above approach was rigorously checked by us for a variety of lithiated intercalation compounds including LixC6 [10], LixCoO2 [5], LixNiO2 [4], LixCo0.2Ni0.8O2 [17], LixMn2O4 [4] and proved completely its validity for calculation of diffusion time constant and chemical diffusion coefficient D. As an example, Fig.7a and b shows Cint vs. E and D vs. E curves obtained for thin LixCo0.2Ni0.8O2 electrode. Fig. 7a demonstrates generally a reasonable agreement between Cint vs. E relationships obtained from SSCV, PITT and GITT 2 measurements. In addition, it is clear that differential (incremental) techniques such as PITT and GITT provide obtaining of more resolved curves compared to SSCV (which is a long-amplitude voltage scanning technique). A pronounced minimum appears on the D vs. E curves (see Fig 5b) corresponding to the maximum in Cint of thin LixCo0.2Ni0.8O2 electrode. Similar observations have been also well-documented for LixC6 [10], LixCoO2 [5], LixNiO2 [4,17], LixMn2O4 [4]. Moreover, the peaks in both Cint vs. E and D vs. E curves were very narrow for those electrodes, in which Li-ion intercalation occurred in the form of first-order phase transition. The above correspondence can be easily substatiated with the use of thermodynamics and simple model of ion diffusion according to which chemical diffusion coefficient is defined by the following equation [2]:
2 D = (a k*)(1-X)X(Li+ /X)(kT) (5)
237 2 Where Mo = a k* denotes ionic mobility of pure phase (X = 1) in terms of the product of the hopping rate constant k* and the nearest neighbor separation a .
-1 The product L = X (Li+ /X)(kT) is usually called the enhancement factor [14,15], which reflects the influence of interactions between intercalation sites on the chemical diffusion coefficient. Of course, this quantity should be specified for each particular form of intercaltion isotherm. It was found [3] that for the Frumkin-type isotherm L = -1 (1-X) [1 + g (1-X) X], hence, the chemical diffusion potential normalized by Mo takes the form:
2 Dnorm/(a k*) = 1 + g (1-X) X (6) General conclusion related to prediction of the shape of D vs. X (or E) curves can be summarized as follows: Eq. 6 describes the shape of log D vs X curves as a function of the interaction parameter g. As g > gcrit D vs. X curve has a single maximum at X = 0.5. At g = 0 (Langmuir isotherm) D is independent of X. At 0 > g > gcrit D vs X plot possesses a minimum at X = 0.5 with D having always positive values. At the critical value gcrit = - 4, both L and D reach zero values. As g < gcrit (non-monotonous charging due to first-order phase transition), L and D imply formally negative values (physically unreasonable) along the unstable branch of the Frumkin isotherm. Experimental plots shown in Fig.5b for LixCo0.2Ni0.8O2 electrode as well as for a variety of lithiated cathode and anode materials [3], reflect this kind of behavior. CONCLUSION In this paper, we presented an integrated view on Li-ion intercalation process into various anode and cathode materials for Li-ion batteries. The intercalation reaction was shown to be dependent, to very much extent, on the short-range interactions amongst intercalation sites. Generally, thermodynamics and kinetics of intercalation reactions can be described in terms of a Frumkin-type intercalation isotherm combined with the Butler-Volmer equation for slow charge-transfer kinetics. Two kinds of species, electronic and ionic, participate in thermodynamic equilibrium resulting, generally, in a complicated potential profile across the cell, including its both interfaces (i.e. current collector / electrode and electrode / electrolyte solution ). Electroanalytical techniques frequently used for the characterizations of intercalation electrodes can be usefully divided into two following groups: large-amplitude technique such as SSCV and differential techniques (PITT, GITT and EIS). For each differential technique we have found his characteristic time-invariant (but potential- dependent) function (E) such that solid-state diffusion time constant (and hence D) can be presented as a unique combination of the related function (E) with the differential intercalation capacity, Cint. In this way, the functions (E) can be inter- related, thus demonstrating equivalence in application of these techniques, provided that the corresponding input signals are small enough.
We have presented clear evidence that both the experimental plots of Cint and log D vs E can be modeled utilizing the Frumkin isotherm and taking into account Li-ion transfer limitations. Theoretical analysis of chemical diffusion coefficient based on the concept of Frumkin intercalation isotherm and simple mechanistic ion diffusion model allowed us to elucidate appearance of narrow minima on D vs. E curves and maxima on Cint vs. E curves as a consequence of short-range attractive interactions between the
238 intercalation sites. Experimental evidence for this phenomenon has been well documented for various lithiated anodes and cathodes: LixC6, LixCoO2, LixNiO2, LixCo0.2Ni0.8O2 and LixMn2O4. REFERENCES
1. Bruce P.G., Ed.: Solid-State Electrochemistry, Cambridge University Press, 1995: (a) Chapter 7 by W.R. McKinnon “Insertion electrodes I: Atomic and electronic structure of the hosts and their insertion compounds “, p.163-199 and (b) Chapter 8 by W. Weppner “Electrode performance”, p.199-228. 2. McKinnon W.R., Haering R.R., Chapter 4 “Physical mechanisms of intercalation” in: Modern Aspects in Electrochemistry; Plenum Press, N.Y. 1987, vol. 15, p. 235-304. 3. Levi M.D.and Aurbach D.: “Frumkin intercalation isotherm – a tool for the elucidation of Li ion solid state diffusion into host materials and charge transfer kinetics. A review” Electrochim.Acta 1999 145 (1-2) 167-85. 4. Aurbach D., Levi M.D., Levi E., Teller H., Markovsky B., Salitra G., Heider U., L.Heider: “Common electroanalytical behavior of Li intercalation processes into graphite and transition metal oxides” J. Electrochem. Soc. 1998 145(9) 3024-34. 5. Levi M.D., Salitra G., Markovsky B., Teller H., Aurbach D. : ”Solid-state electrochemical kinetics of Li-ion intercalation into Li1-xCoO2 : Simultaneous application of electroanalytical techniques SSCV, PITT, and EIS” J. Electrochem. Soc. 1999 146 (4) 1279-89.
6. Yamada S., Fujiwara M., M. Kanda : “Synthesis and properties of LiNiO2 as cathode material for secondary batteries” J.Power Sources, 1995 504 (2) 209- 13 7. Bockris J.O.'M , Khan S.U.H,: Surface Electrochemistry, a Molecular Level Approach, Plenum Press, N.Y., London, 1993. Chapter 3, p. 224. 8. Reimers J.N., Dahn J.R.: “Electrochemical and in-situ X-ray diffraction studies of Li-intercalation in LiXCoO2 “ J. Electrochem. Soc.1992 139(8) 2091-97. 9. Vorotyntsev M.A., Rubashkin A.A., Badiali J.P.: “Potential distribution across the electroactive-polymer film between the metal and solution as a function of the film charging level” Electrochimica Acta, 1996 41 (14) 2313-20. 10. Levi M.D., Aurbach D.: “Diffusion coefficients of lithium ions during intercalation into graphite derived from the simultaneous measurements and modeling of electrochemical impedance and potentiostatic intermittent titration characteristics of thin graphite electrodes” J. Phys. Chem. B 1997) 101 (23) 4641-47 11. Levi M.D., Aurbach D.: “The mechanism of lithium intercalation in graphite film electrodes in aprotic media .1. High resolution slow scan rate cyclic voltammetric studies and modeling” J. Electroanal. Chem.1997 421 (1-2) 79-88. 12. Levi M.D., Aurbach D.: “Simultaneous measurements and modeling of the electrochemical impedance and the cyclic voltammetric characteristics of graphite electrodes doped with lithium” J. Phys. Chem. B 1997 101 (23) 4630- 40.
239 13. Aoki K., Tokuda K., Matsuda H.: “Theory of linear sweep voltammetry finite-diffusion space” J. Electroanal. Chem. 1983 146 417-24. 14. Wen C.J, Boukamp B.A., Huggins R.A., Weppner W.: “Themodynamic and mass transport properties of LiAl” J. Electrochem. Soc. 1979 126 (12) 2258- 66. 15. Weppner W., Huggins R.A.: “Electrochemical methods for determining kinetic properties of solids” Ann. Rev. Mater. Sci. 1978 8 269-311. 16. Ho C., Raistrick I.D.,. Huggins R.A: “Application of A-C techniques to the study of lithium diffusion in tungsten trioxide thin films” J. Electrochem. Soc. 1980 127(2) 343-53. 17. Levi M.D.,.Gamolsky K, Aurbach D., Heider U., Oesten R.: “Determination of the Li-ion chemical diffusion coefficient for the topotactic solid-state reactions occurring via a two-phase or single-phase solid solution pathway” J. Electroanal. Chem. 1999 477 32-40.
240 241 60 infinitely large K 50 400
40 160
30 80 m i
d 40 I 20 16 8 4 1.6 0.8 10
0
-10 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 (E-Eo) / V
60x10-6 Experimental CV Theoretical CV 40x10-6
20x10-6 A
/
I 0
-20x10-6
-40x10-6 3.4 3.6 3.8 4.0 4.2 4.4 4.6 E / V
Fig. 3 Experimental SSCV peak for a thin LiCoO2 electrode ( = 10V/s) fitted
with Eq. 2 with4 K =ko/f = 40 and g=-4.2 (refer to ref. [5]). 10 V s-1 3 20 3 50 -11 Li Co Ni O 100 a x b 0.2 0.8 2 GITT4.0V 2 200 PITT 500 LixCo0.23.62Ni0.8O2
1 3.70 SSCV 2- F 1
s F /
3.57
t 2 / n
i t m
C 0 4.00 n c i -12
/ 3.68V
C
1 D -1
g GITT o l PITT -2 3.57V 0 -3 -13 3.4 3.6 3.8 4.0 4.2 3.6 3.8 4.0 4.2 4.43.4 3.6 3.8 4.0 E / V + E /E V / (Vvs. (vs. LiILi LiILi)+)
Fig. 4 (a) A family of I vs. (E-Eo) curves calculated according to Eq. (2), using different values of the dimensionless constant rate K (as indicated in the figure) and g=-4.2
Fig. 5(a) Comparison between incremental capacity curves, C vs. E obtained for Fig. 4 (b) Experimental plots of differential242 intercalation capacity, C = I/ int Li Co Ni O electrode by SSCV, PITT, and GITT. int vs. electrode potentialx 0.2 measured0.8 2 at different scan rates, as indicated (refer to ref. [5]). Fig. 5(b) Comparison between the plots of log D vs. E obtained for LixCo0.2Ni0.8O2 electrode by PITT and GITT
243