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Anderko A 2010 Modeling of Aqueous Corrosion. In: Richardson J A et al. (eds.) Shreir’s Corrosion, volume 2, pp. 1585-1629 Amsterdam: Elsevier.

Author’s personal copy

2.38 Modeling of Aqueous CorrosionA. AnderkoOLI Systems Inc., 108 American Road, Morris Plains, NJ 07950, USA

� 2010 Elsevier B.V. All rights reserved.

2.38.1 Introduction 1586

2.38.2 Thermodynamic Modeling of Aqueous Corrosion 1587

2.38.2.1 Computation of Standard-State Chemical Potentials 1588

2.38.2.2 Computation of Activity Coefficients 1588

2.38.2.3 Electrochemical Stability Diagrams 1591

2.38.2.3.1 Diagrams at elevated temperatures 1594

2.38.2.3.2 Effect of multiple active species 1594

2.38.2.3.3 Diagrams for nonideal solutions 1595

2.38.2.3.4 Diagrams for alloys 1596

2.38.2.3.5 Potential–concentration diagrams 1597

2.38.2.4 Chemical Equilibrium Computations 1597

2.38.2.5 Problems of Metastability 1599

2.38.3 Modeling the Kinetics of Aqueous Corrosion 1600

2.38.3.1 Kinetics of Charge-Transfer Reactions 1601

2.38.3.2 Modeling Adsorption Phenomena 1604

2.38.3.3 Partial Electrochemical Reactions 1605

2.38.3.3.1 Anodic reactions 1605

2.38.3.3.2 Cathodic reactions 1607

2.38.3.3.3 Temperature dependence 1609

2.38.3.4 Modeling Mass Transport Using Mass Transfer Coefficients 1609

2.38.3.4.1 Example of electrochemical modeling of general corrosion 1611

2.38.3.5 Detailed Modeling of Mass Transport 1611

2.38.3.5.1 Effect of the presence of porous media 1614

2.38.3.6 Active–Passive Transition and Dissolution in the Passive State 1614

2.38.3.7 Scaling Effects 1618

2.38.3.8 Modeling Threshold Conditions for Localized Corrosion 1620

2.38.3.8.1 Breakdown of passivity 1621

2.38.3.8.2 Repassivation potential and its use to predict localized corrosion 1622

2.38.3.9 Selected Practical Applications of Aqueous Corrosion Modeling 1624

2.38.4 Concluding Remarks 1626

References 1626

AbbreviationsCCT Critical crevice temperature

CR Corrosion rate

HKF Helgeson–Kirkham–Flowers equation of state

Me Metal

MSA Mean spherical approximation

NRTL Nonrandom two-liquid (equation)

Ox Oxidized form

Re Reduced form

SHE Standard hydrogen electrode

Shreir’s Corrosion, Fourth Edition

UNIFAC Universal functional activity coefficient

(equation)

UNIQUAC Universal quasi-chemical (equation)

Symbolsai Activity of species i

A Surface area

Aij Surface interaction coefficient for species i and j

b Tafel coefficient using decimal logarithms

1585(2010), vol. 2, pp. 1585-1629

1586 Modeling Corrosion


ci,b Bulk molar concentration of species i

ci,s Surface molar concentration of species i

Cp Heat capacity at constant pressure

d Characteristic dimension

Di Diffusion coefficientsta of species i

Dt Turbulent diffusion coefficient

e� Electron

E Potential

Eb Passivity breakdown potential

Ecorr Corrosion potential

Ecrit Critical potential for localized corrosion

Erp Repassivation potential

E0 Equilibrium potential

f Friction factor

fi Fugacity of species i

F Faraday constant

Gex Excess Gibbs energy

i Current density

ia Anodic current density

ia,ct Charge-transfer contribution to the anodic

current density

ia,L Limiting anodic current density

ic Cathodic current density

ic,ct Charge-transfer contribution to the cathodic

current density

ic,L Limiting cathodic current density

icorr Corrosion current density

ip Passive current density

irp Current density limit for measuring repassivation

potential

i0 Exchange current density

i* Concentration-independent coefficient in

expressions for exchange current density

Ji Flux of species i

kads Adsorption rate constant

kdes Desorption rate constant

ki Reaction rate constant for reaction i

km,i Mass transfer coefficient for species i

K Equilibrium constant

Kads Adsorption equilibrium constant

Ksp Solubility product

li Reaction rate for ith reaction

mi Molality of species i

m0 Standard molality of species i

ni Number of moles of species or electrons

Nu Nusselt number

Pr Prandtl number

R Gas constant

Rk Rate of production or depletion of species k

Re Reynolds number

S Supersaturation

Sc Schmidt number

Shreir’s Corrosion, Fourth Edition

Sh Sherwood number

t Time

T Temperature

ui Mobility of species i

vi Rate of reaction i

V Linear velocity

z Direction perpendicular to the surface

zi Charge of species i

a Thermal diffusivity

ai Electrochemical transfer coefficient for species i

b Tafel coefficient using natural logarithms

gi Activity coefficient of species i

di Nernst layer thickness for species i

DGads,i Gibbs energy of adsorption for species i

DH6¼ Enthalpy of activation

DF Potential drop

e Dielectric permittivity

h Dynamic viscosity

ui Coverage fraction of species i

uP Coverage fraction of passive layer

l Thermal conductivity

mi Chemical potential of species i

mi0 Standard chemical potential

�mi Electrochemical potential of species i

n Kinematic viscosity

ni Stoichiometric coefficient of species i

r Density

F Electrical potential

ji Electrochemical transfer coefficient for reaction i

v Rotation rate

r Vector differential operator

�X Surface species X

2.38.1 Introduction

Aqueous corrosion is an extremely complex physicalphenomenon that depends on a multitude of factorsincluding the metallurgy of the corroding metal, thechemistry of the corrosion-inducing aqueous phase,the presence of other – solid, gaseous, or nonaqueousliquid – phases, environmental constraints such astemperature and pressure, fluid flow characteristics,methods of fabrication, geometrical factors, and con-struction features. This inherent complexity makesthe development of realistic physical models verychallenging and, at the same time, provides a strongincentive for the development of practical models tounderstand the corrosion phenomena, and to assist intheir mitigation. The need for tools for simulatingaqueous corrosion has been recognized in variousindustries including oil and gas production and

(2010), vol. 2, pp. 1585-1629

Modeling of Aqueous Corrosion 1587


transmission, oil refining, nuclear and fossil power gen-eration, chemical processing, infrastructure mainte-nance, hazardous waste management, and so on. Thepast three decades have witnessed the development ofincreasingly sophisticated modeling tools, which hasbeen made possible by the synergistic combination ofimproved understanding of corrosion mechanisms andrapid evolution of computational tools.

Corrosion modeling is an interdisciplinary under-taking that requires input from electrolyte thermo-dynamics, surface electrochemistry, fluid flow andmass transport modeling, and metallurgy. In thischapter, we put particular emphasis on corrosionchemistry by focusing on modeling both the bulkenvironment and the reactions at the corroding inter-face. The models that are reviewed in this chapter areintended to answer the following questions:

1. What are the aqueous and solid species that give riseto corrosion in a particular system? What are theirthermophysical properties, andwhat phase behaviorcan be expected in the system? These questionscan be answered by thermodynamic models.

2. What are the reactions that are responsible forcorrosion at the interface? How are they influ-enced by the bulk solution chemistry and by flowconditions? How can passivity and formation ofsolid corrosion products be related to environ-mental conditions? How can the interfacial phe-nomena be related to observable corrosion rates?These questions belong to the realm of electro-chemical kinetics and mass transport models.

3. What conditions need to be satisfied for theinitiation and long-term occurrence of localizedcorrosion? This question can be answered by elec-trochemical models of localized corrosion.

These models can be further used as a foundation forlarger-scale models for the spatial and temporal evo-lution of systems and engineering structures subjectto localized and general corrosion. Also, they can becombined with probabilistic and expert system-typemodels of corrosion. Models of such kinds are, how-ever, outside the scope of this review, and will bediscussed in companion chapters.

2.38.2 Thermodynamic Modeling ofAqueous Corrosion

Historically, the first comprehensive approach tomodeling aqueous corrosion was introduced by Pour-baix in the 1950s and 1960s on the basis of purelythermodynamic considerations.1 Pourbaix1 devel-oped the E–pH stability diagrams, which indicate

Shreir’s Corrosion, Fourth Editio

which phases are stable on a two-dimensional planeas a function of the potential and pH. The potentialand pH were originally selected because they play akey role as independent variables in electrochemicalcorrosion. Just as importantly, they made it possibleto construct the stability diagrams in a semianalyticalway, which was crucial before the advent of computercalculations. Over the past four decades, great prog-ress has been achieved in the thermodynamics ofelectrolyte systems, in particular for concentrated,mixed-solvent, and high temperature systems. Theseadvances made it possible to improve the accuracy ofthe stability diagrams and, at the same time, increasedthe flexibility of thermodynamic analysis so that it cango well beyond the E–pH plane.

The basic objective of the thermodynamics ofcorrosion is to predict the conditions at which a givenmetal may react with a given environment, leading tothe formation of dissolved ions or solid reaction pro-ducts. Thermodynamics can predict the properties ofthe system in equilibrium or, if equilibrium is notachieved, it can predict the direction in which thesystem will move towards an equilibrium state.Thermodynamics does not provide any informationon how rapidly the system will approach equilibrium,and, therefore, it cannot give the rate of corrosion.

The general condition of thermodynamic equilib-rium is the equality of the electrochemical potential,m�i in coexisting phases,2 that is,

m�i ¼ mi þ ziFf ¼ m0i þ RT ln ai þ ziFf ½1�where mi is the chemical potential of species i; m0i , itsstandard chemical potential; ai, the activity of thespecies; zi, its charge; F, the Faraday constant; and fis the electrical potential. The standard chemicalpotential is a function of the temperature and, sec-ondarily, pressure. The activity depends on the tem-perature and solution composition and, to a lesserextent, on pressure. The activity is typically definedin terms of solution molality mi,

ai ¼ ðmi=m0Þgi ½2�

where m0 is the standard molality unit (1mol kg�1

H2O), and gi is the activity coefficient. It should benoted that the molality basis for species activitybecomes inconvenient for concentrated solutionsbecause molality diverges to infinity as the concentra-tion increases to the pure solute limit. Therefore, themole fraction basis is more generally applicable forcalculating activities.3 Nevertheless, molality remainsthe most common concentration unit for aqueoussystems and is used here for illustrative purposes.

n (2010), vol. 2, pp. 1585-1629



Computation of m0i and gi is the central subject inelectrolyte thermodynamics. Numerous methods,with various ranges of applicability, have beendeveloped over the past several decades for theircomputation. Several comprehensive reviews of theavailable models are available (Zemaitis et al.,4

Renon,5 Pitzer,6 Rafal et al.,7 Loehe and Donohue,8

Anderko et al.3). In the next section, the current statusof modeling m0i and gi is outlined as it applies to thethermodynamics of corrosion.

2.38.2.1 Computation of Standard-StateChemical Potentials

The computation of the standard chemical potentialm0i requires the knowledge of thermochemical dataincluding

1. Gibbs energy of formation of species i at referenceconditions (298.15 K and 1 bar).

2. Entropy or, alternatively, enthalpy of formation atreference conditions.

3. Heat capacity and volume as a function of temper-ature and pressure.

For numerous species, these values are available invarious compilations (Chase et al.,9 Barin and Platzki,10

Cox et al.,11 Glushko et al.,12 Gurvich et al.,13 Kelly,14

Robie et al.,15 Shock and Helgeson,16 Shock et al.,17,18

Stull et al.,19 Wagman et al.,20 and others). In general,thermochemical data are most abundant at near-ambient conditions, and their availability becomesmore limited at elevated temperatures.

In the case of individual solid species, the chemi-cal potential can be computed directly from tabulatedthermochemical properties according to the standardthermodynamics.2 In the case of ions and aqueousneutral species, the thermochemical properties listedabove are standard partial molar properties ratherthan the properties of pure components. The standardpartial molar properties are defined at infinite dilutionin water. The temperature and pressure dependence ofthe partial molar heat capacity and the volume of ionsand neutral aqueous species are quite complex becausethey are manifestations of the solvation of species,which is influenced by electrostatic and structuralfactors. Therefore, the computation of these quantitiesrequires a realistic physical model.

An early approach to calculating the chemicalpotential of aqueous species as a function of temper-ature is the entropy correspondence principle ofCriss and Cobble.21 In this approach, heat capacitiesof various types of ions were correlated with thereference-state entropies of ions, thus making it


possible to predict the temperature dependence ofthe standard chemical potential.

A comprehensive methodology for calculatingthe standard chemical potential was developed byHelgeson et al. (Helgeson et al.,22 Tanger andHelgeson23). This methodology is based on a semi-empirical treatment of ion solvation, and results in anequation of state for the temperature and pressuredependence of the standard molal heat capacitiesand volumes. Subsequently, the heat capacities andvolumes are used to arrive at a comprehensive equationof state for standardmolalGibbs energyand, hence, thestandard chemical potential. The method is referred toas the HKF (Helgeson–Kirkham–Flowers) equation ofstate. An important advantage of the HKF equation isthe availability of its parameters for a large number ofionic and neutral species (Shock and Helgeson,16

Shock et al.,18 Sverjensky et al.24). Also, correlationsexist for the estimation of the parameters for speciesfor which little experimental information is available.TheHKFequation of state has been implemented bothin publicly available codes (Johnson et al.25) and incommercial programs. A different equation of statefor standard-state properties has been developed onthe basis of fluctuation solution theory (Sedlbaueret al.,26 Sedlbauer and Majer27). This equation offersimprovement overHKF for nonionic solutes and in thenear-critical region. However, the HKF remains as themost widely accepted model for ionic solutes.

2.38.2.2 Computation of ActivityCoefficients

In real solutions, the activity coefficients of spe-cies deviate from unity because of a variety ofionic interaction phenomena, including long-rangeCoulombic interactions, specific ion–ion interactions,solvation phenomena, and short-range interactionsbetween uncharged and charged species. Therefore,a practically-oriented activity coefficient model mustrepresent a certain compromise between physicalreality and computational expediency.

The treatment of solution chemistry is a particu-larly important feature of an electrolyte model. Here,the term ‘solution chemistry’ encompasses ionic dis-sociation, ion pair formation, hydrolysis of metal ions,formation of metal–ligand complexes, acid–basereactions, and so on. The available electrolyte modelscan be grouped in three classes:

1. models that treat electrolytes on an undissociatedbasis,

2. models that assume complete dissociation of allelectrolytes into constituent ions, and

n (2010), vol. 2, pp. 1585-1629



3. speciation-based models, which explicitly treat thesolution chemistry.

The models that treat electrolytes as undissociatedcomponents are analogous to nonelectrolyte mixturemodels. They are particularly suitable for supercriti-cal and high temperature systems, in which ion pairspredominate. Although this approach may also beused for phase equilibrium computations at moderateconditions (e.g., Kolker et al.28), it is not suitable forcorrosion modeling because it ignores the existenceof ions. The models that assume complete dissocia-tion are the largest class of models for electrolytesat typical conditions. Compared with the modelsthat treat electrolytes as undissociated or completelydissociated, the speciation-based models are morecomputationally demanding because of the need tosolve multiple reactions and phase equilibria. Anotherfundamental difficulty associated with the use of spe-ciation models lies in the need to define and charac-terize the species that are likely to exist in the system.In many cases, individual species can be clearlydefined and experimentally verified in relativelydilute solutions. At high concentrations, the chemicalidentity of individual species (e.g., ion pairs or com-plexes) becomes ambiguous because a given ion hasmultiple neighbors of opposite sign, and, thus, manyspecies lose their distinct chemical character. There-fore, the application of speciation models to concen-trated solutions requires a careful analysis to separatethe chemical effects from physical nonideality effects.

It should be noted that, as long as only phaseequilibrium computations are of interest, comparableresults could be obtained with models that belong tovarious classes. For example, the overall activity coef-ficients and vapor–liquid equilibria of many transi-tion metal halide solutions, which show appreciablecomplexation, can be reasonably reproduced usingPitzer’s29 ion-interaction approach without takingspeciation into account. However, it is important toinclude speciation effects for modeling the thermody-namics of aqueous corrosion. This is due to the factthat the presence of individual hydrolyzed forms,aqueous complexes, and so on is often crucial for thedissolution of metals and metal oxides. It should benoted that activity coefficients of individual speciesare different in fully speciated models than in modelsthat treat speciation in a simplified way. Therefore, it isimportant to use activity coefficients that have beendetermined in a fully consistent way, that is, by assum-ing the appropriate chemical species in the solution.

The theory of liquid-phase nonideality is well-established for dilute solutions.A limiting law for activity


coefficients was developed by Debye and Huckel30 byconsidering the long-range electrostatic interactions ofions in a dielectric continuum. The Debye–Huckel the-ory predicts the activity coefficients as a function of theionic charge and dielectric constant and density ofthe solvent. It reflects only electrostatic effects and,therefore, excludes all specific ionic interactions. There-fore, its range of applicability is limited to �0.01M fortypical systems. Several modifications of the Debye–Huckel theory have been proposed over the past severaldecades. The most successful modification was devel-oped by Pitzer29 who considered hard-core effectson electrostatic interactions. A more comprehensivetreatment of the long-range electrostatic interactionscan be obtained from the mean-spherical approxima-tion (MSA) theory,31,32 which provides a semianalyti-cal solution for ions of different sizes in a dielectriccontinuum. The MSA theory results in a better pre-diction of the long-range contribution to activity coef-ficients at somewhat higher electrolyte concentrations.

The long-range electrostatic term provides a base-line for constructing models that are valid for elec-trolytes at concentrations that are important inpractice. In most practically-oriented electrolytemodels, the solution nonideality is defined by theexcess Gibbs energy Gex. The excess Gibbs energyis calculated as a sum of the long-range term and oneor more terms that represent ion–ion, ion–molecule,and molecule–molecule interactions:

Gex ¼ Gexlong-range þ Gex

specific þ � � � ½3�where the long-range contribution is usually calcu-lated either from the Debye–Huckel or the MSAtheory, and the specific interaction term(s) repre-sent(s) all other interactions in an electrolyte solu-tion. Subsequently, the activity coefficients arecalculated according to standard thermodynamics2 as

ln gi ¼1

RT

@Gex

@ni

� �T ;P;nj 6¼i

½4�

Table 1 lists a number of activity coefficient modelsthat have been proposed in the literature, and showsthe nature of the specific interaction terms that havebeen adopted. In general, these models can be sub-divided into two classes:

1. models for aqueous systems; in special cases, suchmodels can also be used for other solvents as longas the system contains a single solvent;

2. mixed-solvent electrolyte models, which allowmultiple solvents as well as multiple solutes.

The aqueous electrolyte models incorporate variousion interaction terms, which are usually defined

n (2010), vol. 2, pp. 1585-1629

Table 1 Summary of representative models for calculating activity coefficients in electrolyte systems

Reference Terms Features

Aqueous (or single-solvent) electrolyte models

Debye and Huckel30 Long-range Limiting law valid for very dilute solutions

Guggenheim33,34 Long-rangeþ ion interaction Simple ion interaction term to extend the Debye–Huckellimiting law; applicable to fairly dilute solutions

Helgeson35 Long-rangeþ ion interaction Ion interaction term to extend the limiting law; applicable to

fairly dilute solutionsPitzer29 Long-rangeþ ion interaction Ionic strength-dependent virial coefficient-type ion

interaction term; revised Debye–Huckel limiting law;

applicable to moderately concentrated solutions (�6m)

Bromley36 Long rangeþ ion interaction Ionic strength dependent ion interaction term; applicable tomoderately concentrated solutions (�6m)

Zemaitis37 Long-rangeþ ion interaction Modification of the model of Bromley36 to increase

applicability range with respect to ionic strength

Meissner38 One-parameter correlation as afunction of ionic strength

Generalized correlation to calculate activities based on alimited amount of experimental information

Pitzer and

Simonson,39 Cleggand Pitzer40

Long-rangeþ ion interaction Mole fraction-based expansion used for the ion interaction

term; applicable to concentrated systems up to the fusedsalt limit

Mixed-solvent electrolyte models

Chen et al.41 Long-rangeþ short-range Local-composition (NRTL) short-range termLiu and Watanasiri42 Long-rangeþ short-

rangeþelectrostatic solvation

(Born)þ ion interaction

Modification of the model of Chen et al.41 using a

Guggenheim-type ion interaction term for systemswith two

liquid phases

Abovsky et al.43 Long-rangeþ short-range Modification of the model of Chen et al.41 usingconcentration-dependent NRTL parameters to extend

applicability range with respect to electrolyte

concentration

Chen et al.44 Long-rangeþ short-rangeþ ionhydration

Modification of the model of Chen et al.41 using an analyticalion hydration term to extend applicability with respect to

electrolyte concentration

Chen and Song45 Long-rangeþ short-rangeþelectrostatic solvation

(Born)

Modification of the model of Chen et al.41 by introducingsegment interactions for organic molecules

Sander et al.46 Long-rangeþ short-range Local composition model (UNIQUAC) with concentration-

dependent parameters used for short-range termMacedo et al.47 Long-rangeþ short-range Local composition model (UNIQUAC) with concentration-

dependent parameters used for short-range term

Kikic et al.48 Long-rangeþ short range Local composition group contribution model (UNIFAC) used

for short-range termDahl and Macedo49 Short-range Undissociated basis; no long-range contribution; group

contribution model (UNIFAC) used for short-range term

Iliuta et al.50 Long-rangeþ short-range Local composition (UNIQUAC) model used for short-rangeterm

Wu and Lee31 Extended long-range (MSA) Mean-spherical approximation (MSA) theory used for the

long-range term

Li et al.51 Long rangeþ ion interactionþ shortrange

Virial-type form used for the ion interaction term; localcomposition used for short-range term

Yan et al.52 Long-rangeþ ion interactionþ short

range

Group contribution models used for both the ion interaction

term and short-range term (UNIFAC)

Zerres andPrausnitz53

Long-rangeþ short rangeþ ionsolvation

Van Laar model used for short-range term; stepwise ionsolvation model

Kolker et al.28 Short-range Undissociated basis; no long-range contribution

Wang et al.54,55 Long rangeþ ion interactionþ shortrange

Ionic strength-dependent virial expansion-type ioninteraction term; local composition (UNIQUAC) short-range

term; detailed treatment of solution chemistry

Papaiconomou

et al.32Extended long-range

(MSA)þ short-range

MSA theory used for the long-range term; local composition

model (NRTL) used for the short-range term



Shreir’s Corrosion, Fourth Edition (2010), vol. 2, pp. 1585-1629



in the form of virial-type expansions in terms ofmolality or mole fractions (see Table 1). Amongthese models, the Pitzer29 molality-based model hasfound wide acceptance. Parameters of the Pitzer29

model are available in the open literature for a largenumber of systems.6

The mixed-solvent electrolyte models aredesigned to handle a wider variety of chemistries.They invariably use the mole fraction and concentra-tion scales. A common approach in the constructionof mixed-solvent models is to use local-compositionmodels for representing short-range interactions.The well-known local-composition models includeNRTL, UNIQUAC, and its group-contribution ver-sion, UNIFAC (see Prausnitz et al.56 and Malanowskiand Anderko57 for a review of these models). Thelocal composition models are commonly used fornonelectrolyte mixtures and, therefore, it is naturalto use them for short-range interactions in electrolytesystems. The combination of the long-range andlocal-composition terms is typically sufficient forrepresenting the properties of moderately concen-trated electrolytes in any combination of solvents.For systems that may reach very high concentrationswith respect to electrolyte components (e.g., up to thefused salt limit), more complex approaches havebeen developed. One viable approach is to explicitlyaccount for hydration and solvation equilibria inaddition to using the long-range and short-rangelocal composition terms (Zerres and Prausnitz,53

Chen et al.44). A particularly effective approach isbased on combining virial-type ion interactionterms with local composition models (Li et al.,51 Yanet al.,52 Wang et al.54,55). In such combined models, thelocal-composition term reflects the nonelectrolyte-likeshort-range interactions, whereas the virial-type ioninteraction term represents primarily the specificion–ion interactions that are not accounted for by thelong-range contribution. These and other approachesare summarized in Table 1. Among the models sum-marized inTable 1, the models of Pitzer,29 Zemaitis,37

Chen et al.41 (including their later modifications),44,45

andWang et al.54,55 have been implemented in publiclyavailable or commercial simulation programs.

2.38.2.3 Electrochemical Stability Diagrams

The E–pH diagrams, commonly referred to as thePourbaix1 diagrams, are historically the first, andremain the most important class of electrochemicalstability diagrams. They were originally constructedfor ideal solutions (i.e., on the assumption that gi¼ 1),


which was the only viable approach at the timewhen they were introduced.1 The essence of theprocedure for generating the Pourbaix diagrams isanalyzing all possible reactions between all – aqueousor solid – species that may exist in the system. Thesimultaneous analysis of the reactions makes it possi-ble to determine the ranges of potential and pH atwhich a given species is stable. The reactions can beconveniently subdivided into two classes, that is,chemical and electrochemical reactions. The chemi-cal reactions can be written without electrons, that isX

niMi ¼ 0 ½5�Then, the equilibrium condition for the reaction isgiven in terms of the chemical potentials of individ-ual species by X

nimi ¼ 0 ½6�According to eqn [1], eqn [6] can be further rewrittenin terms of species activities asX

ni ln ai ¼ �P

nim0iRT

¼ ln K ½7�where the right-hand side of eqn [7] is defined as theequilibrium constant because it does not depend onspecies concentrations.

In contrast to the chemical reactions, the electro-chemical reactions involve electrons, e�, as well aschemical substances Mi, that isX

niMi þ n e� ¼ 0 ½8�The equilibrium state of an electrochemical reaction isassociated with a certain equilibrium potential. Sinceelectrode potential cannot be measured on an absolutebasis, it is necessary to choose an arbitrary scale againstwhich the potentials can be calculated. If a referenceelectrode is selected, the equilibrium state of the reac-tion that takes place on the reference electrode is givenby an equation analogous to eqn [8], that isX

ni;refMi;ref þ n e� ¼ 0 ½9�Then, the equilibrium potential E0 of eqn [8] is givenwith respect to the reference electrode as

E0 � E0;ref ¼P

nimi �P

nI ;refmi;refnF

½10�According to a generally used convention, the stan-dard hydrogen electrode (SHE) (i.e., HþðaHþ ¼ 1Þ=H2ðfH2

¼ 1Þ) is used as a reference. The correspondingreaction that takes place on the SHE is given by

Hþ � 1

2H2 þ e� ¼ 0 ½11�

n (2010), vol. 2, pp. 1585-1629



Then, eqn [10] becomes

E0 � E0;ref ¼P

nimi � n m0Hþ � 1

2 m0H2

� �nF

½12�In eqn [12], the standard chemical potentials m0H andm0H2

as well as the reference potential E0;ref are equalto zero at T¼ 298.15 K. For practical calculations attemperatures other than 298.15 K, two conventionscan be used for the reference electrode. According toa universal convention established by the Interna-tional Union of Pure and Applied Chemistry (the‘Stockholm convention’), the potential of SHE isarbitrarily defined as zero at all temperatures (i.e.,E0;ref ¼ 0). When this convention is employed, eqn[12] is used as a working equation with E0;ref ¼ 0. Inan alternate convention, the SHE reference potentialis equal to zero only at room temperature, and itsvalue at other temperatures depends on the actual,temperature-dependent values of m0

Hþ and m0H2. In

this case, it is straightforward to show (see Chenand Aral,58 Chen et al.,59) that eqn [12] becomes

E0 ¼P

niminF

½13�at all temperatures. Equation [13] can be furtherrewritten in terms of activities as

E0¼P

nim0inF

þRT

nF

Xni ln ai ¼ E00 þ

RT

nF

Xni ln ai ½14�

where E00 is the standard equilibrium potential, whichis calculated from the values of the standard chemicalpotentials m0i .

For a brief outline of the essence of the Pourbaixdiagrams, let us consider a generic reaction in whichtwo species, A and B, undergo a transformation. Theonly other species that participate in the reaction arehydrogen ions and water, that is:

aAþ cH2Oþ ne� ¼ bBþ mHþ ½15�If eqn [15] is a chemical reaction (i.e., if n¼ 0), then itsequilibrium condition (eqn [7]) can be rewritten as

mpH ¼ logaaBaaA

� �� logK � logacH2O

½16�

where pH ¼ �logaHþ , and decimal rather than natu-ral logarithms is used. For dilute solutions, it is appro-priate to assume that aH2O ¼ 1, and the last term onthe right-hand side of eqn [16] vanishes. If we assumethat the components A and B are aqueous dissolvedspecies and their activities are equal, eqn [16] definesthe boundary between the predominance areas of spe-cies A and B. If one of the species (A or B) is a pure


solid and the other is an aqueous species, the activity ofthe solid is equal to one and the activity of the aqueousspecies can be set equal to a certain predetermined,typically small, value (e.g., 10�6). Then, eqn [16] repre-sents the boundary between a solid and an aqueousspecies at a fixed value of the dissolved species activ-ity. Similarly, for a boundary between two pure solidphases, the activities of the species A and B are equal toone. Such a boundary is represented by a vertical line ina potential–pH space, and its location depends on theequilibrium constant according to eqn [16].

If eqn [15] represents an electrochemical reaction(i.e., n 6¼ 0), the equilibrium condition (eqn [14])becomes

E0¼ E00 þRT

nFln

aaAabBþRT

nFln aH2Oþ

RT ln 10

F

m

npH ½17�

As with the chemical reactions, the term that involvesthe activity of water vanishes for dilute solutions, andthe ratio of the activities of species A and B can befixed to represent the boundary between the pre-dominance areas of two species. Under such assump-tions, the plot of E versus pH is a straight line witha slope determined by the stoichiometric coefficientsm and n.

Thus, for each species, boundaries can be estab-lished using eqns [16] and [17]. As long as the sim-plifying assumptions described above are met, theboundaries can be calculated analytically. By consid-ering all possible boundaries, stability regions can bedetermined using an algorithm described by Pour-baix.1 Sample E–pH diagrams are shown for iron inFigure 1.

One of the main reasons for the usefulness ofstability diagrams is the fact that they can illustratethe interplay of various partial processes of oxidationand reduction. The classical E–pH diagrams containtwo dashed lines, labeled as ‘a’ and ‘b,’ which aresuperimposed on the diagram of a given metal (seeFigure 1). The dashed line ‘a’ represents the condi-tions of equilibrium between water (or hydrogenions) and elemental hydrogen at unit hydrogen fugac-ity, that is

Hþ þ e� ¼ 0:5H2 ðline ‘a’Þ ½18�The ‘b’ line describes the equilibrium between waterand oxygen, also at unit fugacity,

O2 þ 4Hþ þ 4e� ¼ 2H2O ðline ‘b’Þ ½19�Accordingly, water will be reduced to form hydrogenat potentials below line ‘a,’ and will be oxidized toform oxygen at potentials above line ‘b.’ The location

n (2010), vol. 2, pp. 1585-1629

FeC

l 2

FeCl+

FeOH+

FeCl2+

FeC

l 2

FeOH+2

Fe+2

FeO4–2

Fe(OH)3{aq}

NaF

eO2{

s}

Fe(O

H) 3

{aq

}

Fe(O

H) 2

{aq

}

Fe2O3{s}

Fe{s}

Fe3O4{s}

Fe3O4{s}FeO{s}

Fe{s}

Fe2O3{s}

Fe(OH)–4

Fe(OH)–4

Fe(OH)–4

Fe(OH)–3

Fe(OH)–3

Fe(O

H)+ 2

Fe+2

FeO

H+

Fe+3

FeO–24

2.0

1.5

1.0

0.5

0.0

–0.5

–1.0

–1.5

–2.0

2.0

1.0

1.5

0.5

0.0

–0.5

–1.0

–1.5

–2.00.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

E (S

HE

)E

(SH

E)

+2

a

b

pH

270120

1110

1022

89236

50 25 19

b

a

Figure 1 E–pH (Pourbaix) diagrams for iron at 25 �C (upper diagram) and 300 �C (lower diagram). At 300 �C, the conditions

of the experiments of Partridge and Hall60 are superimposed on the diagram. The vertical bars show the range between the

equilibrium potentials for the reduction of H+ and oxidation of Fe, thus bracketing the mixed potential in the experiments.The numbers under the bars denote the experimentally determined relative attack. The diagrams have been generated

using the Corrosion Analyzer software61 using the algorithm of Anderko et al.62



of the ‘a’ and ‘b’ lines on the diagram indicateswhether a given redox couple is thermodynamicallypossible. For example, in the region between the line‘a’ and the upper edge of the stability region of Fe(s) inFigure 1, the anodic reaction of iron oxidation can becoupled with the cathodic reaction of water or hydro-gen ion reduction. In such a case, the measurableopen-circuit potential of the corrosion process willestablish itself between the line ‘a’ and the equilib-rium potential for the oxidation of iron (i.e., the upperedge of the Fe(s) region). If the potential lies above


line ‘a,’ then water reduction is no longer a viablecathodic process, and the oxidation of iron must becoupled with another reduction process. In the pres-ence of oxygen, reaction eqn [19] can provide such areaction process. In such a case, the open-circuit(corrosion) potential will establish itself at a highervalue for which the upper limit will be defined byline ‘b.’

Pourbaix1 subdivided various regions of the E–pHdiagrams into three categories, that is, immunity,corrosion, and passivation. The immunity region

n (2010), vol. 2, pp. 1585-1629



encompasses the stability field of elemental metals.The corrosion region corresponds to the stability ofdissolved, either ionic or neutral species. Finally,passivation denotes the region in which solid oxidesor hydroxides are stable. In Figure 1, the immunityand passivation regions are shaded, whereas the cor-rosion region is not. It should be noted that thisclassification does not necessarily reflect the actualcorrosion behavior of a metal. Only immunity hasa strict significance in terms of thermodynamicsbecause in this region the metal cannot corroderegardless of the time of exposure. The stability ofdissolved species in the ‘corrosion’ region does notnecessarily mean that the metal rapidly corrodes inthis area. In reality, the rate of corrosion in this regionmay vary markedly because of kinetic reasons. Pas-sivation is also an intrinsically kinetic phenomenonbecause the protectiveness of a solid layer on thesurface of a metal is determined not by its low solu-bility alone. The presence of a sparingly soluble solidis typically a necessary, but not sufficient conditionfor passivity.

Although the E–pH diagrams indicate only thethermodynamic tendency for the stability of variousmetals, ions, and solid compounds, they may stillprovide useful qualitative clues as to the expectedtrends in corrosion rates. This is illustrated in thelower diagram of Figure 1. In this diagram, the verticalbars denote the difference between the equilibriumpotentials for the reduction of water and oxidation ofiron. Thus, the bars indicate the tendency of themetal to corrode in deaerated aqueous solutionswith varying pH. They bracket the location of thecorrosion potential and, thus, indicate whether thecorrosion potential will establish itself in the ‘corro-sion’ or ‘passivation’ regions. The numbers associatedwith the bars represent the experimentally deter-mined relative attack. It is clear that the observedrelative attack is substantially greater in the regionswhere a solid phase is predicted to be stable than inthe regions where no solid phase is predicted. Thus,subject to the limitations discussed above, the stabil-ity diagrams can be used for the qualitative assess-ment of the tendency of metals to corrode, and forestimating the range of the corrosion potential.

Following the pioneering work of Pourbaix andhis coworkers, further refinements of stability dia-grams were made to extend their range of applicabil-ity. These refinements were made possible by theprogress of the thermodynamics of electrolyte solu-tions and alloys. Specifically, further developmentsfocused on


1. generation of diagrams at elevated temperatures,2. taking into account the active solution species

other than protons and water molecules,3. introduction of solution nonideality, which influ-

ences the stability of species through realisticallymodeled activity coefficients,

4. introduction of alloying effects by accounting forthe formation of mixed oxides and the nonidealityof alloy components in the solid phase, and

5. flexible selection of independent variables, otherthan E and pH, for the generation of diagrams.

2.38.2.3.1 Diagrams at elevated temperatures

The key to the construction of stability diagramsat elevated temperatures is the calculation of thestandard chemical potentials m0i of all individual spe-cies as a function of temperature. In earlier studies,the entropy correspondence principle of Criss andCobble21 was used for this purpose. Macdonald andCragnolino63 reviewed the development of E–pH dia-grams at elevated temperatures until the late 1980s.

The HKF equation of state22,23 formed a compre-hensive basis for the development of E–pH diagramsat temperatures up to 300 �C for iron, zinc, chro-mium, nickel, copper, and other metals (Beverskogand Puigdomenech,64–69 Anderko et al.62) An exampleof a high temperature E–pH diagram is shown for Feat 300 �C in the lower diagram of Figure 1. Compar-ison of the Fe diagrams at room temperature and300 �C shows a shift in the predominance domainsof cations to lower pH values and an expansion of thedomains of metal oxyanions at higher pH values.Such effects are relatively common for metal–watersystems.

2.38.2.3.2 Effect of multiple active species

The concept of stability diagrams can be easilyextended to solutions that contain multiple chemi-cally active species other than H+ and H2O. In such ageneral case, the simple reaction eqn [15] needs to beextended as

nXX þXki¼1

niAi ¼ Y þ nee� ½20�

where the species X and Y contain at least one com-mon element and Ai (i¼ 1, . . ., k) are the basis speciesthat are necessary to define equilibrium equationsbetween all species containing a given element. Reac-tion eqn [20] is normalized so that the stoichiometriccoefficient for the right-hand side species (Y ) isequal to 1. Such an extension results in generalized

n (2010), vol. 2, pp. 1585-1629



expressions for the boundaries between predomi-nance regions (eqns [16] and [17]). The equilibriumexpression for the chemical reactions (eqn [20] withne ¼ 0) then becomes

lnK ¼ ln aY � nX ln aX �Xki¼1

ni ln aAi

!

¼ 1

RTm0Y � m0X �

Xki¼1

nim0Ai

!½21�

and the expression for an electrochemical reaction(eqn [20] with ne 6¼ 0) takes the form:

E0¼ E00 þ

RT

Fneln aY � nX ln aX �

Xki¼1

ni ln aAi

!½22�

Thus, the expression for the boundary lines becomemore complicated but the algorithm for generatingthe diagrams remains the same, that is, the predomi-nance areas can still be determined semianalytically.

The strongest effect of solution species on thestability diagrams of metals is observed in the caseof complex-forming ligands and species that formstable, sparingly soluble solid phases other thanoxides or hydroxides (e.g., sulfides or carbonates). Sev-eral authors focused on stability diagrams for metalssuch as iron, nickel, or copper in systems containingsulfur species (Biernat and Robbins,70 Froning et al.,71

Macdonald and Syrett,72 Macdonald et al.,73,74 Chenand Aral,58 Chen et al.,59 Anderko and Shuler75).This is due to the importance of iron sulfide phases,which have a strong tendency to form in aqueousenvironments even at very low concentrations of dis-solved hydrogen sulfide. The stability domains ofvarious iron sulfides can be clearly rationalizedusing E–pH diagrams. Diagrams have also been devel-oped for metals in brines (Pourbaix,76 Macdonaldand Syrett,72 Macdonald et al.,73,74 Kesavan et al.,77

Munoz-Portero et al.78). The presence of halide ionsmanifests itself in the stability of various metal–halide complexes. Typically, the effect of halides onthe thermodynamic stability is less pronounced thanthe effect of sulfides. However, concentrated brinescan substantially shrink the stability regions of metaloxides and promote the active behavior of metals.An example of such effects is provided by the dia-grams for copper in concentrated bromide brines(Munoz-Portero et al.78) Effects of formation of vari-ous carbonate and sulfate phases have also beenreported (Bianchi and Longhi,79 Pourbaix76).

The formation of complexes of metal ions withorganic ligands (e.g., chelants) frequently leads to a


significant decrease in the stability of metal oxides,which may, under some conditions, indicate anincreased dissolution tendency in the passive state(Silverman,80,81 Kubal and Panacek,82 Silverman andSilverman83). An important example of the impor-tance of complexation is provided by the behaviorof copper and other metals in ammoniated environ-ments. Figure 2 (upper diagram) illustrates an E–pHdiagram for copper in a 0.2m NH3 solution. Asshown in the figure, the copper oxide stability fieldis bisected by the stability area of an aqueous com-plex. The formation of a stable dissolved complexindicates that the passivity of copper and copper-basealloys may be adversely affected in weakly alkalineNH3-containing environments. In reality, ammoniaattack on copper-base alloys is observed in steamcycle environments.63 Stability diagrams are a usefultool for the qualitative evaluation of the tendency ofmetals to corrode in such environments.

2.38.2.3.3 Diagrams for nonideal solutions

As long as the solution is assumed to be ideal (i.e.,gi ¼ 1 in eqn [2]), the chemical and electrochemicalequilibrium expressions can be written in an analyti-cal form, and the E–pH diagrams can be generatedsemianalytically. For nonideal solutions, the analyti-cal character of the equilibrium lines can be pre-served if fixed activity coefficients are assumed foreach species (see Bianchi and Longhi79). However,such an approach does not have a general characterbecause the activities of species change as a function ofpH. This is due to the fact that, in real systems, pHchanges result from varying concentrations of acidsand bases, which influence the activity coefficients ofall solution species. In a nonideal solution, the activitiesof all species are inextricably linked to each otherbecause they all are obtained by differentiating thesolution’s excess Gibbs energy with respect to the num-ber of moles of the individual species.2 Therefore, in ageneral case, the equilibrium expressions (eqns [21] and[22]) can no longer be expressed by analytical expres-sions. This necessitates a modification of the algorithmfor generating stability diagrams. A general methodol-ogy for constructing stability diagrams of nonidealsolutions has been developed by Anderko et al.62

In general, the nonideality of aqueous solutionsmay shift the location of the equilibrium linesbecause the activity coefficients may vary by one oreven two orders of magnitude. Such effects becomepronounced in concentrated electrolyte solutions andin mixed-solvent solutions, in which water is notnecessarily the predominant solvent.

n (2010), vol. 2, pp. 1585-1629

2.0

1.5

1.0

0.5

0.0

–0.5

–1.0

–1.5

–2.0

2.0

1.5

1.0

0.5

0.0

–0.5

–1.0

–1.5

–2.0–3.0 –2.0 –1.0 0.0 1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0pH

CuO{s}

Cu(

OH

)– 3

Cu(

OH

)– 42Cu(NH3)4+2

Cu(NH3)4+2

Cu(NH3)5+2

Cu(OH)2{aq}

Cu(

NH

3)3+

2

CuN

H3+

2

Cu(

NH

3)2+

2C

uO{s

} Cu(

OH

) 2{a

q}

b

a

Cu+2Cu+

Cu2O{s}

Cu{s}

CuO{s}

Cu{s}Cu+

Cu2O{s}

E (S

HE

)E

(SH

E)

a

b

Log10 [M(NH3)]

CuC

l{s}

Figure 2 Use of thermodynamic stability diagrams to analyze the effect of ammonia on the corrosion of copper. The upper

plot is an E–pH diagram for Cu in a 0.2m NH3 solution. The lower plot is a potential–ammonia molality diagram.



2.38.2.3.4 Diagrams for alloys

The vast majority of the published stability diagramshave been developed for pure metals. However, sev-eral studies have been devoted to generating stabilitydiagrams for alloys, particularly those from theFe–Ni–Cr–Mo family (Cubicciotti,84,85 Beverskogand Puigdomenech,69 Yang et al.,86 Anderko et al.87).In general, a stability diagram for an alloy is a super-position of partial diagrams for the individual com-ponents of the alloy.69 However, the partial diagrams


are not independent because the alloy componentsform a solid solution and, therefore, their propertiesare linked. The superposition of partial diagramsmakes it possible to analyze the tendency of alloycomponents for preferential dissolution. This maybe the case when a diagram indicates that one alloycomponent has a tendency to form a passivatingoxide, whereas another component has a tendencyto form ions. Also, stability diagrams indicate in asimple way which alloy component is more anodic.

n (2010), vol. 2, pp. 1585-1629



There are two key effects of alloying on thermo-dynamic equilibrium, that is, the formation of mixedsolid corrosion products and the nonideality of alloycomponents in the solid phase. The formation ofmixed solid corrosion products may be particularlysignificant for passive systems. Fe–Ni–Cr alloys mayform mixed oxides, including NiFe2O4, FeCr2O4, andNiCr2O4. Such oxides may be more stable than thesingle oxides of chromium, nickel, and iron, whichappear in stability diagrams for pure metals. E–pHdiagrams that include mixed oxide phases have beenreported by Cubicciotti,84,85 Beverskog and Puigdo-menech,69 and Yang et al.86 for Fe–Ni–Cr alloys inideal aqueous solutions.

Another fundamental effect of alloying on thethermodynamic behavior results from the nonideal-ity of the solid solution phase. In contrast to puremetals, the activity of a metal in an alloy is no longerequal to 1. This affects the value of the equilibriumpotential for metal dissolution, which determinesthe upper boundary of the stability field of an alloycomponent on an E–pH diagram. As with the liquidphase, the activity of solid solution componentscan be obtained from a comprehensive excess Gibbsenergy model of the solid phase. Thermodynamicmodeling of alloys is a very wide area of research,which is beyond the scope of this chapter (seeLupis88 and Saunders and Miodownik89 for reviews).Detailed models have been developed for the ther-modynamic behavior of alloys on the basis of hightemperature data that are relevant to metallurgicalprocesses. These models can be, in principle, extrapo-lated to lower temperatures that are of interest inaqueous corrosion. Despite the inherent uncertaintiesassociated with extrapolation, estimates of activities ofalloy components can be obtained in this way andincorporated into the calculation of stability diagrams.Such an approach was used by Anderko et al.87 in astudy in which solid solution models were coupledwith an algorithm for generating stability diagramsfor Fe–Ni–Cr–Mo–C and Cu–Ni alloys. In that work,the solid solution models of Hertzman,90 Hertzmanand Jarl,91 and Anderson and Lange92 were imple-mented to estimate the activities of alloy components.

It should be noted that the effect of varying activ-ities of alloy components on E–pH diagrams is rela-tively limited. This is due to the fact that alloy phasenonideality, while crucial for modeling alloy metal-lurgy, is only a secondary contribution to the ener-getics of reactions between metals and aqueousspecies. The effect of alloy solution nonideality man-ifests itself in a limited shift of the upper boundary of


the metal stability area in E–pH diagrams. The effectof mixed oxide phases is usually much more pro-nounced on stability diagrams.

2.38.2.3.5 Potential–concentration diagramsOnce the thermodynamic treatment of solutionchemistry is extended to allow for the effect of spe-cies other than H+ and H2O (see eqns [21] and [22]),other independent variables become possible in addi-tion to E and pH. The chemical and electrochemicalboundaries can then be calculated as a function ofconcentration variables other than pH. In particular,the stability diagrams can be generated as a functionof the concentration of active solution species thatmay react with metals through precipitation, com-plexation, or other reactions. Thus, an extension ofE–pH diagrams to E–species concentration diagramsis relatively straightforward. The algorithm devel-oped by Anderko et al.62 for nonideal solutions isequally applicable to E–pH and E–concentration dia-grams. An example of such a diagram is provided inthe lower plot of Figure 2. This plot is an E–NH3

molality diagram and illustrates the thermodynamicbehavior of copper when increasing amounts ofammonia are added to an aqueous environment. Inthis simulation, pH varies with ammonia concentra-tion. However, pH is less important in this casethan the concentration of NH3 because the stabilityof copper oxide is controlled by the formation ofcopper–ammonia complexes. The diagram indi-cates at which concentration of ammonia the copperoxide film becomes thermodynamically unstable,thus increasing the tendency for metal dissolution.

2.38.2.4 Chemical EquilibriumComputations

While the E–concentration diagrams are merely anextension of Pourbaix’s original concept of stabilitydiagrams, a completely different kind of thermody-namic analysis can also be performed using electro-lyte thermodynamics. This kind of analysis relieson solving the chemical equilibrium expressionswithout using the potential as an independent vari-able. Assuming that temperature, pressure, and start-ing amounts of components (both metals and solutionspecies) are known, the equilibrium state of thesystem of interest can be found by simultaneouslysolving the following set of equations:

1. chemical equilibrium equations (eqn [6]) for alinearly independent set of reactions that link allthe species that exist in the system; this set of

n (2010), vol. 2, pp. 1585-1629



equations also includes solid–liquid equilibriumreactions (or precipitation equilibria);

2. if applicable, vapor–liquid and, possibly, liquid–liquid equilibrium equations (i.e., the equality ofchemical potentials of species in coexisting phases),

3. material balance equations for each element in thesystem, and

4. electroneutrality balance condition.

Once the solution is found, the potential can becalculated using the Nernst equation for any redoxpair in the equilibrated system. It should be notedthat procedures for solving these equations are veryinvolved and require sophisticated numerical techni-ques. Methods for solving such electrolyte equilib-rium problems were reviewed by Zemaitis et al.4 andRafal et al.,7 and will not be discussed here.

The solution of the chemical equilibrium conditionsprovides detailed information about the thermodynam-ically stable form(s) of the metal in a particular system.This is in contrast with the information presented onthe E–pH diagrams, which essentially indicate whethera given oxidation reaction (e.g., dissolution of a metal)can occur simultaneously with a given reduction reac-tion (e.g., reduction of water). In the chemical equilib-rium algorithm outlined above, all possible reductionand oxidation equilibria are solved simultaneously.This provides very detailed information about thechemical identity of all species and their concentra-tions. On the other hand, this kind of information is less

0.0

–1.0

–2.0

–3.0

–4.0

–5.0

–6.0

–7.0

–8.0–9.0 –8.0 –7.0 –6.0 –

log

log

[M(H

2S)]

Figure 3 Stability diagram for solid iron corrosion products asnumber of moles of H2S per 1 kg of H2O. The diagram has been

solid–liquid–vapor reactions.93


conducive to gaining qualitative insight into the keyanodic and cathodic reactions that occur in the system.

To visualize the information obtained fromdetailed chemical equilibrium computations, stabilitydiagrams can be constructed by selecting key inde-pendent variables. Such diagrams can be referred toas ‘chemical’ as opposed to ‘electrochemical’ becausethey do not involve the potential as an independentvariable. An example of such a diagram is shown inFigure 3, which shows the stability areas of solidcorrosion products of iron as a function of the amountof dissolved iron and hydrogen sulfide in 1 kg ofwater. Diagrams of this type can be generated byrepeatedly solving the chemical equilibrium expres-sions outlined above and tracing the stability bound-aries of various species (Lencka et al.,94 Sridharet al.93). Such diagrams are useful for illustrating thetransition between various corrosion products (e.g.,iron sulfide and magnetite in Figure 3) as a functionof environmental conditions. Diagrams of this kindhave been generated by Sridhar et al.95 to identify theconditions under which FeCO3, Fe3O4, and Fe2O3

coexist as corrosion products of carbon steel. Thisapproach was used to elucidate conditions that areconducive to intergranular stress corrosion crackingin weakly alkaline carbonate systems, which has beenassociated with the transition between iron carbonate,iron (II), and iron (III) corrosion products.

It should be noted that simplified ‘chemical’ dia-grams can be generated using activities rather than

FeS(mackinawite){s}

Fe3O4{s}

5.0 –4.0 –3.0 –2.0 –1.0[M(Fe)]

a function of the molality of dissolved Fe and the totalconstructed by computing the equilibrium states of

n (2010), vol. 2, pp. 1585-1629



concentrations of species as independent variables.Such an approach makes it possible to obtain the‘chemical’ diagrams in a semianalytical way, as withthe classical Pourbaix diagrams. An example of suchdiagrams is provided by Mohr and McNeil96 forthe Cu–H–O–Cl system. However, such simplified‘chemical’ diagrams suffer from the fundamentaldisadvantage that species activities are not directlymeasurable and are, therefore, much less suitable asindependent variables than concentrations. Compre-hensive ‘chemical’ diagrams can be obtained only bysimultaneously solving the chemical and phase equilib-rium expressions for each set of conditions, preferablywith the help of a realistic model for liquid-phaseactivity coefficients.

2.38.2.5 Problems of Metastability

The main strength of the stability diagrams lies intheir purely thermodynamic nature. Accordingly,they can be generated using equilibrium thermo-chemical properties that are obtained from a varietyof classical experimental sources (e.g., solubility,vapor pressure, calorimetric or electromotive forcemeasurements for bulk systems) without recourse

3

Metastable Fe2O3

Metastable Fe3O4

c

2.0

1.5

1.0

0.5

0.0

–0.5

–1.0

–1.5

–2.00.0 2.0 4.0 6.0

p

E (S

HE

)

Fe+2

NO3–

N2{aq}

DiluteHNO3

Fe+3

NH4+

Fe

ConcentratedHNO3

F

Figure 4 Metastability on the E–pH diagram and the interpret

concentrated nitric acid. The stability fields of the oxides (Fe3O4

dashed lines). An E–pH diagram for nitrogen is superimposed oline for the reduction of HNO3 to HNO2. The ovals marked as ‘c

potential ranges that are expected for iron in concentrated and


to any electrochemical kinetic studies of surfaces.At the same time, the purely thermodynamic natureof the diagrams is also their main weakness. An impor-tant limitation of the stability diagrams is the fact thatthey predict the equilibrium phases, whereas the actualcorrosion behavior may be controlled by metastablephases. The same limitation applies to the detailedthermodynamic computations outlined earlier.

To illustrate the problems of metastability, it isinstructive to examine how E–pH diagrams can beused to rationalize the Faraday paradox of ironcorrosion in nitric acid, which historically played agreat role in establishing the concept of passivity(Macdonald97). Faraday’s key observation was thatiron easily corroded in dilute nitric acid with theevolution of hydrogen. However, it did not corrodewith an appreciable rate in concentrated HNO3 solu-tions despite their greater acidity. When scratched inthe solution, an iron sample would corrode for a shorttime and, then, rapidly passivate. A stability diagramfor this system is shown in Figure 4. This figurepresents a superposition of an E–pH diagram foriron species and another one for nitrogen species.Dilute nitric acid is a weak oxidizing agent and,therefore, the main cathodic reaction in this case is

Fe2O3{s}

Fe3O4{s}

8.0 10.0 12.0 14.0

H

a

{s} NH3{aq}

Fe(OH)3–

eO4–2

FeO

H

b

ation of the Faraday paradox of iron corrosion in dilute and

and Fe2O3) are extended into the metastable region (long

n the diagram for Fe. The line ‘c’ is a metastable equilibriumoncentrated HNO3’ and ‘dilute HNO3’ indicate the mixed

dilute HNO3, respectively.

n (2010), vol. 2, pp. 1585-1629



the common reaction of reduction of H+ ions. Theequilibrium potential for this reaction is given by thedotted line marked as ‘a’ in the diagram. The anodicprocess is the oxidation of Fe, for which the equilib-rium potential is given by the equilibrium linebetween Fe(s) and Fe2+ in the diagram. Thus, thecorrosion potential will establish itself between the ‘a’line and the upper limit of the stability field of ele-mental iron. This corrosion potential range is approx-imately shown by the lower ellipsoid in Figure 4. Inthis region, the stable iron species is Fe2+ and, there-fore, the stability diagram predicts the dissolution ofiron with the formation of Fe2+ ions. Unlike dilutenitric acid, concentrated HNO3 is a strong oxidizingagent. The main reduction reaction in this case is

NO�3 þ 2Hþ þ 2e� ¼ NO�2 þH2O ½23�The equilibrium potential for this reaction is shownby a line marked as ‘c’ in Figure 4. It should be notedthat the nitrite ions (NO2

�) are metastable and, there-fore, a stability field of nitrites does not appear on anE–pH diagram of nitrogen. Nevertheless, the equilib-rium potential for reaction eqn [21] can be easilycalculated as described above. It lies within the sta-bility field of elemental nitrogen N2(aq). The corro-sion potential will then establish itself at a muchhigher potential, relatively close to the dominantcathodic line ‘c.’ The likely location of the corrosionpotential is outlined in Figure 4 by an ellipsoidmarked ‘concentrated HNO3.’ However, the stablespecies in this region are the Fe2+ and Fe3+ ions.Thus, the stability diagram does not explain theFaraday paradox as long as only stable species areconsidered. As indicated by Macdonald,97 the E–pHdiagram becomes consistent with experimentalobservation when the existence of metastable phasesis allowed for. The dashed lines in Figure 4 show themetastable extensions of the Fe2+/Fe3O4 and Fe3O4/Fe2O3 equilibrium lines into the acidic range. Thus, itis clear that the corrosion potential of iron in con-centrated HNO3 is likely to establish itself in a poten-tial range in which metastable iron oxides are stable.In contrast, the metastable solids are not expected toform at the low potentials that correspond to diluteHNO3 environments (i.e., below the Fe2+/metastableFe3O4 boundary). If the surface is scratched, the largesupply of H+ ions near the scratch will create condi-tions under which the potential will be below the ‘a’line and short-term hydrogen evolution will follow.However, this will lead to a rapid depletion of the H+

ions, and the potential will shift to higher values, atwhich metastable phases can lead to passivation.


The formation of metastable solid phases is fairlycommon. The metastability of chromium oxidephases is important for the passivity of Fe–Ni–Cralloys in acidic solutions. Although the E–pH dia-grams indicate that chromium oxides/hydroxidescease to be stable in relatively weakly acidic solu-tions, the practical passivity range extends to a moreacidic range for numerous alloys. Also, the metasta-bility of iron sulfide phases plays an important role inthe behavior of corrosion products in H2S containingenvironments. Anderko and Shuler75 used stabilitydiagrams to evaluate the natural sequence of forma-tion of various iron sulfide phases (e.g., amorphousFeS, mackinawite, pyrrhotite, greigite, marcasite, orpyrite). The sequence of formation of such phasescould be predicted on the basis of the simple butreasonably accurate rule that the order of formationof solids is the inverse of the order of their thermo-dynamic stability. Stability diagrams are then used topredict the conditions under which various metasta-ble phases are likely to form.

Thermodynamics provides a convenient startingpoint in the simulation of aqueous corrosion. Althoughit is inherently incapable of predicting the rates ofinterfacial phenomena, it is useful for predicting thefinal state toward which the system should evolve if itis to reach equilibrium. Furthermore, the computationof the final, thermodynamic equilibrium state can berefined by taking into account the metastable phases.In addition to determining the equilibrium state inmetal–environment systems, electrolyte thermody-namics provides information on the properties of theenvironment. Such information, including speciationin the liquid phase, concentrations and activities ofindividual species, and phase equilibria, is necessaryfor constructing kinetic models of corrosion.

2.38.3 Modeling the Kinetics ofAqueous Corrosion

Aqueous corrosion is intrinsically an electrochemicalprocess that involves charge transfer at a metal–solution interface. Because aqueous corrosion is aheterogeneous process, it involves the following fun-damental steps:

1. reactions in the bulk aqueous environment,2. mass transport of reactants to the surface,3. charge transfer reactions at the metal surface,4. mass transport of reaction products from the sur-

face, and5. reactions of the products in the bulk environment.

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Reactions in the bulk environment (1 and 5) can beconsidered, with some important exceptions, to beequilibrium. As long as they are treated as equilib-rium phenomena, they remain within the domain ofelectrolyte thermodynamics. On the other hand, thecharge transfer reactions at the surface (3), and theircoupling with mass transport (2 and 4) require thetools of electrochemical kinetics. In this section, wereview the fundamentals of modeling approaches toelectrochemical kinetics.

The objective of modeling the kinetics of aqueouscorrosion is to relate the rate of electrochemicalcorrosion to external conditions (e.g., environmentcomposition, temperature, and pressure), flow condi-tions, and the chemistry and metallurgical character-istics of the corroding interface. The two mainquantities that can be obtained from electrochemicalmodeling are the corrosion rate (vcorr, often expressedas the corrosion current density icorr) and corrosionpotential (Ecorr). For practical applications, the calcu-lation of the corrosion rate is of primary interest forsimulating general corrosion and the rate of dissolu-tion in occluded environments such as pits or cre-vices. The value of the corrosion potential (Ecorr) isalso of interest because there is often a relationshipbetween the value of the corrosion potential and thetype of corrosion damage that occurs. In general, ifthe corrosion potential is above a certain criticalpotential (Ecrit), a specific form of corrosion that isassociated with Ecrit can occur, typically at a ratethat is determined by the difference Ecorr – Ecrit.This general observation applies to localized corro-sion including pitting, crevice corrosion, intergranu-lar stress corrosion cracking, and so on. An internallyconsistent model for general corrosion should simul-taneously provide reasonable values of the corrosionrate and corrosion potential. Thus, the computationof the corrosion potential is of interest not so muchfor modeling general corrosion but for predictingother forms of corrosion.

2.38.3.1 Kinetics of Charge-TransferReactions

The theory of charge-transfer reactions is well devel-oped and has been reviewed in detail by a number ofauthors including Vetter,98 Bockris and Reddy,99

Kaesche,100 Bockris and Khan,101 and Gileadi.102

Here, we summarize the key relationships that formthe basis of modeling.

Considering a simple reaction of transfer of nelectrons between two species, a reduced form ‘Red’


and an oxidized form ‘Ox’:

Red�! �ra

rcOxþ ne ½24�

the current density associated with this reaction is,according to Faraday’s law, equal to the differencebetween the anodic rate va and the cathodic rate vc,multiplied by nF:

i ¼ nFðva � vcÞ ½25�According to the theory of electrochemical kinetics,98

the rates of the anodic and cathodic reactions arerelated to the potential and the concentrations ofthe reacting species at the phase boundary, that is

ia ¼ nFva ¼ nFkacx;rr;s exp

aanFERT

� �½26�

ic ¼ �nFvc ¼ �nFkccx;oo;s exp � acnFERT

� �½27�

where ka and kc are the anodic and cathodic rateconstants, aa and ac are the anodic and cathodicelectrochemical transfer coefficients, cr,s and co,s arethe concentrations of the reduced (r) and oxidized (o)forms at the surface, respectively, and x,r and x,o arethe reaction orders with respect to the reduced andoxidized species. For a given individual redox pro-cess, the anodic and cathodic electrochemical trans-fer coefficients are interrelated as ac ¼ 1� aa. Thetotal current density for reaction eqn [24] is then

i¼ nFkacx;rr;s exp

aanFERT

� ��nFkcc

x;oo;s exp �

acnFERT

� �½28�

At the equilibrium (reversible) potential E0, the cur-rent density i is equal to zero. In the absence of a netcurrent, the concentrations of the species at the sur-face are equal to their bulk concentrations (i.e.,cr;s¼ cr;b and co;s¼ co;b). Then, the current density ofthe anodic process is equal to that of the anodicprocess and is defined as the exchange current den-sity i0, that is,

i0¼ nFkacx;rr;b exp

aanFE0RT

� �¼ nFkcc

x;oo;b exp �

acnFE0RT

� �½29�

Using eqn [29], the equation for the current density[28] can be expressed in terms of the exchange cur-rent density and the overvoltage �¼ E�E0:

i¼i0 cr;scr;b

� �x;r

expaanFðE�E0Þ

RT

� �

� i0co;sco;b

� �x;o

exp �acnFðE�E0ÞRT

� �½30�

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The ratios cr;s=cr;b and co;s=co;b depend on the trans-port of reactants and products to and from the corrod-ing interface. If the mass transport is slow compared tocharge transfer, the surface concentrations becomedifferent from those in the bulk. Conversely, if chargetransfer is slow relative to mass transfer, the reaction isunder charge transfer control and the ratios are equalto one not only at the equilibrium potential. In such acase, eqn [30] takes a particularly simple form and isusually referred to as the Butler–Volmer equation forcharge-transfer reactions:

i¼ i0 expaanFðE�E0Þ

RT

� �� i0 exp �acnFðE�E0Þ

RT

� �½31�

The electrochemical transfer coefficient a depends onthe mechanism of the charge transfer reaction. Forsome reactions, its value can be deduced from mecha-nistic considerations. However, in many cases, it needsto be determined empirically. It can be determined inthe form of empirical Tafel coefficients defined as theslope of a plot of potential against the logarithm ofcurrent density, that is

ba¼dE

dln ia; bc¼

dE

d ln ic½32�

which yields

ba¼RT

aanF; bc¼

RT

acnF½33�

or, in a more traditional decimal logarithm form:

ba¼ 2:303RT

aanF; bc¼ 2:303RT

acnF½34�

Then, the Tafel coefficients ba or bc can be used in eqns[28] or [30] instead of the electrochemical transfercoefficient.

The above formalism includes both the cathodic andanodic process for a particular redox couple. However,in practical corrosion modeling, it is usually entirelysufficient to include only either a cathodic or an anodicpartial current for a given redox process. Specifically,the cathodic partial process can be neglected for metal-ion reactions because the deposition of metal ions (i.e.,the reverse of metal dissolution) is typically not ofpractical significance in corrosion. Similarly, the anodicpartial process can be usually neglected for oxidizingagents because it is only their reduction that is ofinterest for corrosion. There are some exceptions tothis rule, for example, in the case of relatively noblemetals whose ions can be reduced under realistic con-ditions. Nevertheless, in the remainder of this review,we will separately consider partial anodic and cathodicprocesses for corrosion-related reactions.


As mentioned above, the concentrations of reac-tants and products at the surface depend on the masstransport to and from the corroding interface. Ingeneral, three mechanisms contribute to mass trans-port, that is, diffusion, migration, and convection. Inmany practical applications, migration can beneglected. This is the case for the transport of neutralmolecules in any environment and, also, for the trans-port of charged species in environments that containappreciable amounts of background electrolytic com-ponents (e.g., as supporting electrolyte). Migrationbecomes important in ionic systems in which thereis no supporting electrolyte. We will return to thetreatment of migration later in this review. If migra-tion is neglected, the treatment of mass transfer bydiffusion and convection can be simplified by usingthe concept of the Nernst diffusion layer. Accordingto this concept, the environment near the corrodingsurface can be divided into two regions. In the innerregion, called the Nernst diffusion layer, convectionis negligible and diffusion is the only mechanism oftransport. In the outer region, concentrations areconsidered to be uniform and equal to those in thebulk solution. Thus, the concentration changes line-arly from the surface concentration to the bulk con-centration over a distance d, which is the thickness ofthe diffusion layer. In such a model, the flux of aspecies i in the vicinity of a corroding interface isgiven by Fick’s law

Ji ¼ �Di@ci@z

� �z¼ 0

½35�

where Di is the diffusion coefficient of species i and zin the direction perpendicular to the surface. Integra-tion of eqn [35] over the thickness of the diffusionlayer gives:

Ji ¼ �Dici;b � ci;s

di½36�

It should be noted that the diffusion layer thicknessd is not a general physical property of the system.Rather, it is a convenient mathematical construct thatmakes it possible to separate the effects of diffusionand convection. It depends on the flow conditions,properties of the environment, and the diffusion coef-ficient of individual species. Thus, it may be differentfor various species. Methods for calculating d will beoutlined later in this review. Equation [36] can beapplied to both the reactants that enter into electro-chemical reactions at the interface and to corrosionproducts that leave the interface. Then, it can becombined with Faraday’s law to obtain the current

n (2010), vol. 2, pp. 1585-1629



density. For an oxidant o, eqn [36] yields an expres-sion for a cathodic partial current density:

ic ¼ nFJo ¼ �nFDoco;b � co;s

do½37�

According to eqn [37], the current density reaches amaximum, limiting value when the surface concen-tration co,s decreases to zero. This condition definesthe limiting current density, that is

ic;L ¼ � nFDoco;bdo

½38�

For a corrosion product (e.g., Me ions), an analogousequation can be written for an anodic current density

ia ¼ nFJMe ¼ �nFDMecMe;b � cMe;s

dMe½39�

In eqn [39], the surface concentration is typicallylimited by the solubility of corrosion products. Thus,a limiting anodic current density can be reached whenthe surface concentration of metal ions corresponds tothe metal solubility, that is

ia;L ¼ �nFDMecMe;b � cMe;sat

dMe½40�

Combination of eqns [26]–[28] for charge-transferprocesses and the simplified mass-transport equations(eqns [37] and [39]) yields a general formalism forelectrochemical processes that are influenced by bothcharge transfer and mass transport. For example, for acathodic process, the current density obtained fromeqn [37] is equal to that obtained from eqn [27]. Thus,the surface concentration of the diffusing species canbe obtained from eqn [37] and substituted into eqn[27]. The resulting equation can be solved analyticallyfor ic for some values of the reaction order x,o. If thereaction order is equal to one (i.e., x; o ¼ 1 in eqn [27],a particularly simple relationship is obtained for ic,that is

1

ic¼ 1

ic;ctþ 1

ic;L½41�

where ic,L is the limiting current density (eqn [38]) andic,ct is the charge-transfer contribution to the currentdensity. The latter quantity is given by eqn [27] withthe bulk concentration co,b replacing the surface con-centration, that is

ic ¼ �nFkccx;oo;b exp � acnFERT

� �with x;o ¼ 1 ½42�

An analytical formula can also be obtained whenx;o ¼ 0:5.103 For an arbitrary value of the reaction


order, the current density can be computed numeri-cally by solving a single equation.

It should be noted that eqns [26] and [27] areparticularly simple forms for reactions of the type[24], in which no species other that Red and Oxparticipate. In general, the preexponential part ofeqns [26] and [27] depends on the mechanism of aparticular electrochemical reaction. In general, thepreexponential terms of eqns [26] and [27] can begeneralized using the surface coverage factors, yi, forreactive species that participate in electrochemicalprocesses, that is,

ia ¼ nFkayx11 yx22 . . . yxmm exp

aanFERT

� �½43�

ic ¼ �nFkcyx11 yx22 . . . yxmm exp � acnFERT

� �½44�

The surface coverage factors, yi, are further relatedto the concentrations (or, more precisely, activities)of individual species at the metal surface throughappropriate adsorption isotherms. In general, analysisof reaction mechanisms on the basis of experimentaldata leads to a substantial simplification of eqns [43]and [44]. In many cases, activities of species can bedirectly used in the kinetic expressions rather thanthe surface coverage fractions.

The above formalism makes it possible to set up amodel of electrochemical kinetics on a corrodingmetal surface by considering the following steps:

1. determining all possible partial cathodic andanodic processes that may occur in a givenmetal-environment combination;

2. writing equations for the partial cathodic oranodic current densities associated with chargetransfer reactions (eqns [43] and [44] or simplifi-cations thereof); and

3. writing equations for the mass transport of thespecies that participate in the charge-transferreactions (eqns [37] and [39]). In some simple,but realistic cases a combination of the charge-transfer and mass-transport equations results inanalytical formulas such as eqn [41] for partialelectrochemical processes.

Once the equations for the partial anodic andcathodic processes are established, the behavior of acorroding surface can be modeled on the basis of theWagner–Traud104 theory of metallic corrosion, oftenreferred to as the mixed potential theory. Accordingto the mixed potential theory, the sum of all partialanodic currents is equal to the sum of all cathodic

n (2010), vol. 2, pp. 1585-1629



currents. Further, it is assumed that the electricalpotential of the metal at an anodic site is equal tothat at a cathodic site. These assumptions follow fromthe requirement that charge accumulation within ametal cannot occur and, therefore, the electrons pro-duced as a result of oxidation processes must beconsumed in the reduction processes. Therefore, ina freely corroding system, we haveX

j

Aaia;j þXj

Acic; j ¼ 0 at E ¼ Ecorr ½45�

where Aa and Ac are the areas over which the anodicand cathodic reactions occur, respectively. The firstsum in eqn [45] enumerates all anodic partial reac-tions and the second sum pertains to all cathodicreactions. Equation [45] is written using the signconventions introduced earlier, in which the cathodicprocesses are written with a negative sign.

Equation [45] can be solved to obtain the corro-sion potential, Ecorr. Then, the corrosion current den-sity and, equivalently, the corrosion rate, can becomputed from the anodic current density for metaldissolution at the corrosion potential, that is

icorr ¼ ia;MeðEcorrÞ ½46�At potentials that deviate from the corrosion poten-tial, the left-hand side of eqn [45] represents thepredicted current. Such a computed current versuspotential relationship can be compared with experi-mentally determined polarization behavior.

In the case of general corrosion, Aa¼ Ac and thesolution of eqns [45] and [46] yields the corrosionpotential and general corrosion rate. For localizedcorrosion, there is usually a great disparity betweenthe areas on which the anodic and cathodic processesoperate.

2.38.3.2 Modeling Adsorption Phenomena

Before discussing partial electrochemical reactions, itis necessary to outline the treatment of adsorptionbecause the presence of adsorbed species is fre-quently assumed to derive expressions for electro-chemical processes.

Adsorption of neutral molecules and ions onmetals has been reviewed in detail by Gileadi,105

Damaskin et al.,106 and Habib and Bockris.107 In gen-eral, adsorption leads to the reduction of the surfacearea that is accessible to electrochemical reactions. Insuch cases, adsorption results in a reduction in therate of both anodic and cathodic processes. Thus, therates of electrochemical reactions become modified


by the factor (1�Syj), in which yj is a coveragefraction by species j. At the same time, adsorptionmay result in the formation of surface complexes thathave different dissolution characteristics. An exampleof such a dual effect is provided by halide ions on Fe-group metal surfaces corroding in the active state. Atrelatively low or moderate halide concentrations,adsorption of halides leads to a reduction in electro-chemical reaction rates. However, at higher halideconcentrations, the adsorbed halide ions interferewith the mechanism of anodic dissolution of iron,which may lead to an increase in the corrosion rate.

A general formalism for modeling the effect ofadsorption on electrochemical reactions is providedby the Frumkin isotherm. The Frumkin formalismtakes into account the interactions between the spe-cies adsorbed on the surface. It results from therequirement that the rate of adsorption is equal tothe rate of desorption in the stationary state,105 that is

vads;i ¼ vdes;i ½47�where the subscript i denotes any adsorbable species.The rate of adsorption is given by

vads;i ¼ kads;i 1�Xj

yj

0@

1Aai exp �b

Xj

Aij yj

0@

1A ½48�

where kads,i is an adsorption rate constant, yi is afraction of the surface covered by species i, ai is theactivity of species i in the solution, b is a transfercoefficient, and Aij is a surface interaction coefficientbetween species i and j. The first term in parentheseson the right-hand side of eqn [48] represents theavailable surface, and the second term representsthe effect of pairwise interactions between adsorbedspecies. The rate of desorption is given by

vdes;i ¼ kdes;iyi exp ð1�bÞXj

Aij yj

0@

1A ½49�

where kdes,i is the desorption rate constant. Combina-tion of eqns [48] and [49] yields the Frumkin iso-therm, that is

Kads;i ai ¼ kads;ikdes;i

ai ¼ yi1�P

j

yjexp

Xj

Aij yj

0@

1A ½50�

where Kads,i is an adsorption equilibrium constant.Equation [50] can be simplified if it is assumed thatthe species are independently adsorbed. Then, theinteractions between the species become zero, andeqn [50] takes the form of the well-known Langmuirisotherm:

n (2010), vol. 2, pp. 1585-1629



Kads;iai ¼ yi1�P

j

yj½51�

The Langmuir isotherm is extensively used in elec-trochemical kinetics. The Frumkin isotherm has beenused in some studies when more accurate modelingof adsorption is warranted by experimental data, forexample, in the case of corrosion in very concen-trated brines (Anderko and Young108).

It should be noted that detailed modeling ofadsorption requires taking into account the effect ofpotential on adsorption. Equations [50] and [51] arestrictly valid only when adsorption is not significantlyinfluenced by metal dissolution. An approach toinclude the effect of dissolution and, hence, potentialon adsorption has been developed by Heusler andCartledge109 who proposed an additional process inwhich a metal atom from an uncovered area (1�Syj)reacts with an adsorbed ion from the covered area yito dissolve as ferrous ion. The adsorbed ion is thenpostulated to leave the surface during the reaction,thus contributing to the desorption process. Accord-ingly, eqn [49] is rewritten by adding an additionalterm, that is

vdes;i ¼ kdiyi exp ð1� bÞXj

Aij yj

!þ ides;i ½52�

where the desorption current ides,i is given by

ides;i ¼ kriyi 1�Xj

yj

!aX exp

bFERT

� �½53�

where aX is the activity of possible additional species(e.g., OH�) that participate in the dissolution. In eqn[53], the desorption current is potential-dependentbecause it involves the dissolution of the metal. Equa-tion [53] can be combined with eqns [48] and [52] toform a system of n equations for a solution with nadsorbable species. This system can be solved numer-ically for the coverage fractions yi of each adsorbedspecies. Because of the potential dependence, themodel predicts that the adsorption coverage rapidlydecreases above a certain potential range.

It should be noted that a detailed treatment ofadsorption is not always necessary for modelingaqueous corrosion. In particular, the potential depen-dence of adsorption can be often neglected. In mostcases, simplified approaches are warranted. Specifi-cally, for low surface coverage, the fraction yi can beassumed to be proportional to the activity of thespecies i as shown by eqn [51]. Thus, eqns [43] and


[44] can often be simplified by using activities ofspecies rather than their surface coverages.

2.38.3.3 Partial Electrochemical Reactions

The behavior of a corroding system results from theinterplay of at least two and, frequently, many partialelectrochemical reactions. Such reactions include:

1. anodic dissolution of pure metals and alloys inboth the active and passive state;

2. reduction of protons, which is usually the primarycathodic reaction in acid corrosion;

3. reduction of water molecules, which is frequentlythe main cathodic reaction in deaerated neutraland alkaline solutions;

4. reduction of dissolved species that can act as pro-ton donors such as undissociated carboxylic acids,carbonic acid, hydrogen sulfide, and numerousions that contain protons (e.g., bicarbonates, bisul-fides, etc.);

5. reduction of oxygen, which is a common cathodicprocess in aerated solutions;

6. reduction of metal ions at high oxidation statessuch as Fe(III) or Cu(II), which can be reduced toa lower oxidation state;

7. reduction of oxyanions such as nitrites, nitrates, orhypochlorites in which a nonmetallic element isreduced to a lower oxidation state;

8. oxidation of water to oxygen, which occurs at highpotentials and, therefore, is rarely important infreely corroding systems; and

9. oxidation of metals to higher oxidation states, forexample, Cr(III) to Cr(VI), which may occur in thetranspassive dissolution region of stainless steelsand nickel base alloys.

In this section, we present illustrative examples ofhow these reactions can be modeled in practice.

2.38.3.3.1 Anodic reactions

The dissolution of several pure metals such as iron,copper, or nickel has been extensively investigated.Thus, it is possible to construct practical equationsfor the partial anodic dissolution processes on thebasis of mechanistic information. For most alloys,detailed mechanistic information is not available and,therefore, it is necessary to establish kinetic expres-sions on a more empirical basis.

For iron dissolution, various multistep reactionmechanisms have been proposed. They have beenreviewed in detail by Lorenz and Heusler,110

Drazic,111 and Keddam.112 From the point of view of

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modeling, particularly important parameters are theelectrochemical transfer coefficient and reactionorders with respect to the ions that participate inanodic dissolution. Although there are substantialdifferences between the various proposed mechan-isms, the dependence of the iron dissolution rate inacidic solutions on the activity of hydroxide ions isgenerally accepted. The mechanism proposed byBockris et al.,113 that is

Feþ OH� ! FeOHads þ e� ½54�

FeOHads ! FeOHþads þ e�

ðrate-determining stepÞ ½55�

FeOHþads ! Fe2þ þOH� ½56�predicts that the reaction order with respect to theOH� ion is one because of the intermediate step ofOH� adsorption. The validity of this prediction hasbeen verified for acidic solutions. Other mechanismsyield reaction orders between one and two. Addition-ally, the current density for iron dissolution has beenfound to depend on the activity of water (Smartand Bockris114). The mechanism of Bockris et al.113

also predicts that the anodic transfer coefficient isaFe¼ 1.5, which is consistent with experimentallyobserved Tafel slopes of 30–40mV. Thus, the currentdensity for Fe dissolution in acidic solutions can beexpressed as

iFe;OH ¼ i�Fe;OHaOHacH2O

expaFeFðE � E0;FeÞ

RT

� �½57�

where i�Fe;OH is a temperature-dependent coefficient,the subscript Fe,OH indicates that the dissolutionreaction is mediated by OH� ions, and c is an empiri-cally determined reaction order with respect to theactivity of water. According to Smart and Bockris114

c ¼ 1.6. The effect of the activity of water on thecurrent density becomes significant for concentratedsolutions, for which the activity of water is usuallysignificantly lower than 1.

Although the reaction order with respect to theOH� ions is valid for acidic solutions, it has beenfound that iron dissolution proceeds with little influ-ence of pH for solutions with pH above �4. Bockriset al.113 explained this phenomenon by assuming acertain nonzero reaction order with respect to Fe2+

and by considering the hydrolysis of the Fe2+ ionsthat result from the dissolution. Alternatively, thechange in the reaction order with respect to OH�

ions can be reproduced by assuming that the exchangecurrent density is proportional to the surface coverage


by OH� ions. This assumption is consistent with thereaction mechanism (see eqns [54]–[56]). Thus,eqn [57] can be generalized as108:

iFe;OH ¼ i�Fe;OHyOHacH2O


RT

� �½58�

Assuming that yOH follows the Langmuir adsorptionmodel, eqn [58] can be rewritten as

iFe;OH ¼ i�Fe;OHaOH

1þ KOHaOHacH2O


RT

� �½59�

Equation [59] reduces to eqn [57] for low activities ofOH�, that is, for acidic solutions. For higher concen-trations of hydroxide ions, the reaction order withrespect to OH� becomes zero. This is consistent withthe lack of a dependence of the Fe oxidation reactionon pH in CO2 corrosion of mild steel, which occurs atpH values above�4 (Nesic et al.,115 Nordsveen et al.116).

The effect of halide ions on the dissolution of ironand carbon steel is of particular interest. Adsorbedhalide ions may accelerate the anodic dissolution,especially in concentrated halide solutions. A numberof reaction mechanisms have been proposed to ex-plain this phenomenon. In particular, Chin andNobe117 and Kuo and Nobe118 developed a mecha-nism that postulates a reaction route that is parallel toeqns [52]–[54]. An essentially identical mechanismhas also been proposed by Drazic and Drazic.119

According to this mechanism, a halide-containingsurface complex is responsible for the dissolu-tion. Thus, eqn [54] is followed by the followingparallel route:

FeOHads þ X� ! FeOHX�ads ½60�

FeOHX�ads ! FeOHXads þ e�

ðrate-determining stepÞ ½61�

FeOHXads þHþ ! Fe2þ þ X� þH2O ½62�The mechanism eqns [60]–[62] results in a dissolu-tion current density that depends on the activities ofboth halide and hydroxide ions. In acidic solutions, anequation analogous to eqn [57] can be written as

iFe;X ¼ i�Fe;X asX�a

tOH�exp

aFeFðE � E0;FeÞRT

� �½63�

where the subscript X indicates the halide ions thatmediate the reaction. For chloride systems, s¼ 0.4and t¼ 0.6 when concentrations are used instead of

n (2010), vol. 2, pp. 1585-1629



activities.118 Since the mechanism described by eqns[60]–[62] is assumed to be parallel to the mechanismunder halide-free conditions, the total current den-sity of anodic dissolution can be assumed to be a sumof the contributions of two mechanisms. Also, eqn[63] can be generalized to neutral solutions in anal-ogy with eqn [59]. Additionally, the desorption cur-rent density (eqn [53]) contributes to the totalcurrent, although it becomes important only at rela-tively high potentials, and its numerical significanceis usually limited. Thus, the expression for the totalactive Fe dissolution current in halide solutionsbecomes108

iFe ¼ iFe;OH þ iFe;X þ ides;i ½64�As with iron, anodic dissolution of copper has alsobeen extensively investigated. Kear et al.120 reviewedthe mechanisms and associated expressions for thecurrent density of anodic dissolution of copper in theactive state in chloride environments. Copper disso-lution is generally thought to proceed through theformation of cuprous chloride complexes and to beunder mixed, charge-transfer and transport, controlclose to the corrosion potential. Several authors (Leeand Nobe,121 Deslouis et al.,122 King et al.123) assumedthe following mechanism:

Cuþ Cl� !k1k�1

CuClþ e� ½65�

Cuþ Cl� !k2k�2

CuCl�2 ½66�

The expression for the anodic current densityderived from eqns [65] and [66] is

iCunF¼ k1k2

k�1a2Cl exp

FðE�E0;CuÞRT

� �� k�2aCuCl�2 ½67�

Since the reaction rate depends on the activity of thereaction products (CuCl2

�) at the surface (see thesecond term on the right-hand side of eqn [67]), thereaction is partially controlled by the mass transportof the CuCl2

� ions, and the anodic current density issimultaneously equal to

iCunF¼DCuCl�2

aCuCl�2dCuCl�2

½68�

Equation [68] is a special case of eqn [39] when thebulk concentration of CuCl2

� is negligible.Much less mechanistic information is available for

the anodic dissolution of alloys in the active state. Inthe case of stainless steels and nickel-base alloys, thisis due to the fact that dissolution of these metals in


the passive state is more important than in the activestate. For these alloys, active dissolution is of impor-tance only in acidic solutions. In this case, expressionsfor anodic dissolution need to be established on anempirical basis. For stainless steels and nickel-basealloys, a positive reaction order between one and twowith respect to hydroxide ions is observed. Whilesuch values are similar to those observed for Fe, theexchange current densities are very different andneed to be determined separately for individualalloys.

2.38.3.3.2 Cathodic reactions

Among the numerous possible partial cathodic pro-cesses, the reduction of protons, water molecules, anddissolved oxygen is ubiquitous in aqueous corrosion.Reduction of protons is an important cathodic processin acidic solutions. The overall reaction is given by

Hþ þ e� ! 0:5H2 ½69�The mechanisms of this reaction have been reviewedby Vetter98 and Kaesche.100 Proton reduction proceedsin two steps according to two alternative mechanisms.The Volmer–Heyrovsky mechanism applies to mostmetals, whereas the Volmer–Tafel mechanism maybe observed on certain noble metals. The Volmer–Heyrovsky mechanism can be represented as asequence of two elementary reactions, that is

Hþ þ e� ! Hads ½70�

Hþ þHads ! H2 ½71�It is generally accepted that the H+ reduction reactionmay proceed under activation or mass transfer control.The cathodic process of H+ reduction can be modeledassuming that the reaction order with respect to theprotons is equal to one. Then, eqns [41] and [42] canbe directly used for modeling. In addition to its depen-dence on the activity of protons, there is empiricalevidence that the H+ reduction depends on the activ-ity of water. According to Smart et al.,124 the reactionorder with respect to water activity is 2.2 on iron. Theelectrochemical transfer coefficient can be assumed tobe equal to�0.5 for carbon steels and many corrosion-resistant alloys, which corresponds to a Tafel slope of118mV at 25 �C.

As the pH of a solution increases, the importanceof the proton reduction reaction rapidly decreases. Inneutral and alkaline solutions, the reduction of watermolecules becomes predominant unless stronger oxi-dizing agents (e.g., oxygen) are present in the system.The water reduction is given by

n (2010), vol. 2, pp. 1585-1629



H2Oþ e� ! 0:5H2 þ OH� ½72�and is thermodynamically equivalent to the reductionof protons. However, its kinetic characteristics aredifferent. Unlike the reduction of protons, the waterreduction reaction typically does not exhibit a limit-ing current density because there are no diffusionlimitations for the transport of H2O molecules tothe surface. This remains true as long as the systemis predominantly aqueous. The water reduction pro-cess can be modeled by assuming the same reactionorder with respect to H2O as that for proton reduc-tion. Also, practically the same value of the electro-chemical transfer coefficient can be assumed.

Reduction of oxygen, that is

O2 þ 4Hþ þ 4e� ! 2H2O ½73�is the predominant cathodic reaction in aerated aque-ous solutions unless the solution contains strongeroxidizing agents such as ferric, cupric, or hypochloriteions. The mechanism of oxygen reduction is substan-tially more complex than the mechanisms of H+ orH2O reduction. Oxygen reduction on iron and car-bon steel has been reviewed by Jovancicevic andBockris,125 Zecevic et al.,126 and Jovancicevic.127 Onstainless steels, it has been analyzed by Le Bozecet al.,128 Kapusta,129 and in papers cited therein. Oncopper, it has been studied by King et al.130 In general,it has been established that the reaction may proceedeither through a four-electron pathway, which leads tothe reduction of O2 to H2O, or through a two-electronpathway, which leads to the formation of H2O2 as anintermediate. An overall reaction scheme may berepresented as

O2,bulk

O2,bulk

O2,surf O2,ads H2O2,ads

H2O2,bulk H2O2,surf

OH–

½74�

where the subscripts ‘surf ’ and ‘ads’ denote the oxygenin the diffusion layer close to the surface and oxygenadsorbed on the surface, respectively. The absorbedintermediate H2O2 can be either further reduced toOH� or desorbed and dissolved in the solution orconverted back to oxygen through decomposition orreoxidation. The actual reaction pathway is influencedby many factors such as the surface treatment of theelectrode.128 However, the four-electron reductionpath from O2 to OH� seems to predominate.129 Oxy-gen reduction may be under charge transfer or masstransfer control, due to the diffusion of dissolved oxy-gen molecules. For passive metals, the process is usu-ally under charge transfer control because the limiting


current density for oxygen reduction is usually greaterthan the passive current density at typical dissolvedoxygen concentrations.

For modeling purposes, the key parameters are theelectrochemical transfer coefficient and the reactionorders with respect to dissolved oxygen and protons.These parameters determine the dependence of thereduction reaction on dissolved oxygen concentra-tion (or, equivalently, the partial pressure of oxygen)and on pH. Once these parameters are known, theoxygen reduction process can be modeled on a semi-empirical basis. The current density for oxygenreduction can be written as:

iO2¼ i�O2

aqO2;s

arHþ;s exp�aO2

F E � E0;O2

� �RT

� ½75�

Equation [75] needs to be coupled with eqn [37] withn¼ 4 for mass-transfer limitations. The reactionorders q and r in eqn [75] are, in general, specific tothe metal surface although they are expected to besimilar within families of alloys. For stainless steels,there seems to be a consensus that the reaction orderwith respect to dissolved oxygen is 0.5 (Kapusta,129

Sridhar et al.131), whereas the order with respect toprotons ranges from 0.5 to 1 (or, equivalently, theorder with respect to hydroxide ions varies from�0.5 to �1). For passive iron or carbon steel, thereaction order with respect to O2 has been reportedas 0.5 (Calvo and Schiffrin132) or 1 (Jovancicevic andBockris,125 Jovancicevic127). For copper, a value of 1has been reported (King et al.130).

Another important cathodic reaction is the reduc-tion of transition metal ions such as Fe3+ and Cu2+ tolower oxidation states, for example

Fe3þ þ e� ! Fe2þ ½76�This process can be modeled as a first-order reac-tion with respect to the activity of ferric ions by takinginto account the mass transport limitations (eqn [37]).

All the cathodic reactions discussed above mayproceed under mass transfer limitations due to thediffusion of reactants to the corroding surface. How-ever, cathodic limiting current densities may alsoarise because of limitations due to homogeneousreactions in the solution. A prominent example ofsuch a reaction is the reduction of carbonic acid,which is the key cathodic process in CO2 corrosionof carbon steel. This reaction accounts for the sub-stantially higher corrosivity of CO2 solutions thanmineral acid solutions at the same pH. Carbonicacid results from the hydration of dissolved CO2,that is

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CO2 þH2O ¼ H2CO3 ½77�Reaction eqn [77] is followed by the reduction ofH2CO3 on the surface, that is

H2CO3 þ e� ! 0:5H2 þ OH� ½78�which is thermodynamically equivalent to the reduc-tion of protons, but is characterized by differentkinetics. The H2CO3 reduction is under activationor chemical reaction control, and can be modeledusing eqn [41]. The charge transfer current isexpressed as (Nesic et al.115):

iH2CO3¼ i�H2CO3

aH2CO3a�0:5Hþ

exp�aH2CO3

F E � E0;H� �

RT

� ½79�

where the transfer coefficient can be assumed to beequal to that for H2O reduction. The limiting currentdensity can be calculated from an equation developedby Nesic et al.115 on the basis of a formula derived byVetter98 for processes with a rate-determining homo-geneous reaction in the solution. Here, the rate-determining reaction is the hydration of CO2 andthe limiting current density is:

iH2CO3;L ¼ FaH2CO3DH2CO3

KH2CO3kfH2CO3

� �1=2½80�

where DH2CO3, KH2CO3

, and k fH2CO3are the diffusion

coefficient of H2CO3, equilibrium constant for thehydration of CO2, and forward reaction constant forthe hydration reaction, respectively.

2.38.3.3.3 Temperature dependence

The rates of the majority of partial anodic andcathodic processes are strongly dependent on tem-perature. This temperature dependence can be mod-eled by assuming that the concentration-independentpart of the exchange current density (here denotedby i*) is expressed as

i�ðT Þ ¼ i�ðTref Þexp �DH 6¼

R

1

T� 1

Tref

� �� ½81�

Equation [81] is equivalent to assuming a constantenthalpy of activation DH 6¼ for each partial process.

2.38.3.4 Modeling Mass Transport UsingMass Transfer Coefficients

To calculate the mass-transport effects on electro-chemical kinetics according to eqns [37]–[40], it isnecessary to predict the diffusion layer thickness di


or, equivalently, the limiting current density. Theo-retical formulas for these quantities cannot beobtained for arbitrary flow conditions and, therefore,empirical approaches are necessary for most practicalapplications.

In the case of a rotating disk electrode, a theoreti-cal solution has been derived by Levich.133 It is note-worthy that Levich’s solution preceded experimentalresults. The thickness of the diffusion layer on arotating disk electrode is

di ¼ 1:61D1=3i n1=6o�1=2 ½82�

where Di is the diffusion coefficient of the reactingspecies, o is the rotation rate, and n is the kinematicviscosity, which is the ratio of the dynamic viscosityand density, that is

n ¼ �=r ½83�In view of the relationships between the thickness ofthe diffusion layer and the limiting current density(eqns [38] and [40]), a physically equivalent predic-tive expression can be written for the limiting currentdensity. For example, the limiting current density of acathodic reaction (eqn [38]) then becomes:

ic;L ¼ �0:6205nFco;bD2=3o n�1=6o1=2 ½84�

For many other flow geometries, mass transport canbe calculated using empirical correlations expressedin terms of the mass transfer coefficient km. In general,the mass transfer coefficient is defined as

km ¼ Reaction rate

Concentration driving force½85�

For an electrochemical reaction, the reaction rate isexpressed using the current density, and eqn [36] fora mass transport-limited reaction can be rewritten interms of the mass transfer coefficient km as

Ji ¼ iiniF¼ �Di

ci;b � ci;sdi

¼ �km;iðci;b � ci;sÞ ½86�

This indicates a relationship between di and km,i, that is

km;i ¼ Di

di½87�

Mass transport rates can be expressed using dimen-sionless groups, for which empirical correlations canbe developed for a number of flow patterns. The masstransfer coefficient km enters into the Sherwood num-ber Sh, which is defined as

Sh ¼ kmd

D½88�

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where d is a characteristic dimension (e.g., a pipe orrotating disk diameter); D, a diffusion coefficient; andthe subscript i has been dropped for convenience as itis understood that eqn [88] is written for individualreacting species. The Sherwood number can be corre-lated with the Reynolds (Re) and Schmidt (Sc) num-bers, which are defined as

Re ¼ Vd

n½89�

Sc ¼ nD

½90�where V is the linear velocity. It can be shown bydimensional analysis that Sh is a function of Re andSc. This function typically has the form134,135

Sh ¼ Const Rex Scy ½91�where x is usually between 0.3 and 1 and y is about 1/3.For example, the theoretically derived results for therotating disk can be recast in terms of the mass transfercoefficient as

Sh ¼ 0:6205 Re0:5Sc0:33 ½92�Empirical expressions for other flow geometries havebeen reviewed by Poulson134,135 for single-phase flowconditions. Equations of the type eqn [91] exist for therotating cylinder, impinging jet, nozzle or orifice, andpipe flow. For the rotating cylinder, the correlation ofEisenberg et al.136 is widely used

Sh ¼ 0:0791Re0:70Sc0:356 ½93�For single-phase flow in a straight pipe, several corre-lations have been developed. Among these equations,Berger and Hau’s137 correlation has found use in anumber of corrosion modeling studies:

Sh ¼ 0:0165Re0:86Sc0:33 ½94�The earlier pipe flow formulas have been reviewed byPoulson,134 and the use of more recent equations hasbeen discussed by Lin et al.138 Equations of this kindare not as well developed for multiphase flow. Corre-lations are available for stratified flow (Wang andNesic139), but a comprehensive treatment is not avail-able for various regimes of multiphase flow. Therefore,a convenient alternative is to base the computation ofmass transfer coefficients on the well-known analogybetween heat and mass transfer.

The analogies between the transport of mass,momentum, and heat can be understood by consideringthe similarity between their respective mathematicalformulations, namely, Fick’s law of diffusion, Newton’slaw of viscosity, and Fourier’s law of heat conduction.


Thus, once a relationship has been established for agiven phenomenon in terms of dimensionless numbers,it can further serve for the calculation of another phe-nomenon that takes place under the same geometricand physical conditions but with different velocities,dimensions, and physical properties of the system. Inparticular, the correlations established for heat trans-fer can be used for mass transfer calculations. Theanalogy between the heat, mass, and momentumtransfer has been stated in a dimensionless form byChilton and Colburn140 as

Sh

Re Sc1=3¼ Nu

Re Pr 1=3¼ f

2½95�

where the Nusselt number, Nu, and the Prandtl num-ber, Pr, are the heat transfer equivalents of the Sher-wood and Schmidt numbers in mass transfer,respectively, and f is a friction factor. Correlationsfor the friction factor are available as a function ofpipe roughness, its diameter, and the Reynolds num-ber (Frank141). The exponent 1/3 in eqn [95] can bereplaced with a generalized exponent n. The Nusseltand Prandtl numbers are defined as

Nu ¼ hd

l½96�

Pr ¼ na

½97�where h is the convection heat transfer coefficient; l,the thermal conductivity; and a, the thermal diffusiv-ity (a ¼ l=rCp).

This relationship makes it possible to determinethe mass transfer coefficient in two-phase flowsystems for which experimental heat transfer correla-tions are available. Heat transfer correlations takethe form

Nu ¼ Const RexPry ½98�They have been reviewed by Kim et al.142 and Adsaniet al.143,144 for annular, slug, and bubbly flow in hori-zontal and vertical tubes. Also, Adsani et al.143,144

developed a correlation for calculating the two-phase Nusselt number, which is a generalization ofthe Chilton-Colburn140 heat transfer expression (i.e.,the second equality of eqn [95]):

Nutwo-phase ¼ C1fC2

L

VLd

n

� �C3

Pr1=3L ½99�

where the liquid-phase friction factor fL and velocityVL are calculated using flow models for annular, slug,and bubbly flow, and C1, C2, and C3 are fitting con-stants. Subsequently, the Sherwood number and the

n (2010), vol. 2, pp. 1585-1629



mass transfer coefficient can be obtained fromthe first equality in eqn [95].

As shown in the above equations, the computationof the mass transfer coefficient requires the diffusioncoefficients, viscosity, and density. Density is a ther-modynamic property and, as such, can be calculatedfrom any comprehensive thermodynamic model forelectrolyte systems. For example, Wang et al.54

describe how to calculate densities in a way that isconsistent with other thermodynamic properties.The computation of viscosities and diffusion coeffi-cients requires separate models, which are beyondthe scope of this chapter. Viscosity and diffusivitymodels have been reviewed and critically evaluatedby Corti et al.145 with particular emphasis on systemsat elevated temperatures.

2.38.3.4.1 Example of electrochemical

modeling of general corrosionTo illustrate the application of the principlesdescribed earlier, Figure 5 shows the computationof the corrosion rate and potential of type 316 stain-less steel in aqueous solutions of HF. These calcula-tions have been made using the model of Anderkoet al.108,146 as implemented in the Corrosion Analyzersoftware.61 The upper diagram shows the partialcathodic and anodic processes in a 2m HF solution.Three cathodic processes are taken into account inthis system: reduction of protons (H+), reduction ofundissociated HF molecules, and reduction of watermolecules. These partial processes are marked inFigure 5(a) as (1), (2), and (3), respectively. The H+

reduction reaction is modeled using eqns [41] and[42] as described earlier. The HF reduction process iscalculated using the same equations, but with a dif-ferent exchange current density. Both the H+ and HFreduction processes show partial current densitiesbecause of mass transport limitations for the trans-port of H+ and HF to the surface. Because of the lowdegree of dissociation of HF, the reduction of HF(line two in Figure 5(a)) plays a much more impor-tant role than the reduction of H+ ions (line 1). Thepartial anodic curve for the oxidation of 316 SS islabeled as (4). It is assumed that the alloy componentsdissolve congruently, and the dependence of thepartial anodic current density on the acidity ofthe solution is analogous to that observed for Fe(see eqn [59]), but with a different exchange currentdensity. The superposition of the partial cathodic andanodic processes yields a predicted polarizationcurve, which is shown by a thick line in Figure 5(a).The location of the mixed potential is calculated


according to eqn [45] with Aa¼ Ac and marked witha triangle. The complete model reproduces theexperimental corrosion rates (Figure 5(b)) and cor-rosion potentials (Figure 5(c)) as a function of HFconcentration and temperature.

2.38.3.5 Detailed Modeling of MassTransport

The treatment of mass transport by the use of masstransfer coefficients is computationally efficient andis capable of reproducing steady-state corrosionbehavior with good accuracy. However, it is subjectto some limitations including:

1. it is suitable for calculating only steady-statebehavior and, therefore, it is not appropriate formodeling time-dependent corrosion,

2. it neglects migration which introduces errorsespecially for dilute systems without a backgroundelectrolyte, and

3. it is not convenient for modeling transport insystems with geometrical constraints, especiallywhen the cathodic and anodic areas are spatiallyseparated; thus, it is not well suited for modelingthe propagation of localized corrosion.

These limitations can be eliminated by a morecomprehensive (but much more computationallydemanding) treatment of transport phenomena. Thistreatment is based on the conservation laws for eachspecies in the solution (Newman150):

@ck@t¼ �rJk þ Rk k ¼ 1; . . . ;K ½100�

where Ck is the concentration of species k, t is the time,Jk is the flux of species k, r is the vector differentialoperator (which reduces to @/@x in a one-dimensionalcase), and Rk is the rate of production (source) ordepletion (sink) of this species as a result of chemicalreactions. In the vast majority of practical applications,the dilute solution theory is used to calculate the fluxof the species, that is

Jk ¼ �Dkrck � zkFukckr’þ ckv ½101�where uk is themobilityof species k,’ is the electrostaticpotential in the solution, v is the fluid velocity, and theother symbolswere defined previously. In eqn [101], thefirst term on the right-hand side represents the contri-butions of diffusion, the second term describes migra-tion, and the third term is a contribution of convection.In the migration term, the mobility can be calculatedfrom the diffusivity using theNernst–Einstein equation:

n (2010), vol. 2, pp. 1585-1629

0.001

0.01

0.1

1

10

100

0.01m HF

Co

rros

ion

rate

(mm

year

–1)

Pawel (1994) 297 K

Schmitt (2004) 298 K

Ciaraldi et al. (1982) 298 K

Pawel (1994) 323 K

Pawel (1994) 349 K

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

0m HF

Eco

rr (V

/SH

E)

Schmitt (2004) 298 K,aerated

Schmitt (2004) 298 K,deaerated

Ciaraldi et al. (1982)366 K, deaerated

0.1 1 10

2 4 6 8 10 12 14

1.5

1.25

1.0

0.75

0.5

0.25

0.0

–0.25

–0.5

–0.75

–1.01.0e - 4 1.0e - 3 1.0e - 2 1.0e - 1 1.0e 0 1.0e 1 1.0e 2 1.0e 3 1.0e 4

Current density (A m–2)

(2)

(3) (1)(4)

Net current density

Corrosion potential

(1) H(+) = 0.5H2–e

(2) HF = 0.5H2+ F(–)–e

(3) H2O = 0.5H2+ OH(–)–e

(4) S31600 = {Fe, Cr, Ni, Mo}(x+) + xe

Pot

entia

l (V

(SH

E))

(a)

(b)

(c)

Figure 5 Application of an electrochemical model of general corrosion108,146 to type 316L stainless steel in aqueous HF

solutions. The upper diagram (a) shows the partial cathodic and anodic processes in a 2m HF solution. The middle (b) andlower (c) diagrams compare the calculated and experimental147–149 corrosion rates and potentials, respectively.



uk ¼ Dk

RT½102�

which is exact for species at infinite dilution and pro-vides a good approximation at finite concentrations.An additional condition for determining the potentialin eqn [101] is given by the Poisson equation:


r2’ ¼ � F

e

Xk

zkck ½103�

where e is the dielectric permittivity of the solution. Itcan be shown150 that, due to the large value of the ratioF/e, even a very small separation of charges (i.e.,P

zkck 6¼ 0) results in a large potential gradient,

n (2010), vol. 2, pp. 1585-1629



which in turn prevents any appreciable separation ofcharge. Therefore, eqn [103] is very often replacedin practice by the simple electroneutrality condition,that is X

zkck ¼ 0 ½104�which is one of the basic equations for computingequilibrium properties of electrolytes.

It should be noted that eqn [101] is rigorous onlyfor dilute solutions. For concentrated solutions, itsmore general counterpart is150

Jk ¼ ckvk ½105�where the velocities vk of species k are determined bythe multicomponent diffusion equations:

ckrmk ¼ RTXj

ckcjctotDkj

ðvj � vkÞ ½106�

where Dkj are the mutual diffusion coefficients, andctot is the total concentration of all components. Theapplication of eqns [105] and [106] is very difficultdue to the lack of a general methodology for com-puting Dkj and computational complexity. Therefore,eqns [105] and [106] have found few applications inpractical models. However, a practical simplifiedform can be obtained for moderately dilute solutionsfor which the concentrations of solute species aresmaller than the concentration of the solvent. Then,eqns [105] and [106] simplify to

Jk ¼ � Dk

RTckrmk þ ckv ½107�

Considering that mk ¼ m0k þ RT ln ckgk þ zkF’, theflux equation becomes150

Jk ¼�Dkrck� zkFukckr’þ ckv�Dkckr ln gk ½108�which is only moderately more complex than thedilute-solution eqn [101], but benefits from the infor-mation on solution nonideality that is embedded inthe activity coefficient and can be calculated from anelectrolyte thermodynamic model.

In the convective term of eqn [101] or [108], theinstantaneous fluid velocity (v) can be calculated, inprinciple, by the methods of computational fluiddynamics. However, such calculations involve a largecomputational effort and are, in practice, limited withrespect to flow geometries and conditions. For turbu-lent flow, a practical approach relies on introducingturbulent diffusion. Accordingly, instantaneous veloc-ity is divided into steady and turbulent components.The steady component is parallel to the surface anddoes not contribute to transport to and from the sur-face. Then, the convection term in eqn [101] or [108],


ckv, is approximated by a turbulent diffusivityterm,�Dtrck , which can be lumped with the molecu-lar diffusion term thus defining an effective diffusioncoefficient (Davis,151 Nordsveen et al.116)

Deffk ¼ Dk þ Dt ½109�

where Dt can be obtained from empirical correlationswith fluid properties.151,152 For example, Davis’s151

correlation has been used in the CO2 corrosionmodel of Nordsveen et al.116:

Dt ¼ 0:18z

d

� �3 �r

½110�

where z represents the distance from the surface (eithera metal surface or a surface covered with corrosionproducts), � is the viscosity, r is the density, and d isthe thickness of the laminar boundary layer, which canbe calculated for a pipe with a diameter d as

d ¼ 25Re�7=8d ½111�It is noteworthy that this formalism can be shown to bephysically equivalent to the treatment of turbulent flowthrough mass transfer coefficients as described in theprevious section. Specifically, Wang and Nesic139

showed a relationship between the mass transfer coef-ficient and Dt :

1

km;i¼Z d

0

dz

Di þ Dt½112�

The computation of the rates of production or deple-tion Rk is necessary in order to apply eqn [100].A general matrix formalism for calculating the Rkterms in a system with multiple reactions has beendeveloped by Nordsveen et al.116 The main limitationhere is the fact that rate data are available only for avery limited number of reactions such as precipitationof common scales (CaCO3, FeCO3) and selectedhomogeneous reactions (e.g., hydration of H2CO3).For the vast majority of reactions, only equilibriumequations are available and, in fact, there is no physicalneed for kinetic expressions for most homogeneousreactions because they are fast relative to mass trans-port. Therefore, arbitrary rate expressions may beassumed as long as they are constrained by the equilib-rium constant (i.e., the ratio of the forward and reverserate constants is equal to the equilibrium constant),give appropriately fast reaction rates and change direc-tion as the equilibrium point is crossed. A convenientexpression for the production or depletion rate forspecies k can be defined in terms of the departure ofthe ionic product from equilibrium (Walton153):

n (2010), vol. 2, pp. 1585-1629



Rk ¼XMm ¼ 1

�rmnkm ln

Qcnkmk

Km

� �� ½113�

where rm is an adjustable numerical rate parameterfor reaction m, Km is the equilibrium constant forreaction m, and nkm is the stoichiometric coefficientfor species k in the mth reaction. Alternatively, thetransport equations can be first solved separatelyfrom the chemical effects and, then, at the end ofeach sufficiently small time step, thermodynamicequilibrium calculations can be performed in eachelementary volume.

To solve the system of transport eqns [100] and[101] or [108], boundary conditions are required. Inthe bulk solution, the equilibrium concentrations arethe natural boundary conditions. For each speciesinvolved in electrochemical reactions, the flux atthe metal surface is determined from

Jk ¼ � iknkF

½114�

where the current density ik is calculated from appro-priate expressions for cathodic and anodic partial pro-cesses as a function of concentrations at the metalsurface. This provides a link to the mechanisticor empirical electrochemical expressions describedabove. For the species that are not involved in thereactions, the flux at the interface is zero. It should benoted that the application of this formalism of masstransport can become quite computationally involved,especially for systems with numerous species, becauseit requires solving a system of differential equations(eqns [100] and [101] or [108]) with constraints (eqns[103] or [104] and chemical terms).

2.38.3.5.1 Effect of the presence of

porous media

The above transport equations need to be modifiedwhen mass transport occurs through porous mediasuch as corrosion products, calcareous deposits, soilor sand, and various man-made environments, includ-ing concrete and ceramics. Such a generalization canbe formulated in terms of two characteristic quantities,porosity and tortuosity. Porosity (e) is the volumetricvoid fraction of the medium, whereas tortuosity (t) isdefined as the ratio of the distance that an ion ormolecule travels around solid particles to the directpath. For practical applications, tortuosity can be cor-related with porosity thus leaving porosity as the onlyparameter to affect transport equations. The generali-zation of transport equations to porous media has been


discussed by Newman150 and Bear.154 A detailed cor-rosion model that includes the transport in porouscorrosion products has been developed for CO2 cor-rosion by Nordsveen et al.116 and Nesic and Lee.155

2.38.3.6 Active–Passive Transition andDissolution in the Passive State

The expressions for anodic partial current densitiesdiscussed above (eqns [54]–[68]) are limited to thedissolution in the active state. However, dissolution inthe passive state and the transition between the activeand passive state are equally important for modelingaqueous corrosion. In fact, passivity is the key to ourmetal-based civilization (Macdonald97) and has beenextensively investigated since the pioneering work ofFaraday and Schonbein in the 1830s. Theories ofpassivity have been reviewed by many investigators(Frankenthal and Kruger,156 Froment,157 Marcus andOudar,158 Natishan et al.,159 Macdonald97,160) and arebeyond the scope of this chapter. In this section, wefocus solely on practical models for calculating theanodic current density in the passive and active–passive transition regions as a function of solutionchemistry.

Passivity manifests itself by a sharp drop in theanodic current density at a certain critical potentialas the metal is polarized in a negative-to-positivepotential direction. For calculation purposes, empiri-cally determined anodic polarization curves can bereproduced using a suitable fitting function. Forexample, such a function has been developed byMacdonald.161 Then, empirical fitting functions canbe used within the framework of the mixed potentialtheory as described above.

A convenient way to introduce the active–passivetransition into a computational model is to consider acurrent that leads to the formation of a passive layerin addition to the current that leads to active dissolu-tion (Ebersbach et al.,162 Anderko and Young108). Forthis purpose, a certain fraction of the surface yP canbe assumed to be covered by a passive layer. Thechange of the passive layer coverage fraction withtime can be expressed as

@yP@t

� �E;ai

¼ ciMeOð1� yPÞ � KyP ½115�

where iMeO is the current density that contributes tothe formation of a passive layer. The second term onthe right-hand side of eqn [115] represents the rateof dissolution of the passive layer, which is

n (2010), vol. 2, pp. 1585-1629



proportional to the coverage fraction. Solution of thisequation in the steady-state limit yields an expressionfor the anodic dissolution current:

iMe;TOT ¼ iMe þ iMeO

1þ ðciMeO=K Þ ¼iMe þ iMeO

1þ ðiMeO=iPÞ ½116�

where iMe is the dissolution current density in theactive state and the ratio iP¼ c/K constitutesthe passive current density. The current iMe is calcu-lated using the active dissolution models describedabove. The current iMeO is expressed using theusual expression for process under activation control,that is

i2 ¼ i02 expa2FðE � EFÞ

RT

� �½117�

in which the parameters can be adjusted to reproducethe observable characteristics of the active–passivetransition including the critical current density (icrit)and Flade potential (EF).

146 Equation [116] can bethen used for the anodic process of metal dissolutionwithin the framework of the mixed-potential theory(eqns [45] and [46].

An example of mixed-potential calculations for apassive metal is shown in Figure 6. This figure illus-trates the computation of the corrosion potential ofalloy 600 in a dilute LiOH solution as a functionof dissolved oxygen concentration. As in Figure 5,the upper and middle diagrams of Figure 6 show thepredicted partial E versus i curves for the anodic andcathodic processes. The upper diagram (Figure 6(a))shows the predictions for a very low O2 concentration(0.013 ppm), whereas the concentration in the middlediagram is somewhat higher (0.096 ppm). In a weaklyalkaline solution, the alloy is passive as indicated bythe vertical portion of the anodic curve (line labeledas (3)). Two main cathodic processes are taken intoaccount in this system, that is, the reduction ofH2O (line (1)) and the reduction of O2 (line (2)). Atthe lower O2 concentration, the limiting current den-sity is lower than the passive current density, and themain cathodic process is the reduction of H2O (i.e.,the mixed potential lies at the intersection of the lines(1) and (3) in Figure 6(a)). As the O2 concentrationincreases, the O2 reduction reaction becomes pre-dominant and determines the mixed potential,which then lies at the intersection of lines (2) and(3). This behavior explains the experimentally deter-mined s-shaped dependence of Ecorr on O2 concen-tration as shown in Figure 6(c). The s-shape is due tothe transition from H2O reduction to O2 reduction asthe dominant cathodic process. The transition


depends on flow conditions because the O2 reductionis partially under mass transport control.

Passive dissolution and active–passive transitionstrongly depend on solution chemistry. In theabsence of specific active ions, the dissolution ofoxide films depends primarily on the pH of thesolution. Appropriate kinetic expressions can be con-structed by considering dissolution reactions betweenthe passive oxide/hydroxide surface layers and solu-tion species (Anderko et al.,146 Sridhar et al.131,164) Inacidic solutions, the key reaction involves the protonsfrom the solution:

� MeOaðOHÞb þ ð2a þ b � xÞHþ

¼ MeðOHÞð2aþb�xÞþx þ ða þ b � xÞH2O ½118�where the symbol ‘�’ denotes surface species. Thecorresponding kinetic equation is

ip;Hþ ¼ kHþaq

Hþ;s ½119�where aHþ;s denotes the surface concentration ofhydrogen ions and q is a reaction order, which is notnecessarily related to the stoichiometric coefficient inthe dissolution reaction. In neutral solutions, the pre-dominant dissolution reaction can be written as

� MeOaðOHÞb þ aH2O ¼ MeðOHÞ0ð2aþbÞ;aq ½120�where the predominant species on the right-handside of eqn [120] is a neutral complex as indicatedby the superscript 0. The corresponding kineticequation is

ip;H2O¼ kH2Oa

rH2O;s

½121�where the reaction order with respect to water indi-cates that dissolution may be affected by water activ-ity. Similarly, the predominant reaction in alkalinesolutions is

� MeOaðOHÞb þ ðx � 2a � bÞOH� þ aH2O

¼ MeðOHÞðx�2a�bÞ�x ½122�with a corresponding kinetic equation given by

ip;OH� ¼ kOH�asOH�;s ½123�

The total passive current density as a function of pHis given by

ip ¼ ip;Hþ þ ip;H2Oþ ip;OH� ½124�

It should be noted that the passive dissolution may beinfluenced by mass transport. For example, aluminumdissolution in alkaline solutions is known to be partlyunder mass transport control due to the transport of

n (2010), vol. 2, pp. 1585-1629

(4)

(2)

(1)

(3)

1.0e - 4 1.0e - 3 1.0e - 2 1.0e - 1 1.0e0Current density (A m–2)


–1.5

–1.25

–1.0

–0.75

–0.5

–0.25

0.0

0.25

0.5

0.75

1.0P

oten

tial (

V (S

HE

))

(a)

(3)

(1)

(2)

(4)

–1.51.0e - 4 1.0e - 3 1.0e - 2 1.0e - 1 1.0e0

–1.25

–1.0

–0.75

–0.5

–0.25

0.0

0.25

0.5

0.75

1.0

Pot

entia

l (V

(SH

E))

(b)

–0.8

–0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

1.E - 07

m O2

Eco

rr (S

HE

)

1.E - 06 1.E - 05 1.E - 04 1.E - 03

(c)

Ecorr, expEcorr, cal

(1) H2O = 0.5H2+ OH(–)–e(2) O2+ 4H(+) = 2H2O-4e(3) Alloy 600={Ni, Cr, Fe}(x+) + xe(4) Cr2O3+ 5H2O = 2CrO4(2–) + 10H(+) + 8eNet current densityCorrosion potential

(1) H2O = 0.5H2+ OH(–)–e(2) O2+ 4H(+) = 2H2O–4e(3) Alloy 600 = {Ni, Cr, Fe}(x+) + xe(4) Cr2O3+ 5H2O = 2CrO4(2–) + 10H(+) + 8eNet current densityCorrosion potential

Figure 6 Modeling of the corrosion potential of alloy 600 in a 0.1 M LiOH solution at 200 �C as a function of dissolved

oxygen concentration. The upper and middle diagrams (a and b) show the calculated partial electrochemical reactions

and predicted polarization curve for solutions containing 4e-7m (0.013 ppm) and 3 10�6 m (0.096 ppm) O2, respectively.The lower diagram (c) compares the calculated corrosion potential with experimental data163 at 200 �C as a function of

dissolved oxygen molality.




(3)

(1)

(2)(4)

(3)

(1)

(2)(4)

1.5

1.25

1.0

0.75

0.5

0.25

0.0

–0.25

–0.5

–0.75

–1.01.0e-4 1.0e-3 1.0e-2 1.0e-1 1.0e-0 1.0e1 1.0e2 1.0e3 1.0e4

Pot

entia

l (V

(SH

E))

1.5

1.25

1.0

0.75

0.5

0.25

0.0

–0.25

–0.5

–0.75

–1.0

Pot

entia

l (V

(SH

E))

(1)(3)

(4)(2)


1.0e-4 1.0e-3 1.0e-2 1.0e-1 1.0e0 1.0e1 1.0e2 1.0e3 1.0e4


(1) H(+) = 0.5H2– e(2) H2O = 0.5H2+ OH(–)–e

(3) O2+ 4H(+) = 2H2O–4e

(4) S30400 = {Fe, Cr, Ni}(x+) + xeNet current density

Corrosion potential

(1) H(+) = 0.5H2– e

(2) H2O = 0.5H2+ OH(–)–e

(3) O2+ 4H(+) = 2H2O–4e

(4) S30400 = {Fe, Cr, Ni}(x+) + xe

Net current densityCorrosion potential

(a)

(b)

–0.20–0.15–0.10–0.050.000.050.100.150.200.250.300.35

0pH

Eco

rr/S

HE

1 2 3 4 5 6(c)

Figure 7 Electrochemical modeling of the depassivation pH and corrosion potential of type 304 stainless steel in aerated

0.1M Na2SO4þH2SO4 solutions. The upper (a) and middle (b) diagrams show the calculated partial electrochemicalprocesses and predicted polarization curve for pH = 0.8 and 1.8, respectively. The lower diagram (c) compares the calculated

corrosion potentials with experimental data.165



OH� ions from the bulk to the interface. Then, thecontributions to the passive current density (eqn[124]) should be coupled with mass-transfer equa-tions such as eqn [36].131


Figure 7 illustrates the electrochemical modelingof the pH dependence of the active–passive transitionof type 304 stainless steel. The upper and middlediagrams in Figure 7 show the partial electrochemical

n (2010), vol. 2, pp. 1585-1629



processes in Na2SO4 solutions with pH¼ 0.8 and 1.8,respectively. The anodic curve (line 4) was modeledusing a model based on eqns [116] and [117]. Themain cathodic process is the reduction of O2. Becauseof the pH effect on the active–passive transition, themixed potential moves from the active dissolutionregion for pH¼ 0.8 to the passive region forpH¼ 1.8. This explains the dependence of the experi-mentally determined corrosion potential on pH(Figure 7(c)). The pH value at which an abruptchange of Ecorr occurs can be identifiedwith the depas-sivation pH.

In addition to pH effects, some active ions mayinfluence the magnitude of the passive current density.The effect of active species on the dissolution in thepassive state can be modeled by considering surfacereactions between the metal oxide film and solutionspecies (Blesa et al.,166 Anderko et al.146):

� MeOaðOHÞb þ ciXi ¼� MeOaðOHÞbXci þ eiOH

� ½125�where Xi is the ith reactive species in the solution, andthe subscripts a, b, ci, and ei represent the reactionstoichiometry. In general, eqn [125] may be writtenfor any active, aggressive, or inhibitive species i in thesolution (i=1, . . ., n). It is reasonable to assume thateqn [125] is in quasi-equilibrium. The surface speciesthat forms as a result of reaction eqn [125] mayundergo irreversible dissolution reactions such as:

� MeOaðOHÞbXci þ aH2O!MeðOHÞ02aþb;aq þ ciXi ½126�

in which dissolved metal species are formed in analogyto those described by eqns [118], [120], and [122].Mathematical analysis of reactions eqns [125] and[126]108,146 yields a relationship between the passivecurrent density and activities of reactive species:

ip ¼ i0pðpHÞ1þP

i

liðaciXi=aeiOH�Þ

1þPi

KiðaciXi=aeiOH�Þ

½127�

where i0pðpHÞ is given by eqn [124], li is the forwardrate of reaction eqn [126], and Ki is the equilibriumconstant of reaction eqn [125].

Figure 8 illustrates the effect of active ions on therate of general corrosion using alloy 22 in mixedHNO3þHF solutions as an example. The upperdiagram (Figure 8(a)) shows the predicted partialelectrochemical processes in a 20% HNO3 solutionand the middle diagram (Figure 8(b)) shows how


these processes change when a moderate amount ofHF (1.57%) is added. A 20 wt.% HNO3 solution isan oxidizing medium, and, therefore, reduction ofNO3

� ions in an acidic environment is the maincathodic process. This results in a high corrosionpotential as shown in Figure 8(a). The corrosionrate is controlled by the dissolution rate of theoxide film. When HF is added, the dissolution rateof the oxide substantially increases even though amoderate amount of HF has practically no effect onthe acidity of the system. This effect is reproduced byeqn [127] and manifests itself by the increased passivecurrent density in Figure 8(b). The predicted effectscan be compared with the observed corrosion rates in20% HNO3 solution as a function of HF concentra-tion (Figure 8(c)).

2.38.3.7 Scaling Effects

In addition to passive dissolution and active–passivetransition, modeling of surface scale formation is ofgreat practical importance. Scales form as a result ofdeposition of corrosion products (e.g., iron carbonateor sulfide) or other solids that reach supersaturationnear metal interfaces (e.g., calcareous deposits).Scales can be distinguished from passive films inthat they do not give rise to the classical active–passive transition such as that shown in Figure 7.Rather, they reduce the rate of dissolution byproviding a barrier to the diffusion of species to andfrom the surface and by partially blocking the inter-face, thus reducing the overall rate of electrochemicalreactions. In general, there may be multiple mechan-isms of scale formation depending on the chemistryof the precipitating solids.

One mechanism of scale formation can be quanti-fied in terms of the competition between the rate ofscale formation, which results in the precipitation of acorrosion product, and the rate of corrosion underthe scale, which leads to the ‘undermining’ of thescale. When the rate of precipitation exceeds therate of corrosion, dense protective films are formed.Conversely, when the corrosion rate is greater thanthe precipitation rate, the scale still forms, but theprecipitation rate is not fast enough to fill the grow-ing voids. Then, the scale becomes unprotective eventhough it may be thick. Nesic and Lee155 developed amodel to represent this phenomenon for FeCO3 scaleformation. In Nesic and Lee’s155 model, the localchange in the volumetric concentration of the scale-forming solid is given by a redefined eqn [100]:

n (2010), vol. 2, pp. 1585-1629

(a)

(b)

0.01

0.1

1

10

100

0wt.% HF

Rat

e (m

mye

ar–1

)

Sridhar et al. (1987)20 wt% HNO3, 353 K

Calculated

1 2 3 4 5(c)

1.5

1.25

1.0

0.75

0.5

0.25

0.0

–0.25

–0.5

–0.75

–1.0

1.5

1.25

1.0

0.75

0.5

0.25

0.0

–0.25

–0.5

–0.75

–1.0

1.0e-4 1.0e-3 1.0e-2 1.0e-1 1.0e0 1.0e1 1.0e2 1.0e3 1.0e4Current density (A m–2)

1.0e-4 1.0e-3 1.0e-2 1.0e-1 1.0e0 1.0e1 1.0e2 1.0e3 1.0e4Current density (A m–2)

Corrosion potential

Net current density

(4) C–22 = {Ni, Cr, Mo, W}(x+) + xe

(3) NO3(–) + wH(+) = NOx + yH2O – ze

(2) H2O = 0.5H2+ OH(–) – e

(1) H(+) = 0.5H2– e

(1) H(+) = 0.5H2– e

(3) H2O = 0.5H2+ OH(–) – e(2) HF = 0.5H2+ F(–) – e

(4) NO3(–) + wH(+) = NOx + yH2O–ze

(5) C–22 = {Ni, Cr, Mo, W}(x+) + xeNet current densityCorrosion potential

(3)

(3)

(4)

(5)

(1)

(1)

(2)

(2)

(4)Pot

entia

l (V

(SH

E))

Pot

entia

l (V

(SH

E))

Figure 8 Electrochemical modeling of the effect of HF concentration on the corrosion rate of alloy 22 in HNO3þHF

solutions. The upper (a) and middle (b) diagrams show the partial electrochemical processes in 20% HNO3 solutions without

HF and with 1.57% HF, respectively. The lower diagram compares the calculated results with experimental data167 as a

function of HF concentration.






@csolid@t¼ Rsolid � CR

@csolid@z

½128�where the first term on the right-hand side representsthe rate of the scale formation and the second term isthe scale undermining rate. In eqn [128], CR is thecorrosion rate and z is a direction perpendicular tothe surface. The rate of formation of the scale is, ingeneral, a product of the scale particles’ surface area-to-volume ratio A/V, a function of temperature, thethermodynamic solubility product Ksp, and an empir-ical function of supersaturation S:

Rsolid ¼ A

Vf ðT ÞKsp f ðSÞ ½129�

where supersaturation is defined as

S ¼Q

aniiKsp¼Q

cniiKsp

½130�

While the f (T) and f (S) functions can be, in principle,derived from precipitation kinetics data that are inde-pendent of corrosion, Sun and Nesic168 have deter-mined that much more reliable precipitation ratescan be obtained from corrosion weight loss and gainmeasurements than from kinetic measurements thatstart from dissolved metal ions. The A/V ratiodepends on the porosity of the scale on the metalsurface. Nesic and Lee155 developed an empiricalfunction of porosity that is consistent with the exper-imental data for FeCO3 scale formation.

A different model is necessary for scales whoseformation does not follow the kinetics of precipita-tion processes. For example, FeS scales form very fastin highly undersaturated solutions, in which theywould be thermodynamically unstable in the bulk,and their formation appears not to be influenced bysolution supersaturation. Thus, the effect of FeSscales can be modeled by assuming a solid-state reac-tion at the metal surface that is mediated by theadsorption of H2S (Anderko and Young,169 Nesicet al.170) The formation of FeS scales is further com-plicated by the existence of an outer layer that resultsfrom the growth, cracking, and delamination of theFeS film. A model that accounts for these phenomenawas developed by Sun and Nesic.171

2.38.3.8 Modeling Threshold Conditions forLocalized Corrosion

Modeling of the evolution of localized corrosion hasbeen the subject of extensive research during the pastthree decades, and a number of important modelshave been developed for the initiation, stabilization,


propagation, and stifling of individual pits, crevices,and cracks and for the statistical behavior of theirensembles in corroding structures. However, thistopic is outside the scope of this chapter and will bereviewed in the chapter ‘Predictive Modeling ofCorrosion’ in this volume.

In this chapter, we focus solely on models thatpredict the conditions for the occurrence of localizedcorrosion without going into the treatment of theevolution of localized corrosion events in time andspace. Such models are designed to find the thresholdcriteria for localized corrosion. In general, localizedcorrosion occurs when the corrosion potential of analloy in a given environment exceeds a critical poten-tial. The meaning, experimental determination, andinterpretation of the key potentials that characterizelocalized corrosion have been reviewed by Szklarska-Smialowska.172 While this general concept is wellaccepted, what constitutes a critical potential con-tinues to be debated. The selection of the criticalpotential depends on the particular phenomenonthat is to be modeled.

The applicability of the critical potential conceptto modeling localized corrosion is qualitatively illu-strated in Figure 9. In this figure, the arrows indicatethe conditions at which localized corrosion isexpected. For a given alloy, the critical potentialdecreases with an increase in the concentrationof aggressive species (e.g., halide ions) as shown inFigure 9(a). The shape of the Ecrit curve correspondsto that of the repassivation potential curve, but thequalitative pattern is more general. Unlike the criticalpotential, the corrosion potential is usually not astrong function of aggressive ion concentration unlesssignificant localized corrosion occurs. The criticalaggressive species concentration for localized corro-sion is observed when Ecorr exceeds Ecrit. Similarly, fora given aggressive chemical environment, a criticaltemperature exists (see Figure 9(b)). The criticalpotential is also strongly affected by the presence ofinhibitors. As shown in Figure 9(c), this gives rise to acritical inhibitor concentration. In many environ-ments, the presence of oxidants may increase Ecorrso that localized corrosion may occur beyond a criti-cal concentration of redox species (Figure 9(d)). Theactual conditions in a system may be a combination ofthe four idealized cases shown in Figure 9. Thus, thekey is to predict both the corrosion potential and therepassivation potential. The corrosion potential can beobtained from a general-corrosion, mixed-potentialmodel for passive metals as described above. Forthe critical potential, separate models are necessary.

n (2010), vol. 2, pp. 1585-1629

(c)

Pot

entia

l

Oxidant

Localized corrosion

Ecrit

EcorrEcrit

Ecorr

Pot

entia

l

Inhibitor

Localized corrosion

(b)

(d)

Pot

entia

l

Temperature

Localized corrosion

Ecrit

Ecorr

(a) Aggressive ionsP

oten

tial

Ecrit

Ecorr

Localized corrosion

Figure 9 A general conceptual scheme of the use of the corrosion potential (Ecorr) and critical potential (Ecrit) to predict

the effect of aggressive ions, temperature, inhibitors, and oxidizing redox species on localized corrosion. The arrows

marked ‘localized corrosion’ denote the potential ranges in which localized corrosion can be expected.



In this chapter, we briefly review the computation andapplicability of the passivity breakdown potential andthe repassivation potential.

2.38.3.8.1 Breakdown of passivity

To predict the initiation of localized corrosion, it isnecessary to calculate the critical passivity breakdownpotential. Several theories have been developed torelate the breakdown potential to the concentrationof aggressive species in the solution (Heusler andFischer,173 Strehblow and Titze,174 Lin et al.,175

Okada,176 McCafferty,177 Haruna and Macdonald,178

Macdonald,97 Yang and Macdonald,179 and paperscited therein). A common theoretical result, confirmedby experimental data, is the linear dependence of thepassivity breakdown potential on the logarithm of theconcentration of aggressive ions. While this observa-tion is generally accepted, its generalization to systemswith multiple aggressive and inhibitive ions is notimmediately obvious.

A particularly comprehensive treatment of passiv-ity breakdown is provided by the point defect model ofMacdonald and coworkers (Lin et al.,175 Haruna andMacdonald,178 Macdonald,97 Yang andMacdonald179).According to the point defect model, passivity break-down results from the condensation of cation or metalvacancies at the interface between the metal and thepassive barrier layer. The vacancies are envisaged to begenerated at the barrier layer–solution interface in anautocatalytic, anion-induced process. For systems con-taining only aggressive (halide) ions X, the criticalbreakdown potential is expressed as


Eb¼ 4:606RT

waFlog

JmaDu�w=2

� �2:303RT

aFlog aX ½131�

where Jm is the rate of annihilation of cation vacanciesat the metal/barrier layer interface, a and u are ther-modynamic parameters related to the absorption of anaggressive ion into an oxygen vacancy, D is the cationvacancy diffusivity, and a is the polarizability of thefilm–solution interface. Yang and Macdonald179

extended eqn [131] to systems containing both aggres-sive ions X� and inhibitive ions Yz�:

Eb¼ E0b�

bapH�2:303

aa0Flog

aX�

aYz�½132�

where the constant E0b is a function of adsorption and

elementary reaction rate parameters that is derivedfrom a competitive adsorption model for the X� andYz� species, b is the dependence of the potential dropacross the barrier layer–solution interface on pH, anda0 is a transfer coefficient. The predictions of the pointdefect model have been found to be in agreement withexperimental phenomena including the linear depen-dence of the breakdown potential on the concentra-tions of aggressive and inhibitive ions, the dependenceof the induction time on potential and chloride con-centration, dependence of the breakdown potential onthe scan rate and the inhibition of pitting by Mo andW in the alloy. It should be noted that the breakdownpotential is a distributed quantity that can be describedwith a normal distribution function. The distributionin Eb has been reproduced by assuming that the cationdiffusivity is normally distributed.97

n (2010), vol. 2, pp. 1585-1629



While the logarithmic dependence of the breakdownpotential on the aggressive species concentration ispredicted by most passivity breakdown models, theinduction time provides a more stringent criterion fortesting alternative models. Accordingly, Milosev et al.180

tested the validity of the point defect model, thetwo-dimensional nucleation model of Heusler andFischer,173 and the halide nuclei model of Okada176 forthe pitting of copper. The point defect model was foundto yield the best agreement with experimental data.

2.38.3.8.2 Repassivation potential and its use

to predict localized corrosion

While the breakdown potential is the critical param-eter for the initiation of pitting, the repassivationpotential (Erp) has been used for predicting thelong-term occurrence of pitting and crevice corro-sion. The repassivation potential (also called protec-tion potential) is the potential at which a stablygrowing pit or crevice corrosion will cease to grow.Thus, localized corrosion cannot occur at potentialsbelow Erp. The use of Erp for engineering predictionscan be justified by the fact that only the fate of stablepits or crevice corrosion is important for predictingthe possibility of failure, and metastable pits do notadversely affect the performance of engineeringstructures. It has been shown by Dunn et al.181,182

that Erp is practically independent of the amount ofcharge passed in a localized corrosion process as longas it is above a certain minimum amount of charge. Asa result, the repassivation potential is relatively insen-sitive to prior pit depth and surface finish. As acorollary, it has been shown that the repassivationpotential for pitting (i.e., measured on an open sam-ple) and the repassivation potential for crevice corro-sion (i.e., measured on a creviced sample) coincide athigh pit depths. This has demonstrated the utility ofthe repassivation potential for engineering design asit provides a reproducible and inherently conserva-tive threshold for the occurrence of localized corro-sion. Thus, the prediction of long-term occurrence oflocalized corrosion can be separated into two inde-pendent parts, that is, the calculation of the repassi-vation and the corrosion potentials. The separation oflocalized corrosion modeling into these two steps isvalid as long as the initial stages of stable localizedcorrosion are considered because the corrosionpotential is not affected at this stage by the progressof the localized corrosion process and the interactionbetween pits can be ignored. The separation remainsvalid as long as significant pit or crevice corrosiongrowth does not occur and the area of an actively


corroding pit does not become significant comparedto the overall area.

A model for calculating the repassivation potentialhas been developed by Anderko et al.183 by consider-ing the electrochemistry of a metal M that undergoesdissolution underneath a layer of concentrated metalhalide solution MX. The concentrated solution mayor may not be saturated with respect to a hydroussolid metal halide. In the process of repassivation, athin layer of oxide forms at the interface between themetal and the hydrous metal halide. The modelassumes that, at a given instant, the oxide layer coversa certain fraction of the metal surface. This fractionincreases as repassivation is approached. Further,the model includes the effects of multiple aggressiveand nonaggressive or inhibitive species, which aretaken into account through a competitive adsorptionscheme. The aggressive species form metal com-plexes, which dissolve in the active state. On theother hand, the inhibitive species and water contrib-ute to the formation of oxides, which induce passivity.The model assumes that the measurable potentialdrop across the interface can be expressed as a sumof four contributions, that is

E ¼ DFM=MX þ DFMX þ DFMX=S þ DFS ½133�where DFM/MX is the potential difference at theinterface between the metal and metal halide, whichmay be influenced by the partial coverage by themetal oxide, DFMX is the potential drop across thehydrous halide layer, DFMX/S is the potential differ-ence across the metal halide–solution interface, andDFS is the potential drop across the boundary layerwithin the solution. Expressions for the potential dropscan be derived using the methods of nonequilibriumthermodynamics.184 In general, these expressions arecomplex and can be solved only numerically. However,a closed-form equation has been found in the limit ofrepassivation, that is, when the current density reachesa predetermined low value irp (typically irp¼ 10�2

A m�2) and the fluxes of metal ions become smalland comparable to those for passive dissolution.Then, eqn [133] can be used to arrive at a closed-form expression for the repassivation potential. Thisclosed-form expression, which can be solved numeri-cally to calculate Erp, is given by:

1þXk

irp

ip� 1

� �l 00kirp

� ynkk exp

xkFErpRT

� �

¼Xj

k00jirp

ynjj expaj FErpRT

� �½134�

n (2010), vol. 2, pp. 1585-1629



where ip is the passive current density, T is the temper-ature,R is the gas constant, and F is the Faradayconstant.The partial coverage fraction of a species j is related tothe activity of this species in the bulk solution by

yj ¼ Kads;j aj1þPk Kads;kak

½135�

where

Kads;j ¼ exp �DGads;j

RT

� �½136�

and DGads,j is the Gibbs energy of adsorption. Theparameters k00j and l 00k in eqn [134] are rate constantsfor surface reactions mediated by the adsorption ofaggressive and inhibitive species, respectively. Theinhibitive species include water, as it is necessary foroxide formation. The parameter nj is the reaction orderwith respect to species j, and aj and xk are the electro-chemical transfer coefficients for reactions mediated byaggressive and inhibitive species, respectively. Someparameters (DGads,i , aj , and nk) can be assigned defaultvalues. The remaining parameters need to be regressedfrom a limited amount of experimental Erp measure-ments. Since Erp data are most abundant for chloridesolutions, the rate constant for the chloride ions (k00Cl),reaction order with respect to chlorides (nCl), rate con-stant for water (l 00H2O

), and electrochemical transfer coef-ficient forwater (xH2O) are determined based on the datafor chloride solutions. The determination of parametersis greatly simplified by the fact that the parameters forFe–Ni–Cr–Mo–W–N alloys can be correlated withalloy composition,185,186 thus enhancing the predictivevalue of the model. The k00j and, if necessary, nj para-meters are determined for other aggressive species j (e.g.,bromide ions) using Erp data for either pure or mixed

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0001

Erp

(SH

E)

m Cl–0.001 0.01

Figure 10 Modeling the repassivation potential of alloy CuNi 7various temperatures.


solutions containing such ions. Finally, the l 00k parametersfor inhibitive ions k are determined on the basis of datafor mixed solutions containing chlorides and inhibitors.Data for mixed systems are necessary because Erp isundefined in solutions containing only inhibitors.

The repassivation potential model has a limitingcharacter, that is, it accurately represents the state ofthe system in the repassivation potential limit. Inaddition to the value of the repassivation potential,the model predicts the correct slope of the currentdensity versus potential relationship as the potentialdeviates from Erp.

183 The current density predictedby the model as a function of potential is given by

i ¼

Pj

k00j ynjj exp

aj FERT

� �þP

j

l 00j ynjj exp

xj FERT

� �1þ 1

ip

Pj

l 00j ynjj exp

xj FERT

� � ½137�

Equation [137] reduces to eqn [134] for E¼ Erp andi¼ irp. Since eqn [137] is a limiting law, its accuracygradually deteriorates as the potential increasinglydeviates from Erp. Equation [137] cannot be regardedas a model for the propagation rate of an activelygrowing pit or crevice because it does not take intoaccount the factors such as the ohmic potential drop,transport limitations, and so on. However, the currentdensity predicted using eqn [137] for E> Erp is usefulbecause it provides an estimate of the maximumpropagation rate of an isolated pit as a function ofpotential. Such an upper estimate is convenientbecause it relies only on parameters that are cali-brated using repassivation potential data.

Figure 10 shows the application of the repassiva-tion potential model to alloy CuNi 7030 in chloride

t = 23C, pH = 9, borate



t = 23C, pH = 11


t = 60C, pH = 11

t = 95C, pH = 11

Cal, t = 23C, borate



Cal,t = 23C, pH = 11

Cal, t = 60C, pH = 11

Cal, t = 95C, pH = 11

0.1 1

030 (UNS C71500).217 as a function of chloride activity at

n (2010), vol. 2, pp. 1585-1629



solutions at three temperatures. As shown in thefigure, the slope of the repassivation potentialchanges as a function of chloride activity. A steeperslope is observed at low chloride concentrations. Thisis a general phenomenon for alloys and becomes morepronounced for more corrosion-resistant alloys.183

The transition between the low-slope and high-slope segments of the curves strongly depends onthe alloy and temperature.

A particularly useful application of the repassiva-tion potential model is for investigating the com-peting effects of aggressive and inhibitive species.For example, Figure 11 shows the inhibitive effectof nitrate ions on localized corrosion of alloy 22in concentrated chloride solutions. The Erp versusNO3

� concentration curves have a characteristicshape with two distinct slopes. As the concentrationof theNO3

� ions is increased, the slope of the Erp versusNO3

� concentration curve initially slowly increaseswith a low slope. At a certain concentration of NO3

�,the slope of the Erp curve rapidly increases and therepassivation potential attains a high value. At NO3

�

concentrations that lie beyond the high-slope portionof the Erp versus NO3

� curve, localized corrosionbecomes impossible even in systems with a high corro-sion potential. Thus, there is a fairly narrow range ofinhibitor concentrations over which the Erp curve tran-sitions from a low-slope region (in which localizedcorrosion is possible depending on the value of thecorrosion potential) to a high-slope region that consti-tutes the upper limit of inhibitor concentrations forlocalized corrosion. The exact location of the transitionregion depends on the temperature and chloride con-centration and can be accurately reproduced using therepassivation potential model.

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0001

Erp

(SH

E)

M NO3–

No localized corrosion(typically, transpassivedissolution)

Transition to of localized c

0.01

Figure 11 Modeling the effect of nitrate ions on the repassiva

The experimental data are from Dunn et al.187


Figure 12 illustrates an application of the corro-sion potential and repassivation potential models topredict the critical crevice temperature. At tempera-tures below critical crevice temperature (CCT), thecalculated corrosion potential (Ecorr) should lie belowthe repassivation potential, whereas it should exceedErp above CCT. Thus, the intersection of the Ecorr andErp curves versus temperature provides an estimate ofCCT. Figure 12 shows the results of such calcula-tions for alloy C-276 in 6% FeCl3 solutions.

188 Therepassivation potential shows an initially steepdecrease followed by a moderate decrease at highertemperatures. On the other hand, the corrosionpotential shows a much weaker temperature depen-dence. The intersection points of the Ecorr and Erpcurves can be compared with experimental criticalcrevice temperatures (Hibner189).

It should be noted that while the approach based oncomputing Ecorr and Erp can predict the long-termoccurrence and maximum propagation rate of loca-lized corrosion, it gives no spatial or temporal informa-tion. For predicting the spatial and temporal evolutionof localized corrosion,models are required that includea detailed treatment of mass transport and take intoaccount the geometric constraints of crevices, pits, andso on. Suchmodels are outside the scope of this chapter.

2.38.3.9 Selected Practical Applications ofAqueous Corrosion Modeling

In this section, we briefly outline selected models thathave been developed for practical applications on thebasis of the principles discussed above.

Extensive efforts have been devoted to the model-ing of aqueous corrosion in oil and gas environments.

95 C, 4M NaCl, exp

80 C, 4M MgCl2, exp

110 C, 4M MgCl2, exp

95 C, 4M NaCl, cal

80 C, 4M MgCl2, cal

110 C, 4M MgCl2, cal

inhibitionorrosion

1

tion potential of alloy 22 in concentrated chloride solutions.

n (2010), vol. 2, pp. 1585-1629

0.00.20.40.60.81.01.21.41.61.82.0

0t (�C)

E (S

HE

)

Erp

Ecorr

exp. CCT

10 20 30 40 50 60 70 80 90

Figure 12 Prediction of the critical crevice temperature(CCT) for various alloys in a 6% FeCl3 solution using

corrosion potential and repassivation potential models.188

The vertical lines show the location of the experimental CCT

values.189 The intersection of the calculated corrosionpotential and repassivation potential lines shows the

predicted critical crevice temperature.



This is due to the great practical importance ofcorrosion in oil and gas production and transmission,and to the fact that the number of key corrosivecomponents in such environments is relatively lim-ited (primarily to CO2, H2S, acetic acid, and O2), thusmaking the modeling task manageable despite theinherent complexity of corrosion mechanisms andtheir dependence of flow conditions. Corrosion mod-eling in this area has been reviewed by Nesic et al.,190

Nyborg,191 and Papavisanam et al.192–194 The CO2/H2S models range from expert systems based onlaboratory and field data to mechanistic models thatrecognize the partial electrochemical processes in anexplicit way. Among the electrochemical models, thetreatment of flow effects using mass transfer coeffi-cients has found wide applicability for both single-phase and two-phase flow (Nesic et al.,115 Dayalanet al.,195 Anderko and Young,169 Pots and Kapusta,196

Deng et al.197). While the methodology for modelingCO2 corrosion is well established, the modeling ofH2S effects is still in a state of flux and is a subject ofsignificant research efforts (Anderko and Young,169

Nesic et al.,170 Sun and Nesic171). A CO2 corrosionmodel based on a detailed treatment of transport(eqns [100]–[103]) has been developed by Nordsveenet al.,116 Nesic et al.,198 and Nesic and Lee.166 Thismodel has been further extended to CO2/H2S corro-sion by considering mechanistic aspects of FeS scaleformation (Nesic et al.,170 Sun and Nesic171). Themodel has been integrated with a multiphase flowmodel, which made it possible to predict the effect ofwater entrainment and water wetting in oil–water


systems (Nesic et al.170). Also, a stochastic algorithmfor predicting localized corrosion due to partially pro-tective FeCO3 scales has been integrated with thismodel (Nesic et al.199).

Another area in which electrochemical modelshave found wide applicability is the prediction ofthe corrosion potential in dilute aqueous solutionsthat exist in power generation industries. In particu-lar, mixed-potential models have been developed forstainless steels in water as a function of oxygen andhydrogen content (Macdonald,161 Lin et al.,138

Kim103). Such models are useful for calculating Ecorrin boiling water reactors and related environments.Ecorr is then further used for localized corrosionmodeling. In such systems, modeling of the initiationand evolution of crevice corrosion, stress corrosioncracking, and corrosion fatigue is of primary impor-tance. For this purpose, several electrochemical mod-els that are suitable for alloys in high temperatureand low-conductivity water have been developed (seeBetts and Boulton,200 Turnbull,201–203 Engelhardtet al.204,205 and references therein).

Corrosion in halide-containing natural and indus-trial environments such as seawater, deliquescingliquids, or production brines is another importantapplication for modeling. For example, King et al.123

developed a model for calculating the corrosionpotential of copper in aerated chloride solutions.Anderko and Young108 modeled general corrosionof steel in concentrated bromide brines used inabsorption cooling. Sridhar et al.131 developed amodel that predicts both the general corrosion andthe occurrence of localized corrosion for stainlesssteels and aluminum in seawater. A mixed-potentialmodel has been developed to predict the behavior ofnuclear fuel in steel containers (Shoesmith et al.206).Models for the initiation, stabilization, and propaga-tion of pitting, crevice corrosion, and cracking inhalide systems have been developed by a number ofauthors (Turnbull and Ferris,207 Turnbull,203,208 Bettsand Boulton,200 Engelhardt et al.,209,210 Cui et al.,211

and references therein).Modeling corrosion in the process industries is a

potentially fruitful area but is subject to great diffi-culties because of the complex and variable nature ofthe chemical environments. In principle, acid systemsare amenable to modeling because of their well-defined chemistry. In particular, corrosion in acidshas been modeled by Sridhar and Anderko212 in themoderate concentration range. Rahmani and Strutt213

developed a model for very concentrated sulfuricacid solutions, in which corrosion can be assumed to

n (2010), vol. 2, pp. 1585-1629



be exclusively under mass transport control. Veawaband Arronwilas214 developed a model for the generalcorrosion in amine–CO2 systems. Models are alsoavailable for the corrosivity in wet porous media(Huet et al.215). Anderko et al.216 applied the localizedcorrosion model based on calculating the repassiva-tion and corrosion potentials to predict the occur-rence of pitting and estimate the worst-casepropagation rates of localized corrosion in a processenvironment.

2.38.4 Concluding Remarks

Over the past three decades, tremendous progress hasbeen achieved in the development of computationalmodels of aqueous corrosion. A number of practicallyimportant models have been developed for applica-tions as diverse as oil and gas production and trans-mission, nuclear and fossil power generation,seawater service, and various chemical processes.

Thermodynamic models of electrolyte systemshave reached a level of sophistication that extendedtheir applicability range from dilute aqueous solu-tions to multiphase, multicomponent systems rangingfrom infinite dilution to solid saturation or puresolute limits. Although they were originally devel-oped mostly for applications other than corrosion(especially chemical processing and geology), theyare increasingly used to predict the solution chemis-try of corrosive environments and to understand theeffect of phase behavior on corrosion. Electrochemi-cal models of corroding interfaces have been devel-oped to predict the kinetics of anodic and cathodicreactions that are responsible for corrosion and torelate them to bulk solution chemistry and flow con-ditions. Semiempirical and mechanistic models ofpassivity have been developed to predict the behaviorof passive metals and the breakdown of passivity.Also, models are available to predict the thresholdconditions for localized corrosion.

The main focus of the models reviewed in thischapter is on relating the chemistry of the environ-ment to electrochemical corrosion phenomena onmetal surfaces. However, this is often only the firststage of corrosion modeling. Beyond this stage, themodels discussed here serve as a basis for simulatingthe spatial and temporal evolution of localized andgeneral corrosion damage in various engineeringstructures subject to localized and general corrosion.This level of modeling will be discussed in otherchapters of this volume.


Acknowledgments

The work reported in this paper has been supportedby the Department of Energy (award number DE-FC36-04GO14043) and cosponsored by Chevron-Texaco, DuPont, Haynes International, MitsubishiChemical, Shell, and Toyo Engineering.

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