Transient Response of the System Al/Al,O,/Electrolyte. Part 11. A.c. Impedance Measurements During Steady-State Film Growth

D. G. W. GOAD' AND M. J. DIGNAM Deparlmenr of C11emisrrj8, Uni~ersi ly of Toronlo, Toronlo 181, Onlario

Received October 13, 1971

Experiments are described in which the a.c. impedance of the system AI/A120,/electrolyte (glycol-borate) is measured under conditions of steady-state film growth. Using an equivalent circuit of a resistive and capacitive element in parallel, the results are presented in the form of the a.c. conductance, u,,, and effective dielectric constant, K,,,, of the oxide film as a function of the d.c. current density, I,, and angular frequency o . The results show that u,,/I, and K,,,vary with (o/I,), but within experimental error are independent o f o and I, separately. The dielectric relaxation model, assuming two ion-current-driven relaxation processes and using the constants determined in Part I, reproduces the data almost within experimental scatter. A poorer fit is achieved if only one relaxation process is assumed.

Nous decrivons des expkriences au cours desquelles ['impedance ac du systeme AI/AI20,/i.lectrolyte (glycol- borate) est mesuree dans des conditions de croissance de film a I'etat stationnaire. Au moyen d'un circuit equivalent d'elements de resistance et de capacitance en parralele, nous prtsentons les resultats sous forme de conductance ac, u,, et de constante dielectrique effective, Kc,,, du film d'oxyde en fonction de la densite du courant d.c., I,, et de la frkquence angulaire o . Les resultats demontrent que uac/I0 et Kc,, varient avec (oil,), mais ils sont independants, a I'intkrieur de I'erreur expkrimentale, de o et I,, separkment. Le modele de relaxation diklectrique, qui suppose deux processus de relaxation produits par un courant d'ion, utilisant les constantes dkterminees dans la partie I, reproduit les donnees presque a I'interieur de l'eparpillement experimental. Un ajustement plus grossier est obtenu lorsqu'on suppose un seul processus de relaxation.

[Traduit par le journal] Canadian Journal of Chemistry, 50, 3267 (1972)

Introduction In Part I of this paper (1) galvanostatic

transient data for the system Al/Al,O,/electro- lyte have been obtained and shown to be in accord with the dielectric relaxation model, provided that one postulates the existence of two independent ion-current-driven relaxation processes. Furthermore, the constants obtained from the analysis agree with those calculated earlier from potentiostatic transient data. The purpose of Part I1 of this paper is to report the results of a.c. impedance measurements made on the same system, during the course of steady- state film growth, and to test whether or not the results are in accord with the dielectric relaxa- tion model.

The method employed is to superimpose a small constant amplitude a.c. current on a constant d.c. formation current. The compo- nents of the a.c. part of the electrode potential which are respectively in and out of phase with the a.c. cell current are then measured, from which the effective a.c. conductivity and dielec-

'Present address: ALCAN International Ltd., Research Centre, Kingston, Ontario, Canada.

'Revision received June 23, 1972.

tric constant of the oxide film can be calculated. Experiments of this kind have been carried out for the tantalum and aluminum systems by Winkel et al., using a bridge circuit to measure the a.c. components (2). They did not, however, report data for the out-of-phase component.

In the present study, a lock-in (i.e. phase sensitive) amplifier, rather than an a.c. bridge, was used to measure the a.c. component, thus eliminating any problem which might otherwise arise from the presence of higher harmonics of the fundamental frequency. (Since the cell volt- age and current are not linearly related, higher harmonics will be generated.)

Experimental Apparatus

A schematic diagram of the apparatus used to maintain a constant d.c. current through the cell and superimpose a small a.c. current is shown in Fig. 1. The cell forms the feedback eiement of an operational amplifier (Philbrick model SP2A with 100 V booster), so that the current through thecell is the sum of the currents through the input resistors. One of the resistors is connected to a 15 V regulated, d.c. supply, the other to the output of the lock-in amplifier reference channel, which provides an a.c. signal continu- ously adjustable from 0 to I V and from 1.5 Hz to 10 kHz. Since the aluminum anode of the cell is connected to the

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3268 CANADIAN J O U R N A L OF CHEMISTRY. VOL. 50, 1972

Relcrenrr Electrode D i q i l o l

..----.. J Lock- I" Ampl i f ier

D.C.

A C. Cnannel Channel

FIG. 1. Schematic diagram of the apparatus for mea- suring the a.c. impedance of an electrode system under galvanostatic conditions.

I I I I I I

\ - Hiqh Impedance Foriawer

amplifier input, it is a virtual ground; thus the anode potential 0s. the reference electrode is just the negative of the reference electrode potential (measured against ground). The reference electrode potential is fed through a high impedance voltage follower to both a d.c. digital voltmeter and the signal channel input of the lock-in amplifier. Thus both the d.c. and a.c. components of the electrode potential are available for recording.

The anodizing cell and electrolyte were the same in all important respects as those described in Part I (I), except that temperature control was maintained by water circula- tion (the cell was of double-walled construction) rather than by an air thermostat.

370-

Results For all the experimental runs, the d.c. elec-

trode potential, V,, and the two a.c. compo- nents Vin and V,,,, were found to increase linearly with the time over the range of 20 to 100 V for V,, a result expected if space charge within the film is either negligible, or confined to zones close to the oxide boundaries (cf: Part 1). For values of V, less than 20 V, the time derivatives of the potentials vary by a few per- cent, a behavior which has been noted before in the case of d Vo/dt (3). Over the region in which steady-state conditions obtain ( i .e . 100 V > V, > 20 V), the data for a given run can therefore be represented by the values of the d.c. cell current, the amplitude and frequency of the a.c. cell current, and the time derivatives of the potentials. The derivatives were determined by linear regression analyses of the primary data. The steady-state formation current density, I,, and corresponding field strength, E,, were cal- culated from d V,/dr as described in Part I. Having obtained I, and knowing the total d.c. cell current, the electrode area, and hence the

I og [(w / I,) / cm2 C-'1 FIG. 2. Graph of (J,,/Io) = (Ii,/E,Io) t's. log (w/Io).

e, 1, - 4 x A/cm2, w varying;O, I, varying, w = 2a x 1.5 s-'. The solid line is calculated from 1121, the broken line from [lo].

amplitude of the a.c. current density, I,, was calculated. Finally, since for steady-state condi- tions there is no d.c. component to the charging current (since Eo and I, are constant) then I, is the d.c. ion current density. Thus from Faraday's law, E, = (d V,/dt)/RI,, Ein = (d Vin/dt)/RIo and E,,,, = (d V,,,/dt)/RI, where Ein and E,,, are the differential field amplitudes corresponding to Vin and V,,,, and R = Wc/Fp, with We being the electrochemical equivalent weight of the oxide, p its density, and F the Faraday. Thus Ein and E,,, may be calculated from Ein = E,(d Vin/dt)/ (d V,/dt) and E,,, = E,(d V,,,/d()/(d V,/dt). The results for a single run may therefore be repre- sented by the values for the six quantities, I,, E,, I,, Ein, E,,,, and o for that run. A further reduction in the data can be achieved by defining a parallel equivalent a.c. conductivity, o;,,, and dielectric constant, Kc,,, according to (see Appendix)

where E, is the permittivity of free space. If the anodic oxide film behaved simply as a lossy dielectric, o,, and Kc,, would be the conduc- tivity and dielectric constant of the film (see

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GOAD AND DIGNAM: ON AI/Al,O,/ELECTROLYTE SYSTEM. PART 11 3269

ciated with the kinetics of ionic conduction will contribute to the current component out of phase with the field, and hence to Kc,,. Thus the behavior of both o, , and Kc,, provides informa- tion on the kinetics of ionic conduction.

In the following sections, expressions for K,,, and o;,, are derived on the basis of the dielectric relaxation model, outlined in Part I, and compared with the experimental results.

FIG. 3. Graph of Kc,, - I,u,/E,nwc~ us. log (w/lo). 2. tion processes (c j . Part I ) . Thus from [5] to [14] 4 x lo-" A/cm2, w varying; 0, I, varying, w = 2n x 1.5 s-'. The solid line is calculated from [13], the broken line from of Part 1,

o 1 I I I I I

Appendix). It should be emphasized, however, that the calculation of o,, and Kc,, from the primary data does not involve assumptions con- cerning the mechanism of anodic oxidation.

In Figs. 2 and 3 the results for all the runs are plotted in the form oac/Io and K,,, cs. log (oil,). A practical lower limit exists on the ratio w : I , which can be used, since for increasing I,, the total time available for measurement decreases, and each measurement takes a minimum of several periods, l / o , to perform. The somewhat larger scatter in the results for small values of w/I , is due to this difficulty.

Reluxatioiz Model

Discussion The results, plotted in the form o,,/I, and

K,,, cs. log (o / I , ) fall on smooth curves within experimental scatter, even though different values for I, have been used. Thus o,lc/Io and K,,, depend on w/I,, but appear to be inde- pendent of w and I, separately, within experi- mental error. Such behavior might have been anticipated from the form of the galvanostatic and potentiostatic transients (cj: Part I ) which were shown not to involve time as a parameter. Thus all three quantities oac/Io, Kc,,, and (oll,), are dimensionally independent of time.

The large negative value which K,,, assumes when w/I , is small appears at first glance to be rather surprising. However, K,,, cannot be interpreted simply as the dielectric constant of the oxide film since relaxation processes asso-

4.0 4.4 4.8 5.2 5.6 60 We employ at the outset the equations which l o g [(w / I,) / cm2 C-'1 involve two ion current driven dielectric relaxa-

PI i = oc exp BE,

[51 P=t.,l,E+ P, + P,

[6] dP,/dt = B,~(E,x,E - P,), k = 2,3

where i, ic, and I are the ion, charging, and total current densities, respectively; E and Ed are the fields in the oxide and oxide/electrolyte double layer, respectively; P is the polarization of the oxide film which is divided into components E,x,E, P,, and P, with corresponding electric susceptibilities I , , I,, and 1,; c, is the per- mittivity of free space; a, B, B,, and B, are constants associated with the kinetics of ion transport (a and b) and dielectric polarization ( B , and B,); and t is time. Next we expand E, i, I, P, P,, P,, and Ed in terms of their d.c. and a.c. components, i.e.,

E = E, + E,, exp ( j o t ) + harmonics

181 i = I, + (iin + jiOul) exp U o r ) + harmonics

I = I, + ( I in + jl,,,) exp Gal) + harmonics

P = Po + (Pin + JP,,,) exp ( j o t ) + harmonics

etc.

where , j = f l and the subscripts in and out refer to the components which are in and out of phase, respectively, with the a.c. field of ampli- tude E,. Finally, on making the substitutions [8] into [3] to 171, and equating corresponding com- ponents, a total of 18 equations are obtained,

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3270 CANADIAN JOURNAL OF CHEMISTRY. VOL. 50, 1972

six in each of the d.c., real, and imaginary a.c. components respectively. Elimination of the d.c. and a.c. components of i, P, P2, P,, and Ed leads to the following three independent equations

where

a a c = 'in 1 E w

Kerf = IoutIEw~~w

(they are also given by 111 and [2], see Appendix),

A = ( Q / B , B , I ~ ) ~ ( B ~ + B: + w2/1i)

and

f i t = (K1 /&)fi

Equation 9 is simply the relationship between the steady-state current density and field strength (see Part I), while [lo] and [ l l ] are the desired expressions for a,, and Kerf when two ion- current-driven dielectric relaxation processes are considered. The equations appropriate for a single such relaxation process are obtained from [lo] and [I :I] on setting x3 = 0, and B2 = B, and are given by

In Table 1, the behavior of a,,/I0 and Kc,, in the high and low frequency limits is given. For both cases Kc,, = K, in the high frequency limit, and a,,/Io = fi, in the low frequency limit, as expected.

For both forms of the dielectric relaxation model, the experimental quantities a,,/I, and Kc,, are predicted to be functions of (w/lo), but not of w or I, independently. (This conclusion is based on the assumption that fis and fi, are independent of Eo, which is not strictly true as noted in Part I, but is a good enough approxima- tion for the present application.) The solid lines in Figs. 2 and 3 were computed from eqs. 12 and 13, by setting K, = 8.5, fit = 129 A/v, and fis/fit = 3.1 (4), then adjusting B to satisfy a weighted least squares criterion for a,,/l0 and Kerf, the weighting factor being taken to be proportional to the square of the reciprocal of the standard deviation of the individual mea- surements of a,,/lo and Kerf. The value for B found in this way is 1.75 x lo4 cm2/C. It is evident that the fit is only fair.

The broken lines of Figs. 2 and 3 were com- puted from [lo] and [ l l ] using the values for Ks/Kl, x2/K1, B2, and B3 determined in Part I of this paper, and adjusting K, and fit to satisfy the same least squares criterion as above. The fit is appreciably better, being within experimental scatter except perhaps for a,,/I0 at low (w/Io) where the experimental measurements are the most difficult to obtain. Furthermore, the values deduced, K, = 10 and 13, = 134 A/v, agree within experimental error with independent measure- ments of these parameters (4). Thus from Fig.

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GOAD A N D DIGNAM: ON AI/AI,O,/ELECTROLYTE SYSTEM. PART I 1 327 1

TABLE I. Expressions for u,,/lo and Kc,, in the high and low frequency limits

Single relaxation Two independent relaxation process processes

Limit (Kc,,) (Ps - PI) + :K,}+ (Ps-lA) x~IA -+- --- + IWIIO I - o COB B E ~ K , (ii iJ 'Ks'*

P I + :EOX*B)* P, + ( ~ o ( ~ z B z + %3B3)}*

Limit Kc,, (cu l lo ) - m

K, K 1

*The terms in parentheses make a negligible contribution Tor the AI/Al,O,/electrolyte system

10, ref. 4, p, has a value of 134 A/v for I, - A/cmZ, which lies approximately mid way

within the range of I. values covered in Figs. 2 and 3 (4 x to 1.5 x A/cmZ). The value of K , , as distinct from the ratios Ks/Kl and xZ/Kl, affects the fit significantly only in the case of Kef,, which approaches K, for large values of (oil,) ( c f f : Table 1). It is apparent from Fig. 3 that a value for K, of 10 is, within experi- mental accuracy, indistinguishable from the value 8.5 (4).

In summary, the dielectric relaxation model, employing two independent relaxation pro- cesses, accounts quantitatively for a.c. imped- ance measurements, as well as for potentiostatic and galvanostatic transient measurements, per- formed on the system Al/A1z03/electrolyte, a single set of constants reproducing the data from all three experiments.

Appendix Defining Iin and I,,, for any circuit element as

the components of the a.c. current density, of amplitude I,, which are respectively in and out of phase with the ax . field of amplitude E,; and Ein and E,,, as the components of a.c. field which are respectively in and out of phase with the a.c.

current density, then

[All Ein = E, cos 6, E,,, = E, sin 6

[A21 Iin = I, cos 6, I,,, = -I, sin 6

where 6 is the phase angle between the a.c. current and field. Solving for Iin/E, and IoUl/E, gives

[A31 Iin/E, = a,, = I,E,,/(E~', + Ek,)

[A41 IoullEm=oc,Keff= -I,Eoull(EL+E&t)

where a,, and Kerf are the parallel equivalent conductivity and dielectric constant for the circuit element. Equations A3 and A4 have been used in writing down [l], [2], [lo], and [I I].

The authors are grateful to the National Research Council of Canada for supporting this research and for a Scholarship (D.G.W.G.). One of us (D.G.W.G.) also wishes to thank Canadian Industries Ltd., for financial assistance in the form of a Fellowship.

1. D. G. W. GOAD and M. J. DIGNAM. Can. J . Chem. This issue.

2. P. WINKEL, C. A. PISTORIUS, and W. CH. VAN GEEL. Phillips Res. Rept. 13, 277 (1958).

3. M. J. DIGNAM and P. J. RYAN. Can. J. Chem. 41, 3108 (1963).

4. M. J. DIGNAM and P. J. RYAN. Can. J. Chem. 46, 549 (1968).

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