J. Phys. Chem. B 2004, 108, 20213-20218
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Oxidation Kinetics in Iron and Stainless Steel: An in Situ X-ray Reflectivity Study D. H. Kim, S. S. Kim, H. H. Lee, H. W. Jang, J. W. Kim, M. Tang,† K. S. Liang,† S. K. Sinha,‡ and D. Y. Noh* Department of Materials Science and Engineering, Gwangju Institute of Science & Technology (K-JIST), Gwangju 500-712, Republic of Korea, Synchrotron Radiation Research Center, Hsinchu, Taiwan 300, Taiwan, and Department of Physics, UniVersity of California at San Diego, San Diego, California 92093 ReceiVed: May 16, 2004; In Final Form: August 24, 2004
The growth kinetics of passive films on iron and stainless steel (Fe-16.31%Cr) substrates in pH 8.4 borate buffer solution was investigated by using in situ specular X-ray reflectivity. The oxide growth rate decays exponentially with increasing oxide thickness consistent with the point defect model in which the electric field in the oxide is maintained constant during growth. In stainless steel, however, the electric field depends strongly on the applied potential, indicating that the oxide properties change as the applied potential varies. Using the electric field in the oxide and the observed saturation oxide thickness in a quasi steady state, we estimate the potential drop at the metal/oxide interface, in the oxide, and at the oxide/solution interface.
Introduction Since Cabrera and Mott’s pioneering theoretical work1 on oxidation in metals, oxidation kinetics has been investigated extensively both experimentally and theoretically.1-8 Understanding the oxidation mechanism is important for practical purpose in many cases, since thin oxide films act as passive layers and protect metals from corrosive environments. Several growth models have been proposed to explain oxidations at low temperatures or under electrochemical environments where thermal diffusion of metallic cations and oxygen anions is limited.1-5 It is now well accepted that the high electric field set across an oxide film assists ionic transport as originally suggested by Cabrera and Mott.1 Theories based on the high electric field predict either inverse logarithmic growth or direct logarithmic growth of oxide films. The inverse logarithmic growth models, called high field model (HFM), assert that the activation energy of metal atoms is inversely proportional to the film thickness when the metal/ oxide system is maintained under constant electric potential. The inverse of the oxide thickness is predicted to decrease logarithmically in time. On the other hand, the direct logarithmic growth models are based on the fact that the electric field in the oxide remains constant during growth due to dielectric breakdown under high electric field. This is called a point defect model (PDM).2,3 Oxide thickness increases logarithmically in time in PDM. Experimentally, however, only limited data are available due to the difficulty in measuring in situ the growth kinetics of very thin passive oxide films a few nanometers thick. In particular, there have been relatively few experimental data on alloys as compared to pure metals. Until now, most studies on the growth kinetics of passive films have been performed by using ellipsometry. Goswami et al. observed the growth kinetics of passive Fe and Fe-based alloy films using ellipsometry. Their data were described better by the inverse logarithmic law. They, * To whom correspondence should be addressed at Gwangju Institute of Science & Technology. † Synchrotron Radiation Research Center. ‡ University of California at San Diego.
however, suggested that it was not definitive.6 On the other hand, Silverman et al.,7 on the basis of ellipsometry data on the growth of passive films of Fe-25Ni-xCr, suggested that although the results fit both the inverse and direct logarithmic laws, the latter deduced from PDM proposed by Chao et al.2 provided a more reasonable representation. Fitting to the Cabrera-Mott’s HFM model resulted in an unrealistic half-jump distance value. Recently, Olsson et al., using an electrochemical quartz crystal microbalance (EQCM), showed that their results fit better to the direct logarithmic growth than the inverse logarithmic growth although it was possible to use both models.8 More direct measurements are on demand to clarify the mechanism of passivity in aqueous solutions. In the present work, we use X-ray specular reflectivity to investigate the growth kinetics of the passive films on iron and stainless steel type 430 (Fe-16.3%Cr). All measurements are carried out in situ. As is evident from previous investigations,9,10 X-ray reflectivity is well suited to determine the thickness of the passive film accurately in the nanometer scale. The evolution of the oxide thickness is one of the key parameters determining the growth kinetics. The data on the thickness evolution are used to test the concepts of the inverse and the direct logarithmic laws, and to deduce the electric field and the potential drop in the oxide. Models on Low-Temperature Oxidation: HFM and PDM In this section, we briefly review two representative theoretical approaches on low-temperature oxidation: high-field model (HFM) and point-defect model (PDM). In both models, it is assumed that electrons from the metal tunnel through oxide to make oxygen (or hydroxyl) anions, which sets up an electric field of order 107 V/cm across the oxide. This high electric field enables the ionic transport necessary for oxide growth even at low temperatures. It is also assumed that the rate-limiting process is the reaction in which metal atoms are oxidized into metal cations and placed in interstitial sites in the oxide close to the interface. The activation energy involved in this reaction can be expressed as ∆Ga ) ∆Ga0 - RnF‚∆φm/f, where ∆Ga0 is the activation energy in the absence of electric potential difference
10.1021/jp0479062 CCC: $27.50 © 2004 American Chemical Society Published on Web 11/26/2004
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at the metal/oxide interface, R is the transfer coefficient, n is the charge of the positive ions diffusing through the film, and F is the Faraday constant. ∆φm/f is the electrical potential drop at the metal/oxide interface, which can be interpreted as the potential difference between the metallic site just inside the metal and the interstitial sites in the oxide just above the interface. In the case where oxide dissolution at the oxide surface is negligible, the oxide growth rate is given by
(
)
(
)
∆Ga dL RnF ) Ωk exp ) Ωk0 exp ‚ ∆φm/f dt RT RT
(1)
where Ω is the oxide volume formed per cation, k0 is the reaction rate, and k is k0 exp(-∆Ga0/RT). In HFM, the potential drop ∆φox set across the oxide film is assumed constant during oxide growth. The electric field is then inversely proportional to the oxide thickness L, and ∆φm/f is given by a∆φox/L where a is the distance from the top of the metal to the nearest interstitial site in the oxide. Applying this to eq 1, one gets the rate for the inverse logarithmic growth,
(
)
RnF a∆φox dL ) Ωk exp ‚ dt RT L
(2)
Meanwhile, in PDM, the generation and annihilation of point defects maintain the electric field in the oxide constant rather than the potential drop across the oxide. In fact, in the sense of the PDM, the electric field is a characteristic of an oxide independent of applied potential and oxide thickness. Numerous studies2-5 have shown that the applied potential Vapp can be expressed as
Vapp ) ∆φm/f - ∆φox - ∆φf/s
(3)
where ∆φm/f, ∆φox, and ∆φf/s represent the potential drops at metal/oxide interface, across oxide film, and at oxide/solution, respectively. The potential drop at the oxide/solution interface is assumed to be linearly proportional to applied potential and to solution pH, i.e.
∆φf/s ) RPDMVapp + βpH + ∆φ0f/s
(4)
where RPDM and β are the proportional constants to the applied potential and pH, respectively, and ∆φ0f/s is the potential drop at the oxide/solution interface at zero applied potential and zero pH. On the basis of this, the potential drop at the metal/oxide interface is expressed as
∆φm/f ) (1 - RPDM)Vapp - βpH -∆φ0f/s - E0L
(5)
where E0 is the electric field in the oxide. By inserting eq 5 into eq 1, the growth rate can be expressed as
dL ) A exp(-B‚L) dt
(6)
where A ) Ωk exp{(RnF/RT)[(1 - RPDM)Vapp - βpH - φf/s0]}, and B ) (RnF/RT)E0 for the direct logarithmic growth. Experimental Setup The oxidation was performed in a 3-electrode electrochemical cell made of Teflon that allows in situ X-ray reflectivity measurement in transmission geometry.9,13 Ultrathin polypropylene served as an X-ray window. The working electrodes were prepared from iron (purity 99.99%) and industrial stainless steel
Figure 1. Cyclic voltammogram of iron and stainless steel 430 in pH 8.4 borate buffer solution measured in the electrochemical cell for X-ray reflectivity. Potential was scanned at the rate of 5 mV/s.
type 430 (Fe-16.31%Cr). They are surrounded by epoxy resin to avoid any current flowing through the sides and the back of the sample. The electric contacts are made by spot welding wires. The electroactive area was 10 × 2 mm2. To achieve a smooth surface, the working electrode was polished to a mirror finish down to 0.1 µm with diamond suspensions, cleaned in an ultrasonic vibrator, and rinsed thoroughly with doubly distilled water. The counter electrode was a Pt wire, and all potentials were measured against an Ag/AgCl reference electrode (in 3.5 M KCl). The electrolyte solution was a pH 8.4 borate buffer solution. The solution was deoxygenated with N2 gas (purity 99.999%) prior to each measurement, and then injected into the cell. During each measurement, the surrounding area near the cell was flushed with N2 gas to prevent oxygen diffusion through the polypropylene window. Cyclic voltammetric and potentiostatic measurements were performed with Model 362 (EG&G Co.) to characterize the electrochemical behavior of the samples. The electrode was reduced potentiostatically at -1.5 V for 10 min, and then maintained at specified potentials for the oxide growth. At each specified potential, the evolution of the oxide thickness was repeatedly measured by X-ray reflectivity for roughly 4 h. The X-ray reflectivity measurements were carried out with the Beamline 5C2 at the Pohang Light Source (PLS) and Beamline 12B2 at SPring-8. In both cases a double bounce Si(111) monochromator is used to achieve high momentum resolution. The X-ray reflectivity data shown in this paper are obtained by using 19.8 keV X-rays. The X-ray photons of high energy have the advantage in transmitting through the electrolyte. Results and Discussion A. Growth Kinetics: Support PDM Model. Figure 1 shows the cyclic voltammograms (CVs) of iron and stainless steel 430 in pH 8.4 borate buffer solution obtained by using the X-ray electrochemical cell. The features shown in the CVs are in good agreement with those published in the literature under similar experimental conditions.10-12 We note two major differences between Fe and stainless steel. The first is that there is a welldefined passivation peak A, originating from the change from active state to passive state in iron, while the CV of stainless steel shows only a broad hump near -200 mV. We speculate that the absence of the anodic peak A in stainless steel is due to the insoluble Cr-oxide film remaining on the surface during the cathodic pretreatment, which partially passivates the surface.11-13 The second is the occurrence of the anodic peak B in stainless steel that is related to the oxidation of Cr3+ to Cr6+, indicating the dissolution of inner Cr-oxide during the breakdown of the passive film.13 The growth kinetics of the
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Figure 3. Oxide layer thickness extracted from X-ray specular reflectivity of iron (a) and stainless steel (b) as a function of the oxidation time under each potentiostatic condition.
Figure 2. Representative specular reflectivity obtained during the oxidation of iron (a and b) and stainless steel (c and d) at 400 mVAg/AgCl and 800 mVAg/AgCl, respectively. Solid lines are the results of fits based on the Parratt’s formalism. Each curve is chosen by using a logarithmic time interval at 500, 1000, 2000, 4000, and 8000 s, upwardly. The curves are offset by multiplying constant factors for clarity. Insets show the reflectivity curves measured in the early stage of oxidation.
oxide on iron and stainless steel has been examined at anodic potentials from 0 to 800 mV relative to the reference electrode. At each potential, we measured the X-ray specular reflectivity repeatedly in situ. Figure 2 shows representative X-ray specular reflectivity curves measured during the oxidation under 400 mVAg/AgCl and 800 mVAg/AgCl potentials, respectively. Diffuse scattering is measured and subtracted in all reflectivity data. The reflected intensity is normalized to the X-ray intensity measured on the monitor incident to the specimen, which is defined by slits. We note that there is a distinct difference between iron and stainless steel just after the cathodic pretreatment at -1.5 V (starting point of the oxidation), indicated by the bottommost curve in each figure. Iron yields a smooth reflectivity curve of a single interface, indicating that the air-formed oxide is completely removed by the cathodic pretreatment. However, in the stainless steel a weak intensity modulation appears, which is indicative of a thin oxide overlayer film on the substrate. This suggests that the oxide on stainless steel is not completely reduced. From our previous work,13 we confirmed that the insoluble oxide was composed of Cr-oxide. The growth of the oxide film is illustrated by the small but clear shift of the intensity minima to lower Qz-values or the decrease of the oscillation period, ∆Qz. In both iron and stainless steel the oxide film grows faster at 800 mV than at 400 mV. The solid lines are the results of a fit to the Parratt’s formalism14
in which we assume a single homogeneous oxide layer of thickness L with a refractive index of n ) 1 - δ - iβ on top of the substrate. The reflectivity is then calculated by solving Maxwell’s equations at each interface yielding the transmission and reflection of the X-rays. The roughness of the substrate/ oxide and the oxide/solution interfaces is included in the modified form of the Fresnel coefficients.15 From the result of the fits, the roughness was evaluated as well as the thickness. However, the roughness of both the substrate/oxide and the oxide/solution changed insignificantly during growth under a given potentiostatic condition and depended little on the applied potential. For example, the roughness of the oxide/solution interface on the stainless steel under 400 mV changes from 8.8 ( 1.9 to 6.3 ( 1.7 Å, while that of the metal/film changes from 3.4 ( 1.4 to 2.8 ( 1.2 Å. As indicated by the large roughness, the quality of the reflectivity, especially on the stainless steel at 400 mV, is rather poor. There were two major experimental difficulties. The first is that we used industrial iron and stainless steel, and polished to obtain the sharp surface. Different from well-defined single crystalline semiconductor surfaces, the roughness is rather large, and resulted in rather poor reflectivity. The second reason is that the oxide thickness is rather thin and the sample is kept under the solution. Although it is rather difficult to reconstruct the density profile of the oxide accurately with the data, our analysis suggests one can obtain thickness evolution with reasonable accuracy, as shown in the error bar in Figure 3. Figure 3 illustrates the change in the oxide thickness L deduced from the fits as the oxidation proceeds. In iron, the oxide grows quite rapidly in the initial stage, which slows down as the oxide thickness reaches ∼20 Å. On the other hand, in the case of stainless steel, the growth starts with the initial thickness of L0 ) 23.4 ( 1.8 Å, due to the nonreducible Cr layer. The thickness increase is more gradual in stainless steel. To compare our data with theoretical predictions, we deduced the growth rate by taking time derivative of the thickness increase numerically. Theoretical models predict the growth rate in relatively simple forms as compared to the thickness itself. As pointed out by Ghez,16 a few mathematical approximations are necessary to obtain the expression for the oxide thickness,
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Figure 5. Electric field strength in the newly formed oxide on iron (open circle) and stainless steel (solid circle) as a function of the applied potential.
Figure 4. Growth rates under each potentiostatic condition plotted as a function of 1/L (a, b) and L (c, d) in a semilogarithmic scale.
which makes it difficult to test theoretical models with experimental data. Figure 4 shows the growth rate thus obtained as a function of inverse thickness 1/L (Figure 4a,b) and thickness L (Figure 4c,d) in a semilogarithmic scale. The solid lines are best fits to eq 2, the inverse logarithmic growth predicted by HFM, and to eq 6, direct logarithmic growth predicted by PDM. Both in iron and stainless steel, the direct logarithmic growth describes the data better than the inverse logarithmic growth. As shown in Figure 4, there is a clear discrepancy between the data and fits in the case of the inverse logarithmic law. The discrepancy is specially pronounced in the stainless steel under relatively high potentials. Under low anodic potentials the change in the oxide thickness is not large enough to distinguish between the models. In the case of iron, the inverse logarithmic growth is not as bad as in stainless steel, although the direct logarithmic law still describes it better. The behavior of the growth rate clearly demonstrates that PDM is a better description of the oxidation. B. Electric Field in the Oxide Growth Front. In terms of PDM, the electric field in the oxide growth front can be estimated from the slopes of the solid lines shown in Figure 4. From the growth rate given in eq 6, we obtain the electric field E0
E0 ) -
{ ( )}
RT d dL ln RnF dL dt
(7)
Under a fixed applied potential, the electric field of the growth front stays constant as indicated by the straight lines in Figure 4 consistent with PDM. Figure 5 shows the quantitative electric field strength in the oxide at each applied potential, obtained from eq 7 assuming R ) 0.5, n ) 3, and RT/F ) 0.023 V. For iron E0 depends little on the applied potential, which supports the assumption of constant field in PDM. The electric field strength E0 ≈ 1.84 × 106 V/cm, which is in good agreement with the values published in the literature.2,17 In stainless steel, however, the electric field strength decreases with increasing applied potential, which is especially pronounced in the passive regime below 500 mV. This behavior seems inconsistent with the assumption of constant field in PDM. However, we note that the constant electric field is based on
the assumption of a homogeneous oxide of fixed composition. In the case of stainless steel the passive film is composed of the inner Cr-rich oxide and the outer Fe-rich oxide.13 Assuming that the electric field in the inner Cr-rich oxide does not change with increasing the oxide thickness as stated in PDM although it might depend on the applied potential, the electric field obtained by eq 7 should be interpreted as the field in the newly formed Fe-rich region. The fact that the electric field depends on the applied potential indeed indicates that the composition of the Fe-rich region depends on the applied potential. As is evident from previous studies,12,18 the ratio of Fe-oxide to Croxide in the passive film increases as the applied potential increases. Therefore we suggest that the steep decrease in the field below 500 mV results from the increased ratio of Fe-oxide to Cr-oxide. Above 500 mV in the transpassive region, the electric field changes significantly less indicating that the composition depends less on the applied potential. The difference between the two regions is associated with the dissolution of the inner Cr-oxide, Cr2O3 f 2Cr6+ + 3O2- + 6e, which begins to occur at a potential near 500 mV.7,12,13,18 After the dissolution of the inner Cr-oxide, the oxide consists mostly of Fe-oxide. However, the electric field in this regime, E0 ≈ 6.1 × 105 V/cm, is much lower than that in the oxide on pure iron. This may indicate that the transpassive oxide on the stainless steel is more defective and cannot support high electric field. Silverman et al.7 reported a similar trend of low electric field in the transpassive regime. C. Potential Drop in the Oxide-Solution Interface and the Inner Cr-Oxide. The oxide growth would eventually reach a steady state where the oxide growth rate becomes equal to the oxide dissolution rate, and dL/dt ) 0. The continuous decrease of dL/dt shown in Figure 4 suggests that the dissolution rate of the oxide is still negligibly small compared to the oxide growth rate. However, the growth rate becomes extremely small in the laboratory time scale after a while as shown in Figure 3. The oxide thickness in the steady state would not be too much different from the final thickness measured in this experiment. For convenience, we define a saturation thickness Lsatu as the thickness where the growth rate reaches 10-4 Å/s. As shown in Figure 6, the saturation thickness of iron increases linearly with an increase in the applied potential. On the other hand, in stainless steel, the slope changes across ∼500 mV. This potential coincides approximately with the potential where the oxidation of Cr3+ to Cr6+ begins as shown in the CV of stainless steel in Figure 1. Above this potential, the saturation thickness increases abruptly due to the dissolution of inner Cr-oxide that acts as a barrier layer at lower potentials.13 In iron, the behavior of the potential drop in the oxide/solution interface, ∆φf/s, can be estimated from the behavior of the
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Figure 6. Oxide thickness, Lsatu, where the growth rate reaches 1 × 10-4 Å/s, plotted as a function of the applied potentials in both iron and stainless steel.
Figure 8. Schematic potential distribution in the passive film on iron (a) and stainless steel (b).
oxide. It is, therefore, necessary to include the potential drop in the Cr-oxide layer, ∆φCrO, separately
∆φf/s ) Vapp - ∆φsatu m/f - E0(Lsatu - L0) - ∆φCrO
Figure 7. (a) Potential drop in the oxide/solution interface in iron. (b) Potential drop in the Cr-oxide in the passive film on stainless steel.
saturation thickness. Defining the potential drop at the metal/ oxide interface when dL/dt ) 10-4 Å/s as ∆φsatu m/f , ∆φf/s is given by
∆φf/s ) Vapp - ∆φsatu m/f - E0Lsatu
(8)
where the potential drop in the oxide is expressed as E0Lsatu, since a homogeneous oxide is formed on the iron surface. We note that ∆φsatu m/f is the same for all applied potential since it is defined to be the potential across the metal/oxide when the growth rate reaches a fixed value of 10-4 Å/s. It is then possible to compare the relative change in ∆φf/s as a function of the applied potential, using eq 8, although we cannot obtain the absolute value. Figure 7a illustrates the relative change in ∆φf/s, which is estimated by using the values of E0 and Lsatu shown in Figure 5 and Figure 6, respectively. The slope of the solid line in Figure 7a, 0.704 ( 0.012, represents the parameter RPDM, the dependence of the potential drop in the oxide/solution on the applied potential. This value of RPDM is in good agreement with those reported by other authors.2 On the other hand, the oxide on the stainless steel consists of nonreducible inner Cr-oxide and the newly formed Fe-rich
(9)
Since both ∆φf/s and ∆φCrO are unknown, it is not possible to determine them from this relation. To estimate ∆φCrO, we make an assumption that ∆φf/s in stainless steel is similar to that in iron, since the same solution is used and the newly formed oxide on stainless steel is iron rich. In Figure 7b, we illustrate ∆φCrO estimated using eq 9 and the thickness of the Cr-oxide inner layer at ∼23 Å as determined previously. In the passive region below 500 mV, ∆φCrO and consequently the electric field increase with increasing the applied potential. This suggests that the nature of the inner Cr-oxide changes as the external potential is applied, which presumably happens in the very early stage. The linear dependence of the saturation oxide thickness on the applied potential has often been quoted as strong evidence supporting the assumption of constant electric field independent of oxide thickness and of applied potential in PDM. In the case of homogeneous oxide such as the one on iron, this is mostly valid. However, when the nature of the oxide depends on the applied potential as in the case of stainless steel, this can be misleading. The saturation thickness in stainless steel increases linearly in the passive regime as illustrated in Figure 6. The electric field in the newly formed oxide shown in Figure 5, however, decreases significantly as the applied potential increases. In the meanwhile, the electric field in the inner Croxide increases with an increase in the applied potential. As illustrated in Figure 7, only relative potential drops in the oxide/solution interface and in Cr-oxide at different applied potentials are available from our experimental data. It is, therefore, difficult to estimate the potential distribution across the metal/oxide/solution system in an absolute scale. As a qualitative reference, we illustrate the potential distribution in the steady state in Figure 8 using the experimentally determined electric field strength in the newly formed oxide and RPDM. The constant term ∆φ0f/s is left unknown. For iron shown in Figure 8a, the potential drops in the oxide and the oxide/solution system increase with an increase in the applied potential. On the other hand, in stainless steel as shown in Figure 8b, the potential drop in the newly formed oxide decreases with an increase in the applied potential because of the increase in the potential drop in the inner Cr-oxide. Conclusion We have observed the oxide growth in stainless steel and in iron directly using in situ specular X-ray reflectivity measure-
20218 J. Phys. Chem. B, Vol. 108, No. 52, 2004 ment for the first time. The oxide thickness evolution and the growth rate are determined unambiguously. In both stainless steel and iron, the oxide growth follows the direct logarithmic growth maintaining a constant electric field in the oxide. This supports the point defect model as opposed to the high field model. In stainless steel, however, the electric field depends strongly on the applied potential in the passive regime indicating that the nature of the oxide changes as the applied potential varies. Using the electric field in the oxide and the observed saturation oxide thickness in a quasi steady state, we estimate the potential drops across the metal/oxide/solution systems. Acknowledgment. We gratefully acknowledge Dr. Du for his help in the experiment at SPring-8. This work was support by MOST through the National Research Laboratory (NRL) program on Synchrotron x-ray and the x-ray/particle-beam nanocharacterization program. The PLS is supported by the Korean Ministry of Science and Technology. References and Notes (1) Cabrera, N.; Mott, N. F. Rep. Prog. Phys. 1948-1949, 12, 163. (2) Chao, C. Y.; Lin, L. F.; Macdonald, D. D. J. Electrochem. Soc. 1981, 128, 1187.
Kim et al. (3) Zhang, L.; Macdonald, D. D.; Sikora, E.; Sikora, J. J. Electrochem. Soc. 1998, 145, 898. (4) Vetter, K. J. Electrochim. Acta 1971, 16, 1923. (5) Kirchheim, R. Electrochim. Acta 1987, 32, 1619. (6) Goswami, K. N.; Staehle, R. W. Electrochim. Acta 1971, 16, 1895. (7) Silverman, S.; Cragnolino, G.; Macdonald, D. D. J. Electrochem. Soc. 1982, 129, 2419. (8) Olsson, C.-O. A.; Hamm, D.; Landolt, D. J. Electrochem. Soc. 2000, 147, 4093. (9) You, H.; Melendres, C. A.; Nagy, Z.; Maroni, V. A.; Yun, W.; Yonco, R. M. Phys. ReV. B 1992, 45, 11288. (10) Bu¨chler, M.; Schmuki, P.; Bo¨hni, H. J. Electrochem. Soc. 1998, 145, 609. (11) Bardwell, J. A.; Sproule, G. I.; Graham, M. J. J. Electrochem. Soc. 1993, 140, 50. (12) Oblonsky, L. J.; Ryan, M. P.; Isaacs, H. S. J. Electrochem. Soc. 1998, 145, 1922. (13) Kim, D. H.; Lee, H. H.; Kim, S. S.; Kang, H. C.; Kim, H.; Sinha, S. K.; Noh, D. Y. Phys. ReV. Lett. Submitted for publication. (14) Parratt, L. G. Phys. ReV. 1954, 95, 359. (15) Tolan, M. X-ray Scattering from Soft-Matter Thin Films; Springer Tracts in Modern Physics; Springer: Berlin, Germany, 1999; Vol. 148. (16) Ghez, R. J. Chem. Phys. 1973, 58, 1838. (17) Macdonald, D. D.; Urquidi-Macdonald, M. J. Electrochem. Soc. 1990, 137, 2395. (18) Bardwell, J. A.; Sproule, G. I.; Macdougall, B. R.; Graham, M. J.; Davenport, A. J.; Isaacs, H. S. J. Electrochem. Soc. 1992, 139, 371.
