Point defect model for the corrosion of steels in supercritical water: Part I, film growth kinetics

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Point defect model for the corrosion of steels in supercritical water: Part I, film growth kinetics Yanhui Lia,b, Digby D. Macdonaldb,*, Jie Yangb, Jie Qiub, Shuzhong Wanga,* a b

Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China Department of Nuclear Engineering, University of California at Berkeley, Berkeley, CA 94720, USA

ARTICLE INFO

ABSTRACT

Keywords: A. Steel C. Oxidation Oxide scale Kinetics Point defect model Supercritical water

A Point Defect Model has been developed to describe theoretically the corrosion of metals and alloys in supercritical aqueous systems. The model, SCW_PDM, accounts for the kinetics of growth of the barrier layer and the total scale thickness and for the oft-reported growth of the barrier layer with the barrier/outer layer interface remaining at the location of the original metal surface. The barrier layer grows into the metal via the production of oxygen vacancies at the metal/barrier layer interface and their annihilation at the barrier layer/outer layer interface. The proposed atomic-level rate laws for the scale growth and oxidation weight gains, not only can act as deterministic equations for extracting values of some fundamental parameters in SCW_PDM, but also can be employed to describe the oxidation kinetics of steels directly and successfully.

1. Introduction The development of supercritical reactors (SCRs) is being actively pursued internationally as part of the program to develop the hightemperature, Generation IV nuclear power reactors following the successful development of super-critical thermal (fossil) power plant technology [1], where the super-critical water [SCW, above the thermodynamic critical point (374.15 °C, 22.1 MPa)] is used as the heat transfer medium. Not only in thermal power plants, SCW is also employed in various other industrial and technological systems, such as supercritical water gasification (SCWG) and supercritical water oxidation (SCWO), which are developing technologies with great potentials in the clean-energy and environmental fields [2,3]. The former aims to convert a variety of wet biomass into hydrogen-rich gases, and SCWO owns the potential for environment-friendly, complete destruction of toxic organic wastes. However, because SCW represents a harsh environment, considerable concern exists concerning the corrosion of structural materials (e.g., Fe-Cr-Ni steels) in these environments. That these environments (SCWs) are corrosive has been demonstrated in innumerable reports in the literature extending back several decades [4–7]. An excellent review of the subject has been published by Kritzer [6]. The current understanding of the evolution of corrosion phenomena in water at high temperatures is linked to the water density and dielectric constant of the medium [8–12]. Extensive work has demonstrated that the corrodent may be oxygen or water with the corrosion

⁎

process being “chemical oxidation, CO” in nature for low density/dielectric constant environments (< 0.1 g/cm3 / < 10) or “electrochemical oxidation, EO” at high density/high dielectric constant environment (> 0.1 g/cm3 / > 10), at least for stainless steels [11,12]. The CO mechanism, not involving partial anodic and cathodic reactions, is described as the direct chemical reaction between the metal and the reactive species (i.e., H2O and O2), while the EO mechanism occurs via the partial anodic and cathodic processes, involving electron transferring among the separated cathodic/anodic sites. Typical scale structures of various steels are displayed in Fig. 1. The scale is commonly a duplex-layer comprising an outer layer and an inner layer (also being called a “barrier layer”) plus some local internal oxidation zones [13–17]. What needs to be pointed out that the oxide scales described above (also the focus of this paper) is for Fe- or Nibased alloys without work-hardened treatment exposed to SCWs. The severe surface plastic deformation derived from the work-hardened treatment, such as shot peening and other mechanical working [17–19], generally favors the formation of a single thinner chromium rich oxide layer and the occurrence of a deeper international oxidation, of which the kinetics determination is not in the scope of this paper. When Pd micrometric markers were deposited on the sample surface prior to oxidation, the original, thin, bright white nano-marker was observed to locate at the barrier/outer layer interface after an exposure [13], as shown in Fig. 1(a). These marker experiments are consistent with the growth of the barrier layer into the metal phase [13,20]. In this

Corresponding authors. E-mail addresses: [email protected] (D.D. Macdonald), [email protected] (S. Wang).

https://doi.org/10.1016/j.corsci.2019.108280 Received 13 May 2019; Received in revised form 15 September 2019; Accepted 9 October 2019 0010-938X/ © 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Yanhui Li, et al., Corrosion Science, https://doi.org/10.1016/j.corsci.2019.108280

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regard, the scale is little different from that which forms on metals in sub-critical, aqueous environments [21,22]. The inward growth of the barrier layer (bl) demonstrates growth at the metal/barrier layer interface (m/bl) via the generation of oxygen vacancies and the inward migration of oxide ions [15,23]. As for the outer layer, in consideration of the high porosity and relatively wide grain boundaries of the columnar oxides [24,25], even the possible occurrence of abundant micro-cracks, there are reasonable reasons to believe that O2 and H2O can penetrate through the outer layer via cracks and pores and hence access the barrier/outer layer interface where the cathodic partial reactions of oxygen reduction or hydrogen evolution in the EO mechanism or the direct reaction of O2 and H2O with species transmitted through the bl (oxygen vacancies and metal interstitials) in the CO mechanism are envisioned to occur. The outer layer grows via the reaction of cations transmitted through the inner layer with species (O2 and/or H2O) in the environment or by transformation (e.g., dissolution/ re-precipitation) of the bl at the barrier layer/outer layer interface (bl/ ol). Generally speaking, for the growth of outer layer in low density SCW, the former is generally predominant compared to the latter (which commonly dominates in condensed aqueous systems), but the dissolution possibility of oxides present at the bl/e interface always exists [16,17]. By means of on-line monitoring the oxide formation and dissolution on alloy 800H, Chouhry et al [16] reported that even at 650 °C in SCW approximate 1/500 of the total amount of oxide formed dissolved. The metal species released into SCW are most likely to be uncharged ion pairs considering the low dielectric constant of SCW, of which some re-precipitate on the outer layer and the other parts are carried away by flowing SCW. Accordingly, except of the variation in relative significance of each microprocess, mainly the processes involved the growth/destruction of the outer layer, the growth of the scale on steels in contact with O-containing SCWs appears to be indistinguishable from that for metals in sub-critical condensed aqueous environments, dry oxygen, and molten metals, all of which can be described by the Point Defect Model (PDM) [15,21,26–30]. This model has evolved considerably over the past near 40 years since it was first proposed in 1981, and the latest Generation III Point Defect Model, PDM-III, is available since 2011 [29]. An important feature is to be noted from Fig. 1, and is one that was

originally noted for the corrosion of iron in high temperature (subcritical) aqueous environments [31]. Thus, it is seen that the boundary between the barrier layer and the outer layer remains located at the original metal/environment interface. This means that the barrier layer grows as a constant volume process even though the molar volume of the oxide occupies approximately 2.5 times the volume of the metal from which it forms. In SCW, several marker experiemnts also reported that the original water-metal interface generally corresponds with the barrier-outer layer interface, as expected [13,32]. Thus, in forming the barrier layer the system appears to “know” how much metal to lose via cation transmission compared to how much to retain in the barrier layer. That is, there must be an appropriate ratio between the outward cation flux and the outward flux of oxygen vacancies such that the bl/ol interface remains fixed in the laboratory frame of reference. We consider explanation of this extraordinary fact to be one of the great challenges in the theory of metal oxidation. The oxide scale formed on steels in SCW generally features an inner iron-chromium spinel layer (barrier layer) and an outer iron-oxide layer as shown in Fig. 2. Cation interstitials are generally predominant at lower oxygen partial pressure within the deeper oxide layer such as the barrier layer [23,33,34]. For the low [O] case, generally less than 1 ppm, the porous outer layer appears to completely comprise columnar magnetite grains. Imposition of a stronger driving force for scale formation (higher [O]) appears to lead to the formation of magnetite outer layer with some Cr content, and some hematite either as a single outmost layer or as discrete phases [16,35]. However, it is still of great challenge to characterize the growth kinetics of each layer, especially at the atomic level. Typical weight gain vs time curves for various ferritic-martensitic steels at 500 °C and for HCM12A steel at 360 - 600 °C and dissolved oxygen of 10 to 25 ppb [36], are presented in Fig. 3(a) and (b), respectively. The weight gain appears to follow a parabolic law but the data are seldom accurate enough to distinguish between the competing rate laws. As for thickness vs. time plots as shown in Fig. 4, this issue is more prominent [13,37,38], primarily because of the difficulty in accurately measuring film thickness when the metal/bl interface is so uneven due to local permeation of the barrier layer into the metal [15,38,39]. Improving the accuracy of such data remains a great

Fig. 1. SEM images of the oxide layers formed on typical steels in SCW [13,14]. (a) marked HCM12A exposed at 500 °C for 6 weeks; (b) polishing 316 L stainless steel exposed to SCW (600 °C, 25 MPa) for 335 h.

2

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Fig. 2. Schematics of oxide scales formed on Fe-based alloys exposed to supercritical water.

experimental challenge. Additionally, the existing cation-diffusion models such as Wagner’s parabolic law have been much-criticized [40,41], because of their failure to account for fundamental quantities (e.g., the “jump distance”), their neglecting the existence of potential drops at the phase boundaries (metal/scale and scale/environment interfaces), and because they fail to account for the behavior at limiting small times (the rate of oxidation must be finite) and limiting long times (steady-state oxidation if the rate of dissolution of the scale is finite). According to the PDM firstly proposed by one of the authors (Digby D. Macdonald, DDM) and the mixed conduction model [42,43], of which all were previously widely used in condensed aqueous systems, Sami Penttilä et all [44,45] developed a kinetics model for scales growth at the atomic level, by which they estimated the kinetic parameters of oxidation of alloys of interest with basically satisfactory accuracy. Although the development and update in assumption for formation of the outer layer, from the dissolution/re-precipitation mechanism to the direct reaction of cations interstitials transported through the inner layer with water, occurs in Sami’s kinetics model, this model neglects the contribution of cations transmitted through the inner layer via cation vacancies mechanism to the growth of the outer layer. Meanwhile,

Fig. 4. Some data on thicknesses of oxide scales formed on HCM12A steel exposed to SCW at 500 °C [13,37,38].

Fig. 3. Comparison of weight gain for common ferritic–martensitic alloys in SCW at various temperatures for up to 3000 h. HT9 and HCM12A are ∼12 wt.% Cr alloys. T91 and NF616 are ∼9 wt.% Cr alloys. After Allen [36].

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Fig. 5. Summary of the defect generation and annihilation reactions envisioned to occur at the interfaces of the barrier oxide layer on a metal, according to SCW_PDM. VM' = cation vacancy, vm = atom vacancy, Mi + = cation interstitial, V O =oxygen (anion) vacancy, OO =oxide ion in anion site on the anion sublattice of the barrier layer, MO /2 . MO /2 is the deposited outer layer, and O2 is oxygen in the SCW environment. and represent the valence of cations in the barrier layer and outer layer, respectively. ki is the rate constant of each reaction. MO /2 (d) is the dissolved or further oxidized outer layer oxide in the SCW. Note that Reaction 3, 10, 10' and 11 are non-conservative (they result in the movement of the interface) whereas the other Reactions are lattice conservative processes [30].

it fails to give a series of specific fundamental microscale reactions determining the growth/destruction of the outer layer; and it ignores the dependences of the relative importance of these reactions on the SCW density and the dependence of their rates on the oxygen content within SCW. Therefore, it is necessary to develop a new model for simultaneously considering the predominant microscale reactions in low/high density SCWs and the effect of oxygen content, accurately describing the “constant volume” growth of the outer layer and then finally completely describing the fundamental atomic-level growth of oxide scales.

exist [26,27,30]. No oxygen interstitials are included, because their energy of formation is sufficiently high so that their concentration is likely to be negligible at the temperatures of interest (T < 600 °C) [46]. The barrier layer exhibits p- or n- type electronic character, depending upon the relative concentrations of cation vacancies (electron acceptors, p-type), oxygen vacancies and cation interstitials (electron donors, n-type). A series of potential reactions occurring at the m/bl interface and the bl/e interface is given in Fig. 5, because it is at these locations that the defects are generated and/or annihilated. In this model, the method of partial charges that is used to derive the rate constants shows that Reactions 1, 2, 3, 4, 4′, 5, 5′, 6, 7, 8, 9, and 10′ are electrochemical processes whereas Reactions 10 and 11 are considered to be chemical processes. Reactions 4′, 5′, and 10′, to a certain degree, represent the one process group for the formation and growth of the outer layer by transformation (e.g., dissolution/re-precipitation) of the bl at the barrier layer/outer layer interface (bl/ol), probably occurring in high density SCW. Another process group describing the outer layer growth are the direct reactions of cations transmitted through the inner layer with species (O2 and/or H2O, the intrinsic role of the later is providing the oxygen by decomposition) in the environment, as given in Reactions 4, 5, and 10. Reaction 10 represents a possible conversion from the barrier layer to the outer layer. Reaction 11 describes the dissolution of the outer layer into the SCW environment, but under some conditions it may be further oxidized into a hematite phase [23,35]. Reactions 3 indicates that the growth of the barrier layer can be attributed to the generation of oxygen vacancies at the metal/film interface, of which the outward transport results in the inward transport (primarily by migration) of oxygen ions; meanwhile the generation and outward migration of metal cations either via a cation vacancy mechanism or as interstitials, as involved in Reactions 1 and 2, respectively, through the barrier layer leading to the growth of the porous, generally water-transmissible outer layer [15]. Relative to

2. Point Defect Model for the corrosion of steels in supercritical water As noted above, the growth of oxide scales on metals and alloys in oxygen-containing SCW is mechanistically-indistinguishable from the growth of passive films on metals in condensed aqueous systems and the growth of scales on metals and alloys in molten metals and in contact with dry oxygen [23]. Over the past about 40 years, one of the authors (DDM) has developed the Point Defect Model [26–30], which has proven to be remarkably successful in describing the growth of passive films and for extracting kinetic data for point defects. Accordingly, below, we extend the PDM to consider scale formation on steels in oxygen-containing supercritical water, concurrently considering both the “chemical oxidation, CO” in low density SCW and “electrochemical oxidation, EO” at high density. We identify this model as the SCW_Point Defect Model (SCW_PDM) to distinguish it from other point defect models. The reaction scheme for the model is depicted in Fig. 5. The origin of the coordinate system is the barrier layer/outer layer (bl/ol) interface and hence that the flux of oxygen vacancies is negative. In the barrier layer, high concentrations of point defects, such as vacancies on the metal and oxygen sub-lattices and metal interstitials 4

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Fig. 6. Postulated distribution of the potential across the barrier layer in the SCW_PDM.

Reactions 4 and 5, Reactions 4′ and 5′ may be more important in high density SCW, whereas, in turn, Reactions 4 and 5 are possibly more important in SCW of low density. It is worthy noting that the relative significance of three annihilation reactions of oxygen vacancies, i.e. Reactions 6, 7, and 8, will also change depending on the SCW density and whether oxygen exists in the environment. Reaction 8, which was included in previous PDMs employed in condensed environments [29,30], is included in SCW_PDM as protons have been demonstrated to exist in high density SCW (through the ability to measure pH [47,48]). While in low density SCW, the direct participation of O2 (Reaction 6, in O2–containing SCW) or that of H2O (Reaction 7, without oxygen or with a very low [O2]), will play a more important role. Reactions 4, 4′, 5, and 5′ contribute directly to the formation and thickening of the outer layer via reactions of the metal cations with the environment. Furthermore, the contribution from Reaction 10 to the growth of the outer layer may be presumed to be negligible, considering the fact that the barrier/outer layer interface generally coincides with the original metal/environment interface [13,32].

barrier layer. On the other hand, if the cathodic reactions occur at the bl/ol interface through the transport of O2 and H2O through the bl and no current flows to a remote cathode no potential correction is necessary and the theory as formulated here is correct. The theory behind the correction for the potential drop across the ol is described elsewhere and has been applied to sub-critical systems [49]. As in the classical PDM, the potential distribution is based upon a number of postulates:

• Potential drops exist at the metal/barrier layer interface ( ) and barrier layer/environment (SCW) (bl/e) interface ( ). • The electric field strength () in the barrier layer is a constant and is independent of the applied voltage. • The fraction of the applied potential (V) that appears across the bl/e m/bl

bl/e

interface is a function of applied voltage and environment pH, but is independent of the thickness of the barrier layer. Therefore, the potential drop at the bl/e interface can be expressed as[27]:

bl/e

=

V + pH+

(1)

0

So that

2.1. Potential drops across the interfaces

V= (

Following the PDM [26–30], the potential is envisioned to be distributed across the inter-phase as depicted in Fig. 6 with potential drops existing at the metal/barrier layer (m/bl) and the barrier layer/outer layer (bl/ol) interfaces. Note that, because the SCW and/or O2 penetrates the porous, non-protective outer layer (ol) in consideration of the high porosity and relatively wide grain boundaries of the columnar oxides (at least the penetration of SCW/O2 is not a rate-limiting step even though the porosity is not totally open) [24,25], the conductivity is high and minimal potential drop is envisioned to exist across the ol. Accordingly, we do not include the potential drop corrections across the outer layer when interpreting the electrochemical properties of the bl. It should be noted, however, that because SCW is a highly resistive medium and if the passive current flows through the ol to a remote cathode a potential drop must exist across the outer layer. That is to say, the static potential of the outer layer along the inner/outer layer interface, noted as bl/ e,ol , is not equal to the environmental static potential ( e ). In present case, a correction for the potential drop across the barrier layer to obtain the potential at the bl/ol interface must be applied, because it is this potential that controls the properties of the

m

–

e)

=

m/bl

+ Lbl +

bl/e

=

m/bl

+ Lbl +

V + pH+

0

(2) Accordingly, the potential drop across the m/bl interface becomes m/bl

= (1

) V

Lbl

pH

0

(3)

The parameter is the polarizability of the bl/e interface (the dependence of the voltage drop across the bl/e interface upon the applied potential V), is the dependence of the voltage drop across the bl/e interface on pH (general, < 0), and 0 is the potential drop across the solid metal/SCW interface in the absence of an applied voltage (V = 0), pH = 0, and in the absence of an oxide scale (Lbl = 0 ). According to a series of experimental observations and evaluations on the breakdown of passive films on steel and alloy surfaces in contact with aqueous solutions, the standard potential drop ( 0 , with V = 0 and Lbl = 0 ), can be defined as 0 = phi+ bet*pH , where phi is always considered as zero, and the common distribution of bet value is -0.01 V to -0.05 V in elevated temperature aqueous systems. 5

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2.2. Rate constants

the fact that the dependence of the potential drop across the bl/e interface on pH, , is commonly negative [29], it can be seen from Eq. (8) and definitions of a10′, b10′, and c10′ given in Table 1 that rate constant k10' decreases with increasing pH value. The value of the rate constant is affected significantly by the magnitude of G0,# . Because SCW has a very low dielectric constant, entities with a high charge tend to have high electrochemical potentials [µ i = µ 0i + RTln(ai) + zi F ], where ai is the species activity, µ 0i is the standard chemical potential, and is the electrostatic potential, and are less stable than species of low charge. Thus, those reactions that generate significant charge, relative to the initial state, in the activation process, particularly those transition states that have components exposed to the environment will tend to be disfavored as the dielectric constant decreases due to increasing temperature and/or decreasing density. Consider for example the simple metal dissolution reaction, ' # + m [(1 Ms + + e'm , where subscripts “m” *) m+ *Ms + *e m] and “s” signify the metal and solution phases, respectively, and noting that the electrons remain in the metal and hence are not exposed to the SCW, it is evident that G0,# will increase (the rate constant will be reduced) as the dielectric constant decreases, because one component of the TS (Ms +) is (partly, at least) exposed to the low dielectric constant medium (SCW). Contrariwise, consider the reaction,

The rate constants of the above mentioned a series of interfacial reaction (1-11) are defined from Activated Complex Theory (ACT), also called Transition State Theory (TST), which relates the rate of a specific reaction to a quasi-equilibrium between the reactants and an activated complex (transition state). Thus, this quasi-equilibrium is envisioned to be as follows:

A+ B

[(1

*)

A+ (1

*) B+

# *C]

(4)

C

where * is the “transfer coefficient” that describes the point along the reaction coordinate (the path between the reactants (A + B) and the product (C)), at which the transition state (TS) occurs. Note that * = 0 describes the initial state and * = 1 corresponds to the final state, and the TS is indicated by the superscript hash mark (#). From the quasiequilibrium, the concentration of the TS can be written as: G0,# /RT [A][B]

[TS]# = K[A][B] = e

(5)

where K is the quasi-equilibrium constant and is the change in standard Gibbs energy of activation. ACT envisions that there exists a mode of vibration in the TS that leads to the formation of the product (C) from those reactants that have sufficient energy to mount the barrier and that the frequency of this vibration is kBT/h, where kB is the Boltzmann's constant (1.381 × 10-23 J∙K-1), h is Planck's constant (6.626 × 10-34 J∙s), and T is the Kelvin temperature. Thus, the rate of the reaction becomes:

G0,#

kBT [TS]# = h

R=

kBT e h

G0,# /RT [A][B]

V O + H2 O+ 2e'

kBT e h

G0,# /RT

(6)

Using this definition of the rate constant, the authors employ the concept of partial charges to apply the ACT to determine which of the reactions listed in Fig. 5 are potential-dependent. This is done by noting that i , the “transfer coefficient”, also describes the extent of charge development in the transition state compared with that in the initial state for a specific reaction involving charge transfer [51]. For the rate constants of the 14 reactions shown in Fig. 5 (ki), they can be defined in the following general form, of which the detailed derivation processes are available in Appendix A.

k 00 i

=

' k 00 i exp

i

0 GR,i 1 R T

1 T0

*)H2

O+ 2(1

' *)e m

+

*OO

+

# *H2]

OO + H2,s

(7)

ki = k 0i exp(aiV )exp(bi Lbl )exp(c i pH)

*)V O +(1

there is a decrease in the activation process of those species, so that G0,# will be lowered and the rate constant is expected to increase with increasing temperature and decreasing density (decreasing dielectric constant). Note that for G0,# =50 kJ∙mol-1at 500 °C, a halving or doubling of the activation Gibbs energy will change the rate constant by factors of 2.2 × 10-2 and 1.84 × 10-7, respectively, over the base value of 4.3 × 10-4, all other factors being equal. Clearly, the rate constant is a sensitive function of G0,# , which in turn is a sensitive function of the extent of charge development in the activation process. The relative significance of all reactions employed in SCW_PDM in various SCW is available in last column of Table 1. As noted above, the point defects in the barrier layer are generated or annihilated by the interfacial reactions shown in Fig. 5, with the concentrations of three types of defects, i.e., cation and oxygen vacancies and cation interstitials, being determined by the kinetics of the reactions, as reflected in the values for the standard rate constants, the transfer coefficients, and the kinetic orders of the reactants. In the steady state, the rates of the reactions at the two interfaces for any given defect must be equal. Thus,

where is the probability that the TS will decompose into Product C. To our knowledge no method is currently available for accurately calculating , but G0,# may be estimated using Density Functional Theory (DFT) [50]. Accordingly, the rate constant is now written as:

k=

[(1

(8)

n k1CLVM = k 4CO + k 4'

(10)

k2 = k5COn C0Mi + k5'C0Mi

(11)

and

(9)

2

where is the standard change of chemical Gibbs free energy of Reaction i (i = 1, 2, 3, 4, 4′, 5, 5′, 6, 7, 8, 9, 10, 10′, and 11) at reference 0 GR,i

p p m 0 k3 = k 6CO C VO + k7CH C0 + k 8CH C0 2 O VO 2 O VO

(12)

where is the concentration of cation vacancies at the m/bl interface, and C0Mi , and C0VO are the concentrations of cation interstitials and oxygen vacancies at the bl/ol (bl/e) interface. The calculation method for concentrations of cation vacancies and metal interstitials will be presented in another paper related to SCW_PDM [53]. CO and CH2O (mol∙cm-3) are the concentrations of oxygen and water, respectively, in the SCW. m is the kinetic order of Reaction 6 with respect to the concentration of oxygen in the supercritical water. It is assumed that the kinetic orders of Reactions 4 and 5 are the same (n), and that for Reactions 7, 8, and 9 the kinetic order with respect to the volumetric concentration of water all equate to p. Because molecular collision frequency is governed by spatial density of reactants (molar concentration), here CO and CH2O are defined as the 0 relative volumetric concentrations (CO and C0H2O = 1 mol∙L-1 of SCW), which are density-dependent, as follows.

CLVM

'

temperature (T0), k i00 is the base standard rate constant of the ith elementary interfacial reaction. ai, bi, and ci are intermediate variables, k 0i is standard rate constant, and k 00 i represents base rate constant. Their definitions are list in Tables 1 and 2. From Table 1, we can see that the rate constants for Reactions 4, 5, 6, 7, 10, and 11 are envisioned to be potential-independent, because there is no compensation between the effects of V and Lbl (note that the coefficient “bi” for these reactions is zero). In low density SCW, the importance of Reaction 10′, is expected to decrease markedly due to low concentration of H+. The lower density of SCW is generally accompanied by the higher pH value, i.e., the lower H+ activity. For neutral SCW, with a density of ∼ 0.1 g∙cm-3, the pH is approximately equal to 11 [52] (vs 7 for pure, condensed water at 25 °C). Considering 6

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Table 1 Coefficients for the rate constants of the reactions in SCW_PDM and their importance in low/high density SCW. ki = k 0i exp(ai V)exp(bi Lbl )exp(c i pH) , k 0i : standard rate constant (see Table 2), = F/(RT) , F: Faraday’s constant, R: molar gas constant, T: temperature in Kelvin. ai (V-1)

Reactions

k1

(1) m+ VM' k2

(2) m Mi

+

MM + vm + e'

+ vm + e'

k3

(3) m MM + V O + e' 2

(4) MM + O2 + e'

k4

VM'+MO

4

(4′) MM

'

k 4'

VM + M

(5) Mi + + O2 + e' (5′) Mi

4 + k5'

+

M

+

(7) V O + H2 O+ 2e

(9) H2 O+2e (10) MO

(11) MO

+

/2

OO

OO + H2

OO + 2H+

2

4

k9

MO

k10' + H+ M

( )O

2

4

k11

1

1

✓✓

✓✓

)

2

2

✓✓

✓✓

3

(1

)

3

✓✓

✓✓

✓✓

✓

3

0 0

+ H2 O+ ( 2

)e'

5'

✓

✓✓

✓✓

✓

✓

✓✓

0

0

0

✓✓

✓✓

0

0

0

✓✓

✓

2

8

✓

✓✓

2

9

2

8

2

9

10' (

0

0

)

10' (

0

/2 (d)

MO

4'

0

0

0

/2

0

0

5'

1/2H2+OH

+

/2

' k7

( )O

+

/2

(10′) MO

' k9

k8

' k6

)

(1

4'

/2

High density

(1

0

)e'

+(

(6) V O + 1/2O2 + 2e

(8) V O + H2 O

MO

Significance in SCW Low density1

2

0

/2

ci

1

)e'

+(

k5

bi (cm-1)

0

)

✓

✓✓

✓

✓

✓

✓✓

2

0

✓✓

✓

Note: 1Here, the critical density between low and high can be defined as 0.1g∙cm-3 [11]. 2 The importance of Reaction 11 becomes marked when the second oxidation of oxides at the outer layer surface occurs.

Table 2

'

Definition of standard rate constants, k 0i , for a series of interfacial reactions employed in the SCW_PDM. T0: reference temperature, k 00 i : base standard rate constant, 0 k 00 : change in standard chemical Gibbs free energy. GR,i i : base rate constant, and Reactions

k0i k1

(1) m+ VM'

MM + vm + e'

k2

(2) m Mi + + vm + e' k3

(3) m MM + V O + e' 2

(4) MM + O2 + e'

k4

4

(4′) MM

'

k 4'

VM + M

(5) Mi + + O2 + e'

+

k5

MO

k5'

M

+

k6

(7) V O + H2 O+ 2e' (8) V O + H2 O

k8

k7

(10′) MO (11) MO

+

/2

/2

+ H+

+

k10'

( )O 4

k9

M

2

0

k300 exp

3 F RT

0

OO

k 600

MO +

k11

+ H2 O+ ( 2

MO

/2 (d)

)e'

'

2

'

4

0

'

5' F RT

0

'

0 5' GR,5' R

'

6

1 T0

1 T

1 T0

1 T

1 T0

'

0 8 GR,8 R

1 T

1 T0

1 T

1 T0

2 9F 0 RT

k 00 9 exp

9

00 k10 exp

10

G0R,9 R

0 GR,10 R

1 T

1 T0

1 T0

'

0 10' GR,10' R

1 T

'

0 11 GR,11 R

1 T

00 k10' exp

00 k11 exp

7

1 T

1 T0

1 T0

k 00 9 exp

00 k11

1 T0

1 T

k800 exp

)F 0

1 T

0 7 GR,7 R

2 8F 0 RT

RT

1 T0

'

k00 7 exp

10'(

1 T

G0R,6 R

k800exp

00 k10' exp

1 T0

1 T

0 5 GR,5 R

k500 1exp

k 600 exp

1 T

0 4' GR,4' R

'

k 00 4 1exp

k500 exp

1 T0

0 GR,4 R

) ) ) ) ) ) ) ) ) ) ) ) ) )

1 T

G0R,2 R

0 3 GR,3 R

k 00 4 exp 4' F RT

( ( ( ( ( ( ( ( ( ( ( ( ( (

'

k300 exp

00 k10

/2

0 1 GR,1 R

k00 2 exp

k700

OO + H2

2

4

2 F RT

00 k5' exp

OO + 2H+

( )O

k200 exp

'

k100 exp

k500

/2

k9

/2

0

k 00 4' exp

(9) H2 O+2e' 1/2H2+OH (10) MO

1 F RT

)e'

+(

(6) V O + 1/2O2 + 2e'

k100 exp

k 00 4

/2

)e'

+(

4

(5′) Mi +

VM'+MO

Units

k00 i

1 T0

cm s mol cm2 s mol cm2 s mol cm2 s mol cm2 s

cm s cm s cm s cm s cm s mol cm2 s mol cm2 s

cm s mol cm2 s

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Y. Li, et al.

CH2O =

CO =

1000 WMH2O C0H2O

[O2] × 10 WMO2 CO0

layer is predicted to vary linearly with the voltage applied at the bl/e interface. Note that the rate constants for Reactions 10 and 10ʹ do not depend upon the potential or upon the thickness of the barrier layer (Table 1), so that the last term in Eq. (20) is a constant. Returning now to Eq. (16), after substitution for k3 [Eq. (8)], we obtain for a constant applied voltage V and dissolution rate Cbl

(13)

3

(14)

where WMH2O and WMO2 are the molecular weights of H2O and O2 in units of g∙mol-1, [O2 ] is the concentration of oxygen (ppm) in water at ambient temperature, and is the water density under the conditions of interest in units of kg∙L-1. Since supercritical aqueous solutions are generally dilute, because of low solubility of solutes, the water density was taken as that for pure water from the NIST steam data [54].

dLbl = (Aexp(b3 Lbl) dt

A=

(

(15)

Lbl (t) = (L0bl

where '

A =

Generally speaking, the smallest units of oxides within the barrier layer are [MM (OO) /2], [MM (VO) /2], and [VM (OO) /2], because the vacancies on the sub-lattices can be considered to be real species. The reactions involving at least one of these smallest units that results in the movement of the original phase interfaces (m/bl, bl/ol, or ol/e interface) are labeled “lattice non-conservative”. Other interfacial reactions are labeled “lattice conservative”, which do not result in the changes of any above interfaces. For the barrier layer, only Reactions 3, 10, and 10′ as shown in Fig. 5, are lattice non-conservative with Reaction 3 resulting in bl formation at the m/bl interface and Reactions 10 and 10′ resulting in bl destruction at the bl/e interface. Accordingly, the net rate of change of the thickness of the bl is written as:

)

s 0 [k10COq + k10' CH+/CH + ]

)s

k3 = k10COq + k10' CH+/C0H+

dL ol = dt

(17)

Substituting for k3 yields

)s

q 0 k30exp(a3V)exp(b3Lbl,ss )exp(c3pH) = k10CO + k10' CH+/CH +

q k10CO + k10' (CH+/C0H+)s

k30

(19)

and hence

1

V

pH

1 3

ln

Cbl t)

0,bl / bl

L0bl

is initial thickness of the barrier layer and

k30exp[a3 (V+

(25)

V)]exp(c3 pH)

Lbl ol [k1C VM

(

)s

q + k2 + k10 CO + k10' CH+/C0H+

r k11CO ]

0,ol / ol

(26)

3.3. Weight gain of steels in low density SCW

k10COq + k10' (CH+/C0H+)s k30

1)

Here, ol is the volume of the oxide per mole of metal ion in MO /2 and is a constant, r is the reaction order of the destruction of the ol at the ol/ e interface (Reaction 11, Fig. 5) with respect to the oxygen concentration in the SCW, and 0,ol and ol are the theoretical- and actualdensity of the outer layer, respectively. Generally, 0,ol > ol due to the unavoidable porosity in the formed outer layer. The porosity develops in the outer layer due to the volume mismatch and the corresponding stresses generated in that layer.

(18)

or

Lbl,ss =

b3Cbl t

The outer layer is envisioned to form via the reaction of the metal cations transmitted through the bl with oxygen or water in the environment or by transformation (e.g., dissolution/re-precipitation) of the bl at the bl/e interface. The cations are transmitted via Reactions 1 & 4 and 2 & 5 in low density SCW, or predominantly by Reactions 1 & 4′ and 2 & 5′ in high density SCW, while transformation is described by Reactions 10 and 10′. Reaction 11 describes the destruction of the outer layer (ol) via dissolution into the supercritical water environment or by being oxidized further to other phases. For the corrosion of steels in SCW, the outward transport of cations through the barrier layer is the rate-limiting step for the outward growth of the outer layer [38,39]. Although Reactions 4, 4′, 5, and 5′ directly results in the growth of the outer layer, the kinetics of Reactions 1 and 2 control the supply of cations transmitted through the bl to the ol, and thus determines the growth rate of the outer layer provide the cations for the outer layer.

where stands for the molar volume of the barrier layer per cation in MO /2 , q is the reaction order of the bl destruction at the bl/ol interface (Reaction 10, Fig. 5) with respect to the oxygen concentration in the SCW, and the other quantities are as defined above. At steady-state, dL/ dt = 0, so that:

a3 V+ b3Lbl,ss + c3 pH= ln

1 A ' b3L0 ln 1 + e bl (e b3 Cbl

3.2. Formation of the outer layer

(16)

(

(23)

(24)

3.1. Steady-state thickness of the barrier layer

(

)s

q k 9 CO + k10' CH+/C0H+

3. Atomic-level characteristics of oxide scales

(

(22)

with the definitions of a3, b3 , and c3 shown in Table 1. The quantity, 0,bl / bl , represents the theoretical/actual density of the barrier layer, in consideration of possible presence of a small amount of pores. But these pores are unlikely to extend through the barrier layer, because if water reached the metal it would react to create a barrier layer within the pores of a thickness that corresponds to the prevailing conditions. Equation (21) is readily integrated [55] in response to a step in the applied voltage from V0 to ΔV + V0 (to a certain degree, ΔV corresponding to a step in [O]), to yield the thickness of the barrier layer as a function of time,

At one particular potential (V = VOCP), ipol = 0, and the steady-state thickness of the barrier layer is LOCP. VOCP is referred to as the “open circuit potential”.

k3

(21)

k30exp(a3V)exp(c3 pH)

Cbl =

As shown in Fig. 5, Reactions 1, 2, 3 produce electron currents while Reactions 4, 5, 6, 7, and 9 consume electron current. For Reactions 4′, ". The partial current 5′, and 10′, they depend on the sign of " provided by each Reaction is available in Appendix B. Under active polarization, i.e., the action of an applied potential, V, the observed current is:

dLbl = dt

0,bl / bl

where

2.3. Interfacial Currents

ipol = i1 + i2 + i3 + i 4 + i 4' + i5 + i5' + i 6 + i7 + i 9 + i10'

Cbl)

(20)

The corrosion behavior of steels exposed to SCW at low density below ∼0.1 g∙cm-3 is similar to that in gaseous environment, where

As with the classical PDM, the steady-state thickness of the barrier 8

Corrosion Science xxx (xxxx) xxxx

Y. Li, et al.

where m is the molar volume of the metal. Likewise, the rate of growth of the barrier layer into the metal is given by:

Table 3 Theoretically-calculated PBR values of oxides of interest. Oxides

FeO

Fe3O4

Fe2O3

FeCr2O4

Cr2O3

NiO

NiCr2O4

PBR Oxides PBR

1.76 NiFe2O4 2.13

2.10 Al2O3 1.29

2.15 SiO2 1.88

2.05 CuO 1.78

2.01 Cu2O 1.68

1.70 TiO2 1.78

2.06 Ti2O3 1.51

dLbl = dt

dL m = dt

x

k10COq +

2

k11COr

bl

dLbl + rol dt

ol

dL ol dt

k2 = ( PBR

(27)

dLbl = dt

(28)

+ k2 + k3)

(31)

q k10CO

(

)

k10' CH+/C0H+ s]

(32)

(33)

1) k3

bl k3

(34)

Actually, for the oxide scales formed on F-M steels in SCW, the bl is n-type and the dissolution of the outer layer is negligible in low-density SCW, so the rate of growth of the outer layer can be expressed as:

dL ol = dt

ol k2

(35)

The thickness ratio between the outer layer and the barrier layer is

T Ratio =

dL ol dLbl / = dt dt

ol k2

(36)

bl k3

Substitution for k2 in Eq. (33), Eq. (36) becomes

TRatio

Examination of Figs. 1 and 2 shows that as the barrier layer grows into the metal the bl/ol interface remains at the location of the original metal surface. This is a ubiquitous observation that has been noted in corrosion studies on iron in high temperature water [31] and in the corrosion of steels in supercritical water [13]. Because the metal and barrier layer oxide have different molar volumes per metal atom, this constant volume growth implies that a definite relationship exists between the flux of cations outward through the barrier layer and the outward flux of oxygen vacancies (inward flux of O2-), which in turn implies a definitive relationship between the rates of the reactions occurring at the m/bl interface. To our knowledge, this constraint has never been imposed or explained in the theory of oxidation. The rate of destruction of the metal due to the generation of cations (including interstitials) due to Reactions 1 and 2, Fig. 5, can be expressed as:

+ k2)

(30)

where PBR = bl / m is the Pilling-Bedworth ratio. Theoretical PBR values of typical oxides are given in Table 3. This provides an important constraint on the values of the rate constants for Reactions 2 and 3 that has hitherto been unrecognized. An additional constraint is that in the steady-state, the rates of Reactions 3 and 6, which account for the generation and annihilation of oxygen vacancies, must be equal, that is m k3 = k 6C0Vo CO in O-containing SCW, but so are the rates of Reactions 3 q and 10 in low density SCW, i.e., k3 = k10CO . These constraints are obviously important in the optimization of the model on experimental data. According to Eq.(16), when neglecting the conversion from the barrier layer to the outer layer, the growth rate of the barrier layer is given by:

3.4. Constant volume growth of the barrier layer

L m (k1C VM

)

If the bl is n-type and the bl does not convert to the ol, then q k1CLVM k2 and k10CO + k10' (CH+/C0H+)s = 0 , in which case we have

Here rbl and rol are the relative mass percentage of oxygen in the barrierand outer- layers, respectively. The obtained SCW_PDM kinetics equation in weight gains, i.e., Eq. (28), derives from the intrinsic understanding on the oxygen combined with ionized metals resulting in the weigh gains; it thus has nothing to do with the morphology of the outer layer. Despite that the actual morphology of the outer later is general not uniform, possibly being very blocky with regions of thick oxide and some very thin especially at early oxidation stage [39,58], the obtained SCW_PDM kinetics equation in weight gains is always effective in the whole oxidation process.

dL m = dt

L m (k1C VM

k1CLVM + k2 + k3 = PBR[k3

where MWO is the molar weight of formed oxides per oxygen atom and the Reaction 11 is considered to be possible second oxidation of oxides at the outer layer surface. In the terms of dLol and dLbl , the rate of the weight gain can also be dt dt given by:

d w = rbl dt

(

k10' CH+/C0H+ s]

For the rate of barrier layer growth to exactly match the rate of dL dL removal of the metal, dtm = dtbl , which yields:

d w p n m 0 = MWO k 4 CO + k COn C0Mi + k 6CO C VO + k7CH C0 2 O VO dt 2 2 5 2

k10COq

However, Reaction 3 not only leads to the growth of the barrier layer, but also can be seen a process that destroys the metal, because it converts the metal into the metal oxides. Accordingly, the total destruction rate of the metal should be Eq. (31), rather than Eq. (29)

only solid growth occurs without any metallic dissolution [56]. At least, we can state that the dissolution of the oxide layer into SCW at low density is negligible, if any [38]. Chouhry et al [16] reported that the released metal were only a small fraction (approximate 1/500) of the total amount of oxide formed on alloy 800H in SCW at 650 °C. For P92 after an exposure in SCW at 500 °C, Yin et al.[57] reported that the calculated amount of oxygen absorbed in the scale was 2.89 mg, very close to the measured weight gain (3 mg). In a whole, we can assume that the weight gains of the samples as a whole (metal plus scale) is predominantly induced by oxygen incorporated in the oxide scales, i.e., all the reactions involving the oxygen capture from O2, H2O, and/or aqueous ions such as OH- should be responsible for the weight gain. Neglecting the relatively lesser interfacial reactions in O-containing SCW at low density, i.e., Reactions 4′, 5′, 8, 9, and 10′, we obtain,

+

bl [k3

dL ol dLbl ( PBR / = dt dt

1) bl

ol

(37)

As for the oxide scales formed on F-M steels, the predominant component of the outer layer is magnetite, while the barrier layer consists of Fe-Cr spinels. The PBR of magnetite is 2.1 and ol is generally similar to or slight higher than bl , because the outer layer is generally porous, thus the calculated outer/barrier layer thickness ratio from Eq. (33) is likely 1 ∼ 1.5, satisfying the experimental phenomenon shown in Figs. 1, 2, and 4. We posit that the constant volume growth constraint of the barrier layer is a consequence of the stresses induced in the bl and metal substrate, as a consequence of the PBR > 1 resulting in a large volume mismatch between the consumed metal and the formed oxides (large growth strain) [59,60]. Thus, the formation of the bl induces a tensile stress in the substrate metal and a compressive stress in the oxide layer. These stresses affect the kinetics of the reactions at both the m/bl and the bl/e interfaces. The impact of stress (or pressure) on the rate constant of a reaction can be expressed as [8]:

(29) 9

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Y. Li, et al.

ln(ki)

=

V 0,# RT

data set, with the aim of developing an atomic-scale, deterministic method for the corrosion kinetics for metals and alloys in low-density SCW. The thickness of the barrier layer as a function of time, is given by Eq. (24) which is reproduced here as Eq. (39)

(38)

where V 0,# is the standard volume of activation and is equal to ¯ 0TS V ¯ 0R , where V ¯ 0TS is the partial molal volume of the TS and V 0,# = V ¯ R0 is the partial molal volume of a reactant with the summation being V taken over all reactants. For the corrosion of Type 304 SS at 350 °C, V 0,# < 0, which is common for reactions in condensed aqueous systems, because of the electrostriction of solvent (water) by the developing charge in the TS. However, at 500 °C for the same system, V 0,# > 0 and decreases with increasing density (pressure) of the environment [8], reflecting the lower role in electrostriction (because of the lower dielectric constant of SCW) in determining the volume change

Lbl (t) = (L0bl

1 A bl b3L0 ln 1 + e bl (e b3 Cbl

b3Cbl t

1)

Cbl t)

0,bl / bl

(39) Note that Abl that equates to A' in Eq. (25) and Cbl are as given previously. Despite the possible presence of small pores in the barrier layer, the linear proportionality between the weight gain and the oxide thickness for P92 exposed to SCW at 500-600 °C and 25 MPa indicates that the scale density is nearly constant [56], implying a constant bl , regardless of the exposure time. In low density SCW, the activities of cations in SCW are negligible, for simplicity, herein we neglect the roles of Reaction 10′, of which the importance is marked only in condensed aqueous systems with a high density. Thus, Eq. (26) for the net growth rate of the outer layer becomes

k2

in the activation process. Thus, for Reaction 2 (m Mi + + vm + e'), V 0,# is expected to be small and positive, corresponding to a small expansion of the lattice to accommodate Mi +, so that the interfacial stress is likely to have only a small, negative impact on the rate constant V 0,# for Reaction 3, of that reaction. On the other hand, k3

m MM + 2 V O + e', is expected to be large and positive, because of the creation of the requisite number of oxygen vacancies, so that the rate constant should be reduced substantially by the compressive stress on the oxide side of the m/bl interface. Accordingly, Reaction 3 is preferentially inhibited with respect to Reaction 2, to the extent that the bl forms in a constant volume process. As for the stress-oxidation interaction in selective oxidation of Cr-Fe alloys, Zhou et al. [59] reported that the compressive stress in the oxide scale significantly slows down the oxidation rate of the metal substrate and has a rather nonlinear distribution across the layer thickness with its maximum at the metal/ scale interface. The stress in the oxide scales can also influence the diffusion coefficients of different point defects and their concentration ratio [61]. However, Yuan et al. [62] reported that the relaxation of the comprehensive stresses generated from the solid-state transformation at the Fe3O4/Fe2O3 interface resulted in the spontaneous formation of Fe2O3 nanowires on the outmost Fe2O3 layer. A more comprehensive analysis on effects of produced stress on the scale growth will be presented in a later paper.

dL ol = dt

Lbl ol [k1C VM

q + k2 + k10 CO

r k11CO ]

0,ol / ol

(40)

Substitution of Eq. (32) into Eq. (40), we have:

dL ol = dt

ol [(PBR

1)k3

PBRk10COq + k10 COq

r k11CO ]

0,ol / ol

(41)

Integrating throughout with respect to exposure time, t, assuming the constant density ( ol ) of the outer layer during growth, we obtain the formula for integration:

Lol (t) =

A ol ln[Abl e b3Lbl (t) A blb3

Cbl]

Col t+ Dol

(42)

where

A ol =

4. Rate laws and determinations

Col =

ol (

PBR

ol [(PBR

1)k30exp(a3V)exp(c3pH) 1)k10 COq

+

0,ol / ol

r k11CO ] 0,ol / ol

(43) (44)

and

Almost any metal or alloy exposed to high temperature environments is always more-or-less covered by loose or compact-protective oxide films. The latter generally occurs on F-M steels, austenitic stainless steels, and nickel-based alloys in SCW and, to a certain degree, they isolate the underlying substrate from the environments, leading to a reduced rate of oxidation. The classical Wagner’s theory proposed that the growth of protective oxide scales follows the parabolic kinetics [63]. At the phenomenological level, this cation-diffusion model relates the growth rate of oxide scales to experimentally measurable transport properties of the oxide, such as the diffusion coefficient, so cation diffusion models are still widely used, somewhat uncritically, despite of their failure to account for some fundamental quantities (e.g., the “jump distance”), and the behavior at limiting small times (the oxidation rate must be finite) and limitingly long times (the rate eventually reaches a steady-state). The above developed SCW_PDM, at the atomic level, illustrates the physico-chemical basis for the growth of oxide scales formed on the metals and alloys in SCW. However, before using it to explain the microscopic corrosion process and oxide properties, and to predict the potential corrosion characteristics of materials after a long-term exposure, it is necessary to obtain a set of the fundamental parameters within the SCW_PDM, such as the rate constants and the transfer coefficients of Reactions (1)-(11). These PDM parameters were ordinarily obtained by optimizing the impedance expression for the PDM on experimental impedance data [64,65]. Now that thickness data of oxide scales has been acquired from physical measurements, we will optimize the SCW_PDM on L(t) to determine the fundamental parameters for the

Dol =

A ol 0 ln[Able b3Lbl Abl b3

Cbl] + ( PBR

1)L0bl

(45)

4.1. Insight into fundamental parameters in SCW_PDM These deterministic equations for the thicknesses of the barrier- and outer- layers are employed below to interpret the kinetics data by optimizing the simplified version of SCW_PDM (only Reactions of 1, 2, 3, 4, 5, 6, 7, 10, and 11 are considered) on the experimentally measured thickness data of oxide scales. Herein, we use the genetically-inspired algorithm, describing biological evolution, IGOR-Pro, to search the optimal set of fundamental parameters [66,67]. Based on three main rules of selection, crossover, and mutation, the genetic algorithm can resolve various issues involving both constrained and unconstrained optimization problems. The optimization program is developed and adapted from the custom software used to extract the key parameters in electrochemical impedance models [68]. The optimization procedure aims to minimize the error between the experimental thickness data and the calculated result based on the extracted parameter values at the same exposure time. During corrosion process, the alloys are oxidized as if under the “open circuit potential”, then the observed polarization current is zero, i.e.,

ipol = i1 + i2 + i3 + i 4 + i5 + i6 + i7 + i10 + i11 = 0

(46)

Thus, the whole optimization procedure is subject to the above 10

Corrosion Science xxx (xxxx) xxxx

Y. Li, et al.

Fig. 7. Comparisons between the experimentally-measured thicknesses and simulated data from (a) and (c) the optimized SCW_PDM for HCM12A F-M steel [13,37,38], (b) and (d) curve fitting for 316 L stainless steel at 500 °C [70].

current constraint. Typical experimental thicknesses of the barrier- and outer- layers of oxide scales formed on HCM12A steel in SCW at 500 °C (see Fig. 4) are again depicted in Fig. 7, in order to compare with the calculated results after optimization of the SCW_PDM on the experimental data to extracting various fundamental model parameters, such as the transfer coefficients ( i ) and the base rate constant (k 00 i ) of the i elementary interfacial reactions. For simplicity, all of the kinetics orders with respect to oxygen content CO are assumed to one half (n = m = r = q =1/2), depending on the previously experimental results from carbon steel exposed to supercritical aqueous systems [69]. During the optimization, the parameters such as temperature (T), oxygen content (DO), pH, the dependence of potential drop at the bl/e interface on pH ( ), and the polarizability of the barrier layer/SCWs interface ( ), are kept constant, because they are known, or assumed in terms of the available literature and previous optimization work in condensed solutions. Optimized values of the fundamental parameters in SCW_PDM for HCM12A steel in SCW at 500 °C are given in Table 4. Good consistency between the experimental results and the thickness data calculated upon the parameter values determined by optimization is observed, as shown in Fig. 7 (a). Fig. 7(a) and (c), to a certain degree, at least at the phenomenal level, already demonstrates the effectiveness of the above optimized fundamental parameters, upon on which the calculated thickness matches the experimental data well. In the following, we will pay more attention to these fundamental parameters to demonstrate that they are physically reasonable. Rate constant k2 reflects the molar growth rate of oxides (MO /2 ) in the outer layer because it determines the transport rate of cations transmitted through the barrier layer outward to form the MO /2 , while the growth rate of the barrier layer is directly determined by k3. Therefore, we can state that the sum of k2 and k3 can effectively embody the total oxidation rate constant of metal atoms,

Table 4 Values of fundamental parameters in the SCW_PDM optimized upon on the experimentally-measured thicknesses of the barrier- and outer layers for 9Cr steel in SCW at 500 °C. Items

Names

Values

Descriptions

T (oC) DO (ppm) pH

Temperate Oxygen content Negative log of H+ Polarizability of bl/ol interface Transfer coefficient Transfer coefficient Transfer coefficient Transfer coefficient Base rate constant

500 0.025 11 0.78

Known Known Evaluated [4] Assumed [26]

0.11 0.12 0.16 0.15 8.93 × 10-12

Optimized Optimized Optimized Optimized Optimized

2

3 9 10 k200 (mol∙cm-2∙s-1) k300 (mol∙cm-2∙s-1) 00 k10 (mol∙cm-2∙s-1) 00 (mol∙cm-2∙s-1) k11 00 q k10 CO (mol∙cm-2∙s-1)

k2 (mol∙cm-2∙s-1) k3 (mol∙cm-2∙s-1)

Epsilon (volt∙cm-1) -3 bl (g∙cm ) ol

(g∙cm-3)

PBR n, m, q, r

Base rate constant Base rate constant Base rate constant

8.00 × 10-12 8.59 × 10-10 3.08 × 10-10

Optimized Optimized Optimized

-

2.40 × 10-13

Averaged rate constant Averaged rate constant Dependence of potential drop at the bl/e interface on pH Field strength Average density of barrier layer Average density of outer layer Average Pilling–Bedworth ratio Kinetics orders with respect to CO (oxygen molar concentration)

1.04 × 10-11 9.32 × 10-12 −0.005

Calculated Calculated Assumed [65]

1.12 × 102 5.01

Optimized Optimized

4.99

Optimized

2.16

Optimized

0.5

Assumed [69]

Optimized

11

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Y. Li, et al.

which is found to be 1.97 × 10-11 mol∙cm-2∙s-1. This value is reasonably consistent with the reported rate constant of oxidation kinetics of F-M steels in SCW at ∼500 °C [37]. An exponential kinetic equation was employed by Tan et al. [37] to fit the experimentally measured weight gains of HCM12A exposed to SCW at 500 °C, extracting a rate constant of 8.31 mg dm-2 h-0.497. All of the oxides within the barrier- and outer layers can be assumed to be in a general formula M3O4 and they are assigned the same molecular mass as magnetite (231.5 g mol-1), which is reasonable, because the predominant components of the barrier- and outer layers, respectively, are Fe3-xCrxO4 and Fe3O4 and the molecular mass of FeCr2O4 (223.8 g mol-1) is close to that of Fe3O4. Therefore, the weight gain rate constant reported by Tan et al can be converted to be an averaged molar rate constant of ∼2.17 × 10-11 mol∙cm-2∙s-1, i.e., oxidation rate constant of the metal atoms. The average density of the outer layer is smaller than the density of magnetite (5.15 g∙cm-3) being almost as the only component of the outer layer, reflecting the porosity of the outer layer which have been reported in a few published papers [24,70,71]. Additionally, the larger PBR (2.16) than the theoretical values (2.05 of FeCr2O4, 2.10 of Fe3O4, see Table 3), to a certain degree, further implies the presence of pores within the outer layer of the oxide scales. The obtained epsilon is smaller than common values (∼106 V∙cm-1) within the films formed on steels in condensed aqueous systems, which derives from that increasing temperature promotes electrons from the valence band to the conduction band, with the result that the bl becomes electronically conductive and the high electric field is suppressed.

Fig. 8. Long-term predictions on the thicknesses of oxide scales formed on HCM12A steel and 316 L stainless steel in SCW at 500 °C.

fundamental parameters from SCW_PDM, as shown in Table 4. Whenever the alloys are exposed to atmospheric environments or the corrosive medium before reaching the desired exposure conditions, they automatically form thin layers of oxides, which may be the basic cause for the presence of initial thickness, i.e., L0bl . > 0. Higher chromium content in the substrate should be beneficial for forming more thin, compact oxide scales, so a smaller initial thickness should be observed on stainless steel 316 L (1.01 nm) than on HCM12A steel (0.49 μm) at same temperature. However, between the two steels at same temperature, there are negligible differences in the values of kinetics parameters, except for b3. The b3 is negative and represents the dependence of the growth rate of the barrier layer on the thickness. The smaller the absolute value of b3 is, stronger is the dependence on the thickness of the barrier layer. That is to say, the growth rate of the oxide scale decreases more quickly with the increasing thickness as the exposure time increases. The composition and structure of alloys likely play a significant role in determining the value of b3. Compared to HCM12A, stainless steel 316 L displays a smaller |b3|, implying the better oxidation resistance of the latter due to its higher Cr content in the substrate. Based on the obtained kinetics parameters for HCM12A and 316 L steels at 500 °C, the thickness data were predicted after long-term oxidation of up to 10 years, as shown in Fig. 8. In this case, exfoliation of the oxide scale does not occur, because of the smaller thickness values. If the scale is sufficiently thick, the growth strain stored in the scales will become greater than that is required for fracture and the scale will crack or partially spall [72,73]. However, a research project performed by Oak Ridge National Lab indicated that, generally, the critical total thickness of oxide scales for 18Cr steels (∼100 μm) and 9Cr steels (approximate 200-500 μm) are necessary to trigger the exfoliation [73]. He et al. [74] investigated the oxide characteristics of T91 steel used as construction material of a superheater in a power plant operating at 541 °C for up to 25762 h, indicating the total thicknesses of oxide scales after exposure of approximate one and three years were ∼125 μm and ∼218 μm, respectively, which are 2.01 and 2.42 times the predicted values for the same exposure time. To a certain degree, this demonstrates the accuracy of the long-term predicted results, because available literature points out that the oxidation rates of F-M steels at 550 °C were 1.5-3 times those at 500 °C [9,23,39,75]. Additionally, the predicted results for 316 L at 500 °C is also in reasonable accord with the experiment test data [73]. After a 3-year exposure, the thickening rate of oxide scales formed on 316 L is almost negligible. Even after an exposure of up to 10 years, the total thickness is about 37 μm and the destruction thickness of the metal substrate is less than 15 μm. As is known, the oxidation kinetics of steels is frequently reported as the weight gain as a function of exposure time. Actually, the kinetics in

4.1.1. Macroscopic kinetics equations Deterministic Eqs. (39) and (42), respectively, for the thicknesses of the barrier- and outer- layers, not only can be employed as the objective function to optimize the SCW_PDM for extracting the fundamental parameters, but also can be used directly to fit the experimental data to obtain the apparent kinetics parameters (L0bl , b3 , Abl , Cbl , A ol , Col , and Dol ). According to the thickness data reported by Behnamian et al. [70] for 316 L stainless steel exposed to deaerated SCW at 500 °C for 500 h, 5000 h, 10000 h, and 20000 h, the fitting curves of Equations (39) and (42) are depicted in Fig. 7(b) and (d). Fig. 7(d) also exhibits the fitting curve optimized by Behnamian et al. [70] based on a power function, and the thickening rates of oxide film predicted by this SCW_PDM and the Behnamian’s equation [70]. After an exposure of approximate one year, the predicted increase rate in film thickness is rather small, and continues to decrease gradually. Obviously, the increase rate in film thickness predicted by the SCW_PDM is smaller than that obtained from Behnamian’s equation. The corresponding apparent kinetics parameters from the curve fitting based on Eqs. (39) and (42) for 316 L stainless steel are available in Table 5. In facilitating comparison, Table 5 also lists the apparent kinetic parameters for the growth of the oxide layers formed on HCM12A steel, which were calculated from the optimized, Table 5 Apparent parameters in Equations (39)* and (42) **, respectively, for oxidation kinetics in the thickness of the barrier- and outer layers. Parameters

Conditions

Symbols L0bl

Units cm

cm-1 cm∙s-1 cm∙s-1 cm∙s-1 cm∙s-1 cm

b3 Abl Cbl A ol Col Dol

* Lbl (t) = (L0bl ** L ol (t) =

HCM12A steel at 500 °C 4.98 × 10-5 −6.08 × 102 3.15 × 10-10 3.66 × 10-12 3.66 × 10-10 4.25 × 10-12 −4.18 × 10-2

( ) ln 1 b3

1+

( ) ln[A e Aol Ablb3

bl

( )e Abl Cbl

b3 Lbl (t)

b3 L0 bl (e b3 Cbl t

Cbl]

1)

316 L at 500 °C 1.01 × 10-7 −7.81 × 102 4.03 × 10-10 2.56 × 10-12 3.28 × 10-10 2.04 × 10-12 −7.83 × 10-3

Cbl t)

0,bl / bl .

Col t+ Dol .

12

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the weight gain can be derived easily from the simplified SCW_PDM. Integrating the increasing rate of the weight gain in Eq. (28), and substituting Equs. (39) and (42) for the thickness of the barrier- and outer- layers, respectively, we have

1 Abl b3L0 ln 1 + e bl (e b3 Cbl

0 bl {Lbl

w= rbl

A ol ln[Able b3Lbl (t) ol { Abl b3

+ rol

b3Cbl t

Cbl]

1)

Cbl t)}

Col t+ Dol }

0,bl /

Table 6 Parameters values (Pi, i = 1,2…7) of oxidation kinetics in the weight gain, obtained by (a) fitting curve and (b) calculation upon optimized fundamental parameters in SCW_PDM shown in Table 3.

bl

(47)

The above equation can be rearranged as an expression dependent of six parameters (P1 to P6),

w= P1

P3 P1/P2 e (e P4

P2ln 1 +

P4 /P2t

1) +

P5 P2 ln(P3 e M P3

ln 1 +

P1 = rbl

0 0,bl Lbl

P2 = rbl

0,bl /b3

=

P3 = rbl

0,bl Abl

= rbl

P4 = rbl P5 = rol

P6 = rol

0,bl Cbl ol A ol

ol C ol

P3 P1/P2 e (e P4

P4 /P2t

1)

P4 /P2 t

(49) (50)

rbl

0,bl

(51)

3

= rbl

= rol

= rol

0,bl

k30exp(a3V)exp(c3 pH)

(52)

0,bl

q k10CO

(53)

0,0l

0,0l

ol (

PBR

ol [(PBR

1)k30exp(a3V)exp(c3 pH) q 1)k10CO

+

r k11CO ]

(54) (55)

and

P7 =

P5 P2 ln(P3 eP1/P2 P3

P4) + P1

(b) Calculateda

P1, mg∙cm-2 P2, mg∙cm-2 P3, mg∙cm-2∙s-1 P4, mg∙cm-2∙s-1 P5, mg∙cm-2∙s-1 P6, mg∙cm-2∙s-1 P7b, mg∙cm-2

0.25 −8.22 1.58 × 10-6 1.83 × 10-8 1.82 × 10-6 2.18 × 10-8 −2.05 × 102

0.223 −7.34 1.41 × 10-6 1.64 × 10-8 1.65 × 10-6 1.97 × 10-8 −1.86 × 102

calculated from their corresponding definitions in Eqs. (50) to (56) and Table 3, upon optimizing SCW_PDM on the experimental scale thickness data. During this calculation, the necessary density ( 0,bl ) and oxygen fraction (rbl ) of the barrier layer were estimated in the following way. It is well known that the outer layer of oxide scales formed on F-M steel in SCW almost completely comprises magnetite, while the component of the inner layer is predominant Fe-Cr spinel, which, for simplicity, can be regarded as a mixture of Fe3O4 and FeCr2O4. In consideration of the Cr- and Fe- contents within the HCM12A substrate, ideally assuming that all oxidized Cr is retained in the barrier layer in the form of FeCr2O4, the oxides in the barrier layer formed on HCM12A are expected to contain 45% Fe3O4 and FeCr2O4 55% in molar fraction. Accordingly, that 0,bl ≈ 4.98 g∙cm-3 and rbl ≈ 0.28 was obtained, depending on the theoretical density of Fe3O4 (5.15 g∙cm-3) and FeCr2O4 (4.79 g∙cm-3). The calculated kinetic parameters given in Table 6(b) are in reasonable accord with the obtained results after the curve fitting, as shown in Table 6(a). This phenomenon again demonstrates the rationality and validity of optimizing the SCW_PDM with the scale thickness as an objective function to extract a series of fundamental parameters. For the sake of comparison, the oxidation rate constants (k) in the frequently-used power law

in which,

M= P1/P2

(a) Fitting curve

a Upon on the obtained fundamental parameters after optimization of SCW_PDM. b P7 depends on parameters P1 to P6.

P4) (48)

(P4 + P6) t+ P7

Parameters

(56)

With the experimentally measured weight gain as an “objective function”, according to the kinetic equation [Eq. (48)], optimization was employed here to obtain the unknown parameters P1 to P6. P7 is dependent of P1 - P6 and can be determined by the definition indicated by Eq. (56). Fig. 9 shows the fitted curve and experimental weight gains as a function of exposure time for HCM12A in SCW at 500 °C, exhibiting a satisfactory agreement between them, at least at the phenomenological level. The obtained parameter values for P1 – P7 are listed in Table 6(a). Additionally, Table 6(b) lists another set of P1 – P7 that are

#

w= k t a

(57)

were fitted and estimated in terms of the reported weight gain kinetic equations, where a# is the time exponent. For 9-12 Cr F-M steels in SCW at 500-600 °C, the average value of rate constant (k) is estimated, based upon the kinetic data reported by Li et al (∼ 6.18 × 10-6 mg∙cm-2∙s-1 at 540 °C) [39] and Tan et al. (∼ 1.1 × 10-6 mg∙cm-2∙s-1 at 500 °C, ∼ 7.58 × 10-6 mg∙cm-2∙s-1 at 600 °C) [37], to be in the range of ∼ 10-6 -105 mg∙cm-2∙s-1. In fact, to a great degree, the kinetics parameters P3 and P5 reflect the increasing rate of weight gain due to the inward growth of the barrier layer and the thickening of the outer layer. The values of these parameters shown in Table 6 are in good agreement with the above estimated rate constants (k), demonstrating the reliability of the kinetic equation [Eq. (48)] in describing the oxidation behavior of steels and alloys in SCW. Actually, Eq. (48) can be expected to characterize the oxidation kinetics of iron/nickel based alloys as reflected in the weight gain in a variety environment such as dry air, high-temperature steam, and other mixed gases systems. On the one hand, that the boundary between the barrier layer and the outer layer remains located at the original metal/environment interface during the corrosion process, has been frequently reported for the various above-mentioned systems. On the other hand, these kinetics parameters (P1 – P7) in Equation (48) are not directly dependent on the fundamental parameters related to specific cation defects, i.e., they exhibit little dependence on the transport mechanism of cations in the barrier layer.

Fig. 9. Expermental and simulated weight gain as a function of time for HCM12A exposed to SCWs at 500 °C [36]. 13

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Y. Li, et al.

5. Conclusion

physical meanings. Based on these rate laws, the microscopic kinetic information on the corrosion of ferritic-martensitic steel HCM12A and macroscopic kinetic parameters of several steels has been obtained successfully. It is also revealed that the model parameters k2 and k3, which are dependent on the composition and structure of the steel substrate determine the corrosion resistance. These kinetic laws may be directly employed in describing various corrosive environments, such as high temperature air and steam, liquid metal, and so forth.

In order to describe theoretically the corrosion behavior of metals and alloys in supercritical aqueous systems, such as the growth kinetics of the barrier layer of oxide scales and the total scale thickness and for the oft-reported growth of the barrier layer with the barrier layer/outer layer interface remaining at the location of the original metal surface, a Point Defect Model for metal oxidation in SCW, SCW_PDM, has been developed. Two atomic-level rate laws, being expressed in the scale thickness and weight gain respectively, are also derived. The principal findings are as follows:

Data availability

(1) A set of interfacial reactions is proposed to simultaneously describe the microscopic corrosion processes in high and low density supercritical water. Based on transition state theory, the kinetic equation of each interfacial reaction and its related parameters are defined. (2) The net growth rate of the barrier layer is the difference between the film growth rate at the substrate/film interface and its destruction rate at the barrier layer/outer layer interface. At the metal substrate/barrier layer interface, the generation rate of cation interstitials and the annihilation rate of cation vacancies together determine the growth rate of the outer layer. (3) It is proposed that the "constant volume" growth of the barrier layer is derived from the stress-related constraint relationship between the Pilling-Bedworth ratio of the produced oxide and the rates of micro reactions occurring at the substrate/film interface, solving one of the great challenges in the theory of metal oxidation. (4) The new-established atomic-scale rate laws in scale thickness and weight gain, reflect clear microscopic processes and possess explicit

The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations. Acknowledgements The authors are happy to acknowledge financial support from the China Postdoctoral Science Foundation [2019TQ0248], DOE-NEUP Award of USA (DE-NE0008541), National Key Research and Development Program of China (No. 2016YFC0801904), the Fundamental Research Funds for the Central Universities [xjj2018201] and [xjj2018006], and the Projects from National Natural Science Foundation of China [51871179]. Yanhui Li also appreciates China Scholarship Council (CSC) to support his study at University of California at Berkeley. Finally, DDM gratefully acknowledges the partial support of this work by FUTURE (Fundamental Understanding of Transport Under Reactor Extremes), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) (neutron scattering studies).

Appendix A For the Reaction 1, as an example, we can write its transition state (TS) in the form:

[ (1

1)

'

m+ (1

1)V M

+

1MM

+

1vm

+

1

]#

e'

(A-1)

of which the electrochemical potential of the TS can be expressed as the sum of the electrochemical potentials of each of the five items in the square brackets of Eq. (A-1): TS,1

0 1)µ m

= (1

+ [(1

1)µ

0

(1

'

VM

1)

F f] +

0 1µ MM

+

0 1µvm

+[

1

µ 0'

1

e

F

m]

(A-2)

Here, µ 0i is the standard chemical potential of a specie i, F is Faraday’s constant, and f represents the electrostatic potential in the barrier layer at the m/bl interface and m is the electrostatic potential in the metal at the m/bl interface. For the change of molar Gibbs free energy from the reactants to the transition state, ‡

G10 =

[µ 0m + µ 0

TS,1

=

0 1 (µ MM

=

1

+ µ0vm + µ 0'

0 GR,1

'

F f]

µ 0m

µ0 ')

VM

e

1

F

1

VM

F(

f)

m

(A-3)

m/bl

Here, m/bl is the potential drop across the m/bl interface ( m/bl = m is the standard change of chemical Gibbs free energy at the f ), reference temperature (T0). Accordingly, for Reaction 1, the rate constant can be expressed by 0 GR,1

k1 =

1

kBT exp h

1

G0R,1

1

F

RT

m/bl

=

1

kBT exp h

Substitution of Eq. (3) into Eq. (A-4), with the terms (A-4) becomes

k1 = k10exp

1

F(1 )V exp RT

1

F Lbl exp RT

1

1

0 GR,1

RT

k10

and

k100

exp

1

F

m/bl

RT

(A-4)

representing standard rate constant and the base rate constant, respectively, Eq.

F pH RT

(A-5)

in which,

k10 = k100 exp

F RT

1

0

(A-5-1) 14

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Y. Li, et al.

'

0 GR,1 1 R T

1

k100 = k100 exp

1 T0

(A-5-2)

where the base rate constant (k100 ) has been written in a form to include the temperature dependence (T vs the reference temperature of T0 ). Accordingly, the rate constants of Reaction 2 and 3, which also occurred at the m/bl interface, are given by

F(1 )V exp RT

i

ki = k 0i exp

i

F Lbl exp RT

i

F pH , i = 2 and 3 RT

(A-6)

where 0

F RT

i

k 0i = k 00 i exp

, i = 2 and 3

(A-6-1)

and '

i

0 GR,i 1 T

1

00 k 00 i = k i exp

1 T0

R

, i = 2 and 3

(A-6-2)

It is important to note that the classical PDM predicts that because of compensation between the effects of V and Lbl in the steady state that F(1 ) V i F Lbl is a constant, so that the reactions at the m/bl interface (Reactions 1, 2, and 3) are potential independent. In a similar way, for all other reactions, the change of molar Gibbs free energy from the reactants to the transition state are given by ‡

G04 =

4

0 GR,4

4'

0 GR,4'

5

0 GR,5

5'

0 GR,5'

6

0 GR,6

(A-11)

7

0 GR,7

(A-12)

8

G0R,8

2 8F

bl/e

(A-13)

9

0 GR,9

2 9F

bl/e

(A-14)

‡

G04' = ‡

G50 = ‡

G5'0 = ‡

G60 = ‡

G07 = ‡

G08 = ‡

G09 = ‡

0 G10 =

10

0 G11 =

11

F

(A-8)

bl/e

(A-9) 5'

F

(A-10)

bl/e

(A-15)

G0R,10'

10'

‡

4'

G0R,10

‡

0 G10' =

(A-7)

10' (

)F

(A-16)

bl/e

0 GR,11

(A-17)

The corresponding rate constant of each reaction can be obtained and expressed as: j

kj = k0j exp

F V RT j

k0j = k00 j exp

F

'

j

kj = k0j exp

0 GR,j 1 R T

RT 2 jF RT

'

j

00 k00 j = k j exp

k10' = k0j exp(

, j = 4' and 5'

(A-18)

, j= 4' and 5'

2 jF V

k0j = k00 j exp

F pH RT

0

RT

00 k00 j = k j exp

j

exp

exp

1 T0

2 jF pH RT

(A-18-1)

, j= 4' and 5'

(A-18-2)

, j = 8 and 9

(A-19)

0

, j = 8 and 9 0 GR,j 1 R T

10' (

)F V RT

1 T0 )exp(

(A-19-1)

, j = 8 and 9 10' (

(A-19-2)

)F pH ) RT

(A- 20)

15

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Y. Li, et al. 10' (

0 00 k10' = k10' exp

)F

0

(A-20- 1)

RT 0 GR,10' 1 R T

1 T0

(A-20- 2)

k s = k 0s , s = 4, 5, 6, 7, 10, and 11

(A-21)

'

10'

00 00 k10' = k10' exp

k s0

=

k 00 s ,

(A-21-1)

s = 4, 5, 6, 7, 10, and 11 '

0 GR,s 1 R T

s

k s00 = k s00 exp

1 T0

, s = 4, 5, 6, 7, 10, and 11

(A-21-2)

Therefore, for the rate constants of the 14 reactions shown in Fig. 5 (ki), they can be defined in the following general form, of which the detailed derivation processes are available in Appendix A. (A-22)

ki = k 0i exp(aiV )exp(bi Lbl )exp(c i pH) '

i

00 k 00 i = k i exp

where

0 GR,i

0 GR,i 1 T

1 T0

R

(A-23)

is the standard change of chemical Gibbs free energy of Reaction i (i = 1, 2, 3, 4, 4′, 5, 5′, 6, 7, 8, 9, 10, 10′, and 11) at reference '

temperature (T0), k i00 is the base standard rate constant of the ith elementary interfacial reaction. ai, bi, and ci are intermediate variables, k 0i is standard rate constant, and k 00 i represents base rate constant. Their definitions are list in Tables 1 and 2. Appendix B The partial current provided by each Reaction i (i = 1, 2, 3, 4, 4′, 5, 5′, 6, 7, 8, 9, 10, 10′11) can be expressed as:

i1 = Fk1CLVM

(36)

i2 = Fk2

(37)

i3 = Fk3

(38) (39)

Fk 4COn

i4 =

i 4' = (

(40)

)Fk 4 n 0 Fk5CO CMi

i5 =

(41)

)Fk5'C0Mi

i5' = (

(42)

i6 =

2Fk 6COm C0VO

(43)

i7 =

2Fk7CpH2 O C0VO

(44)

i9 =

p 2Fk 9CH 2O

(45)

and

i10' = (

(

)s

)Fk10' CH+/C0H+

(46)

References [9]

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