# Generation of chromate diffusion models

Introduction

Chromates (chromium (VI) salts, salts of chromic acid, salts containing the divalent ion, (CrO4)^(–)) are widely used in the aerospace industry for corrosion protection, primarily of aluminium alloys. Although efforts are being made to replace chromates with less harmful alternatives, they are still in widespread use. It is therefore important to understand the role of chromates in corrosion prevention and the mechanisms by which it is achieved.

Corrosion protection schemes for metallic structures, particularly those containing aluminium alloys, imply generally an initial surface treatment (usually involving either chemical conversion or anodising), followed by the application on the resulting oxide layer of a corrosion-inhibiting primer containing a soluble chromate salt. If required, a topcoat can be applied to provide additional protection from harsh environments. Typically, chromate salts of heavy metals such as barium chromate (BaCrO_4) and strontium chromate (SrCrO_4) are used in corrosion-inhibiting aerospace primers.

The corrosion protection system is designed to function in the following manner: when a scratch occurs that penetrates the chromate-loaded primer through to the underlying aluminium alloy substrate, the metal is exposed. Contact with an aqueous medium can then trigger the corrosion process leading to the deterioration of the alloy. However, soluble chromates present in the primer are able to hinder the corrosion process. They dissolve into the aqueous medium and are then transported to the exposed site, where chromium ions react with the aluminium alloy producing a passive layer, arresting further corrosion. Thus, the leaching (or dissolution) of chromate is an important step in corrosion protection process.

Modelling of the leaching process under different conditions may improve understanding of corrosion protection and may lead to a reduction in the amount of testing required for the qualification of new corrosion protection systems.

Research has shown clear time dependency of chromate leaching (Prosek and Thierry 2004, Xia 2000, Scholes et al. 2006): generally, leach rates decrease with time. It has also been reported that the pH of the aqueous medium that comes into contact with the primer has an influence on leaching. High leach rates are found at low pH (Furman et al. 2006). Solubility of chromates has been regarded as not being a controlling factor as the concentration of leached chromate in solution in the immersion medium is low compared to that of a saturated solution (Xia 2000).

The model proposed by Furman et al. 2006 defines the mass of released material per surface unit (Mt, measured in mg/dm2) as proportional to a power of time (the model was found exploiting Fick’s second law of diffusion):

Mt = k Deff (t^n)              (1)

in which Deff is the effective diffusion coefficient, k is a constant and t^n is a power of time.
Furman et al. proposed to use a constant value of 0.25 for the exponent n of the time variable for all primers under various conditions, letting the effective diffusion coefficient Deff be the only varying parameter in the model described in Eq. 1. However, experiments performed on four primers (not described here), whose results are shown in Fig. 1, show that the validity of the Furman’s simplified model is rather limited. Tests proved that the effective diffusion coefficient D_eff has a different value for each primer, and the power of time n cannot be assumed to be equal for all primers either. For example, a basic power fit of the form c(t^n) to the data plotted in Fig. 1 results in different values of the exponent n for primers Aerodur HS 37092 (n=0.33) and Seevenax 313-01 (n=0.12).  This proves that a model using a constant power of time is not appropriate for modelling the leaching process and new models are needed.

Figure 1 :  Leaching curves of different primers in deionised water. The mass of chromate leached per surface unit of the aluminium sample is plotted against time for different primers: $\Diamond$ = Aerodur HS 37092, $\triangledown$ = F580-2080, $\times$ = Seevenax 313-01, $\circ$ = Mapaero P60-A

Methodology

The aim of the research activity here presented is to provide mathematical models of the quantity of chromate salts dissolving into an aqueous solution from an aluminium alloy sample treated with a chromate-loaded primer. The study has focused on the behaviour of three different primers, which will be referred to as A, B and C in the following (the correspondent commercial names are F580-2080 for primer A, Aerodur S15/90 for primer B, Epoxy Primer 37032A for primer C.).

The generation of the models has only been the final stage of a process that involved the following steps:

1. preparatory testing
2. chromate leaching measurement
3. model generation through genetic programming

The first two steps were carried out independently by Mr D. J. Boon, Dr. L. J. Clarke and Mr M. B. Stowe and the gathered data were kindly provided for the metamodelling activity.

During the preparatory stage a sensitivity analysis of the leaching process was carried out to identify the main variables involved. Tests showed that the temperature of the aqueous solution where the treated alloy sample is immersed does not have a significant effect on the leaching over a range of 9° C to 50° C and for a time period of up to two months. As a result, time and pH were identified as the independent variables to use in the design of experiments. In total, 35 sample points were defined for primer A, 72 for primers B and C. The design of experiments (DoEs) used for each primer are shown in Fig. 2.

Figure 2 : Design of Experiments (DoEs) for primers A, B and C

In the second stage, the coated aluminium samples were immersed in an aqueous solution of 300 ml and the total quantity of chromate dissolved into the solution measured at different times and for different initial pH values of the solution, according to the design of experiments for each primer. Over a period of 2 months, 20-ml samples were removed from the medium and chromate content and pH of the samples measured. The chromate concentration in the sample was measured through Atomic Absorption Spectrometry, using an AAnalyst 400 Spectrometer from PerkinElmer, and the total mass of chromate in the solution obtained multiplying the concentration by the volume of the solution. The total volume of the solution was then made back up to 300 ml by adding 20 ml of fresh medium. If the pH of the solution had drifted from the required value, the 20-ml sample that was added to the solution was also used to adjust the pH of the solution back to the desired pH. Because the rate of leaching reduces with time, measurements were taken more frequently at the beginning of the experiment than at the end, as it is shown in the design of experiments for the three primers in Fig. 2.

Modelling using genetic programming

In the third and final stage, data obtained by the measurements on the samples were processed by a HyGP to generate empirical models for the quantity of dissolved chromate as a function of time and pH. Ten independent runs for each input data set were launched to increase the chances of finding acceptable models. A constant population of 200 individuals and 100 generations were used in all the experiments. The functions (primitives) available to the code were the standard algebraic operations (addition, subtraction, multiplication and division) as well as power, sine, cosine, logarithm, reciprocal and the hyperbolic functions. An additional operation to introduce translation by a numerical value was also used.

In the following paragraphs the best models generated for each primer are described. pH is represented by pH; time is measured in hours, and represented in the mathematical expressions with the letter t.

Primer A

The best model found for sheer root mean square error is reported in Eq. 2 :

f(t,pH) = .00245694 t + 462.344 t / (99.2640 t – (95.1374 t/pH) + 203.436) – 0.190638        (2)

Its coefficient of determination is R2 = 0.9958987, the maximum error -2.466175 mg/dm2.
However, the most significant model generated is described in Eq. 3:

f(t,pH) = 4.41652 (1.91811 t)^(1/(4.87395 pH – 3.04801)) – 0.498776                       (3)

having a coefficient of determination (R2 = 0.995062) slightly lower than the previous one’s. The maximum error, 2.341443 mg/dm2, is however smaller. The corresponding plot is shown in Fig. 3, together with a graphical comparison of the estimated response against the measured response. The particular interest in Eq. 3 comes from the striking resemblance to the model proposed by Furman et al., described in the introduction (Eq. 1).

Figure 3 : Generated model for primer A

Primer B

For primer B HyGP was not able to generate models as simple as the ones found for primer A (Eqs. 2 – 3). The extended pH range or the noise in measurements may have forced the algorithm to increase the average size of the mathematical expressions to increase the accuracy. In other words, the models are generally affected by “bloat”, which is a well-known drawback affecting genetic programming techniques (Poli et al. 2008).

The best model found as per root mean square error is:

f(t,pH) =  0.024784 t pH
– tanh(0.125306 t)*(0.0898783 t + 0.36776 pH – 5.1529)
–  0.00318441 pH + (0.25614 t)/(pH^2) – 0.00161642 t pH^2
+ (3.39314 10^(-17)) t pH^14 + 0.00567243                                            (4)

Though complex, the model shows a high coefficient of determination (R2 = 0.970065). The maximum error is -5.889410 mg/dm2. The corresponding plot is shown in Fig. 4.

Actual vs. estimated response

Figure 4 : Generated model for primer B

Primer C

The models found for primer C were neither compact nor easily interpretable. On average, models were larger than in the first case (primer A), being mainly linear combination of non-linear terms.  The model having the best root mean square error is:

f(t,pH) & = 0.243435 pH – 0.00647727 t + tanh(107.85 t)
– (0.000514483/pH)*(210.141 t – 569.132 t/pH + 0.00000156793 t^4/pH)
+ 0.00708876 t pH – 0.0000825987 t pH^3 + 0.000000282282 t pH^5
– 0.0197306 pH^2 – 0.527521                                                                      (5)

The corresponding coefficient of determination is relatively high R2 = 0.983118. The maximum error is -2.542556 mg/dm^2. The plot of the model is shown in Fig. 5.

Figure 5 : Generated model for primer C

Discussion

HyGP has shown the ability to produce high quality models for the three data sets provided by the experiments. The availability of explicit mathematical expressions (Eqs. 2 to 5) eases the interpretation of the data produced by the experiments on the treated aluminium samples.

With regards to the influence of pH value on the leaching, high chromate releases are highlighted at low pH (pH<2) for all three primers, whereas for primer B high releases are detected even for high pH (pH>12). A local maximum is detected near neutral pH for primers B and C. During the experiments on the aluminium samples it was noted interestingly that the pH of solutions having a not neutral pH (pH<=7) had a tendency to move towards neutral. This behaviour, noticed also by Furman et al. 2006, indicates that there are reactions going on between elements from the primer and the acidic or alkaline solution. According to Kondratenko et al. 1986 the drift in pH is the result of four equilibrium reactions going on between the acidic or alkaline solution and chromates.

The general trend of the leaching rate with respect to time has already been discussed in the introduction (see Fig. 1) and the models generated by HyGP are consistent with it. As shown in Fig. 3-4-5 the leaching is initially high and slows down with time. Such reduction in the leaching rate can be explained considering the different times needed by the chromates to diffuse out of the primer, according to their position in the primer film: the chromates near the surface of the primer dissolve rapidly, whereas the chromates sited below the surface need more time to diffuse outwards. Diffusion of chromates out of the primer plays then an important role in corrosion protection, as the efficacy of the corrosion protection process depends on the quantity of chromate leached, and this quantity can only be increased easing the diffusion of the chromates out of the primer.

Focusing instead on the method used to generate the models, the results of the previous paragraph show genetic programming capability to extend or generalise existing chromate leaching models. The model in Eq. 3 is in fact a generalisation of the model found by S.A. Furman et al. 2006 discussed in the introduction (Eq.1), as not only time but also pH is included in the model.  Where simple and interpretable expressions were not found, as for the cases of primers B and C, genetic programming still provided high-quality models.

Conclusion

The results described show that HyGP is able to produce high quality models processing experimental data. The accuracy and the validity of the models is confirmed by the fact that in the case of primer A genetic programming was able to generalise an existing leaching model proposed by Furman et al. 2006. Results show that although the interpretability of the models may be reduced by excessive complexity (see primers B and C), the accuracy is not penalised. Moreover, the current genetic programming implementation has shown an intrinsic smoothing behaviour, filtering out sample points that represent unlikely deviations from the general trend of the data.

The generated models can represent an additional help in the study of the chromate leaching process for the primers analysed. As a result, genetic programming may be considered as a tool mature enough to be applied in industry, at least for low-dimensional problems like the one described here.

References

• S. A. Furman, F. H. Scholes, A. E. Hughes, and D. Lau. Chromate leaching from inhibited primers – part II: Modelling of leaching. Progress in Organic Coatings, 56:33–38, 2006
• N. Kondratenko and V. Sherstyuk. Spectroscopic characteristics of Cr(VI) oxyanions in water solutions. Theoretical and Experimental Chemistry, 22:656–662, 1986
• J. R. Koza. Genetic programming: on the programming of computers by means of natural selection. MIT press, Cambridge, MA, USA, sixth edition, 1992
• R. Poli, W. B. Langdon, and N. F. McPhee. A field guide to genetic programming. Published via http://lulu.com and freely available at http://www.gp-field-guide.org.uk, 2008
• T. Prosek and D. Thierry. A model for the release of chromate from organic coatings. Progress in Organic Coatings, 49:209–217, 2004
• F. H. Scholes, S. A. Furman, A. E. Hughes, T. Nikpour, N. Wright, P. R. Curtis, C. M. Macrae, S. Intem, and A. J. Hill. Chromate leaching from inhibited primers – part I: Characterisation of leaching. Progress in Organic Coatings, 56:23–32, 2006
• L. Xia. Storage and release of soluble hexavalent chromium from chromate conversion coatings. Journal of The Electrochemical Society, 147:2556–2562, 2000