Abstract
In Japan, high level radioactive waste (HLW) produced by reprocessing spent fuels has been planned to be fixed in the glass solid body which is enclosed in the stainless steel canister which is further enclosed in the metal overpack. The overpacks are emplaced in tunnels at a deep underground facility. Carbon steel is the most important candidate material for the overpack, because the corrosion rate of carbon steel is known to be as low as 10 μm/year at an assumed deep underground site which is kept at oxygen depleted condition. The corrosion rate of carbon steel measured in the laboratory was found to be almost 10 μm/year at the initial stage, and decreases to a steady state corrosion rate in the order of 0·1-0·01 μm/year, changing with time according to an inverse parabolic rate law. A corrosion model assuming the diffusion of H2O molecule through the precipitated corrosion film (Fe3O4) has been developed to rationalise the observed corrosion rate law under oxygen depleted environment. In this study, an improved model which takes account of the corrosion film dissolution has been proposed by assuming that the dissolution of precipitated Fe3O4 film is controlled by mass transfer process through adjusting diffusion layer in the solution. Digital simulation based on the improved corrosion model with suitable parameters such as the diffusion constant of H2O in the corrosion film and concentration of dissolved Fe2+ species at the surface of the Fe3O4 corrosion film was found well to simulate the observed corrosion rate change with time at various pH values as a function of partial pressure of hydrogen gas and to predict a low and steady state corrosion rate after long time exposure.
Introduction
High level radioactive waste (HLW) has to be isolated from people and the environment for thousands of years. In Japan, 1 HLW produced by reprocessing spent fuels has been planned to be fixed in a glass solid body which is enclosed in the stainless steel canister which is further enclosed in the metal overpack and stored at the deep underground facility.
Carbon steel is the most important candidate material for the overpacks, because the corrosion rate of carbon steel is known to be less than 10 μm/year at an assumed deep underground site which is kept at oxygen depleted condition. Actual corrosion rate data of the candidate steel at the anaerobic exposure condition have been accumulated in several laboratories. Taniguchi et al. 2 had reported that the average corrosion rates of carbon steel embedded in bentonite immersed in synthetic sea water or bicarbonate solution with chloride ions between 1 and 10 years at 80 or 50°C at anaerobic condition were in the range of 0·055-1·4 μm/year and the initial corrosion rate in the range of 10-25 μm/year decreases gradually with time. They had admitted that the corrosion film was magnetite (Fe3O4) in the case of low bicarbonate solution, while siderite (FeCO3) film was formed in the solution containing high concentration of bicarbonate ions. Kaneko et al. 3 had reported the corrosion rate of carbon steel at 35°C in a calcium hydroxide equilibrated solution at anaerobic condition which is simulated to a concrete contact solution showed the initial high value of 0·2 μm/year which decreased with time and reached to 0·02 μm/year after 900 days.
King 4 had reviewed the anaerobic corrosion behaviour of carbon steel under environmental conditions similar to those expected in the repository. He had pointed out that the corrosion rate in bulk solution decreases to an apparent steady state rate after six months with a long term rate of 0·1 μm/year. In compacted clay systems the rate of decrease in corrosion rate is slower with steady state not being reached after several years exposure and a steady state corrosion rate is in the order of 1-2 μm/year.
For the reliable life prediction of the carbon steel overpack, a rational model for understanding the corrosion behaviour is required with accumulation of exact corrosion data in the laboratory and actual field.
A carbon steel corrosion model 5 assuming the diffusion of H2O molecule through the precipitated corrosion film (Fe3O4) has been developed to rationalise the observed corrosion rate law under oxygen depleted environment.
In this study, an improvement of the original corrosion model 5 is proposed by assuming that the dissolution of precipitated Fe3O4 film takes place simultaneously with the film growth. The dissolution rate is assumed to be controlled by mass transfer process through diffusion layer in the solution. Digital simulation based on the improved corrosion model with suitable numerical values for concentration of dissolved species of Fe3O4 at the surface of the corrosion film and their diffusion constant has been executed to simulate the corrosion rate change with time as a function of pH and partial pressure of hydrogen gas to predict a low and steady state corrosion rate after long time exposure.
Observed corrosion rate of carbon steel under oxygen depleted environment
The corrosion rate of carbon steel under oxygen depleted environment has been reported by using various measurement methods such as the weight loss measurement,
2
the hydrogen gas evolution measurement
3
and the electrochemical hydrogen permeation current measurement.
6
Examples of corrosion rate of carbon steel at 303 K in various pH solutions of (Ca(OH)2+5000 ppm NaCl) under the oxygen depleted environment which simulates the concrete contact solution are shown in Fig. 1,
6
the data in which were measured by the hydrogen gas evolution method
3
as well as the hydrogen permeation current method.
6
Corrosion current density (μA cm–2) can be converted to the corrosion rate expressed by thickness loss per year (μm/year) by multiplying 11·6, which is obtained from the following calculation
Change in corrosion current density, i corr, of carbon steel with time in various pH solutions of (Ca(OH)2+5000 ppm NaCl) under oxygen depleted condition. Corrosion current density up to 105 s was measured by the hydrogen permeation technique (i H), 6 while data from 105 to 107 s were obtained by the gas evolution measurement 3
Thus, 10 (μm/year) shown at the right axis in Fig. 1 is equivalent to 0·86 or approximately 1 (μA cm−2).
As can be seen in Fig. 1,5,6 in the very initial stage from 102 to 105 s, the corrosion current density measured by the hydrogen permeation current method in the solution between pH = 6·6 and 14·4 decreases with time and, in the latter stage from 105 to 107 s, the corrosion current density measured by the hydrogen gas evolution method in the solution between pH = 10·5 to 14·0 decreases with time in a similar slope. It has to be emphasised that continuous decrease in corrosion current was observed in both data measured by different methods3,6 and changes with −1/2 slope in log−log plots. Continuous decrease in corrosion current density was always observed in neutral and alkaline range between pH = 3·5 to 14, whereas in acidic range below pH = 3·5 no time changes were observed as shown in Fig. 2.5,6 Steady corrosion current after a long immersion time shows a concave curve showing a minimum around pH = 10 as can be seen in Fig. 2.5,6
Corrosion current density, i corr, of carbon steel in (Ca(OH)2+5000 ppm NaCl) solution as function of pH under oxygen depleted condition. Steady corrosion current density measured by hydrogen permeation technique 6 was obtained after two weeks and data measured by hydrogen evolution method 3 were obtained after 300-400 days
Corrosion model of carbon steel under oxygen depleted environment
Corrosion process of carbon steel
As shown in Fig. 3, corrosion process of carbon steel under the oxygen depleted environment proceeds by coupling anodic reaction of Fe
Corrosion model of carbon steel under oxygen depleted environment
Fe2+ ion produced by equation (1) combines with OH− ion, forming a precipitated Fe(OH)2 film on the surface.
Precipitated film changes to magnetite by Sickorr reaction
In the solution containing carbonate ion, precipitation of siderite might be possible to take place according to
Solid phase H2O diffusion model 5
In the oxygen depleted environment, corrosion current density of carbon steel, i corr, was found to decrease with time, t, as shown in Fig. 1, according to the following inverse parabolic rate law
5
Since the corrosion current density is expressed as
The parabolic rate law means that the corrosion rate of carbon steel is controlled by a diffusion of chemical species such as Fe2+, O2− or H2O through the precipitated corrosion film or through pores in the precipitated corrosion film.
According to the mass transport model assuming series combination of diffusion of chemical species through corrosion film and boundary layer in the solution,5–7 the corrosion rate or corrosion current density, i corr, is expressed by
is concentration difference and D is diffusion constant. The parameter of 1/k 1 increases with growth of corrosion film formed by precipitation of Fe2+ ions which are produced by the anodic reaction of (1) or i corr. The thickness of the precipitated film is calculated by the amount of dissolved Fe2+ ions which are produced by i corr and a precipitation ratio of Fe2+ ions depending on pH. The corrosion current density changing with time could be calculated as a function of the amount of the precipitated film by using the Excel spread sheet.
By assuming appropriate species with reasonable diffusion constant, a digital simulation using the Excel spread sheet had been executed and compared with the experimental data as shown in Fig. 4. 5 Aqueous diffusion of Fe2+ ions or H2O molecules through pores in the film consisted of Fe3O4 or Fe(OH)2, and solid phase diffusion of H2O molecule through the film of Fe3O4 were compared based on the simulation. After repeated trials, it was concluded 5 that a reasonable simulation could be obtained in the case of the solid phase diffusion of H2O molecule through the corrosion film of Fe3O4 by using the diffusion constant of 8·15×10−17 cm2 s–1 reported for SiO2.
Simulation based on previous corrosion model 5 assuming diffusion of Fe2+ or H2O through film or pores in film. Simulation assuming diffusion of Fe2+ or H2O in aqueous phase in pores which occupy 0·01 fraction in film were plotted as + or ○, simulation assuming solid phase diffusion of H2O in Fe(OH)2 and Fe3O4 film were plotted as Δ and □. The most appropriate curve fitting shown as ⊚ was obtained by assuming solid phase diffusion of H2O through Fe3O4 film using diffusion constant of H2O in SiO2 glass
A modified corrosion model of carbon steel which introduces the corrosion film dissolution
The observed time change in corrosion current density from beginning to years could be simulated by assuming suppression by a precipitated corrosion film, as shown in Fig. 4, 5 although the model does not predict a steady state in corrosion current density after a long immersion time. It is reasonably assumed that the steady state corrosion rate will be obtained when the corrosion film reaches a constant thickness due to film dissolution.
In the case of carbon steel corrosion in high temperature water at depleted oxygen condition, the corrosion rate8,9 has been known to be controlled by the diffusion-limited dissolution of the corrosion film through an aqueous phase boundary layer at the film surface.
Then, a similar dissolution mechanism of corrosion film is introduced for prediction of the steady state corrosion current density after extended time. It is assumed that a corrosion film composed of Fe3O4 dissolves as Fe(OH)b (2–b)+ species which are in equilibrium with solid Fe3O4 by liquid phase diffusion trough the boundary layer of δ.
The dissolution current density, i d, is expressed as
Estimation of concentration of Fe(OH)b (2–b)+ species on surface of corrosion film
Estimation of K b
Fe(OH)b (2–b)+ species (b = 0, 1, 2, 3) which are in equilibrium with Fe3O4 in solid state are Fe2+, Fe(OH)+, Fe(OH)2, and Fe(OH)3 −, equilibrium equations of which can be expressed by
The equilibrium constants for equations (11)–(14) are expressed as K 0, K 1, K 2, and K 3, and concentration of each species can be calculated by
Total concentration of Fe(OH)b (2–b)+ is given by
.
Sweeton and Baes
10
had reported the values of K b (b = 0, 1, 2 and 3) which were determined from the solubility data of Fe3O4 in high temperature water from 50 to 300°C at 
atm. The values of K b are shown in Table 1 and plotted as a function of temperature in Fig. 5.
Equilibrium constant of Fe(OH)b (2–b)+ species as function of absolute temperature 10
K b (b = 0, 1, 2 and 3) for Fe(OH)b (2–b)+ species at various temperatures 10
Estimation of concentration of Fe(OH)b (2−b)+ species
By using equations (15)–(18), concentration of Fe(OH)b (2−b)+ species are calculated and the total concentration (C T) can be obtained by equation (19). In Fig. 6, the calculated concentration of Fe(OH)b (2−b)+ species and C T are plotted with measured values as a function of pH at 298 K and 
atm
Concentration of Fe(OH)b (2−b)+ species and total concentration plotted as function of pH at 298 K and atm
As it can be seen in Fig. 6, almost the same value for calculated and measured values of CT was obtained at pH = 10, although a discrepancy was observed in the range between pH = 4 and pH = 8. This difference might be caused by difficulty to attain the equilibrium condition at 298 K.
Estimation of dissolution current density of corrosion film
As mentioned before, the dissolution current density of the corrosion film can be given by equation (10)
Since Fe(OH)b (2−b)+ species dissolve as Fe2+, n = 2 is into equation (10). D = 7·19×10−5 cm2 s−1 for Fe2+ ion 11 was used instead of D for Fe(OH)b (2−b)+ species which could not be found in the literature. The thickness of Nernst diffusion layer is assumed to be δ = 0·05 cm. 11 It is assumed that the bulk concentration is equal to C 0 = 0 mol cm−3.
Then dissolution current density (A cm−2) is given by
, and is plotted in Fig. 7.
Dissolution current density of corrosion film (Fe3O4) at 298 K as function of pH and
As it can be seen, i d depends largely on pH in the same fashion as C T, showing a minimum at pH = 10. It should be noted that i d decreases with decreasing partial pressure of hydrogen gas, which will decrease in open system due to escape of evolved hydrogen gas to an outside environment.
Corrosion current density based on modified model which introduces corrosion film dissolution
By adding dissolution current density of the corrosion film to the original corrosion model expressed by equation (9), the corrosion current density based on the modified model can be expressed by
For digital simulation using the Excel spread sheet, equation (21) is transformed to the following expression
Constant of A in equation (22) includes 2FD⊿ 
which is the initial corrosion current density depending on corrosion potential, and G is the mass transfer coefficient depending on Reynolds number, Q is the electricity passed through carbon steel surface and H is the precipitation ratio of dissolved Fe2+ ion as corrosion film. Then the thickness of corrosion film is proportional to HxQ(t–1). J d is dissolution current density depending on pH and 
.
Effect of pH on corrosion current density of carbon steel
The corrosion current density based on the modified corrosion model was calculated by the Excel spread sheet using equation (22) and plotted in Fig. 8. The model assumes that corrosion film is Fe3O4, through which H2O diffusses from the solution to the carbon steel surface. D of H2O in SiO2 was used in the calculation, 5 because no reliable data of D of H2O in Fe3O4 have been found. It is clearly seen that corrosion current density decreases with time according to −1/2 slope in the initial stage, after which a steady corrosion current density appears at an extended time. The time of the inflection point, at which the slope changed from −1/2 to zero was affected by pH. In addition the level of corrosion current density after the inflection point was affected by pH. The lowest steady corrosion current density after the longest inflection time was observed at pH = 10 as expected from Fig. 7.
Effect of pH on corrosion current density which was obtained by simulation based on improved model at 298 K and atm
Effect of 
on corrosion current density of carbon steel
The same calculation using the Excel spread sheet was executed at pH = 10 and 298 K with changing the partial pressure of hydrogen gas at 
, 0·01 and 0·001 atm and the results are plotted in Fig. 9. It is interesting to note that the steady corrosion current density decreases with decreasing partial pressure of hydrogen gas. At 
atm, the inflection point to reach the steady corrosion current density is 109 s (about 100 years), at 
atm the inflection point is 1011 s (about 10 000 years), and at 
atm the inflection point is 1012 s (about 100 000 years). With increasing time to reach the inflection point, the corresponding steady corrosion current density decreases. In an open environment, hydrogen gas which is evolved by corrosion of carbon steel will escape to the outside and does not accumulate within the environment, resulted a lower partial pressure of hydrogen gas pressure, which causes the lower steady corrosion current density.
Effect of on corrosion current density which was obtained by simulation based on improved model at 298 K and pH = 10
Comparison of simulation data based on improved model with experimental data
Comparison of simulation data by the previous model with the experimental data is shown in Fig. 4. To this figure, simulation data at pH = 10 and 
atm based on the improved model which introduces the dissolution of the corrosion film are added in Fig. 10.
Comparison of simulation data based on previous 5 and improved model with experimental data. Improved model predicts steady corrosion current density after 109 s (about 100 years)
It is clearly seen in Fig. 10 that the steady and constant corrosion current density of 10−3 μA cm−2 is predicted after 109 s (about 100 years), for which, however, no exact experimental data have been obtained right now because of limited times in the laboratory experiment. It has to be emphasised that the lower constant corrosion current density will be expected at pH = 10, at which the precipitation of Fe2+ ion onto the corrosion film proceeds at a high rate, suppressing the corrosion current density. At another pH such as pH = 9 or pH = 11, the inflection time becomes faster and more higher steady corrosion current density will be obtained as indicated in Fig. 8.
Comparison of simulation data for corrosion loss with laboratory and field data
Simulation data for corrosion loss based on the original model were plotted as (i corr without i d) and data based on the improved model were plotted as (i corr with i d) in Fig. 11. In the same figure, the laboratory data which were measured by Taniguchi et al. 2 in synthetic sea water and aqueous solutions containing bicarbonate ion and chloride ion for 10 years at anaerobic condition were plotted. In addition, field data which were reported by Sumiyama et al. 13 for carbon steel buried in soil for long term were shown. Honda and Yamaguchi 14 reported an empirical relation of H = 0·298Y 0·502 between corrosion loss (H) and time (Y) which was indicated in upper part as a straight line in Fig. 11. It is interesting that the slope of the line ( = 0·502) is almost identical to that of equation (8) which was observed in the laboratory experiment. The experimental data by Taniguchi et al. 2 show lower values of corrosion loss than the field data by Sumiyama et al. 13 and Honda and Yamaguchi, 14 but change with a similar slope in log–log scale plot.
Comparison of simulation data for corrosion loss based on previous and improved model with laboratory data reported by Taniguch et al. 2 and field data reported by Sumiyama et al. 13 and Honda and Yamaguchi. 14 Improved model predicts higher corrosion loss compared with that of previous model after inflection point of 109 s (about 100 years)
Corrosion loss data obtained by the simulation based on the previous model show a lower value, but increase with time according to 0·5 slope in log–log plot. Corrosion loss data based on the improved model, however, increase more rapidly after the inflection point than that predicted by the previous model because of a linear rate law with the constant corrosion rate.
Unsolved issues for improved model on carbon steel corrosion
Simulations were executed by using the diffusion constant of H2O in SiO2 since no exact value of D in Fe3O4 has been reported. If a reliable D in Fe3O4 film were obtained, the proposed model and simulation could be rationalised more correctly.
The model assumes that corrosion film is consisted of Fe3O4, but the corrosion film formed in the compacted bentonite 2 exposed to the solution including the higher concentration of bicarbonate ions has been reported to be FeCO3. King 4 had pointed out that the protective film formed in the compacted clay systems tends to be carbonate-based rather than magnetite based. The same procedure of simulation for the case of FeCO3 film is required for rationalising the improved model.
Conclusions
A previous corrosion model of carbon steel (the solid phase H2O diffusion model) under oxygen depleted environment was modified by introducing dissolution of the corrosion film. The film dissolution rate is assumed to be controlled by boundary layer diffusion of soluble Fe(OH)b (2–b)+ species which are in equilibrium with corrosion film consisting of Fe3O4. The corrosion current density of film dissolution was calculated by i d = nFD(C T–C 0)/δ. The total concentration, C T, of Fe(OH)b (2–b)+ species was estimated by the thermodynamical equilibrium constant and film dissolution current density was calculated as a function of pH and 
. Based on the modified model using the estimated film dissolution current density, the change in the corrosion current density with time was simulated as a function of pH and 
. From simulations, it is concluded that a constant and lower corrosion current density could be reached at pH = 10 and at a lower partial pressure of hydrogen gas.
Footnotes
Acknowledgement
This research is funded by the Secretariat of Nuclear Regulation Authority, Nuclear Regulation Authority, Japan.