1. Introduction.

An important goal of modeling the radiolysis of water in BWR, PWR, and the ITER PHTS, as described in Part I [1] and by Petrov, et.al. [2] is to predict the impact of radiolysis on the corrosion of structural materials.  The most important parameter in defining the evolution of corrosion damage is the electrochemical corrosion potential (ECP) [1].  Many deleterious corrosion phenomena, including general corrosion (GC), pitting corrosion (PC), stress corrosion cracking (SCC), corrosion fatigue (CF), and hydrogen-induced cracking (HIC) occur above (PC, SCC, GC, and CF) or below (HIC) critical potentials (E_crit) and management of coolant radiolysis and chemistry to ensure an ECP that does not lie within a susceptible region is critical to controlling or avoiding corrosion damage in a reactor PHTS  This is the lesson that has been learned in the fission reactor community over the past five decades and it needs to be critically examined by the fusion community, to avoid repeating the same lessons at great cost to the operators and consumers alike.

In Part I [1] and in [2], it is shown that the absorption of ionizing radiation (neutrons, γ-photons, and α particles), generates a variety of radiolysis products, including, H, OH, H₂O₂, HO₂, O₂, O^-, O, H₂, OH^-, H⁺, and possibly others.  These species are either oxidizing agents (e.g., O₂, H₂O₂, OH, O, O^-) or reducing agents (H₂, H,)  some of them being thermodynamically quite powerful, as measured by the standard reduction potential (Table 1 [3]).  Included in this table are data for nitrogen species because nitrogen is formed by the ¹⁶O₈(¹n₀,¹p₁)¹⁶N₇ nuclear reaction and from any ¹⁴N₇ (as dissolved N₂ gas) that might be present in the system. Couples having very negative E⁰ values are strong reducing species while those having very positive values are strong oxidizing species

Selected standard redox potentials for selected radicals [3].

Figure 1 shows a collection of experimental corrosion potential data for stainless steels measured at high temperatures, corresponding roughly to PWR core conditions at the upper temperature extreme.  These data illustrate the strong effect that as little as 20 ppb of oxygen contamination in the feedwater can have on the ECP.  The figure also shows that with no oxygen present in the hydrogenated feedwater, the measured ECP is more negative than -800 mV_she at the highest temperature, depending on the hydrogen concentration, which is slightly more negative than the hydrogen equilibrium electrode potential.  This is so, because the ECP is a mixed potential from a corrosion process comprising a partial anodic process (PAP) and a partial cathodic process (PCP, the HER), with the equilibrium potential for the PAP being more negative than the PCP (HER) equilibrium potential.  The mixed potential falls between the equilibrium potentials for the two partial processes but is closest to the equilibrium potential for the partial process that has the greatest exchange current density (in this case, the HER).  Accordingly, in PWR PHTS, the ECP closely follows the HER equilibrium potential, provided that [H₂] is sufficiently high.

Figure 1. Comparison of calculated electrochemical potentials for Type 304 and 316 SS with experimental data obtained in hydrogenated solutions and deoxygenated solutions [4].

Thus, we can expect that, during normal operation of a reactor with HWC at sufficiently high [H₂], the ECP in the PHTS will closely follow that of the hydrogen electrode, if radiolysis is completely suppressed.  This condition is achieved in the primary circuit of a PWR by operating with [H₂] > 25 cc(STP)H₂/kgH₂O (2.23 ppm) but is not achieved in a BWR operating under a normal HWC regime ([H₂] of < 1.2 ppm.  Part of the problem in BWRs is that boiling in the core strips hydrogen from the coolant, so that much of the added hydrogen is lost to the steam phase and is not present in the liquid water phase to impact the ECP.  In the case of the ITER, which is expected to have a water chemistry not unlike that of a BWR but without boiling, it is specified that 80 ppb [0.89 cc(STP)H₂/KgH₂O] of hydrogen will be added to the PHTS feedwater, which may be compared with about 0.5 ppm [5.5 cc(STP)H₂/KgH₂O] to 2 ppm [22.2 cc(STP)H₂/KgH₂O] employed in BWRs operating under hydrogen water chemistry (HWC), with the upper end of this range corresponding to severe HWC conditions.  The impact that these hydrogen levels have on the electrochemical and corrosion properties of water-cooled reactors, including ITER, is discussed in detail later in this paper but the above establishes the relative hydrogen concentrations employed in the three reactor technologies (PWRs, BWRs, and ITER) of interest.  If the hydrogen concentration is lowered, particularly during the irradiation the suppression of radiolysis is incomplete, the ECP in the radiation zone will rise (i.e., become more positive) due to the generation of small amounts of oxidizing species, such as O₂ and H₂O₂.  The question, then, is how positive is the ECP likely to become compared with E_crit for various forms of localized corrosion damage [note that both E_crit and ECP are also (different) functions of temperature] recognizing also that the rate of propagation of various forms of localized corrosion (PC, SCCC, IGSCC, CF) vary with so that the rate is very sensitive to the overpotential.

Oxygen and hydrogen peroxide generally have very corrosive effects on a reactor coolant system, because they shift the ECP in the positive direction, thereby increasing the driving force for general and localized corrosion processes and in many cases making localized corrosion processes possible by shifting the potential above or below a critical value, E_crit.  Thus, if the ECP is shifted above the critical pitting potential (V_c), pitting will occur, and the resulting pits may act as stress risers for the nucleation of SCC and CF.  For this reason, V_c is also be taken as E_crit for IGSCC, as discussed below.  The critical pitting potential is a sensitive function of chloride concentration, temperature, and pH, such that increasing [Cl^-] and temperature displace V_c in the negative direction while increasing pH displaces V_c in the positive direction.  It is also well known from studies on intergranular stress corrosion cracking (IGSCC) in sensitized Type 304 SS in BWR primary coolant circuits, that cracks will propagate only if the ECP exceeds a critical value (E_IGSCC) that has been set by the NRC to be -0.23 V_she at 288 ^oC, as discussed later in this review.  However, the critical potential shifts in the positive direction with decreasing temperature but so does the ECP, but at a different rate.  While this may seem to be advantageous for the ITER operating at T < 150 ^oC, it is the relative value of the ECP and E_crit that is important, and this relationship has yet to be determined for the ITER.  However, if the potential is displaced too far in the negative direction, HIC may occur in some nickel-based alloys (e.g., Alloy 600 [5]) and in some stainless steels (e.g., sensitized Type 304 SS in acidic solutions [6]), as noted above.  The important point is that these potentially catastrophic failure processes may be avoided by the careful control of the ECP, which is an important lesson learned in the fission reactor community that must be heeded by the fusion reactor community.  Thus, experiences with the impact of radiolysis on the electrochemical and corrosion behavior in water-cooled fission reactors, as discussed briefly above, provide important lessons with respect to the operation of the coolant system proposed for ITER.  These issues are discussed in greater length later in this review.

A simulation of the radiolysis of PWR coolant (1500 ppm B as H₃BO₃ + 1.5 ppm Li as LiOH, + 25 cc/kgH₂O) is shown in Figure 2 [7].  The simulation predicts that the system comes to a steady state within about 1 ms after initiation of irradiation with neutrons and γ-photons.  The reader will note that the most dominant species in the system is H₂ which was added to the coolant (although a small fraction is generated by radiolysis), followed by  , OH, and H.  As we show later, only the species of highest concentration determine the ECP; these being H₂ and possibly and OH, even though the concentrations of the latter two are much lower than that of H₂ by factors of about 200 and 1000, respectively.  However, as shown in Figure 1, even small concentrations of oxidizing species can have a significant impact on the potential, so this “dominant species” rule must be applied with some caution.

Accordingly, when the ECP is calculated around the primary reactor loop of a PWR, it is expected that the ECP will be displaced towards larger (i.e., more positive) values when the feedwater is contaminated with oxidizing species, such as the radiolysis products O₂, H₂O₂, OH, and HO₂ (cf, Figure 1).  However, as noted above, the Mixed Potential Model (MPM) predicts that the contribution that any given species makes to establishing the ECP is roughly proportional to its concentration (see later for a discussion of this topic).  Upon this basis, the most important species are O₂, H₂O₂, and H₂ in the case of a BWR but are H₂, e^-, H, H₂O₂, and OH, in that decreasing order, in the case of a PWR.  Furthermore, because H and are overwhelmed by molecular hydrogen at the concentration typically present in a PWR primary coolant, as shown in Figure 2 [25 cm³(STP)/kg H₂O ≡ 1.17×10^-3 m ≡ 2.23 ppm], they, too, may be ignored to a first approximation, or more appropriately combined with H₂ to form a new “hydrogen species” whose concentration is [H₂*] = [H₂] + 0.5[H] + 0.5[ ], as noted in Part I [2].  Likewise, a new hydrogen peroxide species may be defined as [H₂O₂*] = [H₂O₂] + 0.5[OH] and a new oxygen concentration as [O₂*] = [O₂] + 0.5[O].  These redefined concentrations result in better estimates of the ECP and crack growth rate (CGR).  Thus, it was learned many decades ago in the field of fission reactor technology that, while the formation of O₂ and H₂O₂ could be effectively suppressed by the addition of hydrogen to the primary coolant circuit of a PWR, it is also necessary, from an electrochemical viewpoint, that the other radiolysis products be incorporated into the MPM as indicated above, particularly at low added [H₂].  The defining of new species as described above was necessitated by the lack of kinetic information on the more active radiolysis species, although the standard reduction potentials for some of the species have been estimated [3], as listed in Table 1.  However, because the concentrations of e^-, H, H₂O₂, and OH are so small, their redox currents are mass transfer controlled and hence are insensitive to the kinetic parameters (exchange current density, Tafel constants).

Compared with the work reported on modeling BWR primary coolant circuits [1 and References therein], much less work has been reported on assessing electrochemical effects in PWR primary circuits [1 and References therein], and even less has been reported specifically with regard to the ITER.  This state of affairs reflects the fact that cracking has not been as great a problem in PWR primary coolant circuits as it has been in BWR primary coolant circuits, although the primary water stress corrosion cracking (PWSCC) of mill-annealed Alloy 600 steam generator tubes, pressurizer components, control rod drive tubes, and baffle bolts (highly cold-worked Type 316 SS) have been serious, recurring issues in PWR operation, for example, and because the ITER has yet to operate.  Because of the high hydrogen concentration [typically 25 cc(STP)/kg(H₂O)–50 cc(STP)/kg(H₂O) corresponding to 1.12×10^-3 m to 2.24×10^-3 m or 2.23 to 4.46 ppm] employed in a PWR primary circuit to “suppress radiolysis,” and in view of the lack of sustained boiling, it was generally believed that the ECP is dominated by the hydrogen equilibrium potential and hence that the coolant circuit acts as a “giant hydrogen electrode”, as noted above.  If so, an approximate value of the ECP is readily calculated from the known pH, which, in turn, is easily estimated from the boron and lithium contents of the primary coolant, and the known hydrogen concentration using the Nernst equation.  Considering subsequent modeling, this picture is not entirely accurate; more importantly, though, PWRs are not free from cracking in their primary circuits, and the cracking that is observed is very potential-dependent.  For example, Primary Water Stress Corrosion Cracking (PWSCC) of Alloy 600 steam generator tubes has plagued operators for many years, as noted above, and cracking of core barrel bolts (highly cold-worked Type 316 SS) has also been a recurring problem.  While there are significant materials differences between BWR and PWR primary circuits, in both cases it has gradually become evident that the electrochemistry of the coolant is a prime factor in the nucleation and propagation of corrosion damage [8].  A discussion of the chemistry of the primary coolant circuits of both BWRs and PWRs, and of the proposed chemistry of the ITER coolant, is relevant, because in many respects the envisioned chemistry of the ITER PHTS blends the chemistries of both of those fission reactors, and because there are many lessons that were learned from experiences within the fission reactor community that appear not to have been heeded in ITER.  This parallels the situation that existed in the dawn of the fission reactor age when it was regarded that a fission reactor was like a conventional thermal (fossil-fueled) plant “with a different heat source”.  This led to many of the corrosion problems that have plagued fission plants over the past fifty years.

Predicted concentrations of radiolysis and pH control species in the primary coolant of a PWR as a function of time towards achieving a local steady state for C_B,T = 1500 ppm, C_Li,T = 1.5 ppm, T = 300 °C, [H₂] = 25 cm³/kg STP, G_n = 1´10²⁰ eV/cm³s, and G_g = 3´10²¹ eV/cm³s [7].

2. The Electrochemical Corrosion Potential (ECP).

As noted above, the ECP is the single most important parameter in assessing whether and what type of corrosion can be expected to occur in the PHTS of a nuclear reactor [8].  This is because corrosion processes tend to occur above or below critical values of the ECP, E_crit.  Numerous attempts have been made to measure the ECP in various locations in both BWRs and PWRs by devising reference electrodes that are capable of operating under the harsh thermal/radiolytic conditions that exist within the PHTS of a reactor.  The most successful reference electrode to date is the Ag/AgCl, KCl that has been used by Indig and Nelson [9] to measure ECP in the core of an operating BWR.  As noted above, there has been a trend to use a platinum electrode as a reference in reactor PHTSs, but this is problematic from an electrochemical viewpoint for the following reasons.  First, a noble metal like Pt is an indicator electrode whose potential responds to changes in the very redox conditions that determine the ECP.  Since only differences in potential can be measured (a voltmeter is a two-port device), a Pt electrode cannot yield a reliable measure of the ECP, which ideally is expressed on the universally accepted, the absolute scale of the standard hydrogen electrode (SHE).  The only exception is in an environment in which the hydrogen fugacity is so high, and the concentrations of all oxidizing species are so low that the reference potential of the Pt is governed entirely by the hydrogen electrode reaction.  In this case, the reference electrode potential is readily corrected to the SHE scale.  However, under these circumstances, the indicator electrode potential (i.e., the potential of the component being measured) is also governed by the same factors and hence any measurement is of questionable validity. 

Experience has shown that in-plant ECP measurements are intrusive and can only be made in a few areas of a reactor PHTS.  Fortunately, the ECP can be calculated with acceptable accuracy using a Mixed Potential Model (MPM) [10], so a strategy that has evolved is to calibrate (if necessary) the MPM on the available, measured ECP data and then use the model to calculate the ECP at closely spaced points around the primary coolant circuit using the concentrations of radiolytically generated species using a water radiolysis code [1].  Other electrodes that have displayed acceptable service in high-temperature aqueous environments like reactor coolants are the YSZ(M/MO,O₂), M = Hg, Ni, Cu), where YSZ = Yttria Stabilized Zirconia [11] and W/WO₃ [12], albeit both being more complex than a simple Pt wire.  Importantly, the potentials of these electrodes may be placed on the SHE scale via chemical thermodynamic calculation and they are insensitive to redox potential but are sensitive to pH.  However, the pH may be accurately calculated using appropriate chemical models and hence the potential may be corrected for changes in pH.

The MPM [10] is based upon the fact that charge conservation must be obeyed in a physico-electrochemical system.  Noting that rate of an electrochemical reaction at a metal/solution interface is measured by the partial current density, the conservation of charge constraint requires that the sum of all partial current densities at the interface must be zero.        

where iR/O,j is the partial current density due to the j-th redox couple in the system and i_corr is the metal electrode solution (corrosion) current density.  These partial currents depend on the potential drop across the metal/solution interface (or, practically, the potential difference between the metal and a suitable reference electrode).  In the original version of the MPM [10], as developed for modeling the ECP of Type 304 SS in BWR primary circuits, the steel electrodissolution current density, i_corr, is described by the empirical function of voltage, based on the data of Lee [see Ref. (10)],

where

and

The quantities, b_f, and b_r are the forward and reverse Tafel constants, respectively, for the metal dissolution reaction, with values of 0.06 V being adopted for both.  In fact, they are empirical constants that were assumed a priori in fitting Equation (2) to the current/voltage data of Lee [see Ref. 10].  Note that Equation (2) applies strictly to Type 304 SS in near neutral solutions and, accordingly, it may not be valid for stainless steels in PWR primary circuits, for example.  More recently, the Point Defect Model [13] has been developed for describing the electro-oxidation of a passive metal.  This model yields the passive current density in the form

where the parameters a, b, and c are defined in terms of fundamental parameters [13].  The first term on the right side of Equation (5) arises from the transmission of cations via cation vacancies across the passive film from the metal/barrier layer (m/bl) interface to the barrier layer/solution (bl/s) interface, while the second term reflects the transmission of oxygen ions via oxygen vacancies and/or metal interstitials in the reverse direction.  The values of a, b, and c are determined by optimizing the PDM on wide-band, electrochemical impedance spectroscopic (EIS) data measured on the material under the appropriate conditions [14].  The resulting MPM yields ECP values that are only marginally different (within a few mV) from those calculated using Equation (2).

The current density (iR/O) for a redox couple (e.g. O₂/H₂O, H⁺/H₂, H₂O₂/H₂O), where R is the reduced species and O is the oxidized species, can be expressed in terms of a generalized Butler-Volmer equation as [10]:

where i_0,R/O is the exchange current density, i_l,f, and i_l,r are the mass-transfer limited currents for the forward and reverse directions of the redox reaction, respectively, and ba and bc are the anodic and cathodic Tafel constants.  The parameter  is the equilibrium potential for this reaction as computed from the Nernst equation:

where aR and aO are the thermodynamic activities of R and O, respectively, and is the standard potential, which may be calculated as a function of temperature from electrochemical thermodynamics.  Limiting currents are calculated using the dimensionless mass transfer correlation equation for flow through a pipe as [10]:

where the sign depends on whether the reaction is in the forward (+) or reverse (-) direction, F is Faraday's number, D is the diffusivity of the redox species, is the bulk concentration of O or R, as appropriate, Re is the Reynolds number (Re=Vd/h), Sc is the Schmidt number (Sc=h /D), d is the channel diameter, V is the flow velocity, and h is the kinematic viscosity [10].

In a recent innovation to the MPM [15, 16], a quantum mechanical correction to the exchange current density (  for the presence of the barrier oxide layer of the passive film has been applied to more accurately describe the kinetics of redox reactions on passive metals where the thickness of the barrier oxide layer, through which the charge carriers (e^-, h⁺) must tunnel from the metal Fermi level to an empty acceptor state at the same energy for the oxidized species O at the barrier layer/solution interface.  Thus, the barrier layer represents a tunnel barrier to the transfer of electron charge carriers (electrons and electron holes) and, from quantum mechanical tunneling (QMT) theory [17], the exchange current density can be expressed as:

where ß  is the tunneling constant (≈ 0.6x10⁸ cm^-1 [17]),  is the steady-state thickness of the barrier layer at the HER equilibrium potential, and is the (hypothetical) exchange current density of O/R on the film-free surface.  From the PDM, the thickness of the barrier layer under the HER equilibrium potential is given as:

where  is the polarizability of the bl/ol interface,  is the electric field strength, and

The parameters in G are defined elsewhere [17] but are readily obtained by optimizing the PDM on EIS data [14].  Substitution of Equation (10) into Equation (6) yields:

The bare surface exchange current density is termed “hypothetical” because such a surface cannot be realized in practice on alloys that are of interest in reactor PHTSs (Fe-Ni-Cr alloys) within the kinetic stability range of water since the barrier oxide (typically Cr₂O₃) forms at a potential that is much more negative than the equilibrium potential of the HER.  Any attempt to displace the potential to a sufficiently negative voltage to reduce Cr₂O₃ results in massive hydrogen evolution that renders such experiments impractical.  However, the great advantage of this formulation of electrochemical kinetics of redox reactions on passive surfaces is that a great body of data exists for the exchange current densities on the noble metals, such as Pt and Au, and, because modern computational methods, especially Density Functional Theory, there is a promise for being able to estimate exchange current densities of redox reactions on bare metal surfaces.  This may be a route for obtaining values in the future for the kinetic parameters for the highly reactive redox couples listed in Table 1.  The thickness, L_ss, is found to range from about 0.1 nm to 1.0 nm, depending upon the potential [see Equation (10)] so that the maximum value of  = 2.5x10^-3.  Thus, the QMT correction is significant and should be made where possible.

Because electrochemical kinetic data are available only for the hydrogen electrode reaction (HER, H₂/H⁺), the oxygen electrode reaction (OER, O₂/H₂O), and the hydrogen peroxide electrode reaction (HPER, H₂O₂/H₂O) by assuming the same data as for the OER but with a different standard potential (1.77 V_she vs 1.23 V_she at 25 ^oC, only H₂, O₂, and H₂O₂ can be considered as the redox species in the MPM.  Furthermore, we currently have electrochemical kinetic data for these species only on Type 304 SS, Type 316 SS, Alloy 600, and Alloy 690 over the range of conditions that are of interest in nuclear power reactor technology [14], so that only these substrates can be modeled accurately at this time.  However, significant evidence exists that Types 304 SS and 316 and Alloy 600 and 690 serve as good analogs for other stainless steels and nickel-based alloys.  This is based on the observation that all these chromium-containing alloys form passive films that are essentially substituted, defective Cr₂O₃ and that have the same thickness at any given potential.  Because the exchange current density of a redox species is determined by resonant quantum mechanical tunneling (RQMT) of charge carriers across the barrier layer of the passive film.  Accordingly, the exchange current densities for any given redox reaction on a wide variety of Fe-Cr-Ni alloys are expected to be similar and not to depend on the chemical identity of the barrier layer phase, because QMT is independent of the phase through which tunneling occurs.  This is, indeed, observed.  Furthermore, the electro-oxidation current densities for various Fe-Cr-Ni alloys in the same solutions and under the same conditions are also similar, again reflecting the essentially similar thicknesses of the passive films [14].  Accordingly, the ECP, which reflects a balance between the partial currents for the anodic reactions (substrate oxidation and hydrogen oxidation) and the cathodic reactions (reduction of oxygen and hydrogen peroxide) that occur on the substrate surface, should be similar.  No electrochemical data are available for Zircaloy so the ECP of this substrate cannot be currently modeled.  However, the code has been written so that appropriate values are readily inserted when they become available.

As noted above, the redox reactions of interest in this study are:

as was found in the modeling of ECP in BWRs [10].  Note that the reactant concentrations are redefined, as noted above, as: [H₂*] = [H₂] + 0.5[H] + 0.5[ ], [H₂O₂*] = [H₂O₂] + 0.5[OH] and [O₂*] = [O₂] + 0.5[O].  For simplicity, we drop the asterisk from hereon.  Using the data available from the published literature for the constants and the coefficients [10, 13], the ECP can be calculated by solving Equation (1).

An important point that needs to be emphasized again is that the maximum contribution that any given radiolytic species can make to the ECP is roughly proportional to its concentration.  Thus, in BWR simulations, the concentrations of H₂, O₂, and H₂O₂ are calculated to be orders of magnitude greater than any other radiolytic species, corresponding to the modified Burns and Moore [44] reaction set adopted for the modeling [Ref 1 and citations therein, 18], and hence only these three need to be considered.  In the case of PWR primary HTCs, our previous modeling [1, 8] suggests that aquated electrons, H atoms, and OH radicals may be the most significant radiolytic species in regions of very high-energy dose rate (e.g., near the fuel).  However, no electrochemical kinetic data exist for these species and hence they cannot be directly incorporated at this time, but they are indirectly incorporated by redefining the concentrations, as noted above. 

Some example calculations of the ECP for Type 304 SS in a PWR primary coolant are shown in Figure 3 for the conditions stated in the caption.  The ECP is seen to vary sigmoidally with increasing oxidant concentration and is more positive for H₂O₂ as the oxidant than it is for O₂.  The ECP at the lower oxidant limits corresponds, -0.79 V_SHE, which corresponds closely to the calculated HER equilibrium potential.  The ECP deviates in the positive direction when the oxidant concentration exceeds 1 ppb, demonstrating that, even in a PWR coolant environment, the ECP is sensitive to low concentrations of oxidizing radiolysis products, as noted in Figure 1.  The curves of ECP vs [Oxidant] do not bifurcate until an oxidant concentration of about 10 ppb exists, after which the ECP for H₂O₂ becomes significantly (and increasingly so) more positive than that for O₂.  The critical oxidant concentration of 1 mg/kg (1 ppb, 0,5x10^-6 m) at which the ECP is displaced from the hydrogen electrode behavior is in good semi-quantitative agreement with experimental data obtained from laboratory studies by Bertuch, et.al. [4] (Figure 1).  Because most PWRs operate with high hydrogen levels [20 – 70 cc(STP)/kg H₂O], radiolysis is suppressed and the concentrations of oxidizing species are very low (<< 1 ppb), so the PHTSs of these reactors display ECP values that are about -800 mV_she.

Calculated ECP for Type 304 SS as a function of [H₂O₂] and [O₂] in simulated PWR coolant at 320 ^oC, [B] = 1000 mg/kg, [Li] = 2 mg/kg, [H₂] = 25 cc/kg (2.23 mg/kg), pH = 7.36, flow velocity = 100 cm/s, channel hydrodynamic diameter = 100 cm [19].

One issue that is seldom recognized when assessing the dependence of the ECP on plant operating parameters is the dependence on fluid flow velocity [10].  This dependence arises, because of the dependencies of the limiting currents on flow velocity as expressed by Equation (8).  The importance of this effect is displayed in Figure 5.  The figure also shows the range of ECP that has been reported in practice from both laboratory and in-plant measurements and it is evident that the ECP is hypersensitive to Re (and hence flow velocity) in the transition region between 1 and 20 ppb O₂.  This is the coolant oxygen concentration for BWRs operating on HWC with feedwater hydrogen of about 0.5 ppm.  For NWC operation, [O₂] is about 200 ppb in the recirculation piping system, which places the ECP at 0 ± 0.1 V_she (Figure 4).  An important point from this graph is that the system may change from a non-cracking state where ECP < E_crit (-0.23 V_she at 288 ^oC) to a cracking state of ECP > E_crit merely by changing the flow velocity without changing species concentrations.  Furthermore, the large scatter observed in the laboratory measurements reflects the poor control of most experiments of the mass transfer conditions.  This is a good illustration of the complexity of the factors that control the ECP in operating BWRs.

Figure 4. Calculated ECP vs log [Oxidant] as a function of Reynolds number for Type 304 SS in BWR primary environment under NWC conditions at 288 ^oC.  The Re values correspond to a 50-cm ID pipe with flow velocities ranging from 1 to 10 m/s, which are typical for a BWR recirculation piping system [10].

The accuracy of the Mixed Potential Model in predicting ECP has been evaluated by comparing calculated ECP values for Type 304 SS against measured BWR plant data (Figure 5).  These data are considered to be particularly important because the original authors also reported values for the concentrations of oxygen and hydrogen (but not H₂O₂), and we employ those data that were obtained during a Hydrogen Water Chemistry (HWC) mini-test at the Leibstadt BWR in Switzerland [20, 21].  The vendor retained us to model the reactor chemistry and predict the ECP in a “double-blind” manner (i.e., we did not have access to the ECP data prior to submission of our calculations and the vendor did not have access to our calculations while performing the mini-test).  We were, of course, provided with the required input data, such as the flow velocity, hydrodynamic diameter, [H₂], and [O₂], and temperature data for the test system.  The calculated and measured (plant) ECP data for this case are summarized in Figure 5.  Excellent agreement is obtained in systems to which hydrogen had been added, with the measured and calculated ECP values agreeing within the combined uncertainty levels.  In the normal water chemistry case, the measured ECP is significantly higher than the calculated value.  This is almost certainly due to the presence of hydrogen peroxide in the coolant, which was not measured by the personnel conducting the HWC mini-test.  Accordingly, we were unable to input a value for [H₂O₂] into the MPM.  However, if we use the calculated values for [H₂], [O₂], and [H₂O₂] obtained from our radiolysis code, excellent agreement is obtained over the entire hydrogen concentration range [20, 21].

Figure 5. Comparison of calculated and measured ECP in a test cell attached to the recirculation piping in the Leibstadt BWR as a function of the concentration of hydrogen added to the feedwater.  The concentrations of O₂ and H₂ concentration were supplied from the vendor.  No data for H₂O₂ was available so that this species was not incorporated in the calculation (see text) [20, 21].

Additional studies on modeling the electrochemical and corrosion properties of BWR coolant circuits are presented later in this review in Section 5.

3. Calculation of Crack Growth Rate.

The ultimate goal of modeling the electrochemistry of reactor coolant circuits is to calculate the crack growth rate in structural materials and then to estimate the accumulated damage (i.e., crack length vs time) along the corrosion evolutionary path (CEP, i.e., the operating history) of the reactor.  Various models have been proposed for calculating crack growth rate in PHTS structural materials, such as the austenitic stainless steels (like Type 304 SS, Type 316 SS) and nickel-base alloys, such as Alloy 600.  A comprehensive review of all proposed models is beyond the scope of this review, so attention is focused on describing those models that explicitly incorporate electrochemistry because the ECP is the key parameter in determining the ECP and the CGR.  The only deterministic model (i.e., one whose output is constrained by the natural laws) that has been developed to date is the Coupled Environment Fracture Model (CEFM) [22] and variants thereof [23], so the present discussion is restricted to the CEFM.

The basis of the CEFM is the differential aeration hypothesis (DAH), as illustrated in Figure 6.  Thus, SCC occurs because of the separation of the local anode (in the cavity) and the local cathode (on the external surfaces that have maximum exposure to the cathodic depolarizers (O₂, H₂O₂, H₂).  As a result, a net positive, ionic current flows from the crack tip, where it is produced by the electro-dissolution of the steel to the external surfaces, where it is annihilated by charge transfer reactions [Reactions (13) – (15)] and the electron current flowing through the metal from the crack tip.  That current is termed the “coupling current (CC)” because it couples the internal (crack enclave) and crack external environments, hence the name of the model.  For a model to be deterministic and hence robustly predictive, the system must be constrained by the conservation of charge, which is stated here as [22]:

where dS is an increment the area of the interface between the metal and the environment, including that within the crack. 

Figure 6. Schematic of differential aeration hypothesis for localized corrosion [22].

The integral is evaluated over the entire area, but modeling shows that, because the net current on the external surface decreases exponentially and asymptotically towards zero as the distance on the external surface increases from the crack mouth, the effective distance (“throwing power”) is limited.  Indeed, the integral needs to be evaluated over a region of about ± 10 crack opening dimensions on either side of the crack for most conditions of interest.  Now, mathematically, an infinite number of solutions of the Nernst-Planck equations coupled with Poisson’s equation exist for the distributions of species concentrations and potential, and hence the current, for the crack internal current depending upon the potential in the solution at the crack mouth.  Likewise, an infinite number of solutions also exist for the distributions in the potential and the current on the external surface, depending upon the value of the same potential.   However, there is only one value of the crack mouth potential for which Equation (16) is satisfied and that value imbues the model with determinism.  Once this value is determined by iteration, together with an embedded iteration on the potential in the solution at the crack tip until electroneutrality is achieved, the corrosion current density is calculated and hence the crack growth rate is obtained via Faraday’s law.

The model adopted for describing the events that are envisioned to occur at the crack tip is the periodic slip/dissolution/repassivation model [24] from which the average metal dissolution current is determined.  However, measurement of the micro-crack dimension (MCD) coupled with the known crack growth rate for IGSCC in sensitized Type 304 SS in water at 288 ^oC (simulated BWR primary coolant) shows that the area of metal exposed upon each cycle (the “microcrack dimension”) is too large (≈ 2 µm) to be accounted for by slip alone.   If slip alone was the reason for crack advance, the MCD should be a small multiple of the Burgers vector or a few nm in size and the microfracture frequency (MFF) would have to be in the kHz range, rather than in the observed Hz range, to account for the measured CGR [25].  The strain rate at the crack tip, which is required for calculating the MFF was initially that of Ford and Andresen [26], but later those of Congleton [27] and Shoji [28], and finally that of Hall [29] were used.  The current version of the CEFM employs Hall’s expression for the crack tip strain rate and includes small-scale yielding and strain-rate hardening effects.  The CEFM predicts that the MFF is initially zero for K_I < K_ISCC, corresponding to the absence of environmentally assisted cracking (EAC), but quickly increases to about 2 Hz for K_I > K_ISCC (Figure 7).  The latter condition demonstrates fracture by IGSCC and results in a microfracture dimension of nearly 2 μm; about a factor of 1000 larger than that expected from slip alone.  The same type of analysis was performed on cracking in sensitized Type 304 SS in thiosulfate solution at ambient temperature [30] and in AISI 4340 steel in 7 M NaOH at 70 ^oC [31], both being well recognized as systems subject to hydrogen-induced (HIC), the crack dimensions were calculated to be 42 – 134 μm and ≈ 60 μm, respectively [32].  In the BWR environment, the MCD is sub-grain sized (1/3 to 1/10 the grain size) but for the cases of Type 304 SS in thiosulfate and in AISI 4340 steel in 7 M NaOH at 70 ^oC, the MCD is super grain size.  For this reason, cracking in Type 304 SS in BWR coolant is attributed to slip/dissolution/repassivation augmented by HIC, resulting from the injection of cathodically generated atomic hydrogen into the matrix ahead of the crack tip followed by recombination in creep-induced voids on the grain boundaries.  This recognizes that some cathodic reaction (hydrogen evolution via proton reduction) occurs at the crack tip even though the bulk of the cathodic activity occurs on the external surfaces, as required for maintaining differential aeration.  The resulting pressure within the voids is added to the hydrostatic stress at the crack tip, resulting in the increase in the MFF displayed in Figure 7.  Additional details on the CEFM may be found in [33,34].

Figure 7. Frequency of the brittle micro fracture events versus stress intensity factor for IGSCC in sensitized Type 304 SS in water at 288 ^oC, κ (25 ^oC) = 0.5-1.3 μS/cm, [O₂] = 0.15 x 10^-3 m [25].

A comparison of the CEFM predicted and measured CGR in sensitized Type 304 SS in BWR coolant as a function of the ECP for different values of the ambient temperature conductivity is shown in Figure 8.  Of course, the conductivity at the operating temperature of 288 ^oC was used in the CEFM.  In light of the fact that the CEFM contains several poorly known parameters, such as the strain for fracture of the passive film at the crack tip, the CEFM was calibrated against two measured CGR data at different temperatures, one of which is shown in Figure 8.  Two CGR data at different temperatures were necessary, so as to determine the activation energy [35], which was found to be 1000 kJ/mol when using the Congleton expression for the crack tip strain rate.

of the predicted dependence of the CGR on ECP in BWR coolant (water at 288 ^oC) for different values of the ambient temperature conductivity.  The experimental data are from Ford and Andresen [26].  The citations in the figure are to the original source [18].

As stated previously, the ECP is the single most important parameter in determining the damage accumulation rate for corrosion, including IGSCC in sensitized Type 304 SS, and that position is borne out by the data presented in Figure 8.  Thus, we see that the CGR increases exponentially with ECP [linear log(CGR) vs. ECP] at potentials over which environmental (electrochemical) effects dominate, i.e., at ECP > -0.2 V_she.  Below this ECP, crack advance occurs by creep, which in the CEFM is described by the Wilkinson and Vitek model [see Ref. 22].  The CGR is also predicted to be a sensitive, positive function of the coolant conductivity, a property that is discussed at length by Lu, et.al. [22].  Increasing coolant conductivity increases the throwing power of the coupling current across the surface external to the crack, thereby leading to a larger area for the reduction of the cathodic depolarizer and hence a larger coupling current (CC).  Since the coupling current is linearly related to the CGR by Faraday’s law, the CGR increases correspondingly.  At very high ECP (> 0.2 V_she) the CGR is so large that mass transfer of the cathodic depolarizer exerts an influence on the rate and the predicted log(CGR) vs. ECP correlations deviate negatively from the linear relationship that is apparent at lower ECP.  Finally, it is noted that the Nuclear Regulatory Commission (NRC) sanctions a critical value for the critical potential of IGSCC as E_crit = -0.23 V_she, but we caution that the critical potential is a function of many variables, including temperature, degree of sensitization, and crack length, as discussed below.

The predicted dependence of the CGR for sensitized Type 304 SS in BWR coolant on temperature is displayed in Figure 9 [35] together with experimental data from Andresen [36].  As seen, the CGR is predicted and found to pass through a maximum at about 170 ^oC, which has important implications for the accumulation of IGSCC damage during the start-up and shutdown of BWRs.  The maximum arises from the competing positive effect of temperature on the crack tip strain rate at low temperature and the negative effect of decreasing conductivity and especially decreasing ECP at higher temperatures.

Figure 9. The effect of temperature on crack growth rate in Type 304 stainless steel in dilute sulphuric acid solution having an ambient temperature (25 ^oC) conductivity of 0.27 mS/cm and a dissolved oxygen concentration of 200 ppb.  The experimental data (curves) are taken from Andresen (see [35]).  The model curves are calculated using the CEFM calibrated at 288°C and assuming crack tip strain rate thermal activation energies of 75 kJ/mol or 100 kJ/mol when using the Ford or Congleton crack tip strain rate models, respectively [35, 36].  The citations in the figure are to the original source [35,36].

A particularly important prediction of the CEFM is the dependence of the CGR on the electrochemical crack length (ECL), as shown in Figure 10 [22].  This dependence was missed in past experimental studies of crack growth, because standard fracture mechanics, C(T), specimens tend to be employed, in which the ECL stays essentially constant as the crack grows.  This is because the ECL is the least resistive path through the solution from the crack front to the external surfaces; in the case of a through thickness crack in a C(T) specimen, that path is out through the crack opening on the specimen side rather than the much longer path down the crack and out through the crack opening.  Thus, the CEFM dictates that two crack lengths must be defined if stress corrosion cracking is to be understood mechanistically; a mechanical crack length (MCL), as defined conventionally, as being the distance between the load line and the crack front, and an electrochemical crack length (ECL), which is defined as being the shortest distance (least resistance through the solution) between the crack front and the free external surface upon which the cathodic depolarizer reacts.  It is this latter crack length that is indicated in Figure 10. 

Figure 10. Predicted CGR in Type 304 SS in simulated BWR coolant as a function of ECP for different values of the crack length [22].  T = 288 ^oC. [22].

When CT specimens are employed for measuring CGR, the shortest distance from the crack front is direct to the specimen side surface and this distance tends to be independent of the MCL.  Indeed, the ECL obviously varies with position along the crack front, such that the crack growth rate should be greatest at the crack edges and least in the crack center.  This is predicted to result in a convex crack front, as viewed from the crack mouth, as is commonly observed experimentally [32].  If cracking occurs by creep, the crack growth rate should be lower at the crack edge, where plain strain conditions are no longer fulfilled, than in the crack center and the crack front should appear concave, as is also commonly observed.  An important prediction of the CEFM, as seen in Figure 10, is that the critical potential for IGSCC for a pre-existing crack is also a function of crack length.  Thus, for the stated conditions, E_crit = -0.15 V_she for an ECL of 0.1 cm but is 0.1 V_she for ECL = 60 cm.  The practical implication of this dependence is that all cracks must eventually die (repassivate).  Thus, if the ECP of the steel is 0.1 V_she (typical of the non-irradiated core of an operating BWR), the crack is predicted to die when the ECL reaches about 2 cm, resulting in E_crit becoming equal to the ECP.  Indeed, it is evident that death occurs at a shorter length the more negative the ECP.  For an ECP of the steel of 0.2 V_she, the crack is not predicted to die until the ECL is greater than 60 cm, demonstrating that the critical ECL for crack death (repassivation) is very dependent on the prevailing ECP.  This dependence of the CGR on ECL arises because of the IR potential drop down the length of the crack, due to the flow of the coupling current.  This IR potential drop is subtracted from the driving force for the crack, which is ECP – E_{crach tip} resulting in the potential drop across the external surface that drives the cathodic reaction(s) being reduced.  This results in a smaller throwing power of the CC from the crack mouth and hence a lower CC and CGR.  The authors know of no other model for CGR that makes this important prediction, the consequences of which are discussed further below.

The CEFM predicts that the crack becomes acidified due to the hydrolysis of Fe²⁺, Ni²⁺, and Cr³⁺ at the crack tip and that Cl^- is concentrated within the crack to a factor of up to 10⁶ compared with [Cl^-] in the external environment while Na⁺ is ejected from the cavity [34].  The concentration factors increase with the ECP or more accurately with the magnitude of the coupling current because Cl^- “climbs up” the potential gradient that drives the coupling current as it flows through the solution from the crack tip to the external environment.  These are the expected consequences of the DAH.

The CGR is also predicted to be a function of the coolant flow rate [22].  For a high aspect ratio crack (high crack length/COD), in which flow-induced mixing between the crack internal environment and the external environment is geometrically inhibited, the CEFM predicts that the CGR increases with increasing flow rate, corresponding to the positive shift in the ECP as shown in Figure 6.  As with the shift in the ECP, the increase in the CGR occurs because of the enhanced supply of the cathodic depolarizers to the metal surface external to the crack resulting in a greater CC and hence CGR.  In the case of a low aspect crack, mixing of the crack's internal and external environments destroys the aggressive conditions of high [Cl^-] and low pH at the crack tip, which result from differential aeration and that are necessary to maintain the crack electrochemically active.  Consequently, the dissolution rate at the crack tip decreases, and hence so do the CC and the CGR.  This is expected to be a significant issue in the nucleation of cracks, where nucleation often occurs at stress raisers, such as corrosion pits, but this subject is beyond the scope of the current chapter.

Because of space limitations, the following discussion concerns only Intergranular Stress Corrosion Cracking (IGSCC) in sensitized Type 304 SS in BWRs, Hydrogen Induced Cracking in PWR mill-annealed steam generator tubing, and an assessment of the likelihood of IGSCC in the ITER.  However, we note that the principles underlying these examples are common to all forms of corrosion, including general corrosion (GC), pitting corrosion (PC), crevice corrosion (CC), and corrosion fatigue (CF).

4. The Critical Potential.

It is evident from this discussion that the key parameters in determining the susceptibility of a reactor coolant circuit to corrosion-induced damage are the critical potential (E_crit) and the ECP, such that when ECP > E_crit, PC, SCC (IGSCC), and CF may occur in the PHTS structural materials.  In the case of HIC, the opposite applies, i.e., ECP < E_crit.  As also noted, both ECP and E_crit are functions of other variables, notably temperature, as shown in Figure 11 for sensitized Type 304 SS in dilute (0.01 m) NaCl solution at temperatures between 100 ^oC and 250 ^oC [37 - 39].  The reader will note that the solid line demarcates regions of ductile failure (no SCC) and regions of PC/IGSCC, as determined using constant extension rate tests (CERTs) under potential control.  Extrapolation of that line to 25 ^oC yields E_crit = 0.17 V_she and to 288 ^oC, E_crit = -0.52 V_she.  This latter value is significantly more negative that the -0.23 V_she sanctioned by the NRC for IGSCC in sensitized Type 304 SS in BWR coolant.  Noting that crack initiation must begin with passivity breakdown, for whatever reason (e.g., pitting, surface slip, grain boundary penetration), part of the difference may be attributed to the impact of [Cl^-] on the passivity breakdown potential at which pitting occurs, which coincides with E_crit for IGSCC at T < 120 ^oC (Figure 11). 

Figure 11. Values for  for sensitized Type 304 SS in 0.01 m NaCl solution as a function of temperature as reported by Lin, et.al. [37].

To a good approximation, a correction to the data in Figure 11 can be made by using the Point Defect Model (PDM) [13] for the difference in chloride concentration [10 ppb (3.5x10^-4 m) for BWR coolant vs 0.01 m in the experiments (Figure 11).

where  is the standard breakdown potential,  is the polarizability of the barrier layer/solution (bl/s) interface, and  is the activity (concentration, for our purposes) of chloride.  The correction needed to bring the data plotted in Figure 11 (0.01 m NaCl) into compliance with BWR conditions (3.5x10^-4 m NaCl) at any given temperature is given as:

where the critical potential is designated.  Note that is a function of temperature but not of [Cl^-].  Using parameter values for stainless steel (  = 0.8 [14]), Equation (19) is evaluated at various temperatures, as listed in Table 2 (Row 3), along with the ECP for Type 304 SS in BWR coolant with 200 ppb O₂ (Row 5).  The corrected critical potentials are in reasonable agreement with the value of -0.23 V_she determined in an operating BWR [9] as sanctioned by the NRC.  Unfortunately, there are no data for IGSCC that have been measured for BWR coolant conditions at other temperatures with which to compare.  We also note that the CERT test method tends to “overshoot” the true critical potential, particularly if the reciprocal of the passivity breakdown induction time is small compared with the strain rate employed in the experiment.  Accordingly, the true critical potential is likely to be even more negative than those listed in Rows 1 and 3, Table 2.

An alternate strategy exists for estimating the critical potential and that is based on the concept that a minimum coupling current (CC) is required to maintain the conditions at the crack tip sufficiently aggressive (low pH, high [Cl^-]) to maintain the steel at the crack tip in the active state.  Based upon extensive modeling work and upon experiment [25], for sensitized Type 304 SS, the critical CC for a standard crack (see below) is estimated to be 1 nA.  Thus, since the CC depends upon the ECP, the critical potential ( ) corresponds to the ECP at which the CC = 1 nA.  Values for the estimated critical potential, E_crit,IGSCC are presented in Table 2 as a function of temperature (Row 5).

Table 2. Estimated critical potentials for IGSCC for sensitized austenitic stainless steels as a function of temperature.

The value of E_crit,IGSCC at 288 ^oC (-0.317 V_she) may be compared with that measured by Indig and Nelson [9] (-0.23 V_she, Row 4, Table 2).  However, each order of magnitude decrease in [Cl^-] (e.g. from the assumed 10 ppb to 1 ppb) results in a 0.342 V shift in the correction term ( ) shown in Table 2, increasing the critical potential to 0.025 V_she.  This suggests that E_crit,IGSCC is also a function of [Cl^-]; a relationship that has not been previously detected (or looked for, to our knowledge), but which is understandable if the critical potential is determined by passivity breakdown, as is assumed in the present analysis in determining  [13].  In that case, E_crit,IGSCC will shift in the negative direction by  with each decade increase in the chloride activity (concentration), where α is the polarizability of the barrier oxide layer/solution interface.  The shift is typically 100 mV/decade in [Cl^-].  It is important to note, however, that the sensitivity to chloride activity and other factors (metallurgical, mechanical, geometric, and environmental) indicates that the critical potentials (both  and ) are no more accurate than ± 0.15 V.  Although there are not many experimental data available for the critical potentials for IGSCC in sensitized austenitic stainless steels, those that were available in 1981 are reviewed by Cragnolino and Macdonald [38].  These data mostly refer to 0.01 M NaCl solutions (c.f., Figure 11) but some data for sulfate and borate solutions may be gleaned from the review.  These data are generally in accord with those summarized in Table 2.  Designating the critical potential at which the environmentally assisted CGR as , the data in Figure 10 reveals a dependence of /V_she on ECL that is greater than the difference between the calculated and measured E_crit values.  Additionally, E_crit has long been suspected of also being a function of the degree of sensitization (DoS) of the alloy and of the conductivity of the solution [22].  Accordingly, we conclude that the values of E_crit given in Table 2 as a function of temperature are quite realistic within the bounds of uncertainty, as discussed above, although we opine that it is possibly the most relevant to the initiation of IGSCC in the IBED-PHTS in the ITER. 

As noted above, cracking in mill-annealed Alloy 600 also exhibits a critical potential; in this case for HIC.  This potential has been measured in a typical PWR primary coolant by Totsuka and Smialowska [5] using the CERT technique as shown in Figure 12.  Thus, in this case, HIC is induced for E_crit < -820 V_she.  These data are used later in this review when Primary Water Stress Corrosion Cracking (PWSCC) of Alloy 600 steam generator tubes is discussed.  Unfortunately, E_crit does not appear to have been measured at other temperatures, so, at the current time, our analysis of this problem is restricted to PWR full power operating conditions.

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