Introduction
It is widely known that corrosion of rebars is one of the most important durability issues in reinforced
concrete life time. The corrosion process can be commenced due to chloride ingress in concrete and
attainment of chloride concentration at the rebars some threshold value. The corrosion lowers the
strength of steel and its bond with the surrounding concrete, and in consequence safety of the
structure. In coastal regions with natural high chloride concentrations present, chloride-induced
corrosion is the primary cause of environmental degradation of reinforced concrete construction.
The chloride ion diffusion coefficient is an important indicator of concrete durability. It allows to
evaluate the initiation time to corrosion due to chloride in reinforced concrete structures, and to
estimate the remaining service life (chloride penetration depth as well as the chloride concentration
profile).
Most tests are too time-consuming to satisfy practical requirements in a timely manner. That is why
accelerated methods are becoming increasingly more important.
Transport model of chloride ion in concrete
Concrete is quite complex heterogeneous composite material. Chlorides that reach the concrete
surface are transported in the pore system by diffusion and migration (i.e. movement enforced by
the electric field).
We assume that in concrete only chlorides, Cl  , are moving. The flux is given by the constitutive
Nernst-Planck formula:
J Cl    DCl  c 
F
D  zcE,
RT Cl
(1)
where: DCl   the diffusion coefficient, c  c( x, t )  the concentration of chloride ions in mol·m-3,
E( x , t )  the electric field, z  the charge of the Cl  ion (of course z  1 but we retain symbol for
clarity), and F , R, T have their usual meaning, the Faraday constant, the gas constant, and the
temperature in Kelvins. In one dimension setting and with electric potential,  , expressing the
electric field as E   we simplify (1) to
J Cl    DCl 
c F


DCl  zc .
x RT
x
(2)
The complete set of equations is obtained by mass balance law and Poisson’s equation from
electrostatics which lead to
 c
  2 c F     
z c
  DCl   2 
,
RT x  x  
 t
 x
 2
     F c,
 x 2
 r 0

A Connection Between Anomalous Poisson–Nernst–Planck Model and Equivalent Circuits with Constant
Phase Elements, J. Phys. Chem. 2013.
(3)
where  0 is the electric permittivity of free space (electric constant) and  r is the relative
permittivity of the medium where chloride anions move. Equations (3) are accompanied by the
following boundary conditions
c(0, t )  cL , c(d , t )  cR ,

 (0, t )  0,  ( d , t )  U ,
(4)
where cL , cR are chloride concentrations on the left and right surface, respectively, and U is given
potential difference (applied voltage).
Introducing dimensionless quantities
x  xs x , t  ts t , c ( x , t )  c( xs x , t s t ) / cs ,  ( x , t )   ( xs x , t s t ) / s , s 
RT
,
F
the system is written as
 c
 2c  
 

D
  z Dc
,
 t
2
x
x 
x 

 2
     zc ,
 x 2
where  
(5)
cs  ( F  xs )2
, Di  Di  ts / xs2 for t  0, x [0, d ] and boundary conditions
RT  r  0
c (0, t )  cL , c ( d , t )  cR ,

 (0, t )  0,  (d , t )  U ,
A Connection Between Anomalous Poisson–Nernst–Planck Model and Equivalent Circuits with Constant
Phase Elements, J. Phys. Chem. 2013.
(6)
for t  0.
The first part of (5) can be rewritten using Poisson equation:
 c
 2c
c 

D
 zD
  z 2 Dc 2 ,
 t
2
x
x x
 2
     zc.
 x 2
Discretization of equations (7) using a uniform mesh x0  x1 
(7)
 xn  xn1  d with step h  0
gives differential-algebraic system (DAE)
c  2ck  ck 1
c  c  
 dck
 D k 1
 zD k 1 k 1 k 1 k 1   z 2 Dck2 ,
2

 dt
h
2h
2h

k 1  2k  k 1   zc ,
k

h2

(8)
0=0
n+1=U
c0=cL
c1
c2
ck-1
x0
x1
x2
xk-1
h
ck
ck+1
cn-1
cn
cn+1=cR
xk
xk+1
xn-1
xn
xn+1
Taking into account the boundary conditions (4) c0  cL , cn1  cR and 0  0, n1  U we obtain
the following system
c2  2c1  cL
c c 
 dc1
 zD 2 L 2   z 2 Dc12
(k  1)
2
 dt  D
h
2h 2h

 dck  D ck 1  2ck  ck 1  zD ck 1  ck 1 k 1  k 1   z 2 Dc 2 , (2  n  n  1)
k
 dt
h2
2h
2h

dc
c  2cn  cn 1
c  c U  n 1
 n D R
 zD R n 1
  z 2 Dcn2
(k  n)
2
dt
h
2
h
2
h

0    2  h 2 zc
(k  1)
2
1
L

2
0  k 1  2k  k 1  h  zck (2  k  n  1)

2
( k  n)
0  U  2n  n 1  h  zcR
Radau, która napisana jest dla zagadnienia postaci My  f (t , y ), gdzie y  ( y1 ,
M
N N
, yN )T oraz
(macierz kwadratowa N x N):
Mamy wtedy
A Connection Between Anomalous Poisson–Nernst–Planck Model and Equivalent Circuits with Constant
Phase Elements, J. Phys. Chem. 2013.
 c1 
 
 
c 
y   n 
1 
 
 
n 
1


0
2n
, M 
0


0
0 0
1 0
0 0
0 0
0


0

0


0
2 n2 n
.
Inverse method
Let us denote the measured flux through the left boundary over the period of time [0, t  ] by J  (t ).
Using the model expressed by equations (3) and (4) we can also simulate this flux, J (d , t ). Now we
assume that the diffusion coefficient DCl  is the only parameter which is variable. Thus we arrive at
the problem of finding such DCl  that difference between J (d , t ) and J  (t ) is minimized. If we use
the following measure of difference
t
Err ( D)   | J (d , t )  J  (t ) |2 dt ,
(9)
0
the optimization problem for determination of DCl  reads
DCl  : Err ( DCl  )  min Err ( D),
D
where  is a constraint interval.
A Connection Between Anomalous Poisson–Nernst–Planck Model and Equivalent Circuits with Constant
Phase Elements, J. Phys. Chem. 2013.
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