Storage and Release of Soluble Hexavalent Chromium from Chromate Conversion Coatings on Al Alloys Kinetics of Release

The release of chromate ions from chromate conversion coatings ~ CCCs ! on Al alloys was studied, and the effect of aging of CCCs on the chromate release kinetics was investigated. Chromate release from CCCs into aqueous solutions was monitored by measuring the change in the chromate concentration in solution using UV-visible spectroscopy. Heat-treatment of the CCC greatly reduced the chromate release rate. The chromate release rate also decreased with increasing aging time at room temperature. A diffusion-control model was proposed based on the notion that the CCC in an aqueous solution is a porous, two-phase structure consisting of a solid phase with adsorbed Cr ~ VI ! species that is in local Langmuir-type equilibrium with an interpenetrating solution phase. This model results in a concentration gradient of soluble Cr ~ VI ! in the solution phase of the CCC as chromate is released. The concentration and diffusion coefﬁcients of soluble Cr ~ VI ! in CCC were estimated. The estimated diffusion coefﬁcient tended to decrease with aging time, suggesting that the CCC is modiﬁed with aging time.

For aircraft coating technology, understanding the mechanism of aluminum corrosion and the effect of chromates on corrosion inhibition is important. Chromate conversion coatings ͑CCCs͒ are commonly used on aluminum alloy surfaces to improve the corrosion resistance and to establish an adhesive base for organic coatings. Chromate ions have increased the pitting potential of aluminum alloys in chloride media and reduced the rates of pit nucleation and growth if present in a high enough concentration. 1,2 One of the important aspects of CCCs is the dynamic repair of newly created breaks on coated metals, which is called self-healing. For example, a CCC-treated Al alloy sample with a scribe mark through the coating to the metal exhibits minimal corrosion in salt-spray testing.
Zhao et al. demonstrated the migration of chromates from a CCC to an uncoated metal surface by detecting Cr on the surface of an initially untreated AA2024-T3 sample immersed in chloride solution in close proximity to a CCC-treated sample. 3 The release of Cr͑VI͒ from CCCs was further studied using UV-Visible spectroscopic analysis of solutions exposed to samples with CCCs. 4 The observations were consistent with a release mechanism similar to Langmuir adsorption and desorption of Cr͑VI͒ from a porous, insoluble Cr͑III͒ matrix. To fully understand the inhibition behavior of CCCs, one must fully understand the chromate release phenomenon. In the present study, the release of Cr͑VI͒ from CCCs is examined for samples under a range of conditions, and a model describing the release kinetics is developed.

Experimental
CCCs were prepared on aluminum alloys AA1100 and AA2024-T3 by immersion in Alodine 1200S ͑Henkel Surface Tech-nologies͒ solutions at room temperature. The Alodine bath consisted of 9.0 g L Ϫ1 of a mixture reported to be 5 50-60% CrO 3 , 20-30% KBF 4 , 10-15% K 3 Fe͑CN͒ 6 , 5-10% K 2 ZrF 6 , and 5-10% NaF by weight, which is approximately 0.065 M in Cr. The pH of the bath was adjusted to 1.5 by the addition of nitric acid. The AA1100 samples (3 cm ϫ 8.35 cm ϫ 50 m) were given an acidic pretreatment using a triply diluted, commercially available detergent, Alumiprep 33 ͑Henkel Surface Technologies͒, and the AA2024-T3 samples were prepared by polishing to 1200 grit paper in water, rinsing with deionized ͑DI͒ water and degreasing in alcohol before the Alodine treatment. Most samples were treated in the Alodine bath for 5 min. After Alodine treatment, the samples were rinsed by dipping in DI water, and aged in air at room temperature or at 50, 100, or 200°C. After a certain period of aging time, chromate release was determined during immersion in stagnant DI water or 0.1 M NaCl solution at room temperature in glass beakers or glass vials. To avoid evaporation of water, the beakers were covered with vinyl sheets.
The concentrations of released Cr ions in solutions were periodically measured by UV-visible spectroscopy using a home-built system controlled by a LabVIEW program. A cuvette of 10 mm pathlength containing sample or reference solution was placed in the path of monochromated incident light. The intensity of the transmitted light was measured by a power meter, and the absorption was determined as a function of the wavelength of the incident light. After the measurements, the solution in the cuvette was replaced in the test beaker. Quantitative determination of Cr͑VI͒ concentration was calculated from UV absorption at the isosbestic point of 340 nm, where the absorption is independent of pH and is linearly related to the concentration of total Cr͑VI͒. 4,6,7 The linear calibration relation of Cr͑VI͒ concentration to absorption measured at 340 nm can be written as ͓Cr͑VI͔͒ ϭ (Abs 340 0.00157)/1062. An absorption of 1 is almost equal to 1 mM of total Cr͑VI͒; hereafter, we express the concentration of total Cr͑VI͒ by absorption.

Results and Discussion
Effect of heat-treatment.-The 0.1 M NaCl solution in contact with a CCC-treated AA1100 sample aged for 2 h at room temperature tended to show an increase in Cr͑VI͒ concentration with time, approaching a constant value, as evidenced by an increase in the absorption at 340 nm ͑see Fig. 1͒. The sample heat-treated at 50°C for 2 h showed remarkably less chromate release than the sample aged at room temperature. CCC samples heat-treated at 100 or 200°C did not release detectable amounts of chromates. The CCC sample aged at 200°C lost the brownish golden color characteristic of CCCs as the color of the sample turned to metallic gray. Clearly, heat-treatment suppressed the release of chromates from the CCCs.
Effect of aging and ratio of surface area to solution volume (S/V).- Figure 2 shows the effect of room-temperature aging on chromate release from CCC-treated AA1100. Note that all samples described hereafter were aged at room temperature. In general, the concentration of Cr͑VI͒ in solution increased with immersion time and then reached an almost constant level of concentration. The release rate of chromates from CCC decreased with increasing room-temperature aging time. The release rate in 0.1 M NaCl solution was higher than that in DI water. The samples polished slightly by SiC paper before immersion in water did not behave significantly differently than the unpolished samples, which indicates that aging did not only affect the very surface of the CCCs. For the samples in this figure, the surface area and solution volume were 50 cm 2 and 30 mL, respectively, so that S/V was 1.67 cm Ϫ1 .
The release of Cr͑VI͒ from CCCs on AA1100 aged for 2 h for different S/V ratios is compared in Fig. 3. The Cr͑VI͒ concentration increased with S/V ratio. The steady-state value of Cr͑VI͒ concentration for S/V ϭ 5 was about three times the value for S/V ϭ 1.67, so it varied linearly with this ratio. However, this linear dependence was not maintained for higher S/V ratios. The S/V ratios of 10 and 20 cm Ϫ1 were obtained by coiling the strip sample and immersing it in a glass vial with a relatively little amount of solution. Because of the coiling, it is possible that not all regions of the sample surface had access to the water or that the Cr͑VI͒ concentration was inhomogeneous in the solution. Nonetheless, the measured concentration of Cr͑VI͒ continued to increase with S/V and did not exhibit a limiting value indicative of solution saturation. Accordingly, the plateau of the chromate release curves with time must be determined by something other than saturation of the solution with chromate. One possibility, as described the section on the release model, is that the chromate solution concentration limit is a result of a Langmuir-type adsorption equilibrium with chromate in the CCC. 4 Careful examination shows that the concentration of Cr͑VI͒ in Fig. 3 for S/V ϭ 1.67 cm Ϫ1 is roughly twice that in Fig. 2 for the same S/V. The samples in Fig. 2 were treated in an Alodine solution stored for a long period of time, about 1 year, whereas the samples in Fig. 3 were made in fresh Alodine solutions. The details of sample preparation obviously can greatly affect their behavior. We were unable to investigate this discrepancy further; the samples discussed hereafter were all made in fresh Alodine solutions.
Effect of substrate.- Figure 4 shows Cr͑VI͒ release from CCCs formed on AA1100 and AA2024-T3 substrates. The release of    chromates from CCCs on AA2024-T3 is slightly lower than that from CCCs on AA1100, which indicates that the CCCs on different substrates are different in nature. McCreery and co-workers have shown that the accelerating influence of ferricyanide in Alodine solutions is reduced on Cu and Cu-rich surfaces, such as would be found at the intermetallic particles, as a result of adsorption of Fe͑CN͒ 6 3Ϫ . 8,9 This complex apparently inhibited electron transfer at the surface, thus decreasing Cr͑III͒ generation and CCC formation. The CCC on AA2024-T3 prepared by a 5 min immersion in Alodine bath showed a higher release of chromate than that from a sample given only a 1 min chromate treatment. This suggests that the amount of soluble Cr͑VI͒ increased with increasing immersion time in the Alodine solution.
Release kinetics.-To see the effect of the S/V ratio on release behavior, the UV absorption was normalized. Figure 5 shows a comparison of the chromate release for S/V ϭ 5 cm Ϫ1 and three times the chromate release for S/V ϭ 1.67 cm Ϫ1 . After this normalization, the release curves were almost identical. Therefore, the kinetics of chromate release and the total amount of chromate ͑not the concentration in solution͒ were approximately independent of S/V.
To further investigate the cause of the plateau in the release curves, a second sample made by the same procedure was added to a cell containing a sample that had been exposed for about 400 h of immersion at S/V ϭ 5 cm Ϫ1 ͑see Fig. 6͒. The first sample was kept in the solution, so that the final S/V ratio was 10 cm Ϫ1 . The concentration of Cr͑VI͒ started to increase again after the addition of the second sample. In another experiment, the solution of a sample with S/V ϭ 1.67 cm Ϫ1 was exchanged for fresh DI water after more than 300 h of immersion. A very small amount of Cr͑VI͒ was found in the solution after the exchange. The further increase in chromate concentration with the addition of another sample supports our previous suggestion that the plateau concentration of Cr͑VI͒ does not correspond to saturation. The fact that additional chromate was released when an equilibrated sample was placed in a fresh water solution indicates that the original steady state was not determined by depletion of chromate in the CCC. However, the amount released into the replaced DI water was less than that released into the original solution because the Cr͑VI͒ concentration in the film was lower as a result of the first exposure.
The release curves were normalized in another fashion in Fig. 7. Here, the concentration was normalized by the concentration at the plateau, C(ϱ), to study the release kinetics. Curves of normalized concentration changed considerably with aging time, but they were not strongly influenced by the S/V ratio. This suggests that the release of Cr͑VI͒ was related to the structure of the film, which may have affected diffusion of chromates in the CCC.
Release model.-In a previous paper, the steady-state solution concentrations were shown to be a result of a Langmuir-type equilibrium with Cr͑VI͒ species adsorbed on the CCC. 4 However, the fact that a sample treated for a longer time resulted in a higher solution concentration suggests that the CCC is not a twodimensional surface with chromate adsorption sites, but that Cr͑VI͒ binds to Cr͑III͒ throughout a three-dimensional ͑3D͒ porous Cr͑III͒ film. A diffusion model was developed assuming Langmuir-type equilibrium between dissolved Cr͑VI͒ and a 3D CCC film. This model accounts for all of the observations previously described.
The basis for the model is shown in Fig. 8. The structure of a CCC in contact with an aqueous solution is considered to be a 3D porous Cr͑III͒ matrix or backbone with chromate adsorption sites and an interpenetrating aqueous phase. The chromate concentrations in the aqueous phase within the CCC and the adsorbed Cr͑VI͒, ͓Cr͑VI͒͑aq͔͒ and ͓Cr͑VI͒͑ad͔͒, respectively, can be functions of time and depth from the metal/CCC to the CCC/bulk-solution interfaces. The depth distribution of Cr͑VI͒ is assumed to change with immersion time, as shown by the curves in the figure. Cr͑VI͒ at the CCC/ bulk-solution interface dissolves into the bulk solution and is replen-   ished by outward diffusion of Cr͑VI͒ in the aqueous phase of the CCC. Diffusion in the bulk solution is ignored; the bulk solution is assumed to be homogeneous. Locally within the CCC structure, a Langmuir-type equilibrium is established between the Cr͑VI͒ adsorbed on the CCC backbone and chromate in the aqueous phase within the CCC Cr͑VI͒͑ad͒ Cr͑VI͒͑aq͒ ͓1͔ It is assumed that this Langmuir-type adsorption/desorption equilibrium is fast compared to diffusion in the aqueous phase of the CCC. The binding energy of the adsorbed species is large corresponding to the formation of covalent bonds. 4 This means that the rate constant for adsorption is much higher than that of desorption, so ͓Cr͑VI͒(aq)͔ Ͻ ͓Cr͑VI͒(ad)͔, and the effective diffusivity should be relatively small.
͓Cr͑VI͒͑aq͔͒ and ͓Cr͑VI͒͑ad͔͒ are related locally in the CCC according to the equilibrium of Reaction 1. Because of the interchange of Cr͑VI͒ ions between the adsorption and desorption state, the diffusion equation for the system of chromate release from CCC is more complicated than the usual type of Fickian diffusion.
As discussed in detail in the Appendix, the diffusion of Cr͑VI͒ species in CCC is described by where C f (ϭ͓Cr͑VI͒(aq)͔) is the chromate concentration in the aqueous phase of the CCC film, is the number of the adsorption sites in a unit volume, and is the ratio of desorption to adsorption rate constants, k 2 /k 1 . The concentration of adsorbed chromate, C a (ϭ͓Cr͑VI͒(ad)͔), and the total chromate concentration, C (ϭ͓Cr͑VI͒(aq)͔ ϩ ͓Cr͑VI͒(ad)͔), at a particular x and t are known by The boundary conditions are as follows where C s is the concentration of Cr͑VI͒ in the bulk solution, S is the surface area of the CCC, V is the volume of the bulk solution, and L is the film thickness of the CCC. Due to the time-dependent boundary condition on the right side of Eq. 4, it is difficult to obtain a solution expressed by an equation and to estimate unknown parameters by curve fitting. A program was written to solve these equations and to plot the predicted solution concentration of Cr͑VI͒ as a function of time.
It was previously reported that the chromate release was consistent with a mechanism similar to a Langmuir adsorption isotherm 4 where ⌫ VI is the coverage of Cr͑VI͒ on the Cr͑III͒ oxide in mol cm Ϫ2 , ␤ is a binding constant, and is the fractional coverage of Cr͑VI͒. However, when the rates of adsorption and desorption are the same Note that k 1 is proportional to ͓H ϩ ͔. Owing to its high mobility, it is assumed that ͓H ϩ ͔ is independent of the depth and time. From Eq. 5 and 6 It has been reported that ␤͓H ϩ ͔ ϭ 10 3 mol Ϫ1 L. 4 Therefore, it is predicted that is roughly 10 Ϫ6 mol cm Ϫ3 . Fitting of the model to the experimental data was performed assuming that is about 10 Ϫ6 mol cm Ϫ3 , and the diffusion coefficient, D, the initial concentration of ͓Cr 6ϩ ͔, and the film thickness were estimated. The values of ͑unitless͒ and (mol cm Ϫ3 ) were varied from 1 ϫ 10 Ϫ3 to 2 ϫ 10 Ϫ3 . The unknown parameters are the diffusion coefficient, D, the CCC film thickness, L, the number of the adsorption sites, , and the ratio of rate constants, k 2 /k 1 (ϭ). Sets of values of these parameters were input to the program for obtaining a numerical solution of the differential equation, Eq. 4, and the predicted Cr͑VI͒ concentrations in the bulk solution as a function of time were fitted to the experimental data, providing estimated values of the parameters. L, , and affect the plateau Cr͑VI͒ concentration, but D does not. Figure 9 shows experimental data and curves calculated using a representative set of estimated values of parameters. The model predictions exhibit good agreement with the experimental data, which  indicates that the proposed diffusion model and values of parameters estimated are appropriate. This supports the assumptions that the CCC is a 3D porous Cr͑III͒ matrix or backbone with chromate adsorption sites and an interpenetrating aqueous phase, and that the release kinetics are determined by the diffusion of chromate in the aqueous phase in conjunction with local Langmuir-type equilibrium. The effect of aging is addressed later in this section.
Chromate release curves calculated by using four different sets of estimated parameter values for S/V ϭ 1.67 and 5 cm Ϫ1 are shown in Fig. 10. Changing the parameter values had the largest effect for S/V ϭ 20 cm Ϫ1 , and the values used for the 20 cm Ϫ1 curve in Fig.  10 are those used to fit the data of 2 h aging time in Fig. 9: ϭ 1.0 ϫ 10 Ϫ3 , ϭ 1.5 ϫ 10 Ϫ3 mol cm Ϫ3 , CCC thickness ϭ 1.2 ϫ 10 Ϫ4 cm, and D ϭ 4 ϫ 10 Ϫ11 cm 2 s Ϫ1 . The model predicts that the plateau value of Cr͑VI͒ concentration does not scale exactly linearly with S/V, which was found experimentally, as shown in Fig. 3. The model predicts an increase in release with S/V ratio similar to the experimental data, suggesting that the estimated values of parameters are appropriate and that the Cr͑VI͒ concentration limit results from a Langmuir-type adsorption equilibrium with Cr͑VI͒ in the CCC.
It is possible to use the curves for different S/V ratios to determine which sets of values apply best for each aging time. The estimated diffusion coefficients of CCC samples aged for various periods of time at room temperature are shown in Fig. 11. The bars in this figure ͑as well as in some subsequent figures͒ represent the spread between the highest and lowest values found by fitting the numerical solution to experimental data; different sets of estimated values of parameters result in curves well fitted to the experimental data. The diffusion coefficient tends to decrease with increasing room-temperature aging time, suggesting that the CCC is modified with aging time.
From Eq. 2, the effective diffusivity of Cr͑VI͒ in a CCC is given by D eff is different than D because of the interaction of Cr͑VI͒ with the Cr͑III͒ matrix; D is the diffusion coefficient of Cr͑VI͒ species in the aqueous phase of CCCs with no consideration of the influence of adsorption of Cr͑VI͒ with the Cr͑III͒ matrix. Figure 11 shows that the values for D are on the order of 10 Ϫ11 cm 2 s Ϫ1 , which is significantly lower than that expected for the diffusivity of Cr͑VI͒ ions in bulk solutions. Therefore, the effective diffusivity within a CCC is reduced relative to that in a bulk solution by two factors: the effect of adsorption on the Cr͑III͒ matrix, which is accounted for by the model, and the physical barrier nature of the CCC. The difference between D and typical diffusivities of ions in bulk solutions can be explained by the effect of the network structure of the Cr͑III͒ matrix acting as a diffusion barrier. The CCC can be viewed as a biphase structure consisting of interpenetrating solid Cr͑III͒ matrix and aqueous channels, and the pathways for Cr͑VI͒ diffusion are the aqueous channels. These channels are apparently so tight and convoluted that they impede the transport of the soluble Cr͑VI͒ species through the CCC. This effect is probably larger when the local pH is low and the dominant Cr͑VI͒ species is the larger dichromate anion. Formation of hydrogen bonds may also affect the diffusion of Cr͑VI͒ species. Room-temperature aging probably decreases the Cr͑VI͒ diffusion coefficient in the CCC because of the formation of a multinuclear Cr͑III͒-O-Cr͑III͒ structure 10 that further impedes the diffusion of Cr͑VI͒ species. The structure of CCCs and the effects of aging have recently been studied using extended X-ray absorption fine structure analysis. 11 Aging resulted in a decrease in the Cr͑III͒-Cr͑III͒ nearest-neighbor distances, which is consistent with polymerization associated with dehydration. Note that it is possible to ignore aging effects during the immersion period, because structural rearrangement associated with dehydration would not occur.
It may be more appropriate to compare the diffusivity values to those for ions in polymers, which vary over a wide range, from approximately 10 Ϫ7 cm 2 /s Ϫ1 in polymer electrolytes to near zero for immobile ions or neutrals. [12][13][14] For example, polyethylene oxide electrolytes chosen for high ion mobility exhibit diffusion coefficients for small molecules of 3 ϫ 10 Ϫ7 to 2 ϫ 10 Ϫ8 cm 2 /s Ϫ1 . Much smaller diffusion coefficients are expected for cross-linked polymers such as Cr͑III͒͑OH͒ x , and the values are likely to decrease as the extent of cross-linking increases with time or heat-treatment. Therefore, the D values shown in Fig. 11 are within an acceptable range for cross-linked polymers. Figure 12 shows the effective diffusivity of Cr͑VI͒ in the CCC on AA1100 as a function of room-temperature aging time, position within the CCC, and time of immersion in water. The effective diffusivity decreases with immersion time owing to a decrease in both ͓Cr͑VI͒͑aq͔͒ and ͓Cr͑VI͒͑ad͔͒ as the Cr͑VI͒ is released from the CCC into the bulk solution. The effective diffusivity is lower at the CCC/solution interface than at the Al/CCC interface because of the Cr͑VI͒ concentration gradient within the CCC, as is shown later in this section. The effect of room-temperature aging time is reflected in the change in D shown in Fig. 11.
The estimated initial concentration of soluble Cr͑VI͒ is shown in Fig. 13. The amount of soluble chromate in the CCC does not change markedly with aging time at room temperature, up to 4 days.  This suggests that the Cr͑VI͒ is not reduced to Cr͑III͒ during roomtemperature aging, does not form an insoluble product, and does not form an irreversible bond with the Cr͑III͒ matrix species. Note that the situation changes after very long aging times. A CCC-treated panel of AA3003-H14 contained virtually no Cr͑VI͒ by X-ray absorption near-end structure analysis. 15 The estimated thickness of the CCC is shown in Fig. 14 as a function of aging time. The estimated CCC thickness was almost constant at approximately 1 m, which corresponds with the reported CCC thickness. 5 Figure 15 shows the change in the concentration of adsorbed Cr͑VI͒ on adsorption sites of CCC and free Cr͑VI͒ in aqueous phase in CCC as a function of immersion time. Because the binding of Cr͑VI͒ to Cr͑III͒ is strong, 4 Cr͑VI͒ species exist primarily as adsorbed Cr͑VI͒ in the CCC. The shape of the adsorbed Cr͑VI͒ curves is similar to those of free Cr͑VI͒ because local equilibrium is assumed. The relationship between adsorbed Cr͑VI͒ and free Cr͑VI͒ is expressed by Eq. A-7 in the Appendix. The depth profiles of free Cr͑VI͒ in CCC aged for 2 or 96 h derived from the model calculation are shown in Fig. 16 ͑top͒ and ͑bottom͒, respectively. Because the main difference between 2 and 96 h aged CCCs is in the value of the diffusion coefficient, the shapes of the depth profiles are basically similar to each other, although the time scale is different. The Cr͑VI͒ concentration at the surface of the CCC at the initial stages of immersion is smaller than that at longer times because the Cr͑VI͒ concentration in bulk solution is initially 0 and gradually increases by the Cr͑VI͒ release from CCC. At higher S/V ratios, the change of Cr͑VI͒ concentration at the CCC surface is larger.
Implications of model.-This release model supports the view of a CCC as a 3D porous Cr͑III͒ matrix or backbone with chromate adsorption sites and an interpenetrating aqueous phase. Release of Cr͑VI͒ from this biphase structure is driven by a Langmuir-type equilibrium of the CCC at the film/bulk-solution interface, and the release kinetics are determined by the diffusion of chromate in the aqueous phase in conjunction with a local Langmuir-type equilibrium.
From the corrosion point of view, this model provides important implications as to how Cr͑VI͒ releases in the field. SrCrO 4 pigment in paint does not form a Langmuir-type interaction with the solution in which it is contact; chromate pigment releases Cr͑VI͒ until the solution is saturated. When Cr͑VI͒ in the surface solution reduces to Cr͑III͒ for corrosion protection of a nearby metal, the remaining SrCrO 4 solid releases more Cr͑VI͒ to maintain saturation. The solution remains saturated until the SrCrO 4 is fully consumed. For    Figure 15. Change in the concentration of adsorbed Cr͑VI͒ and free Cr͑VI͒ in aqueous phase at the Al1100/CCC interface and at the CCC/solution interface for CCCs aged 2, 24, or 96 h. The curves were obtained by using the values of parameters used in Fig. 9.
CCCs, the equilibrium Cr͑VI͒ solution concentration depends on the S/V ratio, surface Cr͑VI͒ concentration, and solution pH, etc. Release of Cr͑VI͒ from the CCC film results in a lower surface Cr͑VI͒ concentration. As Cr͑VI͒ in the solution is consumed, the equilibrium Cr͑VI͒ solution concentration decreases. The total amount of chromate in pigmented paint is usually far in excess of the amount in a CCC. However, for a CCC and SrCrO 4 pigment containing the same amount of Cr͑VI͒, the CCC film lasts longer because of the different type of equilibrium reached with the solution.
As shown in Fig. 2, NaCl addition to DI water enhanced Cr͑VI͒ release. It seems that chloride anions inhibit the reversible binding of Cr͑VI͒ to hydrated chromium hydroxide, 16 so that the diffusivity of chromates is enhanced. The higher release rate in the presence of corrosive chloride ions is useful from the viewpoint of corrosion protection.

Conclusions
Release of chromate from CCC was verified by UV absorption measurements. Heat-treatment of CCC markedly reduced the ion release rate. The release rate of chromates from CCC decreased with aging time at room temperature.
A diffusion model was proposed in conjunction with a local Langmuir-type equilibrium with an interpenetrating solution phase. Using this model, the diffusion coefficient of chromates in the CCC, the CCC thickness, and the initial concentration of Cr͑VI͒ were estimated. The diffusion coefficient decreased with aging time at room temperature, presumably because of the development of the CCC matrix.
The total concentration of Cr VI within the coating is a function of the position, x, and the immersion time, t, and is given by C(x,t), with x ϭ 0 at the metal/coating interface and x ϭ L at the coating/solution interface. In general, C is comprised of two contributions, i.e., the concentration C f of Cr VI that is free to diffuse, and the concentration C a that is adsorbed on any of the adsorption sites available per unit volume of the film. We assume that is independent of x and t. Hence Adsorption kinetics-We assume that C f and C a are related by simple Langmuirtype adsorption kinetics. In that regard, consider a thin ''slab'' of coating extending from x to x ϩ dx. The number of adsorption sites per unit area of this slab is dx. Let (x,t) be the fraction of these sites occupied by adsorbed Cr VI , from which we deduce that We consider the insoluble Cr͑III͒ to have many surface hydroxyl groups available for formation of Cr͑III͒uOuCr͑VI͒ bonds. These hydroxyl groups act as sites for adsorption of Cr͑VI͒, accompanied by the addition of a hydrogen ion to produce water. The reaction may be represented by 4 Cr͑III͒-OH ϩ Cr͑VI͒͑aq͒ Cr͑III͒uOH represents insoluble Cr͑III͒ oxyhydroxide, Cr͑VI͒͑aq͒ is solutionphase Cr͑VI͒ species which include oxygen atoms, and Cr͑III͒-O-Cr͑VI͒ is the adsorbed state of Cr͑VI͒ to Cr͑III͒ by oxygen. Note that Cr͑VI͒͑aq͒ has one oxygen atom more than Cr͑VI͒͑ad͒.
The reaction is reversible, so the CCC may release Cr͑VI͒, and the Cr͑III͒ oxyhydroxide may adsorb Cr͑VI͒ from the aqueous phase. Considering the existence of Cr 2 O 7 2Ϫ as a Cr͑VI͒ species, the reaction of insoluble Cr͑III͒ oxyhydroxide with Cr 2 O 7 2Ϫ can be expressed by Reaction A-3-2. 16 Two Cr͑III͒ are connected with each other by two ol-bridges From the stoichiometric point of view, Reactions A-3-1 and A-3-2 are basically the same. The adsorption/desorption reactions within the slab are described as follows where k 1 and k 2 are rate constants, with k 1 being proportional to ͓H ϩ ͔. It is assumed that C a Ͼ C f because the binding energy of the adsorption is high, corresponding to chemisorption. 4 We assume that ͓H ϩ ͔ is independent of x and t. Langmuir-type kinetics involves taking the two rates to be equal We can now establish a relationship between C f and C, i.e.
where ϵ k 2 /k 1 . Note that C a Ͻ , as expected. We now have the following quadratic equation Figure 16. The in-depth profile of free Cr͑VI͒ in aqueous phase in CCC aged at room temperature ͑Top͒ Aged for 2 h. Cr͑VI͒ profile at various immersion times is plotted. The curves were obtained by using the values of parameters used in Fig. 9 and 14. ͑Bottom͒ Aged for 96 h.
only one solution of which is physically acceptable, that being The other solution would give C f Ͻ 0. Thus, we have C f Ͼ 0, and we can show from Eq. A-7 that C f Ͻ C, leading to the conclusion that 0 Ͻ C a Ͻ C, all of which are physically correct.
The corresponding relationship for C a is easily obtained, i.e.
The diffusion equation-The diffusion equation appropriate to this system is understandably more complex than the ''usual'' Fickian-type expression, due to the interchange of Cr VI between the free ͑diffusing͒ and bound ͑adsorbed͒ states. However, it can be derived based on a simple conservation-of-matter argument. Toward that end, consider again the slab of film extending from x to x ϩ dx, taking the surface area of the film to be S. Then The only way this addition of matter can occur is by the diffusive flow of free Cr VI , so that where D is the diffusivity of the free Cr VI . The first term on the right side of Eq. A-12 is the rate of influx of free Cr VI into the slab, and the second term is the rate of egress. Expanding that second term in a Taylor series and combining terms dropping the (x,t) arguments. The second form of Eq. A-13 is recognizable as Fickian diffusion in the presence of a sink. Dependence upon C a in Eq. A-13 can be eliminated using Eq. A-7, which relates C a to C f . We thus obtain Knowing C f at a particular x and t also tells us the values of C a and C at that position and time via Eq. A-7. Equation A-14 can be scaled, and in so doing, all explicit dependence on the parameters and D is eliminated. Toward this end, we define the following dimensionless variables in terms of which Eq. A-14 is reduced to Some of our earlier results can also be scaled if we first define in which case Eq. A-4, A-9, and A-10 become, respectively Initial and boundary conditions.-The following initial conditions are imposed where C s (t) is the concentration of Cr VI in the solution beyond the film, which occupies volume V. Moreover, it is assumed that C f (x,0) and C a (x,0) are related to C(x,0) through Eq. A-9 and A-10, respectively. The boundary conditions are as follows The first of the two boundary conditions is a statement of the assumption that Cr VI does not diffuse backward, into the metal. The second is a statement of the fact that C s (t) results from the diffusive flow of Cr VI from the film into the solution.
The boundary conditions can be scaled to make them consistent with Eq. A-16. Toward that end, we define the following new dimensionless parameters s ϵ Ϫ1 C s , ϵ 2/3 S, ϵ V, ϵ 1/3 L ͓A-23͔ Combining Eq. A-15 and A-23 with A-22, we find that the scaled boundary conditions are Note that there is no explicit dependence on D or in Eq. A-24.
Global conservation of matter.-The diffusion equation is an expression of conservation of matter at the local or microscopic level. A global or macroscopic expression for the conservation of matter is This expression can also be presented in scaled form, first defining But a ϭ f /( f ϩ ), so differentiation with respect to yields Numerical representation.-Equation A-16 represents a partial differential equation with boundary conditions expressed by the first expression in Eq. A-24 and by Eq. A-29. This partial differential equation can be reduced to a system of ordinary differential equations by discretizing the axis and assigning an ordinary differential equation to each of the points considered. To do this, divide the interval from 0 to into 2n equal subintervals. Then there will be 2n ϩ 1 nodes and 2n ϩ 1 ordinary differential equations. Let Let i , f,i , and a,i be the values of , f , and a , respectively, at i , for i ϭ 0, 1, ..., 2n. The quantities i , f,i , and a,i are functions of alone. In terms of these new functions, the initial condition is a,i ͑ 0 ͒ ϩ f,i ͑ 0 ͒ ϭ o for i ϭ 0, 1, ..., 2n Ϫ 1 ͓A-32͔ Given the value of o , the values of f,i (0) can be obtained by applying Eq. A-19. The value of ‫ץ‬ 2 f ‫ץ/‬ 2 can be approximated by taking finite differences Note that C must become independent of x as equilibrium is approached. In addition from which we obtain We can eliminate C a using Eq. A-7 Combining Eq. A-45 through A-47 Dividing through by SL and defining ϵ V/SL Equation A-49 can be expressed as a quadratic expression in C s (ϱ), i.e.