Time−Difference Impedance Spectroscopy of Growing Films

188

J. Phys. Chem. B 2001, 105, 188-194

Time-Difference Impedance Spectroscopy of Growing Films Containing a Single Mobile Charge Carrier, with Application to Surface Films on Li Electrodes Mikhail A. Vorotyntsev,*,† Mikhael D. Levi,‡ Alexander Schechter,‡ and Doron Aurbach‡ UMR 5632 CNRS-LSEO, UniVersite´ de Bourgogne, 21000 Dijon, France, and Bar-Ilan UniVersity, Ramat-Gan, Israel ReceiVed: July 12, 2000; In Final Form: October 9, 2000

In this paper we present a new approach to the analysis of electrochemical impedance spectroscopy (EIS) data for electrodes covered by passivating surface films that grow in time. The impedance of such electrodes increases in time due to the gradual growth of their thickness as well as to the change of their local characteristics. The growing part of the surface films may either be homogeneous in its structure and properties or inhomogeneous, with gradually changing properties as a function of the distance of the newly formed surface species from the metal-film interface. We show herein that the time-difference impedance spectra (DIS), i.e., the difference between two complex impedance curves, Z(ω, t1) - Z(ω, t2), measured after different periods of storage, t1 and t2, as a function of frequency may provide a very useful tool for characterizing the nature of the growing passivating films. These time-difference impedance curves (e.g., plotted in coordinates Zimag vs Zreal) can be simulated both for homogeneous growing surface films on electrodes and for situations in which local properties such as the surface film’s resistivity are nonuniform across the growing film. The prospects of this approach have been demonstrated in the analysis of the surface films at lithium electrodes freshly prepared and stored in alkyl carbonate mixture (ethylene-dimethyl carbonates) with a Li salt, and measured periodically by EIS. The analysis of DIS of this system has shown that during a period of several hours after the electrode’s preparation in solution, the process of the surface film deposition is very complicated. Properties of the new portions of the film are varying in time so that the deposited film possesses a highly nonuniform structure. Moreover, the local characteristics of earlier formed portions are modified gradually inside the film. However, as the storage time is longer, the growth of the surface films on the lithium electrodes becomes more homogeneous, probably due to the increased selectivity of the Li-solution reactions when the surface films are thicker.

Introduction It is generally known that the electrochemical behavior of lithium electrodes in all commonly used nonaqueous electrolytes suitable for Li batteries is controlled by passivating surface films, which are unavoidably present on the active metal. Li metal is always covered by native films comprised of its reaction products with the atmospheric components O2, N2, CO2, and H2O. The native surface films usually contain Li2O in the inner layer and LiOH, Li2CO3 (resulting from the reaction of CO2 with Li2O) in the outer layer.1 When such a lithium surface is introduced into any nonaqueous, polar aprotic system, a variety of reactions takes place. These include partial dissolution of the components of the native films, nucleophilic reactions between Li2O and LiOH and electrophilic solvent molecules such as esters or alkyl carbonates, and direct reactions between the active metal and solvent molecules, salt anions, and trace water that percolate to the Li surface through the native surface films. These complicated situations form complex, highly nonuniform surface films (at the nanoscopic and microscopic levels) with a mosaic-type structure, as already discussed.2 When fresh Li is exposed to * Corresponding author: UMR 5632 CNRS-LSEO, Universite´ de Bourgogne, 21000 Dijon, France. Tel: +33 3 80 39 60 64. Fax: +33 3 80 39 60 86 or 60 98. E-mail: [email protected]. † UMR 5632 CNRS-LSEO, Universite ´ de Bourgogne. ‡ Dept. of Chemistry, Bar-Ilan University, Ramat-Gan, 52900 Israel.

solution (e.g., during Li deposition), we anticipate the following scenario: 1. Li initially attacks most of the solution species nonselectively, including solvent molecules, salt anions, trace water, CO2, and O2, etc., because it is reactive with all polar aprotic solvents, commonly used salt anions (e.g., ClO4-, AsF6-, PF6-, BF4-, N(SO2CF3)2-, SO3CF3-, etc.), and atmospheric contaminants. These first reactions form an initial surface layer since most of the Li salts produced by these reactions are insoluble in polar aprotic solvents. 2. As the surface film grows due to continuous reactions with the solution species, the reactions between the active metal and the solution components become more and more selective. This is due to the fact that further surface reactions proceed via electron tunneling through the existing surface films. Hence, the reactivity of the active metal toward solution species decreases as the surface films grow and thicken. 3. Finally, the film becomes sufficiently thick to avoid any pronounced electron tunneling through it. Because the insoluble Li salts that comprise the surface films are electronically insulating, the active metal reaches passivation, after which the surface reactions can proceed via scattered electron tunneling through defects in the surface films. 4. In parallel to the above processes, there is a continuous reduction of surface species within the surface film, and, therefore, the inner part of the surface films becomes comprised

10.1021/jp002486b CCC: $20.00 © 2001 American Chemical Society Published on Web 12/06/2000

DIS of Growing Films Containing a Mobile Charge Carrier of species of a low oxidation state. In addition, there are secondary reactions of the surface species with solution components, as well as diffusion of trace water through the surface layer. The above description has a very solid basis, based on extensive XPS spectroscopy of Li electrodes, which also included depth profiling.3-8 It is well-known that the surface films formed on Li electrodes in all polar aprotic electrolyte systems behave like solid electrolyte interphases, because most of the Li salts formed on lithium by the above surface reactions (e.g., Li halides, Li2CO3, Li alkoxides, Li alkyl carbonates, Li carboxylates, Li2O, Li3N, LiOH, etc.) are Li-ion conductors when precipitated as thin films, under an electrical field.9 The way in which these surface films are formed, as described above, leads to the obvious conclusion that their electrical properties change in a perpendicular direction to the electrode surface. Electrochemical impedance spectroscopy represents one of the principal sources of information on transport characteristics of mobile charge species inside solid films, as well as on their interfacial exchange properties, see e.g., refs 10-15. However, this information can only be extracted from experimental data with the use of a theoretical model. The most popular approach is based on the choice of an equivalent circuit analogue representing the system under study. Indeed, over the past three decades of extensive studies of Li electrochemistry, many papers dealing with impedance properties have appeared in the literature.16-21 Besides, several models have been proposed for the understanding of impedance spectra of lithium electrodes.22,23 However, careful examination of the majority of these papers reveals that most of these studies suffered from the fact that the complicated Li surfaces were not well defined. For instance, the use of Li electrodes covered by pristine surface films is almost impossible to analyze due to their complex, mosaic-type structure described above. In addition, Li surfaces prepared by anodic (Li dissolution) or cathodic (Li deposition) processes have a poorly defined surface morphology. We have developed a method of in situ preparation of fresh and highly uniform Li surfaces in solutions, which enables us to obtain reliable and reproducible impedance spectra of Li electrodes in a wide variety of electrolyte solutions.24-27 As described in detail,26,27 with such a preparation mode, we were able to probe the formation of multilayer surface films on Li electrodes, and to further probe their aging and growth processes during storage. Our approach for simulating the impedance spectra obtained from Li electrodes freshly prepared in solution was based on the understanding that the surface films possess a multilayer structure due to their history of formation (as described above). As a first step, we used a relatively simple model based on the “Voigt”-type equivalent circuit analog26,28 composed of R|C circuits in series, which represent different layers in the surface film (hence, the R values correspond to the resistance to Li-ion migration across the corresponding layer, and the C values represent the layer’s capacitance), and interfaces with charge-transfer resistance in parallel to the double layer capacitance. One of the R|C circuits in these models that related to the solution-film interface also included a “Warburg”type element in series with the resistor, which accounted for Li-ion diffusion in the solution phase. The number of R|C circuits in the model, as well as the R and C values, was determined from the simulation procedure, see description in.25 This approach implies that the film consists of several uniform layers without extended transition regions between them. However, there are usually no independent data on the reality of such layered structures. The spectroscopic and depth profiling

J. Phys. Chem. B, Vol. 105, No. 1, 2001 189 data coming from XPS can only prove that the chemical structure of the surface films changes considerably on going from the Li to the solution side, in terms of the oxidation states of the surface species. (As the species are closer to the Li metal, their oxidation state is generally lower and they are more inorganic in nature).3-8 In addition, morphological studies by in situ ellipsometry29 and AFM30 suggest that the inner part of the surface films is usually compact, while the outer part is porous. It was therefore clear from the beginning that the description of the surface films on Li electrodes by a “Voigt”type analogue that reflects a distinctly layered structure of the Li-solution interphase, is only an approximation of the possibly more complicated situation in which the surface films’ structure changes gradually in a perpendicular direction to the Li surface (one can prove that the impedance spectra of such smoothly varying distributions can always be approximated by a finite sum of Voigt elements in series28,31). Another shortcoming of the use of a simple “Voigt”-type analogue in describing the impedance behavior of Li electrodes is usually the need for a great number of fitting parameters. Moreover, when it is applied to the characterization of a growing film, the impedance data for each time interval has to be treated in the same way, thus making the overall number of parameters very high. In this paper we describe a simple alternative approach to the interpretation of impedance data obtained for Li electrodes covered by surface films which grow during storage and contain a single type of mobile charge carrier, namely, Li ions. It is based on the time-difference impedance spectra, DIS, which we expect to be easier to interpret. The theoretical background of the approach is described below. Impedance of Nonuniform Films with a Single Type of Mobile Charge Carrier The surface films that cover Li electrodes in solutions contain a single type of ionic species, namely, Li-ions, which can move with respect to the matrix. If the latter is taken as a reference, its flux is equal to zero, while the flux density of the mobile charges and the gradient of their electrochemical potential µ are proportional to each other. The local current density i is given by the following equation:32-34

i(r,t) ) - σ(r,t) ∇ (µ(r,t)/zF)

(1)

where the local conductivity, σ(r), is generally a tensor in an anisotropic medium, z is the charge valence of the mobile species, and F is Faraday number. The simpler case of an isotropic, but generally inhomogeneous medium, where the conductivity is scalar, is examined below. In the bulk medium, the local electroneutrality condition necessitates that the total charge density of the mobile species does not change in time. For the systems under consideration in this paper, it means that the variation of the local concentration of the mobile species is zero at any point in the medium. As a result, the local characteristics, e.g., the concentration or the conductivity in eq 1, do not depend on time (within the period of EIS measurements while they are generally functions of time at the scale of the film-aging process), and the thermodynamic force in eq 1 is reduced to the gradient of the perturbed part of the Galvani potential, φ′(r,t). Then, for the case of a nonstationary, one-dimensional transport, eq 1 takes the form:

i(x,t) ) -σ(x) ∇φ′(x,t)

(2)

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Vorotyntsev et al.

Equation 1 should be combined with the general “continuity equation” for mobile species. In the case of the one-dimensional transport the latter results in the constancy of the total current density itot(t) across the system for any time moment, itot being the sum of the densities of the ion current and of the “displacement current”:

itot(t) ) i(x,t) - o (x) ∂∇φ′(x,t)/∂t

(3)

where o is the dielectric constant of the vacuum, and (x) is the local value of the (relative) dielectric constant of the medium. The combination of eqs 2 and 3 leads to a relation between ∇φ′(x,t) and the total current density:

itot(t) ) - σ(x) ∇φ′(x,t) - o (x) ∂∇φ′(x,t)/∂t

(4)

This formula may be easily integrated along coordinate x within the bulk phase (outside the electrical double layer regions near the interfacial boundaries) for the case of a sinusoidal variation of all characteristics in time, e.g., itot(t) ) Re[itot exp (jωt)]. The complex impedance of the bulk phase is then given by the formula

Zbulk(ω) ) S-1

∫OL [σ(x) + jω o (x)]-1 dx

(5)

where L ) thickness of the film and S ) surface area. In the simplest case of a uniform medium, the local characteristics (σ and ) are constant across the film, and the bulk-film impedance is reduced to a “‘Voigt’-type” element, i.e., a parallel combination of the bulk-film resistance, Rf, and the geometrical capacitance of the film, Cf:

ZRC(ω) ) S-1 [1/Rf + jωCf]-1,

Rf ) L/σ, Cf ) o /L (6)

which corresponds to a semicircle in the complex impedance plane. Nonuniformity of the film means a spatial variation of the local properties in eq 5. Their simplest approximation for this situation is a film separation into several layers, which have uniform properties. Then, the bulk-film impedance is given by the sum of “Voigt”-type elements6 with RC parameters characteristic for each layer, see refs 25 and 26. An alternative treatment should take into account a continuous variation of the surface film properties. Both approaches allow the simulation of the characteristic shape of the complex impedance plots, namely, depressed semicircles. Impedance of Growing Surface Films As described in the introductory section, one should generally expect two different phenomena during the lithium electrode exposure to the solution, the film growth (increase of the thickness) and its aging (gradual change in time of the local properties inside the earlier-formed film). The properties of newly deposited portions also depend on time, an obvious consequence of the fact that the reactive ability of the Li surface lowers gradually so that the processes on the Li surface become more selective as the surface films become thicker. The film aging may be related to the decrease of the average oxidation state of the surface species close to the active metal, as a function of time, due to secondary reactions (further reduction) between the lithium and the constituents of the surface films. One may expect that this phenomenon is especially pronounced at initial stages of the film formation. On the other hand, after the end

of this primary period the further process may be restricted to the addition of new layers outside the previously formed film, i.e., merely to the increase of the film thickness L in eq 5. Another important question is related to the film’s inhomogeneity problem. One expects that the portion of the film formed during the primary period should be nonuniform, with a significant variation of the properties even within a sufficiently thin layer, due to the low selectivity of the primary formation processes. However, the new layers deposited at later stages may be much more homogeneous. One may hope to verify these expectations and, under favorable circumstances, to distinguish between the above three situations with the use of time-difference impedance spectra (DIS). For this, one needs to measure the impedance of the growing film as a function of time for the same set of frequencies. Then, the time-difference spectra are determined as differences between two consecutive impedance spectra of the same electrode (measured during its aging in solution), calculated separately for each frequency. The time intervals between the impedance spectroscopic measurements should be sufficiently small to minimize the inhomogeneity effects inside the newly formed portion of the film, but not too short to ensure the acceptable precision of the difference data. One should mention an approach with a similar name, “differential impedance analysis/spectroscopy”, derived by Stoynov et al.35-37 which represents a general method of extracting the system parameters from experimental impedance data, i.e., addressed to quite a different problem. One can expect three different types of the resulting plots for the time-difference impedance spectra (DIS) for the time moments t1 and t2: (1) Having a complicated shape, e.g., loops in both the upper (i.e., negative sign for Z′′) and the lower (i.e., positive sign for Z′′) parts of the complex impedance plan which would be interpreted as the “disappearance” of a certain part of the film, and its replacement by a layer having different properties. The latter includes the part of the “old” film with modified properties and a newly formed layer; (2) Having a regular shape, like a “depressed semicircle”, which would demonstrate that the film has already been relaxed, i.e., its local properties inside the earlier formed parts (at t < t1) are time independent within the interval t1 < t < t2, but the deposition process has not been stabilized so that the layer formed during the interval t1 < t < t2 between the neighboring measurements is essentially inhomogeneous. In this case, the integrals over the “initial” portion of the film formed before the time interval under study (at t < t1), 0 < x < Lini, Lini ) L(t1), are canceled in the course of the subtraction of the “initial” impedance, Zini(ω), from the “final” one, Zfin(ω), so that the time-difference impedance of the bulk film is given by the formula

dZbulk(ω) ≡ Zfin(ω) - Zini(ω) ) S1-

∫LL

fin

ini

[σ(x) + jω o (x)]-1 dx (7)

(3) The plot is close to a semicircle, which would mean that the film has been relaxed and the deposition process produces a uniform additional layer between the two sets of measurements. As an illustration of this method, we have analyzed the complex impedance spectra of a growing surface layer on the metallic lithium electrode, freshly prepared in an ethylene carbonate (EC)-dimethyl carbonate (DMC) solution.

DIS of Growing Films Containing a Mobile Charge Carrier

J. Phys. Chem. B, Vol. 105, No. 1, 2001 191

To avoid being misleading, we are to emphasize that all this reasoning corresponds to systems containing a single type of mobile charge carriers. If the medium contains mobile charges of different types, e.g., for the case of a mixed electron-ion conductor (see, e.g., refs 34,38-47 and references therein) the simple additiVe type of the contributions from different parts of the films does not hold, and the subtraction procedure defined by eq 7 does not remove the contribution from the earlier deposited portion of the film, even for the period where the film has already “relaxed”, see above. Experimental Section Impedance spectra of a Li electrode were measured in a threeelectrode cell of a nearly ideal parallel plate geometry, which was previously described in detail.24-26 Li rods (with a crosssection of 0.785 cm2) served as working and counter electrodes, which were freshly prepared in solution by shearing a Li rod with the stainless steel wire (0.5 mm thick), thus cutting it into two identical pieces, as already described.24 Li-wire placed between the two electrodes, close to the surface of the working electrode, served as a reference electrode. The measurements were performed in a solution containing 1 M LiAsF6 (Lithco) in a 1:3 mixture (by volume) of ethylene carbonate (EC) and dimethyl carbonate (DMC) obtained from Tomiyama Inc., Japan (Li battery grade). Both salt and solvents could be used as received without further purification. The electrolyte solutions usually contained 20-30 ppm of water. The cell was assembled in a glovebox filled with highly pure Ar, (VAC Inc., equipped with trace H2O and O2 absorbers and monitors), then hermetically sealed in a specially designed aluminum container, which was placed in a thermostat. Electrochemical impedance spectra were measured using a Schlumberger Model 1255 Frequency Response Analyzer (FRA) combined with the Schlumberger Model 1286 Electrochemical interface. FRA is able to provide a sufficiently high precision, the dispersion being well below 1% in measuring the real and imaginary components of the complex-plane impedance in a wide range of frequencies from 106 to 10-4 Hz.47 The data were acquired with the use of the Zplot software from Scribner and a Pentium-II PC. The amplitude of the ac voltage was 5 mV, whereas the range of frequencies extended from 100 kHz to 10 mHz. Impedance spectra were measured after different storage times of the Li electrode in the electrolyte solution. The duration of measurement of each Nyquist plot was ca. 20 min, i.e., much shorter than the intervals between different storage times. It allowed us to conclude on the maintenance of the stationary state of the films on the surface of Li electrode within the whole time domain of each Nyquist plot measurement. All measurements were performed at 25 °C ( 0.1 °C. Results and Discussion Figure 1 shows a family of Nyquist plots obtained for the Li electrode freshly prepared and stored in the LiAsF6 1M ECDMC 1:3 solution, with the exposure periods of 3, 7, 23, 48, and 168 h after the electrode is prepared in solution, as indicated. The “time-difference impedance spectra” (DIS) were obtained from all couples of the neighboring spectra, 1-2, 2-3, 3-4, and 4-5, in Figure 1. One has to keep in mind that these plots can only be informative for systems with a single type of mobile charge carriers. Since the low-frequency part of the spectra (the “tails” in the plots in Figure 1) corresponds to the contribution of the outer porous layer of the surface films,24-26 which possesses complicated conductivity features (lithium ion

Figure 1. Experimental impedance plots measured for lithium electrodes freshly prepared and stored in LiAsF6 1M EC-DMC 1:3 solution after 3, 7, 23, 48, and 168 h of storage (curves 1-5, respectively).

Figure 2. Time-difference impedance plot for the first time interval, between 3 and 7 h, corresponding to curves 1 and 2 in Figure 1: experimental points for the high- and medium-frequency ranges. Characteristic frequency values are indicated in Figures 2-5.

conduction inside the film and the electrolyte, both cation and anion conduction in pores), it was ignored in the subsequent analysis. The subtraction operation should lead to the increase of the relative dispersion of experimental data. Therefore, to achieve a sufficient precision the measurements within the medium- and high-frequency ranges (used to obtain Figures 2-5) were realized within numerous cycles for each frequency. Figure 2 (similar curves can also be obtained by the subtraction of plot 1 in Figure 1 from the other plots) shows the difference plot for the time interval which is the closest to the electrode’s preparation, and thus, to the beginning of the film formation (between 3 and 7 h of exposure). The curve consists of two loops of different diameters above and below the real axis, as indicated. This shape definitely reflects a rapid evolution of local characteristics of the film within this time interval. For example, a possible gradual increase of the conductivity in time gives a counter-effect with respect to the growth of the film thickness, the former factor playing the greater role within a lower frequency range. As a result, the imaginary part of impedance may change in the opposite directions at high and medium frequencies, thus leading to the two depressed semicircles in

192 J. Phys. Chem. B, Vol. 105, No. 1, 2001

Figure 3. Time-difference impedance plot related to curves 2 and 3 in Figure 1. Experimental points (circles) are compared with the bestfitting theoretical approximations: (1) semicircle, dRinf ) 0.43 Ω cm2, dRf ) 3.1 Ω cm2, ωo ) (dRf dCf )-1 ) 1.8 × 104 s-1, dispersion ) 0.22; (2) linear model, dRinf ) 0.35 Ω cm2, Lfin - Lini ) 0.32 nm, dω ) 4.7 × 104 s-1, ωini ) 5.3 × 104 s-1, dispersion ) 0.07; (3) simple exponential model, dRinf ) 0.32 Ω cm2, Lfin - Lini ) 0.36 nm, xvar ) 0.13 nm, ωini ) 1.2 × 105 s-1, dispersion ) 0.05. The value of  was always taken as equal to 5, see refs 18- 20.

Figure 4. Time-difference impedance plot related to curves 3 and 4 in Figure 1 (i.e., to the third time interval): (1) semicircle, dRinf ) 0.41 Ω cm2, dRf ) 1.3 Ω cm2, ωo ) (dRf dCf )-1 ) 2.5 × 104 s-1, dispersion ) 0.14; (2) linear model, dRinf ) 0.34 Ω cm2, Lfin - Lini ) 0.22 nm, dω ) 1.1 × 105 s-1, ωini ) 1.1 × 105 s-1, dispersion )0.05; (3) simple exponential model, dRinf ) 0.32 Ω cm2, Lfin - Lini ) 0.29 nm, xvar ) 0.07 nm, ωini ) 6.9 105 s-1, dispersion ) 0.05. One should pay attention to very small amplitudes of the data in this figure, compared to the other time-difference plots, so one can expect a greater experimental error here.

the upper and lower halves of the complex impedance plane, as observed in Figure 2. Quite different behavior of the time-difference plots is seen within the subsequent time interval (difference of curves 2 and 3 in Figure 1). As demonstrated in Figure 3, the experimental plot demonstrates a regular loop, close to a depressed semicircle without any points in the lower half of the complex plan (below the real axis). According to the above analysis, this feature indicates a sufficient stabilization of the film in terms of its local properties. On the other hand, the experimental curve is far from that corresponding to a simple R|C couple:

dZRC(ω) ) S-1 dRinf + S-1 [1/dRf + jω dCf]-1, dRf ) (Lfin - Lini)/σ, dCf ) o /(Lfin - Lini) (8) which should take place for the case of the deposition of a homogeneous layer within this time interval. Here and below, dRinf, i.e., the change of the time-dependent high-frequency resistance, Rinf, which reflects the total resistance of the solution, the interfacial charge-transfer process and the porous layer

Vorotyntsev et al.

Figure 5. Time-difference impedance plot related to the last (the longest) time interval, curves 4 and 5 in Figure 1: (1) semicircle, Rinf ) 2.2 Ω cm2, dRf ) 26 Ω cm2, ωo ) (dRf dCf )-1 ) 0.49 × 104 s-1, dispersion ) 0.58; (2) linear model, Rinf ) 2.2 Ω cm2, Lfin - Lini ) 0.56 nm, dω ) 1.9 103 s-1, ωini ) 5.9 × 103 s-1, dispersion ) 0.57; (3) simple exponential model, Rinf ) 2.2 Ω cm2, Lfin - Lini ) 0.58 nm, xvar ) 0.008 nm, ωini ) 2.8 × 105 s-1 (the last two parameters are not well defined here, only their combination may be determined from the fitting procedure), dispersion ) 0.62. Labels 1, 2, and 3 are not indicated in the figure since all theoretical curves are practically coincident.

within this frequency interval, has been added to the bulk-film impedance. The resulting time-difference curve corresponds to the dashed line 1 in Figure 3. It should be emphasized that thus analyzed homogeneity (or inhomogeneity) in the surface film properties relates only to the specific part of the surface film that was added to the Lisolution interphase between the two relevant impedance measurements. In the case of the curve in Figure 3 the marked deviation of the experimental data from the semicircle line means a significant inhomogeneity even within a sufficiently thin “new” portion of the film. It is interesting to note that the degree of its heterogeneity is comparable with that of the curves in Figure 1 representing the properties of the whole film including its parts formed during the initial period. Figure 3 also shows simulations of DIS with the use of two model approximations48 for the dependence of the surface film’s conductivity within the added part of the film as the function of the normal coordinate, either in the linear or exponential way (curves 2 and 3 in Figure 3, respectively). The dielectric constant of the surface species was assumed to be constant with a value of 5, because this value corresponds to the dielectric constant of the major surface species formed on lithium in alkyl carbonate solutions, including Li2CO3, ROCO2Li, and LiF.24-26 The corresponding equations for the local conductivity in these two models are as follows:

(x) )  ) constant, σ(x)/o  ) ωini - dω (x - Lini)/(Lfin - Lini) (“linear model”) (9) (x) )  ) constant, σ(x)/o  ) ωini exp(-(x - Lini)/xvar) (“simple exponential model”) (10) The overall impedance of the surface films in these calculations was considered to be the sum of the bulk film contribution and a “high-frequency contribution”, Rinf, see above. Figure 4 represents the time-difference impedance plots for curves 3 and 4 in Figure 1, which correspond to impedance measurements after 23 and 48 h of storage. It is significant to notice an obvious similarity between the plots for Li electrodes in Figures 3 and 4 characterizing the change of the film impedance within the time intervals of 16 and 25 h long,

DIS of Growing Films Containing a Mobile Charge Carrier respectively. In particular, experimental data in Figure 4 represent again a depressed semicircle that cannot be simulated by eq 8, i.e., the new portion of the film added during this time interval is still essentially inhomogeneous. Once again, similar to Figure 3 experimental data in Figure 4 can be simulated quite nicely by both models 9 and 10, which suggest a linear or exponential (respectively) variation of the surface film conductivity across the additional layer. It should be noted a relation of these models with the information delivered by measurements of the same films with the use of independent physical techniques. As is generally known,49,50 ion conductance in solid ionic media such as the surface films formed on lithium, depends on points of disorder, defects in the lattice, grain boundaries, etc. One can assume that in the points inside the film closer to the metal-film interface, the surface layer is formed under less selective conditions, and therefore it is more heterogeneous in nature, and hence, more conductive. Our earlier analysis25,26 of surface films formed on Li surfaces in a variety of electrolyte solutions based on the series of “Voigt” elements showed that the closer a surface layer is to the lithium, the lower its resistivity. The difference between the last two curves, 4 and 5, in Figure 1 corresponds to a much longer time interval (from 48 to 168 h of storage), compared with those related to Figures 2-4. Figure 5 shows the relevant time-difference impedance plot to this time interval, together with the corresponding simulations. One may expect that after the previous extended period of the film exposure to the solution all processes have been stabilized, in particular the interaction between the active metal and solution species takes place under highly selective conditions so that there are already practically no further changes of the local film properties in time. Moreover, one may expect that under such circumstances, the additional layer that precipitates further on top of the existing film should be homogeneous in its properties. This all enables one to expect that the timedifference impedance plots should be close to a semicircular (category 3 above). The experimental data in Figure 5 correspond nicely to these expectations. The experimental time-difference plot is close in shape to an ideal semicircle. The use of more general “nonuniform” models, eqs 9 and 10, (see the relevant curves in Figure 5) does not improve the quality of simulation: the deviation of experimental points from any of the theoretical curves in Figure 5 is very small, and all three theoretical plots are practically coincident. With the use of eq 8 one can calculate the corresponding values of the parameters corresponding to this time interval (relative dielectric constant of the film being taken as 5): - change of the film thickness, Lfin - Lini ) 0.55 nm; - film conductivity, σ ) 2.1 10-9 S/cm. These values correspond well to those found for one of the layers for the films grown at similar conditions.24-26 This analogy allows us to give a substantiated attribution of this layer as located at the outside boundary of the film, in conformity with the heuristic conclusion in refs 24-26. We have not made calculations of these parameters for other time intervals since even the additional layers during those time intervals were strongly inhomogeneous, i.e., the conductivity varies across the film. A detailed discussion of this problem will be given in another publication.48 Conclusions General theoretical expression for the complex impedance of the film possessing a single type of mobile charge carriers

J. Phys. Chem. B, Vol. 105, No. 1, 2001 193 has been used to analyze the time-difference impedance spectra (DIS) representing the change of the complex impedance characteristics of the growing film during the time interval between their subsequent measurements. A formula for this quantity has been derived for the case where the local properties of the earlier-formed portions of the film do not change during the subsequent deposition, i.e., this part of the film has already relaxed. The analysis has been carried out for the general case where the film is essentially nonuniform in its chemical and physical properties. Three qualitatively different situations have been outlined: (1) An initial stage where the already formed part of the film is subject to the subsequent modifications (“film aging” or “relaxation”); (2) A longer period, where local properties of the surface film have already stabilized so that the timedifference impedance spectra (DIS) are only determined by the portion of the film deposited during the corresponding time interval under consideration, but the properties of the additional surface layers deposited at each moment within this period still change rapidly as a function of time so that this added part of the film as a whole is essentially nonuniform; (3) Where the film has stabilized and the new deposited layer is sufficiently uniform. For these three cases qualitative predictions on the shape of the time-difference impedance plots have been made. This analysis has been applied to the interpretation of the impedance data for surface films on a lithium electrode freshly prepared and stored in a Li salt anion. We have found a complete confirmation of the above predictions, with a gradual transition in the sequence (1) f (2) f (3). For the latter two cases the local parameters of the uniform or nonuniform layer have been determined by fitting the corresponding theoretical curves to the experimental ones. Acknowledgment. The authors express their deep gratitude to the bilateral France-Israeli collaboration program, Arc-enCiel 1999, for the support of the authors’ travels. References and Notes (1) Fugieda, T.; Yamamoto, N.; Saito, K.; Ishibashi, T.; Hanjo, M.;. Koike S.; Wakabayashi, N.; Higuchi, S. J. Power Sources 1994, 22, 99. (2) Aurbach, D. Nonaqueous Electrochemistry; Marcel Dekker: New York, 1999; Chapter 6. (3) Kanamura, K.; Tamura, H.; Takehara, Z. I. J. Electroanal. Chem. 1996, 333, 127. (4) Aurbach, D.; Weissman, I.; Schechter, A.; Cohen, H. Langmuir 1996, 12, 3991. (5) Kanamura, K.; Tamura, H.; Shiraishi, S.; Takehara, Z. I. J. Electroanal. Chem. 1995, 394, 49. (6) Kanamura, K.; Shiraishi, S.; Takehara, Z. I. J. Electrochem. Soc. 1996, 143, 2187. (7) Schechter, A.; Aurbach, D. Langmuir 1999, 15, 3334. (8) Kanamura, K.; Shiraishi, S.; Takehara, Z. I. J. Electrochem. Soc. 1994, 141, L108. (9) Peled, E. Lithium Batteries; Academic Press: London and New York, 1983; Chapter 3. (10) Macdonald, D. D. Transient Techniques in Electrochemistry; Plenum: New York, 1977. (11) Gabrielli, C. Use and application of electrochemical impedance techniques; Solartron: Farnborough, 1990. (12) Buck, R. P. Electrochim. Acta 1990, 35, 1609. (13) Macdonald, D. D. In Techniques for Characterization of Electrodes and Electrochemical Processes; Varma, R., Selman, J. R., Eds.; Wiley: New York, 1991. (14) Inzelt, G. In Electroanalytical Chemistry; Bard, A. J., Ed.; Dekker: New York, 1994; Vol. 18; p 89. (15) Gabrielli, C.; Keddam, M.; Khalil, A.; Rosset, R.; Zidoune, M. Electrochim. Acta 1997, 42, 1207. (16) Takami, N.; Ohsaki, T. J. Electrochem. Soc. 1992, 139, 7. (17) Churikov, A. V.; Niman, E. S.; Lvov, A. L. Electrochim. Acta 1997, 42, 179. (18) Sloop, S. E.; Lerner, M. M. J. Electrochem. Soc. 1996, 143, 3.

194 J. Phys. Chem. B, Vol. 105, No. 1, 2001 (19) Jamnik, J.; Gaerscek, M.; Medem, A.; Pejovnik, S. J. Electrochem. Soc. 1991, 138, 6. (20) Morita, M.; Aoki, S.; Matsuda, Y. Electrochim. Acta 1992, 37, 119. (21) Liebenow, C.; Luhder, K. J. Appl. Chem. 1996, 26, 689. (22) Thevenin, J. G.; Muller, R. J. Electrochem. Soc. 1987, 134, 273. (23) Peled, E. Rechargeable Lithium and Lithium-Ion Batteries; The Electrochem. Soc. Inc.: Pennington, N.J., 1995: PV 94-28, p 1. (24) Zaban, A.; Aurbach, D. J. Electroanal. Chem. 1993, 348, 155. (25) Aurbach, D.; Zaban, A. J. Electroanal. Chem. 1994, 367, 15. (26) Aurbach, D.; Zinigrad, E.; Zaban A. J. Phys. Chem. 1996, 100, 3098. (27) Aurbach, D.; Zaban, A.; Gofer, Y.; Abramson, O.; Ben-Zion, M. J. Electrochem. Soc. 1995, 142, 687. (28) Orazem, M. E.; Agarwal, P.; Garcia, L. H. J. Electroanal. Chem. 1994, 378, 51. (29) Schwager, F.; Geronov, Y.; Muller, R. H. J. Electrochem. Soc. 1985, 132, 285. (30) Aurbach, D.; Cohen, Y. J. Electrochem. Soc. 1996, 143, 3525. (31) Frateur, I.; Deslouis, C.; Orazem, M. E.; Tribollet, B. Electrochim. Acta 1999, 44, 4345. (32) Kornyshev, A. A.; Vorotyntsev, M. A. Electrochim. Acta 1978, 23, 267. (33) Kornyshev, A. A.; Vorotyntsev, M. A. Electrochim. Acta 1981, 26, 303. (34) Vorotyntsev, M. A.; Badiali, J. P.; Vieil, E. Electrochim. Acta 1996, 41, 1375. (35) Stoynov, Z. Polish J. Chem. 1997, 71, 1204.

Vorotyntsev et al. (36) Vladikova, D.; Stoynov, Z.; Ilkov, L. Polish J. Chem. 1997, 71, 1196. (37) Vladikova, D.; Stoynov, Z. Proc. 4th International Symposium on Electrochemical Impedance Spectroscopy, Rio de Janeiro, 1998; p 309. (38) Gabrielli, C.; Takenouti, H.; Haas, O.; Tsukada, A. J. Electroanal. Chem. 1991, 302, 59. (39) Mathias, M. F.; Haas, O. J. Phys. Chem. 1992, 96, 3174. (40) Vorotyntsev, M. A.; Daikhin, L. I.; Levi, M. D. J. Electroanal. Chem. 1994, 364, 37. (41) Deiss, E.; Sullivan, M.; Haas, O. J. Electroanal. Chem. 1994, 378, 93; 1996, 403, 261. (42) Buck, R. P.; Mundt, C. J. Chem. Soc., Faraday Trans. 1996, 92, 3947. (43) Buck, R. P.; Mundt, C. J. Chem. Soc., Faraday Trans. 1996, 92, 4987. (44) Ren, X. M.; Pickup, P. G. J. Electroanal. Chem. 1997, 420, 251. (45) Vorotyntsev, M. A.; Badiali, J.-P.; Inzelt, G. J. Electroanal. Chem. 1999, 472, 7. (46) Lefebvre, M. C.; Martin, R. B.; Pickup, P. G. Electrochem. SolidState Lett. 1999, 2, 259. (47) Impedance Spectroscopy; Macdonald, J. R., Ed.; John Wiley & Sons: New York, 1987; p 152. (48) Vorotyntsev, M. A.; Levi, M. D.; Aurbach D. In preparation. (49) Young, L. Anodic Oxide Films; Academic Press: New York, 1961. (50) Ashcroft, N. W.; Mermin, N. D. In Solid State Physics; Saunders, W. B., Ed.; Philadelphia, 1976; p 616.