impedance measurements during anodization of aluminum

Impedance Measurements during Anodization of Aluminum H. J. de Wit, C. Wijenberg, and C. Crevecoeur

Philips Research Laboratories, Eindhoven, The Netherlands

ABSTRACT

The impedance of a growing anodic oxide layer is induct ive in a rcertain f requency range. This effect is related to the occurrence of t ransients after changes in current densi ty or applied electric field. Measurements on a lu- m i n u m samples in the frequency range 5 mHz-10 kHz and the cur ren t density range 3 X 10-6-5 • ]~O -3 A /cm 2 and in various electrolytes are presented and compared with a theory in which the relaxat ion effect is ascribed to a bui ldup of surface charge at the metal-oxide interface.

I t has long been recognized that the s tudy of non- s teady-state situations dur ing anodization is helpful to an unders tanding of the mechanism of anodization. A very early measurement was made by Bauma nn in 1939 (1) who studied the f requency dependence of the impedance dur ing porous oxide formation on a luminum. The techniques used most commonly are: a sudden change of the applied field, which is kept constant af terward (potentiostatic t r ans ien t ) ; a sudden change of the current densi ty which is kept con- s tant af terward (galvanostatic t r ans ien t ) ; and the superposit ion of a small a-c signal on the d-c signal.

Potentiostat ic t ransients have been measured on t an ta lum by Vermilyea (2), Young (3), and Taylor and Dignam (4), and on a luminum by Vermilyea (2) and Dignam and Ryan (5). Galvanostat ic t ransients have been measured on t an ta lum by Dewald (6), Vermilyea (2), and Young (3) and on a luminum by Goad and Dignam (7). A-C measurements have been performed on a l u m i n u m by Baumann (1), Winkel, Pistorius, and v a n Geel (8), and Goad and Dignam (9), and on t an ta lum by Taylor and Dignam (4). In all these exper iments it was shown that the system after a change does not immediate ly assume the new steady-state values for current or overpotential , but that it relaxes to the new steady state in a cer- ta in time. This re laxat ion t ime is inversely propor- t ional to the d-c ionic current density.

Several theories have been put forward to explain this relaxation. Bean, Fisher, and Vermilyea (10) ascribe it to a sluggishness in the response of the n u m b e r of charge carriers, assumed to be metal in ter - stitials, to the var ia t ion of the applied field. Their approach has come in for criticism, on the one hand, from Young (3), who showed that it does not satisfactorily describe the acceleration of the ionic current which is found when a potential step is applied, and, on the other hand, from Dignam (11), who challenged it on the grounds of free path considerations. Dignam and co-workers (4, 7, 11) consider a relaxat ion of the dielectric polarization of the oxide to be respon- sible for the effects. Their approach leads to a set of equations which satisfactorily accounts for the experiments and which is used as a s tar t ing point for the theoretical considerations of the present paper. The mechanism, however, by which these polarization changes occur is unclear.

The present work presents a cont inuat ion and ex- tension of the cited work. It gives the results of impedance measurements dur ing anodization of a lu- m i n u m in much wider f requency and current ranges than used hitherto. A theory is presented in which the re laxat ion phenomena are ascribed to t ime var ia- tions of a surface charge near the metal-oxide in te r - face.

Firs t we describe how the experiments are performed, how the specimens are prepared, and how the results are analyzed and presented. Then the

Key words: amorphous, capacitance, oxidation, stoichiometry.

exper imental results are given and compared wi th those of other workers. In the theoretical par t exist ing theories are discussed and finally the new model is presented.

Experimental GeneraL--In an exper iment an a l u m i n u m specimen

is placed in a vessel containing electrolyte and a p la t inum counterelectrode. All exper iments have been carried out at room temperature. A l inear ly increasing voltage, on which a sinusoidal signal of small ampl i tude and constant f requency is superimposed, is applied to this cell in series with a measur ing resistance. The increasing d-c component causes anodie growth, and the a-c component is used to measure the i m p e d a n c e of the growing anodic oxide layer. The a-c signals are measured with a Solar t ron 1170 Frequency Response Analyser which determines the real and imaginary parts of the ratio of the impedances of the cell and the measur ing resistor. The d-c voltage and cur ren t signals are fed into analog-digi tal con- verters which, together with the Solar t ron apparatus, are coupled to a Philips PC2200 computer. The computer starts a measurement at regular t ime intervals and after its completion stores the values of the real and imaginary parts of the impedance ratio as well as the d-c current and voltage signals and the f requency of the signal. After a run the complete data of the complex impedance as a funct ion of cell voltage and layer thickness are punched on paper tape for fur ther evaluation.

Specimens.--Samples were punched from a luminum foil 0.3 m m thick, supplied by Merck, pur i ty 99.95%, the main impur i ty being Si. The samples were spoons with a circular area of 15 mm diam and a handle 5 mm wide and 2 cm long. The anodized surface used was about 3.7 cm 2. Before anodization the a luminum was cleaned, electropolished, annealed at 600~ electropolished once more, and etched in a HF-H2SO4 solution in water, all as described in Ref. (12).

Some anodizations were performed after hydra t ing the specimens for 4 min in boil ing deionized water in a quartz vessel.

Analysis of the measurements.DA par t icular measurement r un gives the complex impedance as a funct ion of bias voltage at a certain frequency. We represent the impedance of the cell formal ly by Z -: Rs ~- 1/Cs~i, where Rs is the series resistance, Cs the series capacitance, ~ the angular frequency, and i ---- ~/--1. R~ and Cs are dependent on frequency. Figure 1 gives an example obtained in ammon ium pentaborate (apb)-glycol at 10 Hz. Anodizat ion in this electrolyte yields layers of a homogeneous amorphous anodic oxide, the properties of which do not depend on the thickness of the layer. This is reflected in the l inear dependence of the impedance on cell voltage. For this electrolyte this behavior is found at all frequencies of interest. To obtain the impedance as a funct ion of frequency, the values obtained in dif-

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780 J. Electrochem. Soc.: ELECTROC H EMI C A L SCIENCE AND TECHNOLOGY May I979

5! Rs

l/Cs~ (10h_0.) 4

3

2

1

0

0 e

0

I I I

j =0"075mA/cm 2 f=10Hz

O

0

0

o O

O O

R s ~ o

O O

O o I/Cs%

O Q e

0 6 e

0 I) l e Q

0 �9

0 �9

e o

I I I

50 100 150 200 )Vdc(Vl

Fig. 1. Absolute values of the real and imaginary part of the impedance of a growing oxide layer on aluminum as a function of the cell voltage. Surface area, 3.6 cm2; electrolyte, 17g ammonium pentaborate in 100 ml ethylene glycol.

ferent runs at a par t icular bias voltage are compared with each other. The results of such a series of measurements are presented in Fig. 2 where the negative of the imaginary part of the impedance is plotted vs. the real part with the frequency as param- eter as measured at Vd~ ---- 100V on specimens being anodized in apb-glycol. The plot can be divided into three regions, in regions I and III 1/Cs~ > 0, the system behaves as a capacitance, in region II 1/Cs~ < 0, the behavior is inductive, the system has a certain amount of inertia. The behavior in region II is the main interest of the present paper , but we first deal with I and IIL

In region I, at high frequencies, the "ordinary" capacitance of the anodic layer determined by its dielectric permit t iv i ty is measured. Since ionic current is flowing dur ing anodization, this capacitor is leaky and the impedance can be represented by a capacitance Cp parallel with a resistance R,. If the impedance of such a circuit is calculated we obtain

Rp 1 Rp2Cp~ Rs = '; - -

1 + Rp2Cp2w 2 Csw 1 --~ Rp2Cp2w 2

If 1/Cs~ is plotted vs. Rs a semicircle is obtained with Rs ---- Rp/2 on the abscissa as midpoint. Quali ta- t ively this describes the behavior found in region I.

&= p~

q

1 1/Cst~

(104n)

T

_ - - o ~

\ 01o

I \ \ LIII \

~ ..... \ o~ 21 /1311 ~;)0'2

\ O0.s 1 0 / "~ ~ _ 1 / Vdc = 100V

l'f J = 0.075 mAicm 2

Fig. 2. Impedance of growing aluminum oxide layers. The experimental conditions are the same as Fig. |. The numbers at the experimental points indicate the frequency in Hz.

The value of Rp is discussed in connection with region II, The electrolyte resistance in the cell is of the order of 20012, which is negligible compared with the impedances measured in the case presented in Fig. 2. At higher currents and lower bias voltages the measured impedances are corrected for the series resistance of the electrolyte.

In region III the frequency is so low that the thickness of the layer follows the voltage fluctuations to a certain extent. A voltage increase hV causes a thickness increase hL -- arAV, where ar is the anodizing ratio (~.13 A/V) . To form an amount ~L of anodic oxide, a charge ~Q - AL/aX per un i t area is needed, where ~ is the current efficiency (~1 for apb-glycol) and ~. is the ratio of the equivalent weight of a luminum oxide to the product of Faraday number and density of the oxide (~5.7 X 10 -5 cm3/C).

The quotient hQ/hV -- ar/~h is a faradaic capacitance per un i t area of ~2.3 X 10 -8 F / c m 2 which, in series with the capacitance of the counterelectrode, determines the capacitive behavior in region III. The measurements cannot be used to obtain this faradaic capacitance quanti tat ively. The measur ing method is based on the assumption that dur ing a few periods of the applied frequency the thickness of the layer remains approx'imately constant. In regions I and II this assumption is fulfilled, but at the very low frequencies of region III it is no longer correct.

Region / / . - -P lo t t ing the imaginary vs. the real par t of the impedance, as was done in Fig. 2, is not very suitable for presenting the results in region II. Be- cause the d-c current density range investigated is 10-6-10 -2 A/cm~ and the voltage range is 10-200V, d-c resistances between 103 and l0 s 12 cm 2 can occur. Plots obtained at different d-c resistances cannot be compared directly with each other, whereas the l inear behavior shown in Fig. 1, which is also found in region II, suggests that the unde~rlying process is the same to a large extent. To overcome this difficulty the results are normalized, which is performed by dividing the measured complex a-c admit tance by the d-c conductance. The dimensionless complex quan- t i ty obtained is called BF. The reason for this will become clear from an examinat ion of Fig. 3. This gives schematically the relat ion between ionic current density j and applied electric field F dur ing anodization. In practice this is a highly nonl inear dependence, even more than indicated in the figure. I t can be represented with an exponential relat ion j -- jo exp (BF), where Jo and B are constants. A constant field causes a constant current density. Because ionic flow leads to oxide formation the thickness and the voltage

l T / h, iJ(If) i j(hf}

/

/ /

- - ' ~ F

j = jo exp (BF)

&j = joexp(BF}-BAF

= jBAF = j lF - (BF)AF

I~'ac/C'dc = BF 1

Fig. 3. Schematic dependence of the ionic current density j on the applied electric field F. A fluctuation AF superimposed on F causes a fluctuation ~j of j. Division of the a-c admittance ~ac = ~j/AF by ~dc ~ j/F results in a dimensionless quantity, which is equal to BF if the re[Qtion between F and i is given by i = Io exp BF,



Vol. 126, No. 5 I M P E D A N C E M E A S U R E M E N T S 781

increase l inear ly wi th time. When an a-c component AF with ve ry low f requency is superimposed on the d-c field F the cur ren t density j follows the d-c dependence and the response a j ( l f ) is measured. When the ratio be tween ~ac = Aj/AF and ~ c " - j /F is calculated for the relat ion j -- Jo exp (BF) it is found that ~aJ~ac -- BF. Figure 2 shows that at h igher f requencies the impedance increases. In Fig. 3 this means a decrease of Aj and also of O'ac/O'dc. We define BF -- ~ac/r162 at all frequencies. This means that f rom now on BF is considered to be an exper imenta l quan- t i ty which at ve ry low frequencies becomes equal to the a rgument of the exponent ia l re la t ion be tween j and F, but which at h igher frequencies becomes a complex quanti ty, the absolute value of which decreases.

Exper imenta l results in terms of BF are shown in Fig. 4, which gives the values obtained in three series of measurements , all in apb-glycol. The filled circles are the same results as g iven in Fig. 2 obtained at a current densi ty of 75 # A / c m 2. The region below the real axis corresponds to par t II in Fig. 2. The open circles were obtained at a current density of 700 ~A/cm~. The crosses are data taken f rom the work of Goad and Dignam (9) replot ted in the way we use here.

It is clear that the data obtained in different s i tua- tions coincide in this plot. For points lying close together the data obtained at a current density a factor of ten h igher were measured at a f requency which was also a factor of ten higher. All three series of points show in region 1I an increase of Re (BF) wi th decreasing frequency, wi th a simultaneous emergence of a negative, induct ive Im (BF).

As discussed with reference to Fig. 3 the low frequency l imit of Re (BF) can also be found from the d-c current- f ie ld dependence. In Fig. 4 we also give values calculated f rom the data of Bernard and Cook (13) for j = 75 ~A/cm 2 and j = 700 ~A/cm 2. The points lie close together and are sl ightly h igher than the values obtained here f rom a-c measurements . This could be due to the fact that we use here an ammonium pentabora te concentrat ion which is different f rom that used by Bernard and Cook (17% vs. 30%). Goad and Dignam also used 30% apb and this could explain why their ]Im BF] is somewhat la rger than ours.

Quant i ta t ive ly the induct ive phenomenon can be character ized wi th three quantit ies: the high f requency l imit of Re (BF) , which is designated BiF; the difference be tween the high f requency and low

Table I

Mark in F~g. 5 Electrolyte BiF BrF

O

[]

~7

ZX

17g ammonium pentaborate/100 12 25 ml ethylene glycol

25 g/t ammonmm pentaborate 12 22 in water

14.7 g/l adipic acid and 7.0 12 18 g/1 ammonium pentaborate in water

Capacitor electrolyte consisting 13 21 of d,methylacetamide, picrie acid, ethylene glycol, and additives

Capacitor electrolyte consisting 13 22 of ethylene glycol, boric acid, acetic acid, and additives

f requency limits of R e ( B F ) , which is designated BrF; and the f requency ]r at which the t ransi t ion f rom (Bi H- Br)F to BiF occurs. For the la t ter quant i ty we use the ~requency at which I m ( B F ) / R e ( B F ) has a minimum. (See the insert of Fig. 4.)

Experimental results.--The impedance measurements were carr ied out in a var ie ty of electrolytes including aqueous and organic solutions. The current density ranged f rom 3 X 10 -6 to 5 X 10 -3 A / c m 2. The r e - s u l t s , all obtained at room temperature , are presented in Table I and Fig. 5. T a b l e I gives the values for B;F and BrF for different electrolytes de termined in the current range 0.1-1 m A / c m 2. It shows that, whereas the value of BiF is essential ly the same for all e lectro- lytes, the values of BrF vary f rom 25 for apb-glycol to 18 for adipic acid in water. F igure 5 gives the values of the re laxat ion t ime �9 defined as z -- 1/(2~fr). TWO points are obvious: The re laxat ion t ime and current density are str ict ly inversely proport ional in the whole range, and all the points obtained on different electrolytes lie on the same curve. For the product j~ we obtain j'~ _-- (3.8 +_ 0.4) X 10 -5 C /cm 2. The measurement indicated wi th a 4- was determined f rom the exper imenta l points of Goad and Dignam, presented in our Fig. 4. Their point lies close to our results. The black squares in Fig. 5 were obtained on specimens held in boiling water before anodization. Figure 6 gives the d-c current voltage curves of these exper iments compared with the curves for nonboiled specimens in the same electrolyte. Af te r the boiling t rea tment the current is lower in the whole region, which is one of the reasons why this t rea tment is used industrially. The current decreases f rom 0.6 m A / c m 2 at 50V to 0.4 m A / c m 2 at 180V. It

+10 Im(BF) .2o

-10

eO.O05

o0.01 elO

o52

0 101 s~ 201 301 .o.o240..= o Re(BF) 0

2+~+,o5..0 5 o2 ,O5 ~

~2 oi o0.1 00.5 �9 +

%

Fig. 4. Imaginary part of BF vs. real part of BF for layers anodized in apb-glycol at room temperature. O Present work, current density 700 /~A/cm2; �9 present work, current density 75 /~A/cm2; 4- points calculated from data given by Goad and Dignam (9); �9 [ ] points calculated from the steady-state values for/o and B given by Bernard and Cook (13) for j = 75 /~A/cm 2 and j ---- 700 /~A/cm 2, respectively. The insert shows the definition of BIF, the instantaneous part of BF, BrF the relaxing part of BF; and fr the relaxation frequency.

100

"C=ll2Tcf r

1 \ 0.1 Goad~-~

Dignam

0.01 Bouman~ / "

0.001 0.01 0.1 1 10 j (mA/cm 2 )

Fig. 5. Relaxation time vs. d-c current density. Markings: see Table I. 4- Point calculated from data given by Goad and Dignam (9); X point calculated from data given by Baumann (1) (porous oxide); �9 points measured during anadization in the adipic acid- apb-water solution after hydration of the aluminum specimens.



782 J. Electrochem. Soc.: E L E C T R O C H E M I C A L SCIE N CE A N D T E C H N O L O G Y May 19 79

J

(mA/cm 2

0.!

0 0

AI

o ~ ~o,~ AC~o~c~

~oz~o hydrated AI %o~ ~ %o

adip. +apb 25~ 0.34 V/s

50 100 150 200 ~V{V)

Fig. 6. Current density vs. cell voltage during anodization in the adipic acid-apb-water solution without ( + • and with (Z~ 0 ) previous hydration.

is known tha t this decrease is connected wi th the deve lopment of c rys ta l l ine oxide inside the anodic layer . The r e l axa t ion t imes indica ted wi th the b lack squares were obta ined at 50 and 180V. I t appears tha t the re laxa t ion mechanism is independen t of the changes occurr ing dur ing this mode of anodization.

Comparison with experimental results oJ other workers.--A r e m a r k a b l e achievement was Baumann ' s measurement in 1939 (1) of the impedance dur ing porous anodizat ion of a luminum. He showed tha t an induct ive behavior occurs and tha t the f requency at which tan 8 has a negat ive m a x i m u m increases wi th cur ren t density. For a cur ren t densi ty of 2 m A / c m 2 he found an 1r of 14 Hz giving a j X �9 of 2.26 • 10 -5 C /cm ~, which is less than a factor 2 lower than we found in the presen t work. In a few p re l im ina ry exper iments on porous anodizat ion of a luminum we found a j~ produc t equal to tha t obta ined by Baumann. T h i s is poss ibly connected wi th cur ren t inhomogenei t ies at pore bases.

Winkel , Pistorius, and van Geel (8) repor ted in 1958 tha t the rea l pa r t of the impedance of a growing a luminum ba r r i e r oxide layer , anodized in an aqueous solut ion of borax and boric acid, increases wi th increasing f requency in a cer ta in range. The changes of the rea l and imag ina ry components of the impedance have been measured in apb-g lyco l by Goad and Dig- nam (9), whose measurements have been incorpor- a ted in Fig. 4 and 5. Impedance measurements dur ing anodizat ion have also been per formed o n t an ta lum by Winkel et al. (8) and by Tay lo r and Dignam (4) wi th qua l i t a t ive ly the same results.

Closely re la ted to the impedance effects are the effects occurr ing dur ing galvanosta t ic and poten t io- s tat ic t rans ient measurements (2-7). Al l these mea - surements show the presence of a r e l axa t ion effect, the r e l axa t ion t ime of which is inverse ly propor t iona l to the d-c ionic cur ren t density. For a discussion on the exper imen ta l connection be tween t rans ient and impedance measurements we refer to the papers by Goad and Dignam (7, 9), Taylor and Dignam (4), and to the rev iew by Dignam (11).

Theory The exper imen ta l resul ts on impedance and t rans ient

measurements t r ea ted in the previous section show tha t the anodizat ion process as a whole does not fol low changes in the appl ied field ins tantaneously . The cur ren t response can be d iv ided into two components, one fol lowing the change in the field d i rec t ly and the o ther a r e laxa t ion component. Thus we no longer have only two variables , to ta l cur ren t and field, but a th i rd var iab le enters into the pic ture and an explana t ion of the phenomena resolves i tself into re la t ing this va r iab le wi th a pa r t i cu l a r phys ica l mechanism. This p rob lem is discussed in the present section.

Jus t as the expe r imen ta l work presented here is a cont inuat ion of the work done by others, so too the theore t ica l work rel ies to a large ex ten t on previous work.

As ment ioned ear l ie r the two theories on these effects tha t have a t t rac ted most a t ten t ion are the "high field F renke l theory" pu t fo rward by Bean, Fisher , and Vermi lyea (10) and Dignam's "dielectr ic polar iza t ion theory" (11). Both theories have been ex tens ive ly discussed in the l i t e ra tu re [see Dignam (11)] wi th notable expe r imen ta l and theore t ica l con- t r ibut ions by Young and co -worke r s (3, 14, 15). Young (15) has also proposed a theory cal led the "channel model ," in which i t is assumed tha t the mobile ions move through channels in the r a the r open s t ruc ture of the anodic oxides. The re laxa t ion effects a re ascr ibed to b locking and unblocking of these channels. This theory too can account qua l i t a t ive ly for the exper i - ments, bu t i t has not been worked out in mathemat ica l detail .

The theore t ica l discussion is p resented in the fol - lowing way. F i r s t we show how a set of equat ions put fo rward by Dignam (11) to descr ibe the r e l axa t ion effect can account for the observed phenomena. Then we discuss the phys ica l mechanism which, according to Dignam, under l ies this set of equations. F ina l ly we propose a different model which leads t o the same equat ions and thus expla ins the phenomena.

Dignam's equations.--In its most s imple form, Dig- nam's equat ions read l ike

J = Jo exp (BiF + R) [1]

dR -- j /q . (BrF -- R) [2]

dt

Here j is the total cur ren t measured and F the appl ied e lect r ic field. The th i rd va r i ab le is in t roduced by spl i t t ing the a rgument of the exponent ia l function, at this s tage quite formal ly , into two terms, the t e rm BiF where Bi is a constant, which fol lows the field d i rec t ly and the t e rm R. The t ime dependence of R follows the second equation, in which Br is a constant and q has the dimension of a surface charge density. Now we consider the s t eady-s ta te and a -c behavior pred ic ted by this set of equations. For the cases of t ransients see Dignam (11).

Steady state.--With dR/dr : 0 in Eq. [2], R = BrF is obtained, and inser t ion in [1] gives j : Jo exp (Bi -~ B r ) . F , which is the exponent ia l re la t ion found expe r imen ta l ly to a first approx imat ion be tween cur - ren t and appl ied field.

A-C behavior.--To obta in the a -c proper t ies of the sys tem of equations, we assume tha t the s t eady-s ta te field Fs has super imposed upon it an a-c component AFei~t: F : Fs + AFe u~ where AF < < F. The resu l t - ing R and j a re wr i t t en as: R -- Rs + are ~ and J = Js + AJe ~t. The a-c components of R and j may or may not be in phase wi th AF, AR and aj therefore contain phase factors and are complex quanti t ies.

Inser t ing R, j, and F in Eq. [2] gives

i~ARei~t = (is -t- Aj" e ~ t ) / q

�9 {Br(F, + hFe i~t) -- (Rs + ARe u~ }

Equat ing the terms independent of t ime yields BrFs = Rs the s t eady-s t a t e re la t ion be tween R and F. If this is inser ted i t is found tha t the factor a j then only occurs mul t ip l ied by a F or hR. These second-order quant i t ies are lef t out of account. Af te r division by e ~ t and solving for AR we obtain: AR = (js/q)BrAF/ (Js/q + i~).

Inser t ing R, j, and F in Eq. [1] and proceeding in the same way, af ter e l iminat ion of the s t eady-s ta te solution, and discarding higher o rder terms, we obtain: aj = js(BihF + AR). Inse r t ing AR obta ined above gives



VoL 126, No. 5 I M P E D A N C E M E A S U R E M E N T S 783

BrJs/q ) Aj/~XF = ~ac = Js Bi "~ Js/q + i~,

I f we d iv ide this b y ~dc : is~F, as we have done for the presen ta t ion of the expe r imen ta l results , we ob ta in

def js'] qBrF ~j/~F'F/js =--- BF = BiF ~-

i~ + Js/q

When real and i m a g i n a r y pa r t s a re spl i t we get

R e ( B F ) -- BiF § (js/q)2BrF/(~ ~ § ( i s /q) 2) [3]

Ira(BE) = - - ~ ( j s / q ) B r F / ( ~ § ( i s /q) 2) [4]

These equat ions a re s imi lar to those descr ibing a Debye r e l axa t ion wi th a single r e l axa t ion t ime (16). For low frequencies ~ ~ < is /q : R e ( B F ) --> (Bi § Br)F, the s t eady- s t a t e va lue of BF, and I m ( B F ) --> 0. Fo r

> > is /q : R e ( B F ) --> BiF and I m ( B F ) -~ 0. At in t e r - media te f requencies a nega t ive i m a g i n a r y pa r t of BF occurs. We have defined i r as that f requency at which t an 8 -- I Im(BF)/Re(BF)] has a max imum. Eva lua t ing this quan t i ty gives

1 ~ B i "~ - - - - q / J s " [5]

o~aax Bi ~- Br

As found exper imenta! ly , the r e l axa t ion t ime is in- ve rse ly p ropor t iona l to the cur ren t dens i ty is. The b roken l ine in Fig. 2 has been ca lcula ted wi th Eq. [3] and [4] for region II, supp lemented wi th the known dependences at h igh and low frequencies ( re - gions I and I l I ) : I t shows tha t the behav ior in region II is reasonably wel l descr ibed wi th a s ingle r e l a x a - t ion t ime.

Dignam's theory.~When one a t t empts to give the ma thema t i ca l equat ions [1] and [2] phys ica l content, two questions are impor tan t : (i) What is the r a t e - con - t ro l l ing step for ion t r anspor t th rough the meta l ox ide and in pa r t i cu l a r is the cu r ren t contro l led at one of the interfaces or in the bu lk of the oxide? (ii) With which phys ica l process can the r e l axa t ion of the quan t i ty R be identified?

Dignam pos tu la ted tha t defect in ject ion at one of the in terfaces const i tutes the ra te -con t ro l l ing step. The a rgumen t is as fol lows: If conduct ion is bu lk control led, the process of annih i la t ion and creat ion of charge car r ie rs and the i r mobi l i ty wi l l cause a pa r t i cu l a r flow of charge car r ie rs a t a pa r t i cu la r bu lk electr ic field. To main ta in the same charge flow across both in terfaces different fields at these in terfaces wi l l be required. To match the bu lk and in ter face fields space charges are set up.

Young (17, 18) and Vermi lyea (19, 20) showed f rom the independence of the cur ren t - f ie ld re la t ion on l aye r thickness tha t if the cur ren t is bu lk con- t ro l led the charac ter i s t ic space charge length is of the o rde r of 10A or less.

Dignam ca lcula ted [see also F romho ld (21)] tha t if this holds at a low cur ren t density, say 10 -6 A / c m 2, then at 10 -2 A / c m 2 the concentra t ion of mobile ionic defects must be of the o rde r of the la t t ice ion concen- t ra t ion. Dignam considers this to be improbable , and there fore he discards the poss ibi l i ty of bu lk control.

Fo r the cur ren t - f ie ld re la t ion at the in ter face he assumes

j = a exp ~Fd [6]

where Fd is the field at the in ter face and a and ~ are constants. Fd is connected wi th the field in the bu lk of the oxide F th rough eoKdFd ---- eoF § P where Kd is the suscept ib i l i ty of the oxide of the in ter face and P is the polar iza t ion of the bu lk oxide. Surface charges be tween in ter face and bu lk a re lef t out of account. The polar iza t ion P is spl i t in a fast component P1 = �9 oxlF, which follows the field direct ly , and one or more s low compounds Pk SO tha t P = eoxiF § ~kPk.

In the fol lowing we wi l l confine ourselves to one component a l though wi th 2 or more components a be t t e r fit wi th the expe r imen ta l resul ts can be obtained. Inser t ing this in Eq. [6] Dignam obta ins

{ s o ( l § ,P~ } j ---- a exp ~ ,~Kd + ~ [7]

Compar ison of Eq. [1] and [7] shows tha t in Dignam's model the role of the r e l ax ing quan t i ty R is p layed b y the slow polar iza t ion component P2: R = ~/(eoKd) P2. Consequent ly Dignam takes for the kinet ics of the polar iza t ion dPk/dt : 1/'Ck(eoXkF -- Pk) and on wr i t ing ad hoc 1/Tk : Bk ' j Dignam finds dPk/dt : B k ' j ( e o X k F - - P k ) , which is comple te ly equiva len t to Eq. [2] ).1

In this way Dignam obtains a set of equat ions equiv- a lent to [1] and [2]. In fact he der ived them in this way. However i t is not c lear wha t the phys ica l mechanism is by which the ionic cur ren t br ings the po la r iza - t ion changes about. Moreover the expe r imen ta l resul ts (in our n o t a t i o n Br ~ 3 • Bi) r equ i re ve ry la rge c u r r e n t - d r i v e n components which are much l a rge r than the "normal" polar izat ion. Because changes of the s l o w polar iza t ion are on ly caused by ionic flow, they can only be measured dur ing apprec iab le ionic flow for which the i r presence has been pos tu la ted and i t is not possible to de te rmine the slow components by independen t means. This leaves Dignam's theo ry in a somewhat unsa t i s fac to ry state.

Surface charge.--The most in teres t ing fea tu re of the expe r imen ta l resul ts is the appearance of a quan- t i ty q having the d imension of a surface charge density, which is independen t of specimen thickness. In the theo ry to be developed a surface charge ( in- s tead of a polar izat ion, as used by Dignam, which has of course the same dimension) p lays an essential role. The poss ib i l i ty of such an approach has been ment ioned brief ly b y Young and by Dignam. Young [Ref. (15), p. 577] states: "As regards o ther possibi l - i t ies i t is possible tha t a model could be de r ived in te rms of space charge effects due to oxygen ion mo- tion." Dignam [Ref. (1 t ) , p. 272] r emarks : "I t is apparen t tha t a t ime var ia t ion in the b o u n d a r y charge � 9 would behave in a l l impor t an t respects as a po la r - izat ion process." In the model we specifical ly use the expe r imen ta l fact tha t t r anspor t numbers of about 0.5 are found and we assume tha t this is due to the fact tha t both oxygen and meta l ions move th rough the bu lk of the a luminum oxide.

We adopt Dignam's view tha t ionic conduct ion is de te rmined at one of the interfaces. An addi t iona l a rgumen t resul t ing f rom our own exper iments is tha t the r e l axa t ion t ime of the process is independen t of the e lec t ro ly te used. This holds even for the case of anodizat ion af te r boiling, a l though in tha t case the bu lk of the oxide, being p a r t l y amorphous and p a r t l y c rys ta l l ine and consist ing fu r the r of hydroxide , is comple te ly different f rom the homogeneous amorphous oxide which is fo rmed in apb-glycol . In view of this a rgument we consider the me ta l -ox ide in ter face r a the r than the e l ec t ro ly te -ox ide in ter face to be ra te de te r - mining. The model is sketched in Fig. 7. We divide the meta l plus the oxide into th ree regions, the metal , a bounda ry l aye r which is a few atomic layers thick, and the bu lk of the oxide. The essential assumpt ion is tha t in the bu lk of the oxide both posi t ive and

1 N o t e : In f a c t a p a r t f r o m the current driven terms Pk Dignam a lso c o n s i d e r s t h e r m a l l y a c t i v a t e d t e r m s P ' j f o l l o w i n g t h e equa - t i on d P ' j / d t = 1 /T j (eox ' jF -- P'J) so that the t o t a l r a t e of c h a n g e of P becomes

dP dF = r + k ~ Bkj(eox~F - Pk) + Y~ I / r j ( e o x ' j F - - P ' $ )

d t dt l

As t h e measured re laxat ion t ime r e m a i n s i n v e r s e l y proport ional to t h e c u r r e n t d e n s i t y j , e v e n a t t h e l owes t c u r r e n t dens i t i e s (see Fig . 5) a n d as t he i m p e d a n c e o f F ig . 2 is r e a s o n a b l y described w i t h a single re laxat ion t ime, w e wiU discard all t erms except Ps f o r simplicity.



784 J. E~ectrochem. Soc.: ELECTROCHEMICAL SCIENCE AND TECHNOLOGY Ma~ I979

METAL

G

�9 O X I D E

)oundary ayer

Jpl

--~F 1

G

bulk

ii_

Jp2

.--~ F 2

q

Jn2

~n Fig. 7. Model for relaxation phenomena. Between the aluminum

metal and bulk of the oxide a boundary layer is assumed which is impervious to negative charges. The negative charge flow in the bulk causes o surface charge an to be formed. The relaxation phenomena are ascribed to time variations of an.

negative charges move but that the boundary layer is impervious to negative charges. There the total cur rent is equal to the positive current , which we assume exponent ia l ly dependent on the local field FI: j = joexp (BplF1) where Bpl is a constant. Be- cause the boundary layer does not t ransmi t negative charges, a negative surface charge an is bui l t up at the interface between boundary layer and bulk. The field F1 is equal to the field in the bulk, F, plus the surface charge an divided by the permit t iv i ty eoer (which is assumed to be the same in the bulk and in the boundary layer) . If this is substituted, we obtain

j = Jo exp (Bpl (F 4- an/eoer) ) [8]

On comparison with Eq. [1] it becomes clear what we have in mind: We in tend to ascribe the re laxat ion to variat ions of an. It is assumed that the s teady-state v a l u e o f an increases with F. However, Vn cannot follow rapid variations of the field, so that the high frequency admit tance is lower than the low frequency admittance, an is increased by the negative current in the bulk jr~. We assume for the t ime being that Jn~ < < j. (This assumption wil l have to be abandoned for a quant i ta t ive comparison of theory and expert- ment ; it is not at all essential for the model but at this stage it great ly facilitates the derivation.) Cn decreases if incoming positive charges which have crossed the boundary layer are captured by the negative charges Cn. We assume that the probabi l i ty of such an event is proport ional to the surface charge ~rn and to a constant W which is a capture cross section per negative charge. This results in the equat ion

dCndt -" in2-- gnW~ -- j W ~ jW -- crn [9]

The positive charge flow in the bu lk is given by jp~ : j(1 -- cnW). Because we have assumed that in2 < < j we can take jp2 ~-- J. Inser t ing this we obtain

den (in2 1 ) dt = jW J~ W an [10]

In order to go fur ther we make the ad hoc assumption that the positive and negative charge flows in the bu lk are coupled by some process at the oxide-electrolyte interface, and write: Jn2/Jp2 = gF, where g is a constant. That is to say, we assume that the probabi l i ty that a negative ion enters the oxide out of the electrolyte is determined both by the field and by the arr ival of positive charges at this interface. It is known exper imenta l ly that the t ranspor t numbers of oxygen and metal ions are about 0.5 in a wide current density range (23). We do not, however, have any physical model for such a behavior. If we insert

this we arr ive at, or ra ther have worked toward

d ~ . / d t = j W ( g r ' / W - - an) [11]

which is completely equivalent with Eq. [2]. The charge q in Eq. [2] is here equal to 1/W, the reciprocal of the capture cross section.

Quali tat ively the theory has the at tract ive property that it can be directly understood t h a t the re laxat ion t ime is inversely proport ional to the current j. Say that we have a high steady-state surface charge and decrease the applied field F. The rate of decrease of ~n is then determined by the rate of capture of in - coming positive charge carriers by an, which is pro- portional to the total cur ren t j.

Quantitative comparison of theory and experiment.-- Star t ing from Eq. [10] and [11] for this simple case we can obtain numer ica l values for the different quantities. W can be obtained by insert ing the measured quanti t ies j.~, Bi, and Br in Eq. [5], taking into account that W = 1/q. The value 1.5 • 104 cm~/C is obtained. This is equivalent to a cross section for capture of a moving positive by a fixed negative (single) charge of 25A e, which seems quite reasonable. The equi l ibr ium surface charge follows by put t ing dan/dr -= 0 in Eq. [11] which gives ~n = gF/W. If this is inserted in [8] we obtain j = jo exp (BplF(1 -~ g/Weoer)). It follows that in this theory Bi = Bpl and Br = Bplg/Weeer. So we obtain an ( = gF/W) = FeoerBr/Bi. We know Br/B~ from experiment , and on taking for eoer the values for bu lk A1203 we arr ive at: Vn ~- 22 ~C/cm 2. If we compare this value with the charge necessary to form anodically one atomic layer of a luminum oxide, 530 #C/cm 2, we see that a nonstoichiometry of about 4% in one atomic layer is sufficient to explain the relaxation, which also seems quite reasonable. F ina l ly the product anW gives the probabi l i ty that an incoming positive charge will be captured by the negative surface charge, and this quant i ty is identical with the fraction of oxide which is formed at the A1 side of the oxide layer or with the t ransport number of oxygen. Exper imental values for anW lie a round 0.3. The t ransport numbers found in practice for oxygen (22, 23) lie between 0.37 and 0.76 and ~nW lies reasonably close to this range. How- ever, the equations have been derived for the case where in2 < < J~2 ~ J, whereas our exper imental result and the known t ranspor t numbers point to in2 ~ Jp2. Therefore, we have to derive a new set of equations for this more general case.

Derivation for the case where the positive and negative currents in the bulk of the oxide have the same mag- nitude.--If we re tu rn to Eq. [9], insert Jp2 -- j (1 -- vn" W) and again make the assumption Jn2/jp2 "-" gF we obtain

-- j ( g r ( 1 -- cnW) -- anW) [12] dt

This is a more complicated equation because of the product F 'an in the first te rm on the r igh t -hand side. In equi l ibr ium we find ~n'W = gF/(1 + gF). There- fore in this case too the equi l ibr ium surface charge increases with the field, which is t an tamoun t to in - ductive behavior. To derive the impedance we use again that the field is varied only in a very small region around the steady-state field Fs. The derivat ion is completely analogous to the one carried out under "a-c behavior" above. The results are compared in Table II with the results obtained in the case in2 < < ~p2.

Comparing the equations the following can be re- marked.

1. Quant i ta t ive ly the differences are not very large. If ~n2 =- ~p2 the calculated surface charge density is a factor 1.5 larger. The calculated W is lower by the same factor.



Vol. I26, No. 5 I M P E D A N C E M E A S U R E M E N T S

T.ble II.

785

Quantity Equation for j.~ << jp~ Equation for jn~ ~ b~

Steady-state current

Relaxing part of B

Steady-state surface charge density

Steady-state surface charge density in terms of Br/Bl

Relaxation time

Fraction of incoming charge captured at metal-oxide interface

( ( :w)) ( ( )) J = joeXp BiFs 1 + ~: j = j o e x p BiF. 1 + g F . (

g g Br = Bi Br = Bl

eoerW eoerW(l + gF.) s gF. gF.

W W(I + gF.)

o~. = F.eoer Br/B~ ca. = F.(I + gF.) eoer B./BI

1 1

j , W ~ j ,W(1 + g F . ) - ~ F.eoer Fseoer

~n. " W = ~ ~ ~ " Br/Bi ~a.'W = j.r~/l + Br/Bi Br/B,

2. The p roduc t ~nW is not dependen t on g. This is as i t should be: The f rac t ion of the incoming charge cap tu red at the me ta l side is not dependent on the processes occurr ing at the o ther interface, which de te rmines the quan t i t y gF.

3. The field dependence of the s t eady-s t a t e cur ren t dens i ty in the case in2 ~ Jp2 is weake r than in the case in2 < < Jp2. That is to say, if the a rgumen t of the exponent is developed a round F -- Fs a negat ive quadra t ic t e rm appears , which is in accordance wi th the expe r imen ta l resul ts (11) and of the same o rde r of magni tude .

Al toge the r the observat ions made above on the reasonableness of the magn i tude of the canstants following f rom the exper imen t s r ema in valid. The con- clusions about ~nW in pa r t i cu l a r r ema in unchanged.

Discussion and Conclusion The theory presented in the previous section, in

which the r e l axa t ion effects occurr ing dur ing anodiza- t ion are ascr ibed to t ime var ia t ions of a surface charge nea r the me ta l -ox ide interface, has a number of drawbacks . The m a i n one is tha t the theo ry is incom- plete. An ad hoc assumpt ion had to be made about the mu tua l dependence of posi t ive and negat ive charge cur ren ts in the bu lk of the anodic oxide. A second d r a w b a c k is tha t the effects of different e lec t ro ly tes cannot be accounted for completely. A change of e lec t ro ly te shows up in the value of BF, if the to ta l cu r ren t dens i ty is wr i t t en as j - - Jo exp (BF). In the theo ry BF is the only var iab le quan t i ty and one would expect tha t i f BF changes, the to ta l cur ren t dens i ty j would change. In fact, however , these different values for BF are found at about the same cur ren t densi ty so tha t a p p a r e n t l y the va lue of Jo changes too, which implies tha t condit ions at the me ta l -ox ide in ter face are changed. This weakens the a rgumen t tha t the r e l axa t ion occurs at the me ta l -ox ide interface, because the r e l axa t ion t ime is independen t of the electrolyte .

In view, however , of the facts that , first, the theory a t t r ibu tes the r e l axa t ion to a quan t i t y easi ly visual ized physical ly , second, i t gives a na tu r a l exp lana t ion for the inverse dependence of the re laxa t ion t ime on the cur ren t density, and that , third, i t resul ts in reasonable numer ica l values for the cross sect ion W, for the amount of surface charge Cn, and, in par t icu lar , for the amount of oxide formed at the meta l side, we feel tha t this model deserves at tent ion.

Acknowledgment We are ve ry much indeb ted to J. M. G. Reemers a n d

A. C. P. van de Bosch for the au tomat ion of the appara tus , which made i t possible to do ~ the la rge number of measurements r equ i red for this inves t iga- tion.

Manuscr ip t submi t t ed Aug. 21, 1978, rev ised m a n u - scr ipt received Nov. 27, 1978.

A n y discussion of this paper wi l l appea r in a Dis- cussion Section to be publ i shed in the December 1979 JOURI~AL. Al l discussions for the December 1979 Dis- cussion Sect ion should be submi t t ed by Aug. 1, 1979.

Publication costs ol this article were assisted by Philips Research Laboratories.

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impedance measurements during anodization of aluminum

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