Corrosion of steel reinforcement can seriously compromise the service life of reinforced concrete structures. Hence, service life prediction and enhancement of concrete structures under corrosion attack are of significant importance. As a result, numerical methods that can reliably predict the service life of concrete structures have attracted increasing attention. In this, the first of two companion papers, a simple and significantly improved inverse relation relating the current density with potential for the cathodic reaction is proposed. This enables the current densities to be determined accurately from the measured potentials. Equally importantly, the proposed inverse relation also enables the efficient and straight-forward nonlinear algorithm for modeling of steel corrosion in concrete structures to be developed. Such an algorithm is presented in the companion paper of this. © 2010 by ES

Ann, Ki Yong

Dao, Luc The Ngoc

Dao, Vinh The Ngoc

Kim, Sang-Hyo Kim

University of Queensland eSpace

  
Int. J. Electrochem. Sci., 5 (2010) 302 - 313 
 
International Journal of 
ELECTROCHEMICAL 
SCIENCE 
www.electrochemsci.org 
 
 
Modeling Steel Corrosion in Concrete Structures - Part 1: A 
New Inverse Relation between Current Density and Potential 
for the Cathodic Reaction 
 
Luc The Ngoc Dao1,*, Vinh The Ngoc Dao2, Sang-Hyo Kim1 and Ki Yong Ann1 
1
 School of Civil and Environmental Engineering, Yonsei University, Seoul 120-749, Republic of 
Korea 
2
 School of Civil Engineering, The University of Queensland, Australia 
*E-mail: lucdao@yonsei.ac.kr  
 
Received:  6 January 2010  /  Accepted:  15 March 2010  /  Published: 31 March 2010  
 
 
Corrosion of steel reinforcement can seriously compromise the service life of reinforced concrete 
structures. Hence, service life prediction and enhancement of concrete structures under corrosion 
attack are of significant importance. As a result, numerical methods that can reliably predict the service 
life of concrete structures have attracted increasing attention. In this, the first of two companion 
papers, a simple and significantly improved inverse relation relating the current density with potential 
for the cathodic reaction is proposed. This enables the current densities to be determined accurately 
from the measured potentials. Equally importantly, the proposed inverse relation also enables the 
efficient and straight-forward nonlinear algorithm for modeling of steel corrosion in concrete 
structures to be developed. Such an algorithm is presented in the companion paper of this. 
 
 
Keywords: Corrosion, Concrete, Numerical modeling, Inverse relation, Cathodic reaction 
 
 
1. INTRODUCTION 
Corrosion of steel reinforcement is a well-known and well-documented phenomenon, and is 
considered the most prevalent form of deterioration of reinforced concrete structures [1, 2]. Initially, 
due to the highly alkaline nature of the concrete, a passive protective oxide film is formed on the 
surface of the steel reinforcement, effectively preventing the steel surface from being corroded. The 
passive protection layer, however, may be seriously compromised when the chemical composition of 
the pore solution is altered by carbonation or chloride contamination of the concrete cover. As a 
consequence, corrosion begins, resulting in a reduction in steel cross-sectional area, cracking, and 
spalling as well as loss in bond between steel and concrete. 
Int. J. Electrochem. Sci., Vol. 5, 2010 
  
303
In recent years, numerical methods that simulate the corrosion processes of reinforcing steel in 
concrete and allow parametric studies in addition to laborious experimental investigations have gained 
increasing attention [3-6]. This is mainly due to the considerably improved understanding of the basic 
processes underlying reinforcement corrosion and the significantly increased possiblities for modelling 
complex electrochemical processes involved in corrosion. However, much remains to be investigated 
further to fully realise the potentials of current numerical models. Specifically, because of the 
concentration polarization, the polarized potential of the cathodic reaction expressed as a function of 
the current density is nonlinear and cannot be solved for explicit inverse relations, which have to be 
approximated instead. Current approximate inverse relations, however, do not represent well the exact 
solution. This paper thus aims to propose a new inverse relation between the current density and 
potential for the cathodic reaction. 
 
 
 
 
2. KINETICS OF CORROSION 
2.1. Potential-current density relations for anodic and cathodic reactions 
The potential-current density relations for the anodic and cathodic reactions of steel corrosion 
are well-established [7, 8].  
The corrosion of steel in concrete is caused by the dissolution of iron into the pore water at the 
anode [9], which can be represented by the following half-cell reaction  
 
−+ +→ eFeFe 22
      (1)  
 
In order for electrical neutrality to be preserved, the electrons produced in this anodic reaction 
must be consumed at the cathodic sites on the steel surface [9]. The cathodic reaction is given by  
 
−− →++ OHeOHO 442 22
             (2)   
    
At equilibrium, the rate of the forward reaction in Eqs. 1 and 2 is equal to the reverse one. The 
potentials and current densities at these states are called equilibrium potentials and exchange current 
densities, respectively. When the equilibrium is disturbed, there is a net current between the anodic and 
cathodic areas on the steel, and the equilibrium potentials are changed to new potentials at the 
electrodes. The extent of potential change caused by the net current at the electrodes, measured in 
volts, is termed polarization. There are three causes of polarization, namely activation, concentration 
and Ohmic potential drop. For steel corrosion in concrete structures, activation is considered as the 
only cause of the anodic polarization while activation and concentration are the causes of cathodic 
polarization [8]. 
 
 
Int. J. Electrochem. Sci., Vol. 5, 2010 
  
304
2.1.1. Polarization of anodic reaction 
The activation polarization of the anodes can be determined by  
 
0
log
a
a
aa i
iβη =
      (3) 
 
where βa is the Tafel slope of the anodic reaction (V/dec), ia is the anodic current density (A/mm2) and 
ia0 is the exchange current density of the anodic reaction (A/mm2).  
Hence the polarized potential φa (V) of the anodic reaction can be written as [10] 
 
0
00 log
a
a
aaaaa i
iβφηφφ +=+=
    (4) 
   
where φa0 is the equilibrium potential of the anodes (V). 
 
2.1.2. Polarization of cathodic reaction 
If the amount of oxygen at cathode is not sufficient, the concentration polarization controls the 
polarization of cathodes. Concentration polarization at the cathodes can be calculated as  
 
cL
L
cc ii
i
zF
RT
−
−= log303.2η
     (5)   
 
where R is the universal gas constant (8.314 J/K.mol), T is the absolute temperature (K), F is Faraday’s 
constant (9.65.104 C/mol), z is the number of electrons exchanged in the cathodic reaction, ic is the 
cathodic current density (A/mm2) and iL is the limiting current density of the cathodic reaction 
(A/mm2).  
The limiting current density iL of the oxygen reduction at the cathodic sites can be calculated as 
follows [7] 
310
2
2 −
= O
n
O
L Ct
zFD
i δ
      (6) 
 
where 
2OD  (m2/s) is the effective oxygen diffusion coefficient in concrete, δ (= 0.005 mm) is the 
thickness of the stagnant layer of electrolyte around the steel surface, tn (=1) is the transference number 
of all ions in the solution except for the reduced species, and 
2O
C
 (mol/l of pore solution) is the 
concentration of oxygen around the steel. 
The activation polarization of the cathodes can be determined as  
0
log
c
c
cca i
iβη −=
      (7)   
Int. J. Electrochem. Sci., Vol. 5, 2010 
  
305
where βc is the Tafel slope of the cathodic reaction (V/dec) and ic0 is the exchange current density of 
the cathodic reaction (A/mm2).  
Thus the polarized potential of the cathodic reaction can be written as [10] 
 
0
00 loglog
303.2
c
c
c
cL
L
ccacccc i
i
ii
i
zF
RT βφηηφφ −
−
−=++=
  (8) 
  
where φc is the potential (V), and φc0 is the equilibrium cathodic potential (V) under a certain 
environment. Fig. 1 illustrates the relation between potentials and the current densities for anodic and 
cathodic reactions given in Eqs. 4 and 8. 
 
 
Figure 1. Potential-current density relations for anodic and cathodic reactions. 
 
2.2. Inverse relations for anodic and cathodic reactions 
In Eqs. 4 and 8, the polarized potentials φa and φc of the anodic and cathodic reactions are 
expressed as functions of the current densities. However, in reality, the potentials are usually measured 
and inverse relations are therefore required to enable the current densities to be determined from the 
obtained potentials.  
Equally importantly, suitable inverse relations are also needed to allow the efficient and 
straight-forward nonlinear algorithm for modeling of steel corrosion in concrete structures to be 
developed. The modeling of steel corrosion in concrete structures involves solving the governing 
equation in Laplace form that satisfies the two boundary conditions of potential and current density at 
the steel-concrete interfaces [11, 12]. Currently available models often use only one of the above two 
boundary conditions, with the other satisfied by iteration to convergence. In contrast, with a suitable 
inverse relation, the two boundary conditions can be combined and satisfied simultaneously, resulting 
Int. J. Electrochem. Sci., Vol. 5, 2010 
  
306
in a straight-forward algorithm for modeling of steel corrosion in concrete structures. In addition, the 
incorporation of a suitable inverse relation also enables a unified algorithm for different types of 
corrosion modeling, e.g. to solve macro-cell modeling [12-14] and macro-and-micro-cell modeling 
[11]  in one single algorithm. This is presented in detail in the companion paper [15].  
 
2.2.1. For the anodic reaction 
The inverse relation of Eq. 4 is the same as the Butler-Volmer relation for the anodic reaction 
a
aa
eii aa
β
φφ )(303.2
0
0−
=
      (9)   
 
2.2.2. For the cathodic reaction 
Without the concentration polarization term ηcc, the inverse relation of Eq. 8 is similar to the 
Butler-Volmer relation for the cathodic reaction 
c
cc
eii cc
β
φφ )(303.2
0
0 −
=
      (10)   
 
However, the effect of concentration polarization on the cathodic reaction often cannot be 
ignored due to the low oxygen concentration around the cathodic sites on the steel surface. As a result, 
Eq. 8 becomes nonlinear and cannot be solved for explicit inverse relations, which have to be 
approximated instead.  
Kranc and Sagues [16] attempted to incorporate the concentration polarization into Butler-
Volmer equation as follows 
 
c
cc
ei
C
C
i cS
O
O
c
β
φφ )(303.2
0
0
2
2
−








=
     (11) 
   
where 
2OC , 
S
OC 2  are the concentrations of oxygen in the concrete pore solution and at the steel surface, 
respectively. This relation, however, is overly simplistic. Among various shortcomings, the relation 
fails to reflect the asymptotic nature of the curve at limiting current density iL, an important 
characteristic of the current density-potential curve for cathodic reaction.  
An improved approximation was proposed by Gulikers [3], which was also based on  the 
Butler-Volmer relation 
L
c
c
c
i
ei
eii
c
cc
c
cc
β
φφ
β
φφ
)(303.2
0
)(303.2
0
0
0
1
−
−
+
=
     (12)   
Int. J. Electrochem. Sci., Vol. 5, 2010 
  
307
A comparison of Gulikers’ proposed relation and the exact solution for iL between 0.2x10-8 and 
60x10-8 (A/mm2) is given in Fig. 2. The exact solution for the current density is obtained by solving 
Eq. 8 numerically with known values of ic0, φc0, βc, iL, φc using Brent method [17]. It is clear from Fig. 
2 that Gulikers’ relation overcomes the non-asymptotic shortcoming of that proposed in [16]. 
However, Gulikers’ relation still does not completely represent the inverse relation of Eq. 8, especially 
in the regions of significant change of slope in the current density-potential curve for cathodic reaction. 
A better relation that represents more fully the inverse relation of Eq. 8 is therefore needed. 
 
 
Figure 2. Gulikers’ relation with exact inverse relation. 
 
 
 
3. NEW INVERSE RELATION BETWEEN CURRENT DENSITY AND POTENTIAL FOR 
THE CATHODIC REACTION 
 
Upon close examination of the potential-current density relation for cathodic reaction in Eq. 8 
and Fig. 2, the following features can be observed 
 
i. When the limiting current density iL of the cathodic reaction is significantly larger than the 
cathodic current density ic, i.e. iL/(iL-ic)≅ 1, the concentration polarization is negligible, 
resulting in the familiar Butler-Volmer relation for cathodic reaction as in Eq. 10. 
ii. When the polarized potential of the cathodic reaction φc approaches the equilibrium cathodic 
potential φc0, the cathodic current density ic approaches the exchange current density of the 
cathodic reaction ic0. 
iii. When the polarized potential of the cathodic reaction φc approaches negative infinity, the 
cathodic current density ic reaches the limiting current density iL (asymptotic nature of the 
curve). 
Int. J. Electrochem. Sci., Vol. 5, 2010 
  
308
Based on the above observations, a number of modified current density-potential relations for 
cathodic reaction have been investigated, among which the following relation provides the desired 
shape of the cathodic curves 
γ
γ
β
φφ
γ
β
φφ
1
)(303.2
0
)(303.2
0
0
0
1
































+








=
−
−
L
c
c
c
i
ei
ei
i
c
cc
c
cc
       (13) 
  
where γ is a curvature-defining constant. Fig. 3 illustrates the change of curvatures of the cathodic 
curves with different values of γ.  
 
 
Figure 3. Effect of γ constant on the curvature of cathodic curves. 
 
 
In order to determine the appropriate value for the constant γ, a sensitivity analysis is carried 
out by comparing the current densities determined by the exact relation and the proposed relation for a 
typical input data set and for a range of γ. Again, the exact solution for the current density is obtained 
by solving Eq. 8 numerically with known values of ic0, φc0, βc, iL, φc using Brent method. 
Based on the common values of parameters ic0, φc0, βc, iL and φc for steel corrosion in concrete 
structures (Table 1), the input parameters for the sensitivity analysis are selected and given in Table 2. 
The resulting current densities determined by the exact and proposed relations for different values of γ 
are presented in Fig. 4. By observation, the current densities determined by the two relations correlate 
very well for γ of about 3, but differ considerably for γ of less than 1.  
 
 
Int. J. Electrochem. Sci., Vol. 5, 2010 
  
309
 
a) Correlation with γ = 0.2 
 
b) Correlation with γ = 0.5 
 
c) Correlation with γ = 1 
 
d) Correlation with γ = 3 
 
e) Correlation with γ = 5 
 
f) Correlation with γ = 10 
 
Figure 4. Current density determined by exact and proposed relation for different values of γ. 
 
Int. J. Electrochem. Sci., Vol. 5, 2010 
  
310
Table 1. Review of parameters for cathodic curve. 
 
 ic0(A/mm2) φc0(mV) βc(mV/dec) 
Kim and Kim [11] 0.0006x10-8 160 176.3 
Isgor and Razaqpur [12] 0.0006x10-8 160 160 
Ghods, Isgor and Pour-Ghaz [18] 0.001x10-8 160 160 
Kranc and Sagues [16] 0.000625x10-8 160 160 
Pour-Ghaz,  Isgor and Ghods [13] 0.001x10-8 160 180 
Warkus, Raupach and Gulikers [19] - - 200 
 
Table 2. Selected input parameters for sensitivity analysis. 
(Note: the combination of different values of ic0, φc0, βc, iL, φc provided in following table                
gives 891 data points) 
 
Input parameters Values 
φc (mV) -800,-700,-600,-500,-400,-300,-200,-100,0,100,200 
ic0(A/mm2)  0.0001x10-8, 0.00055x10-8, 0.001x10-8 
φc0(mV) 100,150,200 
βc(mV/dec) 100,150,200 
iL(A/mm2) 0.2x10-8,30x10-8,60x10-8 
 
 
Figure 5. Variation of Root-Mean-Square error with γ. 
 
In order to find the optimal value of γ, the variation of Root-Mean-Square error, defined in Eq. 
14, with γ is plotted in Fig. 5. 
 
( )
n
ii
errorRMS
n
proposed
c
exact
c∑ −
=
1
2
     (14)   
Int. J. Electrochem. Sci., Vol. 5, 2010 
  
311
where exactci  and 
proposed
ci  are current densities determined by the exact and proposed relations, 
respectively, and n is the number of data points (n=891).  
It is clearly evident from Fig. 5 that the optimal value of γ is 3, and hence the proposed relation 
becomes 
3
1
3)(303.2
0
3)(303.2
0
0
0
1
































+








=
−
−
L
c
c
c
i
ei
ei
i
c
cc
c
cc
β
φφ
β
φφ
      (15)  
 
  
 
a) Changing iL 
 
b) Changing ic0 
 
c) Changing φc0 
 
d) Changing βc 
Figure 6. The current density-potential curves for cathodic reaction. 
Int. J. Electrochem. Sci., Vol. 5, 2010 
  
312
The validity of this proposed relation is further demonstrated by the following illustration. 
Let’s consider a typical base case (Table 1) of ic0 of 0.001x10-8 (A/mm2), φc0 of 160 (mV), βc of 180 
(mV/dec), and iL of 10x10-8 (A/mm2). Each of these parameters is varied in turn within its common 
range (Table 1), while the others remain unchanged. The resulting potential-current density curves are 
presented in Fig. 6. In all cases, the prediction by the proposed relation correlates very well with the 
exact solution, confirming the capability of the new relation between current densities and potentials 
for cathodic reaction. 
 
 
 
4. CONCLUSIONS 
In this paper, a new inverse relation that relates the current density with potential for the 
cathodic reaction has been proposed. Besides its desirable simple nature, the proposed inverse relation 
has been clearly shown to correlate very well with the exact solution. 
The significantly improved inverse relation proposed enables the current densities to be 
determined accurately from the measured potentials. Equally importantly, the proposed inverse relation 
also enables the efficient and straight-forward nonlinear algorithm for modeling of steel corrosion in 
concrete structures to be developed. Such an algorithm is presented in the companion paper of this. 
 
ACKNOWLEDGEMENTS 
The authors would like to thank Concrete Corea for the support in finance for this research. Mr L.T.N. 
Dao specially thanks Prof H.W. Song for his sincere support and valuable advice in setting up this 
study. 
 
 
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(1998) 205. 
2. K.Y. Ann and H.-W. Song, Corrosion Science,. 49 (11) (2007) 4113. 
3. J. Gulikers, Corrosion in reinforced concrete structures, CRC Press (2005) 71. 
4. S.C. Kranc and A.A. Sagüés, Corrosion Science, 43 (7) (2001) 1355. 
5. H.-W. Song, H.-J. Kim, V. Saraswathy and T.-H. Kim, Int. J. Electrochem. Sci., 2 (2007) 341. 
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© 2010 by ESG (www.electrochemsci.org) 
 


Modeling steel corrosion in concrete structures - part 1: A new inverse relation between current density and potential for the cathodic reaction

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Modeling steel corrosion in concrete structures - part 1: A new inverse relation between current density and potential for the cathodic reaction

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