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Modelling of crack chemistry in sensitized stainless steel in boiling water reactor environments
CorrosionScience,Vol. 39, No. 4, pp. 78!%805,1997 Crown copyright 0 1997Published by Elwier Science Ltd Printed in Great Britain. 001&938X/97 $17.00+0.00
PII: soo1&938x(%)00171-0
MODELLING STAINLESS
OF CRACK CHEMISTRY IN SENSITIZED STEEL IN BOILING WATER REACTOR ENVIRONMENTS A. TURNBULL
Centre for Materials Measurement and Technology, National Physical Laboratory, Teddington, Middlesex, TWll OLW, U.K. Ah&ret--An advanced model has been used to predict the chemistry and potential in a stress corrosion crack in sensitized stainless steel in a boiling water reactor (BWR) environment. The model assumes trapezoidal crack geometry, incorporates anodic reaction a&cathodic reduction within the crack, and takes into account the limited solubility of cations in high temperature water. The results indicate that the crack tip potential is not independent of the external potential, and that the reactions on the walls of the crack must be included for reliable prediction. Accordingly, both the modelling assumptions of Ford and Andresen and of Macdonald and Urquidi-Macdonald, whilst having merit, are not fully satisfactory. Extended application of the model for improved prediction of stress corrosion crack growth rate is constrained by limitations in electrochemical data which are currently inadequate. Crown copyright 0 1997 Published by Elsevier Science Ltd
INTRODUCTION
Life prediction of components, structures or products in which environment assisted cracks can develop and propagate is a major challenge which has been met primarily by establishment of empirical models based on laboratory data and/or field observation. This approach has some inherent limitations, which can be critical in particular applications. These limitations are associated with the short term nature of the data and with the complexity of service conditions which may involve both short and long term variations in temperature, environment and stresses. Deterministic models of environment assisted cracking have greater potential, in principle, since they provide insight into the controlling variables enabling, for example, an improved assessment of the impact of fluctuations in operating conditions. However, confidence in the use of such models within the engineering community is not enhanced by the continuing scientific debate as to the applicability of different models and of the different assumptions adopted. The major factor hindering progress in many cases remains the difficulty of quantifying, to a sulhcient extent, the local environment and the kinetics of reaction processes at the tip of a crack. In the application of models to prediction of crack propagation rates in BWR conditions in particular, there has been controversy between groups led by Andresen and Ford1*2 at GE and Macdonald and Urquidi-Macdonald3 at Pennsylvania State University. The reason for the difficulty is that both groups make apriori assumptions regarding the factors controlling crack tip electrochemistry, primarily because of limitations in modelling the interactive effects of mass transport, chemistry and electrochemistry in cracks in an effective manner. In Manuscript received 29 July 1996; in revised form 8 October 1996. 789
A. Turnbull
790
this paper, advanced modelling is utilised as a first stage in resolving this issue. The present paper should be considered as an update on the extended abstract published previously4 in which a computational error resulted in the generation of unreliable data and erroneous conclusions. MATHEMATICAL
MODEL
Background
The model of Ford and Andresen1*2 for cracking of sensitised 304 SS in BWR environments is based on crack advance by the slip oxidation mechanism. The key input is the transient electrode kinetics following a strain rupture event. These data can be obtained by various methods but the important requirement is to simulate conditions at the tip of the crack in relation to the local environment and electrode potential. Experimental measurement of the chemistry and electrode potential in cracks at 288°C is particularly difficult and modelling has been the primary approach to defining the local environment. Ford and AndresenlV2 make the assumption that the crack tip, in terms of the local potential, is essentially decoupled from the external driving force of oxygen reduction on the surface external to the crack. Macdonald and Urquidi-Maedonald have developed a more elaborate model of crack chemistry (the CEFM model) which takes into account mass transport by diffusion and ion migration in the crack, electrochemical reactions at the crack tip, and hydrolysis of ferrous ions. A feature of the model is that it takes account also of the potential drop local to the crack mouth. The latter authors argue that all of the current for crack advance is supported by reaction on the external surface and that the crack tip reactions and external reactions are intrinsically coupled. The model of Macdonald appears complex but simplifying assumptions are made which are critical: the crack is assumed to be parallel sided; the walls of the crack are assumed to be inert; the solubility of metal cations is not considered, despite their very low solubility at 288°C. Advanced model
More rigorous models of the chemistry in cracks and crevices have been developed previously 5-7 in which the importance of these factors has been demonstrated, but have not been applied to this specific problem. These models have now been adapted for prediction of the chemistry and electrochemistry in cracks in BWR environments. The primary features included are: (a) transport by diffusion and ion migration; (b) anodic and cathodic reactions at the tip and on the crack walls; (c) hydrolysis of metal cations; (d) precipitation to account for the limited solubility of metal cations; (e) trapezoidal crack geometry. In addition, the potential drop in the environment outside the crack is evaluated from solution of Laplace’s equation. Crack geometry.
A trapezoidal crack geometry was assumed with the width of the crack
tip defined by CTOD = 0.6( I - d)K2/Eo,
Modelling of crack chemistry in sensitized stainless steel
791
where CTOD is the crack-tip opening displacement, v is Poisson’s ratio, K is the stress intensity factor, E is the modulus and o,, is the yield strength. The crack mouth opening displacement (6) was calculated, using standard linear elastic fracture mechanics expressions’, assuming a single-edged notched (SEN) specimen with a/ b=0.3 and with the crack length, a, equivalent to the crack depth, i.e. the notch depth is negligible. The parameter b is the width of the SEN specimen. Typical values are shown in Table 1. The reason for adopting this configuration was to allow sensible application of Laplace’s equation outside the crack without having to account for the geometry of the notch and reactions on the walls of the notch. The crack length was assumed to be 1 cm. Note that the calculated crack mouth displacements for this specimen geometry are very similar to the values for a compact tension specimen for equivalent a/W, where W is the width of the specimen. The half-width of the crack is defined by: h(x) = CTOD/2 + x0 where 8 = (6 - CTOD)/2a To allow assessment of the effect of crack geometry, calculations were performed also with a parallel-sided crack with a width equal to the mean width of the trapezoidal crack. In this preliminary analysis, calculations were performed mainly at low K values, typically 10 MPa m ‘/* . At these Kvalues the crack tip opening displacement is 0.76 pm. The width of the chromium depleted zone in a sensitised 304 SS is,9 at most, about 0.8 pm. At higher K values, contributions to the crack tip current from the exposed adjacent matrix should be incorporated into the calculations. In the absence of quantitative expressions for the transient current for initially bared matrix material the values for the depleted zone were adopted at all K values. Chemical and electrochemical reactions Anodic reactions and hydrolysis. The solubility of metal cations is so low at BWR
temperatures that the anodic reaction process can be considered as simply a source of hydrogen ions described by: 3Fe + 4H20 + Fe304 + 8H+ + 8e-
(1)
Ni + Hz0 + NiO -I- 2H+ + 2e-
(2)
2Cr + 3H20 + Cr2Os + 6H+ + 6e-
(3)
Table 1. Variation of crack mouth opening diplacement (6) with stress intensity factor K(MPam”*) 10 20 30
6 (cm) 1.43 x 10-3 2.86 x 1O-3 4.28 x 1O-3
A. Turnbull
192
Calculations were performed on this basis, i.e. every electron produces a H+ ion, following other work,6,7 but further confirmation of the validity of this assumption was made by considering the hypothetical situation that all of the anodic current was generating the more soluble nickel ions in solution, with the attendant reactions of hydrolysis and precipitation: Ni -+ Ni2 + 2e-
(4)
Ni2’ + Hz0 * NiOH+ + H+
(5)
Ni2 + 20H-
(6)
+ NiO + Hz0
with the kinetics of precipitation given by
where KSis the solubility constant. There are no specific data for the rate constant for precipitation of NiO. Consequently, a value for the precipitation rate constant, k,, was chosen which is sufficiently large to ensure a minimum of supersaturation. For the range of parameters investigated, there was no significant difference ( < 0.5%) in calculated potential or pH when assuming all of the anodic current goes to hydrogen ions (eqnuatios (I), (2) and (3)) or when using the full hydrolysis and precipitation reactions (equations (4) (5) and (6)). In the latter case, the metal ion concentration increases with time until the solubility limit is reached. No account has been taken of the oxidation of hydrogen atoms on the metal surfaces within the crack. Molecular hydrogen may diffuse into the crack from the bulk solution if significant in the bulk solution, and may be generated by internal cathodic reaction. Oxidation of hydrogen molecules produced initially from cathodic reactions in a crack was indicated previously as a factor of relevance in low temperature applications.5 It is not difficult to incorporate this reaction in modelling provided the relevant diffusion and kinetic data are available, and will be considered for future development. Cathodic reactions in the crack. It is readily possible to demonstrate that the oxygen concentration in the crack is low using the analytical relationships developed previously” and correspondingly this reduction of oxygen was neglected (see later). The cathodic reduction of water and of hydrogen ions in the crack is described by
HzO+e-
+ OH-+H
H’+e--+H
(8) (9)
Other cathodic reduction reactions in the crack can be important in some cases such as reduction of sulphate to hydrogen sulphide and subsequent reactions of this species.6 Reductive dissolution of magnetite can also be a factor. However, the state of development of data for the simplest reactions, reduction of hydrogen ions and water, is so poor that adding a series of additional reactions would be premature. Nevertheless, the possible impact of the production of reduced sulphide species in the crack should not be ignored. The final reaction process in the crack is the ionisation of water described by
Modelling of crack chemistry in sensitized stainless steel
193
Hz0 = H+ + OH-
(10)
Mm conservation equation. All models represent simplifications of reality. Hence, it is important to clarify the assumptions and simplifications made. The crack is considered to be sufficiently narrow that concentration and potential gradients will not develop between the walls, allowing appropriate averaging of the two-dimensional equations. Dilute solution theory is used. Also, convective mass transport in the crack is neglected which implicitly assumes that the external flow direction is across the narrow mouth opening so that through-thickness convection can be neglected. Diffusion in the through-thickness direction is not considered. It should be noted that the use of fracture mechanics specimens can introduce artifacts such as through-thickness diffusion which are not representative of real cracks. Using the simplifying assumptions, the mass conservation equations can be expressed in the general form:
where Ci represents the concentration of dissolved species i, Di is the diffusion coefficient, zi is the charge, Fis Faraday’s constant, R is the gas constant, Tis the absolute temperature, I$ is the difference in potential between the value at the crevice mouth (E,) and that at a position in the crevice, Rij,hom represents homogeneous equilibrium reaction process that can occur in the solution in the crack, Rij,het represents heterogeneous processes that can occur on the walls of the crack (including electrochemical reactions and precipitation). Equations of the type described by (11) are required for each of the dissolved species in the crack. An additional equation is needed to solve for the potential. The equation used is the total current density equation given by i,=FCziJi=-F i
~ziDi~+~~C’i~ i
(12) I
where id is the current density in the crack solution and Ji is the flux of species i. Initial conditions. It is not possible to represent the detailed history of the crack, which includes plant start-up and shut-down processes, and crack growth. Thus, the initial conditions are inherently arbitrary. It is assumed that the initial concentrations in the crack are uniform, the solution is at the correct temperature and the initial solution chemistry in the crack is assumed to be similar to the bulk but without the oxidising species. The initial potential distribution is calculated using Ohm’s law which is valid in this specific initial situation only, i.e. when the concentrations are uniform. Boundary conditions. It is assumed that convective motion external to the crack sustains bulk water chemistry very close to the crack mouth. The assumed boundary condition at the mouth is then: ci
=
Ci,bulk
(13)
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A. Turnbull
The time evolution of the crack mouth potential can be accounted for only by coupling the conservation equations in the crack with the solution of the continuity equations for the bulk conditions. This can lead to considerable complexity and the simpler approach adopted here is to assume that the mouth potential is constant in time, i.e. Em = constant
(14)
Since only the steady-state solution is of primary interest in this analysis, this is not a critical assumption. The key issue then is to relate the potential at the crack mouth to the corrosion potential for the system, remote from the mouth. In the case of a net anodic current flowing from the crack, the potential local to the crack mouth will decrease with respect to the corrosion potential remote from the mouth in order to imbalance the external anodic and cathodic currents, thus providing a net cathodic current on the external surface. This is evaluated through the solution of Laplace’s equation for the potential distribution outside of the crack. In applying Laplace’s equation it is necessary to define the reactions occurring on the external surface, the corrosion current density remote from the crack and the potential dependence of the reaction kinetics. The features of this problem are illustrated in Appendix A. In view of the fact that the crack mouth opening displacement is small compared with surface length of the crack, Laplace’s equation can be reduced to two dimensions. Since the distribution of potential is symmetric about the mid-line of the crack only one half of the system is illustrated. The boundary conditions are generalised and make no a priori assumptions about mass transport control of the external electrode kinetics, this being reflected in the choice of transfer coefficient. The spatial parameters, I and d, in Appendix A are chosen to be sufficiently large so that increasing their magnitude further has no effect on the potential drop calculated for the crack mouth region. Macdonald and Urquidi-Macdonald3 have solved this problem by directly coupling external and internal processes in the crack (at steady state) but, at steady state, it is unnecessary to couple this directly and it adds to the complexity of the basic numerical analysis. The approach we have adopted is to solve the crack chemistry problem for a defined crack mouth potential. The magnitude of the current emerging from the crack is also predicted and is then used as input data for the solution of Laplace’s equation to enable calculation of the associated corrosion potential, i.e. ECOrf= &l + (P,,,
(15)
An alternative way of understanding this approach is to consider that there is a specific set of external conditions and we are required to define the mouth potential and crack chemistry, In essence, we make an initial guess for the mouth potential, run through the calculations using the crack chemistry model and Laplace’e equation and determine the associated corrosion potential. This is then carried out iteratively (i.e. varying mouth potential) to converge on the corrosion potential of interest. In essence, this is the basis of showing the relationship between corrosion potential, mouth potential and crack-tip parameters (see later). Since a computer run takes 10 min this is not an inefficient process. At the crack tip, the flux of all non-reactive species is zero. The flux of hydrogen ions at the tip is defined by the net anodic reaction: FJ = i&
(16)
Modelling of crack chemistry in sensitized stainless steel
195
Alternatively, this flux could be perceived as the source of metal cations when adopting equations (4), (5) and (6). The mass conservation of each dissolved species in solution is defined based on equation (1 l), S7 with equation (12) providing the additional relationship necessary for solving for the potential. Thus, for a model system involving n dissolved species, there are n + 1 nonlinear differential equations that must be solved as a function of position and time subject to the initial and boundary conditions above. The numerical method adopted is described elsewhere.7 Electrode kinetics. The kinetics of reactions at the crack tip were based on timeaveraging the current density transient at the crack tip for a specific K value, which is essentially the approach of Ford and Andresen and following this, of Macdonald and Urquidi-Macdonald.3 Thus, iav = s(4.11
x
lo-“K4)”
(17)
where Kin units of MPa m1’2, i0 is the bare surface current density, to is the time of exposure of the bare surface prior to refilming and n is the current decay constant. The use of time-averaged data effectively excludes examination of the time variation in local potential and chemistry in detail. If the new crack-tip surface is assumed to be produced instantaneously, there would be an initial transient drop in potential at the tip. Nevertheless, the magnitude would decrease very quickly as refilming occurred. Thus, the impact would be limited and the time-averaging can be considered a reasonable approach. In principle, such transient behaviour could be evaluated using this model but, inherently, the time steps would need to be very small to resolve the details of the transient and this could lead to very long computational times. There would also be the issue of accommodating the potential dependence of the decay constant (see below) as it changed during the transient period. The data of Ford et ~1.~do not show much variation in the bare surface current density and the experimental data could be represented by io = 0.52 exp
(18)
No explicit expression for the variation of the transient current density with potential has been given by Ford et aL9 although incorporation as a dependency of the decay constant is implicit. In the majority of the present calculations, the decay constant has been assumed to be 0.5, and the time prior to refilming assumed to be 0.01 s. These data were used for the calculations unless otherwise stated. Nevertheless, complementary calculations were made using the relationship adopted by Macdonald and Urquidi-Macdonald, which at 561 K can be expressed in the form i0=1.74x
102exp
(19)
However, the latter assume a significant potential dependence only of the bare surface current density which is not supported by experimental observation. The potential dependence of the current is inadequately expressed by either equation
196
A. Turnbull
(18) or equation (19) but will be adopted herein because of the limited objective of this preliminary analysis. The electrode reaction kinetics on the walls of the crack are also uncertain in their definition. The anodic current was assumed to be given by the passive current. The majority of calculations were performed assuming a value of 1.5 x 10e6A cm-‘, which would correspond to short-term exposure conditions, but were complemented by calculations corresponding to long-term exposure conditions. assuming a value of 1.5 x 10-7Acm-2 The kinetics for reduction of water and of hydrogen ions were assumed to be given by:9 iH20 = 7.5 x 10-i2exp[-l.14FE/RT]A in+ = 3.2 x lo-‘[H+]exp[-
l.l4FE/RT]A
cme2 cmp2
(20) (21)
where [H +] is in units of moles cme3 and E is the electrode potential. The kinetics of water reduction were derived from the data of Kim and Niedrach.’ ’ The kinetics of hydrogen ion reduction were derived assuming the same transfer coefficient with the added assumption that the respective reaction kinetics were equivalent at neutral pH. In addition, a first order dependence on hydrogen ion concentration was assumed. In modelling the conditions external to the crack using Laplace’s equation, we used the data from Kim and Niedrach in relation to the hydrogen oxidation kinetics and oxygen reduction kinetics with the passive current the value utilised for the crack walls. The major process supporting the anodic current from the crack was reduction of oxygen defined by: io, = 1.3 x 102[02] exp
A cmw2
(22)
where [O,] is in units of moles cme3. It was assumed that the flow rate is high so that the external cathodic current was described by Tafel kinetics. Bulk solution chemistry. The solution in the bulk was taken to be that of BWR water at 561 K with a pH561 of 5.7. The solution was considered to be contaminated with NaCl of concentration 1.9 x 10v6 M to give a conductivity (o) at 298 K of 0.3 pS cm-’ corresponding to a value of 5.7 uS cm-’ at 561 K. The other data used in the analysis are summarised in Appendix B.
RESULTS
AND DISCUSSIONS
To confirm the depletion of oxygen in the crack a calculation was performed for K = 20 MPa rn112and the resulting concentration profile is shown in Fig. 1. The potential assumed was -0.2 V (SHE) which is the highest that might be reasonably achievable in the crack (see later) and would be non-conservative in so far as it should show the minimum extent of depletion. At more negative potentials the depletion would be even more marked. Accordingly, as indicated previously, oxygen reduction in the crack is not considered further. Initial calculations were performed to compare results from a parallel-sided crack and from a trapezoidal crack. In both cases the crack mouth potential was maintained constant at -0.1 V (SHE) so that the comparison was not invalidated by the variations in potential
Modelling of crack chemistry in sensitized stainless steel
0,
_ ............
0 0
.I ................
i
0.1
. ................
j
0.2
. ................
0.3
i.. ..............
. ................
0.4
0.5
. ................
0.6
i.. ..............
0.7
197
i ................
0.8
. .........
0.9
I
X
Fig. 1. Normal&d oxygen concentration profile in crack for K= 20 MPa rn’12assuming a potential in the crack of -0.2 V SHE.
drop outside the crack associated with the different values of the crack mouth opening displacement. The passive current (ir) was assumed to be 1.5 x iOe6 A cm-*. The results are summarised in Tables 2 and 3 where the former represents the results in the absence of enhanced crack-tip reaction kinetics associated with the crack-tip. It is quite evident that the adoption of a parallel-sided crack geometry is not a good approximation to a trapezoidal crack if based simply on the mean width. Interestingly, polarisation is more easily attained with enhanced crack-tip reaction although the impact is not large for either geometry. This is presumably due to the influence of the increased reaction kinetics on the local pH and, correspondingly, on the local solution conductivity. The isolated potential (corresponding to the crack being decoupled from the external surface) in the absence of enhanced crack-tip reaction kinetics would be about -0.490 V (SHE) which would indicate that there was not much polarisation of the parallel-sided crack, the assumption adopted by Andresen and Ford.‘*2 The dependence of crack-tip potential on the external potential at steady state is shown in Fig. 2. The dot-dash line represents the condition for no potential drop in the system. Also shown is an indication of the maximum potential expected if there is no effective coupling of the crack-tip to the external surface. The actual potential in the latter case would depend on the magnitude of the crack-tip current with the predicted potential being lower
Table 2. K (MPa ml’*) 10 10
Effect of geometry (unstrained crack tip)
Geometry
Crack width (cm)
ParalleI-sided Trapezoidai
7.2 x 1O-4 from 7.6 x lo-’ to 1.4 x 10e3
o(pScm-‘) 0.3 0.3
E,(V) -0.1 -0.1
&p(V) -0.478 - 0.307
pH,i, 5.3 3.4
A. Turnbuil
798 Table 3. K (MPa ml’*) 10 10
Effect of geometry
(strained
cr (f&cm-‘)
Geometry
Crack width (cm)
Parallel-sided Trapezoidal
7.2 x 10V4 from 7.6x 10m5 to 1.4x 10-j
0.0
crack tip) E, (V)
0.3 0.3
-0.1 -0.1
Tip 0’)
PHrir
-0.444 -0.292
4.7 3.3
‘. ,
E]
Y
‘. .
-0.1 -
\
.. ,
.
-0.2 -
\
. ‘\
-0.3 -
‘\.,
0 <\ x. ‘x,
-0.4 -
0
....
‘\
. ‘L.
___________
-0.5
----_-------_--__------_---------__________~~
isolated and neutral/alkaline pH i 0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
E (V SHE) Fig.
2. Crack tip potential as a function of crack K=10MPam’~2;a293=0.3pScm~‘;i,=l.5x10~6Acm~2;
mouth
potential and corrosion T=561K;pHburk=5.7.
potential
with increasing value of the tip current (because of the need to get enough cathodic current to balance the anodic current). The overall potential drop has two components, the drop in the crack and that outside the crack. The latter becomes significant at corrosion potentials greater than about -0.2 V (SHE) and can be distinguished in Fig. 2 from the difference in full and open points at the same value of crack-tip potential. There is considerable uncertainty in the values of the parameters adopted in this analysis, principally the electrochemical data. For that reason it is important to explore the sensitivity of the results to the particular choice. The predictions of crack-tip potential and pH for different values of key parameters are shown in Fig. 3. It is evident that decreasing the efficiency of the cathodic reduction reactions in the crack by decreasing the rate constant for hydrogen ion and water reduction by an order of magnitude (represented in this figure by k, x 0.1) results in a larger potential drop in the crack. The potential in the crack decreases in order to generate sufficient cathodic reaction to balance the anodic current. The results demonstrate the intrinsic importance of the internal cathodic reactions, which is in contrast to the assumption of Macdonald and Urquidi-Macdonald.3
Modelling of crack chemistry in sensitized stainless steel
199
-0.2 2 %
-0.3 -
L .a LLi-
-0.4 -
‘_ _______________________-_--_--
-0.5 --
-0.6 0.3
_________(ulj____________L._
isolatedand neutral/alkalinepH 1’1’1’ 0.2
I 0.1
0.0
-
-0.1 E,,,
I -0.2
’
1 -0.3
’
1 -0.4
’ -0.5
0’St-W
Fig. 3. Crack tip potential (pH,ip in parenthesis) as a function of corrosion K=10MPam”2;a~~~=0.3~Scm~‘;i,=1.5x10~6Acm~2;T=561K;pH~ul~=5,7.
potential
It is important to understand initially what controls the pH in the crack. Clearly, the steady state pH will depend on the balance between the rate of production of hydrogen ions and their removal by mass transport and by internal cathodic reduction. In the case of a long crack, mass transport is not efficient. Accordingly, when the crack mouth potential is made more noble (i.e. with respect to the corrosion potential in deaerated solution of neutral pH) the pH will fall until essentially the anodic production of hydrogen ions is balanced approximately by the cathodic reduction in the crack. When the solubility of metal cations is low the production of hydrogen ions via the anodic process is approximately equivalent to the anodic reaction current in the crack. In this situation, the crack is very nearly at the free corrosion potential in the local crack environment and the net current from the crack can be small. Measurements of corrosion potential in the simulated crack environment would be expected to give values similar to the predicted crack potentials. The concept that the crack may be polarised into a regime where it is very close to the corrosion potential in the local environment is not unusual and applies to crevices in high temperature water12 and also to crevices of cathodically protected steels in sea-water.13 In the latter case, corrosion potentials have been measured in the alkaline environments predicted and measured in the crevice and are indeed very close to the potentials in the crevice. The concept is sometimes not readily grasped, principally because of neglect of the environmental changes in the crack or crevice. The effect of decreasing the passive current is to result in a decreased potential drop in the crack. Intuitively, this seems readily apparent but it should be set in contrast to the opposite impact of the crack tip current indicated by comparison of Tables 2 and 3. This contrast is not easy to rationalise but part of the difficulty may be our inability to conceptualise the relative impact of transport processes, crack geometry, local pH and conductivity changes. It is important to try to isolate simple explanations for the behaviour,
A. Turnbull
800
because this provides additional confidence in the predictions, but there is a risk of oversimplification and intuition may be inadequate. A crude analogy would be trying to analyze the taste of a soup in terms of the particular ingredients. The effect of stress intensity factor on crack-tip potential is shown in Fig. 4. At low potentials, the potential drop decreases with increasing stress intensity factor which would tend to be expected based on the greater crack opening displacement. The inverse behaviour occurs at more noble potentials which reflects primarily differences in the magnitude of the potential drop external to the crack. The evaluation of the effect of corrosion potential on crack-tip potential should be treated with some caution in view of the assumed limited dependence of the crack tip current on potential. Although the impact of crack tip current on the local potential is modest at a mouth potential of - 0.1 V (SHE) (compare Tables 2 and 3), this would not be expected to prevail as the potential is lowered. The logical approach would be to utilise the predicted dependence of the current decay constant on potential from the work of Ford et a1.9 However, the data relate to solutions of lower conductivity than would be predicted within a crack. It is intended at a future date to utilise a potential-dependent decay constant as a basis of an improved model, although with the uncertainty in the dependence of the decay constant on pH also to consider. In the interim, we have utilised the potential dependence of the bare surface current density assumed by Macdonald and Urquidi-Macdonald,3 despite its limited basis. The results are shown in Fig. 5 and compared with the “standard” data and predictions assuming no crack tip straining. The results are again apparently unusual insofar as the potential drop is greater than the “standard” data at the more negative potentials despite the lower value of the crack tip current. It is at least consistent with previous observations
Etip -“.3 ‘..
-0.4
-0.6 .
I 0.2
I 0.1
I 0.0
I -0.1
I -0.2
I -0.3
-0.4
E corr Fig. 4. Crack tip potential intensity factor (MPam”*);
as a function of corrosion potential for different values of the stress o293=0.3 pScm_‘; ip= 1.5 x 10-6Acm-2; T= 561 K; pHbulk =5.7.
Modelling of crack chemistry in sensitized stainless steel
-.
N\ .\.
\.
\.
801
3-A. .. . \.
-0-
~---
‘L.
standard no straining modifieditiP
EUp= Eoon
I. ‘L.
\.
‘.\. \. . . ‘...
w_*
‘.\._
.\
l
.\..
.\.\ -0.5 -
.\
_ _ _ :::.,,
--___________-______~_~~__~__~~_~-__-_~~~_~~____ -A isolated and neutral/alkaline pH
-0.6
1 0.1
I
I
I
I
I
I
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
E,,,
Fig. 5.
0’ SW
Crack tip potential as a function of corrosion potential with and without crack tip reaction: K=10MPam”z;~2~3=0.3~Scm~‘;i,=1.5x10~6Acm~2;T=56lK;pH~Y~~=S.7.
that increasing the crack tip current would appear to increase the local conductivity, by pulling in chloride ions, to the extent that the increased conductivity becomes the dominant factor. The pH of 5.2 predicted at - 0.3 V (SHE), based on the relationship of Macdonald and Urquidi-Macdonald, should be compared with pH 4.0 obtained using the “standard” data. Nevertheless, both sets of predictions will be inaccurate and probably represent the bounds of predicted potential in the crack. It would be premature to discuss in detail the implications of this work for the prediction of crack growth kinetics in view of the uncertainty in input parameters and the inadequate representation of the potential dependence of crack-tip current. The conceptual perspective that there is coupling between the external surface and crack tip, supported by this model, seems reasonable in view of the dependence of crack growth rate on potential. Oxides within the crack could restrict transport and in the limit prevent coupling of the tip with the external surfacei but this would not be compatible with the observed potential dependence of crack growth. CONCLUSIONS (1) The crack-tip potential is not independent of the external potential in BWR water (conductivity 0.3 uS cm-’ at 298 K). (2) Reactions on the crack walls cannot be ignored in predicting the chemistry and potential in a crack. These reactions include the cathodic reduction of hydrogen ions and water which are too often erroneously neglected in modelling of crack and crevice chemistry. (3) The results demonstrate the need to utilise crack chemistry models of adequate
802
A. Turnbull
quality if there is to be confidence in the predictions and this becomes especially important when validation by measurement of local crack chemistry and potential is difficult. (4) The predictions of this model also have uncertainty in relation to the details because of the paucity of reliable electrochemical data. For that reason more extensive evaluation of parameters, such as solution conductivity and crack size, has not been undertaken at this stage. Acknowledgements-The gratefully acknowledged.
constructive
comments
of Peter Andresen
and Peter Scott in applying
this model are
REFERENCES 1. P.F. Ford and P.L. Andresen, Development and use of a predictive model of crack propagation in 304/316L, A533B/A508 and Inconel 6001182 alloys in 288°C water, Environmental Degradation of materials in Nuclear Power Systems, 789 (ed. G.J. Theus and J.R. Weeks, (The Metallurgical Society, 1988). 2. P.L. Andresen, Conceptual similarities and common predictive approaches for SCC in high temperature waters, Corrosion ‘96 (Houston, TX: NACE International, 1996). 3. D.D. Macdonald and M. Urquidi-Macdonald, Corros. Sci. 32(l), 51 (1991). 4. A. Turnbull, Resolution of key issues in deterministic models of environment assisted cracking. Life prediction of structures subject to environmental degradation (eds. P.L. Andresen and R.N. Parkins (Houston, TX: NACE International, 1996). 5. A. Turnbull and D.H. Ferriss, Corros. Sci. 27(12), 1323 (1987). 6. M. Psaila-Dombrowski, A. Turnbull and R.G. Ballinger, Implications of crevice chemistry for cracking of BWR recirculation inlet safe-ends, Life prediction of corrodible structures (ed. R.N. Parkins (Houston, TX: NACE International, 1994). 7. M. Psaila-Dombrowski, Ph.D Thesis, MIT (1990). 8. H. Tada, P. Paris and G. Irwin, The Stress Analysis of Cracks Handbook. (Del Research Corporation: 1973). 9. P.F. Ford, D.F. Taylor, P.L. Andresen, and R.G. Ballinger, Corrosion-assisted cracking of stainless and low alloy steels in LWR environments. EPRI Report NP-5064M (1987). 10. A. Turnbull, British Corrosion Journal M(4), 162 (1980). Il. Y.-J. Kim and L.W. Niedrach, Electrochemistry of Hz, 02 and Hz02 in high temperature high purity water, II: On stainless steel., GE Corp, to be published (1996). 12. A. Turnbull and M. Psaila-Dombrowski, Modelling of crevice chemistry in the secondary side of nuclear steam generators, Improving the understanding and control of corrosion on the secondary side of steam generators (ed. R.W. StaehIe (Houston, TX: NACE International, 1994). 13. A. Turnbull, Corrosion Science 23(8), 833 (1983). 14. R. Parsons, Handbook of Electrochemical Constants, Academic Press, New York, (1959). 15. H.E. Barnes and R.V. Scheueman, Handbook of Thermodynamieal Datafor Compounds and Aqueous Species, John Wiley and Sons, New York, (1978). 16. P.R. Tremaine and J.C. Leblanc, J. Chem. Thermodynamics 12, 521 (1980).
Modelling of crack chemistry in sensitized stainless steel
APPENDIX
803
A
Geometry and boundary conditions for solution of Laplace’s equation for potential drop outside crack. 2
24
vs$ = 0
-*-0
ax
0
-
fh
X
k, = i,Jk k, = iJk k=F’C$UiCi i
where i. and i, are, respectively, the anodic and cathodic current densities, which by de&&on are identical at the free corrosion potential, i,,, is the current density at the crack mouth, k.‘“’ and kclirnare the limiting anodic and cathodic currents, and w is the ion mobility. a. and cr, are parameters which detine the potential dependence of the anodic and cathodic current densities (e.g. from equation (22), a, = 0.77F/R7’).
APPENDIX
B - DATA USED IN ANALYSES
The diffusion coefficients used are listed below in Table Al .14
Homogeneous reactions
Very few homogeneous reaction rate constants are known, especially at the temperature of interest. The important assumption made is that the rate constants are sutTrcientlylarge to ensure extremely rapid attainment of equilibrium following any perturbation of individual species concentration by reaction. In essence, a very large value of one of the rate constants is assumed. Tbe only restriction is that the ratio of the forward reaction rate constant to the backward rate constant (kf and kb, respectively), as given in the following reaction
A. Turnbull
804 Table Al. Species
Diffusion
coefficients
coefficient
(cm’/s)
I .76 x 10-j
Hi OHNa ’ Cl Ni* ’ NiOH
must be equal to the equilibrium
Diffusion
9.8x 10-j 7.5 X lom~3 3.8 x 10-j
1.3x IOP ’
constant
1.76x IO -4
(K.& as shown below &s=$
(A3)
Thus, in order to assume values for the forward and backward rate constants, the equilibrium constant for each reaction must be known. Equilibrium constants can be calculated from the free energy change for the reaction using the following relationship AGo = -RT
InK,,
(A4)
where A@ is the free energy change for the reaction, R IS the gas constant and Tis the temperature standard free energy change for the reaction is given by the sum of the free energies of formation minus the free energies of formation of the reactants, or AGL,,,” The free energies at the temperature Solubility Limits The precipitation
reaction,
= c
I
A%
products - F
in Kelvin. The of the products
AC;., reactants
(A5)
of interest are listed below in Table A2.”
which is assumed
to take place on the walls of the cavity region, is given by
M2+ + 20H-
4
MO + Hz0
(A6)
where k, is the rate constant for the precipitation reaction. The rate constant is not known and is chosen to be very large to insure that the solution does not become supersaturated. The solubility of the metal ion can be expressed as
Table AZ. Free energies of formation relevant species used in model Species H+ OHH20
Ni2 + NiOH ’ NiO
for
Free energy (kJ mole-‘) - 12.25 - 72.38 - 195.81 -40.16 - 179.91 - 186.61
Modelling of crack chemistry in sensitized stainless steel
805
KS = [M*+][OH-]*
647)
The solubility limit assumedI for Ni2+ was 2.25 x IO-*’ (molesdm-3)3.
PII: soo1&938x(%)00171-0
MODELLING STAINLESS
OF CRACK CHEMISTRY IN SENSITIZED STEEL IN BOILING WATER REACTOR ENVIRONMENTS A. TURNBULL
Centre for Materials Measurement and Technology, National Physical Laboratory, Teddington, Middlesex, TWll OLW, U.K. Ah&ret--An advanced model has been used to predict the chemistry and potential in a stress corrosion crack in sensitized stainless steel in a boiling water reactor (BWR) environment. The model assumes trapezoidal crack geometry, incorporates anodic reaction a&cathodic reduction within the crack, and takes into account the limited solubility of cations in high temperature water. The results indicate that the crack tip potential is not independent of the external potential, and that the reactions on the walls of the crack must be included for reliable prediction. Accordingly, both the modelling assumptions of Ford and Andresen and of Macdonald and Urquidi-Macdonald, whilst having merit, are not fully satisfactory. Extended application of the model for improved prediction of stress corrosion crack growth rate is constrained by limitations in electrochemical data which are currently inadequate. Crown copyright 0 1997 Published by Elsevier Science Ltd
INTRODUCTION
Life prediction of components, structures or products in which environment assisted cracks can develop and propagate is a major challenge which has been met primarily by establishment of empirical models based on laboratory data and/or field observation. This approach has some inherent limitations, which can be critical in particular applications. These limitations are associated with the short term nature of the data and with the complexity of service conditions which may involve both short and long term variations in temperature, environment and stresses. Deterministic models of environment assisted cracking have greater potential, in principle, since they provide insight into the controlling variables enabling, for example, an improved assessment of the impact of fluctuations in operating conditions. However, confidence in the use of such models within the engineering community is not enhanced by the continuing scientific debate as to the applicability of different models and of the different assumptions adopted. The major factor hindering progress in many cases remains the difficulty of quantifying, to a sulhcient extent, the local environment and the kinetics of reaction processes at the tip of a crack. In the application of models to prediction of crack propagation rates in BWR conditions in particular, there has been controversy between groups led by Andresen and Ford1*2 at GE and Macdonald and Urquidi-Macdonald3 at Pennsylvania State University. The reason for the difficulty is that both groups make apriori assumptions regarding the factors controlling crack tip electrochemistry, primarily because of limitations in modelling the interactive effects of mass transport, chemistry and electrochemistry in cracks in an effective manner. In Manuscript received 29 July 1996; in revised form 8 October 1996. 789
A. Turnbull
790
this paper, advanced modelling is utilised as a first stage in resolving this issue. The present paper should be considered as an update on the extended abstract published previously4 in which a computational error resulted in the generation of unreliable data and erroneous conclusions. MATHEMATICAL
MODEL
Background
The model of Ford and Andresen1*2 for cracking of sensitised 304 SS in BWR environments is based on crack advance by the slip oxidation mechanism. The key input is the transient electrode kinetics following a strain rupture event. These data can be obtained by various methods but the important requirement is to simulate conditions at the tip of the crack in relation to the local environment and electrode potential. Experimental measurement of the chemistry and electrode potential in cracks at 288°C is particularly difficult and modelling has been the primary approach to defining the local environment. Ford and AndresenlV2 make the assumption that the crack tip, in terms of the local potential, is essentially decoupled from the external driving force of oxygen reduction on the surface external to the crack. Macdonald and Urquidi-Maedonald have developed a more elaborate model of crack chemistry (the CEFM model) which takes into account mass transport by diffusion and ion migration in the crack, electrochemical reactions at the crack tip, and hydrolysis of ferrous ions. A feature of the model is that it takes account also of the potential drop local to the crack mouth. The latter authors argue that all of the current for crack advance is supported by reaction on the external surface and that the crack tip reactions and external reactions are intrinsically coupled. The model of Macdonald appears complex but simplifying assumptions are made which are critical: the crack is assumed to be parallel sided; the walls of the crack are assumed to be inert; the solubility of metal cations is not considered, despite their very low solubility at 288°C. Advanced model
More rigorous models of the chemistry in cracks and crevices have been developed previously 5-7 in which the importance of these factors has been demonstrated, but have not been applied to this specific problem. These models have now been adapted for prediction of the chemistry and electrochemistry in cracks in BWR environments. The primary features included are: (a) transport by diffusion and ion migration; (b) anodic and cathodic reactions at the tip and on the crack walls; (c) hydrolysis of metal cations; (d) precipitation to account for the limited solubility of metal cations; (e) trapezoidal crack geometry. In addition, the potential drop in the environment outside the crack is evaluated from solution of Laplace’s equation. Crack geometry.
A trapezoidal crack geometry was assumed with the width of the crack
tip defined by CTOD = 0.6( I - d)K2/Eo,
Modelling of crack chemistry in sensitized stainless steel
791
where CTOD is the crack-tip opening displacement, v is Poisson’s ratio, K is the stress intensity factor, E is the modulus and o,, is the yield strength. The crack mouth opening displacement (6) was calculated, using standard linear elastic fracture mechanics expressions’, assuming a single-edged notched (SEN) specimen with a/ b=0.3 and with the crack length, a, equivalent to the crack depth, i.e. the notch depth is negligible. The parameter b is the width of the SEN specimen. Typical values are shown in Table 1. The reason for adopting this configuration was to allow sensible application of Laplace’s equation outside the crack without having to account for the geometry of the notch and reactions on the walls of the notch. The crack length was assumed to be 1 cm. Note that the calculated crack mouth displacements for this specimen geometry are very similar to the values for a compact tension specimen for equivalent a/W, where W is the width of the specimen. The half-width of the crack is defined by: h(x) = CTOD/2 + x0 where 8 = (6 - CTOD)/2a To allow assessment of the effect of crack geometry, calculations were performed also with a parallel-sided crack with a width equal to the mean width of the trapezoidal crack. In this preliminary analysis, calculations were performed mainly at low K values, typically 10 MPa m ‘/* . At these Kvalues the crack tip opening displacement is 0.76 pm. The width of the chromium depleted zone in a sensitised 304 SS is,9 at most, about 0.8 pm. At higher K values, contributions to the crack tip current from the exposed adjacent matrix should be incorporated into the calculations. In the absence of quantitative expressions for the transient current for initially bared matrix material the values for the depleted zone were adopted at all K values. Chemical and electrochemical reactions Anodic reactions and hydrolysis. The solubility of metal cations is so low at BWR
temperatures that the anodic reaction process can be considered as simply a source of hydrogen ions described by: 3Fe + 4H20 + Fe304 + 8H+ + 8e-
(1)
Ni + Hz0 + NiO -I- 2H+ + 2e-
(2)
2Cr + 3H20 + Cr2Os + 6H+ + 6e-
(3)
Table 1. Variation of crack mouth opening diplacement (6) with stress intensity factor K(MPam”*) 10 20 30
6 (cm) 1.43 x 10-3 2.86 x 1O-3 4.28 x 1O-3
A. Turnbull
192
Calculations were performed on this basis, i.e. every electron produces a H+ ion, following other work,6,7 but further confirmation of the validity of this assumption was made by considering the hypothetical situation that all of the anodic current was generating the more soluble nickel ions in solution, with the attendant reactions of hydrolysis and precipitation: Ni -+ Ni2 + 2e-
(4)
Ni2’ + Hz0 * NiOH+ + H+
(5)
Ni2 + 20H-
(6)
+ NiO + Hz0
with the kinetics of precipitation given by
where KSis the solubility constant. There are no specific data for the rate constant for precipitation of NiO. Consequently, a value for the precipitation rate constant, k,, was chosen which is sufficiently large to ensure a minimum of supersaturation. For the range of parameters investigated, there was no significant difference ( < 0.5%) in calculated potential or pH when assuming all of the anodic current goes to hydrogen ions (eqnuatios (I), (2) and (3)) or when using the full hydrolysis and precipitation reactions (equations (4) (5) and (6)). In the latter case, the metal ion concentration increases with time until the solubility limit is reached. No account has been taken of the oxidation of hydrogen atoms on the metal surfaces within the crack. Molecular hydrogen may diffuse into the crack from the bulk solution if significant in the bulk solution, and may be generated by internal cathodic reaction. Oxidation of hydrogen molecules produced initially from cathodic reactions in a crack was indicated previously as a factor of relevance in low temperature applications.5 It is not difficult to incorporate this reaction in modelling provided the relevant diffusion and kinetic data are available, and will be considered for future development. Cathodic reactions in the crack. It is readily possible to demonstrate that the oxygen concentration in the crack is low using the analytical relationships developed previously” and correspondingly this reduction of oxygen was neglected (see later). The cathodic reduction of water and of hydrogen ions in the crack is described by
HzO+e-
+ OH-+H
H’+e--+H
(8) (9)
Other cathodic reduction reactions in the crack can be important in some cases such as reduction of sulphate to hydrogen sulphide and subsequent reactions of this species.6 Reductive dissolution of magnetite can also be a factor. However, the state of development of data for the simplest reactions, reduction of hydrogen ions and water, is so poor that adding a series of additional reactions would be premature. Nevertheless, the possible impact of the production of reduced sulphide species in the crack should not be ignored. The final reaction process in the crack is the ionisation of water described by
Modelling of crack chemistry in sensitized stainless steel
193
Hz0 = H+ + OH-
(10)
Mm conservation equation. All models represent simplifications of reality. Hence, it is important to clarify the assumptions and simplifications made. The crack is considered to be sufficiently narrow that concentration and potential gradients will not develop between the walls, allowing appropriate averaging of the two-dimensional equations. Dilute solution theory is used. Also, convective mass transport in the crack is neglected which implicitly assumes that the external flow direction is across the narrow mouth opening so that through-thickness convection can be neglected. Diffusion in the through-thickness direction is not considered. It should be noted that the use of fracture mechanics specimens can introduce artifacts such as through-thickness diffusion which are not representative of real cracks. Using the simplifying assumptions, the mass conservation equations can be expressed in the general form:
where Ci represents the concentration of dissolved species i, Di is the diffusion coefficient, zi is the charge, Fis Faraday’s constant, R is the gas constant, Tis the absolute temperature, I$ is the difference in potential between the value at the crevice mouth (E,) and that at a position in the crevice, Rij,hom represents homogeneous equilibrium reaction process that can occur in the solution in the crack, Rij,het represents heterogeneous processes that can occur on the walls of the crack (including electrochemical reactions and precipitation). Equations of the type described by (11) are required for each of the dissolved species in the crack. An additional equation is needed to solve for the potential. The equation used is the total current density equation given by i,=FCziJi=-F i
~ziDi~+~~C’i~ i
(12) I
where id is the current density in the crack solution and Ji is the flux of species i. Initial conditions. It is not possible to represent the detailed history of the crack, which includes plant start-up and shut-down processes, and crack growth. Thus, the initial conditions are inherently arbitrary. It is assumed that the initial concentrations in the crack are uniform, the solution is at the correct temperature and the initial solution chemistry in the crack is assumed to be similar to the bulk but without the oxidising species. The initial potential distribution is calculated using Ohm’s law which is valid in this specific initial situation only, i.e. when the concentrations are uniform. Boundary conditions. It is assumed that convective motion external to the crack sustains bulk water chemistry very close to the crack mouth. The assumed boundary condition at the mouth is then: ci
=
Ci,bulk
(13)
194
A. Turnbull
The time evolution of the crack mouth potential can be accounted for only by coupling the conservation equations in the crack with the solution of the continuity equations for the bulk conditions. This can lead to considerable complexity and the simpler approach adopted here is to assume that the mouth potential is constant in time, i.e. Em = constant
(14)
Since only the steady-state solution is of primary interest in this analysis, this is not a critical assumption. The key issue then is to relate the potential at the crack mouth to the corrosion potential for the system, remote from the mouth. In the case of a net anodic current flowing from the crack, the potential local to the crack mouth will decrease with respect to the corrosion potential remote from the mouth in order to imbalance the external anodic and cathodic currents, thus providing a net cathodic current on the external surface. This is evaluated through the solution of Laplace’s equation for the potential distribution outside of the crack. In applying Laplace’s equation it is necessary to define the reactions occurring on the external surface, the corrosion current density remote from the crack and the potential dependence of the reaction kinetics. The features of this problem are illustrated in Appendix A. In view of the fact that the crack mouth opening displacement is small compared with surface length of the crack, Laplace’s equation can be reduced to two dimensions. Since the distribution of potential is symmetric about the mid-line of the crack only one half of the system is illustrated. The boundary conditions are generalised and make no a priori assumptions about mass transport control of the external electrode kinetics, this being reflected in the choice of transfer coefficient. The spatial parameters, I and d, in Appendix A are chosen to be sufficiently large so that increasing their magnitude further has no effect on the potential drop calculated for the crack mouth region. Macdonald and Urquidi-Macdonald3 have solved this problem by directly coupling external and internal processes in the crack (at steady state) but, at steady state, it is unnecessary to couple this directly and it adds to the complexity of the basic numerical analysis. The approach we have adopted is to solve the crack chemistry problem for a defined crack mouth potential. The magnitude of the current emerging from the crack is also predicted and is then used as input data for the solution of Laplace’s equation to enable calculation of the associated corrosion potential, i.e. ECOrf= &l + (P,,,
(15)
An alternative way of understanding this approach is to consider that there is a specific set of external conditions and we are required to define the mouth potential and crack chemistry, In essence, we make an initial guess for the mouth potential, run through the calculations using the crack chemistry model and Laplace’e equation and determine the associated corrosion potential. This is then carried out iteratively (i.e. varying mouth potential) to converge on the corrosion potential of interest. In essence, this is the basis of showing the relationship between corrosion potential, mouth potential and crack-tip parameters (see later). Since a computer run takes 10 min this is not an inefficient process. At the crack tip, the flux of all non-reactive species is zero. The flux of hydrogen ions at the tip is defined by the net anodic reaction: FJ = i&
(16)
Modelling of crack chemistry in sensitized stainless steel
195
Alternatively, this flux could be perceived as the source of metal cations when adopting equations (4), (5) and (6). The mass conservation of each dissolved species in solution is defined based on equation (1 l), S7 with equation (12) providing the additional relationship necessary for solving for the potential. Thus, for a model system involving n dissolved species, there are n + 1 nonlinear differential equations that must be solved as a function of position and time subject to the initial and boundary conditions above. The numerical method adopted is described elsewhere.7 Electrode kinetics. The kinetics of reactions at the crack tip were based on timeaveraging the current density transient at the crack tip for a specific K value, which is essentially the approach of Ford and Andresen and following this, of Macdonald and Urquidi-Macdonald.3 Thus, iav = s(4.11
x
lo-“K4)”
(17)
where Kin units of MPa m1’2, i0 is the bare surface current density, to is the time of exposure of the bare surface prior to refilming and n is the current decay constant. The use of time-averaged data effectively excludes examination of the time variation in local potential and chemistry in detail. If the new crack-tip surface is assumed to be produced instantaneously, there would be an initial transient drop in potential at the tip. Nevertheless, the magnitude would decrease very quickly as refilming occurred. Thus, the impact would be limited and the time-averaging can be considered a reasonable approach. In principle, such transient behaviour could be evaluated using this model but, inherently, the time steps would need to be very small to resolve the details of the transient and this could lead to very long computational times. There would also be the issue of accommodating the potential dependence of the decay constant (see below) as it changed during the transient period. The data of Ford et ~1.~do not show much variation in the bare surface current density and the experimental data could be represented by io = 0.52 exp
(18)
No explicit expression for the variation of the transient current density with potential has been given by Ford et aL9 although incorporation as a dependency of the decay constant is implicit. In the majority of the present calculations, the decay constant has been assumed to be 0.5, and the time prior to refilming assumed to be 0.01 s. These data were used for the calculations unless otherwise stated. Nevertheless, complementary calculations were made using the relationship adopted by Macdonald and Urquidi-Macdonald, which at 561 K can be expressed in the form i0=1.74x
102exp
(19)
However, the latter assume a significant potential dependence only of the bare surface current density which is not supported by experimental observation. The potential dependence of the current is inadequately expressed by either equation
196
A. Turnbull
(18) or equation (19) but will be adopted herein because of the limited objective of this preliminary analysis. The electrode reaction kinetics on the walls of the crack are also uncertain in their definition. The anodic current was assumed to be given by the passive current. The majority of calculations were performed assuming a value of 1.5 x 10e6A cm-‘, which would correspond to short-term exposure conditions, but were complemented by calculations corresponding to long-term exposure conditions. assuming a value of 1.5 x 10-7Acm-2 The kinetics for reduction of water and of hydrogen ions were assumed to be given by:9 iH20 = 7.5 x 10-i2exp[-l.14FE/RT]A in+ = 3.2 x lo-‘[H+]exp[-
l.l4FE/RT]A
cme2 cmp2
(20) (21)
where [H +] is in units of moles cme3 and E is the electrode potential. The kinetics of water reduction were derived from the data of Kim and Niedrach.’ ’ The kinetics of hydrogen ion reduction were derived assuming the same transfer coefficient with the added assumption that the respective reaction kinetics were equivalent at neutral pH. In addition, a first order dependence on hydrogen ion concentration was assumed. In modelling the conditions external to the crack using Laplace’s equation, we used the data from Kim and Niedrach in relation to the hydrogen oxidation kinetics and oxygen reduction kinetics with the passive current the value utilised for the crack walls. The major process supporting the anodic current from the crack was reduction of oxygen defined by: io, = 1.3 x 102[02] exp
A cmw2
(22)
where [O,] is in units of moles cme3. It was assumed that the flow rate is high so that the external cathodic current was described by Tafel kinetics. Bulk solution chemistry. The solution in the bulk was taken to be that of BWR water at 561 K with a pH561 of 5.7. The solution was considered to be contaminated with NaCl of concentration 1.9 x 10v6 M to give a conductivity (o) at 298 K of 0.3 pS cm-’ corresponding to a value of 5.7 uS cm-’ at 561 K. The other data used in the analysis are summarised in Appendix B.
RESULTS
AND DISCUSSIONS
To confirm the depletion of oxygen in the crack a calculation was performed for K = 20 MPa rn112and the resulting concentration profile is shown in Fig. 1. The potential assumed was -0.2 V (SHE) which is the highest that might be reasonably achievable in the crack (see later) and would be non-conservative in so far as it should show the minimum extent of depletion. At more negative potentials the depletion would be even more marked. Accordingly, as indicated previously, oxygen reduction in the crack is not considered further. Initial calculations were performed to compare results from a parallel-sided crack and from a trapezoidal crack. In both cases the crack mouth potential was maintained constant at -0.1 V (SHE) so that the comparison was not invalidated by the variations in potential
Modelling of crack chemistry in sensitized stainless steel
0,
_ ............
0 0
.I ................
i
0.1
. ................
j
0.2
. ................
0.3
i.. ..............
. ................
0.4
0.5
. ................
0.6
i.. ..............
0.7
197
i ................
0.8
. .........
0.9
I
X
Fig. 1. Normal&d oxygen concentration profile in crack for K= 20 MPa rn’12assuming a potential in the crack of -0.2 V SHE.
drop outside the crack associated with the different values of the crack mouth opening displacement. The passive current (ir) was assumed to be 1.5 x iOe6 A cm-*. The results are summarised in Tables 2 and 3 where the former represents the results in the absence of enhanced crack-tip reaction kinetics associated with the crack-tip. It is quite evident that the adoption of a parallel-sided crack geometry is not a good approximation to a trapezoidal crack if based simply on the mean width. Interestingly, polarisation is more easily attained with enhanced crack-tip reaction although the impact is not large for either geometry. This is presumably due to the influence of the increased reaction kinetics on the local pH and, correspondingly, on the local solution conductivity. The isolated potential (corresponding to the crack being decoupled from the external surface) in the absence of enhanced crack-tip reaction kinetics would be about -0.490 V (SHE) which would indicate that there was not much polarisation of the parallel-sided crack, the assumption adopted by Andresen and Ford.‘*2 The dependence of crack-tip potential on the external potential at steady state is shown in Fig. 2. The dot-dash line represents the condition for no potential drop in the system. Also shown is an indication of the maximum potential expected if there is no effective coupling of the crack-tip to the external surface. The actual potential in the latter case would depend on the magnitude of the crack-tip current with the predicted potential being lower
Table 2. K (MPa ml’*) 10 10
Effect of geometry (unstrained crack tip)
Geometry
Crack width (cm)
ParalleI-sided Trapezoidai
7.2 x 1O-4 from 7.6 x lo-’ to 1.4 x 10e3
o(pScm-‘) 0.3 0.3
E,(V) -0.1 -0.1
&p(V) -0.478 - 0.307
pH,i, 5.3 3.4
A. Turnbuil
798 Table 3. K (MPa ml’*) 10 10
Effect of geometry
(strained
cr (f&cm-‘)
Geometry
Crack width (cm)
Parallel-sided Trapezoidal
7.2 x 10V4 from 7.6x 10m5 to 1.4x 10-j
0.0
crack tip) E, (V)
0.3 0.3
-0.1 -0.1
Tip 0’)
PHrir
-0.444 -0.292
4.7 3.3
‘. ,
E]
Y
‘. .
-0.1 -
\
.. ,
.
-0.2 -
\
. ‘\
-0.3 -
‘\.,
0 <\ x. ‘x,
-0.4 -
0
....
‘\
. ‘L.
___________
-0.5
----_-------_--__------_---------__________~~
isolated and neutral/alkaline pH i 0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
E (V SHE) Fig.
2. Crack tip potential as a function of crack K=10MPam’~2;a293=0.3pScm~‘;i,=l.5x10~6Acm~2;
mouth
potential and corrosion T=561K;pHburk=5.7.
potential
with increasing value of the tip current (because of the need to get enough cathodic current to balance the anodic current). The overall potential drop has two components, the drop in the crack and that outside the crack. The latter becomes significant at corrosion potentials greater than about -0.2 V (SHE) and can be distinguished in Fig. 2 from the difference in full and open points at the same value of crack-tip potential. There is considerable uncertainty in the values of the parameters adopted in this analysis, principally the electrochemical data. For that reason it is important to explore the sensitivity of the results to the particular choice. The predictions of crack-tip potential and pH for different values of key parameters are shown in Fig. 3. It is evident that decreasing the efficiency of the cathodic reduction reactions in the crack by decreasing the rate constant for hydrogen ion and water reduction by an order of magnitude (represented in this figure by k, x 0.1) results in a larger potential drop in the crack. The potential in the crack decreases in order to generate sufficient cathodic reaction to balance the anodic current. The results demonstrate the intrinsic importance of the internal cathodic reactions, which is in contrast to the assumption of Macdonald and Urquidi-Macdonald.3
Modelling of crack chemistry in sensitized stainless steel
199
-0.2 2 %
-0.3 -
L .a LLi-
-0.4 -
‘_ _______________________-_--_--
-0.5 --
-0.6 0.3
_________(ulj____________L._
isolatedand neutral/alkalinepH 1’1’1’ 0.2
I 0.1
0.0
-
-0.1 E,,,
I -0.2
’
1 -0.3
’
1 -0.4
’ -0.5
0’St-W
Fig. 3. Crack tip potential (pH,ip in parenthesis) as a function of corrosion K=10MPam”2;a~~~=0.3~Scm~‘;i,=1.5x10~6Acm~2;T=561K;pH~ul~=5,7.
potential
It is important to understand initially what controls the pH in the crack. Clearly, the steady state pH will depend on the balance between the rate of production of hydrogen ions and their removal by mass transport and by internal cathodic reduction. In the case of a long crack, mass transport is not efficient. Accordingly, when the crack mouth potential is made more noble (i.e. with respect to the corrosion potential in deaerated solution of neutral pH) the pH will fall until essentially the anodic production of hydrogen ions is balanced approximately by the cathodic reduction in the crack. When the solubility of metal cations is low the production of hydrogen ions via the anodic process is approximately equivalent to the anodic reaction current in the crack. In this situation, the crack is very nearly at the free corrosion potential in the local crack environment and the net current from the crack can be small. Measurements of corrosion potential in the simulated crack environment would be expected to give values similar to the predicted crack potentials. The concept that the crack may be polarised into a regime where it is very close to the corrosion potential in the local environment is not unusual and applies to crevices in high temperature water12 and also to crevices of cathodically protected steels in sea-water.13 In the latter case, corrosion potentials have been measured in the alkaline environments predicted and measured in the crevice and are indeed very close to the potentials in the crevice. The concept is sometimes not readily grasped, principally because of neglect of the environmental changes in the crack or crevice. The effect of decreasing the passive current is to result in a decreased potential drop in the crack. Intuitively, this seems readily apparent but it should be set in contrast to the opposite impact of the crack tip current indicated by comparison of Tables 2 and 3. This contrast is not easy to rationalise but part of the difficulty may be our inability to conceptualise the relative impact of transport processes, crack geometry, local pH and conductivity changes. It is important to try to isolate simple explanations for the behaviour,
A. Turnbull
800
because this provides additional confidence in the predictions, but there is a risk of oversimplification and intuition may be inadequate. A crude analogy would be trying to analyze the taste of a soup in terms of the particular ingredients. The effect of stress intensity factor on crack-tip potential is shown in Fig. 4. At low potentials, the potential drop decreases with increasing stress intensity factor which would tend to be expected based on the greater crack opening displacement. The inverse behaviour occurs at more noble potentials which reflects primarily differences in the magnitude of the potential drop external to the crack. The evaluation of the effect of corrosion potential on crack-tip potential should be treated with some caution in view of the assumed limited dependence of the crack tip current on potential. Although the impact of crack tip current on the local potential is modest at a mouth potential of - 0.1 V (SHE) (compare Tables 2 and 3), this would not be expected to prevail as the potential is lowered. The logical approach would be to utilise the predicted dependence of the current decay constant on potential from the work of Ford et a1.9 However, the data relate to solutions of lower conductivity than would be predicted within a crack. It is intended at a future date to utilise a potential-dependent decay constant as a basis of an improved model, although with the uncertainty in the dependence of the decay constant on pH also to consider. In the interim, we have utilised the potential dependence of the bare surface current density assumed by Macdonald and Urquidi-Macdonald,3 despite its limited basis. The results are shown in Fig. 5 and compared with the “standard” data and predictions assuming no crack tip straining. The results are again apparently unusual insofar as the potential drop is greater than the “standard” data at the more negative potentials despite the lower value of the crack tip current. It is at least consistent with previous observations
Etip -“.3 ‘..
-0.4
-0.6 .
I 0.2
I 0.1
I 0.0
I -0.1
I -0.2
I -0.3
-0.4
E corr Fig. 4. Crack tip potential intensity factor (MPam”*);
as a function of corrosion potential for different values of the stress o293=0.3 pScm_‘; ip= 1.5 x 10-6Acm-2; T= 561 K; pHbulk =5.7.
Modelling of crack chemistry in sensitized stainless steel
-.
N\ .\.
\.
\.
801
3-A. .. . \.
-0-
~---
‘L.
standard no straining modifieditiP
EUp= Eoon
I. ‘L.
\.
‘.\. \. . . ‘...
w_*
‘.\._
.\
l
.\..
.\.\ -0.5 -
.\
_ _ _ :::.,,
--___________-______~_~~__~__~~_~-__-_~~~_~~____ -A isolated and neutral/alkaline pH
-0.6
1 0.1
I
I
I
I
I
I
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
E,,,
Fig. 5.
0’ SW
Crack tip potential as a function of corrosion potential with and without crack tip reaction: K=10MPam”z;~2~3=0.3~Scm~‘;i,=1.5x10~6Acm~2;T=56lK;pH~Y~~=S.7.
that increasing the crack tip current would appear to increase the local conductivity, by pulling in chloride ions, to the extent that the increased conductivity becomes the dominant factor. The pH of 5.2 predicted at - 0.3 V (SHE), based on the relationship of Macdonald and Urquidi-Macdonald, should be compared with pH 4.0 obtained using the “standard” data. Nevertheless, both sets of predictions will be inaccurate and probably represent the bounds of predicted potential in the crack. It would be premature to discuss in detail the implications of this work for the prediction of crack growth kinetics in view of the uncertainty in input parameters and the inadequate representation of the potential dependence of crack-tip current. The conceptual perspective that there is coupling between the external surface and crack tip, supported by this model, seems reasonable in view of the dependence of crack growth rate on potential. Oxides within the crack could restrict transport and in the limit prevent coupling of the tip with the external surfacei but this would not be compatible with the observed potential dependence of crack growth. CONCLUSIONS (1) The crack-tip potential is not independent of the external potential in BWR water (conductivity 0.3 uS cm-’ at 298 K). (2) Reactions on the crack walls cannot be ignored in predicting the chemistry and potential in a crack. These reactions include the cathodic reduction of hydrogen ions and water which are too often erroneously neglected in modelling of crack and crevice chemistry. (3) The results demonstrate the need to utilise crack chemistry models of adequate
802
A. Turnbull
quality if there is to be confidence in the predictions and this becomes especially important when validation by measurement of local crack chemistry and potential is difficult. (4) The predictions of this model also have uncertainty in relation to the details because of the paucity of reliable electrochemical data. For that reason more extensive evaluation of parameters, such as solution conductivity and crack size, has not been undertaken at this stage. Acknowledgements-The gratefully acknowledged.
constructive
comments
of Peter Andresen
and Peter Scott in applying
this model are
REFERENCES 1. P.F. Ford and P.L. Andresen, Development and use of a predictive model of crack propagation in 304/316L, A533B/A508 and Inconel 6001182 alloys in 288°C water, Environmental Degradation of materials in Nuclear Power Systems, 789 (ed. G.J. Theus and J.R. Weeks, (The Metallurgical Society, 1988). 2. P.L. Andresen, Conceptual similarities and common predictive approaches for SCC in high temperature waters, Corrosion ‘96 (Houston, TX: NACE International, 1996). 3. D.D. Macdonald and M. Urquidi-Macdonald, Corros. Sci. 32(l), 51 (1991). 4. A. Turnbull, Resolution of key issues in deterministic models of environment assisted cracking. Life prediction of structures subject to environmental degradation (eds. P.L. Andresen and R.N. Parkins (Houston, TX: NACE International, 1996). 5. A. Turnbull and D.H. Ferriss, Corros. Sci. 27(12), 1323 (1987). 6. M. Psaila-Dombrowski, A. Turnbull and R.G. Ballinger, Implications of crevice chemistry for cracking of BWR recirculation inlet safe-ends, Life prediction of corrodible structures (ed. R.N. Parkins (Houston, TX: NACE International, 1994). 7. M. Psaila-Dombrowski, Ph.D Thesis, MIT (1990). 8. H. Tada, P. Paris and G. Irwin, The Stress Analysis of Cracks Handbook. (Del Research Corporation: 1973). 9. P.F. Ford, D.F. Taylor, P.L. Andresen, and R.G. Ballinger, Corrosion-assisted cracking of stainless and low alloy steels in LWR environments. EPRI Report NP-5064M (1987). 10. A. Turnbull, British Corrosion Journal M(4), 162 (1980). Il. Y.-J. Kim and L.W. Niedrach, Electrochemistry of Hz, 02 and Hz02 in high temperature high purity water, II: On stainless steel., GE Corp, to be published (1996). 12. A. Turnbull and M. Psaila-Dombrowski, Modelling of crevice chemistry in the secondary side of nuclear steam generators, Improving the understanding and control of corrosion on the secondary side of steam generators (ed. R.W. StaehIe (Houston, TX: NACE International, 1994). 13. A. Turnbull, Corrosion Science 23(8), 833 (1983). 14. R. Parsons, Handbook of Electrochemical Constants, Academic Press, New York, (1959). 15. H.E. Barnes and R.V. Scheueman, Handbook of Thermodynamieal Datafor Compounds and Aqueous Species, John Wiley and Sons, New York, (1978). 16. P.R. Tremaine and J.C. Leblanc, J. Chem. Thermodynamics 12, 521 (1980).
Modelling of crack chemistry in sensitized stainless steel
APPENDIX
803
A
Geometry and boundary conditions for solution of Laplace’s equation for potential drop outside crack. 2
24
vs$ = 0
-*-0
ax
0
-
fh
X
k, = i,Jk k, = iJk k=F’C$UiCi i
where i. and i, are, respectively, the anodic and cathodic current densities, which by de&&on are identical at the free corrosion potential, i,,, is the current density at the crack mouth, k.‘“’ and kclirnare the limiting anodic and cathodic currents, and w is the ion mobility. a. and cr, are parameters which detine the potential dependence of the anodic and cathodic current densities (e.g. from equation (22), a, = 0.77F/R7’).
APPENDIX
B - DATA USED IN ANALYSES
The diffusion coefficients used are listed below in Table Al .14
Homogeneous reactions
Very few homogeneous reaction rate constants are known, especially at the temperature of interest. The important assumption made is that the rate constants are sutTrcientlylarge to ensure extremely rapid attainment of equilibrium following any perturbation of individual species concentration by reaction. In essence, a very large value of one of the rate constants is assumed. Tbe only restriction is that the ratio of the forward reaction rate constant to the backward rate constant (kf and kb, respectively), as given in the following reaction
A. Turnbull
804 Table Al. Species
Diffusion
coefficients
coefficient
(cm’/s)
I .76 x 10-j
Hi OHNa ’ Cl Ni* ’ NiOH
must be equal to the equilibrium
Diffusion
9.8x 10-j 7.5 X lom~3 3.8 x 10-j
1.3x IOP ’
constant
1.76x IO -4
(K.& as shown below &s=$
(A3)
Thus, in order to assume values for the forward and backward rate constants, the equilibrium constant for each reaction must be known. Equilibrium constants can be calculated from the free energy change for the reaction using the following relationship AGo = -RT
InK,,
(A4)
where A@ is the free energy change for the reaction, R IS the gas constant and Tis the temperature standard free energy change for the reaction is given by the sum of the free energies of formation minus the free energies of formation of the reactants, or AGL,,,” The free energies at the temperature Solubility Limits The precipitation
reaction,
= c
I
A%
products - F
in Kelvin. The of the products
AC;., reactants
(A5)
of interest are listed below in Table A2.”
which is assumed
to take place on the walls of the cavity region, is given by
M2+ + 20H-
4
MO + Hz0
(A6)
where k, is the rate constant for the precipitation reaction. The rate constant is not known and is chosen to be very large to insure that the solution does not become supersaturated. The solubility of the metal ion can be expressed as
Table AZ. Free energies of formation relevant species used in model Species H+ OHH20
Ni2 + NiOH ’ NiO
for
Free energy (kJ mole-‘) - 12.25 - 72.38 - 195.81 -40.16 - 179.91 - 186.61
Modelling of crack chemistry in sensitized stainless steel
805
KS = [M*+][OH-]*
647)
The solubility limit assumedI for Ni2+ was 2.25 x IO-*’ (molesdm-3)3.