Impedance of a reaction involving two adsorbed intermediates: aluminum dissolution in non-aqueous lithium imide solutions

www.elsevier.nl/locate/jelechem

Journal of Electroanalytical Chemistry 482 (2000) 125–138

Impedance of a reaction involving two adsorbed intermediates:aluminum dissolution in non-aqueous lithium imide solutions

Laszlo Peter 1, Juichi Arai *, Haruo AkahoshiDepartment of 1-Material, Hitachi Research laboratory, MD no. 260, 1-1 Omika-cho 7-chome, Hitachi-shi, Ibaraki-ken 319-1292, Japan

Received 10 February 1999; received in revised form 13 January 2000; accepted 14 January 2000

Abstract

The model presented considers the dissolution of a trivalent metal in three consecutive steps involving two adsorbedintermediates. If mass transport effects are negligible, it is possible to construct equivalent circuits in which adsorption-relatedelements are doubled compared to the case of a single adsorbate. In the case where mass transport affects the dissolution, theFaradaic admittance can be evaluated as a fraction of two power series and no simple equivalent circuit can be constructed fromconventional circuit elements. Depending on the mechanism assumed, the low-frequency behavior can be either similar to aWarburg impedance or different fundamentally. The impedance of aluminum dissolution is discussed in the case of insignificantmass transport. The Langmuir isotherm is supposed to hold for intermediate adsorption, and only anodic partial reactions areaccounted for. It has been concluded that the second step is rate-determined and that solvent takes part in the desorption of theproduct only. An empirical correlation was found between the dipole moment of the solvent used and the ratio of the rateconstants of non-rate determining steps. © 2000 Elsevier Science S.A. All rights reserved.

Keywords: Impedance; Al dissolution; Non-aqueous solution; Imide anion; Lithium battery

1. Introduction

Electrochemical impedance spectroscopy (EIS) is avery effective tool to analyze multistep electrochemicalreactions [1–5]. In the course of the development of thetheory of EIS, treatments of reactant and productadsorption [6,7], intermediate adsorption [8–15], diffu-sion [16] and the combination of the above processes[12,17,18] have been well established (see also Refs.[1–5]). The interest of the authors was focused mainlyon practically important reactions. The theory of theadsorption of intermediates and impedance of systemsinvolving them has developed in line with the investiga-tion of reactions composed of two consecutive steps.The family of such reactions includes, among others,hydrogen evolution, chlorine evolution, oxalic acid re-duction, electrochemical dissolution of a number of

divalent metals and electrocrystallization of several di-valent transition metals, as summarized by Diard et al.[15]. The above mentioned reactions all include oneadsorbed intermediate only. Electrochemical reactionswith several intermediates occur when parallel reactionstake place. A typical example is the electrodeposition ofzinc where hydrogen evolution is an unavoidable sidereaction and coupling of the two reactions also occurs[13]. A brief discussion of transpassive nickel dissolu-tion was published by Epelboin et al. [11] with noanalysis in detail. The latter reaction is composed ofthree consecutive steps and involves two adsorbedintermediates.

In general, electrochemical dissolution or depositionof trivalent metals has attracted little interest. Alu-minum and chromium exhibit protective native surfaceoxide layers which make investigation of their dissolu-tion in acidic aqueous media rather difficult, and thefirst step of the oxidation is usually very fast. Thus,adsorption of different intermediates cannot be detectedby EIS. Reduction of bismuth ions (Bi3+) on mercurycan be carried out under appropriate conditions, butonly one intermediate can be detected with the help of

* Corresponding author. Fax: +81-294-527636.E-mail address: [email protected] (J. Arai)1 Present address: Research Institute for Solid State Physics and

Optics, Hungarian Academy of Sciences, H-1525 Budapest, POB 49,Hungary.

0022-0728/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved.PII: S 0 0 2 2 -0728 (00 )00028 -0

L. Peter et al. / Journal of Electroanalytical Chemistry 482 (2000) 125–138126

EIS [19]. Therefore, examples of electrochemical metaldissolution or deposition in three consecutive stepswhere each partial reaction can be distinguished fromthe others are very scarce.

An impedance analysis of aluminum dissolving inconcentrated alkaline solution was published recentlyby Macdonald et al. [20]. In this work, the dissolutionof aluminum was analyzed in terms of a three-stepmechanism, removing one electron from the aluminumatom in each step. Formation of adsorbed hydrogenatoms and their electrochemical desorption had to beunavoidably assumed due to the presence of water andthe relatively negative potential applied. Since bothanodic and cathodic reactions occurred in the entirepotential range studied, the separation of the kineticconstants of the cathodic and anodic partial reactionswas not possible. These authors considered no compli-cation related to diffusion.

Recently, aluminum dissolution in non-aqueous solu-tions containing bis(perfluoroalkylsulfonyl) imides wasdiscovered [21] and later characterized as a reactiontaking place in three steps involving two adsorbedintermediates [22]. Investigation of the reaction is ofgreat practical importance because imide-type lithiumsalts are very promising candidates as electrolyte com-ponents in lithium ion batteries [21,23] where aluminumis used almost exclusively as the current collector forthe positive electrode. Details of dc analysis of alu-minum dissolution using several solvent mixtures andlithium bis(trifluoromethylsulfonyl) imide (Li(CF3-SO2)2N, LiTFSI) and/or lithium bis(pentafluoroethyl-sulfonyl) imide (Li(C2F5SO2)2N, LiBETI) was pub-lished by Peter and Arai [22]. The reaction was for-mulated as follows:

AlS+TFSI−SOLV� [Al(TFSI)]ADS+e− (R1)

[Al(TFSI)]ADS+TFSI−SOLV� [Al(TFSI)2]ADS+e−

(R2)

[Al(TFSI)2]ADS+nSoSOLV� [Al(TFSI)2(So)n ]+SOLV+e−

(R3)

where So denotes the solvent. The same reaction pat-tern is valid for LiBETI with the appropriate substitu-tion of the anion.

The aim of this communication is to establish animpedance model of the above mechanism for bothnegligible and significant contributions of the diffusionimpedance and to analyze EIS data obtained for alu-minum dissolution under different conditions.

2. Theoretical

The reaction mechanism is considered as describedby reactions R1, R2, and R3. The partial currentdensity of each reaction is a function of the electrode

potential, the partial coverage of the surface with thetwo intermediates and the concentration of the anion inthe solution at the metal surface if it is also a reactant:

j1= j1(E, u1, u2, c) (1)

j2= j2(E, u1, u2, c) (2)

j3= j3(E, u1, u2, cp) (3)

Evaluation of the expressions for the Faradaicimpedance of the electrode from this point depends onthe assumptions made about the concentration of theanion and the product.

2.1. Impedance of the dissol6ing aluminum without theinfluence of mass transport

This section deals only with the case where theconcentration of the complexing anion close to thesurface is constant and the concentration of the productcan be taken as zero (or reaction R3 is totallyirreversible).

If a sinusoidal potential perturbation with smallenough amplitude is superimposed onto the dc biaspotential, the resulting perturbation of the current den-sity can be evaluated with a Taylor expansion followedby truncation of the higher-order terms, hence lineariz-ing the current density perturbation:

Dj= %3

m=1

Djm= %3

m=1

�DE#jm#E

+Du1

#jm#u1

+Du2

#jm#u2

�(4)

The change in the surface coverages is associatedwith some of the partial current densities:

b#ul

#t= jl− jl+1 (5)

where l=1, 2. In Eq. (5), b is a proportionality factorwhich is necessary because of the arbitrary choice of theunits of partial coverage (05ul51) and the currentdensity. For sake of simplicity, b for the two differentadsorbed species is taken equal, although in principlethey can be different. If the amplitude DE of thepotential perturbation DE exp(jvt) is small enough, theperturbation of the surface coverages can also be evalu-ated with the help of harmonic functions (ul=ul,dc+Dul exp(jvt)) and therefore the derivation in Eq. (5)can be performed, resulting in

b jvDul=DE�#jl#E

−#jl+1

#E�

+Dul

� #jl#u1

−#jl+1

#u1

�+Du2

� #jl#u2

−#jl+1

#u2

�(6)

Thus, the following system of linear equation isobtained for the relationship of the coverage and poten-tial perturbations:

L. Peter et al. / Journal of Electroanalytical Chemistry 482 (2000) 125–138 127�b jv+

�#j2#u1

−#j1#u1

�nDu1+

�#j2#u2

−#j1#u2

�Du2

=�#j1#E

−#j2#E

�DE (7a)�#j3

#u1

−#j2#u1

�Du1+

�b jv+

�#j3#u2

−#j2#u2

�nDu2

=�#j2#E

−#j3#E

�DE (7b)

The equation system Eqs. (7a) and (7b) can besolved, for instance, with the help of Cramer’s method.It is difficult to keep track of each partial derivative, sothe following simplification is made: by solving theequation system by Cramer’s method, the terms for thetwo coverage perturbations have identical denomina-tors in which there are terms comprising the second andfirst powers of jv in addition to a term which isindependent of jv. The terms in the numerators canalso be classified in the same manner except for thesecond power of jv which does not appear. By theappropriate substitution into Eq. (4):

Dj=� K %1jv+K %0

N %2(jv)2+N %1jv+N %0+ %

3

m=1

#jm#E

nDE (8)

where K %1, K %0, N %2, N %1, N %0 stand for the sum of thepartial derivative products originating from Eqs. (7a)and (7b). It can be seen that the fraction in Eq. (8) canbe simplified by dividing each coefficient with N %0 whichgives the exact number of the independent coefficients.

Dividing both sides of Eq. (8) by the amplitude of thepotential perturbation, the Faradaic admittance isevaluated:

YF=DjDE

=K1jv+K0

N2(jv)2+N1jv+1+ %

3

i=1

#ji#E

=K1jv+K0

N2(jv)2+N1jv+1+RCT

−1 (9)

It is easy to prove that several circuits exhibit suchadmittance functions that are equivalent to Eq. (9). Aselection of the appropriate circuits is shown in Fig. 1,and a few equivalence conditions are given in Table 1.It is worth mentioning that circuit elements related tothe adsorption of the intermediates can be simply dou-bled as compared to the analogous process involvingone intermediate [14], regardless of which symmetriccircuit is applied (see Circuits 1, 5 and 6). Interestingly,Circuits 5 and 6 are difficult to treat because theevaluation leading to the relationship of the circuitparameters as a function of parameters in Circuit 1requires solution of second-order equations. This prob-lem originates from the symmetry of the circuits, i.e. theidentical subcircuits responsible for the two adsorptionprocesses. Moreover, the assumption of the equality ofthe time constants of Circuit 1 and those of Circuit 5 or6 is incorrect. In this aspect, symmetric circuits in Fig.1 are similar to those reported by Harrington andConway [Ref. [14], fig. 1(a,d)].

Fig. 1. Equivalent circuits for the Faradaic impedance obtained for surface reaction controlled aluminum dissolution (Eq. (9)).


Table 1Parameters of the circuits shown in Fig. 1 as a function of the parameters of Circuit 1

Circuit 4Circuit 2 Circuit 5Circuit 3 Circuit 6

R41=RCT Simple transformation is1/R31=1/RCT+1/RAR21=RCT R61=RCT

not availableR32=R2

CT/(RA+RB)R22=R2CT/(RA+RB) R42= (RAL2

B+RBL2A)/(LA+LB)2 Simple transformation

for the rest of the parameters isnot available

L32=LAR2CT/(RA+RCT)2C22=LA/R2

CT L42=LALB/(LA+LB)1/(R42+R43)=1/RA+1/RBR33=RBR23=RB

L23=LB C43= (LA+LB)/[R43(RA+RB)]L33=LB

The choice of an equivalent circuit for further analy-sis is somewhat arbitrary, but a convenient choice canfacilitate finding the relationship between equivalentcircuit parameters and primary kinetic variables. In ourcase, the occurrence of the inverse charge transfer resis-tance (R−1

CT ) as an additive term in the expression of theFaradaic admittance indicates that a parallel connec-tion of the charge transfer resistance with the otherreactive elements is a natural choice. Hence, Gerischer-type (or Voight-type) equivalent circuits (i.e. Circuits 2,3, 5 and 6 that include all resistors connected only inseries with other elements) are omitted. The admittanceof these circuits does not provide the separation ofvariables that is obtained naturally from Eq. (9). Also,it appears to be convenient to treat the two adsorbedintermediates equally and to choose a circuit in whichparameters for the two adsorption processes are sym-metric. It is reasonable in the chemical sense as wellconsidering that: (i) both disappearance of [Al(-TFSI)]ADS as a result of reduction and that of [Al(-TFSI)2]ADS as a result of oxidation make a free surfacesite available; (ii) disappearance of [Al(TFSI)]ADS as aresult of oxidation and that of [Al(TFSI)2]ADS as aresult of reduction increase the partial coverage of thesurface with the other intermediate in the same way.Thus, Circuit 1 was chosen for further analysis, thoughit is recognized that any of the circuits displayed in Fig.1 can yield a mathematically correct description for theFaradaic impedance (or admittance) of the systemconsidered.

Efforts were made to find the relationship betweenthe system parameters and the time constants of theequivalent circuit in the same way as presented byArmstrong and Henderson (Eq. (6) in Ref. [12]). Thiscan be carried out by calculating the admittance ofCircuit 1:

YC1=(tBRA

−1+tARB−1)jv+RA

−1+RB−1

tAtB(jv)2+ (tA+tB)jv+1+RCT

−1 (10)

With the comparison of the terms in the denomina-tors of Eqs. (9) and (10) and evaluating N1 and N2 withthe derivatives of the partial current densities one canobtain

tA,B=b!#dj21

#u1

+#dj32

#u2

9��#dj21

#u1

�2

+�#dj21

#u1

�2

−2#dj21

#u1

#dj32

#u2

+4#dj21

#u2

#dj32

#u1

n1/2",�

2�#dj21

#u1

#dj32

#u2

−#dj21

#u2

#dj32

#u1

�n(11)

Eq. (11) shows that change in any partial derivativemodifies the time constant of both adsorption pro-cesses, hence relaxation times depend on the kineticconstants related to both intermediates.

Eq. (11) can be simplified considerably by assumingthat the Langmuir isotherm holds for both adsorptionprocesses and that only anodic reactions need to betaken into account. (Cathodic processes are unlikely tobe of importance in the case of aluminum dissolutionsince it is transpassive in nature in the media used[21,22]. This assumption appears to be quite general inthe literature of aluminum dissolution [20].) Eqs. (1)–(3) can be rewritten according to the assumptionsmade:

j1=k1(1−u1−u2) (12)

j2=k2u1 (13)

j3=k3u2 (14)

The rates of reactions R1 and R2 are proportional tothe anion concentration. The Tafel relationship is as-sumed to hold for each partial reaction, and thereforethe electrochemical rate constants can be expanded asfollows:

k1=ck10 exp(b1E) (15)

k2=ck20 exp(b2E) (16)

k3=k30 exp(b3E) (17)

Substitution of Eqs. (12)–(14) into Eq. (11) leads to thefollowing relationship:

tA,B=b{k1+k2+k39 [k12+k2

2+k32−2k1k2−2k1k3

−2k2k3]1/2}/(2k1k2+2k1k3+2k2k3) (18)

L. Peter et al. / Journal of Electroanalytical Chemistry 482 (2000) 125–138 129

The time constants can be obtained in a very simpleform if the rate constants differ from each other consid-erably, i.e. kp�kq�kr. Thus, after neglecting the ap-propriate terms in Eq. (18) one obtains:

tA=bkr−1 (19)

tB=bkq−1 (20)

If the same conditions are considered for an electro-chemical reaction involving one intermediate, the rela-tionship for the time constant of adsorption can begiven in an analogous form [Ref. [12], Eq. (6)]. Theratio of the adsorption–related time constants yieldsthe ratio of the appropriate rate constants:

r=tB/tA=kr/kq (21)

If the difference between the individual rate constants islarge enough, the partial surface coverage with thereactant involved in the rate determining step (i.e. freesurface site for reaction R1 or the corresponding inter-mediate for reactions R2 and R3) is approximatelyequal to 1. Thus, the rate constant of the rate-determin-ing step can be calculated from the net current density:

kp= j/3 (22)

If the potential dependence of the adsorption–relatedtime constants is known, it is easy to calculate the Tafelconstants for the corresponding partial reaction:

br= −# ln tA

#E(23)

and

bq= −# ln tB

#E(24)

while the Tafel constant for the rate-determining reac-tion step can be calculated from the potential depen-dence of the net current density:

bp= −# ln j#E

(25)

as well as by an alternative method by using thepotential dependence of the charge transfer resistance:

bp= −# ln RCT

−1

#E(26)

The method described above illustrates the advantagethat adsorption–related resistors, RA and RB, are notused in the analysis. It can be shown that RA and RB

are a function of terms composed of various differencesof the Tafel constants. In contrast with the rate con-stants, Tafel constants are usually of the same order ofmagnitude for consecutive reaction steps, therefore ne-glecting any term involving their difference may resultin a substantial error. (The latter statement is particu-larly true for Tafel constants of the steps of aluminumdissolution both in aqueous solutions [20] and in the

media applied in this work. For data, see followingsections.)

It is also possible to estimate the order of magnitudeof b. If kp�kq and tB\tA as assumed above,

b�kptB (27)

2.2. Impedance of the dissol6ing aluminum with theinfluence of reactant diffusion

If the concentration of the anion in the solution closeto the electrode surface is not constant, Eq. (5) has tobe modified as follows:

Dj= %3

m=1

Djm= %3

m=1

�DE#jm#E

+Du1

#jm#u1

+Du2

#jm#u2

+Dc#jm#c

�(28)

The diffusion of the reactant is characterized withFick’s equation:

#Dc(x, t)#t

=D#2Dc(x, t)#x2 (29)

If the amplitude of the potential perturbation is smallenough, the time-dependence of the concentration per-turbation can also be described by a harmonic function,and thus the concentration perturbation as a functionof x can be determined from the following equationwith the help of the boundary conditions:

Dc(x)=A exp[(jv/D)1/2x ]+B exp[− (jv/D)1/2x ]

=A exp[sx ]+B exp[−sx ] (30)

For a semi-infinite diffusion layer, the concentrationperturbation is zero at an infinite distance from theelectrode, and therefore A=0. The other constant inEq. (30) can be elucidated from the concentration per-turbation gradient at the electrode surface that is equalto an appropriate linear combination of the partialcurrent density perturbations:

−Dj1+Dj2

F=J= −D

#Dc#x

)x=0

=sBD (31)

Now B can be eliminated:

B= −Dj1+Dj2

sFD(32)

and hence the concentration perturbation of the reac-tant at the electrode surface is

Dc= −Dj1+Dj2

sFD(33)

Perturbations of the corresponding partial current den-sities have to be substituted into Eq. (33), thus arelationship between the concentration, coverage andpotential perturbations can be obtained:


�−sDF−

#j1#E

−#j2#E

�Dc+

�−#j1#u1

−#j2#u1

�Du1

+�

−#j1#u2

−#j2#u2

�Du2=

�#j1#E

+#j2#E

�DE (34a)

Two additional equations can be derived similarly tothe method by which Eqs. (7a) and (7b) were obtained,with the difference being that the resulting equationsinvolve an additional term related to the concentrationperturbation:�#j2#c

−#j1#c�Dc+

�b jv+

�#j2#u1

−#j1#u1

�nDu1

+�#j2#u2

−#j1#u2

�Du2=

�#j1#E

−#j2#E

�DE (34b)

�#j3#c

−#j2#c�Dc+

�#j3#u1

−#j2#u1

�Du1

+�

b jv+�#j3#u2

−#j2#u2

�nDu2=

�#j2#E

−#j3#E

�DE (34c)

The equation system (Eqs. (34a), (34b) and (34c)) issolved with Cramer’s method, and the result is trans-formed by using the coefficients of different powers ofjv. Following the same rearrangement that led to Eq.(9), one obtains

YF=1

RCT

+1+L1(jv)1/2+L2(jv)+L3(jv)3/2+L1(jv)2

M0+M1(jv)1/2+M2(jv)+M3(jv)3/2+M4(jv)2+M5(jv)5/2

(35)

The power series in Eq. (32) contains apparently tenindependent variables. Unfortunately, all attempts toreduce the number of variables as well as to find anequivalent circuit containing conventional circuit ele-ments (including Warburg impedance) failed.

Two remarks of theoretical importance should bemade here:1. If v�0, Eq. (32) leads to a finite and real Faradaic

impedance ZF(v�0)= (R−1CT +M−1

0 )−1 even ifsemi-infinite boundary conditions prevail. In thissense, Eq. (32) is different fundamentally from rela-tionships hitherto suggested for mechanisms withadsorption and mass transport (Eq. (21) in Ref. [12]and Eq. (97) in Ref. [3]).

2. If reaction R3 is modified and involvement of addi-tional anions in the product desorption is permitted,the character of the results obtained above may ormay not change. Expressions obtained for forma-tion of complexes with four or more complexinganions are the same as Eq. (32), with differentmeanings of the parameters accordingly. If the reac-tion product contains exactly three anions (in otherwords, the product is not charged), the number ofindependent parameters to be taken into account in

the expression of the Faradaic impedance is re-duced to seven and the low frequency impedanceis dominated by a Warburg-type function (i.e.limv�0

(#ZF/#v−1/2)= (1− j)s).

3. Experimental

Solvents such as ethylmethyl carbonate (EMC),ethylene carbonate (EC), propylene carbonate (PC) anddimethoxy ethane (DME) of battery grade with watercontent typically less than 20 ppm were purchased fromTomiyama Yakuhin Kogyo K.K. and used as received.Trifluoropropylene carbonate (TFPC) was the productof Japan Energy K.K. and of the same purity as theother solvents. Solvent mixing ratios are given in mol/mol% throughout this paper. The purity of LiTFSI andLiBETI (both from 3 M) was better than 99%. Bothlithium salts were used after vacuum drying. Electrodespecimens were made of commercial 99.5% pure alu-minum. Previous results confirm that the purity ofaluminum beyond this level has very little impact onthe electrochemical behavior in the media used [24].Chemicals were stored in a glove box with a constantlypurified argon atmosphere. The frost point of H2O wasmaintained below −80°C.

The electrochemical cell used for impedance measure-ments has already been described in a previous publica-tion [22]. Cells were mounted and later stored under theargon atmosphere of the glove box in order to eliminatecontamination from ambient oxygen and water. Elec-trochemical instruments were connected to the cell byusing the leak-proof interface of the glove box. Poten-tials are referred to the rest potential of the lithiumreference electrode immersed into the same solution(Li � Li+).

Electrochemical data were obtained by using aHokuto Denko workstation including a HB-105 func-tion generator and a HA-501G potentiostat–galvano-stat. EIS measurements were performed with the samesystem completed with a NF 5080 frequency responseanalyzer. The workstation was controlled with an IBM-compatible computer and software provided by HokutoDenko. Impedance spectra were recorded in potentio-static mode with a 4 mV ac perturbation amplitudefrom 100 kHz to 6 or 30 mHz in the descendingfrequency direction. Impedance measurements werestarted at least 1 h after polarization when the currentno longer varied. Constancy of the current and repro-ducibility of the spectra were adopted as stabilitycriteria.

The optimized structure of the solvent moleculeswere calculated by using semi-empirical molecular or-bital simulation (MOPAC version 94 program) with thePM3 method. Dipole moments of the solvent moleculeswere derived from these simulations.


4. Experimental results

4.1. Dc beha6ior of aluminum and potential regions forEIS measurements

Typical dc polarization data are plotted in Fig. 2.For most of the solutions studied, aluminum activationand repassivation potentials can be established as thepotentials of current onset and current cut-off, respec-tively. Potentiostatic polarization (and therefore EISexperiments) lead to different results according to thedc bias potential. Aluminum electrodes are passive ifthe dc bias is lower than the potential of currentcut-off. Steady-state dissolution cannot be achieved ifthe dc bias is between the potentials of current cut-offand current onset. In this potential interval, potentio-static polarization results in a slow current decay and

impedance spectra are irreproducible due to the occur-rence of both dissolution and reactions leading to re-passivation. Stable dissolution conditions can beachieved if the dc bias potential is higher than thepotential of current onset. The current increase in thispotential interval is quite abrupt and dissolution be-comes affected by anion diffusion at higher potentials.However, it is possible to attain conditions such thataluminum dissolution is unaffected by anion diffusionbut rather controlled by the surface process. The higherthe potential of current onset and the lower the slope ofthe current–potential curve, the wider is the potentialinterval of the surface reaction controlled aluminumdissolution.

However, potentials of current onset and currentcut-off cannot be established for some solutions sincebreakpoints between linear regions in the semilogarith-mic current–potential plot are missing. Such solutionsare more appropriate to investigate the potential-depen-dence of the surface reaction controlled dissolutionthan systems exhibiting activation and repassivation.

Detailed discussion of the dc behavior of aluminumcan be found in an earlier communication [22].

4.2. Impedance spectra measured for passi6e aluminum

A typical EIS spectrum measured for passive alu-minum is shown in Fig. 3. All spectra measured forpassive aluminum are essentially featureless and com-posed of a depressed semicircle with a diameter rangingfrom several tens to a million kV cm2. The dc currentobserved in the course of the EIS measurement decaysin time, and spectra measured consecutively exhibit asemicircle with larger and larger diameters, as a resultof the aging of the passive layer. Therefore, no correla-tion between measurement conditions and spectraparameters could be found, although parameter estima-tion by using a circuit made up of a resistor connectedparallel with a constant phase element (CPE) was usu-ally successful.

4.3. Impedance spectra measured for surface reactioncontrolled dissolution of aluminum

Spectra obtained for surface reaction controlled dis-solution of aluminum are composed of three overlap-ping semicircles if the solution contains only one typeof imide anion. The high-frequency semicircle is alwayscapacitive and its time constant is thought to be deter-mined by the charge transfer resistance and the doublelayer capacitance. The medium-frequency and low-fre-quency semicircles are assumed to account for theadsorption processes. The sequence of the adsorption-related semicircles can be either capacitive–capacitive,capacitive–inductive or inductive–inductive with in-creasing relaxation time (or decreasing frequency). No

Fig. 2. Typical potentiodynamic curves obtained for a 5 mV s−1

sweep rate. Solid line, 1 M LiTFSI+EMC+3% EC; broken line, 1M LiBETI+EMC. Arrows along the potentiodynamic curves showscan direction.

Fig. 3. Typical impedance spectrum measured for passive aluminum.Solution, 1 M LiTFSI+EMC+1% EC. Potential, 3.7 V versusLi � Li+. Frequency increment between data labeled with filled dia-monds, 1 decade.


Fig. 4. Impedance spectra measured for aluminum dissolution in 1 MLiBETI+EMC solution. Potential: squares, 4.50 V; circles, 4.60 V;triangles, 4.70 V. Frequency increment between data labeled withfilled symbols, 1 decade.

example was found for an inductive–capacitive se-quence, though such spectra can be simulated easilywith the help of either Circuit 1 or Eq. (9) by using anappropriate set of parameters. In contrast, the induc-tive–capacitive sequence of the adsorption-relatedsemicircles is typical in aqueous solutions at somepotentials [20]. A few selected experimental spectra areplotted in Figs. 4 and 5.

Several impedance spectra were measured by usingsolutions containing both LiTFSI and LiBETI. A typi-cal (and remarkably reproducible) spectrum obtained isshown in Fig. 6. The character of the spectra obtainedwith imide mixtures is different from those measuredwith a single salt in the sense that the number ofoverlapping semicircles cannot be established easily.These systems can probably be described by using morethan three time constants.

4.4. Impedance spectra measured for diffusioncontrolled dissolution of aluminum

Diffusion-controlled dissolution of aluminum can beattained easily in the potential interval of the anodiccurrent plateau. For most of the solutions made withEMC+EC solvent blends, a dc bias of 4.5 V is highenough so that aluminum dissolution is substantiallyaffected by the diffusion of the anion. A representativecurve can be seen in Fig. 7.

Impedance spectra measured for diffusion controlleddissolution of aluminum exhibit unusual features. First,spectra are composed of one capacitive high-frequencysemicircle, a single medium-frequency inductive loopand an ascending capacitive line. The medium-fre-quency loop is always inductive, regardless of whetherany of the semicircles is inductive when the dissolutionis not diffusion controlled. Second, no Warburg regimecan be found in the low-frequency domain. The low-frequency part of the spectra never tends to approach aline inclining at 45° to the real axis, even at the lowestfrequency measured. Data obtained for frequencieslower than 10 mHz are usually very scattered, and thisis attributed to natural convection of the electrolytesolution and thermal instability. The character of thespectra is the same for solutions of LiTFSI, LiBETI orboth.

5. Discussion

5.1. Parameter estimation

Delineation of impedance spectra for surface reactioncontrolled dissolution was carried out with the help ofparameter estimation. Circuit 1 (see Fig. 1) or Eq. (35)was chosen as the equivalent for the Faradaic process.The double layer capacitance was taken into account as

Fig. 5. Impedance spectra for aluminum dissolution in 1 MLiTFSI+EMC+EC solutions at 4.25 V. EC concentration in thesolvent: squares, 1%; circles, 3%; triangles, 10%. Frequency incrementbetween data labeled with filled symbols, 1 decade.

Fig. 6. Impedance spectrum for aluminum dissolution. Solution, 0.3M LiTFSI+0.7 M LiBETI+EMC+30% EC; potential, 4.25 V.


a capacitor connected in parallel with the set ofFaradaic elements. Replacement of this capacitor witha CPE did not improve the fits. The impedance of thesolution was accounted for by adding a resistor to theequivalent circuit. Data given below were calculated asan average of parameters estimated for several experi-

mental spectra. Though individual parameters showsome variation, values of the charge transfer resistanceand adsorption–related relaxation times are typicallywithin a 6% interval. The standard deviation of thedouble layer capacitance was about 15%.

The accuracy of complex non-linear least-squaresfitting is demonstrated in Fig. 8 with two examples.

5.2. Surface reaction controlled dissolution with 1 MLiBETI+EMC electrolyte solution

Data obtained for the 1 M LiBETI+EMC systemare summarized in Table 2. Both adsorption time con-stants decrease with electrode potential. The decrease intime constants can be characterized by an exponentialdecay as a function of potential. Therefore, it is as-sumed that the conditions considered for the establish-ment of the relationship of electrochemical rateconstants and adsorption time constant hold for alu-minum dissolution in the media applied. Tafel con-stants calculated with Eqs. (23) and (24) are br=7.48V−1 and bq=8.34 V−1 (tABtB). The Tafel constantfor the rate-determining step was estimated by usingEqs. (25) and (26), resulting in 7.01 and 6.78 V−1,respectively. The difference between the two values isnot significant (less than 4%) and can be attributed tothe approximations applied in the model. It can be seenthat the Tafel constant of the rate-determining step islower than the other Tafel constant. Assuming that therate-determining step is identical in the potential inter-val studied, it can be concluded that the rate-determin-ing step cannot change with an increase in potential.(The hypothesis about which a partial reaction can berate-determining is explored in the discussion of the 1M LiTFSI+EMC+EC system where it is morerelevant.)

Knowledge of both the Tafel constants and the ratioof the reaction rate constants (calculated with Eq. (21))of the non-rate determining steps at a particular poten-tial enables one to determine how much the potentialshould change so that the relative ease of the reactionsinverts. If the Tafel relationship holds for the partialreactions (Eqs. (15)–(17)) and each Tafel constant canbe considered constant, a simple calculation yields thefollowing equation:

lnr(E1)r(E2)

= (br−bq)(E1−E2) (36)

By using br=7.48 V−1 and bq=8.34 V−1, E1=4.6V, r(E2)=1 and r(E1)=20 (calculated from datashown in Table 2), E2=8.08 V, an extremely highpositive potential. This result indicates that the relativehindrance of the partial reactions changes little withpotential. r calculated with the help of data in Table 2also confirms that ratio of the rate constants changes

Fig. 7. Typical impedance spectrum obtained for diffusion-limiteddissolution of aluminum. Solution, 1 M LiTFSI+EMC+3% EC;potential, 4.50 V. Frequency increment between data labeled withfilled symbols, 1 decade.

Fig. 8. Typical fits of EIS data performed with non-linear leastsquares fitting and displayed in the Bode plot. (a) Solution, 1 MLiTFSI+PC+50% EC; potential, 3.78 V; equivalent circuit, Circuit1 completed with double layer capacitance and solution resistance. (b)Solution, 1 M LiTFSI+EMC+30% EC; potential, 4.50 V (same asdata shown in Fig. 7), impedance function, Eq. (35) completed withdouble layer capacitance and solution resistance.


Table 2Result of the parameter estimation for impedance spectra measured with 1 M LiBETI+EMC electrolyte solution

CDL/mF cm−2 RCT/V cm2 tA/s tB/sE/V versus Li � Li+ j/mA cm−2

6.7 2304.70 0.05380.777 0.9544.60 0.433 7.9 404 0.105 2.10

9.0 8924.50 0.240.191 5.06

Table 3Result of the parameter estimation for impedance spectra measured with 1 M LiTFSI+EMC+x% EC solutions at 4.25 V

x CDL/mF cm−2j/mA cm−2 RCT/V cm2 tA/s tB/s

20 1102 0.2810.0 20.70.5265 6400.56 0.1811.0 11.17

0.943.0 169 382 0.180 7.5710.0 4581.63 219 0.180 7.19

only from 21.1 to 17.7 despite the fact that the netcurrent, the charge transfer resistance and the individ-ual time constants differ by a factor of about four.

Calculation of b with the help of Eq. (27) yieldsb�1.45×10−5 C cm−2. The maximum surface con-centration of the intermediates is obtained by dividingb by the Faraday constant. This results in the followingrelationship:

G=bF−1�1.5×10−10 mol cm−2 (37)

Earlier assumptions concerning the maximum surfaceconcentration of the intermediates are based on eitherthe number of metal atoms per surface unit for aparticular crystal plane or somewhat arbitrary assump-tions, setting the surface concentration of the intermedi-ate in the 9×10−11 to 1.36×10−7 mol cm−2 interval[3,9,13]. In general, the surface concentration assumedis inversely proportional to the size of the intermediate.In our case, it appears to be reasonable that the surfaceconcentration of the intermediates is much lower thanthat of the metal atoms at a particular crystal plane.Imide anions are rather bulky and once they are ad-sorbed, they may hinder the adsorption of anotheranion to adjacent metal atoms. Repulsion of the per-fluoroalkyl chains is expected to have the same impact.Therefore it is obvious that expected values of themaximum surface concentration of [Al(BETI)]ADS and[Al(BETI)2]ADS intermediates must be lower than dataobtained for adsorption of smaller anions. (Data pub-lished for metal deposition or dissolution in aqueoussolutions usually account for halogenide or hydroxideanions.)

5.3. Surface reaction controlled dissolution in 1 MLiTFSI+EMC+EC electrolyte solutions

Results of the parameter estimation for the 1 MLiTFSI+EMC+EC system are summarized in Table3. Surface process controlled dissolution can beachieved at 4.25 V in the 0–10% EC concentrationrange. The potential interval in which aluminum disso-lution without any appreciable influence of the aniondiffusion can be observed is too narrow to investigatethe potential dependence of the parameters. Instead,the effect of the solution composition was studied atconstant potential. The current increase for theEMC+30% EC solvent blend beyond the potential ofcurrent onset is too fast, and the dissolution of alu-minum is diffusion limited at any potential at whichrepassivation no longer interferes with the dissolution.

The maximum surface concentration of the adsorbedspecies was obtained in the same way as for the 1 MLiBETI+EMC electrolyte solution. The result for the1 M LiTFSI+EMC+EC system is that G�1.05×10−9 mol cm−2 which is an order of magnitude higherthan for the previous case. Although the value for G

has to be taken as the possible maximum in both cases,the difference indicates that the size of the anion is asubstantial factor in the dissolution kinetics amongother effects discussed earlier [21,22].

The data in Table 3 show that the dependence of tA

on solution composition at 4.25 V is negligible. Accord-ing to Eq. (19) and assuming that b is independent ofthe solution composition, it can be seen that the rateconstant kr is fairly independent of the molar fractionof EC in the solvent. Hence it is concluded that kr

belongs to a partial reaction which does not involve the


solvent, i.e. reaction R1 or R2. In contrast, tB shows astrong dependence on the solution composition. Theonly reaction step which involves the solvent is reactionR3, thus, tB and kq are assigned to reaction R3 (q=3).

The data presented above indicate that EC acceler-ates reaction R3 at 4.25 V. The explanation of thiseffect may be associated with the large dipole momentof EC and thus the change in the Gibbs energy ofsolvation of the product. There are indications in theliterature that EC can complex Li+ ions selectively viathe most negative double-bonded oxygen atom [25].Since the interaction is likely to be mostly Coulombicrather than specific, it can be assumed that the sameselectivity applies for other compact cations as well. Ithas also been demonstrated that solvation of cationstakes place via the ether-like oxygen atom in linearcarbonates such as EMC [25]. The latter has by far thelarger steric effect than complexing with the �C�Ooxygen, hence it plays a minor role in solvation of[Al(TFSI)2]+. This is in good agreement with both dcpolarization data [22] and the result of the presentimpedance analysis.

It cannot be decided unambiguously merely on thebasis of data measured whether tA and kr belong toreaction R1 or R2. Thus, the following hypothesis isconsidered: the reaction step in which aluminum atomsleave the crystal lattice can be either reaction R2 or R3.If aluminum atoms leave the crystal in the course ofreaction R2, it is certainly more hindered than reactionR1. If aluminum atoms leave the crystal lattice whenthe third electron is released, reaction R2 is probablyvery slow because of the steric effects of the anions andthe unfavorable N�Al�N bond angle. Both consider-ations lead to the conclusion that R2 is more likely tobe the rate-determining step than R1 and therefore tA

and kr can be assigned to Reaction R1 (r=1).Another interesting result of the EIS study of the 1

M TFSI+EC+EMC system is that the double layercapacitance at 4.25 V is a linear function of the ECconcentration. The increase of the electrode capacitywith EC concentration is well expected because addi-tion of EC raises the dielectric constant of the solvent.

However, the influence of EC seems to be an order ofmagnitude larger than the expectation. It is interestingto note that similarly high double layer capacitancescan be determined with 1 M LiTFSI+DME or 1 MLiTFSI+TFPC electrolyte solutions (for instance, 300mF cm−2 at 4.5 V for the former), and therefore thisobservation cannot be attributed to experimental error.The large value of the double layer capacitance can beassigned to the coupling between the double layercharging and Faradaic process. This effect can be usu-ally observed for species adsorbing very strongly at theelectrode surface [7]. Since such an anomaly does notoccur for solutions containing LiBETI as a single elec-trolyte, one can conclude that adsorption of TFSI− ismuch stronger than that of BETI−. The latter conclu-sion is supported strongly by the structural data of bothanions [22] and limits of the surface concentration ofthe intermediate discussed above. The difference be-tween the strength of adsorption of the imide anionsmay also account for their different corrosivity towardsaluminum.

5.4. Effect of the sol6ent polarity on the surfacereaction controlled dissolution of aluminum

Medium-rate dissolution for carbonate blends con-taining 1 M LiTFSI could be attained at differentpotentials and therefore direct comparison of the timeconstants calculated is not meaningful. Instead, theratios of the adsorption time constants (r as defined byEq. (21)) have been calculated and summarized inTable 4. As a consequence of the results discussed inthe previous sections, we presume that k2�k3�k1 ineach case studied. While r does depend on the electrodepotential, the dependence originates from the differencein the Tafel constants b1 and b3. This difference is lessthan 1 V−1 for the 1 M LiBETI+EMC solution and isthought to be of the same order of magnitude forsolutions with 1 M LiTFSI. Hence the variation of r isless than 10% in the potential interval of surface pro-cess controlled dissolution. Thus, r is taken as a char-acteristic variable of the solvent applied and could be

Table 4Average dipole moment of some carbonate blends and r calculated from impedance spectra measured at the potential indicated a

Weighted average of dipole moments/D Electrode potential/V versus Li � Li+Solvent r

EMC 73.04.2500.8900.927EMC+1% EC 4.250 64.6

4.2501.001EMC+3% EC 42.139.94.250EMC+10% EC 1.261

TFPC+50% EMC 2.332 4.335 19.8TFPC 3.773 4.320 18.1

11.3PC+50% EC 3.7504.705

a 1 D=3.336×10−30 cm.


Fig. 9. Ratio of the adsorption time constants against the weightedaverage of dipole moments of the solvent components. Circles, exper-imental data. The broken line is shown as a guide to the eye.

The quality of the fit can be improved by adding atleast two more parallel branches to Circuit 1, eachcontaining a resistor and an inductor connected inseries.

The mechanism of aluminum dissolution in the pres-ence of both TFSI− and BETI− anions is more in-volved than in the case of solutions containing a singleelectrolyte. Possible reaction pathways are summarizedin the scheme shown in Fig. 10. As many as fiveintermediates can form and therefore the number oftime constants should also be enhanced accordingly.

An increase in the number of time constants as aresult of electrolyte mixing is strong evidence for themechanism suggested. If solvent molecules were in-volved in the formation of the intermediates, the sameeffect should be observed in the case of solvent mixing.However, solvent mixing always leads to spectra thatexhibit no more than two time constants related tointermediate adsorption, while enlargement of the num-ber of time constants is seen as a result of electrolytemixing.

5.6. Diffusion controlled dissolution of aluminum

Quantitative delineation of impedance spectra forhigh-rate aluminum dissolution was not performed dueto the mathematical complications and the large num-ber of simplifications required. However, fitting Eq.(35) completed with a double layer capacitance and asolution resistance could be carried out with excellentresults, as shown in Fig. 8b. Though charge transferresistance and double layer capacitance can be calcu-lated, the diffusion coefficient of the anion is toodifficult to elucidate. The latter can be obtained byapplying appropriate hydrodynamic conditions, for in-stance, rotating disc electrodes.

6. Summary

1. The impedance for a reaction involving metal disso-lution in three consecutive steps and two adsorbedintermediates has been evaluated. Several equivalentcircuits for the Faradaic admittance (impedance)have been given and it has been established thatadsorption-related circuit elements are doubledcompared to the case of one adsorbate. A symmetriccircuit with a parallel connection of the chargetransfer resistance and two branches standing foradsorption has been proposed to calculate kineticdata. It has been shown that the time constants ofthe equivalent circuit proposed can be correlated tothe rate constants of some partial reaction if the rateconstants are of a different order of magnitude.Cathodic reactions are omitted and the Langmuirisotherm holds for intermediate adsorption.

Fig. 10. Reaction mechanism of aluminum dissolution in the presenceof both TFSI− and BETI− anions.

correlated to the weighted average of the dipole mo-ments of the solvent components. The larger the aver-age dipole moment of the solvent, the smaller is the r

parameter, as shown in Fig. 9. It has been concludedthat r indicates the relative ease of reactions R1 and R3and hence characterizes the ability of the solvent tostabilize the reaction product. The smallest value of r

obtained is 11.3, which confirms that the separation ofthe rate constants is sufficient and the distortion due tothe contribution of neglected terms in the time con-stants is less than 10% for all data presented. It alsoshould be stressed that the empirical correlation foundfor carbonate blends as solvent cannot be extendedautomatically to other solvents. For instance, DME hasa very low dipole moment, 0.016 D (1 D=3.33×10−30 Cm), but r for 1 M LiTFSI+DME electrolytesolution is as low as 4.7. The latter data indicate thatthe chemical environment of the solvent is also ofimportance and the trend established can be applied tosolvents with similar chemical environments, i.e. withidentical functional groups.

5.5. Surface reaction controlled dissolution in thepresence of both LiTFSI and LiBETI

Impedance spectra for surface reaction controlledaluminum dissolution in solutions with mixed elec-trolyte cannot be evaluated with the model accountingfor two intermediates. Fits obtained with Circuit 1(completed with non-Faradaic elements) are rather in-accurate with far larger standard deviation of parame-ters than for solutions containing a single electrolyte.


2. The impedance of the three-step dissolution with theinfluence of the anion diffusion has also been calcu-lated. An equivalent circuit for this process couldnot be constructed. The equation obtained for theFaradaic admittance (impedance) is different fundamentally from those calculated earlier in the sensethat low-frequency behavior cannot be described bya Warburg-type function if the number of partici-pating anions and the charge of the central metalion are different.

3. Experimental spectra have been measured for alu-minum dissolving in several solutions. If the dissolu-tion is slow enough so that the anion concentrationat the electrode surface can be taken constant andone sort of imide anion is only present, the spectracan be analyzed with the help of the modelsuggested.

4. Tafel constants have been estimated for each partialreaction, and it has been concluded that the relativeease of the partial reaction changes very little in thepotential region of the dissolution. The order ofmagnitude of the surface concentration of the inter-mediates has been estimated for dissolution in solu-tions containing LiBETI or LiTFSI. The dataobtained correlate well with the size of the anions.

5. A hypothesis has been made on the relative hin-drance of the partial reactions, and it has beenconcluded that the second step is rate-determining.Data obtained for the 1 M LiTFSI+EMC+ECsolutions prove that the rate of the first step isunaffected by the solvent at a particular potentialand that solvent molecules are involved in the des-orption of the product only.

6. An empirical correlation has been found betweenthe relative rates of the non-rate determining stepsand the polarity of the solvent for LiTFSI solutionswith carbonate blends. This effect has been at-tributed to the change in the solvation state of theproduct as a result of the change in solvent polarity.

7. It has been shown that aluminum dissolution in thepresence of more than one type of anion is moreinvolved than dissolution in the solution of a singleelectrolyte. The number of time constants to betaken into account in the equivalent circuit to fitEIS data must be at least four. A reaction patternhas been suggested which accounts for the forma-tion of intermediates involving different anions.

7. List of symbols

constants to be determined fromA, Bboundary conditions in the expressionof the concentration perturbation

surface concentration of the imidecanions

cP surface concentration of the dissolu-tion product

D Diffusion coefficient of the anionelectrode potential versus Li � Li+E

F Faraday constant, 96 500 C mol−1

imaginary unit (j= (−1)1/2)jtotal current densityjpartial current density of the reactionjmRmdifference of the corresponding currentdjkl

densities (djkl= jk−jl))flux of the anion at the electrodeJsurface

ki rate constant for reaction Ri (in cur-rent/surface area unit)

ki0 rate constant independent of reactant

concentration and electrode potential(in current/surface area unit)constants calculated from the partialKi, Ni, K %i,

N %i derivatives of partial current densitiesin case of surface process controlleddissolution

Li, Mi constants calculated from the partialderivatives of partial current densitiesin case of diffusion controlleddissolution

p, q, r index variables which denote the num-ber of the reaction steps (1, 2 or 3)

s (jv/D)1/2

timetdistance from the electrode surfacexcomplex amplitude of the sinusoidDXperturbation of the quantity Xcharge passing the unit surface area ofb

the electrode when Dui=1G surface concentration of the adsorbed

species (in mole/surface area)angular frequency of the potentialv

perturbationratio of adsorption time constantsr

defined as r=tB/tA

Warburg parameters

partial surface coverage with theu1

[Al(TFSI)]ADS (or [Al(BETI)]ADS)intermediate

u2 partial surface coverage with the[Al(TFSI)2]ADS (or [Al(BETI)2]ADS)intermediate

tA time constant in Circuit 1 (tA=LARA

−1)tB time constant in Circuit 1 (tB=

LBRB−1)


References

[1] M. Sluyters-Rehbach, H. Sluyters, in: A.J. Bard (Ed.), Electro-analytical Chemistry, vol. 4, Marcel Dekker, New York, 1970, p.1.

[2] D.D. Macdonald, Transient Techniques in Electrochemistry,Plenum, New York, 1977.

[3] D.D. Macdonald, M.C.H. McKubre, Impedance measurementsin electrochemical systems, in: J. O’M. Bockris, B.E. Conway,R.E. White (Eds.), Modern Aspects of Electrochemistry, vol. 14,Plenum, New York, 1982, p. 61.

[4] I. Epelboin, C. Gabrielli, M. Keddam, Non-steady state tech-niques, in: E. Yeager, J. O’M. Bockris, B.E. Conway, S.Sarangapani (Eds.), Comprehensive Treatise of Electrochem-istry, vol. 9, Plenum, New York, 1984 Ch. 3.

[5] M. Sluyters-Rehbach, J.H. Sluyters, A.C. Techniques, in: E.Yeager, J. O’M. Bockris, B.E. Conway, S. Sarangapani (Eds.),Comprehensive Treatise of Electrochemistry, vol. 9, Plenum,New York, 1984 Ch. 4.

[6] M. Senda, P. Delahay, J. Phys. Chem. 65 (1961) 1580.[7] B. Timmer, M. Sluyters-Rehbach, J.H. Sluyters, J. Electroanal.

Chem. 18 (1968) 93.[8] H. Gerischer, W. Mehl, Z. Electrochem. 59 (1955) 1049.[9] I. Epelboin, M. Keddam, J. Electrochem. Soc. 117 (1970) 1052.

[10] I. Epelboin, R. Wiart, J. Electrochem. Soc. 118 (1971) 1577.

[11] I. Epelboin, M. Keddam, Electrochim. Acta 17 (1972) 177.[12] R.D. Armstrong, M. Henderson, J. Electroanal. Chem. 39 (1972)

81.[13] I. Epelboin, M. Ksouri, R. Wiart, J. Electrochem. Soc. 122

(1975) 1206.[14] D.A. Harrington, B.E. Conway, Electrochim. Acta 32 (1987)

1703.[15] J.-P. Diard, B. Le Gorrec, C. Montella, J. Electroanal. Chem.

326 (1992) 13.[16] J.E.B. Randles, Discuss. Faraday Soc. 1 (1947) 11.[17] R.D. Armstrong, M.F. Bell, J. Electroanal. Chem. 55 (1974) 201.[18] I. Epelboin, C. Gabrielli, M. Keddam, Electrochim. Acta 20

(1975) 913.[19] G. Salie, J. Electroanal. Chem. 273 (1989) 31.[20] D.D. Macdonald, S. Real, S.I. Smedley, M. Urquidi-Macdonald,

J. Electrochem. Soc. 135 (1998) 2410.[21] L.J. Krause, W. Lamanna, J. Summerfield, M. Engle, G. Korba,

R. Loch, R. Atanasoski, J. Power Sour. 68 (1997) 320.[22] L. Peter, J. Arai, J. Appl. Electrochem. 29 (1999) 1053.[23] A. Webber, J. Electrochem. Soc. 138 (1991) 2586.[24] J. Braithwaite, G. Nagasubramanian, A. Gonzales, S. Lucero,

W. Cieslak, The Electrochemical Society Proceedings Series, PV96–17, Pennington, NJ, 1996, p. 44.

[25] J. Arai, K. Nishimura, Y. Muranaka, Y. Ito, J. Power Sour. 68(1997) 304.

.

Documents

Impedance of a reaction involving two adsorbed intermediates: aluminum dissolution in non-aqueous lithium imide solutions