Rebar corrosion in carbonated concrete exposed to variable humidity conditions. Interpretation of Tuutti’s curve

Corrosion Science 51 (2009) 1747–1756

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Rebar corrosion in carbonated concrete exposed to variable humidity conditions. Interpretation of Tuutti’s curve P. Dangla a,*, W. Dridi b a b

Université Paris-Est, Navier, UMR CNRS 113, 2 allée Kepler 77420 Champs sur Marne, France CEA, DEN, DPC, SCCME, Laboratoire d’Etude du Comportement des Bétons et des Argiles, F-91191 Gif-sur-Yvette, France

a r t i c l e

i n f o

Article history: Received 15 September 2008 Accepted 28 April 2009 Available online 19 May 2009 Keywords: A. Concrete B. Modelling studies C. Atmospheric corrosion C. Oxygen reduction C. Oxidation

a b s t r a c t This paper deals with the modelling of the rebar corrosion kinetics in unsaturated concrete cover. The concrete is assumed carbonated resulting in an active corrosion of steel. The corrosion kinetics is coupled with the ionic transport processes. Free corrosion in different concretes is studied in terms of ambient relative humidity. The numerical results obtained by the finite volume method are discussed and compared with a reported experiment performed on a carbonated concrete. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Corrosion of steel in reinforced concrete is one the major cause of the deterioration of today’s structures, especially for those exposed to aggressive environments like seawater or deicing salts. As a result, the repair costs nowadays constitute a major part of the current spending on infrastructure. Steel embedded in concrete usually remains passive due to the formation of a layer of iron hydroxides. The characteristics of this passive layer depend on the pH value in the alkaline range typical of concrete and on the electrode potential of the rebar which, in turn, also depends on the oxygen content in the concrete pore solution [1]. This layer dissolves due to action of chlorides or a drop in pH induced by carbonation of the concrete cover [2–4]. These conditions result in active corrosion of steel providing the concrete is moist enough. The further evolution of the corrosion rate have been studied by means of experimental technics [5,6]. It is well established that the moisture content of the concrete is the main parameter controlling the rate of the process [7]. Thus, when the concrete is dry, the corrosion shows negligible values (below 0.1 lA/cm2 ). These values increase when the humidity goes up to maximum values of around 1 lA/cm2 [8]. Temperature also seems to have some influence on the corrosion rate. Some references can be found in [5]. A lot of experimental data are available on the corrosion rate of steel reinforced concrete. In a well-known report [9], experiments * Corresponding author. Tel.: +33 1 40 43 54 44; fax: +33 1 40 43 54 50. E-mail addresses: [email protected] (P. Dangla), [email protected] (W. Dridi). 0010-938X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.corsci.2009.04.029

performed in carbonated concrete exposed to different humidity have shown that the corrosion rate exhibits a maximum value in the high range of humidity and a discontinuity in the low range of humidity. The modelling of such experiments is the purpose of this paper. To this end we propose to model the couplings between the corrosion kinetics at the steel–concrete surface and the transport of the reactive species in the porosity of the concrete. This model has identified the predominant mechanism in the whole range of humidity thereby resulting in a pattern similar to that observed by Tuutti. Also the question of how the concrete properties affect this behaviour has not been addressed. The modelling of such experiments is the purpose of this paper. 2. Corrosion kinetics The corrosion is an electrochemical process consisting of anodic and cathodic reactions occurring at the steel–concrete surface. The anodic reaction is the process of iron dissolution with emission of electrons inside the metal grid (iron oxidation)

Fe ! Fe2þ þ 2e

ð1Þ

The cathodic reaction is the consumption of electrons by water and oxygen under the form (oxygen reduction)

1 O2 þ H2 O þ 2e ! 2OH 2

ð2Þ

In both reactions, reverse reactions can be neglected in problems of practical interest. This assumption has been adopted here. Furthermore, it is assumed that the steel–concrete surface is an active

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region resulting e.g. from the carbonation of the concrete. As a result it has been assumed that there is no layer of iron hydroxides at the steel–concrete surface and that the steel is in contact with the pore solution of the concrete. Thus only the oxygen concentration, the Galvani potential difference, g (i.e. the difference between the metal potential and the concrete potential: g ¼ wm  wc ), and the moisture content at the steel–concrete surface are expected to control the kinetic of the reactions. The kinetic laws are formulated in terms of the local current densities at the steel–concrete surface, ia and ic , which accompany the anodic (oxidation) and cathodic (reduction) reactions. Furthermore, the oxygen reduction is assumed to follow a first order kinetic law with respect to the oxygen concentration. Because both processes are activated, the Butler–Volmer activation law has been employed for both reactions under the form [10,11] 0

ia ¼ ia eg=ba A 0

ic ¼ ic

qO2 g=bc e A q0O2

ð3Þ ð4Þ 0

These equations are functions of the exchange current density i and the Tafel slope b. qO2 and q0O2 are the oxygen concentrations of the pore solution which is in contact with the rebar and the air, respectively (see Section 3). 0 0 In the computations below we used ia ¼ 19 lA/cm2 , ic ¼ 48 lA/ cm2 [12,13]. The Tafel slopes were chosen as ba ¼ 60 mV, bc ¼ 160 mV [14]. These Tafel slopes are not well defined in concrete. Under different conditions these slopes can range within a factor 10 or even more [15–17]. A sensitivity analysis for the Tafel slopes and the exchange current densities have been proposed in the end of this paper. 0 Unlike i which refers to the full unit area of the rebar, the current density i refers to a fractional area due to the porosity and the partial saturation of the concrete. This fractional area is taken into account through the ratio A in Eqs. (3) and (4). Actually if the steel–concrete contact is perfect, A is the product of the surface porosity and the surface degree of saturation which may differ from their analogous volume quantities. In the followings the surface porosity and the volume porosity are assumed to be the same while the surface decan differ from the volume degree of saturagree of saturation Ssurf l is assumed to be a function of the capillary tion Sl . Thus Ssurf l pressure of the same form as that used for the volume degree of saturation (see (6) in Section 3) but with its own parameter m. Let us consider a reference electrode in contact with the pore solution of the surface concrete. The metal itself being considered as an equipotential surface, by virtue of its high electric conductivity, there is a potential difference between the rebar and the reference electrode currently defined as the electrode potential of the rebar: E ¼ wm  wr . As a standard hydrogen electrode (SHE) is assumed throughout the paper it is relevant to consider that wr ¼ 0:058pH V, where pH refers to the pore solution of the surface concrete. During the free corrosion process, a voltameter connecting the rebar and the reference electrode will record an electrode potential defined as the corrosion potential: Ecorr . Due to the high resistance of the voltameter no electric current would be recorded, hence i ¼ 0 during the free corrosion process. More generally we can consider a test, known as a polarisation test, in which an electric current i is generated giving rise to an electrode potential change E  Ecorr . In fact the numerical method presented here can simulate this general test by imposing a constant electric current i resulting in an electrode potential E. The free corrosion simulations presented here are obtained by imposing i ¼ 0 resulting in the corrosion potential Ecorr . The continuity of the charge flux through the interface metal– solution imposes that

i¼

Z

ðia  ic Þds

ð5Þ

Srebar

where, by arbitrary convention, the electric current is taken as positive from the rebar to the pore solution. In case of homogeneous free corrosion the anodic and cathodic reactions must have the same rate, ia ¼ ic . The local anodic current density is also known as the corrosion current density icorr ¼ ia . Section 4 is devoted to the simulation of the free corrosion in variable relative humidity conditions. Before to perform these calculations, the modelization of ion transport in the pore solution of the unsaturated concrete is presented now. 3. Ion transport in unsaturated concrete The concrete is considered as an undeformable and unsaturated porous material. The gas pressure is neglected and assumed null throughout the material. The saturation degree, Sl , is experimentally determined as a function of the capillary pressure pc ¼ pl . This Sl  pc relationship is represented by the van Genuchten’s model [18]

 m 1 Sl ¼ 1 þ ðpc =p0 Þ1m

ð6Þ

The parameters p0 and m are fitted from isotherm sorption experiments, in main wetting (w) and drying (d) paths, and the Kelvin’s law, pc ¼ qRT ln hr where q is the molar concentration of water and hr the relative humidity. Three concretes have been studied. Each of them is characterized by a water to cement ratio (w/c) and noted M25, BO, BH. Some parameters given in Table 1 were taken from [19,20]. The main wetting and drying curves, Sl ðhr Þ, are plotted in the left hand side of Fig. 1. In the computations of the next section two examples of curve for the surface degree of satura¼ Sl or Ssurf < Sl (m ¼ 0:7 for wettion have been chosen, either Ssurf l l ting and m ¼ 0:8 for drying). These last curves are plotted in the right hand side of Fig. 1. The oxygen O2 is present in gas and liquid phases. The thermodynamic equilibrium of O2 in the two phases follows from the Henry’s law which states the proportionality of the molar concentrations in the two phases:

qlO2 ¼ kH qgO2

ð7Þ

with kH ¼ 3% (see for example [21,22]). Other molecules or ions are considered in the liquid phase. The ferrous ion Fe2þ resulting from the oxidation of iron is assumed to be in equilibrium with the dissolved ferrous-hydroxide FeðOHÞ2 (l). The latter precipitates as it reaches the solubility of the solid ferrous-hydroxide FeðOHÞ2 (s). Other cations (Cþ ) and anions (A ), not involved in the corrosion, are considered to match the electric property of the concrete. The liquid phase resulting from the mixture of these five species and the solvent H2 O with its associated ions OH and Hþ flows into the concrete with a volumetric flux q given by the Darcy’s law:

q¼

k

ll

krl ðSl Þrpl

ð8Þ

Table 1 Material data of 3 concretes. BH has 10% of silica fume. In this table w/c is the water– cement ratio, / is the porosity, k is the intrinsic permeability, p0 and m are the parameters found in the Sl  pc relationship (6). Concrete

M25 BO BH

w/c

0.84 0.487 0.267

/

0.148 0.12 0.082

k (m2 )

2:1019 3:1021 2:1022

p0 (MPa)

m

(d)

(w)

(d)

(w)

13.11 35.43 106.6

1.11 3.47 5.44

0.437 0.47 0.64

0.32 0.28 0.25

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100

100 drying exp model

wetting exp model

60 M25 40 20

100

20

40 60 hr (%)

80

100

0

Sl (%)

BO 40 20 0

100

40 60 hr (%)

80

0

20

40 60 hr (%)

100

80

100

100 wetting drying

80

Sl (%)

BH 40 20

40

60

80

60 BH 40 20

wetting exp model 20

80

BO 40

0

60

0

100

60

100

drying exp model

80

80

20

wetting exp model 20

40 60 hr (%)

wetting drying

80

60

0

20

100

drying exp model

80

Sl (%)

M25 40

0 0

Sl (%)

60

20

0

0

wetting drying

80

Sl (%)

Sl (%)

80

100

hr (%)

0

0

20

40

60

hr (%)

Fig. 1. Sorption curves for BO, M25 and BH. The left hand side curves represent the saturation degree in the bulk while the right hand side curves represent the surface < Sl is considered (in that case the parameter m is taken as 0.7 or 0.8 for wetting or drying, respectively). saturation degree at the steel–concrete surface when the case Ssurf l Experimental results are taken from [20].

where ll is the liquid viscosity, k the intrinsic permeability (Table 1). The relative permeability krl ðSl Þ is assumed to follow the Mualem’s model [23]

krl ¼

  2 pffiffiffiffi 1 m Sl 1  1  Sml

ð9Þ

Since the permeability of Mualem’s approach is determined by the full pore radii, m has to be taken in Table 1 as the wetting parameter whatever the path followed. Hysteresis of the relative permeability is, therefore, neglected. Let us denote by qi the molarity (molar concentration) of i in the solution. The molar flux of i, denoted by wi , is the sum of an advective motion represented by qi q and a diffusive motion ji :

wi ¼ qi q þ ji

ð10Þ

Noting M i the molar mass of i, the fluxes ji are constrained by the P condition i Mi ji ¼ 0 resulting from the assumption that the total P mass flux, namely w ¼ i Mi wi , is a pure advective motion. This condition serves to express the diffusive flux of the solvent H2 O as a combination of the diffusive fluxes of other species, namely O2 (l), Fe2þ , FeðOHÞ2 (l), Cþ , A , Hþ , OH . These latter are given by the Nernst–Planck equation [24]:

  F ji ¼ Di rqi þ zi qi rw RT

ð11Þ

In Eq. (11) Di is the effective diffusion coefficient of i in the porous medium and zi the valence of i with zi ¼ 0 in case of a neutral molecule. F is the Faraday’s constant, R the gas constant and T the temperature assumed constant in this work. The effective diffusion coefficient is classically expressed as the product of the molecular

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diffusion coefficient of i in water (given in Table 2) and the tortuosity coefficient, namely Di ¼ sl Dli . The tortuosity is given by the following expression, found in [25] and [26] 3 9:95/

sl ¼ 0:296  10 e

4

=ð1 þ 625ð1  Sl Þ Þ

ð12Þ

As the oxygen diffuses both in the two phases, its diffusion coefficient is actually the sum of two terms as follows:

DO2 ¼ sl DlO2 þ sg DgO2 =kH

ð13Þ

DgO2 is 3

where the molecular diffusion coefficient of oxygen in air (DgO2 ¼ 10 cm2 /s) and sg the tortuosity coefficient in the gas phase. Eq. (13) holds because the advective motion of the gas phase is assumed to be negligible. The tortuosity relative to the gas phase is given by the following expression proposed by Millington [27]

sg ¼

/a Sbg

ð14Þ

with Sg ¼ 1  Sl and where a and b have been fitted from oxygen and carbon dioxide diffusion tests performed by Papadakis on mortars [28]. The results obtained by Thiery [29] have yielded a ¼ 2:74 and b ¼ 4:20. Apart from the autoprotolysis of water,

H2 O  Hþ þ OH

ð15Þ

two other chemical reactions are considered: (i) the dissociation of the dissolved ferrous-hydroxide FeðOHÞ2 (l)

FeðOHÞ2 ðlÞ  Fe2þ þ 2OH

ð16Þ

and (ii) its precipitation in solid ferrous-hydroxide FeðOHÞ2 (s)

FeðOHÞ2 ðsÞ  FeðOHÞ2 ðlÞ

ð17Þ

The chemical equilibrium of the dissociation of FeðOHÞ2 (l) yields

aFe2þ a2OH ¼ K FeðOHÞ2 ðlÞ aFeðOHÞ2 ðlÞ

ð18Þ

qeðOHÞ2 ðsÞ

o

n o ¼ Max qFeðOHÞ2  SFeðOHÞ2 ; 0

ð22Þ ð23Þ

The differential equations governing the fields of concentrations can be obtained from the mass balance equations written for O2 , Cþ , A and the atoms O, H, Fe found in the species considered previously. These mass balance equations take the following form

@na ¼ divwa @t

ð24Þ

where the subscript a stands for the three species and the three atoms mentioned above. In Eq. (24) na represents the molar content of a per unit volume of the porous medium. For hydrogen, a ¼ H,

  nH ¼ /Sl qHþ þ qOH þ 2qH2 O þ 2qFeðOHÞ2

ð25Þ

Obviously the molar fluxes wa can be decomposed in the same manner. The six equations (24) and the three chemical equilibriums give nine equations for the 11 unknown fields consisting of the eight liquid molarities qi , the solid molar content qFeðOHÞ2 ðsÞ , the liquid pressure pl and the electric potential w. Two more equations are then necessary. The first one is given by the definition of the partial molar volume V i yielding

X

qi V i ¼ 1

ð26Þ

i

This relation is used to express the molarity of water, qH2 O , as a function of other molarities present in solution. However, as the solutes are diluted in the solution, they occupy a negligible volume compared to that of the water. Hence the molarity of water is approximated to a constant equal to 55.5 mol/L. The last equation comes from the fundamental equation of electrostatic

divðrwÞ ¼ F

X

zi q i

ð27Þ

i

where ai denotes the activity of i given, in the ideal-dilute approximation, by the relative molality bi =b (b ¼ 1 mol/kg) or qi =q (q  1 mol/L). The constant of dissociation is K FeðOHÞ2 ðlÞ ¼ 107:4 [30,31]. The precipitation of FeðOHÞ2 (l) takes places only if its concentration reaches the solubility SFeðOHÞ2 estimated at 107:7 mol/L [31]. Denoting by qFeðOHÞ2 ðsÞ the molar content of solid per unit volume of pore solution (a kind of solid molar concentration introduced for conveniency), the precipitation process is mathematically written by the set of equations:

qFeðOHÞ2 ðlÞ 6 SFeðOHÞ2 qFeðOHÞ2 ðsÞ P 0 ðqFeðOHÞ2 ðlÞ  SFeðOHÞ2 ÞqFeðOHÞ2 ðsÞ ¼ 0

n

qFeðOHÞ2 ðlÞ ¼ Min qFeðOHÞ2 ; SFeðOHÞ2

ð19Þ ð20Þ ð21Þ

The resolution of this set of inequalities can be performed by introducing the total molar concentration of ferrous-hydroxide qFeðOHÞ2 ¼ qFeðOHÞ2 ðlÞ þ qFeðOHÞ2 ðsÞ , thereby resulting in the description of qFeðOHÞ2 ðlÞ and qFeðOHÞ2 ðsÞ in terms of qFeðOHÞ2 : Table 2 Molecular diffusion coefficients in water (here the cation Cþ and the anion A stand for Naþ and HCO 3 , respectively) [22]. Molecule

Dli (cm2 /s)

O2 OH Hþ Fe2þ FeðOHÞ2 (l) Cþ A

2.51  105 5.24  105 9.32  105 0.72  105 0.72  105 1.33  105 1.18  105

with  the permittivity of the (dielectric) medium. With typical val3 w ¼ 1 V, the characteristic ues such as z ¼ 1, q ¼ 100 pmol/m ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiand ffi length of Eq. (27), namely w=ðFzqÞ, lies in the range typical of Debye length i.e. few nanometers. As a result Eq. (27) reduces, at the scale of interest (i.e. around 1 cm) and with a very good approximation, to the electroneutrality of the medium

X

zi qi ¼ 0

ð28Þ

i

The validity of the electroneutrality approximation has been discussed in several contexts [32,33]. It is pointed out that electroneutrality entails that the current, unlike the electric field, is a divergence free vector. Because Eq. (28) applies in place of Eq. (27), the electric potential does not obey Dw ¼ 0 as still assumed in some papers, e.g. [14]. Eq. (28) serves to eliminate one of the molar concentration qi . Finally the closed set of equations composed of the six mass balance equations and the chemical equilibrium (18) associated with the seven principal unknowns pl , w, qFe2þ , qFeðOHÞ2 , qO2 , qCþ , qA is solved by using the finite volume method [34]. The principle of this method is recalled in Appendix. This model is applied to a 1-cm width concrete cover exposed to a variable relative humidity condition on the external surface. Example of initial conditions is given as follows: p0l ¼ qH2 O RT ln hr , q0Fe2þ ¼ q0FeðOHÞ2 ¼ 0, q0O2 ¼ 0:25 mol/ m3 (kH times the concentration of oxygen in atmosphere), q0A ¼ 100 mol/m3 , q0Cþ ¼ 100 mol/m3 , giving rise to a saturated BO concrete resistivity of 711 Xm. On the external surface of the concrete cover, the boundary conditions are given by: pl ¼ qH2 O RT ln hr , w ¼ 0 (the potential of the pore solution is 0 by arbitrary convention), qO2 ¼ q0O2 , no fluxes of Cþ , A and Fe (i.e. sum of that of Fe2þ and FeðOHÞ2 ).

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The opposite surface of the concrete cover is in contact with the rebar where the boundary conditions account for the oxido-reducing reactions of iron given by (3) and (4) with q0O2 ¼ 0:25 mol/m3 . Therefore the boundary conditions at the metal surface of unit outward normal n are given by

ia ; 2F

wO2  n ¼

ic 4F

others wa  n ¼ 0

ð29Þ

4. Simulation of free corrosion A e ¼ 1 cm thick concrete cover is submitted to variable humidity conditions hr . The computations are performed over a period of 1 year. As an example, Fig. 2 displays the evolution of some quantities at the steel–concrete surface for BO, wetting curve and ¼ Sl . For example it can be seen that the corrosion potential Ssurf l is stable after only few days. This result holds in all calculations. The evolution of the (surface) saturation degree shows that due to the cathodic reaction, concrete dries at the steel surface for weak enough humidities because for such level of humidities the permeability of the concrete is not high enough to supply the steel surface with water. This drying phenomenon appears only for ¼ Sl but not for Ssurf < Sl (see Fig. 5). The decrease of the corroSsurf l l sion current is a direct consequence of this drying since in this range of humidity the corrosion rate is controlled by the anodic and cathodic reactions as seen after. The same comment applies for the smooth increase of the oxygen concentration taking place after an abrupt decrease not visible in the figure. After 1 year of corrosion it is assumed that the corrosion current has reached a quasistatic state for most of humidities except for 60% (40% for the drying case) for which this is not exactly true (but this is exactly true in the all range of hr for the case

0.6

1

hr 100 % 90 % 80 %

icorr (μA/cm2)

0.8

hr 60 % 40 % 20 %

0.6 0.4

0.5 Ecorr vs SHE (V)

wFe  n ¼ 

Ssurf < Sl ). However, the results presented below have been obl tained after a period of 1 year of corrosion for each imposed humidity. As ferrous ions and dissolved ferrous-hydroxide diffuse from the rebar towards the concrete surface, the formation of the solid ferrous-hydroxide concentrate at the steel–concrete surface, namely at the first node of the mesh as illustrated in the right hand side of Fig. 3. It is pointed out that the model does not account for the reduction of the transport properties due to the clogging of pores by the precipitation of FeðOHÞ2 (s). Of course this mechanism could play an important role. However, considering that some of the properties of the steel–concrete surface is already taken into , we thought account through the liquid surface fraction, A ¼ /Ssurf l that taking the clogging of pores in the modelling is out of the scope of this paper. In these quasistatic states, the profiles of oxygen concentration are linear through the concrete cover as illustrated in the left hand side of Fig. 3. The concentration of oxygen at the rebar surface is shown in Fig. 4 for all the cases studied. In the low range of humidity oxygen concentration decreases for increasing humidity because oxygen diffuses faster in the gas phase than in the liquid phase. For humidity greater than a critical value, the concentration of oxygen falls to zero, actually a tiny but positive value related to the flux of oxygen at the rebar surface. It results that in the range of humidity greater than this critical value the corrosion rate is diffusion-controlled. As the gradient of concentration of oxygen is constant (equal to q0O2 =e =0.25 103 M/cm) the flux of oxygen is controlled by the transport properties of the concrete only. Fig. 5 represents the corrosion current vs. humidity. For each case two values for m are employed in the expression of the surface ¼ Sl or Ssurf < Sl (m ¼ 0:7 for wetdegree of saturation, either Ssurf l l ting and m ¼ 0:8 for drying). All theses curves exhibit a maximum

0.4 hr 100 % 90 % 80 %

0.3 0.2

hr 60 % 40 % 20 %

0.1

0.2

0

0 0

100

200 time (d)

300

0

400

0.25

1

0.2

0.1

hr 60 % 40 % 20 %

300

400

hr 60 % 40 % 20 %

0.6 Sl

ρO

2

hr 100 % 90 % 80 %

200 time (d)

hr 100 % 90 % 80 %

0.8

0.15

100

0.4

0.05

0.2

0 0

100

200 time (d)

300

400

0 0

100

200

300

time (d)

Fig. 2. Evolution of some quantities at the steel–concrete surface for BO, wetting and Ssurf ¼ Sl . l

400

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P. Dangla, W. Dridi / Corrosion Science 51 (2009) 1747–1756

0.25

2.5e-05 Solubility 2e-05

3

ρFe(OH) (mol/m )

3

ρO (moles/m )

0.2 0.15

2

0.05

0

0.2

0.4

0.6

0.8

2

0.1

0

hr 100 % 90 % 80 % 60 % 40 % 20 %

1.5e-05

hr 100 % 90 % 80 % 60 % 40 % 20 % 1

1e-05 5e-06 0

0

0.2

distance (cm)

0.4

0.6

0.8

1

distance (cm)

Fig. 3. Profiles of some quantities in the concrete cover after 1 year of corrosion for BO, wetting and Ssurf ¼ Sl . l

Fig. 4. Oxygen concentration at the rebar surface.

corrosion current for a critical value of the relative humidity separating a high range of humidity in which the corrosion current is diffusion-controlled from a low range of humidity in which the corrosion current is controlled either by the transfer of charges or by the water transport. Evidence of the first mechanism is confirmed by plotting DO2 q0O2 =e as a function of hr . Assuming a fast transport of oxygen and water through the concrete cover, the corrosion current can thus be approximated by 0

0

ba

0

bc

icorr ¼ ia ba þbc ic ba þbc /Ssurf ðhr Þ l

ð30Þ

However, as shown by plotting expression (30) in Fig. 5, the corro0 ¼ Sl , lower than icorr because of the sion current is, in case of Ssurf l drying phenomena mentioned above. In this case the corrosion current is therefore controlled by the transport of water rather than the

< Sl , the corrosion current is transport of charges. For the case Ssurf l controlled by the transfer of charges because the flow of water is so low that the permeability is high enough to supply the steel surface with water. However, this result depends on the parameter m departs from Sl . For concrete BH and on drydescribing how far Ssurf l ing curve, simulations have shown that the corrosion current is diffusion-controlled above 30% of hr , namely for practical range of humidity. The measure of the cell current in a carbonated concrete of w=c ¼ 0:9 performed by Tuutti [9] and reproduced in Fig. 6 exhibits a similar pattern. It is worth mentioning that the water consumed by the reduction of oxygen is eventually captured in the ferrous-hydroxide crystals formed at the steel–concrete surface. As shown in Fig. 7, computations together with the Tuutti’s experiment show that the critical value of hr increases with the w/c ratio and moreover it is lower in drying than in wetting. Tuut-

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P. Dangla, W. Dridi / Corrosion Science 51 (2009) 1747–1756

M25 wetting 10

10

0

DO ρO /e

icorr

10 1

2

2

0 icorr

10

10 0 10 -1

surf

=Sl

Sl

10 -2 10 -3

2 icorr (μA/cm )

2 icorr (μA/cm )

M25 drying

2

drying

0

20

40

10 10 10

surf Sl
10 -4

10

60

80

10

100

2

0 surf

-1

Sl

=Sl

-2 -3 surf

Sl

-4

0


20

BO wetting 2

0

10 2

icorr (μA/cm )

2

icorr (μA/cm )

2

icorr Ssurf l =Sl

10 -1

drying

10 -2 surf

10 -3

Sl


20

10 10 10 10

10 -4 0

40

60

80

10

100

2

Ssurf l =Sl

-2 surf

-3

Sl


-4

0

2

10

10 0

2

Ssurf l =Sl drying

10 -2 surf

10

Sl

-3


20

10 10 10 10

10 -4 0

40

60

2

0

-1

10

0

DO ρO /e

0 icorr

10 -1

2

0

icorr

20

40

60

80

100

BH drying

icorr (μA/cm )

2

icorr (μA/cm )

10

100

DO ρ0O /e

icorr

1

BH wetting

1

80

hr (%)

2

icorr

60

2

hr (%)

10

40

BO drying 10

DO ρ0O /e

icorr

10 0

2

hr (%)

10 2 10

2

0 icorr

hr (%)

1

0

DO ρO /e

icorr

1

80

100

10

2 0

DO ρO /e

icorr

1

2

2

0 icorr

0

-1 -2

Ssurf l =Sl surf Sl
-3 -4

0

20

hr (%)

40

60

80

100

hr (%)

Fig. 5. Corrosion current vs. hr for M25, BO and BH.

ti’s experiment seems to agree better with the assumption Ssurf ¼ Sl l than with Ssurf < S . l l From his experimental result, Tuutti observed a jump in the cell current (see Fig. 6) for a so-called critical degree of saturation. From the previous analysis we can claim that this jump lies in the range of humidity in which the corrosion current is controlled by the transfer of charges or the transport of water resulting in a corrosion induced drying at the steel surface. In both case the surðhr Þ play an important role. As face degree of saturation curve Ssurf l the experiment is performed by wetting the sample, it is possible that this jump might be the consequence of the wetting of the steel–concrete surface during the general wetting process from

the external surface of the concrete. Because the surface degree of saturation is controlled by the capillary pressures related to the size of pores in contact with the steel surface, it is suggested that the pore size distribution at the surface cannot be identified exactly with the pore size distribution of the bulk material. Since we thought that only the biggest pores of the bulk can ‘‘touch” , compared to that of Sl , the steel surface, a steepest curve for Ssurf l has been chosen, resulting in the previous choices for m. However, these considered values of m (0.7 or 0.8 for wetting or drying case, respectively) have served as illustrative examples only since, to the authors’ knowledge, no experimental data is available to fit m. The critical degree of saturation as introduced by Tuutti suggest to

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P. Dangla, W. Dridi / Corrosion Science 51 (2009) 1747–1756

examine the pore radius filled with this water content. By using the Kelvin–Laplace law we find a pore radius of about 6 nm. The previous analysis suggest that pores in the range lower than 6 nm do not ‘‘touch” the steel–concrete surface. The electric field in the concrete is found to be negligible (less than 1.5 mV/cm). The corrosion potential is plotted in Fig. 8. The bounding values E0corr and E100 corr obtained, respectively, at 0% and 100% of relative humidity can be estimated by using the approximation qO2 ¼ q0O2 and qO2 ¼ 0, respectively: 0

E0corr ¼

100

I 0

ia

100

80

80

Tuutti’s exp.

Critical hr (%)

Critical hr (%)

E100 corr ¼ ba ln

60 40 20

0

0.2

0.4

0.6

0.8

60 40 20

surf

S =S Ssurf
wetting

Ssurf=S Ssurf
drying 0

1

0

0.2

w/c

0.4

0.6

0.8

1

w/c Fig. 7. Critical humidity vs. w/c.

0.6 Corrosion potential vs SHE (V)

Corrosion potential vs SHE (V)

0.6 0.5 0.4 0.3 0.2 wetting 0.1 BO M25 BH 0 0 20 40 60 hr (%)

80

0.4 0.3 0.2 wetting BO M25 BH

0.1

0

20

40 60 hr (%)

80

100

40 60 hr (%)

80

100

0.6 Corrosion potential vs SHE (V)

Corrosion potential vs SHE (V)

0.5

0

100

0.6 0.5 0.4 0.3 0.2 drying 0.1 BO M25 BH 0 0 20

40 60 hr (%)

 wr q0O 2

ð31Þ

where I is a material constant given by 4FDlO2 e sl =/ and practically constant in the range typical of concrete porosity, namely I 

Fig. 6. Corrosion current vs. hr from Tuutti [9].

0

ba bc i ln c  wr ; ba þ bc i0a

80

100

0.5 0.4 0.3 0.2 drying BO M25 BH

0.1 0

0

Fig. 8. Corrosion potential vs. hr .

20

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P. Dangla, W. Dridi / Corrosion Science 51 (2009) 1747–1756

0.7 Corrosion potential vs SHE (V)

Corrosion potential vs SHE (V)

0.7 0.6 0.5 0.4 0.3 0.2

i0a = 19 μA/cm2

0.1

0 ia 0 ia

0

= 190 μA/cm = 1.9 μA/cm

2

2

0.6 0.5 0.4 0.3

20

40 60 hr (%)

80

100

0

0

Corrosion current (μA/cm2)

10

2

Corrosion current (μA/cm )

0

ic = 4.8 μA/cm

2

2

0 0

10

0

ic = 480 μA/cm

0.1

-0.1

10

i0c = 48 μA/cm2

0.2

-1

i0a = 19 μA/cm2

10-2

i0a = 190 μA/cm2 i0a = 1.9 μA/cm2

10-3 0

20

40 60 hr (%)

10

40 60 hr (%)

80

100

80

100

0

-1

i0c = 48 μA/cm2

10-2

i0c = 480 μA/cm2 i0c = 4.8 μA/cm2

10-3 80

20

100

0

20

0

40 60 hr (%) 0

Fig. 9. Sensitivity analysis with respect to ia and ic .

0.02 lA/cm2 whatever the concrete. It results that the bounding values are independent of the concrete as shown in Fig. 8. In the previous equation the potential wr , about 0.493 V, results from the simulations in which the pH has approached 8.5. Fig. 9 shows the sensitivity of the corrosion potential and the 0 0 corrosion current with the reference current densities ia and ic (for concrete BO, wetting curve and Ssurf ¼ S ). Factors 10 and 0.1 l l have been applied to the reference values. As expected the corro0 0 sion current is insensitive to ia and ic in the high range of humidity since it is diffusion-controlled. Unexpectedly the corrosion current is also insensitive to these parameters in the low range of humidity because it turns out that due to the control by the transport of 0 0 water the higher ia and ic the drier the steel–concrete surface. 0 The corrosion potential is a decreasing function of ia in the whole 0 range of humidity while it increases with ic in the low range of humidity only.

Similarly Fig. 10 shows the sensitivity with the Tafel slopes. A factor 0.5 has been applied to the reference values. Only the corrosion potential is affected by a change in the Tafel slopes. 5. Conclusion By modelling the corrosion kinetics and the transport of ions in the concrete, we have simulated the corrosion of rebars in different concrete subjected to variable humidity conditions. The results have allowed interpreting the mechanisms controlling the corrosion current in the whole range of humidity. These mechanisms result in the observation of a maximum corrosion current for a critical value of humidity depending on the concrete properties. In the high range of humidity the corrosion current is diffusioncontrolled. In the low range of humidity and in some cases a corrosion induced drying phenomena has been observed. In these cases 10 Corrosion current (μA/cm2)

Corrosion potential vs SHE (V)

0.6 0.5 0.4 0.3 0.2 βa = 60 mV ; βc = 160 mV

0.1

βa = 30 mV ; βc = 80 mV 0 0

20

40 60 hr (%)

80

100

0

10

-1

10

-2

βa = 60 mV ; βc = 160 mV 10-3

βa = 30 mV ; βc = 80 mV 0

20

40 60 hr (%)

Fig. 10. Sensitivity analysis with respect to ba and bc .

80

100

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P. Dangla, W. Dridi / Corrosion Science 51 (2009) 1747–1756

the corrosion current is controlled by the transport of water rather than by the transport of charges because the permeability of the concrete is not high enough to supply the steel–concrete surface with water. Introduction of the surface degree of saturation as an independent property can modify these conclusions in the low range of humidity. Appendix. The finite volume method For a mass balance equation

@n ¼ divw @t

ð32Þ

and a Fick’s law

w ¼ Drq

ð33Þ

the finite volume method is obtained by a discrete balance in the elements of a mesh. It can be written as

XK

nnþ1  nnK X K wKK 0 ¼ 0 þ Dt K0

where wKK 0 expresses an approximation of block K and any neighbor K 0 of K:

wKK 0 ¼ j@ XKK 0 jD

qK 0  qK dKK 0

ð34Þ R @ XKK 0

w  n ds for any grid

ð35Þ

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