CORROSION SCIENCE SECTION
Application of a General Reactive Transport Model
to Predict Environment Under Disbonded Coatings
N. Sridhar, D.S. Dunn,* and M. Seth**
ABSTRACT
Understanding the evolution of the chemical environment and
potential inside a disbonded region is essential to a quantitative risk assessment of corrosion and stress corrosion cracking of pipelines. A general reactive transport model is
presented in this paper that enables calculation of the time
evolution of chemistry and potential under disbonded coating
exposed to a variety of solutions under an applied potential.
Model predictions are compared to a number of experimental
observations reported in the literature. It is shown that the
predicted pH and potential gradients in the disbonded region
are the results of competing anodic dissolution and cathodic
reactions. The pH and potential gradients are influenced by
exposure time, applied potential, solution conductivity, and
crevice geometry. The presence of carbon dioxide (CO2) results in a lower pH at short time periods, but after a few
thousand hours, the pH is determined by the cathodic potential. Further improvements to the model are identified.
KEY WORDS: cathodic protection, crevice corrosion,
disbonded coating, models, pipelines, stress corrosion
cracking
INTRODUCTION
According to the Office of Pipeline Safety statistics, in
the years 1994 through 1998, the average percentage
of total annual failure incidents attributable to exterSubmitted for publication January 2000; in revised form,
February 2001. Presented as paper no. 366 at CORROSION/
2000, March 2000, Orlando, FL.
* CNWRA, Southwest Research Institute, 6220 Culebra Road, San
Antonio, TX 78238-5166.
** Technical Software and Engineering, Inc., 2506 Springwood Lane,
Richardson, TX 75082.
598
nal corrosion was 9.5% for gas transmission and
gathering lines and 15.7% for liquids pipelines. While
accurate statistics regarding the proportion of external corrosion failures attributable to disbonded coatings are not available publicly, disbonded coatings
present a great uncertainty to pipeline companies in
terms of risk management. These uncertainties arise
from not only the possibility of deep localized corrosion under the disbonded region, but also stress corrosion cracking (SCC). Both intergranular (high pH)
and transgranular (near-neutral pH) SCC have been
observed mainly under disbonded coatings.1-4 If the
disbonding of coating is detected or suspected, pipeline companies want to assess the risk of continued
operation and the effectiveness of mitigation measures. The mitigation measures can include additional cathodic protection (the application of more
negative potentials in a continuous or pulsed fashion), application of additional layers of a different
type of coating on top of the disbonded region, or
removal of the old coating and application of a new
coating.
Disbonding may be created either by an inadequate coating process, naturally existing crevices in
certain coating geometries (e.g., spiral-wrapped coatings have crevices under overlaps and tenting may
occur when taping over weld crowns), differential soil
stresses, or cathodically induced reduction in bonding between coating and substrate. If a defect (or
coating holiday) exists at one end of the disbonded
region, groundwater penetrates the disbonded region
and a crevice cell is established. The disbonding of
coating is believed to be detrimental because ca-
0010-9312/01/000123/$5.00+$0.50/0
© 2001, NACE International
CORROSION–JULY 2001
CORROSION SCIENCE SECTION
thodic protection (CP) may not be able to penetrate
deep into the crevice from the coating defect at the
mouth of the crevice. This effect, called shielding, is
believed to be more severe for certain types of coating
(e.g., polyolefin tape) and low-conductivity groundwater. Quantitative prediction of CP penetration in crevices has been attempted by a number of researchers
over the last 30 years, but there are apparent inconsistencies in the findings. Toncre partially reflected
this view by concluding that the theoretical analyses
and laboratory studies showed that CP can penetrate
adequately through the disbonded region given sufficient time,5 whereas field studies indicated that some
disbondments exhibited significant corrosion including through-wall pitting. Nevertheless, Toncre recommended that application of –1 V vs copper-copper
sulfate (Cu-CuSO4) would achieve adequate CP under
disbonded coating.5
Even among laboratory tests and analyses, there
are significant discrepancies. In natural seawater
and brackish water (1,090 ppm chloride), Toncre and
Ahmad found that adequate CP was achieved at a
distance of 45.7 cm (18 in.) within a crevice.6 However, they found a potential difference between the
interior and exterior to be on the order of 500 mV,
even for a sandblasted surface. The potential was applied by a direct current (DC) power source between
the steel and an anode in the same tank, and tests
were conducted for 10 days. In contrast, Turnbull
and May found no significant gradient in the potential within crevices made between two steel plates
immersed in 3.5% sodium chloride (NaCl), natural
seawater, and ASTM(1) synthetic seawater, unless the
applied potential was > –700 mV vs saturated
calomel electrode (SCE) or < –1,000 mVSCE.7 Turnbull
and May found that the pH inside the crevice increased with more negative potential and was not
significantly affected by the crevice gap.7 It should
be noted that Turnbull and May performed their
tests for time periods ranging from 2 days to 10 days.
Peterson and Lennox found similar dependence of
crevice pH on applied potential for Type 304 (UNS
S30400)(2) stainless steel exposed to 3.5 wt% NaCl at
room temperature.8 They conducted their applied
potential tests for time periods up to 75 h. Fessler, et
al., found that the potential gradient inside a crevice
between steel and polymer immersed in a moderately
concentrated solution of 1 N sodium bicarbonate
(NaHCO3) + 1 N sodium carbonate (Na2CO3) decreased with time in an exponential fashion given by:9
 −x 
E( x ) = A 0 + E applied − A 0 exp

 A1σt 
(
(1)
(2)
)
ASTM, 100 Barr Harbor Drive, West Conshohocken, PA 19428.
UNS numbers are listed in Metals and Alloys in the Unified
Numbering System, published by the Society of Automotive
Engineers (SAE) and cosponsored by ASTM.
CORROSION–Vol. 57, No. 7
(1)
where E(x) is the potential at any distance inside the
crevice, A0 and A1 are constants to be determined experimentally, σ is the solution conductivity, and t is
time. Equation (1) suggests that the crevice potential
will approach the externally applied potential at long
time periods. They arrived at Equation (1) by assuming that there was no chemical gradient. This is obviously inconsistent with experience since the pH has
been demonstrated to increase inside the crevice.
Later, Markworth suggested that chemical changes
may be responsible for such a potential distribution,
but did not provide a specific model.10 Parkins suggested that the rate of change of potential is a function of steel surface condition and the residence time
in a critical potential regime for SCC may be responsible for the observed SCC in some pipelines.3 Gan,
et al., examined the potential distribution under
crevices in NaCl solutions ranging in conductivity
from 0.24 mS/cm to 6.3 mS/cm (0.002 M to 0.06 M
NaCl).11 They showed that the potential at a distance
of 7.4 cm from the mouth of the crevice became continually more negative, in qualitative agreement with
Fessler, et al.9
Jack, et al., conducted disbonded coating experiments using a thermoplastic coating and disbondment gaps ranging from 1 mm to 5 mm.12 Test
solutions were varying concentrations of potassium
chloride (KCl) to obtain conductivities ranging from
0.56 mS/cm to 4.18 mS/cm. They found that the
potential distribution inside the disbonded region fit
the equation:
(
E( x ) = E corr + E corr − E applied


−x
exp 

2
.
086
0
.
826
G
S
+
(
)


)
(2)
where Ecorr is the corrosion potential of the steel in
the solution, Eapplied is the applied cathodic potential
at the mouth of the disbondment, G is the crevice
gap in centimeters, and S is the conductivity of the
solution in mS/cm. Their results, obtained over a
relatively short period of time (24 h), indicate that in
these relatively low-conductivity solutions, the potential at the far end of the crevice never approaches the
protection potential. Brusseau and Qian also measured the potential distribution under an artificial
disbondment exposed to a relatively dilute solution
(5 × 10–4 M NaHCO3 + 5 × 10–4 M calcium chloride
[CaCl2] + 5 × 10–4 M tricalcium phosphate [Ca3(PO4)2])
and came to the conclusion that the potential at the
deepest point in the crevice (38 cm) remains at corrosion potential after 250 h of cathodic polarization.13
However, the depth of penetration of adequate CP
(–950 mVSCE) increased with greater externally applied CP (more negative potential). They also found
that the pH at the deepest point remained near neu-
599
CORROSION SCIENCE SECTION
tral, while locations closer to the mouth attain quite
alkaline pH values.
Lara and Klechka discussed the evolution of
potential under a polymethyl methacrylate (PMMA)steel crevice exposed to saturated soil with conductivity ranging from 5.56 µS/cm to 0.17 mS/cm.14 The
crevice gap was 1.1 mm, the total length of the crevice region was 24 cm, crevice width was 10.8 cm, and
there was a rectangular holiday region with an area
of 48.4 cm. They concluded that regardless of the
conductivity, the potential 17.8 cm deep into the
crevice was close to the externally applied potential,
and the pH attained a value close to 12. The pH inside the crevice only depended on the potential at the
holiday, not on solution conductivity. The total duration of their test was 162 days, but the CP “on” time
ranged from 21 days to 61 days.
The apparent divergence in the potential and
pH distribution within the disbonded crevice indicates that:
—Solution conductivity is not the only parameter
governing the evolution of conditions inside the
disbonded region.
—The change in pH within the crevice has a
complex dependence on external environment and
crevice geometry.
—Test time is an important factor in assessing
the changes inside the disbonded region.
The other factors that are important in quantitatively modeling the chemistry inside the
disbonded region are the temperature, chemistry of
the bulk solution (cationic and anionic concentrations and gases), the permeability of the coating
material to gases such as carbon dioxide (CO2)
and O2, steel surface condition, and the size of the
holiday region.
Since solution conductivity is not the only (nor,
perhaps, even the most important) parameter dictating the evolution of conditions inside the disbonded
region, a quantitative model is needed that can consider the chemical species and homogeneous and
heterogeneous reactions explicitly. Since test time
is an important factor in assessing the effect of
disbondment on corrosion or SCC, a transient rather
than a steady-state model is essential. A computer
model, Transient Electrochemical Transport
(TECTRAN) code, that satisfies these requirements
is described in this paper.
Modeling crevice corrosion by solving a coupled
reactive transport equation is not new in corrosion
science.15-21 However, the following are some of the
features that distinguish the present model from previous computer models of crevice corrosion:
—The chemical species and the kinetic reactions
can be specified through an input file rather than
requiring modification of the code.
—A large number of chemical species, limited
only by the computer memory and thermodynamic
600
data, can be specified, enabling a wide variety of
problems to be solved.
—A wide range of electrochemical and non-electrochemical reactions can be included, with several
parallel steps. Different electrochemical reactions
can be specified at different spatial locations of a
system as desired by the user.
—The code considers mineral precipitation, but
the current version does not include change in crevice geometry as a result of mineral formation.
—Equilibrium between the gas and aqueous
phases is considered explicitly, although the code in
its present form does not permit separate permeation
of gas through the coating.
The technical basis for the code is described first
and is followed by comparisons to experimental investigations. While the computer code is applicable to
a wide variety of reactive-transport problems involving electrochemical reactions, the focus in this paper
is on application of this model to disbonded coating
corrosion.
MODELING APPROACH
Model Formulation
The basis for reactive transport modeling has
been described in detail elsewhere.22 In any system of
Nc chemical components, the reactions can be represented in the following forms:
Homogeneous Reactions:
∑ ν ji A j ↔ A i
j
Gaseous Equilibrium Reactions:
∑ νgji A j ↔ A ig
j
Mineral Precipitation Reactions:
(3)
(4)
∑ ν jm A j ↔ M m (5)
j
Electrochemical Reactions of Dissolved Species:
∑ νejk A j + n k e(−s ) ↔ A ek
(6)
j
Electrochemical Reactions Involving a Solid:
∑ νejm A j + n m e(−s ) ↔ Mem
(7)
j
In Equations (3) through (7), the species with subscript “j” are called primary or basis species and the
others called secondary species. The choice of primary and secondary species is a matter of convenience, although primary species are generally
chosen such that they may be present throughout
the spatial domain. Equation (3) is exemplified by the
hydrolysis reactions of dissolved ferrous ions. Equation (4) illustrates the dissolution of oxygen in the
CORROSION–JULY 2001
CORROSION SCIENCE SECTION
aqueous environment. Equation (5) describes the
formation of various iron oxides and carbonate.
Equation (6) is exemplified by the cathodic reduction
of dissolved oxygen and hydrogen ion. Equation (7)
represents the anodic dissolution of iron.
In the computer model presented in this paper,
the reactions represented by Equations (3) and (4)
are considered to be in equilibrium and the secondary species are calculated from the primary species
through a thermodynamic model using an appropriate database. For relatively dilute solutions typically
encountered in groundwater, a Debye-Hückel activity
coefficient correction taken from the EQ3/6 database, data 0.com.R16,23 is used as a reasonable
approximation. For estimating activity coefficients
in more concentrated solutions, a Bromley and
Meissner approach for ion-ion interaction and Pitzer
approach for ion-molecule and molecule-molecule
interactions can be invoked by the user.24 In this
case, the code accesses a thermodynamic speciation
module derived from a commercial package.24
The reactions represented by Equations (5), (6),
and (7) are treated as kinetically controlled reactions.
The relationship between the primary and secondary
species in these reactions are represented in terms of
various kinetic rate laws appropriate for the species.
For non-electrochemical reactions, a transition-state
rate law is used:
I m = − k m s m 1 − (K m Qm )

σm

(8)
where Im is the molar rate of formation or dissolution (mol/cm3-s), km is the reaction rate constant
(mol/cm2-s), sm is the specific surface area (cm–1),
Km is the equilibrium constant for the reaction written in Equation (5), Qm is the ion activity product,
and αm is a constant (sometimes called the Tempkin
constant) for a given reaction. When the reaction
is at equilibrium, the product (KmQm) = 1 and the
reaction rate vanishes. The value of Im is positive if
mineral precipitation occurs and negative if dissolution occurs. For a homogeneous reaction (Equation
[3]), a similar formulation can be used, without the
specific surface area. For an electrochemical reaction, a generalized Butler-Volmer formulation can
be used:

e − α m ηm − eβm ηm
Iem = s m ∑ Pml k ml 
s
P
k
l
1 + m ml ml e − α m ηm + eβm ηm
r

lim
(
)




(9)
where Pml is the prefactor consisting of the product
of concentrations of species considered to affect the
kinetics (user defined), rlim is the limiting rate of
reaction, αm and βm are transfer coefficients, and the
dimensionless overpotential is defined as:
CORROSION–Vol. 57, No. 7
ηm =
(
nmF
E – E eq
m
RT
)
(10)
where nm is the number of electrons involved in the
reaction, F is Faraday’s constant, R is the gas constant, T is absolute temperature, E is the potential at
eq
any spatiotemporal point, and Em
is the equilibrium
potential that is dependent upon the concentrations
of species involved in the electrochemical reaction. It
can be shown that:
E eq
m =
RT
ln[k m Qm ]
nmF
(11)
For the disbonded coating application, where the
reactions occur generally far from equilibrium, a
rate-limiting Tafel relationship is more appropriate:
Iem
ηm


+–
0.43 ba ,c


±e
= s m ∑ Pml k ml 

ηm


l

s m Pml k ml  0.43 ba ,c  
e

1 +
rlim


(12)
where ba,c represent the Tafel slope in V/decade. In
addition to these reaction laws, the code allows input
of a constant reaction rate as well as experimental
polarization curves in the form of tabular data. Note
that the polarization curve represented by Equation
(12) exhibits dependence on solution chemistry from
the prefactor and equilibrium potential. The prefactor
is the product of concentrations of species involved
in the electrochemical reactions raised to a power
that depends on the order of the reaction, with respect to that species. The species for the prefactor
can be chosen by the user.
The overall transport equations at any point can
be written as:
( )
(
)
∂
φΨj + ∇ • Ωdj + Ωej = − ∑ νeji Iei − ∑ νejm Iem − ∑ ν jm I j (13)
∂t
i
m
m
where ␾ is the porosity, ν are the stoichiometric coefficients, Ie are the electrochemical reaction rates, and
Ij is the non-electrochemical reaction rate. The generalized concentration (ψj) is given in terms of the primary species (Cj) and secondary species (Ci) by:
Ψj = C j + ∑ ν ji C i
i
(14)
The diffusive flux (Ωdj) is given by:



Ωdj = − φ ∇ • D jC j + ∑ ν ji Di C i 

i


(15)
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CORROSION SCIENCE SECTION
i 0 = F ∑ z jΩdj
The electromigration flux (Ωej) is given by:

 F
Ωej = − φ D jC jz j + ∑ ν ji Di C i z i 
∇•Φ

 RT
i
(16)
The Φ in Equation (16) is the potential in solution,
whereas the E in Equation (10) is the potential difference between the metal and solution. If the two sides
of Equation (13) are multiplied by zj, and taking advantage of electroneutrality:
∑ z jΨj = 0
(17)
j
Then, an independent equation results:
∑ z j∇ • (Ωdj + Ωej ) = − ∑ z ei Iei − ∑ z em Iem
j
i
m
(18)
In Equation (18), the right-hand side involves only
electrochemical reaction rates multiplied by the net
charge involved in each reaction. One method of
solving the reactive transport problem is to solve
Equations (13) and (18) simultaneously to yield concentrations of primary species and potential. This
is done using an implicit finite difference approach.
The boundary conditions are specified in terms of
constant concentration (total or individual species)
or zero flux. In the former case, the applied potential
is considered, whereas for zero flux boundary, the
applied potential is not considered in the calculation.
Reaction kinetics can be specified at each node as
well as the crevice gap. Once the primary concentrations are known, the secondary species concentrations can be solved using appropriate equilibrium
or kinetic expressions. This is the preferred method
in the code. The total current in the solution is
given by:
(
i = F ∑ z l Ωdl + Ωel
)
(19)
From Equations (15), (16), and (19), a relationship
between the potential and the total solution current
can be obtained:
 i − i0 
∇•Φ = 

 κ 
(20)
where κ is the conductivity defined by:
κ=φ
F2
RT




∑ z j  z jD jC j + ∑ ν ji DiCi 
j
i
(21)
The term, i0, in Equation (20) can be considered as a
diffusion current given by:
602
(22)
where Ωdj is defined by Equation (15). Note that Equation (20) is a modified form of Ohm’s law. Finally,
from Equations (18) and (19), the divergence of the
total current is related to the net reaction:
(
)


∇ × i = F • ∑ z j∇ • Ωdj + Ωej = − F •  ∑ z ei Iei − ∑ z em Iem  (23)


j
i
m
Hence, another method to solve the reactive transport problem is to solve Equation (23) first, assuming
an initial potential, then solve Equation (20) by iteration until the potential converges, and finally solve
Equation (13) iteratively to obtain new concentrations for the next time step. The code allows the user
to chose either method, but the second method, at
present, can be used only for 1-D problems.
Prior to simulation of CP systems, two benchmark problems were analyzed using the computer
code. These benchmark problems yield simple analytical solutions, which then can be used to check
the numerical solution. Good agreement was obtained between numerical and analytical solutions.
The benchmark problems are discussed in the
Appendix.
Electrochemical Reaction Rates
The four most important reactions for modeling
the behavior of steel under disbonded coatings are
the iron dissolution, oxygen reduction, hydrogen ion
reduction, and water reduction reactions. In some
cases, the dissolved CO2 (carbonic acid [H2CO3]) may
also contribute significantly to the cathodic reactions.25 While the code does not place any limitation
on the number of electrochemical reactions, the carbonic acid reduction reaction is ignored in the simulations discussed here. The range of parameters for
these reactions, obtained from the literature, is
shown in Table 1. The highlighted values are used
in the code calculations presented in this paper. As
shown in this table, the cathodic reduction reaction
of oxygen depends on the concentration of dissolved
oxygen and pH. The equilibrium potential is assumed
to be explicitly dependent on the concentration of
dissolved species appropriate for the reaction under
consideration. However, for iron dissolution, the
equilibrium potential is considered to be fixed at
–0.458 VSHE. This is consistent with the experimental
polarization curves.25-26 The rationalization provided
by Nesic, et al., for this assumption is that the
exchange current density also depends on the
dissolved ferrous ion concentration; and hence,
the assumption of a fixed equilibrium potential compensates for the lack of assumed dependence of exchange current density on ferrous ion concentration
(Table 1).25
CORROSION–JULY 2001
CORROSION SCIENCE SECTION
TABLE 1
Selected Values of Electrochemical Reaction Parameters from the Literature
(A)
Species
n
io (A/m2)
ilim (A/m2)
Tafel (V/decade)
Species Order
Reference
O2
O2
O2
H2O
H2O
H+
Fe
Fe
Fe
4
4
4
1
1
1
2
2
2
1.24 × 10–20
1 × 10–6
—
8.9 × 10–7
3 × 10–5
5.0 × 10–2
0.1 to 1.0
2.6 × 104
3.8 × 103
—
—
1.5 × 10–1
No limit(A)
No limit
3 × 10–1
No limit
No limit
No limit
0.12
0.14
0.06
0.118
0.12
0.12
0.04
—
0.04
None
[O2]0.5; [OH–]–1
[O2]1.0; [OH–]0.6
None
None
[H+]0.5
None
None
[Fe2+]0.7; [OH–]0.5
30
31
32
26
25
25
25
26
33
Highlighted values are used in the code calculations.
No limit indicates that the limiting current was either not measured or was very high. A default value of 1 × 106 A/m2 was used to prevent
calculated current from becoming singular.
An example of the electrochemical reaction kinetics represented by Equation (10) is the reduction
of oxygen. Based on the kinetic relationships shown
in Table 1, the oxygen reduction kinetics can be
written as:
I O2
 Φ − E O2  

eq
−

 0.43 bc  

1.0


− 0.6 O2
s
O
OH
i
e

 O2 [ 2 ]
o
=

 Φ − E O2  

eq

−
 




.
0
6
.
0
43
b
1.0
c 



−
2
iO

o e
 1 + [O2 ] OH




i lim


[
]
[
]
(24)
where the equilibrium potential is determined by the
pH and concentration of dissolved oxygen. The surface area per unit volume (sO2) can be specified for
different reactions at different locations. In the 1-D
simulation, the species arising from a heterogeneous
reaction within an element is averaged over the volume of that element. Hence, the specific surface area
(sO2) is equivalent to the inverse of the crevice gap.
RESULTS
The model predictions are compared to experiments conducted for the present study and others
reported in the literature.
Three sets of experiments were studied:
—Experiments conducted by Turnbull and May7
and Parkins and Liu27 used a steel-to-steel crevice
with various crevice gaps in 3.5 wt% NaCl as well as
seawater. Various cathodic and anodic potentials
were applied at the mouth of the crevice solution and
the pH was measured over a 2-day to 10-day time
period. The steel surfaces were ground and polished.
—Experiments conducted by Brousseau and
Qian involved a steel-PMMA crevice of variable crevice gap ranging from 8 mm at the mouth to an unknown crevice gap,13 which may be a few microns
CORROSION–Vol. 57, No. 7
dictated by surface roughness of steel at the tip. The
steel surfaces were grit-blasted and machined.
—Experiments conducted by the authors of this
paper (referred to in the rest of the paper as SwRI
experiments) relied on a metal-to-polyethylene tape
crevice to better simulate actual coating. The steel
surface had an undisturbed oxide layer to simulate
exposed pipeline surface conditions.
In all these experiments, the external potential
was held constant over the experimental period.
Turnbull and May7
and Parkins and Liu27 Experiments
The Turnbull and May experiments were performed in either artificial seawater or a 3.5% NaCl
solution.7 Experiments performed in 3.5% NaCl solution are simulated here for simplicity. The crevice
was created by abutting two steel plates with polytetrafluoroethylene (PTFE) spacers and ensuring that
solution penetrates through only two ends and not
the sides. The crevice assembly was completely immersed in 14 L of solution, which was replenished
midway through the test. Cathodic potentials, ranging from –0.7 VSCE to –1.1 VSCE were applied. Typically, the experiments started by applying a potential
at the high end of this range for up to 10 days
and stepping down the potential every 2 days. The
potential just outside the crevice was controlled and
the pH and potential at two locations were monitored. The total crevice length was 240 mm and
gaps ranged from ~ 0.75 mm to ~ 0.002 mm. In the
Parkins and Liu experiment, a segmented crevice
was used with a gap of 0.25 mm and a length of
70 mm.27
These experiments were simulated using a 1-D
geometry. Because of the symmetry, the simulation
assumes a crevice length of 120 mm (with one closed
end) and a gap of ~ 0.4 mm. It was assumed that the
end open to bulk solution was maintained at a constant potential and concentration for each simulation. The pH of the bulk solution was determined by
603
CORROSION SCIENCE SECTION
charge balance and therefore is not constant. The
temperature was assumed to be 25°C, although the
experiments were performed at temperatures of
18°C and 5°C. Equilibrium of the bulk solution with
atmosphere (0.21 atm O2 and 10–3.5 atm CO2) was
assumed. The end opposite to the open end was
assumed to be a zero-flux boundary.
As expected, the predicted pH inside the crevice
after 1,000 h increases as the externally applied
potential becomes more negative (Figure 1). It can be
seen that at more positive potentials, the agreement
between experiments and calculation is quite good.
At externally applied potentials more negative than
~ –0.9 VSCE, the calculated pH reaches a maximum
value of ~ 11.2, whereas Turnbull and May observed
that the pH in the crevice attain a constant value at
lower potentials (–1.1 VSCE) and the value is higher
(~ 12.5).7 The calculated pH at more negative potentials is determined by the exchange current density
of water reduction because the anodic current density is negligible and the oxygen is consumed rapidly,
thus reducing its limiting current density. As the
water is reduced and the pH rises, the equilibrium
potential for water reduction becomes more negative,
resulting in the water reduction rate to be close to
the exchange current density.
The dependence of crevice pH on external potential is also consistent with the results of Lara and
Klechka in groundwater of varying conductivity.20
The spatial distribution of pH, total Fe2+, and dissolved O2 and H2 concentrations after 1,000 h of
simulation time are shown in Figure 2. It can be
seen that oxygen concentration decreases within the
crevice because of consumption through cathodic
processes and dissolved hydrogen concentration
increases as a result of the reduction of water. The
pH increase with depth in Figure 1 may be, at first,
counter intuitive because the cathodic polarization is
greatest at the mouth of the crevice, and, hence, the
highest pH should occur there. However, the mouth
pH is affected by diffusion of lower pH solution from
the bulk electrolyte, which is assumed to be at constant concentration. These two opposing phenomena,
lowering of pH caused by diffusion from the bulk and
increase because of the cathodic polarization results
in a slow increase in pH with increasing depth. In a
stagnant bulk solution, there will be a small diffusion
region outside the crevice, which will affect the pH
gradient in the crevice. This will be especially important near disbonded coatings on pipelines where
there may be limited groundwater.
The potential gradient in the crevice is shown in
Figure 3. There is no significant potential gradient at
the long time periods. At short time periods, a signifi-
(a)
(b)
FIGURE 1. Comparison between calculated and measured pH at
the deepest point in the crevice. Experiments were performed by
Turnbull and May7 and Parkins and Liu.27
2+
FIGURE 2. Predicted distribution of pH, total Fe , O2(aq), and H2(aq) concentrations for a crevice controlled at –0.8 VSCE
(–0.558 VSHE) after 1,000 h. Note that the distance axis is plotted on log scale in Figure 2(b) to show the oxygen curve
better. External solution composition was 0.6 M NaCI.7
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CORROSION SCIENCE SECTION
(a)
(b)
FIGURE 3. Distribution of potential in the same crevice at various externally applied potentials for two time periods.
Simulation of Turnbull and May experiments.7 The applied potentials are given by y-intercepts.
cant potential gradient is observed only at the high
applied potentials. This is consistent with the experimental observation of Turnbull and May.7 Lack of potential gradient in such a highly conductive solution
is not surprising. Turnbull and May7 and Parkins
and Liu27 reported a significant negative potential
gradient at the high applied potentials, which is consistent with the predicted results at short time periods. It should be noted that the experimental time
periods were much shorter than 1,000 h.
Effect of solution resistivity on the predicted
potential gradient is shown in Figure 4. In this case,
the same crevice geometry as in the Turnbull and
May experiments was assumed, but the solution
composition was varied from 0.6 M to 0.0002 M
NaCl, with solution resistivity varying from ~ 16 Ω-cm
to 46,700 Ω-cm. As expected, the potentials deep inside the crevice tended to be more anodic with higher
solution resistivity. However, the potential gradient
tended to decrease with time, suggesting that the
empirical equation (Equation [2]) developed by Jack,
et al., is not valid over long periods of time.12 However, Equation (1), which predicts a progressive decrease in potential gradient, is only partially valid
because it fails to consider the effects of reaction
rates and the changes in the solution conductivity
with time as the concentration of dissolved species
increases in the crevice.
Brusseau and Qian13 Experiments
Their experiments involved a crevice between
PMMA and steel such that one end of the crevice was
closed and the other end had a single hole through
CORROSION–Vol. 57, No. 7
FIGURE 4. Effect of solution resistivity on potential drop and pH in a
disbondment. The crevice gap was assumed to be 0.04 cm and the
applied potential was –0.958 VSHE.
which the crevice region communicated with an external reservoir that was ≈ 1.6 L in volume (Figure 5).
The dimension of the hole was not provided by the
authors, but was assumed to be ≈ 2 cm long. The
crevice gap varied from 8 mm to a few microns over a
length of ~ 480 mm (the crevice gap at the tip of the
crevice is unknown, but is assumed to be very small,
perhaps determined by the surface finish of the
substrate). The bulk solution contained 5 × 10–4 M
sodium bicarbonate (NaHCO3), 5 × 10–4 M calcium
chloride (CaCl2), and 5 × 10–4 M tricalcium orthophosphate (Ca3[PO4]). A constant external potential,
ranging from –1.06 VSCE to –1.5 VSCE was applied with
the reference electrode placed at the hole near the
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CORROSION SCIENCE SECTION
FIGURE 5. Illustration of the experimental arrangement used by
Brusseau and Qian.21
mouth of the crevice. The initial resistivity of the
solution was calculated to be ≈ 17,000 Ω-cm.
This experiment was simulated using a 1-D
simulation. Two different crevice profiles were assumed and the potential and solution concentration
at the mouth of the crevice were held constant. To
model the crevice in Figure 5, a stepwise decrease in
crevice gap was assumed as shown by Curve A and
Curve B in Figure 6. The concentration of ionic
species in the above solution was first determined
through an equilibrium calculation using OLI Systems ESP† code. The ionic composition was then used
as the initial concentration in further calculations.
The initial composition of the solution was assumed
to consist of: 5 × 10–4 M Na+, 1 × 10–3 M Cl–, 4.1 ×
10–4 M HCO3–, 1.15 × 10–3 Ca2+, and 1.49 × 10–4 HPO42–
in equilibrium with atmospheric oxygen and CO2. A
†
Trade name.
(a)
constant external potential of –0.818 VSHE was assumed at the mouth of the crevice.
The predicted potential gradient in the crevice
was compared to the experimental values in Figure 6
for an applied external potential of –0.458 VSHE. The
predicted gradient depended on the assumed crevice
profile. For the profile (Curve B), the predicted gradient approached the measured gradient more closely
than for the assumed profile (Curve A). Note that the
experimental profile involves a very narrow crevice
gap of unknown dimension at the tip. Further reductions in the assumed crevice gap resulted in larger
potential gradient, but also increased the computation time considerably. The predicted pH for the two
assumed crevice profiles can be seen in Figure 7. The
pH deep in the crevice for the assumed profile (Curve
B) was slightly acidic, consistent with experimental
results. However, experiments showed a much higher
pH near the mouth of the crevice than predicted by
the model.
SwRI Experiments
These experiments were conducted with PMMA
crevice former and tape coating. In the case of PMMA
crevice, disbondment was created by a spacer between the steel and PMMA plates. Crevice gaps of
0.0005 cm and 0.005 cm were used, and the crevice
length was 25 cm. The holiday area was 12.25 cm2.
In the case of polyethylene tape crevice, a polyethylene tape coating was obtained from one of the pipeline companies and several layers were applied in a
longitudinal fashion on a steel plate. Then, a rectangular section was cut out from the middle of these
tape layers, creating a depression equal to the number of layers multiplied by the tape thickness. A final
layer of tape was applied, creating a rectangular
disbondment. Holes (≈ 1 mm in diameter) were
(b)
FIGURE 6. Potential distribution inside the disbonded region in the Brusseau and Qian experiment. Also shown is the
crevice gap profile used in the experiment and the two profiles used in the simulations.
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punched into the top layer at three locations along
the length of the tape to insert commercial micro pH,
chloride, and reference electrodes. The whole assembly was suspended in a tank, which contained the
bulk solution such that the solution level was above
the tape coating, at approximately the mid-level of
the upper PMMA block. The top of the tank was
closed, but the solution was not deaerated. In this
paper, the PMMA crevice was simulated because of
its better-controlled geometry and impermeability to
atmospheric CO2. The crevice gap assumed in the
model was 0.001 cm, which was twice the size of the
smaller crevice gap used in the experiments.
In the experiments reported in this paper, a mixture of 10–2 M NaCl + 10–2 M sodium sulfate (Na2SO4)
and 1/10th dilution of this solution were used as test
solutions. The disbondment gap was maintained at
0.005 cm, with a length of 25 cm. The potential at
the entry hole was fixed by a potentiostat at –0.8 VSCE
(–0.558 VSHE). The anode compartment was separated
from the cathode by a porous frit such that the hydrogen peroxide (H2O2) generation and pH change in
the anode compartment did not influence the measurements. Additionally, the steel surface was not
polished to leave the mill scale and atmospheric oxidation product intact. Parkins has shown that the
presence of mill scale on steel surface results in secondary redox reactions arising from Fe3+/Fe2+ species
present in the oxides.3 Because the kinetics of these
secondary reactions are not known sufficiently to include in the model at this time, the equilibrium potential for the iron dissolution reaction was shifted in
the positive direction by 100 mV. In future calculations, a more rigorous expression for the kinetics of
secondary redox reactions will be incorporated.
Evolution of pH and potential through a 1-D
simulation is shown in Figures 8 and 9 and are
FIGURE 8. Comparison of experimental and modeled values of pH
in the SwRI study. PMMA crevice device was used with an applied
potential of –558 VSHE. Steel surface was not ground.
CORROSION–Vol. 57, No. 7
FIGURE 7. Predicted pH in a disbonded coating at an external
potential of –0.818 VSHE after 240 h compared to experimental results
of Brusseau and Qian. Curve A and Curve B refer to the two crevice
profiles assumed in Figure 5.
compared to experimental results. It can be seen
from Figures 8 and 9 that there is reasonable
agreement between predicted and measured potentials. In the case of the experiments, the specimen
was maintained at open-circuit potential for several
hours prior to the application of cathodic polarization. This is reflected in the initial drop in the
crevice potential. In contrast, the model assumes
a fixed cathodic potential at all times and therefore
shows an initial increase in the crevice potential.
It should be noted that simulation of a ground steel
surface by using a lower equilibrium potential
resulted in substantially lower potentials and pH
values.
FIGURE 9. Comparison of experimental and modeled values of
potential at the tip of the crevice in the SwRl study. PMMA crevice
device was used with an applied potential of –558 VSHE. Steel surface
was not ground.
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CORROSION SCIENCE SECTION
(a)
(b)
FIGURE 10. Effect of crevice gap on pH and potential. A 1-D crevice geometry and 0.6-M NaCl solution are assumed.
DISCUSSION
Effect of Applied Potential
and Solution Conductivity
The model predicts that increased cathodic
potential results in increased pH within the crevice.
This is consistent with experimental observations.
The model also predicts that, provided a sufficient
cathodic potential is applied, the pH in the crevice
eventually reaches high values regardless of the solution conductivity. However, short-term results may
show a substantially less negative potential within
the crevice. In other words, shielding in a dilute
groundwater environment is a transient phenomenon
that is also dependent on factors, such as the
disbondment geometry. Given the presence of adequate CP, the near-neutral pH within the disbonded
region is not sustainable over long periods of time.
This is consistent with the results of Lara and
Klechka, who showed that the pH inside a crevice
exposed to solutions with conductivity ranging from
5.56 µS/cm to 0.17 mS/cm only depended on the
holiday potential.14 This result also suggests that the
transients in the pH and potential in the disbonded
region are important in predicting the occurrence of
corrosion and SCC. The potential and pH gradients
are, however, also dependent on the crevice geometry. In a highly resistive solution, disbondments
with narrow crevices exhibited acidic pH values over
long periods of time. As shown in Figure 7, the predicted pH values in the Brusseau and Qian experiments were not as high as the measured values near
the mouth of the crevice. One factor in the difference
between predicted and observed pH values is the
diffusion boundary layer present near the mouth. In
the 1-D simulation, no diffusion boundary layer was
assumed. In reality, the solution just outside the
mouth will undergo significant alkalinization as a
608
result of cathodic polarization, which will tend to
shift the pH inside the disbondment near the mouth.
Effect of Crevice Geometry
As shown in Figure 6, crevice gap has a significant effect on the potential and pH distribution inside a disbonded region. The effect of crevice gap
depends on the external potential and solution resistivity. The effect of crevice gap for a low-resistivity
solution (0.6 M NaCl) is shown in Figure 10. Decreasing the gap resulted in an increase in pH (Figure 11).
Reducing the gap affects three processes:
—The cathodic reduction of water, which is at its
assumed limiting value, produces OH– ions into a
smaller volume.
—The rate of oxygen reduction increases initially,
but O2 is consumed rapidly and results in a large
decrease in O2 reduction kinetics. It is further decreased since it is dependent on OH– concentration
(Table 1).
—Anodic dissolution kinetics is increased, but
the effect is more pronounced at less negative cathodic potentials.
The above results, which need to be verified experimentally, suggest a much more complex role of
crevice geometry than envisioned by existing models.
Extremely long crevices can be found under
overlapping wraps of spiral-wound tape coating or
under concrete weights placed on coated pipelines.
The effect of crevice length on the pH and potential
distribution in a low-resistivity solution (0.6 M NaCl)
is shown in Figure 11. For lengths ranging from
0.25 m to 1.2 m (corresponding to length/gap ratios
of 625 and 3,000), no significant effect of crevice
length was observed in this solution. For extremely
long crevices (length to gap ratio of 12,500), even
such a conductive solution may result in significant
gradient in potential. These model results are consis-
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CORROSION SCIENCE SECTION
(a)
(b)
FIGURE 11. Effect of assumed crevice length on predicted pH and potential inside a crevice.
tent with the predictions by Chin and Sabde, although they examined much smaller length/gap
ratios in highly resistive solutions.21 Note that the pH
change is much less significant because cathodic
processes occur even at the more positive potentials
found inside the crevice. The combined effect of solution resistivity, crevice length, and gap need to be explored further.
Effect of CO2
Presence of CO2 has been cited as the cause of
both intergranular stress corrosion cracking (IG) and
transgranular stress corrosion cracking (TGSCC). In
the case of TGSCC, CO2 can oppose the alkalization
of the environment by CP and also increase the
hydrogen generation rate.28-29 Most of the TGSCC
experiments simulate the environment under the
disbonded coating by adopting a dilute, near-neutral
solution purged with CO2. Hence, it is important to
determine whether the presence of CO2 causes the
near-neutral pH within the disbonded region. The
simulation of CO2 effect was done by varying the CO2
partial pressure in the gas phase and assuming that
this is in equilibrium with the dissolved CO2 in solution. The solution composition and applied potential
were the same as those used in the SwRI experiments
(10–2 M NaCl + 10–2 M Na2SO4; –0.9 VSCE). A 1-D simulation was conducted and the results are shown in
Figure 12. The atmospheric partial pressure of CO2 is
≈ 3.16 × 10–4 atm. If this partial pressure is increased
by 1 order of magnitude, no substantial change in pH
at the tip of the crevice is predicted, despite an increase in the dissolved CO2. However, if the partial
pressure is increased by 3 orders of magnitude, then
the crevice pH at the tip remains near neutral for a
considerable length of time before attaining an alkaline value attributable to CP. This suggests that the
transient effects are important to consider in under-
CORROSION–Vol. 57, No. 7
standing the role of environmental factors on SCC.
These results are consistent with the experimental
findings reported by Parkins in bulk solutions purged
with CO2.3
It should be noted that the partial pressures of
CO2 used in the simulations were much higher than
ordinarily observed in soils. However, it has been
stated that in some geographical locations, near
pipeline rights-of-way, unusually high concentrations
of CO2 have been observed.1-2 The selective permeation of CO2 through the coating may result in its
increased concentration within the disbonded region.
In its present stage of development, the model does
not consider diffusion of gases through the coating
separately from that of solution. This can be a serious limitation in simulating tape coatings, which are
permeable to gases such as O2 and CO2, but not to
water.28 To a certain extent, the effect of permeation
of O2 through the coating can be simulated by increasing the cathodic current for oxygen reduction.
This will result in an increase in the predicted pH.
However, a more rigorous approach is needed to
evaluate the effect of these gases. Furthermore, the
movement of the water table may affect the cathodic
potential at the holiday and affect the pH. Although
the pH is shown to increase despite the presence of
elevated levels of CO2, low pH environments can
result from the change in potential to more positive
values at the holiday during periods of low water
table. Therefore, the ability to model the transient
effects caused by changes in potential at the mouth
of the crevice will be important in accurately simulating field observations.
CONCLUSIONS
❖ A generalized reactive transport model, which permits comparison of the model predictions with ex-
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CORROSION SCIENCE SECTION
(a)
(b)
FIGURE 12. Effect of increasing CO2 in the gas phase on the environment inside the crevice. Solution was 10–2 M Cl– +
10–2 M SO42–. Potential: –0.9 VSCE.
perimental observations, was discussed. Benchmark
studies and comparison to experiments with relatively simple crevice geometry and boundary conditions showed good agreement between the model and
experiments.
❖ The pH evolution in the disbonded region is the
result of competition between anodic dissolution of
iron resulting in hydrolysis of ferrous species and
cathodic reduction of oxygen and water.
❖ The pH evolution inside a disbonded coating is
affected by the potential at the mouth of the
disbondment and solution conductivity. This is consistent with experiments where these boundary conditions are maintained. In such cases, a simple 1-D
model will be sufficient.
❖ The effect of crevice gap on pH in the crevice is
more complex. Generally, pH increases with a decrease in crevice gap under cathodic polarization.
The effect is less pronounced at high cathodic potentials. These effects are a result of the effect of crevice
gap on various reaction rates and the consumption of
oxygen. Increasing the crevice length results in a significant gradient in potential, but a less-pronounced
effect on pH.
❖ In experiments with more complex boundary
conditions, the pH evolution may be dependent upon
solution composition, applied potential, and the holiday size. The last factor is important because the
external solution outside the crevice can no longer
be assumed to be constant in composition.
❖ The presence of CO2 results in a decrease in pH
at short times, but after a few thousand hours, the
cathodic polarization effect dominates. CO2 partial
pressures < 3.16 × 10–1 atm do not have a significant
610
effect on pH in crevices, provided CP is maintained.
❖ The measured pH and potential depend on the surface condition of the steel. The effect of surface scale
on crevice pH and potential was successfully simulated by assuming a higher equilibrium potential for
iron dissolution reactions. However, this may not be
mechanistically reasonable. At present, the kinetics
of various reactions at the surface oxides have not
been incorporated in the model. Significant uncertainties exist in the electrochemical parameters.
While a more thorough review of the literature improved the estimation of these parameters, the sensitivity of these parameters to surface condition and
solution composition may necessitate a semi-empirical approach to modeling where the model is first
calibrated against known data. Such data may be obtained for a given site through the use of disbonded
coupons. The model then serves as a method to extrapolate the coupon results in time and space.
❖ The model also needs to be improved to consider
gas phase boundary conditions separately from the
aqueous boundary conditions, incorporate constant
flux as another boundary condition, and consider
open-circuit potential conditions.
ACKNOWLEDGMENTS
The code described in this paper was originally
developed by P.C. Lichtner. The technical discussions
with G.A. Cragnolino, O. Moghissi, and K. Krist
helped in the identification of test cases and improvements in conceptual models. The financial support of
GRI and project management support by P. Dusek,
K. Krist, and K. Leewis are gratefully acknowledged.
CORROSION–JULY 2001
CORROSION SCIENCE SECTION
REFERENCES
1. T.R. Jack, M.J. Wilmott, R.L. Sutherby, MP 34 (1995): p. 19.
2. National Energy Board, Report MH-2-95, “Public Enquiry
Concerning Stress Corrosion Cracking on Canadian Oil and
Gas Pipelines” (Calgary, Canada: National Energy Board,
Regulatory Support Office, 1996).
3. R.N. Parkins, “Environmental Aspects of the Stress Corrosion
Cracking of Carbon Steels,” Report no. 172 (Arlington, VA:
Pipeline Research Committee International, 1987).
4. R.N. Parkins, W.K. Blanchard, Jr., B.S. Delanty, Corrosion 50,
5 (1994): p. 394-408.
5. A.C. Toncre, MP 23, 8 (1984): p. 22-27.
6. A.C. Toncre, N. Ahmad, MP 19, 6 (1980): p. 3,943.
7. A. Turnbull, A.T. May, MP 22, 10 (1983): p. 34-38.
8. M.H. Peterson, T.J. Lennox, Corrosion 29, 10 (1973): p. 406410.
9. R.R. Fessler, A.J. Markworth, R.N. Parkins, Corrosion 39, 1
(1983): p. 20-25.
10. A.J. Markworth, Corrosion 47, 3 (1991): p. 200-201.
11. F. Gan, Z.-W. Sun, G. Sabde, D.-T. Chin, Corrosion 50, 10
(1994): p. 804-816.
12. T.R. Jack, G. Van Booven, M. Willmott, R.L. Sutherby, R.G.
Worthingham, MP 34 (1994): p. 17-21.
13. R. Brousseau, S. Qian, Corrosion 50, 12 (1994): p. 907-911.
14. P.F. Lara, E. Klechka, MP 38, 6 (1999): p. 30-36.
15. P.O. Gartland, “Modeling Crevice Corrosion of Fe-Ni-Cr-Mo
Alloys in Chloride Solutions,” Proc. 12th Int. Corrosion Cong.,
vol. 3B (Houston, TX: NACE International, 1993), p 1,9011,914.
16. S.M. Sharland, Corros. Sci. 33, 2 (1992): p. 183-201.
17. M. Watson, J. Postlethwaite, Corrosion 46, 7 (1990): p. 522530.
18. K. Stewart, “Crevice Corrosion by Cathodic Focusing” (Ph.D.
diss., University of Virginia, 1999).
19. J.C. Walton, G. Cragnolino, S.K. Kalandros, Corros. Sci. 38, 1
(1996): p. 1-18.
20. S.M. Gravano, J.R. Galvele, Corros. Sci. 24, 6 (1984): p. 517534.
21. D.-T. Chin, G.M. Sabde, Corrosion 56, 8 (2000): p. 783-793.
22. P.C. Lichtner, “Modeling Reactive Flow and Transport in
Natural Systems,” Proc. Rome Seminar on Environmental
Geochemistry, L. Martini, G. Ottonello, eds. (Rome, Italy:
University of Genoa, 1996), p. 5-72.
23. T.J. Woolery, “EQ#NR, A Computer Program for Geochemical
Aqueous Speciation-Solubility Calculations: Theoretical
Manual, User’s Guide, and Related Documentation, ver. 7.0,”
UCRL-MA-110662 (Livermore, CA: Lawrence Livermore National
Laboratory, 1992).
24. A. Anderko, S.J. Sanders, R.D. Young, Corrosion 53, 1 (1997):
p. 43-53.
25. S. Nesic, J. Postlethwaite, S. Olsen, Corrosion 52, 4 (1996): p.
280-294.
26. A. Turnbull, M.K. Gardner, Corros. Sci. 22, 7 (1982): p. 661673.
27. R.N. Parkins, Y. Liu, Report RD4649, “Effects of Dynamic
Strain on Crack Tip Chemistry, vol. 1: Tests Using Segmented
Artificial Crevice” (Palo Alto, CA: Electric Power Research
Institute, 1986).
28. J.A. Beavers, N.G. Thompson, MP 36, 4 (1997): p. 13.
29. J.A. Beavers, N.G. Thompson, “Effects of Coatings on SCC of
Pipelines: New Developments,” paper no. 95-886, 14th Int.
Conf. Offshore Mechanics and Arctic Engineering (Copenhagen,
Denmark: OMAB, 1995).
30. J.F. Yan, T.V. Nguyen, R.E. White, R.B. Griffin, J. Electrochem.
Soc. 140, 3 (1993): p. 733-742.
31. E.J. Cavo, D.J. Schiffrin, J. Electroanal. Chem. 243 (1988):
p. 171-185.
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APPENDIX A
Benchmark problems are solved using the code
to determine whether the computer code functions
properly. In these benchmark problems, the numerical results generated by the code are compared to
CORROSION–Vol. 57, No. 7
analytical solutions to partially evaluate the numerical algorithms. Because of the need to obtain analytical solutions, the benchmark problems are relatively
simple. Further, the benchmark problems may be
purely mathematical constructs and may not be realized physically. For example, a purely binary system
seldom exists in aqueous environments because of
the formation of various aqueous complexes.
NONREACTING, 1-D, BINARY SYSTEM
This system consists of two species with different
diffusivities that do not react chemically. Under these
constraints, the transport equations may be written as:
∂C1
∂2C1
z D ∂  ∂Φ 
− D1
−F 1 1
=0
C1
∂t
∂x
RT ∂x  ∂x 
(A-1)
∂C2
∂ 2C 2
∂Φ 
z D ∂ 
− D2
−F 2 2
=0
C2
∂t
∂x
∂x 
RT ∂x 
(A-2)
z1C1 + z 2C2 = 0
(A-3)
The concentrations of Species 1 and 2 are given by:
 xt 
C1,2 ( x, t ) = C1i ,2 − C1i ,2 − C10,2 Erfc 

 2 Deff t 
(
)
(A-4)
i
where C1,2
are the initial concentrations of Species 1
and 2, and C01,2 are the bulk concentrations. The effective diffusivity is given by:
Deff =
(z1 − z 2 )D1D2
z1D1 − z 2D2
(A-5)
The potential is given by:
Φ( x ) = Φ 0 −
RT D1 − D2
F z1D1 − z 2D2
  C1 ( x )  
ln
0 
  C1  
(A-6)
The diffusive current density is given by:
i 0 = Fz1 (D1 − D2 )
∂C1
∂x
(A-7)
These results are applied to a binary system consisting of Na+ and Cl– with equal initial concentrations of
10–4 M and a bulk concentration of 0.1 M. The applied potential is fixed at zero. The diffusivities of Na+
and Cl– are 1.334 × 10–9 m2/s and 2.032 × 10–9 m2/s,
respectively. The analytical solutions for the concentration and potential are compared in Figures A-1
and A-2.
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CORROSION SCIENCE SECTION
FIGURE A-1. Analytical vs numerical solution of concentration in a
binary system.
FIGURE A-2. Analytical vs numerical solution of potential.
REACTING BINARY SYSTEM
In this case, a binary system with one of the
species undergoing a reaction is considered. A single,
potential-independent electrochemical reaction rate
is considered. The transport equations in this case
can be written as:
∂C1
∂2C1
z D ∂  ∂Φ 
− D1
−F 1 1
= − Ie
C1
∂t
∂x
RT ∂x  ∂x 
(A-8)
∂C2
∂ 2C 2
∂Φ 
z D ∂ 
− D2
−F 2 2
=0
C2
∂t
∂x
∂x 
RT ∂x 
(A-9)
where the reaction rate is given by:
Ie = − ks
FIGURE A-3. Numerical and analytical calculations of steady-state
concentrations.
(A-10)
where k is the reaction rate constant and s is the
specific surface area (surface area per unit volume).
The two transport equations are coupled through the
potential-dependent term. By assuming charge balance at all points, the potential can be eliminated.
Detailed derivation is not given here for the sake of
brevity. The steady-state concentrations of the two
species are given by:
C1 ( x ) = C1l −
1
a1 ( x − l )( x + l )
2
(A-11)
C2 ( x ) = C2l −
1
a 2 ( x − l )( x + l )
2
(A-12)
with:
FIGURE A-4. Numerical and analytical calculations of potential and
current density.
612
a1 =
ks
ks
(1 − n eω l ) and a 2 = D n eω 2
Deff
eff
(A-13)
CORROSION–JULY 2001
CORROSION SCIENCE SECTION
and:
ω1 =
z1D1C1
1
and ω 2 =
(1 − z1ω1 )
z2
z12D1C1 + z 22D2C2
(A-14)
These calculations are applied to a binary system of
Fe2+ and Cl–. Concentrations at the right-hand side of
a 1-D column at a distance of 1 are assumed to be
0.1 M for Fe2+ and 0.2 M for Cl. The diffusivities are
0.8 × 10–9 m2/s and 2.032 × 10–9 m2/s for Fe2+ and Cl,
respectively. The dissolution rate constant of Fe is
fixed at 10–9 mol/cm2s. A zero flux boundary is assumed at x = 0. Analytical and numerical solutions
are compared in Figures A-3 and A-4. Since these
two solutions agree exactly, it is hard to distinguish
them in the figures.
As an alternative, a transport-limited Tafel relationship was assumed with a large tafel constant of
10. This is unrealistic from a mechanistic point of
view, but yields a potential-independent reaction rate
similar to Equation (A-10). Again, the numerical and
analytical solutions matched exactly.
CORROSION RESEARCH CALENDAR
CORROSION is a technical research journal devoted to furthering the knowledge of corrosion science and engineering. Within
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considered for publication.
2001
July 16-20 — 2001 U.S. Navy and
Industry Corrosion Technology
Information Exchange and Exhibits —
Louisville, KY; Contact Don Hileman,
Phone: 502/364-5231; Fax: 502/364-5354;
E-mail: hilemande@nswcl.navy.mil.
* August 5-9 — 10th International
Conference on Environmental
Degradation of Materials in Nuclear
Power Systems—Water Reactors —
Lake Tahoe, NV; Contact NACE, 281/
228-6200.
* August 5-11 — 8th Annual International
Conference on Composites Engineering — Tenerife, Spain; Contact David
Hui, Phone: 504/280-6652; Fax: 504/2805539.
August 12-16 — Sea Horse 2001: 52nd
Sea Horse Institute Meeting, “A Marine
Corrosion Conference” — Myrtle Beach,
SC; Contact Carolyn Bancroft, Phone:
910/256-2271, ext. 200; E-mail:
carolyn@laque.com.
* August 21-24 — 10th International
Symposium on Corrosion in the Pulp
and Paper Industry — Helsinki, Finland;
Contact Tero Hakkarainen, Phone: +358 9
456 5410; Fax: +358 9 456 7002; E-mail:
tero.hakkarainen@vtt.fi.
* August 26-29 — NACE Northern Area
Eastern Conference — Halifax, NS,
Canada; Contact Don Marchand, Phone:
902/883-2220; Fax: 902/758-3622.
* Sponsored or cosponsored by NACE
International.
CORROSION–Vol. 57, No. 7
September 3-6 — European Coatings
Conference — Zurich, Switzerland;
Contact Amanda Zilic, Phone: +49 (0) 511/
99 10271; Fax: +49 (0) 511/99 10279,
E-mail: zilic@coatings.de.
* September 9-14 — NACE Fall Committee
Week — Phoenix, AZ; Contact NACE,
281/228-6200.
September 16-21 — International
Conference on Hydrogen Effects on
Material Behavior and Corrosion
Deformation Interaction — Moran, WY;
Contact Gary Was, Phone: 734/763-4675;
Fax: 734/763-4540; E-mail: gsw@
umich.edu.
* September 18-20 — NACE UK Section
Joint Conference with ICORR —
Edinburgh, Scotland; Contact Institute of
Corrosion, Phone: +44 (0) 1525 851771;
Fax: +44 (0) 1525 376690; E-mail:
admin@icorr.demon.co.uk; Web site:
www.corrosionconf.co.uk.
September 30-October 4 — EUROCORR
2001, The European Corrosion
Congress, European Federation of
Corrosion — Lake Garda, Italy; Contact
AIM, Phone: +39 02 7639 7770; Fax: +30
02 76020551; E-mail: aim@fast.mi.it.
* October 1-3 — NACE Western Area
Corrosion Conference 2001 — Portland,
OR; Contact Roy Rogers, Phone: 503/
226-4211; E-mail: rfr@nwnatural.com.
* October 7-10 — NACE Central Area
Conference — Corpus Christi, TX;
Contact Rick Underwood, Phone: 361/
242-5520; Fax: 361/241-6940; E-mail:
underwor@kochind.com.
October 9-13 — AWT’s Water
Technologies and Exposition 2001 —
Dallas, TX; Contact AWT, Phone: 800/
858-6683; Web site: www.awt.org.
October 29-31 — 14th International
Conference on Pipeline Protection —
Barcelona, Spain; Contact Tracey
Wheeler, Phone: +44 (0) 1234 750422;
Fax: +44 (0) 1234 750074; E-mail:
twheeler@bhrgroup.com.
November 4-7 — ICE 2001, International
Coatings Technology Conference
and the FSCT Annual Meeting and
International Coatings Expo —
Atlanta, GA; Contact FSCT, Phone: 610/
940-0777; Fax: 610/940-0292; E-mail:
fsct@coatingstech.org.
November 11-15 — SSPC Annual
Conference — Atlanta, GA; Contact Rose
Mary Surgent, Phone: 412/281-2331; Fax:
412/281-9993; E-mail: surgent@sspc.org.
November 13-15 — Stainless Steel
World 2001 Conference and Expo —
The Hague, the Netherlands; Contact
Sjef Roymans, Phone: +31 575 585 286;
E-mail: s.roymans@kci-world.com.
November 18-21 — 41st Annual
Conference, Corrosion and Prevention
2001 — Newcastle, NSW, Australia;
Contact Sally Nugent, Phone: +61 (0)3
9874 0800; Fax: +61 (0)3 9874 4800.
E-mail: aca@corrprev.org.au; Web site:
www.corrprev.org.au/caphome.htm.
* December 4-6 — 8th Annual New
Orleans
Offshore
Corrosion
Conference — Kenner, LA; Contact Bill
Grimes, Phone: 504/728-4145; E-mail:
wdgrimes@shellus.com.
613
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