Mathematical Modeling of Rebar Corrosion in Seawater
V. Ayala, A. Ingalls and J. Genesca
Dpto. Ingenieria Metalurgica. Facultad Quimica.
UNAM. Ciudad Universitaria. 04510 Mexico D.F.
email questions or comments to the author(s) at genesca@servidor.unam.mx
ABSTRACT
The aim of this work was to obtain the electrode potential profile of steel in a reinforced concrete structure of cylindrical size and immersed in sea water, as a function of surface concrete chloride concentration and pH, taking into account Green Rust (GR1, Fe4(OH)8Cl) formation as a corrosion product.
The proposed model is based on a bi-directional chloride flux through the concrete thickness. It allows to predict the chloride concentration profile along metal/concrete interface, as well as, when the critical concentration (necessary to cause passive film breakdown) has been reached and the time necessary to (at fixed concrete thickness and chloride effective diffusion coefficient). With these values, the redox potential may be estimated.
Through a series of field data, the capability of
the model was tested, finding good correlation.
INTRODUCTION
Concrete is a ubiquitous material. Concrete, the combination of cement acting as a binder and nonreactive or partially reactive aggregate fillers, is normally considered protective to the reinforced steel embedded in it because of the alkalinity produced during the hydration reactions and contained in the pore solution. Nevertheless, corrosion of reinforcing steel in concrete occurs as a result of many factors, including chloride ion contamination, carbonation of the concrete, etc. These result in a build-up of corrosion products which, being more voluminous than the embedded metal, introduce significant tensile and compressive loads on the concrete leading to cracking, disbondment and, ultimately, spalling of the concrete cover. Cracking and disbondment lead to further corrosion, which can compromise the life of the entire structure. The problem involves reinforced concrete structures such as bridges, etc.
The corrosion of reinforcing steel bars (rebars) in concrete is a growing problem affecting the integrity of a vast number of structures. The reinforcing steel is embedded in the concrete, which initially provides an alkaline environment conducive to surface passivation. Under these conditions, metal dissolution takes place at an extremely low rate. However, depassivation of the steel surface can take place if chlorides from seawater penetrate through the concrete cover and reach the rebar. Depassivation can also result from penetration of a carbonation front through the concrete as a result of exposure to atmospheric carbon dioxide. The locally active steel surface behaves predominantly as an anode while the entire bar may serve as a cathode. The main cathodic reaction is thought to be the reduction of oxygen, which is transported to the metal surface through the concrete cover. Metal ions dissolved at the anodic reaction form corrosion products which are expected to occupy a volume significantly larger than that of the initial metal. Cracking and spalling of the concrete cover eventually follow, requiring expensive repairs or replacement of the structure.
Substructure members and pilings supporting marine bridges are frequently constructed using steel reinforced concrete. In typical installations, the columns are partially submerged in seawater, so that a region of high chloride ion concentration builds up in the splash zone just above the high water line. Passivity breakdown at the surface of the steel embedded in this region and below water results, with subsequent active corrosion of the steel, shortening the useful life of the element (1).
There are few quantitative studies aimed to predicting the distribution of potential in rebars. Moreover, these predictions cannot be confirmed easily by experimental measurements. This is because the polarization conditions at the steel surface are complicated by slow transport of oxygen and corrosion products. There is a need for potential prediction of steel in concrete.
Steel in concrete is a clear example of a half-cell, a metal surrounded by an electrolyte. The potential of such a half-cell can only be measured relative to another half-cell, which is known as a reference half-cell or a reference electrode. In 1980, ASTM issued its standard C876-77 describing the test procedure for measuring the potentials of reinforcing steel in concrete. The placement of a reference electrode on the concrete surface and the measurement of the potential difference between the reference electrode and the embedded steel allows to measure the potential difference, which is indicative of the state of corrosion of steel.
The potential of steel reinforcement can be used to asses the probability of corrosion at the time of measurement.
It is important to remark that half-cell potential
data taken in this way can only be used to asses the probability
for corrosion and not the rate of corrosion. Interpretation of
such data can be made according to the following table:
| >90% probability of no corrosion occurring | |||
| Probability of corrosion uncertain | |||
| > 90% probability of corrosion occurring |
The table above is not valid for cases where corrosion is cathodically
controlled, that means when oxygen availability is limited, such
as underwater, or where the concrete has remained saturated for
a long period of time.
APPROACH
Reinforced Concrete Configuration
The system chosen consisted of a reinforced concrete cylinder.
The configuration of the computational model is shown in Figure
1, consisting of a column of a reinforced concrete cylinder, with
a single axial rebar, similar to the piles of a bridge. The system
is cylindrically symmetric. All concrete and rebar dimensions
are finite.
The following assumptions are made:
1. The element is assumed to be a vertical, circular section reinforced concrete column, totally immersed in seawater. Under these conditions, the pores are saturated and the Cl- transport is controlled only by diffusion.
2. Only chloride ions, Cl- , are considered as aggressive agents, being negligible all other ions.
3. Due to Cl- transport is very slow (D Cl- = 9.10-11 m2/s) chemical composition inside pores is considered in equilibrium during the diffusion process. As a result [Cl-] ad = [Cl-] free.
Before Cl- ions can reach the metal/concrete interphase,
the pH value in this area is considered constant, because de diffusion
coefficient of OH- ions is larger than Cl-
ions. Once Cl- ions reach the metal/concrete interphase,
the pH becomes more acid. This can be explained because OH-
diffuse to outside, plus the H+ production by means
of the reaction
By the geometry employed, Figure 2, the Cl- flux is considered bi-directional (in two directions), axial, z, and radial, r. The values of effective diffusion coefficient are assumed to be equal in the both directions considered.
The effective diffusion coefficient does not depend of [Cl-],
position and temperature.
4. It is assumed that chloride contamination has reached the threshold for active corrosion for rebar. The rest of the rebar is treated as a passive surface.
The corrosion rate of structures immersed in seawater is very little, due to the fact that [O2] is low and corrosion is cathodically controlled, that means when oxygen availability is limited, such as underwater, or where the concrete has remained saturated for a long period of time. In our model, once the critical chloride concentration to produce depassivation of steel is reached, pitting formation is favored. The potential at this point will be a function of [Cl-] and pH only.
5. The electrochemical reactions take place in the concrete/metal interphase, after reaching the critical Cl- concentration. The formation of Green Rust 1 (GR1) is assumed to be the main corrosion product on steel. This compound has been identified in seawater environments (2,3).
6. The reinforcement consists of one steel bar placed as shown in Figure 1.
7. The concrete is treated as homogeneous conducting electrolyte
with variable conductivity and oxygen diffusivity along the column
axis. The effective oxygen concentration is assumed constant along
the outer surface of the column.
GOVERNING EQUATIONS
With the assumptions listed above, the equations governing the system are as follows:
The diffusion process of chloride ions in unstable state through
the concrete can be mathematically represented by application
of Fick's second law
where C Cl- is the chloride concentration
(mole/m3) and D Cl- is the effective
diffusion coefficient of chloride ions on concrete (m2/s),
with the following initial and frontier conditions:
where N and j are the molar flux under static and dynamic conditions respectively.
To calculate the potential value as a function of chloride concentration,
the Nernst equation corresponding to the formation of GR1 can
be employed
Such equation is similar than the reported by Davies (4) to correlate the potential values of passivity breakdown as a function of pH and chloride concentration.
To evaluate the potential and concentration everywhere in the column, the volume was divided into a rectangular grid coincident with the boundaries and the rebar. The horizontal plane was divided into 3 by 3 nodes, as shown in Figure 3, so the total number of balance equations to define the system are nine.
The governing equations were solved by the implicit alternating
direction method as explained in Reference 5. This chosen method
has no restrictions on the time intervals as a cause of unstability,
as in the case of the explicit finite difference method.
EXPERIMENTAL RESULTS AND DISCUSSION
A predictive mathematical model has been developed. This model is purely diffusional and is based on Fick's second law. The software model package has been developed using Turbo Pascal.
The experimental data presented in this paper were obtained from Castro (6) and corresponds to concrete cylinder samples manufactured with a Portland I cement immersed in seawater, Figure 3, during 1 month, after that, the chloride concentration, [Cl-], in the radial direction was analyzed.
With the concentration profile obtained, it was possible to experimentally calculate an effective diffusion coefficient, which takes into account the concrete heterogeneity, as well as, a mass transfer coefficient. The consumption rate of chloride ions for the electrochemical reaction is defined by the stoichiometry, once known the data relative to the corrosion rate of rebar structures immersed in seawater.
For the mathematical simulation a chloride flux of 6 E-10 mole/m2.s was employed, considering the high corrosion rate (400 µm/year) (7) under aggressive conditions.
The algorithm works as follows.
The input data required for the computation are:
System dimensions, chloride concentration in concrete and seawater,
effective diffusion coefficients in radial and axial direction,
critical chloride concentration, pH and time interval.
TABLE 1.- Calculation parameters
|
|
The progress of chloride isoconcentration line (0.18 mole/m3) as a function of time is shown in Figure 4. One can see that after 0.5 months, such concentration has been reached on the concrete surface and that 5.5 months are necessary for this concentration to be present at the metal/concrete interphase. For this time, the chloride concentration profiles inside the concrete cover are given in Figure 5. It is clearly observed that a chloride flux exist both in radial and axial directions. Moreover, the upper part of the cylinder is affected by both chloride fluxes, reflecting this in the curvature of isoconcentration lines.
Thus, the model is capable to predict the time necessary to reach a determined value of chloride concentration in the metal/concrete interphase of a reinforced concrete structure immersed in seawater, as a function of the experimental conditions prevailed: concrete thickness, rebar radius, effective diffusion coefficient, chloride concentration on seawater and temperature.
Seeing the graph of Figure 5, the model also allows to predict the chloride concentration profile both in radial and axial directions, to a fixed time. It is then possible to decide which of both fluxes has a higher influence in the diffusion of chloride ions to the metal/concrete interphase.
The good response of the proposed predictive model in the simulation of the chloride concentration profile through the concrete cover is showed in Figure 6, as compared with the experimental results obtained by Castro (6).
Once the model has been able to predict the distribution of chloride
ions, the next step is to calculate the redox potential, taking
into account the GR1 formation as a corrosion product. At Figure
7 one can see the values of this potential at the metal/concrete
interphase as a function of chloride concentration. The potential
becomes more negative as chloride concentration increase, according
to the Nernst equation
and the Pourbaix diagram (3). This model is of thermodynamic nature and under prestablished conditions, basically pH and chloride concentration, it may be possible to obtain information about the thermodynamic factibility of pitting generation (localized corrosion). Analyzing the graph on Figure 7, obtained at pH 7, the potential values obtained are not in the potential range (40 to -260 mV vs enh) in which the probability for pitting is higher (6). From Figure 8, one can observe that for a fixed concentration, 0.18 mole/m3, there is an influence of pH on the potential, and as decreases the pH, more negative is the potential and approaches to the critical range.
Finally, the model allows to make predictions on the system response
when is subjected to external changes. Figure 9 shows the result
of an increase on the thickness of concrete cover to 0.08 m over
the time necessary to reach the critical chloride concentration
at the metal/concrete interphase. Comparing with Figure 4, this
increase in the thickness of concrete can be reflected in a significant
rise to 14.5 months. Another interesting aspect may be to study
the effect of decreasing the effective diffusion coefficient to
a value of 1E-11 m2/s, on the time necessary to reach
the critical concentration at the metal/concrete interphase. The
result is presented in Figure 10, showing that the time to reach
this concentration has been increased to 27 months, much more
higher that the time predicted for an increment to 0.08 m of the
concrete cover. This last results seem to be interesting from
a practical point of view. An increase on the concrete thickness,
but specially, diminishing the effective diffusion coefficient
of chloride ions, that means a less porous concrete, can result
in a higher durability of the structures.
CONCLUSIONS
In spite of the simplicity of the predictive model proposed, some
important conclusions can be made, some of them of practical importance.
This diffusion model for concrete structures immersed in seawater
can help not only to calculate the profile concentration of chloride
ions inside the concrete cover and the corresponding potential
from a thermodynamic point of view, but specially the consequences
of changing the concrete thickness and diminishing the effective
chloride diffusion coefficient on the life of concrete structures.
REFERENCES
1. A. A. Sagues, S.C. Kranc and B.G. Washington, in Corrosion 2000, edited by R.K. Dhir and M.R. Jones, E.&F.N. Spon, London (1993), ISBN 0 419 18120 2, p: 1275-1284.
2. J.J. Carpio, G. Hernandez-Duque, L. Martinez and T. Perez, CORROSION'94, Paper # 288, NACE, Houston (1994).
3. P.H. Refait and M.R. Genin, Corrosion Science 34 (5) 797 (1993).
4. J.A. Davies and P.A. Brook, Corrosion Science 33 (2) 315 (1992).
5.B. Carnahan, "Applied Numerical Methods" John Wiley & Sons, New York (1969), p: 452.
6. P. Castro, PhD Thesis, Faculty of Chemistry, National University of Mexico (UNAM), Mexico D.F. (1995).
7. K. Tuutti, Swedish Cement and Concrete Research 4, 17
(1982).
FIGURE 4.- ISOCONCENTRATION LINES THROUGH THE CONCRETE COVER.
FIGURE 5.- CONCENTRATION PROFILES OF CHLORIDE IONS THROUGH CONCRETE COVER FOR 5.5 MONTHS.
FIGURE 6.- RESPONSE OF MODELED SYSTEM TO SIMULATION.
FIGURE 7.- POTENTIAL DISTRIBUTION ON THE METALLIC SURFACE AS A FUNCTION OF CHLORIDE CONCENTRATION (pH = 7).
FIGURE 8.- POTENTIAL DEPENDENCE WITH pH AT A CHLORIDE CONCENTRATION OF 0.18 MOLE/M3.
FIGURE 9.- EFFECT OF AN INCREASE OF CONCRETE COVER THICKNESS ON THE ISOCONCENTRATION LINES OF CHLORIDE IONS (0.18 mole/m3).
FIGURE 10.- EFFECT OF DIMINISHING THE EFFECTIVE DIFFUSION COEFFICIENT
TO 1E-11 m2/s ON THE ISOCONCENTRATION LINES OF CHLORIDE
IONS (0.18 mole/m3).