J. Phys. Chem. B 1997, 101, 4641-4647
4641
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 M. D. Levi and D. Aurbach* Department of Chemistry, Bar-Ilan UniVersity, Ramat-Gan 52900, Israel ReceiVed: January 13, 1997; In Final Form: March 28, 1997X
The solid state diffusion of lithium into graphite during electrochemical intercalation processes was investigated using potentiostatic intermittent titration (PITT) and impedance spectroscopy (EIS). The diffusion coefficient (D) as a function of the intercalation level (X) and the electrode potential (E) was calculated on the basis of both methods and gave similar results. The D Vs X or E plots were found to be nonmonotonous, with three pronounced minima at the same potentials in which the cyclic voltammetry of these systems shows the peaks related to the phase transition between the lithium-graphite intercalation stages. It was found that the critical diffusion length for these electrodes relates to the graphite particles’ dimensions along their basal planes and not to the electrode thickness. The reason for the peaklike dependence of the D on X and E is discussed in light of the nature of the intercalation processes.
Introduction The development of novel Li ion batteries based on insertion/ intercalation electrodes is one of the most important areas of modern electrochemistry today. One of the important kinetic characteristics of an electrochemical intercalation process in ioninsertion compounds which are in use as electrode materials in these batteries is the corresponding ionic chemical diffusion coefficient, D. The solid state diffusion of ionic species characterized by rather low values of D may control the ratedetermining step of the intercalation process, although there may be particular cases where the rate-determining step is diffusion in the electrolyte solution within the pores of those kinds of composite electrodes. Several relaxation techniques have been proposed for the evaluation of D, among them the potentiostatic and galvanostatic intermittent titration technique (PITT and GITT, respectively),1,2 electrochemical impedance spectroscopy (EIS),3a,4 and current pulse relaxation.5 Using these techniques, the plots of the diffusion coefficient of lithium ions Vs their intercalation level X (0 e X e 1) have been obtained for graphite and other carbonaceous materials,6-10 transition metal oxides,11-15 and sulfides,16,17 etc. However, the values of D for the same materials, reported by different research groups, may differ by several orders of magnitude. One of the common difficulties in comparing D values obtained by different groups or with different samples is the uncertainty in the true cross-sectional surface area for the insertion process, A, which, however, should influence the absolute value of D but should not significantly distort the D Vs X relationship. However, it is expected that when different techniques are applied for studying the diffusion process in the same system, their D values would be similar, although this is not always the case in practice. Thus, it seems that there are problems in the correct interpretation of the equations for the determination of D. In this paper we report on simultaneous measurements of the diffusion coefficient (D) of lithium into graphite, using impedance spectroscopy (EIS) and potentiostatic intermittent titration (PITT) applied to thin electrodes in which the graphite platelets are highly oriented. Using such electrodes enabled us to obtain well-resolved and reproducible dependence of D on X (the X
Abstract published in AdVance ACS Abstracts, May 1, 1997.
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intercalation level) and E (electrode potential). The features of this dependence, as well as the significance of the calculated values of D, are discussed. Experimental Section Two types of graphite electrodes, to be referred to as thin and ultrathin coatings of the active mass on the current collector (Cu foil), were used in this study. The former was ca. 10 µm thick, whereas the latter was about 20 times thinner. The procedure of their preparation and their morphological characterization have been reported elsewhere.18-20 The electrodes were 1.1 × 1.1 cm squares containing ca. 4.1 and 0.18 mg of graphite, respectively (KS-6 graphite flakes from Lonza was used as the graphite source). About 10% (by weight) of PVDF was added into the active mass as a binder. A three-electrode cell was used for the electrochemical measurements.18-20 All potentials are measured and quoted with respect to the Li/Li+ electrode in the same solution. The electrolyte solution was 1 M LiAsF6 (Lithco) in an ethylenecarbonate-dimethylcarbonate mixture (1:3) purchased from Tomiyama (highly pure Li battery grade), initially containing 20 ppm of water. The glovebox operation has been previously described.21,22 Electrochemical impedance spectra, cyclic voltammograms, and chronoamperograms were measured using Schlumberger’s 1286 electrochemical interface and 1255 FRA driven by Corrware software from Scribner Associates (486 IBM PC). Around the voltammetric peaks, the PITT was performed with a step of ca. 1-2 mV, whereas, beyond the peaks, steps of 1020 mV could be applied without any deterioration of the resolution. Typically, 30-40 titration points covering the whole range of intercalation potentials of Li into graphite (0.3 to 0.0 V Vs Li/Li+) were obtained. Impedance spectra were measured at all potentials of interest, from 200 kHz to 5 mHz, with a 2.8 mV peak-to-peak alternative voltage after a full equilibrium at OCV was reached. Results A critical issue with regard to the study of diffusion coefficients of intercalating species is the choice of the ap© 1997 American Chemical Society
4642 J. Phys. Chem. B, Vol. 101, No. 23, 1997
Levi and Aurbach Vs log t at different potentials are very useful in visualizing the potential dependence of the characteristic diffusion time, τ:
τ ) l2/D
(1)
where l is the characteristic diffusion length and t the measuring time. According to the theory of PITT, τ is connected with the amount of the charge injected during the potential step, ∆Q ) Qt∆X, by the simple formula1
τ ) [Qt∆X/π1/2It1/2]2
at t , τ
(2)
where Qt is the maximum (limiting) amount of the charge that may be involved in the entire intercalation process of the particular electrode and X is the intercalation level (for a fully intercalated electrode X ) 1). Upon continuous chronoamperometry at a given potential step at the long time domain (t . τ), an exponential dependence of the current (I) Vs time is observed:1
ln
Figure 1. Comparison between the experimental and theoretical (a) It1/2 Vs log t and (b) ln I Vs t plots.
propriate geometric model for the diffusion process. On the basis of rigorous studies of our electrodes by SEM, it is clear that our electrodes comprise highly oriented layers of graphite flakes whose average dimensions are 6 × 6 µm (basal planes) and their average thickness is submicronic. As a reasonable approximation we model a single graphite particle as a prism of 6 × 6 × (0.1 f 0.5) µm. The cross-section for the diffusion is therefore an envelope of facets perpendicular to the basal planes 4 × 6 × (0.1 f 0.5) µm2. Since the electrode’s thickness is precisely measured, the overall cross-section for the diffusion could be calculated: 3.3 and 0.15 cm2 per one geometric cm2 for the thin and ultrathin electrodes, respectively. Since the diffusion process advances from the sides to the center of the particles, its cross sectional area contracts during the process. However, as shown in ref 25, the basic equations, and thus the expected dependence of the diffusion coefficient in different experimental conditions (e.g., potential, intercalation level), may be quite similar for both diffusion with a constant cross-sectional area and diffusion into cylindrical geometry of reducing cross-sectional area. As further explained, we derived the values of D from the “Cottrell” region, meaning measurements of short diffusion processes. Therefore, it seems to be legitimate to approximate our diffusion processes as a line diffusion from each side of the particle to its center, and thus the diffusion length should be taken as half of the dimension of the basal planes of the particles (3 µm). A further refinement of this model in order to take into account the change in the diffusion’s cross-section during the process, as well as the complicated true geometry of the graphite particles, is beyond the scope of this paper. However, on the basis of ref 25, we assume that the basic dependence of D on the intercalation potentials as found in this work should not change due to such a refinement. We have shown elsewhere19 that the chronoamperograms measured from graphite electrodes by applying potential steps during intercalation with lithium and presented in plots of It1/2
(
)
Iτ ) -(π2/4τ)t 2Qt∆X
(3)
Both eqs 2 and 3 follow from the more general expressions for the long and short time domains obtained by differentiation of the local concentration of species C(x,t) with respect to the distance x.1 (By definition, the current is proportional to the concentration gradient at x ) 0.) They have the following form: ∞
I ) (2Qt∆X/τ)
exp[-(2n - 1)2π2t/4τ] ∑ n)0
if t > τ (4)
∞
I ) (2Qt∆X/τ)
∑(-1)n(2τ1/2/(πτ)1/2{exp[-(n + 1)2τ/t] n)0 exp(-n2τ/t)}
if t < τ (5)
where n are integer numbers. Equations 4 and 5 correspond to the case of a one-dimensional diffusion transport of a species under finite-space conditions.1 Numerical calculations using these equations allow for the simulation of the chronoamperometric response within the entire time domain. As an example, Figure 1a compares the experimental It1/2 Vs log t curve for the thin electrode (potential step from 0.23 to 0.22 V) with the theoretical one calculated according to eqs 4 and 5. A value of τ ) 38 s was obtained from the experimental curve using eq 2, which was then introduced as a numerical parameter in eqs 4 and 5. It is seen that a rather close agreement between both curves is observed in the range of t < τ (the Cottrell region). At the very beginning of the potential step, the deviation of the experimental curve from the theoretical one is clearly connected with the fast relaxation processes related to the charging processes of interfacial capacitances, etc., which can be better studied by EIS.20,23 At longer time intervals, the discrepancy between both curves is much larger, as evidenced from the corresponding ln I Vs t plots (Figure 1b). This figure shows that application of the long time approximation (eq 3) for the determination of τ is only possible for the ideal finitespace diffusion behavior whose corresponding impedance spectroscopic response behaves like a finite-space Warburg element3b (see Discussion below). Figure 2 shows as an example a typical potential dependence of the experimental It1/2 Vs log t plots for the thin electrode in the vicinity of the first voltammetric peak (diluted phase I to
Lithium Ions during Intercalation into Graphite
Figure 2. It1/2 Vs log t plot for a thin graphite electrode (intercalating with lithium). The values of potentials at which the titration has been performed are indcated on each curve. Each potential step was about 2 mV.
J. Phys. Chem. B, Vol. 101, No. 23, 1997 4643 of the curves of Figure 2), which corresponds to processes reflected at relatively higher frequencies in the impedance spectra, to the faster lithium ion diffusion process within the pores of the electrode; the contribution of this process to the total diffusion mechanism is more important at large ∆Q, i.e., at the large potential step which leads to pronounced deviation from the initial equilibrium). This explanation is in line with the observation that, for the ultrathin electrodes which are much less porous (due to the thinner active mass), only a single relaxation process has been observed for the same time domain, and the corresponding Nyquist plots do indeed have a very narrow “Warburg” domain (i.e., the low-frequency domain is dominated by the bulk capacitance of graphite.20 Thus, we assume that the τ Vs E dependence for the solid state diffusion of lithium ions always corresponds to the largest value of τ if several relaxation processes are observed. The Warburg slope, AW is defined as3c
AW ) ∆Re/∆ω-1/2 ) ∆Im/∆ω-1/2
(6)
where ∆Re and ∆Im are the differences of the real and imaginary components of the impedance, respectively, corresponding to a finite variation in the angular frequency of the alternative current, ∆ω. The time constant for the finite-space solid state diffusion process can be derived from the equation3a,4
τ ) [x2QtAW dX/dE]2
Figure 3. Voltammetric peak corresponding to the diluted phase I f phase IV transition measured with the thin electrode with ν ) 4 µV/s. The arrows indicate the potentials at which the titration has been performed. Hence, the steps were as follows (V): 0.23 f 0.22, 0.22 f 0.21, 0.21 f 0.208, 0.208 f 0.206, and 0.206 f 0.204.
phase IV transition). The pattern of the corresponding cyclic voltammogram (CV) related to diluted phase I is shown in Figure 3, with the arrows indicating the values of potentials at which the PITT has been performed. We observed a peculiar potential dependence of the It1/2 Vs log t curves. They have a single flat minimum (approaching the Cottrell behavior) at potentials at the feet of the CV peak and two minima at potentials close to the peak potential. Similar behavior has been observed around the other two voltammetric peaks (corresponding to phase III f phase II and phase II f phase I transitions).19 The reason for the appearance of the two Cottrell regions can be understood by comparing the curves of Figure 2 with the corresponding impedance spectra measured with the same electrode. Parts a-e of Figure 4 present a family of Nyquist plots at the beginning of intercalation (a), in the vicinity of the three voltammetric peaks (b, d, and e), and in the wide potential range corresponding to solid solutions comprising phases IV and III (c) (see typical slow scan rate voltammograms of the lithium-graphite intercalation process appearing in ref 20). Parts b-d of Figure 4 show clearly the transition from the essentially capacitive behavior to a Warburg-type response when passing from the low- to the medium-frequency domain. A nonmonotonous peak-shaped dependence of the Z′′ on E in the lowfrequency domain corresponding to the CV peak is clearly seen. (Note the Z′′ values at 5 mHz as a function of the electrode’s potential in the plots of Figure 4b-d). It is remarkable that a slight S-shaped curvature appears at medium frequencies and is especially pronounced in Figure 4e. Thus, we ascribe the short-time minimum in the It1/2 Vs log t curves (the first in part
(7)
where dX/dE is the derivative of the intercalation isotherm with respect to E and Qt is the total charge involved in the specific intercalation process. In linear sweep voltammetry, the potential is a linear function of time: E ) E0′ - νt, with ν denoting the potential scan rate. Hence, Qt(dX/dE) ) ICV/ν, where ICV is the voltammetric current. If ν is sufficiently small, the corresponding CV response is considered to be close to the equilibrium one, and thus τ is readily calculated using the Warburg coefficient AW and the voltammetric current in the whole range of the intercalation potentials. Figure 5 compares the log D Vs X plots obtained from PITT data, using eq 2, and derived from the Warburg region of the Nyquist plots, using eq 7. Both curves are in satisfactory agreement with each other. On the other hand, we have also calculated the diffusion coefficients using the classical Warburg formula:
τ ) 2(QtXAWF/RT)2
(8)
Comparison between the D Vs X relationships calculated using eq 8 for the EIS data and eq 2 for the PITT shows a completely different shape for both of these curves (Figure 6). Finally, Figure 7 compares the D Vs X plots obtained for the thin and ultrathin electrodes according to eqs 1 and 2, with the same value of the characteristic diffusion length l ) 3 µm (i.e., half of the average size of the graphite platelets as explained above). A reasonable coincidence between both of these two curves is observed, despite the fact that Qt for the latter electrode is ca. 20 times smaller than that for the former one. Discussion We begin the discussion with the comparison between the classical Warburg formulas (8) for the semiinfinite bulk redoxspecies diffusion and that for the finite-space diffusion eq 2. In the former case, any harmonic variation of the potential with a
4644 J. Phys. Chem. B, Vol. 101, No. 23, 1997
Levi and Aurbach
Figure 4. Nyquist plots measured with a thin electrode (a) at the beginning of the intercalation and (b) around the peaks corresponding to the following phase transitions: (a) diluted phase I, (b) diluted phase I f IV, (c) solid solution IV and III, (d) III f II, and (e) II f I (condensed).
small amplitude results in the corresponding harmonic oscillation of lithium ion concentration near the electrode surface, whereas its bulk concentration remains constant. On the contrary, in the case of the finite-space diffusion, the potential dependence
of the concentration of lithium ions is controlled by the intercalation isotherm for the specific redox system. Application of the Nernst equation in its linearized form in the range of X , 1 results in the corresponding harmonic
Lithium Ions during Intercalation into Graphite
J. Phys. Chem. B, Vol. 101, No. 23, 1997 4645
Figure 5. Comparison between the log D Vs X plots calculated using PITT and EIS, eqs 2 and 7, respectively (a thin graphite electrode intercalating with lithium).
Figure 7. Comparison between the log D Vs X plots for the thin and ultrathin electrodes calculated.
constant within the small variations in X and in E during the potential step. For example, for the Frumkin-type isotherm with a strong attractive interaction between the intercalated species, ∆E should be smaller than for the Langmuir-type isotherm. In ref 4, the “Warburg” slope for solid state diffusion processes is presented as
AW ) Vm(dE/dX)/(FAx2D)
Figure 6. Comparison between the log D Vs X plots calculated using PITT and EIS, eqs 2 and 8, respectively.
oscillation of the concentration of species near the electrode surface ∆Cx)0:
∆Cx)0 ) (F/RT)C∆E ˜ sin(ωt)
(9)
where C is the bulk concentration of lithium ions and ∆E ˜ is the amplitude of the alternative voltage. Application of a very small alternative voltage of the form
(11)
where Vm is the molar volume of the intercalation compound and A is the cross-sectional area for the diffusion flux. Substitution of Vm/FA for l/Qm in eq 11 results immediately in eq 7. (The specific surface area was estimated as about 0.16 m2/(g of graphite) or 3.3 and 0.15 cm2 per cm2 of the geometric area of the thin and the ultrathin electrodes, respectively). The approach in ref 4 was to express the interactions of the diffusing species in the bulk in terms of activity coefficient, while in the present work such interactions are treated in terms of parameters of the intercalation isotherm (e.g., attraction interactions in a Frumkin-type intercalation isotherm). However, it can be seen that in both approaches the linearized equations which control the behavior at small potential steps (or amplitude) give the same formula for the Warburg slope, eq 11. It can be shown that both eqs 7 and 2 for EIS and PITT, respectively, equiValently describe the finite-space diffusion situation. In fact, assuming a potential step in the PITT experiment (∆E) to be very small and dividing both the numerator and denominator in eq 2 by ∆E result in the following expression:1/2
τ ) [Qt(dX/dE)∆E/π1/2It1/2]2
(12)
∆E ) ∆E ˜ sin(ωt)
Comparison between formulas 12 and 7 results in the following equation:
∆E ≈ (dE/dX)∆X
AW ) (2π)-1/2∆E/It1/2
results in the sinusoidal variation of ∆X irrespectiVe of the particular type of the intercalation isotherm:
which connects the Warburg slope with the corresponding parameter ∆E/It1/2, characterizing the PITT response. The difference between eq 12 for the PITT of the intercalation process and the classical Cottrell equation24 for the semiinfinite bulk diffusion is that, in the former case, τ1/2 is proportional to the amount of the injected charge, Qt(dX/dE)∆E, whereas for the latter one, the same quantity is proportional to the bulk concentration of the electroactive species. The validity of eq
such that
∆X ) (dX/dE)∆E ˜ sin(ωt)
(10)
This equation is valid for any kind of intercalation isotherm. However, the required applied values of ∆E for the PITT should depend on the shape of the isotherm, in order to keep dX/dE
(13)
4646 J. Phys. Chem. B, Vol. 101, No. 23, 1997
Levi and Aurbach
TABLE 1: Potential Dependence of the Characteristic Diffusion Time τ (Ultrathin Electrode) Obtained by Fitting the Experimental Impedance Spectra with Theoretical Curves, Based on the Model Combining Voigt and FMG Equivalent Circuit Analogs and from Analysis of Simultaneous PITT Measurementsa τ/s
τ/s
E/V
EIS
PITT
E/V
EIS
PITT
0.050 0.070 0.075 0.080
75 155 1460 4700
75 153 1457 5240
0.085 0.090 0.095
148 280 440
154 278 442
a The error in the measurements and calculations is estimated as (5%.
13 is confirmed by the results of Figure 5, comparing the log D Vs X plots calculated with use of eqs 2 and 7 which show good agreement between both plots. It is important to note that τ and hence D are independent of the electrode thickness δ, since, in both terms, Qt and I in eq 12 are proportional to δ (as δ is proportional to the active mass). In a parallel study it was shown that the impedance spectra measured with the present electrode solution systems can be fitted precisely with an equivalent circuit analog based on the Frumkin-Melik-Gaykazyan model for adsorption processes.20 This analog contains a finite-length Warburg element (describing diffusion) in series with a capacitor which reflects the bulk capacity of the electrode for the adsorption or intercalation process. An interesting aspect of the comparison between PITT and EIS is that the values of τ derived from eq 2 and from the parameters of the Frumkin and Melik-Gaykazyan model fitted to the impedance spectra obtained are practically identical. This is demonstrated in Table 1, which provides (as an example) the τ values calculated by two methods for an ultrathin electrode at potentials around the major peak in the cyclic voltammograms of these electrodes, which relate to the phase II f phase I transition. Comparison between the τ Vs E dependence and the corresponding CV peak20 shows pronounced maxima in τ and thus minima in D around the peak potentials (Figures 5 and 7). It has been previously shown by strict thermodynamic base that the minima in D (Vs X or E) corresponds approximately to equal molar amounts of the coexisting phases, whereas D approaches maximal values for the pure phases (e.g., X ) 0, X ) 1).25 This shape of the D Vs X or E curves may simply be a result of the phase transition which characterizes the lithium graphite intercalation processes. In fact, it was previously shown that the very narrow peaks of the cyclic voltammograms measured at slow scan rate with thin graphite electrodes (highly oriented graphite particles on the current collector) intercalating with lithium, may be modeled by CV curves of an adsorption process with high attractive interactions between adsorption/intercalation sites (a Frumkin-type isotherm), in which slow charge transfer is also involved.18-20 A combination of the equation defining D25
D ) U(1 - X)X(∂µ/∂X)
(14)
with a Frumkin-type intercalation isotherm
µLi+ ) µ°LI+ + RT ln X/(1 - X) + RTgX
(15)
results in a simple equation:
D/URT ) 1 + g(1 - X)X
(16)
Here, U is the mobility of lithium ions at X ) 0, µLi+ is its
Figure 8. Dimensionless D0/URT Vs X plots calculated according to eq 16 with different values of g.
chemical potential (µ°Li+ ) standard value), and g is the interaction parameter between intercalation sites (negative for attraction). Figure 8 shows the D Vs X relationship calculated according to eq 16 at four different values of g: 0, -2, -4, and -6. At negative g values, these curves have a minimum which falls at X ) 0.5, and when g approaches the critical value -4, D drops to zero. Hence, from an electroanalytical point of view, it may be possible to model the intercalation of lithium into graphite in terms of the diffusion of lithium into the existing LixC phase, which forms clusters of intercalation sites with high attraction interactions among them. These clusters are, in fact, the new phase which is formed uniformly in droplets within the old phase. Modeling of the corresponding CV response and the charging curve26 shows their qualitatiVe similarity to that expected in terms of Frumkin-type intercalation isotherms under interfacial kinetic control. In fact, the D values practically measured should be viewed only as effective values, because the real situation is more complicated, and there are several other relaxation process that are involved in addition to solid state diffusion (e.g., slow charge transfer, Li+ diffusion in pores, complicated Li+ migration processes through surface films, etc.). This explains the fact that even for high attractive forces between intercalation sites, when the theory behind eqs 14-16 may predict D values close to zero, the actual D values measured are always higher. Hence, it should be noted that in a case where the intercalation is controlled by a Langmuir-type isotherm (g ) 0), the factor ∂µLi+/∂X should completely compensate the product (1 - X)X in eq 14, and as a result, D becomes independent of X (Figure 8). For the repulsive interaction, eq 16 predicts a maximum at X ) 0.5. Similar results have been obtained when, instead of eq 15, there are more complicated equations on the basis of the mean-field and exact Ising model25 or modified Frumkin isotherm, taking into account short-range interactions between electronic and ionic species.26 Experimental evidence for the validity of such theoretical predictions concerning the D Vs X relationship has been reported for some conventional redox polymers27,28 and in the case of adsorption of molecules on the separate faces of monocrystals.29,30 Graphite presents an interesting particular case because of its crystalline nature, which makes it possible to observe directly the phase transition during the intercalation of lithium ions. Corresponding in situ XRD studies have provided direct evidence for the two-phase coexistence during the intercalation processes.19,31,32 For each of the phase transitions along the lithium-graphite intercalation, the intensity of the XRD peaks of the two phases is approximately the same at the halfway point of the transition from one phase to another (which corresponds to X ) 0.5 for a single process, in reference to eqs 14-16). Figure 9 shows the XRD peak intensity for phases II and I (LiC12 f LiC6) as an example (X ) 0 and X ) 1
Lithium Ions during Intercalation into Graphite
J. Phys. Chem. B, Vol. 101, No. 23, 1997 4647 Conclusion
Figure 9. Intensity of the (002) XRD peaks of graphite related to phase II and phase I (LiC12, LiC6, respectively) measured during in situ experiments, in which Li was intercalated into graphite in 1 M/1 M LiAsF6/EC-DMC solution.
correspond to the initial and final phases, respectively). Hence, the minima in D, that is, the equality of the XRD peaks of the corresponding two phases and the peak currents of the slow scan rate CV curves of these electrodes, appear at the same X and E values, which correspond to the midpoint of the specific phase transition. This, in fact, corresponds well with the response of D Vs X predicted by eqs 14-16 when the intercalation sites are attractive. This is well-explained by the fact that all of these functions have a similar dependence on X. D(X) predicted by eq 16 is proportional to the factor (1 - X)X, whereas the intensity of the XRD peaks for phases II and I is also proportional to (1 - X) and X, respectively. The theoretical voltammetric current predicted by the same model, based on the Frumkin-type intercalation isotherm with attractive interactions, is also a function of X and 1 - X. Equation 17 presents the dimensionless voltammetric current as a function of X.
ICV/Qtν ) dX/dE ) F/RT[1/X(1 - X) + g]-1
(17)
This equation also predicts a maximum in I at X f 0.5 (the halfway point of the process). Equations 16 and 17 provide theoretical support for our earlier observation that, within a certain range of X, D is inversely proportional to ICV and thus to dX/dE.19 It is important to emphasize that the thermodynamic approach for understanding the nonmonotonous dependence of D on X and E is relevant to X and E values close to the voltammetric peaks, where a coexistence of the corresponding lithiated graphite phases in equilibrium is observed. On the other hand, as was shown in refs 19, 31, and 32, at least two potential ranges from 1.5 to 0.22 V and from 0.195 to 0.135 V are characterized by a continuous shift of the XRD peaks which had been ascribed to a continuous increase of the diluted phase I and phase II, respectively. In parallel, the corresponding Nyquist plots show a monotonous decrease of Z′′ at the small frequencies as a function of X and E (see Figure 4a,c). These two relatively wide ranges of potentials beyond the voltammetric peaks are clearly related to the ranges of X values where the determination of the diffusion coefficient D is strictly in accordance with all of the assumptions of the PITT technique. Figures 5-7 show a monotonous and rather steep decrease of D in the range of X around 0.03-0.04 and from 0.12 to 0.21 for these two regions, respectively.
We have shown that the shape of D Vs X plots for the intercalation of lithium ions into thin graphite electrodes derived using PITT or EIS strongly correlate with the corresponding voltammetric response I(E,X) and the dependence of the XRD peak intensity on X. This correlation is based on the fact that D and XRD peak intensity are dependent on the mole fractions of the available sites (1 - X) and X, whereas the current is normally defined as their derivative with respect to time. Using a simple analysis of the dependence of the Warburg slope and the PITT parameter (∆E/It1/2) on the intercalation isotherm, it was shown that both techniques should lead to the same D Vs X relationship. The minima in the values of D obtained at the peak potentials of the CV peaks, at the halfway point of the intercalation process and its related phase transition, may be explained by a model which predicts strong attractive interactions between the intercalated species (a Frumkin-type isotherm with negative g values). Acknowledgment. This work was partially supported by the Israeli Ministries of Science & Technology and of Absorption. References and Notes (1) Wen, C. J.; Boukamp, B. A.; Huggins, R. A.; Weppner, W. J. Electrochem. Soc. 1979, 126, 2258. (2) Weppner, W.; Huggins, R. A. Annu. ReV. Mater. Sci. 1978, 8, 269. (3) Impedance Spectroscopy. Emphasizing Solid Materials and Systems; MacDonald, J. R.; Ed.; Wiley: New York, 1987; (a) p 252, (b) p 90, (c) p 23. (4) Ho, C.; Raistrick, I. D.; Huggins, R. A. J. Electrochem. Soc. 1980, 127, 343. (5) Basu, S.; Worrell, W. L. In Fast Ion Transport in Solids; Vashishta, Mundy, Dhenoy, R. V., Eds.; Elsevier: Amsterdam, 1979; p 149. (6) Morita, M.; Nishimura, N.; Matsuda, Y. Electrochim. Acta 1993, 38, 1721. (7) Uchida, T.; Morikawa, Y.; Ikuta, H.; Wakihara, M. J. Electrochem. Soc. 1996, 143, 2606. (8) Takami, N.; Satoh, A.; Hara, M.; Ohsaki, T. J. Electrochem. Soc. 1995, 142, 371. (9) Guyomard, D.; Tarascon, J. M. J. Electrochem. Soc. 1992, 139, 937. (10) Zaghib, K.; Tatsumi, K.; Abe, H.; Ohsaki, T.; Sawada, Y.; Higuchi, S. In Rechargeable Lithium and Lithium-Ion Batteries; Megahed, S., Barnett, B. M., Xie, L., Eds.; Battery Division, The Electrochemical Society, Inc.: Pennington, NJ, 1995; p 121. (11) Baddour-Hadjean, R.; Farcy, J.; Pereira-Ramos, J. P. J. Electrochem. Soc. 1996, 143, 2083. (12) Farcy, J.; Messina, R.; Perichon, J. J. Electrochem. Soc. 1990, 131, 1337. (13) Bach, S.; Pereira-Ramos, J. P.; Baffier, N.; Messina, R. J. Electrochem. Soc. 1990, 137, 1042. (14) Groult, H.; Devilliers, D.; Kumagai, N.; Nakajima, T.; Matsuo, Y. J. Electrochem. Soc. 1996, 143, 2093. (15) Barker, J.; Pynenburg, R.; Koksbang, R.; Saide, M. Electrochim. Acta 1996, 41, 2481. (16) Skundin, A. M.; Stefanovskaya, E. E.; Egorkina, O.; Yu, J. Power Sources 1993, 43-44, 301. (17) Kumagai, N.; Tanno, K. Electrochim. Acta 1991, 36, 935. (18) Levi, M. D.; Aurbach, D. J. Electroanal. Chem. 1997, 421, 79. (19) Levi, M. D.; Aurbach, D. J. Electroanal. Chem. 1997, 421, 89. (20) Levi, M. D.; Aurbach, D. J. Phys. Chem. B 1997, 101, 4630. (21) Aurbach, D. J. Electrochem. Soc. 1989, 136, 906. (22) Aurbach, D.; Gottlieb, H. E. Electrochim. Acta 1989, 34, 141. (23) Aurbach, D.; Levi, M. D.; Levi, E. A.; Schechter, A. J. Phys. Chem. B 1997, 101, 2195. (24) Bard, A. J.; Faulkner, L. R. Electroanalytical Methods, Fundamentals and Applications; Wiley: New York, 1980; p 143. (25) McKinnon, W. R.; Haering, R. R. Modern Aspects in Electrochemistry; Plenum Press: New York, 1987; Vol. 15, p 235. (26) Vorotyntsev, M. A.; Badiali, J. P. Electrochim. Acta 1994, 39, 289. (27) Inzelt, G. In Electroanalytical Chemistry; Bard, A. J., Ed.; Dekker: New York, 1994; Vol. 18, p 90. (28) Chidsey, C. E. D.; Murray, R. W. J. Phys. Chem. 1986, 90, 1479. (29) Butz, R.; Wagner, H. Surf. Sci. 1977, 63, 448. (30) Zwerger, W. Z. Phys. B-Condens. Matter 1981, 42, 333. (31) Dahn, J. R. Phys. ReV. B 1991, 44, 9170. (32) Ohzuku, T.; Iwakoshi, T.; Sawai, K. J. Electrochem. Soc. 1993, 140, 2490.
