A Theoretical Study of Crevice and Pitting Corrosion in

A Theoretical Study of Crevice and Pitting Corrosion in Steels.

Submitted by

Susan Margaret Sharland

for the degree of Doctor of Philosophy

to the University of London

1988

Department of Materials,

Royal School of Mines,

Imperial College of Science and Technology,

London.

1

Abstract

One of the most destructive modes of corrosion of metals is localised corrosion.

This can take many forms depending on the chemical and electrochemical

environment of the metal. It includes pitting corrosion which is characterised by

isolated cavities on the metal surface, and also crevice corrosion which occurs in

situations where two or more surfaces in close proximity lead to the creation of a

locally occluded region.

Localised corrosion is generally characterised by an intrinsic unpredictability, both

in the time and place of initiation and in the rate and direction of the

propagation of established cavities. This unpredictability has made the process very

difficult to investigate experimentally and despite much effort, it is still not

adequately understood.

The aim of this research is to improve the understanding of the physical

mechanisms controlling pitting and crevice corrosion and to aid the prediction of the

occurence and intensity of these forms of localised corrosion by developing a

number of mathematical models. The models described in this thesis are of three

types:

(1) Initiation models which predict when localised corrosion will occur relative to

the environmental conditions of the metal. These include a model of the

transport of oxygen to and its consumption at a metal embedded in a porous

medium which yields the maximum period in which localised corrosion can be

sustained, and a model of the evolution of the solution chemistry within

a passively corroding crevice in stainless steel until critical conditions are

achieved, whereupon the crevice will activate.

(2) Initiation models which investigate the underlying dynamics of the seemingly

random electrochemical behaviour associated with the initiation of localised

corrosion. These studies suggest that the process is deterministic and controlled

by a small number of variables rather than being truly random, as has

been assumed in many statistical models of initiation.

Propagation models which predict the rate of enlargement of active pits and

crevices. These involve both a steady-state and time-dependent description of

the solution chemistry and electrochemistry within an active pit or crevice

in carbon steel.

2

Contents

Acknowledgments. 15

1. Introduction. 16

2. Corrosion Processes. 22

2.1 Equilibrium processes and the Nernst equation. 22

2.2 Why corrosion happens. 23

2.3 Kinetics of corrosion processes. 24

2.4 Relationship of thermodynamic equilibria to corrosion cells. 26

2.5 Formation of passive films. 27

2.6 Localised corrosion mechanisms. 28

2.7 Susceptibility of iron and steels to localised attack. 29

I THE INITIATION OF LOCALISED CORROSION. 39

3. The Chemical Environment of a Passively Corroding Metal. 40

3.1 Introduction. 40

3.2 Environmental limitations to occurrence of localised attack. 40

3.3 Localised corrosion in a diffusion limited environment. 41

3.3.1 Description of model of oxygen concentration inconcrete pore water. 42

3.3.2 Parameters used for calculations of carbon steel inconcrete. 43

3.3.3 Results from model. 44

3

3.3.4 Discussion. 44

3.4 Initiation of crevice corrosion in stainless steel. 45

3.4.1 Literature survey. 45

3.4.2 Calculation of the crevice solution with the computer program CHEQMATE. 47

3.4.3 Comparison of CHEQMATE model with the Bernhardsson model. 49

3.4.4 Comparison of CHEQMATE model with experimental data. 52

3.4.5 Discussion. 55

3.5 Summary. 56

4. A Microscopic View of the Initiation of Localised Corrosion. 69


4.2 Passivation of metal surfaces. 70

4.3 The role of halide ions in pit nucleation. 74

4.4 Electrochemical fluctuations in passive systems. 76

4.5 Pit nucleation- probability event or dynamical process? 80

The Initiation of Localised Corrosion: A Process Governed by

a Strange Attractor? 89


5.2 Instabilities in non-linear dissipative systems. 89

5.3 Characterisation of strange attractors. 91

5.3.1 Reconstruction of dynamics of a system from a

time series. 91

4

Applications of this method and discussion of results. 93

5.4 Application of the method to current oscillations frompassive metal surfaces. 94

5.4.1 Results. 94

5.4.2 Discussion of results. 95

5.5 Implications of results to the construction of a mechanistic model of pit initiation. 96

• 5.6 Summary. 98

II. THE PROPAGATION OF LOCALISED CORROSION. 113

A Literature Survey of the Modelling of the Propagation of Pits

and Crevices. 114


6.2 Methods of modelling the propagation of localised corrosion cavities. 114

6.2.1 Transport by electromigration only. 117

6.2.2 Transport by diffusion only. 119

6.2.3 Transport by diffusion and electromigration. 121

6.2.4 Transport by convection. 128

6.2.5 Moving boundary formulations. 132

6.3 Summary. 133

5

A Mathematical Model of the Steady-State Propagation of Localised

Corrosion Cavities. 139


7.2 Description of the preliminary steady-state model. 139

7.2.1 Electrochemical reactions and data. 140

7.2.2 Solution chemistry. 141

7.2.3 Governing mass-transport equations. 142

7.2.4 Solution of mass-transport equations. 143

7.3 Results from preliminary model. 144

7.4 Precipitation of ferrous hydroxide. 146

7.5 Comparison of the preliminary model with experiment. 147

7.6 Sensitivity tests. 148

7.6.1 Addition of a moving boundary representation. 148

7.6.2 Addition of ferrous chloride. 149

7.6.3 Sensitivity to diffusion coefficients in cavity. 149

7.7 Summary. 150

8. A Finite-Element Model of the Propagation of Localised Corrosion

Cavities. 165


8.2 A finite-element model of corrosion cavity propagation. 165

8.2.1 The TGSL subroutine library. 165

8.2.2 Model of cavity propagation using TGIN. 166

8.3 Applications of the finite-element model. 167

6

8.3.1 Addition of ferrous chloride. 167

8.3.2 Comparison of finite-element model with experiment.

8.4 Summary.

168

172

9. Conclusions and Future Work. 178

References 183

Nomenclature 187

Appendix 1 190

Appendix 2 192

Appendix 3 199

Appendix 4 205

Appendix 5 210

Appendix 6 213

Appendix 7 216

Appendix 8 221

Appendix 9 223

7

List of figures

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.82.9

A schematic Pourbaix diagram for an iron/water system.

Schematic illustration of charge flow during aqueous

corrosion.

A typical Tafel plot of electrode potential against logarithm

of anodic current density.

A typical Evans diagram for corrosion of iron in which anodic and

cathodic areas are equal and electrolyte resistance is neglected.

A schematic plot of electrode potential against solution pH

for aqueous corrosion of a metal, indicating the environmental

conditions for the different modes of corrosion.

A schematic polarization curve of a corroding metal, illustrating the

effect of passive film formation on corrosion current density.

A Pourbaix diagram for 0.22% carbon steel in a solution

of 0.01 M H C O f /CO}~ with various concentrations of Cl~

ions present5.

A typical Pourbaix diagram for a ferrous-chromium alloy.

Schematic polarisation curves for various ferrous chromium alloys

to show the effect of increased chromium percentage.

30

31

32

33

34

35

36

37

38

Schematic Evans diagram of iron in a chloride solution. The

anodic curve is marked by the bold line and the cathodic curve by

the dashed three different rates of oxygen flux to the metal. 57

8

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

Schematic illustration of the model of a section of concrete with

a passive metal surface at one end and an infinite supply of oxygen

at the other. 58

Predicted oxygen concentration in concrete pore water across the

section with time for a diffusion coefficient 10-11 m2 s_1, leakage current

0.01 piA cm and concrete thickness 1 m. 59

Predicted oxygen concentration in concrete pore water across the

section with time for a diffusion coefficient 10-11 m2 s-1, leakage current

_o0.01 fiA cm and concrete thickness 0.3 m. 60

Predicted variation of guaranteed passive period of metal with leakage

current and diffusion coefficient. 61

Predicted variation of minimum concrete thickness to maintain passivity of

metal with oxygen diffusion coefficient and leakage current. 62

Flow chart of CHEQMATE program to indicate structure16. 63

Schematic illustration of CHEQMATE model of solution chemistry within

a passively corroding crevice in stainless steel. 64

Predicted change in pH at crevice base with time and chromium content

using parameters of Bernhardsson et al.14. 65

Comparison of predicted pH with chromium content from CHEQMATE

model with that of Bernhardsson14 and various experimental data from

the literature. 66

Comparison of predicted evolution of pH in a crevice solution using

CHEQMATE model for two sets of thermodynamic data. 67

Comparison of predicted pH with chromium content from CHEQMATE

against data of Bogar at al. . 68

9

4.2

4.3

Schematic anodic polarisation curve showing the relation of the passivation

and activation potentials to the dissolution current density. 86

o r

Current density-potential curves calculated by Griffin for different values

of the cation interaction parameter 87

An ensemble of current-time transients for l8Cr13Ni1Nb steel in

0.028 M NaCI polarized to 50 mV SCE36. 88

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

Time series of z variable in Lorenz equations calculated with R=15

and time delay r=0.25. 99

Time series of z variable in Lorenz equations calculated with R=28

and r=0.25. 100

Schematic representation of Lorenz attractor in (x,y,z) phase space. 101

Correlation function against distance for 15000 points of z variable time

series from Lorenz equations, calculated with R=28 and embedding

dimension n=3. 102

Corrosion current time series recorded at 5 Hz from stainless steel at

potential +200 mV in a 1000 ppm chloride solution. 103

Corrosion current time series recorded at 35 Hz from stainless steel at

potential -200 mV in a 1000 ppm chloride solution. 104

Corrosion current time series recorded at 35 Hz from dummy cell

representing instrument noise. 105

Correlation integrals calculated for corrosion current recorded at +200 mV

for embedding dimension 2, 4, 6 and 8. 106

Correlation dimension in figure 5.8 against log r for

embedding dimension 2, 4, 6 and 8. 107

Correlation integrals calculated for corrosion currents recorded at -200 mV

10

5.12

5.13

5.14

for embedding dimensions 2 to 8. 108

Correlation integrals calculated for data collected from dummy cell for

embedding dimension 2 up to 5. 109

Correlation dimension, v, against embedding dimension, n , for

+200 mV, -200 mV and dummy cell data. 110

Correlation integrals for z variable of Lorenz attractor sampled 8 times

less frequently than in figure 5.2 (r=4.0). 111

(a) Regular current oscillations from iron at 0.26 V in 1M H 2SOA .

(b) Oscillations from stainless steel in 1 M H 2S04 ,0.22 M Cl~ with

varying imposed potential34.

(c) Current oscillations measured from 18Cr13Ni1Nb steel in 0.028M NaCl

polarised to 50 mV36. 112

6.1

6.2

6.3

6.4

Illustration of the crack shape predicted by Bignold53. 135

Schematic representation of the experimental system used by Tester and

Isaacs54. 136

Variation of the concentration of ferrous and ferrous hydroxide ions with

distance from crack tip(x=0) as predicted by the model of Turnbull and

Thomas60. 137

Variation in pH in an artificial crevice with external potential at distance x

from the crevice tip as measured by Turnbull and Thomas60 138

7.1 Schematic illustration of the processes included in the

cavity propagation model.

Concentration profiles along the cavity length for a crevice with

passive walls.

152

153

11

7.4

7.5

• 7.6

7.7

7.8

7.9

7.10

7.11

7.12

7.13

Electrostatic potential drop along the cavity length for a crevice with

passive walls at various metal potentials. 154

Variation of corrosion current with crevice length for a crevice with

passive walls. 155


active walls. 156

Comparison of the potential drops along the crevice length for

corroding and non-corroding walls. 157

Corrosion current against cavity length for a crevice with

active walls. 158


active walls (with a precipitation reaction included but no change to

diffusion coefficients). 159

Experimental variation of corrosion current density with crevice

length5. 160

Comparison of Turnbull and Thomas's experimental71 and theoretical60

results of the variation of pH within a crevice with metal potential

and the predictions of the present model. 161

Comparison of potential drops along crevice length for calculations

carried out in a static and moving geometry. 162

Comparison of predicted concentration profiles in a crevice with a

static and non-static geometry. 163

Corrosion current against diffusion coefficient of Fe2+, FeOH+ , Na +

and Cl~ for a crevice with active walls. 164

Comparison of variation in predicted corrosion current density with crevice

12

length with and without ferrous chloride present as a solid phase in

the crevice. 173

8.2

8.3

8.4

8.5

Comparison of predicted corrosion current density with crevice length

from finite-element model with ferrous chloride and experimental data

of Marsh et al.5. 174

Comparison of predicted steady-state potential drop in a

crevice with experiment of Alvali and Cottis77. 175

Experimentally determined corrosion current densities at base of

an artificial crevice with passive and active walls by Beavers

and Thompson73. 176

Comparison of predicted corrosion current density from time-dependent

finite-element model and data of Beavers. 177

A4.1 Concentration profiles along the cavity length for a crevice with

corroding walls calculated using an approximation technique. 208

A4.2 Comparison of potential drops along cavity length for a crevice with

corroding walls using the approximation technique and the full solution. 209

A comparison of the relation between the guessed crevice tip

potential and the calculated value for a convergent case, (j)M = — 0.11 V

and a bifurcating system, (pM = — 0.09 V. 215

A7.1 a) A plot of the single-humped mapping function,/(jc) = Ajc(1— jc) with

A = 0.7, showing iterations to the fixed point jc*.

b) A plot of the mapping function applied twice, f 2(x). 218

A7.2 a) A plot of the single-humped mapping function,/(jc) with A = 0.785,

13

showing the iterations oscillating in a stable cycle of period 2.

b) A plot of this mapping function applied twice.

A7.3 A schematic illustration of a hierachy of unstable cycles in a

bifurcating system and the onset of chaos.

219

220

14

Acknowledgments.

I would first like to express my very sincere thanks to my supervisor at Harwell

Dr.P.W.Tasker for all his guidance and encouragement during the course of this work.

His invaluable help in directing my mathematical skills towards other

branches of science such as electrochemistry and materials science will always be

greatly appreciated. I would also like to thank my supervisors at Imperial

College, Professor D.lnman, who has provided me with much guidance on some of

the more fundamental aspects of electrochemistry, and Professor

B.C.H.Steele.

The research has been funded by the United Kingdom Atomic Energy Authority. I

am grateful to the Authority for this support and, in particular, to

DrAB.Lidiard, Head of Theoretical Physics Division at Harwell Laboratory, for

providing me with the opportunity to carry out the work. I would also like to thank a

number of my collegues from the Authority for their contributions;

Dr.C.M.Bishop from Theory Division, Culham Laboratory for his assistance in the

analysis of experimental current transients, Dr.CJ.Tweed of Theoretical Physics

Division, Harwell for her collaboration in the development of the CHEQMATE

computer program, Dr.C.PJackson of Theoretical Physics Division for his help in the

development of the finite-element program. Their assistance is very much

appreciated. I am also very grateful to Dr.M J.Norgett of Theoretical Physics Division

for so many helpful comments on the completed thesis. This thesis was

produced on Typesetting System for Scientific Documents (TSSD), a typesetting

system developed by Mr.MJ.Hopper of Computer Science and Systems Division,

Harwell.

Finally, I would like to thank my parents for so much support and encouragement

throughout my education and all my family and friends (especially Simon and

Sara) for suffering me while I have worked on this thesis.

1. Introduction.Most naturally occurring metals are recovered from the earth in an oxidised state

and occur in the earth's crust in ores. There are exceptions to this, such as

gold and platinum and these elements are noted for their ability to remain

untarnished. However, most other metals on exposure to the atmosphere tend to

return to an oxidised form. This process is known as corrosion and can take many

forms depending on the precise nature of the environment and the composition of

the metal or alloy. Corrosion of metal structures can lead to their premature

failure and be very damaging economically. It is estimated that 2-3 % of the Gross

National Product of industrialised countries of the West is consumed by the

replacement or repair of metallic structures through corrosion. Consequently, it has

been a subject of much research. Greater understanding of the physical and

chemical processes involved in corrosion may lead to the development of techniques

to minimise its occurrence and effects.

Corrosion takes many forms, the simplest and best characterised of which is

uniform corrosion. It is distinguished by a progressive and uniform thinning of the

metal surface. Under certain chemical and electrochemical conditions, this process

leads to the deposition of an oxide film on the metal surface which tends to

inhibit the corrosion. The rate of corrosion though such a film depends on many

factors including the composition of the film (determined by the type of metal and

the composition of the environment), the film thickness and its atomic

structure. Generally the rate of metal wastage in this passive state is less than that of

a bare, active metal surface. However, flaws in this film can lead to the

exposure of small parts of the metal surface to the corrosive environment. In some

cases the passive film can repair itself and re-cover the metal but under

certain conditions, these bare sites can activate and become areas of localised corrosion . This can take several forms, but is generally characterised by much higher

dissolution rates than those associated with uniform corrosion1. The high

penetration rates make localised corrosion a particularly damaging form of

degradation and consequently, this particular mode is of great interest to corrosion

scientists and engineers in a wide range of practical situations.

One of the most destructive forms of localised corrosion is pitting which is

characterised by small cavities on the metal surface, initiated where the passive film

has been locally removed. The shape of these pits depends on many factors

such as metal composition and surface orientation. Another mode of attack is crevice

corrosion which occurs in situations where two or more surfaces in close

proximity lead to the creation of a locally occluded region where enhanced

dissolution can occur. The intensity of attack is a function of the width and length of

the crevice. It is generally favoured by conditions of stagnant solution in the

gap. The restricted geometry in the crevice makes an exchange of solution between

the interior and the bulk difficult, so changes in the composition of the

electrolyte in a crevice occur much more quickly than for an exposed pit.

Consequently, crevice corrosion tends to initiate more rapidly than pitting. There are,

however, some close interrelations between pitting and crevice corrosion.

Some authors have suggested that pitting is a special form of crevice corrosion and

that micropores present in a metal surface act in a manner similar to gaps

between two metal parts where local acidification occurs . Another assumption is

that crevice corrosion starts from pits formed within the crevice . Both of these views

may be valid under special chemical conditions. From the electrochemical

point of view, it has been suggested that both processes are identical but crevice

corrosion involves longer ionic diffusion paths4.

Localised corrosion appears to be characterised by an intrinsic unpredictability,

both in the time and place of initiation of the process and in the rate and direction of

the propagation of established cavities. This unpredictability has made

localised corrosion a very difficult phenomenon to investigate and despite much

effort it is still not adequately understood. The difficulty of both achieving controlled

localised corrosion under laboratory conditions and measuring and

interpreting experimental results has led to the development of mathematical models

of the process. This type of mathematical modelling has several important

functions:

(1) Models may be used to interpret experimental data and relate empirical results

to various physical and chemical mechanisms. They can also be used to

identify which processes are controlling the behaviour of the system at various

stages.

(2) In some cases, new effects can be predicted by the model, which are later

confirmed by experiment.

(3) Mathematical models provide a means of extrapolating empirical results over

longer timescales than the duration of the experiment (provided, of

course, that the model is validated by the short-term results). They may also be

used to extrapolate to other physical conditions, such as different

external chemistries.

Such models may be broadly divided into two categories; mechanistic models and

empirical data-fitting models. A mechanistic model is constructed by firstly

identifying the aspect of a system of interest and then the important physical

processes in the system that contribute to the development of this phenomenon.

These physical processes and their interactions must then be translated to a set of

17

governing mathematical equations. The equations are solved with parameter

values inferred from experimental data. The interpretation of the solutions of the

equations form the model's predictions which must be compared with further

empirical data for proper validation of the model. In most systems, there are many

levels of approximation for the various physical processes and it is not always

true that the most complicated models provide the most realistic predictions; if the

approximations are based on sound physical arguments, then often a

simplified representation can provide a very reasonable simulation of a process. It is

quite difficult to assess immediately the level of approximation necessary, and

it often requires some investigation of the sensitivity of the model to the

approximations once the model is completed. In this way, mechanistic models are

often constructed as an iterative process, refining assumptions and

approximations at each stage. The predictions from the final model should be

insensitive to more complex quantifications of the physical processes of the system,

or indeed to the mathematical solving techniques employed. A data-fitting

model relies on a more direct usage of empirical data. Generally these models are

used in situations in which the physical and chemical processes controlling

the system are not so readily identifiable. They involve a large number of parameters

which are calculated with various statistical manipulations of experimental

data. It is generally more difficult to use this type of model in a predictive role.

In this thesis, examples of mathematical models of both types of the various

stages of pitting and crevice corrosion will be developed. A single model of the

whole process would not only be impractical but smaller, more detailed models

should give a better insight into understanding the important physical and chemical

processes in each stage. At particular stages, pitting and crevice corrosion are

closely interrelated so that many of the models will be applicable to both. The

predictions from these models are presently used to assess the important

mechanisms during the process, but ultimately these models may extrapolate

experimental results to aid the prediction of the occurence and intensity of localised

corrosion. The models may be roughly divided into three types:

(1) Initiation models which allow the prediction of when localised corrosion will

occur, based on mechanistic arguments. Localised corrosion can only initiate

and be sustained under fairly specific chemical and electrochemical

conditions, so these models predict the environment of the metal in various

situations. In particular, a simple but novel method of predicting the period in

which localised corrosion can be initiated and sustained on a passive

metal in a diffusion-limited environment such as concrete is developed. A

second model is developed to predict the time from exposure to an electrolyte

to the onset of crevice corrosion in stainless steel by considering the

evolution of the solution chemistry within a passively corroding crevice. This

problem involves the coupling of ionic transport within the crevice and the

complex solution chemistry arising from the release of various cations from the

steel. Some models reported in the literature neglect some of the

important physical phenomena and their predictions are not in accord with

experimental data.

Initiation models which involve a more microscopic perspective of the process

and include phenomena such as passive-film formation, film rupture

under mechanical stresses, film penetration by aggressive anions etc. The

microscopic mechanisms involved in initiation are extremely complex, so this

area of our modelling will rely more on the analysis of empirical data. It

has been widely assumed that the initiation of localised corrosion is a random

event since many of the measures of the process (such as the corrosion

current) display highly irregular oscillations and are extremely sensitive to the

initial conditions applied to the system. A number of models which draw

on the statistics of stochastic processes have been developed with reasonable

success. However, recent developments in the theory of non-linear

dynamics have demonstrated the existence of very complex (so-called chaotic)

solutions to very simple deterministic equations. The state of the system

(i.e. the values of the variables in the equations) at any time may be represented

as a point in an abstract space whose coordinates are the variables

('phase space'). As time evolves, the solution of the equations traces out a

'trajectory' in this phase space. In the case of dissipative systems, these

trajectories eventually remain confined to a subset of the phase space (known

as an attractor). Chaotic behaviour occurs where there exists a 'strange'

attractor (i.e. one having non-integral dimension) and the apparent randomness

of the solutions arises from an extreme sensitivity to initial conditions. A

novel and relatively simple method of data analysis which allows deterministic

chaotic behaviour to be distinguished from truly random fluctuations is

applied to some typical oscillatory current transients associated with the

initiation of localised corrosion.

(3) Propagation models describing the enlargement of cavities with active walls.

The rate of corrosion of the cavity in this case is strongly dependent on the

composition of the solution in the crevice (unlike the initiation stage when the

metal is screened by a passive film). This leads to some rather complex

non-linearities in the mathematical description of this process. Again, several

models have been developed in the literature which involve various

19

simplifications to these non-linearities. The validity of these approximations will

be discussed and a number of mathematical techniques developed to

provide a more rigorous model of cavity enlargement which yields a closer

agreement with experimental data over a wider range of conditions.

The various mathematical models developed in this thesis will principally be

applied to localised corrosion in iron and steels, although most of the methods are

applicable to other materials with appropriately altered parameter values. In

Chapter 2, some of the basic chemistry and electrochemistry implicated in corrosion

processes is outlined and a brief description of some of the mechanisms

associated with pitting and crevice corrosion is given. Some of the differences

between iron, carbon steels and stainless steels are discussed and these are related

to their susceptibility to localised corrosion.

The main part of the thesis will be divided into two sections. The first of these is

entitled The Initiation of Localised Corrosion and contains Chapters 3, 4

and 5. In Chapter 3, some of the fundamental environmental limitations to the

occurrence of localised corrosion are outlined and a number of models of the various

changes in the solution chemistry surrounding a corroding metal as described

in (1) above, are constructed. Chapters 4 and 5 concentrate mainly on models of type

(2) which involve a more microscopic perspective of the initiation process. In

Chapter 4, some models and theories from the literature are described and discussed

and the need for a more fundamental investigation of the source of the

unpredictability of the initiation phase is highlighted. In Chapter 5, recent

developments in the theory of non-linear dynamics are applied to the corrosion

situation and the assumption of a stochastic (i.e. truly random) process made in

many statistical interpretations is investigated.

The second section of the thesis (Chapters 6, 7 and 8) considers The Propagation

of Localised Corrosion and includes models of type (3). Chapter 6

comprises a survey of modelling of the propagation of active cavities from the

literature. The strengths and deficiencies of these models are discussed. In Chapter

7, a preliminary steady-state model of the solution chemistry and

electrochemistry within a pit or crevice is developed and in Chapter 8 a more

sophisticated model which allows both a better representation of the solution

chemistry and a time-dependent description of the system is described. This model

uses a finite-element numerical technique. The predictions of the models are

compared with various experimental data from the literature.

Chapter 9 summarises the results of the various models and the main conclusions

that have been reached through these studies. Directions of future work which

either follow up new approaches to modelling a particular phase of the localised

corrosion process, or generally improve the accuracy of the models' predictions are

also outlined in this final chapter.

21

2. Corrosion Processes

2.1 Equilibrium Processes and the Nemst Equation.

When metals other than noble metals are exposed to water at room temperature,

there is a tendency for them to undergo a reaction such as the following,

This spontaneous reaction is accompanied by a negative change of free energy, AG.

Initially the forward (anodic or dissolution) reaction dominates, but as it

and negative charge is left on the metal. (The arrangement of charges

constitutes an electrical double layer similar to a capacitor). This gradient tends to

slow down the dissolution reaction until a potential is reached at which the

equilibrium occurs is the equilibrium potential, E eq(=(f>M—(ps, where <pM is

the inner potential of the metal and (ps is the potential in the solution immediately

beyond the electrical double layer).

At equilibrium the electrochemical potentials of the ions in the metal and the ions

in solution are equal. The electrochemical potential, p is related to the

chemical potential, p by the following,

M(s) + m H20(ads) M z+ .mH20(aq) + ze . (2.1.1)

proceeds an adverse potential gradient is set up as positive metal ions enter solution

rates of dissolution and deposition are equal and there is no net charge transfer. The

resulting equilibrium is a dynamic equilibrium; charge continues to be

transferred across the double layer but at equal and opposite rates. The flux of

charge is known as the exchange current density, iQ and the potential at which the

p = p + zF(p (2.12 )

where (p is the electrostatic potential. At equilibrium,

f*S ~ l*M' (2.1.3)

therefore,

t*s - Pm = zF(<Pm - <Ps) = zFE‘q ■ (2.1.4)

Now, ft = ft0 + RTlna, where ft0 is the standard chemical potential and a is the

ionic activity, therefore

where E° is the standard electrode potential. This is the Nemst equation and it may

be applied to any process in which charge transfer takes place e.g. oxidation

and reduction. In this case, the equation may be written

E 'q = E ° + £ £ ln — , (2.1.5)zF “,'d

where aox is the activity of the oxidised species (e.g. the metal ions) and ared is the

activity of the reduced species (e.g. the metal itself, in which case the activity

is unity).

A Pourbaix diagram is a plot of equilibrium potential against pH using the Nemst

equation and indicates regimes of thermodynamic stability of various phases

at a particular temperature. Figure 2.1 shows an example of a schematic Pourbaix

diagram for the iron/water system. Horizontal lines indicate that no hydrogen ions are

involved in the equilibrium (such as in reaction (2.1.1)) and vertical lines

indicate that no oxidation is involved. In all cases, the activities of all ions except H + are taken as 10” 6.

2.2 Why Corrosion Happens.

The spontaneous dissolution of metal in water leads to the release of cations into

the solution and an excess of electrons being left in the metal, forming an

electrical double layer, as described in section 2.1. The potential gradient across this

double layer is such that the rate of metal dissolution is exactly balanced by

the rate of metal deposition in equation (2.1.1). Let us now consider the effect of

removing these electrons from the metal by providing a cathodic reactant with which

they spontaneously combine. Examples of possible cathodic (i.e. electron

consuming) reactions in corroding systems include the reduction of oxygen,

0 2 + 2H 20 + 4e" 4 0 / / " , (22.1)

the reduction of water

H20 + e " H + O H ~, (222)

and the reduction of hydrogen ions

H + + e " H. (22.3)

The consumption of the electrons in the metal perturbs the equilibrium within the

electrical double layer by reducing the potential gradient across it. This shift in

potential increases the forward reaction of equation (2.1.1) so more positive ions are

released into solution and more electrons are left on the metal, available for

cathodic reduction reactions. This is a simple description of a corrosion cell, in which

anodic and cathodic reactions tend to act together to produce a self-sustaining

system with the overall effect of a net metal dissolution. The process is schematically

illustrated in figure 2.2. The sites of these reactions may be less than a

millimetre apart in the case of uniform corrosion, or may be separated by as much as

several metres for certain specific localised corrosion systems. In the next

section, the kinetics of the reactions within a corrosion cell are considered.

2.3 Kinetics of Corrosion Processes.

The rate at which corrosion occurs is determined by kinetic factors such as the rate

of electron transfer across the double layer and mass transport of ions in

solution. In ordinary chemical kinetics, where there is no electrochemical component,

the rate constant of a reaction, k , is related to the height of the energy barrier,

Ea, according to a relation

An electrochemical rate constant requires an additional term to reflect the additional

electrostatic potential barrier.

where / is a function of the potential, E, the charge number, z and Faraday's

constant, F, concerned with the passage of an ion across the double layer. At the

simplest level

with ft = 0.5 for a symmetrical energy barrier. For a cathodic reaction (such as the

reverse reaction in (2.1.1)) the rate constant k_ is given by

with k+ = A aexp(—Ea/RT).

At equilibrium, the flux of charge across the electrical double layer is the same in

each direction, i.e.

k = Aexp(-EjRT). (2.3.1)

k = Aexp(—(Ea + f(EzF))/RT). (2.3.2)

f(EzF) = PEzF (2.3.3)

k_ — k_exp{—fiEzF/RT) (2.3.4)

with k_ the chemical rate constant,

k_ = A cexp(—EjRT).

For an anodic reaction,

k+ = k+exp((l—p)EzF/RT) (2.3.5)

24

(2.3.6)

where i+ and /_ denote the 'partial current densities' in the forward and reverse

directions of the electrode reaction. If the electrode potential is shifted from

equilibrium in the cathodic direction the same equations apply with

i_ > i+ (2.3.7)

and the net cathodic current density is given by

0 ic = / _ - / + . (2.3.8)

If the electrode potential is shifted in the opposite direction from equilibrium then the

net anodic current density is

ia = i+ ~ L (2.3.9)

where

i_ = nFCik_exp(—pEzF/RT) (2.3.10)

i+ = rtFk+ exp ((l—p)EzF/RT). (2.3.11)

Rewriting E as

E = E eq + q

where E eq is the equilibrium potential of the system, these partial current densities

become

/_ = i0exp(—pqzF/RT) (2.3.12)

i+ = i0exp ((l—P)rjzF/RT). (2.3.13)

with

i0 = nFCtk_exp(—pE eqzF/RT) = nFk+exp((l - p )E eqzF/RT). (2.3.14)

(The C, term reflects the dependence of the cathodic current on the concentration of

the cathodic reactant.) Thus the net current density in an electrode reaction, i is given by

i = i + - L = i0(exp((l - P)Fr]/RT) - exp(~PzFr]/RT)). (2.3.15)

This is the Butler-Volmer equation. The quantity q in an electrochemical cell is

known as the overpotential. It is literally the extra potential which one must apply to

25

an electrode reaction to make it occur at a certain net velocity. If the

overpotential is zero, the anodic and cathodic partial electrode rates are still occuring

but at equal and opposite rates, as we have discussed previously. For a

positive overpotential, the anodic process dominates whereas for a negative

overpotential, there is a net cathodic current. At large overpotentials (associated with

corrosion cells), this equation may be approximated by

i = i0exp((l - P)zFrj/RT). (2.3.16)

This is the Tafel equation. Figure 2.3 shows a schematic plot of anodic polarization

under such conditions. The polarization characteristics of a metal i.e. the

parameters /0 and P can be determined experimentally by applying a series of

potentials to a specimen, measuring the steady-state current density and plotting the

Tafel curves.

The Evans Diagram.

An Evans diagram is an experimentally determined plot of both anodic and

cathodic polarization curves for a corroding system. Figure 2.4 shows an example for

the uniform corrosion of a metal in an oxygenated environment where the

dominant cathodic reaction is the reduction of oxygen. The potential and rate of

corrosion of the system, Econ and icorr are marked by the point of intersection of the

polarization curves. At this point, the total number of electrons generated by

the metal dissolution exactly balances the number of electrons being consumed by

the cathodic reduction process. This potential is positive to the equilibrium

potential associated with the metal dissolution, therefore giving a positive

overpotential.

2.4 Relationship of Thermodynamic Equilibria Calculations to Corrosion Cells.

It is important to appreciate that the behaviour of a corrosion cell, i.e. the

steady-state current, is governed both by kinetic parameters and by thermodynamics.

The thermodynamic calculations can also provide an extremely useful basis

for understanding the conditions under which corrosion can occur. If the free energy

change of an electrochemical reaction is negative, then the reaction occurs

spontaneously. The free energy change of an electrochemical reaction is given by

AG = - zFE (2.4.1)

where E is the cell e.m.f. corresponding to the reaction associated with the reaction.

Thus, for a spontaneous reaction, E is positive. Let us consider the

thermodynamic equilibria of the overall corrosion reaction for iron in a deoxygenated

environment where the dominant cathodic reaction is the reduction of

hydrogen, i.e.

(2.4.2)Fe + 2H + ^ Fe2+ + H 2. The Nernst equation for this reaction is

E ( = Ecorr) = £ ° + (2.4.3)

E = £ ° - - ^ F l o g C ^ . (2.4.4)

From this equation, one can establish the conditions at which corrosion should begin

by plotting the reversible potential of hydrogen as a function of pH (on a

Pourbaix diagram) and finding where the free energy of the reaction for the

dissolution of iron becomes negative, for example one can calculate the pH for E to

be positive assuming a certain ferrous ion concentration. In this way, Pourbaix

diagrams become an extremely useful device in aiding the understanding the

corrosion behaviour of metals. They are commonly simplified in terms of domains of

corrosion behaviour, for example figure 2.5 shows a schematic Pourbaix

diagram of iron in sea water. Three basic domains of behaviour are indicated; the

metal can be either thermodynamically stable, corroding uniformly or covered by a

passivating surface film. However, if suitable conditions for corrosion do exist

in a given system, this approach yields no information on the magnitude of the

corrosion current. To obtain a full view of a corrosion process, one must study both

thermodynamic equilibria (with Pourbaix diagrams) and electrode reaction

rate data (using Evans diagrams, for example).

2.5 Formation of Passive Films.

During uniform corrosion of a bare metal, a solid oxide or hydroxide will

precipitate at the metal surface when its solubility product is exceeded. If the

conditions of potential and pH are appropriate, this surface film will hinder and may

even stop corrosion. At a fairly simple level this may be explained as follows.

The film slows the migration of metal ions away from the surface, leading to a build

up of positive ions near the surface and an increase in electrode potential.

This tends to reduce the metal dissolution current eventually to a value governed by

the ionic conductivity of the film. This steady-state current is known as the

leakage current. The 'appropriate conditions' of potential and pH may be identified

by the passive region on a Pourbaix diagram (figure 2.5). The passivation of a

metal also has an effect on the form of the polarization curve. In figure 2.6, this

schematic polarization diagram shows the effect of the surface film on the corrosion

current. The current density reaches a maximum with increased electrode

potential marking the point at which the migration of metal ions becomes hindered

2.6 Localised Corrosion Mechanisms.

The mechanism whereby localised corrosion develops on passive surfaces of

metals has been a subject of considerable research. It is generally agreed that there

are two main stages to the process, the initiation of a site and its propagation

into the metal, each governed by different mechanisms. The initiation stage is related

to flaws in the passive film, but there is much debate into the way in which

these flaws occur and their consequences on a microscopic scale. However, one of

the critical environmental parameters determining the resistance of a passive

film to breakdown is the concentration of halide ions in solution. This may be

illustrated by the empirically determined Pourbaix diagram in figure 2.7. This shows

the regions of stability for carbon steel in a solution of 0.01 M H C O f /CO$~ with various concentrations of chloride ions present5. As the concentration increases

the region of passivity becomes smaller and the electrode potential necessary

to sustain localised corrosion extends to more cathodic values. Some current

theories of the microscopic interactions of halide ions with passive films are

discussed in Chapter 4.3. There is generally more agreement, however, regarding the

physical mechanisms of the cavity propagation stage. The essential features

are as follows:

(1) At the end of the initiation stage, the environment is sufficiently acidic to cause

the metal to corrode actively. This may be illustrated on the Pourbaix

diagram in figure 2.5. The conditions over the bulk of the surface film may be

represented at point A (high pH, high potential), whereas those at the

corrosion site are represented at point B (lower pH and potential). This results

in the development of a potential gradient between the active sites and

the remaining passive surface6.

(2) The cathodic reactions required to sustain the dissolution at the active sites take

place at this surface and as a result, anions such as OH~ and Cl~

migrate into the active sites.

(3) The migration of the anions into the pit results in the formation of species such

as FeCl2 which hydrolyse to produce HCl. This maintains the acidity of

the sites, thus sustaining the localised corrosion process.

This mechanism has been supported by measurements of the pH in localised

corrosion sites, which in the case of carbon steel is in the region 3-47. The electrode

potential within localised corrosion sites has also been measured and has

been shown to be more active than the surrounding surface.

by the presence of the film. In reality the creation of passive films involves a

complex and interrelated set of processes. Some current theories are described and

discussed in Chapter 4.

2.7 Susceptibility of Iron and Steels to Localised Attack.

The susceptibility of different metals to corrosion is closely related to the quality of

the passive film. The chemical composition of the film, its structure, chemical

and physical properties, coherence and thickness are all of great significance. The

presence of chromium in steel is known to be beneficial in reducing both the

likelihood of initiation of localised corrosion and its subsequent rate of development.

This arises from the high stability and low ionic conductivity of Cr20 3 which

exists as a passive film on chromium itself and also on alloys in which it is present,

even at low concentrations. Figure 2.8 shows a schematic Pourbaix diagram

for a ferrous-chrome alloy in dilute chloride solution. Comparison with the Pourbaix

diagram in figure 2.5 for pure iron in a similar solution shows that the passive

area has been extended and the area of pitting susceptibility reduced. Figure 2.9

shows the influence of chromium content on the polarization curves of the alloys in a

chemically aggressive solution. The leakage current is clearly reduced with

the addition of further chromium indicating the increased integrity of the passive film.

Other alloying elements in stainless steel e.g. nickel and molybdenum further

improve the tenacity of the film. There are other methods of improving the quality of

passive films on metals including heat treatments and removal of impurities

such as carbon and sulphides which can accumulate at the surface in small areas

and cause local thinning of the film.

u>o

Figure 2.1 A schematic Pourbalx diagram for an iron/u/ater system.

METAL WATER

Cathode 0 2 —► reduced products

\

f low in metal

\Positive

ionic flow in w ater

Anode ® M — Mn* ( a q )

Figure 2.2 Schematic illustration of charge flow during aqueous corrosion.

u>N3

Figure 2.3 A typical Tafel plot of electrode potential against logarithm of anodic

current density.

Figure 2.4 A typical Evans diagram for corrosion of iron in which anodic and

cathodic areas are equal and electrolyte resistance is neglected.

0 7 UpH

Figure 2.5 A schematic plot of electrode potential against solution pH for aqueous

corrosion of a metal, indicating the environmental conditions for the different

modes of corrosion.

Ele

ctro

de

po

ten

tial

Pi t t ing

Di s s o l u t i on through film

P a s s i v e film formation i nh i b i t s corrosion c ur r e n t

D i s s o l u t i o n of bare metal

Corrosion current d e n s i t y

Figure 2.6 A schematic polarization curve of a corroding metal, illustrating the effect

of passive film formation on corrosion current density.

Met

al

po

ten

tial

,

O r

pH

Figure 2.7 A Pourbaix diagram for 0.22% carbon steel in a solution of 0.01 M

H C O ^ fC O \" with various concentrations of C/~ ions present5.

Elec

trod

e po

tent

ial

Figure 2.9 Schematic polarisation curve for various ferrous-chrome alloys to show

the effect of increased chromium percentage.

38

PART I:

THE INITIATION OF LOCALISED CORROSION.

39

3. The Chemical Environment of a Passively Corroding Metal.

3.1 Introduction.

In this chapter, some fundamental environmental limitations to the occurrence of

localised corrosion are outlined and several models of various changes in the

solution chemistry surrounding a corroding metal are constructed. These changes are

related to the subsequent behaviour of the system. Two aspects of the

environment are considered; the concentration of oxygen at the metal surface

(critical to the initiation and the persistence of both pitting and crevice corrosion) and

the solution chemistry within a passively corroding crevice (critical to the

breakdown of the oxide film and the onset of crevice corrosion).

In section 3.2, it will be demonstrated that localised corrosion is only possible if

there is sufficient oxygen (or some similar oxidising agent) over the bulk

metal surface to maintain a positive potential gradient between the active area

(whether an isolated patch on the metal surface or within an existing cavity) and the

passive film covering the bulk of the surface. In a diffusion limited

environment, the flux of oxygen may become restricted to a point where passivity can

no longer be maintained. In section 3.3, a model of the transport of oxygen to

a metal surface in such an environment is presented. This model considers the

length of the time in which pitting or crevice corrosion may be initiated and sustained

in terms of the total supply of oxygen present and the transport properties and

thickness of the material covering the metal. The second model concerning the

chemical environment of the metal is described in section 3.4; this model calculates

changes in the solution within a passively corroding crevice and the length of

time until a 'critical solution chemistry' is attained. At this point, the oxide film on the

cavity walls can no longer be assumed stable and activation of the metal may

occur, leading to crevice corrosion. (The attainment of this critical solution chemistry

may be regarded as a second necessary condition for the initiation of crevice

corrosion, in addition to the requirement of an adequate flux of oxygen to the bulk

metal surface).

3.2 Environmental Limitations to Occurrence of Localised Attack.

If a passive metal surface is locally activated by some mechanism and a localised

corrosion site is nucleated, then the mechanism for cavity propagation

outlined in section 2.6 indicates a fundamental limitation to the persistence of such

attack; it ceases when the potential at the surface governing the rate of

cathodic reaction is no longer more positive than the potential at the active sites.

Without a supply of cathodic current from the external surface all the cathodic

reactions required to sustain the dissolution will have to occur within the active site.

This will result in the consumption of acidity which will no longer be

regenerated by hydrolysis, because there is no potential gradient to cause migration

of anions into the active sites. An example of such a limitation in the supply

of cathodic charge is a restricted supply of oxygen to the metal surface, as may occur

in reinforced concrete, for example. The effect may be demonstrated with a

schematic Evans diagram of the polarization of iron in a chloride solution in figure

3.1. The solid line represents the anodic polarization and shows regimes of

uniform corrosion, passivity and pitting. The broken line represents the cathodic

reaction rate, assuming that the main cathodic reactants are oxygen and water. The

potential of the metal is given by the point of intersection of the anodic and

cathodic curves. At high oxygen concentrations the corrosion potential is given by

point A' in the figure. However, if there is insufficient supply of oxygen than the

potential drops to point B' and the potential drops below that at the base of the

cavity. The minimum requirement for maintaining the metal at a passive or possible

pitting potential (e.g. A') is that the oxygen reduction rate is at least equal to

the rate of metal dissolution through the passive film. Comparison of corrosion

potentials A' and B' with the electrode potential in a localised corrosion site (such as

B in figure 2.5) shows that A' is more positive and B' more negative. This

leads to the conclusion that a metal can only be subject to localised corrosion when

in a passive state. There are, however, situations in which the above analysis

no longer applies, where there is a driving force for migration against a concentration

gradient such as a temperature gradient. An example of this occurs in the

corrosion of boiler tube steels in highly deoxygenated water. The key difference here

is that the high heat-flux causes the concentration of corrosive salts in pores

and fissures in the protective Fe3 0 4 layer to increase. However, in most cases of

metal corrosion in a diffusion limited environment, the maximum period for localised

corrosion can be estimated by calculating the rate of transport of oxygen to

the metal surface. Once the supply of oxygen does decrease to this critical level and

the cathodic reaction switches to the reduction of water, then the stability of

the passive film can no longer be guaranteed. The mechanism by which the film

actually breaks up is not certain, but eventually the metal will return to a state of slow

active corrosion.

3.3 Localised Corrosion in a Diffusion Limited Environment.

There are many situations in which the supply of oxygen to the metal surface can

become sufficiently low that localised corrosion is no longer possible by the

mechanisms described in section 3.2. This may arise in systems where the oxygen

supply is not replenished as it is consumed by the cathodic reactions, or

where the transport of oxygen to the metal surface is restricted by a porous medium

such as concrete. The ability to predict the period after which localised

corrosion can no longer initiate or continue to propagate, is potentially very useful,

especially in situations where it is necessary to provide a corrosion allowance

in the thickness of the metal. (The metal dissolution rate associated with localised

corrosion can be many orders of magnitude faster than uniform corrosion, so

a prediction of the period in which each mode of corrosion is possible could

significantly affect the overall corrosion allowance). This period may be calculated

using relatively simple mechanistic arguments.

A model of the transport and consumption of oxygen in a diffusion limited

environment is described in section 3.3.1 and applied to carbon steel embedded in

concrete in section 3.3.2. However, this model is readily applicable to other

materials with appropriately modified input data. A realistic range of physical

parameters for this system are identified and concentration profiles of dissolved

oxygen are presented for two different concrete thicknesses. In one case, the

diffusion of oxygen is not fast enough to initiate or sustain localised corrosion

beyond a finite period, and in the other a steady-state profile is achieved with

adequate oxygen transport to maintain pitting or crevicing 'indefinitely'. The passive

period and the concept of a minimum concrete thickness below which there is

no 'switch off' of localised corrosion are explored further as functions of diffusion

coefficient and leakage current in section 3.3.3.

3.3.1 Description of Model of Oxygen Concentration in Concrete Pore Water.

In the model, a one-dimensional section of concrete with a passive metal surface

at one end and an infinite supply of oxygen at the other is considered. This

system is schematically illustrated in figure 32. The concrete is assumed totally

saturated with water and initially this pore water is saturated with oxygen. However,

the effect of the existence of any fast migration pathways for the oxygen will

be investigated. Further, it is assumed that there is no fluid flow through the system,

so the only transport mechanism for the oxygen is diffusion. The

concentration of dissolved oxygen in the concrete pore water, C(x,t) is then

governed by the diffusion equation,

D d2C(x,t) _ a BC(x,t)Bx2 &

0 b , (3.3.1)

where D is the intrinsic diffusion coefficient of oxygen in concrete and oc is the

capacity factor. This parameter reflects both the physical retention of the diffusing

42

ion in the porosity and the bulk chemical retention due to equilibrium sorption. In this

case, the sorption of dissolved oxygen on concrete is assumed negligible and

oc is equal to the porosity. The thickness of the concrete is given by b. The boundary

conditions to the problem may be assumed as follows:

(1) At time zero (the time at which the corrosion starts), a uniform distribution of

oxygen exists through the concrete pore structure. As a 'worst case' which

would tend to overestimate the period in which localised corrosion is possible, it

is assumed that the initial concentration of dissolved oxygen, C0 is 8.9 x

10 3 mol dm 3. This repês'e^cts "the. concerirtra.ilribri ©J-p r e s e v i j t lVv. t;ke_ cerwe-wt p o re s .

(2) At the metal surface, the flux of oxygen remains equal to the leakage current, iL (multiplied by the appropriate charge and unit conversion factors ), i.e.

D a c( o,o = (l

dx 4 F

(3) The concentration of oxygen in the pore water at the other end of the concrete

surface remains constant at value C0 , i.e.

C ( M = c 0

Equation (3.3.1) and boundary conditions (1)—(3) are solved using a Laplace

transform technique (Appendix 1) to yield oxygen concentration profiles as a function

of time.

3.3.2 Parameters Used for Calculations of Carbon Steel in Concrete.

Leakage currents of carbon steel in concrete have been measured in the range

0.01-1.0 juA cm .There is much variation in the diffusive properties of oxygen in

concrete, depending on the degree of saturation; Tuutti9 reported that the

diffusivity in the gas phase may be between 104 and 105 times greater than in water

so the diffusion coefficients in different systems can vary by many orders of

magnitude. There is also much variation in diffusive transport rates through different

'mixes' and grades of concrete. As a reasonable estimate for dissolved

oxygen in a concrete of water/concrete mixing ratio of about 0.5, a diffusion

coefficient of 1 x 10-11 m2 s-1 is used10, but the sensitivity of the model to this

parameter will be tested. The porosity is taken as 0.1.

3.3.3 Results from Model.

Figure 3.3 shows the depletion of dissolved oxygen in a concrete of thickness 1 m

assuming a leakage current of 0.01 fiA cm-2. At 29 years, the pore water at

the surface becomes totally anaerobic and localised corrosion of the metal is no

longer possible. However, for a concrete covering of 30 cm, this situation is not

reached and a steady-state profile is achieved (figure 3.4). In this case, the

steady-state flux of oxygen is sufficient to balance the metal dissolution and the

surface can remain passive indefinitely (assuming no other film breakdown

mechanism).

The passive period (i.e. the period in which localised corrosion is possible) is

plotted in figure 3.5 as a function of leakage current for a concrete thickness of 1.0 m

for different oxygen diffusion coefficients. A prediction of this period is useful

in combination with a corrosion rate for localised corrosion, for allowing adequate

corrosion thickness in a metal structure. For a totally saturated 'typical'

concrete (with diffusion coefficient 1 x 10-11 m2 s-1), the passive period decreases

from 29 years at a low leakage current to a period of a few months at the

higher end of the range. For larger diffusion coefficients which represent a situation

in which the oxygen in the system exists in the gaseous phase (2* 10 m

s"1), the passive period becomes infinite over the range of leakage currents.

In figure 3.6, the minimum concrete thickness required to prevent an 'infinite'

passive period is plotted against diffusion coefficient on a logarithmic scale for two

leakage currents (0.01 and 1.0 juA cm ). In both cases, the relation is linear.

At the smaller current value, a larger thickness of concrete is necessary to ensure

localised corrosion is only possible for a finite period.

3.3.4 Discussion.

A model of the transport and consumption of oxygen through concrete

surrounding a passively corroding metal which yields the period in which localised

corrosion of the metal is possible has been constructed. This model is based

on fairly simple mechanistic arguments. The predictions have not yet been validated

by experimental data, but they are potentially very useful, especially in

situations where it is necessary to provide a corrosion allowance in the thickness of a

metal. The sensitivity of the model to the various physical parameters of the

system may be easily tested. Generally, the period of localised corrosion decreases

with increasing the leakage current of the metal or decreasing the diffusion

coefficient of oxygen in the concrete pores. Also, the minimum thickness of concrete

required to ensure that localised corrosion is only possible for a finite period

decreases with increased leakage current and decreased diffusion coefficient.

3.4 Initiation of Crevice Corrosion in Stainless Steels.

It is generally agreed that the initiation of crevice corrosion in stainless steels must

be preceded by the attainment of a certain critical chemistry in the solution to

break down the passive film on the cavity walls. There have been a number of

models in the literature of the evolution of the chemistry within a passive crevice.

These models aim to predict whether the critical cavity solution will be obtained in

various situations, such as in different metals and for different crevice

dimensions. The critical crevice solution is usually defined in terms of a pH and

chloride concentration. For stainless steels in general, the critical pH values have

been found in the range 1 to 311. In section 3.4.1, two such models are described and

their predictions compared with experimental data. A number of possible

sources of inaccuracies in the models' formulations and input parameters are

identified, and these are investigated with an alternative model of passive crevice

solution chemistry in section 3.4.2. This model is further tested against other

experimental data.

3.4.1 Literature Survey

One of the first models of passive crevice solution chemistry was developed by19

Oldfield and Sutton . In this model, crevice corrosion was treated as a four stage

process:

(1) The environment becomes de-oxygenated due to restrictions on the transport of

oxygen.

(2) The cathodic reaction switches to the outside of the crevice and changes occur

in the crevice solution e.g. an accumulation of hydrogen ions.

(3) The crevice solution becomes sufficiently aggressive for permanent breakdown

of the passive film and the onset of rapid corrosion.

(4) The crevice begins to propagate.

The duration of the first stage was assessed by assuming oxygen reduction was

constant and determined by the anodic current. A relationship for the distance

oxygen can diffuse in a given time was also given. This latter parameter gives the

minimum crevice depth required for oxygen depletion to be effective i.e. the

minimum depth for crevice corrosion. The time scales involved in the first stage were

calculated as being relatively short. A model of the second stage which aimed

to predict the pH in the crevice as a function of time with respect to attaining either a

limiting value as a result of mass transfer considerations or the critical pH

value which causes film breakdown and onset of rapid crevice corrosion was

presented. The model involves some severe simplifications to the set of mass

transport and chemical reaction equations that govern the concentration and

distribution of ionic species within the crevice. It is likely that these approximations

lead to inaccuracies in the predictions, although experimental comparison is

limited in this paper. The pH fall characterising the second stage was determined by

considering the hydrolysis of the metal ions. The total time to decrease the

pH from its initial value to the value at which passive film breakdown occurred was

found by evaluating the time for reductions in steps of ApH. A ApH was set

and the amount of Cr(OH)3 which would have to be precipitated to produce this pH

fall was calculated. At the new pH the total concentration of chromium

species was calculated. The time taken to achieve the pH fall was then evaluated

using this concentration and the rate of production of chromium species. The

changes in other species over this time interval were also calculated. This process

was repeated to give the progress of the solution chemistry with time. At each stage,

however corrections were made to allow for diffusion, electromigration and

the effect of chloride ions on the pH. No consideration was given to the effect of

potential drops down the crevice although it was felt that this would not change

significantly the model's predictions in relation to practical situations. In an

accompanying paper, describing experimental studies13 associated with the initiation

of crevice corrosion, comparisons were made with the model. However, these

comparisons were somewhat limited with regard to a thorough assessment of the

model's validity; only the times to passivity breakdown within a crevice in 316 steel

were given and the experimental values covered a large range (for example,

the breakdown times were given in the range 5-20 hours). Although the theoretical

times fell within these ranges, the model should perhaps be tested further

against other measures of this system, or indeed other systems before being used in

a totally predictive role.

Bernhardsson et al.14 produced a similar model using Oldfield and Sutton's four

stage formulation but treated both the solution chemistry and ionic transport

in a more rigorous manner. Their time-dependent numerical model combined

diffusion and migration effects with a more complex formulation of the chemical

equilibria in the crevice. This was done using a chemical equilibrium program,

HALTAFALL. The mass transport and chemical equilibrium equations are solved

simultaneously in this program. The system Fe-C r-N i-M o-Cr-O H - -e - was

studied. Data was available for about 50 soluble complexes and 15 solids. Changes

in activity coefficients were taken into account by using two sets of

equilibrium constants- one for the ionic medium of the bulk solution and another for

concentrated crevice solutions (of ionic strength 3 M).

The model was compared with a number of experimental data and although some

reasonable qualitative agreements were obtained, the quantitative

comparisons were less favourable. In general, the pH predicted by the model was

higher than those measured in crevices. There is a wide range of scatter of the

empirical data which may be partly attributed to the different conditions (bulk

solution compositions and crevice geometries) under which they were obtained.

However, the trend of overestimating the pH is quite marked. There are several

possible sources for this error. Firstly, the thermodynamic data used in this model

differ quite markedly from those reported elsewhere in the literature15 (because it has

been corrected for 3 M strength solutions). Also the transport part of the code

may contain certain invalid approximations under some conditions such as in

concentrated solutions. The model does not account for corrosion of the base of the

cavity but this will only be significant for wide, shallow crevices. The authors

also use different diffusion coefficients for the dominant ionic species, which have

been estimated using approximate ionic radii of the metal complexes. The

influence of some of these factors on the predictions will be assessed in section

3.4.3, in which an alternative crevice solution model is developed.

3.4.2 Calculation of the Crevice Solution in Stainless Steel with the Computer Program CHEQMATE.

In this section, a model of the evolution of the solution within a passive crevice in

stainless steel is described. The model is based on the four stage Oldfield

and Sutton formulation for the early stages of crevice corrosion . The crevice

solution is assumed to be exhausted of oxygen through consumption in the cathodic

reactions, but there is sufficient oxygen over the bulk surface to sustain the

leakage current within the crevice. This model treats both the chemical equilibria

within the crevice and the ionic transport in a more rigorous manner thar. Oldfield

and Sutton. The construction and results from the model are compared with those of

Bernhardsson et al.14 and experimental data.

The model makes use of the computer program CHEQMATE (CHemical

EQuilibrium with Migration and Transport Equations)16. This program combines

one-dimensional ionic diffusion and electromigration with chemical equilibration via

the PHREEQE code17 and predicts the solution chemistry and solid phases as

a function of time and position. Unlike the model of Bernhardsson et al. which

simultaneously solves the chemical reaction and migration equations, the two

processes are iteratively coupled in CHEQMATE in such a way that chemical

equilibrium is re-established in the system at the end of each migration timestep. The

PHREEQE program calculates the equilibrium water chemistry for a particular

chemical inventory and associated minerals. It draws on a database of

thermodynamic data and its results are as reliable or otherwise as the data. It

includes various correction methods for the thermodynamic data in different

environmental regimes such as concentrated solutions or different temperatures

which can be chosen as appropriate to the particular system being modelled. Some

of these approximations are outlined briefly in Appendix 2. CHEQMATE also

includes an automatic mineral accounting procedure so that solid phases may be

added or removed from the system as precipitation or dissolution occurs.

The concentrations of aqueous species in the dilute solution filling the crevice in

the metal are governed by the following mass-conservation equation,

SC- j f = D ,V2Ci + z,U,V(C,V4>) + R, (3.4.1)

where Cx is the concentration of species /, D( is the apparent diffusion coefficient, zi is the charge number and <p is the electrostatic potential in solution. is the

mobility given by the expression

u‘- w (3.4.2)

The first term on the right hand side of (3.4.1) describes the rate of transport of ions

by diffusion under concentration gradients. The second term represents the

migration of charged species under electrostatic potential gradients and

represents the rate of production or depletion of species i by chemical reaction. In

CHEQMATE, chemical equilibration is assumed to occur on a much faster

timescale than the ionic migration processes. In this way, the equations may be

decoupled and the migration part of equation (3.4.1) solved without the chemical

reaction but at each timestep the solution is re-equilibrated by PHREEQE.

Electrochemical reactions can be included in the model as sources and sinks of the

ions involved in these reactions. Each timestep begins with appropriate

amounts of these ions being added or removed from particular cells. A more detailed

description of the CHEQMATE program is given in Appendix 2 and the

program structure is illustrated in the flow chart in figure 3.7.

In the model, a rectangular crevice in stainless steel of length / and width w, with

passive walls, corroding at a constant rate iL A m-2 is used. This assumes

that the rate of dissolution through the passive film is unaffected by changes in

solution composition. This may not be strictly valid, but there is scope to improve on

this as data become available. This system is schematically illustrated in

figure 3.8. The steel is assumed to contain chromium and nickel and these elements

dissolve in the same molar ratio as they are present in the alloy. For

comparison with experimental data and other models from the literature, the model

48

crevice is initially filled with a dilute sodium chloride solution and anaerobic

conditions are assumed to exist. The CHEQMATE model of this system is

constructed by firstly dividing the crevice into discrete cells. Each timestep begins

with a charge neutral, chemically-equilibrated solution in each cell. An amount of

ferrous, chromium and nickel ions, given by the expression below, is added to the

solution in each cell. The fraction of the leakage current corresponding to the

production of ions of metal /, /*), is given by the expression

This step perturbs the solution chemistry in each cell from equilibrium so the

program re-equilibrates the solution in each cell using the PHREEQE program, whilst

maintaining the charge imbalance. The program then takes the second part of

concentration and potential gradients with the constraint that the current flow

between each cell is proportional to the leakage current. This step again perturbs the

chemistry from equilibrium and PHREEQE is once more called for each cell.

This procedure is repeated until the required time has been reached. In this way, the

program produces the variation in solution chemistry within the crevice and its

evolution in time.

3.4.3 Comparison of CHEQMATE Model with the Bemhardsson Model14.

The predictions of the model of Bernhardsson et al. were found to overestimate

the pH within the crevice as discussed in section 3.4.1. The two most likely

sources for this error are the handling of the ionic migration and chemical

equilibration within the crevice by the model and the thermodynamic data used for

the chemical equilibrium. These will now be investigated using the

CHEQMATE model described in the previous section. The parameters for the

comparison run are as follows:-

(3.4.3)

where M j is the molar fraction of element j and n is the number of alloying elements.

The rate of production, Pt is then

mol m 3 s (3.4.4)

the timestep which involves migration of the ionic species between each cell under

Bulk pH

Bulk Cl~

49

7

0.52 M

Crevice depth

Crevice width

Steel composition

Corrosion current density

1.0 x 10~2 m1.0 x 10"4m9% nickel, 2-25% chromium

1.0 A m- 2

1) Comparison of Coupling of Ionic Migration and Chemical Equilibrium.

The thermodynamic data and diffusion coefficients used for the initial CHEQMATE

runs are the same as those used by Bernhardsson. Figure 3.9 shows the

change in pH at the crevice base as a function of time and chromium content of the

steel with the above parameters and the thermodynamic data given in Table

values associated with steels of higher chromium content. The HALTAFALL

model has been tested against a range of empirically determined pH values in

passive crevices reported by various workers in the literature (figure 3.10). The wide

scatter in these results gives an indication of the experimental difficulties in

making such measurements, although generally, the model tends to overestimate the

pH. It was suggested that part of the inaccuracy was due to invalidity of the

transport equations in the regime of concentrated solutions. However, at this stage,

CHEQMATE also has no such corrections, so this exercise will test only the

accuracy of the approximations used in the method of coupling and solving the

migration and equilibrium equations. Comparison with results from CHEQMATE with

the same input parameters shows that a generally closer agreement to the

empirical data is obtained (figure 3.10). This would suggest that the transport code

used in CHEQMATE is more accurate although the details of the

approximations used in the HALTAFALL code are not clear in the paper, so it is

difficult to identify the precise area of improvement.

2) Thermodynamic Data

Using the thermodynamic data of Bernhardsson which has been corrected for high

ionic strength solutions (no correction method is given), no solid corrosion

products are precipitated in the crevice. However, running the CHEQMATE model

with the thermodynamic data of Oldfield and Sutton (as given in Table 3.1),

chromium hydroxide is precipitated throughout the crevice and the pH is lower,

improving agreement with experimental data. Figure 3.11 shows a comparison of pH

at the crevice tip for 25%Cr 9%Ni steel for the two different sets of data,

indicating that the predictions of these models depend very strongly on the accuracy

of the thermodynamic data for the chemical equilibria reactions.

3.1. These figures indicate a rapid decrease in pH within the crevice, with lower pH

Reaction Bemhardsson log K Oldfield log Kj

Aqueous species

Fe{OH)+ ^ Fe2+ + H 20 - H + 9.8 8.3

C r(O H )1+ C r3+ + H 20 - H + 4.84 3.8

Cr(OH)2 C r3+ + 2H 20 - 2H + 11.19 10.0

N i(O H )+ ^ M 2+ + H 20 - H + 10.3 9.5

Solids

Fe(OH)2 ^ Fe2+ + 2 0 / / " -14.62 -15.01

Fe30 4 ^ 3Fe2+ + 4 //zO - 8H + - 2e~ -36.06

FeO(OH) ^ Fe2+ + 2 H20 - 3H * - e~ -16.02

Cr{O H)3 ^ C r3+ + 3 0 / / ' -28.62 -37.39

Cr(O H)2 ^ C r3+ + 2 H 20 - 2H + - e~ -17.9

N i(O H )2 M 2+ + 2 0 / / ' -16.44 -14.86

NiO (O H) ^ M 2+ + 2 H 20 - H * - e~ -3 5 2

Table 3.1 Thermodynamic data for metal hydrolysis reactions used by Bernhardsson

el al.14 and Oldfield et al.12.

It would therefore seem that a combination of inaccurate thermodynamic data and

simplifications within the transport part of the code lead to inaccurate

predictions of crevice solution chemistry (particularly the pH) in the Bernhardsson

model. A more general discussion of the importance of the thermodynamic data is

given in section 3.4.5.

3.4.4 Comparison of CHEQMATE model with Experimental Data

The CHEQMATE model is now tested further against experimental data. The first

test involves a study by Suzuki et al. in which the acidity and concentrations

of metal and chloride ions were measured in an artificial pit in austenitic stainless

steel maintained under galvanostatic control. Although the pit was actively corroding

in the experiment, the constant rate of dissolution of the metal makes this a

suitable test for the model. The parameters used in this comparison are as follows

Bulk C7~

Bulk pH

Crevice depth

Crevice width

Steel composition


0.5 M

7.0

5 x 10“ 3 m

2.5 x 10~3 m

19% chromium, 9% nickel

15 mA cm -2

The experiment consisted of an artificial cylindrical pit of diameter 5 mm, which is

approximated in the model as a rectangular crevice of width 2.5 mm. It can

be shown that such approximations in the geometry have a much less significant

effect on the model's predictions than inaccuracies in the thermodynamic data for

the chemical equilibration. Three types of steel were used in the experiment

(304L, 316L and 18Cr-l6Ni-5Mo) but only the 304 will be considered in this

comparison due to uncertainty in the molybdenum equilibrium data. The experiment

was performed at 70 0 C. The pit assembly was activated by applying a

current of about 15mA cm until the potential reached a steady-state. For 304 steel,

this took approximately 1 hour. The pit solution was sampled after a further 30

minutes. The results of the analysis are shown in Table 3.2. Separate tests were also

performed to simulate the solution chemistry within cavities in pure iron,

chromium and nickel and these suggested that the chromium hydrolysis dominates

the chemistry in the steel pit. The authors also compared the observed pH

values with values calculated using thermodynamic data assuming hydrolysis of

dissolved metal ions to form hydroxides or oxides. These calculations gave hydrogen

ion concentrations about an order of magnitude higher than the measured

ones, suggesting the formation of not only simple hydroxides in the pit, but also

hydroxy-chloro complexes. The solution was a dark green colour and contained

colloidal particles which became brown precipitates when stored in a glass bottle for

a long period.

52

The chemical analysis of the pit solutions has yielded some fairly high

concentrations of aqueous species, so in the validation exercise, the model is run

with several different correction methods for the thermodynamic data. These

methods are outlined in Appendix 2. The comparison runs may be summarised as

follows

1) Data used by Oldfield (Table 3.1) and two additional equilibria reactions for the

formation of chromium chloride species given below19, and using the

Debye-Huckel formulae for the activity coefficients of all species

2) The same set of thermodynamic data as case 1) but with the Davies formula for

the activity of all species.

3) The same set of thermodynamic data as case 1) but with the Debye-Huckel

activity correction for all species and the Van't Hoff correction for the

and chloride ion concentration when the Debye-Huckel formula is used

(Case 1) ), as shown in Table 3.2. However, the pH is almost two points higher than

the measured values. This may be a function of incorrect data for a

precipitated in the crevice which is consistent with the observations made

during the experiments.

The model was next run with the Davies formula for all species (Case 2)). This

approximation is reported as being valid in solutions of strength less than 0.5 M,

whereas the Debye-Huckel gives good agreement with experiment up to 0.1 M.

However, after a few timesteps the equilibration step in PHREEQE failed to converge

when the ionic strength of the crevice solution reached about 0.8 M. The

analytic expression for logarithm of activity coefficient against ionic strength given in

Appendix 2 contains a minimum at about 7=0.7 M, after which the activity

increases with ionic strength. Clearly then, for highly concentrated solutions this

approximation breaks down and is not suitable for use in this model. In contrast, the

Debye-Huckel approximation tends to a limit as the ionic strength increases

and although it may become less accurate as the solution becomes increasingly

concentrated, the method does not break down in the same way. The validity of the

various approximations is discussed further in section 3.4.5.

Cr3+ + c r ^ CrCl2+ log K = -1.1

Cr3+ + 2Cl~ ^ CrCl+ log K = -0 .4

equilibrium constants for 70° C.

The model gives some good agreements to the experimental values of the metal

temperature of 70° C. The model also predicts that solid chromium hydroxide will be

53

The 25° C data gives fairly good agreements for the ionic concentrations when

Debye-Huckel is used. The model is next run with a correction for temperature in the

thermodynamic data, using the Van't Hoff isotherm (Appendix 2). The results

are again given in Table 3.2. This correction improves the pH but it is still about 0.5

points higher than the experimental value. Suzuki's suggestion that this is due

to the complexation of metal ions with chloride and hydroxyl does not seem valid

since both ferrous and chromium chloride complexes have been included. However,

the accuracy of the thermodynamic data for these is questionable and there

may even exist other complexes for which data is not available. The temperature

correction seems least accurate for the chromium reactions.

Experimental

Concentrations,M

Predicted

Case 1)

(Debye-Huckel)

Predicted

Case 3)

(Van't Hoff)

p H 0 .6 - 0 . 8 228 1 . 1 1

Fe2+ 1.15 1.27 1.25

m 2+ 0.17 0.16 0.16

C r3+ 0.35 024 0.14

c r 3.87 3.64 3.39

Na + 0.06 0.04 0.04

Table 3.2. A comparison of the experimentally determined solution composition

within an artificial pit measured by Suzuki et al. 18 and the predictions of the

CHEQMATE model with two different approximations for the determination of the

activity coefficients for the chemical equilibration.

20The CHEQMATE approach is next tested against an experiment by Bogar et al.

which demonstrates the dominance of chromium on the solution chemistry

within a corroding crevice. In this experiment, the potential on samples of various

ferrous chromium binary alloys was kept at a constant value of +300 mV Ag/AgCl and the pH of the solution within the crevice measured at various intervals.

Although, the experiment was run with a constant potential imposed on the metal, it

is likely that during the experiment the corrosion cunent would have varied

during the time as the passive film formed and thickened on the metal surface. There

is no data on this given in the paper, so the model is run at two different

currents which would be typical of those from steel under such conditions. The

Debye-Huckel approximation is used for the activity coefficient in all cases. The

parameters for this comparison are given below

Bulk Cl~Bulk pH

Crevice depth

Crevice width

Steel composition


1.28 M

5.7

10~2m3.8 x 1<r4 m

Fe/Cr binary alloys 2-25 % chromium

10- 3 -1 0 ~ 2 A m- 2

Figure 3.12 shows a comparison of the experimental results and the model's

predictions at 5 hours. The broad trend of a decrease in solution pH with increasing

chromium content is apparent. The agreement is better for a lower current at

low chromium content and higher current at higher chromium content. This would

again suggest that the greatest inaccuracies are in thermodynamic data for

the chromium equilibria reactions.

3.4.5 Discussion

The various comparisons with experimental data made during this study suggest

that the method of modelling the transport and chemical equilibria reactions

within a passively corroding crevice (or one under galvanostatic control) may be an

improvement on existing models in the literature. However, these

comparisons have underlined a fundamental limitation to using such models in a

totally predictive role, namely the extension of the thermodynamic data for the

equilibrium constants to other environmental regimes, such as concentrated solutions

and different temperatures. The accuracy and applicability of this data is

central to the accuracy of the predictions.

The Davies equation is most commonly used of the approximations for activity

coefficient (given in Appendix 2) since it is applicable over the widest range of ionic

strength. However, it is only really strictly valid in cases where the ionic

strength is less than about 0.5 M and breaks down in the regime above this. (In the

examples given in section 3.4.4, the ionic strengths were greater than 1 M).

The Debye-Huckel equation may not be accurate in concentrated solutions but the

method does not break down at high ionic strengths and the predictions of

the model do give reasonable agreement to experimental data. An alternative

approach would be the 'specific interactions' method originally developed by

Pitzer14, in which dissolved components are assumed to be fully dissociated. All

interactions between ions are treated as factors affecting activity coefficients of the

individual ions. The solution is considered to comprise only of ions. This

approach is most successful when only weak ion-pairing and no strong complexing

occurs.

3.5 Summary

In this chapter, a number of models of various changes in the solution chemistry

surrounding a corroding metal are presented. Some of the consequences of

these changes with respect to the future behaviour of the metal are discussed.

The model of oxygen supply to a metal embedded in a porous medium aims to

predict the period in which localised corrosion is possible on the metal, as a

function of the oxygen content of the system and the transport properties and

thickness of the medium. This model is based on fairly simple mechanistic

arguments. At this stage, the model requires validation with experimental data, but

the predictions are potentially very useful especially in situations where it is

necessary to provide a corrosion allowance in the metal thickness.

The model of the solution chemistry within a passively corroding crevice in a metal

ultimately aims to predict the time at which a 'critical crevice solution'

(determined experimentally) is achieved. If these conditions are achieved and

assuming there is a sufficient supply of oxygen to the bulk metal surface, crevice

corrosion may begin. This model has generally improved agreement with various

experimental data over several models of similar systems in the literature. However,

the results produced by this type of model rely fairly heavily on empirical

input data in the form of equilibrium constants for chemical reactions, and the overall

predictions can only ever be as accurate as this data.

Ele

ctro

de

p

ote

nti

al

Log current density

Figure 3.1 Schematic Evans diagram of iron in a chloride solution. The anodic curve

is marked by the bold line and the cathodic curve by the dashed(three

Hiffprpnt rafps of oxvaen flux to the metal).

U l00

Metal

l/l/ ^ — Passive Film// 0 o 0// o 0 o/ O o/ o//

O o o

/ O/ o ° o

// 0 Po rous o

o/ Medium o/ o u/ o Q o/ o// o O

ooO

/ o/ O 0/ O/V o 0

Atmosphere (constant supply

of oxygen)

oc = 0 oc= b

Flux of oxygen to metal to balance leakage current

Figure 3.2 Schematic illustration of the model of a section of concrete with a passive

metal surface at one end and an infinite supply of oxygen at the other.

Figure 3.3 Predicted oxygen concentration in concrete pore water across the section

with time for a diffusion coefficient 10_n m2 s-1, leakage current 0.01 juA cm-2

and concrete thickness 1 m.

10 —

Figure 3.4 Predicted oxygen concentration in concrete pore water across the section

with time for a diffusion coefficient 1 0 " n m2 s_I, leakage current 0 . 0 1 fiA cm- 2

and concrete thickness 0 . 3 m.

Figure 3.5 Predicted variation of passive period of metal with leakage current and

diffusion coefficient.

61

Lo

g10

min

imu

m

co

nc

rete

th

ick

ne

ss

.m

-11-0 -10-0 -9 .0 -8*0 -7 .0Log 10 d i f f us i on c oe f f i e n t m2 s ~1

Figure 3.6 Predicted variation of minimum concrete thickness to maintain passivity

of metal with oxygen diffusion coefficient and leakage current.

Figure 3.7 Flow chart of CHEQMATE program.

63

Figure 3.8 Schematic illustration ol CHEQMATE model of solution chemistry within

a passively corroding crevice in stainless steel.

Figure 3.9 Predicted change in pH at crevice base with time and chromium content

using parameters of Bemhardsson et al.14.

65

6

U

ax

0 E x p e r i m e n t a l v a l u e s

0 - 5 2 M NaCl

1 = 1 cm

w =0 - 1 mm

i =1 Am " 2

O B e r n h a r d s s o n

“ O

O

oo

oCHEQMATE

O

o

oooo

o

10 20Cr % w e i g h t

30

Figure 3.10 Comparison of predicted pH with chromium content from CHEQMATE

model with that of Bernhardsson et al14 and various experimental data from

the literature.

66

pH

at

Cre

vice

B

ase

Figure 3.11 Comparison of predicted evolution of pH in a crevice solution using CHFOMATF mndpl fnr tu/n nf thermodv/namir data.

pH

at

crev

ice

ba

se

6

1-28M NaCi pH 5 7Held at 0 3V A g / A g C l for 5 h o u r s Crevice width = 3 8x10~^m Crevice depth = 1 0 x 10’ 2m

U

3

2

1

00

o Experimental- 3 ~2x L e a k a g e c ur r e nt = 10 Am

£ L e a k a g e c u r r e n t = 10~2Arrf2

__________ L_____________L10 20

Chromium % weight

30

Figure 3.12 Comparison of predicted pH with chromium content from CHEQMATE20against data of Bogar et al. .

68

4. A Microscopic View of the Initiation of Localised Corrosion.

4.1 Introduction.

Localised corrosion requires certain specific chemical and electrochemical

conditions as indicated in Chapter 3. However, even if these conditions exist, then

the initiation of localised corrosion is generally an unpredictable event. Two

'identical' metal surfaces exposed to the same conditions will neither corrode at the

same time or in the same places. This intrinsic unpredictability has led to

much controversy over the mechanisms controlling the process. However, several

clear stages are generally recognised:

(1) The metal surface is covered by an oxide film and the leakage current is

balanced by cathodic reactions over the whole surface.

(2) A small area of this film is removed and this part of the surface activates. A

current flows between this active site and the remainder of the passive surface.

Halide ions play an important role in film breakdown although the

microscopic mechanisms are not well understood.

(3) Processes act to prevent repassivation of the exposed patch and a localised

corrosion site is established. It is thought that hydrolysis reactions during the

film removal process lead to an increasingly acidic environment at the active

site. This encourages further film breakdown and eventually accelerated metal

dissolution. Again, halide ions have an important role in preventing

repassivation.

In this chapter, various models and theories related to the initiation of localised

corrosion are discussed and the direction of the modelling of the process in this

thesis is outlined. In section 4.2, a number of theories from the literature regarding

the formation of oxide films on metal surfaces and passive corrosion are

discussed. There is generally little experimental data available for validation, so many

of these theories are purely speculative. The microscopic mechanisms of

halide interactions with films are discussed in section 4.3 again with reference to

various models from the literature. In section 4.4, some of the experimental evidence

of the unpredictability of initiation events is described. This takes the form of

oscillations in both current and electrode potential measured from metal samples

exposed to a wide range of conditions. Both regular and irregular oscillations have

been observed in various systems. A number of models have been developed

to help understand and explain this oscillatory behaviour. These include both

mechanistic interpretations and data-fitting models and generally take a broader

perspective of the process than the microscopic models of film breakdown described

in section 4.3. The mechanistic models, for example, make certain more

global assumptions about the process, for example that ion transport to and from the

metal surface dominates the system. The other type of modelling avoids

mechanistic interpretations as far as possible and uses empirical data analysis

methods to identify various trends in behaviour. In this case, the initiation of

corrosion is regarded as a stochastic process and statistical models are constructed.

In contrast, the more mechanistic approach assumes that pit initiation is a

deterministic system governed by a series of ordinary differential equations and the

oscillatory behaviour of the dynamical measures of the system is related to

non-linearities in these equations. However, the exact equations have not been

identified, nor even the variables governing the system. Both approaches are

discussed in section 4.4.

It is worth noting at this stage several parameters characterising the initiation of

localised corrosion which are widely recognised and are discussed at length

in the literature. These are as follows,.

(1 ) The pitting potential, i.e.the electrical potential of the metal at which areas of

localised corrosion are nucleated, EP

(2) The passivation potential, Epass, the potential of the metal at which the corrosion

current passes through a maximum i.e. the lowest potential at which the

metal is covered by an oxide film which restricts the dissolution current (figure

4.1).

(3) The activation potential or Flade potential, Eact, the potential of the metal at

which any flaw in the oxide film results in the consumption of the entire surface

covering and a switch to general corrosion, rather than the nucleation of

a localised corrosion site at the flaw.

(4) The induction time, r — the experimentally measured time between the injection

of agressive anions, or change of potential and the first evidence of pit

nucleation.

4.2 Passivation of Metal Surfaces.

There have been many papers which attempt to describe and/or quantify the

observed features of passive film formation and repassivation. Hoar21 presents a

review of the basic processes involved in the passivation of a metal surface by an

oxide layer. The primary act of passivation- the formation of an oxide monolayer

firmly attached to the metal and forming a compact barrier between the metal and

solution- can be expected often to be a very fast reaction perhaps limited

only by the rate of transfer of protons from adsorbed water molecules to those

immediately outside them. This act can be hindered if the Helmholtz double layer

contains an appreciable density of adsorbed anhydrous anions. Chloride ions have to

be displaced by water molecules or hydroxyl ions before the reaction

M + H 20 MO(s) + 2I t + 2e~

can occur. This displacement may be slow and difficult. Thus a metal that passivates

rapidly in say an acid sulphate solution may passivate slowly and with

difficulty in a similar solution containing chloride ions. Once the primary act of

passivation has occured the anodic current density at any potential is limited by one

or more of three consecutive processes, 1 ) movement of cations from the

metal into the oxide, 2 ) movement of cations and (or oxide ions) through the oxide

and 3) movement of cations from the oxide into solution. The rate of each

depends on the electric field in the region where it occurs, which is a function of the

total potential drop between the metal and solution. If the anode potential is

raised from one steady value to another the current density increases momentarily

and then falls as the film thickens by ion transport, reducing the field across

the film at the new higher potential. Metals forming several oxides may be better

passivated by some oxides than by others. Generally sesquioxides are reckoned

better passivating materials than oxides of divalent cations since they dissolve more

slowly under many conditions. When the potential of the passivated anode is

lowered, reactivation occurs at a potential usually a little below the passivation

potential and the passivating film often disappears rapidly. Figure 4.1 shows a

schematic illustration of the polarization curve representing this sequence.

Reactivation may also occur if the film contains sufficient weak places where local

anodes can develop. If there is insufficient oxidising agent present, the film

itself can become a cathodic reactant locally and disappear rapidly. ( The 'activation'

potential is known as the Flade potential but it is commonly confused in the

literature with the passivation potential).

In a paper on the kinetics of repassivation, Ambrose agreed that passivity results

from the development of a high interfacial resistance for which the driving

force for metal dissolution is consumed by the potential drop across the passive film.

This drop can be as much as 107 V cm-1. The author described a model of

repassivation events following some mechanical depassivation of a metal surface,

such as a scratch in the oxide film. He asserted that once the coverage kinetics have

been determined, it is possible to characterize the effect which repassivation

rates have on the morphology of the corrosion process. An equation describing the

kinetics of repassivation, i.e. the time dependence of 6, the fraction of the

area covered by anodic film, was derived,

(42.1)

where i T is the total current from the metal surface. It was suggested that the time

time for complete repassivation. In practice experimental determination of the

current dependence would be extremely difficult since there is much evidence that

the total current from a metal under pitting conditions is highly oscillatory.

However, the author pointed out that this formulation represents a simplification of

the system anyway since although 0 could increase to 1 with respect to the

original area, the area available to metal dissolution increased in proportion to the

depth of crevice, x. The model was improved by assuming 0 has the form

0 = exp(—k/t). He gave no evidence to support this assumption but by substituting

cli tthis expression into (4.2.1) and setting equal to zero (for complete

atrepassivation) an expression was obtained for the penetration rate of the metal. This

agreed favourably with many reported stress-corrosion crack propagation

rates. However, the author pointed out that under most circumstances 0 will be a

complex function of t. For example, Beck derived an expression of the form

0 = 1 — exp(— Ct2) using a Fleischmann-Thirsk2 4 mechanism for oxide patch

nucleation and two-dimensional growth coverage. Ambrose compared this

expression with experimental data. There were some good correlations in the

intermediate regions of the repassivation transient but there were deviations in other

regions. Although Beck attempted to explain these deviations, the breakdown

of either the Fleischmann and Thirsk mechanism or the expression for the current

density derived by Beck would spoil the fit. Unless either of these can be verified

then this modelling can only be regarded as speculative.

Griffin proposed a fairly simple model of the passivation process shown in figure

4.1 which nevertheless accounts for all the characteristic features of this

active-passive transition region. The model contains only two elementary rate

processes; the oxidative hydrolysis of surface metal atoms to produce adsorbed

cations

with rate constant kox and the dissolution of these cations away from the electrode

surface

dependence of the total current could be obtained from experiment and the

equation solved to yield the evolution of the surface coverage and, in particular, the

M — ne —» M n+ (a) (4.22)

M n+ (a) —> M n+ (aq) (42.3)

as well as cations incorporated in the oxide layer

M n+ (ox) —> M n+ (aq) (42.4)

with rates kdis2 and kdis3 respectively. Passivation is effected by assuming that the

rate of cation dissolution decreases as the cation coverage increases, due to the

only feature which distinguishes an isolated adsorbed cation from a cation in

the oxide layer is the presence of a full complement of nearest neighbour cations.

This permits treatment of the surface adsorbate layer as a lattice gas of interacting

cations, for which isolated cation behaviour is obtained in the coverage limit,

6 —> 0 and oxide-like behaviour as 0 —> 1. It is also assumed that the dissolution

rate constants in (4.2.3) and (42.4) can be thought of as the two limiting extremes of

a single coverage-dependent dissolution rate constant, kdis

The coverage dependence of kdis(6) is assumed to arise from the fact that the

difficulty in dissolving a cation will increase with the number of nearest-neighbour

be described by a single rate constant (which depends on 6 ). The average

associated desorption energy, Ed, is linearly proportional to the cation coverage

where rjs is the saturation number of the nearest cation neighbours surrounding a

surface cation in the oxide lattice and a) the effective interaction energy between

neighbouring cations. The latter includes the decrease in solvation energy produced

by exclusion of water ligands from a cation when a new oxide bond is formed

as well as the metal-oxygen bond energy. Other factors which influence passive layer

stability (e.g. anion concentration, pH etc.) are also included in the effective

value of a). The rate constant for cation dissolution is

At steady state this equals the rate of cation formation, assumed to show Tafel

behaviour,

stabilizing influence of the oxide-lattice bond formation. The model assumed that the

kdis2 = ^ ( 0 = 0 ) and kd i!3 = * ^ ( 0 = 1 ). (42.5)

cations, due to the formation of lattice-oxide bonds. Thus the cation dissolution may

Ed = Ed + Vs010 (42.6)

kdis = kdis exp(—r]s 0)6jR T) (4.2.7)

where kdis is the dissolution rate constant for an isolated adsorbed cation. The

dissolution current i is then

i = kdis 6exp(r]s md/RT) (42.8)

73

i = k Qox( 1 - 6)exp(aFV/RT) (42.9)

Equating these expression yields a relation between V and 0. For certain values of

= rjs(o/RT (5s 4) multiple steady-state solutions for 6{V) exist. The

behaviour represents the mean-field approximation to a phase transition between

isolated cations and a continuous oxide layer. Plots of ( i —V) were produced for

various values of (figure 4.2). For = 2 a current maximum is observed. At = 4

the maximum value of ^ in the passivating region reaches infinity. For

= 6 the sigmoidal shape of 0 = V curve results in a multi-analytic form of the

i —V curve in the active-passive transition region. If a nucleation process is required

to convert between isolated cations and an oxide layer then the i —V curve

will follow the metastable branch dictated by the direction of the potential sweep.

The model was futher used to examine the limiting cases of low and high metal

potentials and major structural features of the i —V curves. Clearly the use of a

two-step, variable-rate constant model based on a mean-field treatment is a gross

approximation to the complex molecular behaviour associated with the formation

and destruction of the passive layer at a real metal surface. However the model does

reproduce the main characteristics of the active-passive transition phenomena

fairly successfully.

4.3 The Role of Halide Ions in Pit Nucleation.

There is much discussion in the literature on the role of halide ions in the initiation

of localised corrosion. They seem to have two main functions in the process;

they encourage the local breakdown of the passive film and exposure of parts of the

metal surface, and also they tend to prevent repassivation of the exposed

area. The exact nature of the first mechanism is particularly controversial. From

experimental evidence it seems likely that the ions encourage breakdown in two

ways. Firstly, they increase the conductivity of the solution surrounding the metal.

This prevents excessive potential build up without requiring diffusion of negatively

charged ions from cathodic sites to anodic sites and hence enhances the

dissolution process. Also, it is likely that there is some chemical interaction between

the halide ions and the film structure which leads to local thinning and

sometimes eventual removal. Several workers have produced evidence which

supports this hypothesis, in the form of a dependence of the pitting induction time on

the particular halide ion present in solution. For example, Janik-Czachor2 6

reported that the minimum induction period was higher for solutions containing

iodide ions than those containing the same concentration of chloride or bromide

ions. Hoar and Jacob2 7 quantified this phenomenon and reported

74

(4.3.1)r

with n between 2.5 and 4.5 for chloride solutions and n between 4.0 and 4.5 for

bromide solutions, where r is the minimum induction period and C the halide ion

concentration. Although the difference for these particular ions is not very large, the

results suggest that there is some chemical interaction between the solution

and the film. It seems widely accepted that the penetration process is preceded by

the interfacial tension at the oxide/solution boundary is lowered so far by mutually

repulsive forces between such adsorbed anions that a kind of peptization by

interfacial charge occurs. The adsorbed anions push apart one another and the oxide

to which they are strongly attached. Further anions are adsorbed onto the

sides of any crack or split thus produced in the films, which so become progressively

degraded. No direct experimental evidence was produced for this theory

although he argued that such a mechanical breakdown would be consistent with the

observation of preferential film breakdown at singularities in the surface such

as grain boundaries. However, Ambrose and Kruger2 8 discounted this theory since

their experiments indicated a dependence of the induction time on such

phenomena as film thickness and annealing, which they could not explain in terms of

mechanical rupture. From their optical observations of passive film growth on

both bare and film-covered iron surfaces, they concluded that chloride ions must

penetrate to the metal surface before breakdown can occur. They suggested the

following mechanism of local electrochemical depassivation. An increase in the

conductivity of the film caused by chloride penetration would encourage diffusion of

ferrous ions up to the film/solution interface. These ions would then undergo

hydrolysis to aqueous ferrous hydroxide which eventually precipitates as a porous

hydrated oxide, y—FeOOH. In this manner a plug of non-protective oxide grows

down to the metal surface and allows rapid localised diffusion of metal ions leading

to pitting. Implicit in such theories of transport of aggressive species through

the film to the metal surface is the assumption that the transport is the

rate-determining step. This may be appropriate for pitting in chloride solutions but in

solutions containing larger, less mobile anions such as sulphate there may be

some doubt about its validity. Also, as Strebhlow29 points out penetration of a film by

anions would have to be balanced by the transport of 0 2~ against the electric

field to the film/solution interface to maintain electrical neutrality.

A more popular theory involves a high-energy complexation between metal ions at

the surface of the film and a number of chloride ions. Hoar and Jacob

postulated that 3 or 4 halide ions would jointly adsorb on the oxide film surface

around a lattice cation. The probability of formation of such a complex at any instant

the adsorption of anions at the film/solution interface. Hoar21 suggested that

75

is small but once formed it is assumed readily separable from the oxide ions

At the film/solution interface it does not find a stabilising oxide ion but

instead several more halide ions. Thus once begun, this process has a strong

probability of repeating itself and of accelerating due an increasing electrostatic field

whole surface. Several other authors suggest similar mechanisms by which halide

ions remove passive films to explain their experimental results. For example,

film repair after removal of a soluble chloride-metal complex. He concluded

film, in that hydrated oxide films have a strong buffer action against

breakdown due to the presence of plenty of water molecules in the film itself. A

well-developed oxide which has lost protons has less capacity to repair the broken

sites produced by chloride ion attack.

There is also evidence that chloride ions play an extremely important role in the

prevention of an exposed surface becoming repassivated11. It has been

suggested that this is due to the formation of complexes with cations and hydroxides,

the increase of activity of hydrogen ions with chloride ions present and the

readily formed salt layers on the base of a pit at low pH. All of these factors prevent

oxide film formation.

4.4 Electrochemical Fluctuations in Passive Systems.

Localised corrosion is generally quite unpredictable with regard to the time of

initiation and seemingly often the place of attack. Experimental observations of

initiation events are characterised by a strong degree of randomness and

irreproducibility, for example there have been many observations of electrochemical

oscillations in a wide range of passivated systems. These fluctuations take the

form of either current oscillations and/or potential oscillations depending on the

conditions of the experiments. Both regular and irregular oscillations have been

observed in various systems. They reflect the trend of a corroding metal to vary

between two situations, kinetically stable, i.e. passivity, and instability in the form of

anodic dissolution over all or part of the surface. These fluctuations are

in the lattice leading to local thinning of the film. Under the anodic field a further

cation is then assumed to come up through the film to replace the dissolved cation.

in the film. However, as Westcott3 0 points out, while the formation of such

complexes is probably feasible it is difficult to visualise how localisation might be

achieved with a frequency necessary to observe pitting. The suggested mechanism of

increased film strength at the thinned portion of the film where the cation

resided may be negligible compared with the non-uniformity of the field over the

Valverde and Wagner found an increase in the dissolution rate of iron oxide with

increasing chloride concentration. Okamoto discussed in his paper mechanisms of

the key role in controlling the nature of the film is played by the bound water of the

76

correlated to the resistance of the metal to localised corrosion. In this section, several

of these experiments are reviewed and the relationships between the nature of

resulting in a near sinusoidal zinc concentration wave in the electrolyte. (The

in the range 0.05-0.1 Hz). The oscillations were characterised by a rapid

increase in current, followed by a fairly slow decrease and then a rapid tail off. The

region of metal potential in which this instability occurred was found to be

strongly influenced by electrolyte temperature and convection.

solution interfaces as functions of electrolyte concentration, hydrogen ion activity,

stirring conditions and temperature. Periodic current fluctuations were initiated by

first polarising the metal to a potential within the passive region of the

potential/current curve of the system (schematically illustrated by figure 3.1).

Application of a potential close to the passivation potential (about 0.27 V SCE) led to

regular current oscillations of constant amplitude. With all of the solutions

used, the oscillations exhibited in each period a lapse with relatively large currents

and another with relatively low values. These were associated with the active

and passive switch over. They were however, only clearly recorded when the applied

potential was within 0.05 V of the passivation potential. Typical magnitudes of

the frequencies and amplitudes were as follows. For a solution of 1M H 2S04 and

metal held at potential 026 V (SCE) the oscillations were of amplitude

approximately 10 mA and frequency 0.4 Hz and for a solution of 2M H 2S04 and

metal potential 0.28 V the amplitude increased to 30 mA and frequency to about 0.5

Hz. The shape of the oscillations were very similar to those of McKubre and

Macdonald33 with zinc in sodium hydroxide. The rapid current increase and slow fall

off was correlated to the part of the cycle in which the surface was active and

the rapid decrease in current was associated with the re-formation of the passive film

and return to the leakage current. The sensitivity to other environmental

conditions was also tested with the following conclusions. The frequency of the

oscillations was proportional to the hydrogen ion concentration in the solution and

related to the angular velocity of the rotating disc electrode co by the

expression / = A + Bco, where A and B are constants.

In a later study, Podesta et al. considered the effect of chloride ions on different

the oscillations and the chemical and electrochemical environment of the metals and

their intrinsic properties are discussed.

McKubre and Macdonald33 observed oscillatory activation/passivation processes in

zinc electrodes in concentrated NaOH solution. These oscillations were

regular in frequency and amplitude and could effectively be sustained indefinitely

amplitude of the oscillations were between about 10 and 100 mA and the frequencies

Podesta et. al.3 4 studied potentiostatic current oscillations at iron/sulphuric acid

77

austenitic stainless steels immersed in 1M H 2SOA. The electrochemical

oscillations are quite different on stainless steel since the passive film is much more

fairly large amounts of carbon and sulphur and were exposed to concentrations of

chloride ions up to about 1 M. The metals were polarised to a potential near

the passivation potential, about -0.0825 V SCE for a solution of 022 M chloride ions

although this was found to become more positive as the chloride

concentration in solution increased. Regular current oscillations were observed

although only in a fairly narrow potential range of the passivation potential, about

0.01 V. The maximum and minimum current values associated with these oscillations

(IMAX an< m in ) ar|d the frequencies (f) depended on electrolyte

composition and the hydrodynamic conditions of the system: IMAX*\og[Cl~] and

fe[Cl~]0 25. The amplitude of the oscillations were of the order of mA and

the frequencies of the order of seconds. For more negative potentials, damped

current oscillations were recorded. The results indicated a clear correlation between

surface inhomogeneities due to minor components of the alloys and the

oscillating current effect. Inclusions of oxy-sulphides, carbide precipitates etc. act as

preferential sites for corrosion since a passive film will have difficulty in

forming and be thinner at these sites than over the bulk alloy. A qualitative model of

the processes leading to the observed electrochemical fluctuations was

suggested, which involved competitive adsorption between water, Cl~ ions and

HS04 ions. The regularity of the oscillations is quite interesting in this experiment

since the passive film is only breaking down at isolated sites. The breakdown

is encouraged by the potential being fixed very near the passivation potential, the

high chloride concentration and the high percentage of carbon and sulphur in the

stainless steels in chloride solutions are different from those of Podesta et al.

They also occur over a much wider range of potentials. The main differences are that

the metal is polarised to a potential within the passive regime of the

polarization curve (in the range -0.075 to -0.205 V SCE) so that the passive film is

more resistant to breakdown. Also the alloy used is of a higher purity with a

smaller carbon content and the chloride concentrations in the electrolytes are about

an order of magnitude lower. (The passivation potentials for such solutions

are lower than those used by Podesta et al., so although the steel samples are

polarised to similar potentials in these experiments, the surface films are much more

stable and the metals are well within the passive region of the polarization

curves). These oscillations are illustrated in figure 4.3. The oscillations are much

effective at protecting the surface. The steels used in this study contained

alloy.

The potentiostatic current oscillations observed by Williams et. al3 6 in systems of

in that they show a high degree of randomness in both amplitude and frequency.

78

smaller in amplitude (of the order 77A) since the surface film is more resistant to

attack and occur less freqently.

polycrystalline F<?16Cr steel (low carbon) in chloride concentrations of 0.3 M, but

passive regime and the amplitude of the oscillations was of the order 0.5 V.

Irregular oscillations have also been observed on passive surfaces of iron.

exposed to various sulphate solutions with low chloride content (about 0.03 M). The

potential of the samples was controlled at about 1 V SHE (again well within

the passive region) and irregular current oscillations of amplitude approximately

10mA were observed.

potential close to the passivation potential may be correlated with the repeated

activation and repassivation of the whole surface. One proposed mechanism for this

behaviour involves the formation of a salt film at the metal surface. Firstly,

ions released into solution from the active metal precipitate as a salt film at the

surface. There is a large potential gradient across this film and hydrogen ions tend to

be driven away from the surface, increasing the pH locally. As time evolves a

passive oxide gradually builds up between the salt film and the bulk solution which

tends to decrease the potential driving force eventually to a point where the

flux of hydrogen ions reverses. The pH at the surface decreases again and the oxide

film dissolves under conditions of increasing acidity until only the salt film is

left and the cycle is repeated. Regular oscillations may only be achieved within a

narrow range of this potential and they require high chloride concentrations in

solution which is consistent with the proposed salt film mechanism. Similarly, regular

oscillations may be obtained for stainless steel with a polarisation close to the

passivation potential. However, it is unlikely that the same mechanism would occur

on a steel and one explanation may be that there exist a number of

preferential sites on the surface that activate and repassivate with each cycle. These

have been identified as the positions of inclusions and carbides where the

surface film is thinner.

Irregular oscillations have been obtained on both iron and steel polarised to a

potential well within the passive regime of the polarization curve. The surface films

are much more resistant to attack and the irregularity of the oscillations may

Szlarska-Smialowska and Janik-Czachor observed similar behaviour on

7monitored irregular oscillations in potential of the samples . The system was

_ Qgalvanostatically controlled at 10 mA cm , so that the metal was well within the

noPickering and Frankenthal conducted several experiments in which iron was

Summary.

The regularity of electrochemical oscillations from iron surfaces polarised to a

79

be associated with different parts of the film breaking and healing as time elapses.

For stainless steels, it is therefore important that there are few impurities

otherwise breakdown of the film at these sites will dominate the electrochemical

behaviour of the surface. Also, the chloride concentration in the solution must not be

too high, particularly with iron, otherwise film breakdown may be too

extensive.

4.5 Pit Nucleation — Probability Event or Dynamic Process?

localised corrosion on a larger scale than the microscopic models discussed in

in experimentally observed phenomena such as the oscillatory and often

seemingly random current or potential transients described section 4.4. There are two

main approaches to this problem; the first treatment considers the initiation of

a localised corrosion site on a passive surface as a rare event. The time of initiation,

the induction time and often the place of attack are unpredictable, resulting in

the observed current fluctuations. This approach leads to the construction of

statistical models to describe the processes. The alternative approach regards the pit

initiation process as a dynamical system governed by a series of non-linear

ordinary differential equations. It is suggested that the highly oscillatory nature of the

observed current transients is related to oscillatory or multiple steady-state

solutions of these equations. In this case, the underlying dynamics of the system

would be perfectly deterministic.

Statistical Models.

In a series of papers, Williams et al. have developed a stochastic model for pit

initiation. Their work aims to provide a rigorous definition of the parameters

of a statistical model of pit initiation. Their studies also define and validate methods

of data analysis for obtaining such parameters from experimental data. The

modelling is based on nucleation-type theory in conjunction with statistical methods

used to describe rare event processes. Agreement with experiment was

obtained with a model having the following features:

1 ) events are nucleated with a frequency A (s cm ),

2 ) events have a probability ju (s-1) of dying,

3 ) events which survive beyond a critical age, rc (s) do not die ,

4) each event has an induction time, r, (s), during which the local current does not

increase but during which the event may die.

The model assumed that pits are unstable when first nucleated and become stable

only after they have reached a critical age. The rate of birth of a stable pit, A,

on a specimen of surface area a is

The theories outlined in this section consider the kinetics of the initiation of

section 4.3. These global models attempt to explain and identify trends in behaviour

80

A = a A e x p (—p rc) (4.5.1)

and the probabilites P(n,t) of generating n stable pits at time t are governed by the

set of simple differential equations

By defining a probability generating function, the set of equations can be solved to

yield the expected number of stable pits at any time ( > tc),

While these probabilities could be derived under favourable circumstances, the

directly measured variable in the experiments was the current and the processes were

more correctly modelled using this as a stochastic variable. Figure 4.3 shows

some examples of the current against time plots obtained from the experiments. It

was concluded that

1 ) the nucleation frequency varies from 0 to about 0 . 0 1 s cm over a narrow range

of potential and the limiting value of the nucleation rate, A, does not depend

on the alloy, chloride concentration or the electrode potential.

2 ) the death probability is also not sensibly dependent on electrode potential, nature

of alloy or chloride concentration.

3) the critical age depends on the electrode potential.

The addition of a buffer and of a supporting electrolyte reduced the nucleation rate

markedly. Also, stirring of the solution affected both the nucleation rate and

the death probability. These results were interpreted in terms of a simple microscopic

model of pit initiation. It was suggested that initiation requires the production

and persistence of gradients of acidity and electrode potential on a scale of the

surface roughness of the specimen. Fluctuations in these gradients, leading to the

birth and death of events, could arise because of fluctuations in the boundary layer of

the liquid at the metal surface ; a pit becomes stable when its depth

significantly exceeds the thickness of the solution boundary layer. The solution

boundary layer was considered to be made up of two parts, one part defined by the

roughness of the surface and the other being the hydrodynamic boundary

layer. The initiation of a pit was considered related to the local attainment of some

critical solution composition. The parameters of the general statistical model

were reconcilable with this microscopic model.

= A P ( n - l ) - A P(n) at

(4.52)

(n ) = A a (t - Tc)exp(-/iTc) (4.5.3)

and the probability that no stable pit is formed within a time t > tc

ln(P(0)) = -A a { t -T c)e xp (-p rc). (4.5.4)

81

Dynamical Models

There has been a limited amount of work done on the nature of the equations

governing the initiation of localised corrosion. In a simple model, Franck and

Fitzhugh39 attempted to explain some of the periodic electrochemical oscillations

observed for iron in terms of the behaviour of the solutions of a specific set of

non-linear equations. They formulated a mathematical model of iron polarized at a

constant potential and found oscillatory solutions of the two system variables,

the surface coverage and an overpotential, E = Eappl — Epass , where Eappl is the

applied potential and Epass is the passivation potential (although it is referred

to as the Flade potential). The model described the following sequence of steps. The

iron is assumed to be in an active state with a prescribed current density.

Because of the presence of Fe1+ ions, hydrogen ions migrate away from the active

surface, increasing the pH here. The passivation potential is assumed to

depend on the pH in such a way that this migration increases the overpotential and

thus reduces the anodic current density. As passivation occurs the flux of

hydrogen ions reverses and eventually the pH at the electrode decreases once again.

Hence, the overpotential decreases and the electrode reactivates. This cycle

of events was said to produce the observed electrochemical oscillations. The basic

assumptions in this model are a discontinuity in the kinetics at the passivation

potential and a dependence of this potential on the pH of the system. The model

was simplified by linearising the current-potential curve, the passivation potential-pH

relation and the concentration profiles in the electrode diffusion layer. Two

equations for the overpotential and surface coverage were derived and solved

numerically. For certain sets of parameters these were found to contain periodic

solutions. It was concluded that the discontinuity in the overpotential was essential

for periodic solutions. However, such a discontinuity is not defined in the

differential equations and this must be regarded as a weak point in the model. It was

also inferred that the assumed linearizations in the model caused it to be

invalid, since initial values of E and d far outside the limit cycle resulted in solutions

which headed off to infinity rather than approaching the cycle.

Talbot and Oriani4 0 used a technique of analysing the stability and multiplicity

behaviour of the chemical dynamics of reaction networks which follow mass-action

kinetics to determine whether certain passivation mechanisms proposed in

the literature could reproduce in any way the type of oscillatory behaviour

characteristic of these systems. The mathematical method used in this study had the

advantage that complicated reaction networks can be investigated but it could

ascertain only that a system of differential equations describing the reaction kinetics

does not give multiple steady-states or unstable solutions41.

More recently, Talbot et al.42 demonstrated the power of techniques such as linear

stability and bifurcation analysis in determining the conditions for multiple

steady states and periodic solutions of various non-linear models. Their analysis was

applied to two models from the literature. Firstly Griffin's one-dimensional

phase-transition model25 was examined. In this model he derived values for the

parameter for which multiple steady states of the solution exist for one particular

set of rate constants. Talbot et al. reproduced this result by firstly deriving a

differential equation for the rate of change of surface coverage using the proposed

mechanism, and then by applying a small perturbation to the solution and

examining for instability and hence multiple steady states. This analysis provided

much more information about the system; a range of values of the rate constants kox and kdis where the system exhibited the active-passive transition feature of the

polarization curve. Talbot noted that such a model could not predict any oscillations

in the solution since a system of equations must be of dimension greater than

one to contain any form of periodic solution. The second model under consideration

by Talbot et al. was that of Franck and Fitzhugh39 who had concluded that a

discontinuity in the overpotential of the metal was necessary for there to exist

oscillatory solutions. However, the linear stability analysis by Talbot produced a more

comprehensive set of parameter values for such a state, which included this

discontinuity in E. The analysis suggested that the other conclusion drawn by Franck

and Fitzhugh, that the model did not represent the real system far outside the

limit cycle, was incorrect. From initial conditions far from this cycle another steady

state would be approached. This steady state is an unstable saddle point in

the phase plane but it corresponds to conditions beyond the scope of this model. As

a saddle point, the trajectories near it would eventually wind back to the

stable steady state. This paper demonstrates very well the power of these methods in

the search for a set of physically sensible parameters which produce the

desired dynamical behaviour. In some cases, the necessary conditions imposed upon

the model may prove not to be physically realistic, in which case serious

doubt would be cast on the physical reasonableness of the model itself.

The modelling of Franck and Fitzhugh, and Griffin provides a useful introduction to

the search for instabilities in some non-linear description of the pit initiation

process. Although these models are fairly simple they do predict certain

experimentally verifiable relationships which characterise the onset of some unstable

regime associated with pitting corrosion. However, neither of these models

would be able to reproduce such erratic electrochemical oscillations as those

observed by Williams et al.36, for example. If the process is governed by dynamic

instabilites, a system of at least three non-linear differential equations would be

necessary to produce such behaviour. Okada43 constructed a model for pitting

83

corrosion with three variables and searched for unstable solutions using linear

stability analysis. This model offers an alternative microscopic interpretation of the

parameters in William's models. Okada suggested that the nucleation

frequency and death probability should, in fact, be functions of potential and chloride

ion concentration. The variables of the model were the electrode potential,

the chloride ion concentration and the metal ion concentration. He assumed a

mechanism by which halide ions affect the liberation of metal ions from an oxide film

into solution and derived a relationship between the flux of metal ions and

chloride ion concentration. Mass-balance equations were derived for the ionic

species and the potential was assumed to obey Poisson's equation. Linear

perturbation theory was applied to these equations and conditions for the

perturbations to grow were obtained. These included a relationship between the

electrode potential and halide ion concentration at the edge of the passive film,

which contradicts the conclusions of Williams et al. However, such a relationship has

associated with a perturbation, Okada also illustrated that the cathodic current flows

in the surrounding surface preserving passivity here and localising the pit

nucleation site. This model does not, however, predict the effects of buffer capacity

and electrolyte conductivity that Williams et al. found from their experiments.

Although this model goes further towards the structure of the equations necessary to

describe the initiation of pitting, and another experimental result is predicted,

Okada does not unfortunately proceed to investigate the nature of this instability. The

analysis involved in the perturbation method is fairly complex so this may not

be feasible.

In conclusion, there have been several attempts to construct the dynamical

equations associated with the breakdown of passivity of a metal and the onset of

pitting. Several of these proposed mechanisms predict periodic solutions in systems

of two or less variables. A system of at least three variables would be

necessary to produce such highly irregular instabilities as those seen in the

experimental current transients reported by Williams et al. Several of the models also

predict experimentally observed relationships between certain parameters

associated with the onset of pitting corrosion.

Summary.

There is a hint of some controversy in the literature regarding the question of

whether the initiation of localised corrosion is truly a 'rare event', in the statistical

EP = const. — InC<xF(4.5.5)

been reported experimentally44. By evaluating a local dissolution current

84

sense. Although experimental observations of initiation events are characterised by a

strong degree of randomness and lack of reproducibility, it is possible that

this behaviour might arise from certain instabilities in the dynamics of the system.

Statistical models have been used to model initiation processes very successfully in

many ways, but it has not been shown how random this 'random' process is.

This question will be addressed in some detail in the next chapter.

85

00ON

Figure 4.1 Schematic anodic polarisation curve showing the relation of the

nacsivatinn and artiuatinn nnfpntiak tn thp dissolution current densitv.

1 1000C u r r e n t ( m A / c m 2 )

10 100

o

■ 8 -1250o

o< v

LlI

-1300

Figure 4.2 Current density against potential curves calculated by Griffin2 5 for

different values of cation interaction parameter jB.

87

5 0

T im e (s)

Figure 4.3 An ensemble of current transients for lSCrl3NHNb steel in 0.028 M

NaCl polarised to 50 mV36

88

5. The Initiation of Localised Corrosion; A Process Governed by a Strange Attractor?

5.1 Introduction.

It has been widely assumed that the initiation of a pit on a passive surface is a

stochastic event. Several models which draw on the statistics of random processes

have been fairly successful in increasing the understanding of the many

complex phenomena involved in the initiation phase. (Some of these models are

discussed in Chapter 4). However, it has not been shown that the assumption of a

random process is strictly valid. The aim of this part of the project is to

investigate this assumption.

Recent developments in the theory of non-linear dynamics have demonstrated the

existence of very complex (so called chaotic) solutions to certain very simple

deterministic differential equations. In the case of dissipative systems, the solution

curves (i.e. the trajectories) eventually remain confined to a subset of the

phase space. This subset is known as an attractor. Chaotic behaviour in the solutions

occurs when there exists a 'strange' attractor (i.e. one having non-integer

dimension). The apparent randomness arises from an extreme sensitivity to initial

conditions. Many examples of experimental systems governed by strange attractors

have been discovered in a number of different branches of physical sciences.

In section 52, the theory of instabilities in non-linear dissipative systems is

introduced. An example is given of a system which displays highly oscillatory,

apparently random behaviour, but which is governed by a relatively simple set of

non-linear differential equations. In section 5.3, a method of data analysis which

allows the reconstruction of the dynamics of a system from the time series of a single

variable is described and demonstrated on a system of known dynamics. This

method is used to analyse some experimental current transients showing unstable

pitting events recorded from passive stainless steel exposed to a solution

containing chloride ions (section 5.4) and in section 5.5 the implications of the

results are discussed in the context of the construction of a deterministic model of pit

initiation.

5.2 Instabilities in Non-Linear Dissipative Dynamical Systems.

Consider a set of n non-linear ordinary differential equations

dX.dt

(5.2.1)

89

where Fi are a set of non-linear functions of the variables X} and control parameters

Ht. The solutions of these equations, given a set of initial data {AT,(0)}, may

be regarded as curves in an abstract multi-dimensional phase space whose

coordinates are X x ,....9X n. Each point in this phase space represents a possible

instantaneous state of the system. A solution of equations (5.2.1) is represented by a

moving point in phase space which traces out a trajectory. For a range of

initial conditions, the trajectories converge to some subset of the phase space. This

subset is known as an attractor. If the long-time behaviour of the system is

time-independent, the attractor of the system will be a point. However, if the system

performs sustained oscillations in time then the attractor is represented as a

closed curve in the phase space. The amplitude and period of these oscillations will

be intrinsically determined by the form of the governing equations of the

system. More complex attractors include toroidal surfaces in three dimensions and

other smooth topological manifolds in higher dimensions. It is possible to

construct systems in which there exist attractors of non-integer dimension. These are

known as strange attractors. They can arise from seemingly simple sets of

non-linear equations. Strange attractors are particularly interesting since they model

irregular, time-dependent phenomena characterised by two features- a

marked sensitivity to initial conditions and the appearance of irregular fluctuations

similar to a stochastic process even though the underlying dynamics are

perfectly deterministic.

A well known and much studied system which possesses a strange attractor is the

set of Lorenz equations4 5

equations represent cellular convection in a forced dissipative hydrodynamic

flow. The basic properties of the solution of this set of equations are as follows.

1) For 0 ^ R < 1 , there is one equilibrium point (i.e. where x = y = z = 0 ) and the

system is globally stable, and for values of a and b such that

—{a + b — 1 ) < 0 the variables tend monotonically to the equilibrium point.

2) For 1 < R < Rc, (with a=10,fc=8/3, the most studied case, Rc=28), there are

two further equilibrium points, Pl 2 = (± (b (R —1 ))*, 1))*, R —1).

Figure 5.1 shows the time series of the z variable for R = 15. The irregular

x = o(y — x) y = —y — xz + Rx

z — xy — bz

(5.22)

where o, R and b are positive constants ('control parameters' of the system). These

90

oscillations decay to a steady state of z— 14 and the attractor is represented by

one of the equilibrium points in the phase space.

For R > Rc, the time series of the variables are quite different in character.

Figure 52 shows a sample of 100 points of the z component of a trajectory (with

sampling period, r=0.25). The oscillations appear to be random in both

frequency and amplitude. However, in phase space (i.e. the (x,y,z) space) the

attractor shows a quite remarkable structure as illustrated in figure 5.3. A typical

trajectory spirals out from one of the equilibrium points until a certain

instant at which it is attracted to a region close to the other critical point. It then

spirals out from this point until again it jumps back to the first point; the

time at which the trajectory switches between the two lobes of the attractor

appears random. Also, two trajectories starting arbitrarily close to each other on

the attractor diverge exponentially. These characteristics are associated

with the fractal nature of the attractor surface and lead to the chaotic nature of the

solutions.

The problem is then how to reconstruct the dynamics of the system from the time

series of a single variable. A relatively simple method of such data analysis

has recently been developed by Grassberger and Procaccia46. This method

determines firstly whether the long-term behaviour of a system is confined to a

subset of the multi-dimensional phase space (i.e. whether an attractor exists) and

secondly, the dimension of this subspace which yields the minimum number of

variables that govern the system. The method will be applied to some seemingly

random corrosion current measurements from stainless steel associated with

unstable pitting events.

5.3 Characterization of Strange Attractors.

5.3.1 Reconstruction of the Dynamics of a System from Time Series.

Strange attractors are typically characterised by a fractal dimensionality, D, which

is smaller then the number of degrees of freedom, F. This fractal dimension

has been the most common measure of the strangeness of attractors but it is

impractical to compute D from a single time series of any observable for D > 2.

Grassberger and Procaccia4 6 have recently developed a different measure for the

strangeness which can easily be obtained from a single time series of any observable

and is closely related to the fractal dimension. The measure v is obtained by

considering correlations between points of a long time series on the attractor. Let the

m variables of the system be denoted {X k} for k = 1 ...m and let {A ^(r)} be

the time series of variable k extracted from experiment. The analytical procedure

involves constructing a set of N. points,

91

{ £ W , = * 1 ('/+*), JT1(f/+2r),...:., ^ fo + O i- l ) ! ) } (5.3.1)

where r is an arbitrary but fixed increment and t( i= l. . .N s are evenly spaced time

intervals. A reference point is chosen and all distances — Yj\ from the

N — 1 remaining points are computed. The following quantity can then be calculated.

N.C(r) = E 0 ( r - |£ - 1 J|)

N s

(5.3.2)

where 9 is the Heaviside step function. C(r) gives a measure of the number of data

points within distance r of X j and is known as the correlation function of the

attractor. If we fix a small parameter e and use it to define the site of a lattice which

approximates a line attractor then the number of data points within distance r from a prescribed point should be proportional to r/e. If the attractor is a surface, this

number should be proportional to (r/e) 2 and more generally if it is a

d-dimensional manifold it should be proportional to (r/e)d. We would expect,

therefore, that for a range of r, e < r< L (where L is some length scale of the lattice,

C(r) should vary as

C(r) = r d (5.3.3)

i.e. the dimensionality of the attractor is given by the slope of the log C(r) vs log r line for a certain range of r . For attractors of fractal dimensionality, there

exists a range of r over which C(r) scales like r v , where v is the correlation

dimension. It has been shown that v is closely related to D and

v D (5.3.4)

with v = D if there is uniform coverage of the attractor.

For large values of r, all points of the attractor are correlated, i.e. each hypersphere

of radius r contains almost the entire attractor, and C(r) tends to 1. For small

r, the statistics are poor and a large scatter in the values of C(r) often occurs. Values

of C(r) are therefore usually only meaningful over a limited range of r. The

correlation integral C(r) cannot increase faster than r n for a given choice of n, where

n is the chosen value of the embedding dimension, that is the dimension of

the vectors { ^ } l=i ^ as defined in equation (5.3.1). As a result, for n < v, the slope

of log C(r) against log r will converge to n rather than v. If an attractor is

present, then the value of the slope will remain equal to v for n > v. The search for

an attractor therefore involves calculating C(r) at successively greater values

of n, beginning typically at n = 2. For truly random data the slope of logC(r) against

logr increases as n.

92

5.3.2 Applications of this Method and Discussion of Results.

In this section, some examples of the application of this method both to time

series generated from systems that are known to possess attractors and to those

generated from experimental systems with unknown underlying structure are

discussed. Some of this work has both highlighted limitations of the method and

defined some of the requirements for the accuracy and amount of data necessary for

this analysis. In this way, it is possible to obtain a better insight into both the

accuracy and relevance of the analysis of the corrosion data presented in this

chapter.

Grassberger and Procaccia demonstrated application of their method to several

well-known chaotic systems including the Lorenz equations. The correlation integral

is shown in figure 5.4 as a function of r with embedding dimension /i=3,

calculated from 15000 points from the z variable time series (generated by a

numerical solution of the equations). The straight-line portion of this curve at small r

has slope v which agrees closely with the fractal dimensionality, £>, of these

systems (computed previously with box counting algorithms). The value of v is given

as 2.05 ± 0.01 with D = 2.06 ± 0.01. The errors quoted are called

'educated guesses' but within the error bounds, v is for all cases examined less than

or equal to D. (The agreement improves as the trajectories cover the attractor

more evenly). A very large number of data points in the time series (15000) was used

to calculate the correlation integral. However, it was stressed that the

algorithm for v converged quite rapidly and reasonable results (v ± 5%) were

obtained in most cases with only a few thousand points.

Before this method is applied to corrosion systems, a number of restrictions on

analysis of empirical time series will be considered briefly. These may be

summarised as follows:

1) The number of data points in the sample must be great enough that there is an

adequate coverage in the phase space of any attractor. If there are too few, then the

correlation integral will be dominated by spurious correlations especially at

large values of n. This can be avoided by a modification to the algorithm for

calculation of the correlation integral (5.32)47,

ck (r) = £ V . S 0 (r - IY, - Y, |) (5.3.5)

and choosing k = k such that Ck(r) remains constant for k>k\

2) The sampling frequency must not be so great compared with the timescales of the

3) The experimental noise on the signal should be kept as low as possible.

Instrument noise will not, however, completely destroy the fractal structure, but will

cause fuzziness on length scales that are smaller than or equal to the noise

amplitude48 i.e.

oscillations that consecutive values in the time series are essentially

uncorrelated. The resulting signal would then appear random in the analysis method;

C(r) « r n fo r r < rnoise (5.3.6)

C(r) r v for r > r„oUe. (5.3.7)

5.4 Application of the Method to Current Oscillations from Passive Metal Surfaces.

The methods of Grassberger and Procaccia are now applied to experimental

current transients recorded from passive stainless steel exposed to a solution of 1 0 0 0

ppm chloride. Currents were recorded for potentials in the range -300 mV to

+200 mV at frequencies in the range 2-35 Hz. Also, a dummy cell was set up in

which the instrument noise was monitored and recorded49.

5.4.1 Results

The currents recorded in these experiment show a range of behaviour. Figure 5.5

shows a sample of 2000 current values recorded at 5 Hz from a sample

polarised to a potential of +200mV. The peaks in current may be associated with

breaks in the film and local activation of the metal surface, followed by rapid

repassivation of the active patch. The current is highly oscillatory under these

conditions and the fluctuations appear random in both frequency and amplitude. At

lower potentials, any breaks in the film lead to much smaller currents. Figure

5.6 shows 1000 points taken from a sample polarised to -200mV and recorded at a

rate of 35 Hz. The current in this case seems to be characterised by irregular

oscillations on two different timescales. There are extremely rapid fluctuations which

may be associated with instrument noise and slower oscillations of frequency

of the order 1 Hz. Figure 5.7 shows a plot of the current from the dummy cell

(measured at 35 Hz), which gives a good representation of the instrument noise. The

amplitude of these oscillations is of the order 1 0 - 3 /iA and again they appear

random.

It will now be demonstrated using the correlation integral analysis that the

initiation of pitting corrosion is governed a small number of variables. The corrosion

current sample collected with a specimen at potential +200 mV SCE (Figure

5.5) is considered initially. The data consists of 2500 observations recorded at 5 Hz.

The correlation integral C(r) is calculated for a range of embedding

dimension n—2— 8 (with k — 20 in equation (5.3.5)). Log-log plots of four of these

curves are shown in figure 5.8. For large r, we see that the correlations tend to 1 as

expected. For r between about 10 “ 1 0 and 10 _2-5, the slopes of the lines are

reasonably parallel for large n, indicating that the process is indeed governed by an

attractor. Plotting the slope of these curves against r (figure 5.9), shows a very

reasonable saturation in slope at around 1.9 for n greater than about 4. At small r

(C IO” 2-75), the slopes show a sharp increase in value. This may be correlated with

the scale of the instrument noise, as explained in the last section.

The currents recorded at a lower potential (-200 mV, figure 5.6) are next

considered. The correlations calculated with 5000 data points (recorded at 35 Hz)

are shown in figure 5.10 for n = 2— 8. The slopes do not show the same

saturation at large n as those at the higher potential but instead increase quite

significantly. For small r, the slopes increase as n , which again may be associated

with random instrument noise. Figure 5.11 shows the correlation integrals

calculated from 5000 points recorded at 35 Hz from the dummy cell. The slopes of

these curves are very close to the values of the embedding dimension,

confirming that the instrument noise is truly random at sampling frequencies slower

than 35 Hz.

These results may be summarised by figure 5.12 on a plot of correlation dimension

v against embedding dimension n.

5.4.2 Discussion of Results.

The results from the analysis of the data at +200 mV suggest that the process of

depassivation and repassivation of steel leading to pit nucleation is

deterministic in nature, and is not a stochastic process, as has been widely assumed.

This implies that the process can be described by a set of well-behaved

differential equations. The system appears to possess a strange attractor of

non-integral dimension, which is consistent with the chaotic nature of the current

from the metal surface. The first set of analyses, presented in this chapter, suggests

that the dimension of this attractor is approximately 1.9 and the minimum

number of variables needed to describe the system (i.e. the minimum embedding

dimension of the attractor) is 4. (The significance of these values is discussed in

section 5.5) These results were obtained using various modifications to the original

method of Grassberger and Procaccia for the analysis of real experimental

data, such as equation (5.3.8). There are a number of further checks that will be

carried out to confirm the accuracy of the fractal dimension of the attractor v and the

embedding dimension n as more experimental data becomes available.

However, the correlation dimensions calculated from existing limited data saturate at

a small value as the embedding dimension is increased and the identification

of an attractor of low dimension from this behaviour is an extremely important result.

In the case of the current recorded at lower potentials, it has not been possible to

resolve an attractor in the same way. It may be that there is no attractor

characterising the system at this potential. However, if there is, two possible

explanations for the failiure of the method to identify the structure are suggested.

Firstly, the signal-to-noise ratio is poor at low potentials since breaks in the passive

film lead to much smaller current bursts. The structure of the attractor may in

that case be lost underneath the instrument noise signal. The signal-noise ratio is

limited with the present experimental data collection technique but it is hoped to

improve on this at a later stage with different equipment. The other possible cause

for the poor correlation is that the sampling frequency is not high enough.

The effect of this may be demonstrated with the Lorenz attractor. If a time signal is

generated and sampled eight times less frequently than the example in figure

52 and the same analysis is performed, the slope of the correlation integrals show

no sign of saturation as the embedding dimension is increased (figure 5.13).

In the case of the low potentials then, it is possible that a combination of the two

limitations on the data yield the poor correlation. Further experimental data are

required.

5.5 Implications of Results to the Construction of a Mechanistic Model of Pit Initiation.

Two major points have come out of this preliminary analysis of corrosion current

data; firstly that the depassivation and repassivation of metal surfaces leading

to the nucleation of pits is a deterministic process and secondly, that the system

seems to be governed by a strange attractor of low dimension. This contradicts the

popular assumption that pit initiation is a stochastic, i.e. truly random, event.

The method of analysis used in this study does not unfortunately give any indication

of which variables are governing the system, nor the precise nature of their

interaction, although with a little physical intuition into the process, it should be

possible to make reasonable guesses at least to the identity of the variables.

However, even if it is possible to construct and solve the mathematical equations

describing the initiation of pitting corrosion, it is important to be clear about the role

of these equations. We have identified the existence of a strange attractor

governing the system and this imposes a fundamental restriction on the use of the

equations in a predictive role. The solutions of equations possessing such an

attractor are characterised by an extreme sensitivity to their initial conditions and

display the apparently random oscillations associated with chaos. In other words, for

a genuinely chaotic process one cannot specify the initial conditions to a

sufficient degree of precision to use the results for a prediction of the state of the

system at any time in the future. However, in this context, it is unlikely that one would

wish to predict the precise current at a particular instant; rather, such

phenomena as the increase or decrease in pitting events with some parameter

describing the environment would perhaps be of more interest and practical use.

Such predictions of the relation between environmental parameters and the general

behaviour of the system also form one of the principal applications of the

statistical models of the process. However, the microscopic models derived from

stochastic interpretations of the process depend to a large extent on the precise

parameters defined by the statistical analysis, and the predictions of these models do

vary accordingly3 6 43. The definition of environmental conditions from

deterministic equations would seem to be a more rigorous method.

With regard to the possibility of the construction of such equations, there have

already been several studies reported in the literature which search for instabilities in

mathematical formulations of pit initiation processes4 0 42 43. The variables

used in the equations include the metal potential, pH of the solution and the chloride

and metal ion concentrations, although the most used in a single model is

three. The mathematical instabilities in the equations have again principally been

interpreted in terms of environmental conditions (with varying degrees of success).

Generally, however, the models have not been sufficiently reconciled with the

direct measures of the system such as the corrosion currents. The preliminary

analysis of corrosion current data suggests that a model containing at minimum four

independent variables is necessary to describe the system.

Many chaotic systems are characterised by 'control parameters' in their equations

which determine regimes of different solution characteristics. For example,

the general behaviour of the solution of the Lorenz equations (5.2.2) are determined

by the parameter R. For R <R C, (a critical value of approximately 25), the

solutions show decaying oscillations (figure 5.1), whereas for R >R C, the behaviour is

chaotic (figure 5.2). Other systems are characterised by transitions from

regular periodic oscillations to the irregular fluctuations associated with chaos

according to some control parameter within the equations50. There have been many

observations of both regular and irregular electrochemical oscillations in a

wide range of passivated systems. Podesta et al3 4 3 5 observed regular oscillations on

both iron and stainless steel in sulphuric acid (Figures 5.14(a) and (b)). In

both cases, these oscillations could only be obtained for potentials close to the

passivation potential for each system. For stainless steel, a small decrease in

potential led to decaying oscillations. Figure 5.14(c) shows a series of highly irregular

oscillations observed on stainless steel36. These are associated with unstable

pitting events and occur at potentials well within the passive region of the

polarisation curve. These observations suggest that if the process of pit initiation is

indeed deterministic as our analysis has implied, then the metal potential may

be a control parameter within the equations describing the system. This may be

useful in any attempt to construct these equations.

The system of equations describing the depassivation and repassivation of a metal

will, however, be considerably more complex than a chaotic system such as

the Lorenz equations because they will involve mass transport of ions in solution (i.e.

a spatial variation in the system in addition to the temporal variations through

chemical reaction and dissolution kinetics). The governing equations will then be a

set of partial differential equations rather than ordinary differential equations.

It seems likely, however, that the non-linearities producing the chaotic behaviour

arise from the complex dependence of electrode reaction kinetics on the solution

chemistry rather than the mass transport processes, but these will clearly add

complexity to the problem. Unfortunately, the characterisation of chaotic processes in

systems of partial differential equations is limited at present.

5.6 Summary.

The initiation of a pit on a passive metal surface is generally regarded as a rare

event and one approach to modelling the process has involved the use of the theory

of stochastic (i.e. truly random) processes. However, by using a novel method

of data analysis developed by Grassberger and Procaccia46 on some seemingly

random current records from stainless steel, it is suggested that the process of pit

initiation is deterministic, i.e. that the process can be described by a set of

well-behaved differential equations. Further, the analysis suggests that the long-time

behaviour of the system is governed by a strange attractor (leading to the

chaotic behaviour of the direct measures of the system) and this attractor is

embedded in a phase space of small dimension i.e. the system could be modelled by

a small set of differential equations. Preliminary results give the dimension of

the attractor at around 1.9 and the minimum number of variables needed to model

the system at 4. However, further applications of the method will be

performed to confirm the accuracy of these values. The method does not

unfortunately give any indication as to which variables are governing the system, but

with a little physical insight into the problem, it may be possible to construct a

deterministic model of pit initiation.

8

Figure 5.1 Time series of z variable in Lorenz equations calculated with R=15 and time delay t=0.25.

8

5 0 1—

AO

28

0 L 0 20 AO 60 80

Number of timesteps

Figure 5.2 Time series of z variable in Lorenz equations calculated with R=28 and

r=0.25.

Figure 5.3 Schematic representation of the Lorenz attractor in (x,y,z) phase space.

101

T

Log ( r /rQ) ( rQ a rb itra ry )

Figure 5.4 Correlation function against distance for 15000 points of z variable time

series from Lorenz equations, calculated with R= 28 and embedding

dimension n —3.

102

Corr

osio

n cu

rren

t de

nsity

jA

c

3.6

2 A

1.2

0

Metal potential =+ 200 mV

Sampling rate = 5 Hz

i i i i i i i i i i i i___ 1___ i___ i___ i___ 180 160 240 320 400

Time, seconds

Figure 5.5 Corrosion current time series recorded at 5 Hz from stainless steel at potential +200 mV in 1000 ppm chloride solution.

103

Cor

rosi

on

curr

ent

dens

ity

x 10

/j

A cm

.fvl 4.72

4.60

4.48

4.36

4.24I

4,12

4.0 J____L0

Metal potential = - 200 mV

Sampling rate » 35 Hz

I i » i 1 i i i I ■___i__ i__ I4 6 8 10

Sample number ( x 100 )

Figure 5.6 Corrosion current lime series recorded at 35 Hz from stainless steel at potential -200 mV In a 1000 ppm chloride solution.

orro

sion

cu

rren

t de

nsit

y x

10

/uA

c

roi •E 2 7 2 1—

2.68 —

2.64 -

280

u2.56 1

0 2

Dummy cellSampling frequency » 35 Hz

Figure 5.7 Corrosion current time series recorded at 35 Hz from dummy cell representing instrument noise.

s

1.0

0.0

- 1.0

u - 2 0

cjcn -3.0 o _j

-4.0

-5.0

- 6.0

Metal potential = ♦ 200mV Sample size = 2500

y ; / Key

i

-4 .0

^ / / /

/ /

-3.0

n = 2 n e 4 n = 6 n = 8

- 2.0 - 1.0 0 . 0 1.0Log r

Figure 5.8 Correlation Integrals calculated for corrosion current recorded at +200

mV for embedding dimension 2 ,4 ,6 and 8.

o

Figure 5.9 Correlation dimension in figure 5.8 against log r for embedding

dimension 2, 4, 6 and 8 .

Log

C Cr

)0 0

- 1 . 0

- 2.0

-3.0

- 4 0

-5.0

- 6.0

-7.0

Figure 5.10 Correlation integrals calculated for corrosion currents recorded at -200

mV for embedding dimensions 2 to 8.

/

Log

C (r

)

Figure 5.11 Correlation integrals calculated for data collected from dummy cell for

embedding dimension 2 up to 5.

v

7

6

5

3

2

0

oo Instrum ent noise

□ -2 0 0 mV 35 Hz

A 4-200 mV 5 Hz

o□

□o

□

o□

o□

o□A

O

A A A A A A *

J_____________ 1_____________ |_____________ |_____________ I_____________ 1_____________ |_____________ !_____________ |

1 2 3 4 5 6 7 8 9

Embedding dimension n

Figure 5.12 Correlation dimension v against embedding dimension n for +200 mV,

-200 mV and dummy cell data.

110

• •

Figure 5.13 Correlation integrals for z variable of Lorenz attractor sampled 8 times

less frequently than In figure 52 (r=4.0).

(a) Time

(b)

Figure 5.14 Different modes of current oscillation from various passive iron and steel systems.

112

PART II:

THE PROPAGATION OF LOCALISED CORROSION

113

6. A Literature Survey of the Modelling of the Propagation of Pits and Crevices.

6.1 Introduction.

The difficulty in measuring the chemistry and electrochemistry within the corrosion

cavity has led to the development of a number of mathematical models of the

process. In this chapter, the physical mechanisms that are generally thought to

control the propagation of pits and crevices are outlined, and the approximations

involved in the translations of these mechanisms to mathematical equations

discussed. A selection of models from the literature are next described and

compared. These models are classified according to the complexity of the physical

and mathematical approximations made during their construction. The merits

and limitations of these approximations are evaluated and the necessity for more

sophisticated modelling is highlighted. In this way, it is hoped to demonstrate both

the achievements and 'weak spots' of the current state of corrosion

propagation modelling.

6.2 Methods of Modelling the Propagation of Localised Corrosion Cavities.

At the 'end' of the initiation stage of localised corrosion, it is generally agreed that

the solution at the base of a pit or crevice is sufficiently acidic to cause active

corrosion of the metal, whereas over the bulk metal surface outside the cavity, the

surface remains passive. This results in the development of a potential gradient

between the base and the mouth of the cavity which causes anions such as OH~ and Cl~ to migrate into the cavity. The hydroxyl ions combine with the metal ions

from the corrosion, releasing hydrogen ions. This process maintains the

acidity of the sites, thus sustaining the localised corrosion. The relationship between

the solution chemistry and the corrosion rate in the pit is quite complex; the

rate at which the metal corrodes at the pit base depends on the potential of the

solution. This is determined by the solution chemistry which in turn depends on the

rate of metal corrosion.

The common aim of the variety of mathematical models that have been developed

is the prediction of the solution chemistry and electrochemistry within the

restricted geometries of the cavities as a function of parameters such as cavity

dimensions, bulk solution composition etc. Such information yields metal penetration

rates. A full description of the problem should include the following;

(1 ) The migration of ions under both concentration and potential gradients and the

advection of ions if the electrolyte is flowing;

An account of the complex chemical equilibria reactions within the crevice,

including the precipitation of any solid corrosion products;

(3) Electrochemical reactions including the dependence of reaction rates on

parameters such as electrostatic potential and solution pH;

(4) The effect of the changing shape and dimensions of the crevice as propagation

proceeds;

(5) The effect of blocking of the crevice with any solid corrosion products.

The complexity of the problem has led to a wide variety of approximations in the

models in the literature. The majority of these models are based on the same

fundamental equations governing the mass transport of aqueous chemical species in

electrolytic solutions. In brief, the transport of a given species i is controlled

by three mechanisms : diffusion, electromigration and convection. The flux of the

species in a dilute solution, / , is given by

L = -D,VC, - + uCj (62.1)

where C, is the concentration of the ion, D, is the diffusion coefficient, is the

charge, <f> is the electrostatic potential and u is the velocity field describing the

motion of the electrolyte . By conservation of mass

9 £dt - V - l + R, (6.2.2)

where Rt represents the rate of production or depletion of species i by chemical

reaction. The electrostatic potential is governed by Poisson's equation,

V 2 0 = £- (6.2.3)

where p is the charge density. However the magnitude of £o- 1 is sufficiently large that

any slight deviations of the system from electroneutrality result in very large

electrical restoring forces. This causes many numerical problems when modelling the

systems and virtually all of the models approximate Poisson's equation with

the equation of local electroneutrality,

E z /Q fe ) = 0. (6.2.4)

The other important equation used in many of the models relates the flux of species

to the current density in the solution by a simple application of Faraday's law;

115

(62.5)F E z. i -iThe specification of boundary conditions of the problem is given in terms of

concentrations in the bulk solution outside the crevice and prescribed fluxes of

species on the active metal surfaces, usually dependent on values of several of the

variables at these points. The resulting problem is highly non-linear.

Equation (6.2.1) is only strictly valid in the case of dilute solutions. There are very

few models which consider corrosion in concentrated solutions. In these

cases the diffusion is driven by activity rather than concentration gradients and

equation (6.2.1) should be replaced by

l = C,v, (6.2.6)

where v, is the velocity of species i and is related to the chemical potential gradient,

V/i, by

c T ^ i = - v.).i

K «is the drag coefficient and is given by

RTCi CjCTDlt

(6.2.7)

(6.2.8)

where Di} is a 'diffusion coefficient' describing the interaction of species / and j and

CT is the sum of all the concentrations including the solvent. Equation (62.8)

represents the balance between the driving force and the total force exerted by the

other species. The system of equations is completed with an equation

governing the concentration of solvent using an appropriate conservation law.

Solutions of the problem have been obtained using a variety of approximations to

the equations themselves, the boundary conditions and the geometry of the

crevice. The most common approximations are

(D

(2)

(3)

(4)

(5)

Adoption of simplified geometries representing the pit or crevice.

Reduction of the number of dimensions of the problem,usually to one.

Neglect of diffusion of species under concentration gradients.

Neglect of migration of charged species under potential gradients.

Neglect of convection of species by a moving electrolyte.

116

(6 ) Neglect of the moving boundaries that may characterise the system.

(7) Simplification of chemical and electrochemical reactions.

(8 ) Assumption of the steady state.

The complexity of the various models in the literature seems largely dependent on

their required application. Some of the simpler models serve to aid the

understanding of results from a particular experimental system, with little

consideration paid to validity of the approximations made, or their implications with

respect to other systems. The more complex models allow assessments of the

sensitivity of the predictions to the various approximations made during their

construction. A distinction can also be made between models which are based on

large quantities of empirical data, and those which are constructed from more

rigorous physical arguments and are generally more predictive. The former group, the

'engineering' models, are most commonly used to provide specific answers

relating to some particular aspect of a corroding system, for example defining

environment limitations for the use of a given alloy. These include the models of the

initiation of crevice corrosion described in section 4.3.1.

In this review, the simpler models are described first. These consider ionic

migration by one transport mode only. The merits and limitations of these are

discussed and the necessity for more sophisticated modelling is highlighted.

Similarly, the validity and flexibility of the models which include two transport

processes are assessed.

6.2.1 Transport by Electromigration only.

The simplest models are those that consider transport by electromigration only.

These use Ohm's law to describe the potential distribution in the cavity and assume

uniform concentrations of species throughout. Melville51 developed a model

to assess the variation in potential in a stress-corrosion crack but his analysis is

applicable to crevice and pitting situations. A second-order ordinary differential

equation relating the potential of the specimen relative to the solution, E(x), to the

current density, /(£ ), was derived using Ohm's law:

cPE _ 2i(E) (62.9)dx2 w>c

where w is the crack width and c the conductivity (both assumed independent of x). This conductivity was chosen at values between 10 and 100 Q _ 1 m-1. (This

would correspond to solutions of, say, sodium chloride at concentrations between

about 0.02 and 0.12 M). In a similar formulation, Doig and Flewitt52 used boundary

dEconditions — = 0 at E = Er (free corrosion potential), but Melville pointed ax

out that this is only valid for semi-infinite cracks. In his model, he first considered

conditions of potentiostatic control setting the potential at the crack mouth equal to

some applied potential and then used the value of the current at the crack tip

in the case of small potential drop (e.g. for a short crack). An analytic solution

free corrosion potential for the passivated sides of the crack. An analytic expression

was again obtained for the potential and current down the crack as a function

of position. The current was non-zero at the crack mouth, from which Melville

charge generated on the surface of the specimen, in addition to cathodic

reactions on the sides. A further analysis suggested estimates for crack lengths at

which all the crack tip reactions are balanced on the crack sides. The predictions

were compared with one set of experimental data and agreement is reasonably good

but the author admitted that this may have been fortuitous since the

assumption that the current density was constant over the crack length was not

strictly valid for a crack with such large potential drops.

Bignold developed a time-dependent model for the propagation of a

stress-corrosion crack with consideration of passive film growth, and also derived

expressions for the potential and current flow in the crevice. The latter analysis may

again be applied to other forms of localised corrosion. Anodic processes were

assumed to occur only at the crack tip and cathodic processes only on the walls i.e

the walls were assumed passive. An expression for the rate of propagation of

the crack was derived in terms of the corrosion current and active surface area. This

equation was integrated to give a relation describing the shape of the crack

with time;

the potential-dependent rate of propagation of the passive film, N the

passive-layer nucleation rate, p the density of the metal and M the molecular weight.

This crack shape is shown in figure 6.1 He also derived expressions for the

potential in the crack with respect to the crack tip and the cathodic current down the

dEto determine at this point. An approximate expression was derived for the current

was obtained for E , and i(E ) was derived. He predicted a drop in potential and

decrease in current down the crack. The model was then used to consider conditions

where there was no applied potential and the potential remained close to the

concluded that the anodic reactions at the crack tip have to be balanced by cathodic

(6.2.10)

where v = -J p due to yielding at the crack tip, i is the anodic current density, k isdt

118

pit at the free corrosion potential using Ohm's law. These relations indicate

that the cathodic current is distributed over the whole surface but most occurs

outside the pit. No experimental verifications of the model's predictions were

reported.

The distinguishing feature of these simple models is the assumption that the

concentration of species is uniform throughout the cavity. This is only strictly valid at

time zero, the instant at which the metal is exposed to the electrolyte and the

solution within the crevice is the same as that outside. A large potential drop in the

crevice implies some restricted transport between the cavity and the bulk

which would lead to concentration gradients. Thus the validity of any such models

describing the steady state is questionable. The potential drops calculated

using Ohm's law must be regarded as the maximum possible in the steady state

since diffusion tends to level out any concentration gradients. Thus, such potential

gradients may be very much greater than in reality.

6.2.2 Transport by Diffusion Only.

Several models consider ionic transport by diffusion only. The most common

justification of this approximation is that the potential drop associated with a

particular system has been shown experimentally to be small. Other authors avoid

the electromigration by considering the migration of a neutral complex such

as a metal ion combined with a halide.

Tester and Isaacs5 4 developed a fairly simple model of an experimental pitting

system by considering transport by diffusion only. Figure 6.2 shows a schematic

representation of their experiment. The potential gradient in such a system had been

shown experimentally to be insignificant. Two distinct phases of pit growth

were distinguished; the early stage in which the dissolving wire electrode was flush

with the surface of the cavity and the quasi-steady-state period in which the

pit growth was under diffusion control. In the first phase time-dependent diffusion

equations were solved in a hemispherical and cylindrical geometry to yield the

concentration of dissolved metal ions at the electrode surface. A mass-transfer model

was also developed for the second phase which equated the flux of metal

ions to the anodic current density. A linear concentration gradient was assumed and

a simple analytic expression relating the current density to time was obtained.

The movement of the dissolving electrode was not considered specifically. Some

predictions from the model gave good agreement with experimental results but

others were poor. This was attributed to the influence of salt concentration on the

diffusion coefficient. Agreement with experimental data was improved by replacing

concentration gradients by activity gradients and correcting the diffusion

coefficients for viscosity effects.

Alkire et al. 5 5 neglected electromigration effects in a model developed to verify

electrochemical and chemical variations found experimentally within artificial pits

undergoing dissolution. The pit was modelled as a circular cylinder filled with a

binary solution of electrolyte containing a soluble salt of the corroding metal. No

cathodic processes were considered. An electromigration term in the transport

equation was avoided by considering the motion of the neutral salt. The governing

equation used was simply a one-dimensional time-dependent diffusion

equation with boundary conditions of constant concentration at the pit mouth (equal

to the bulk value) and a concentration gradient proportional to the

potential-dependent current density at the pit base. This latter condition assumes

that the metal ions combine with the halide ions instantaneously on entering the

solution to produce the metal salt (since it is assumed that the flux of each is equal).

The potential between the dissolving anode and reference electrode was

resolved into four component overpotentials arising from charge transfer,

concentration differences, resistance effects within the pit and resistance effects

outside the pit. Expressions were given for each relating them to the current density

and concentrations. The concentration and potential profiles were obtained

numerically using an iterative technique. The results gave good agreement with

experimental data and it was concluded that an estimate of the conditions likely to

initiate the propagation stage of pitting (i.e. after which time the pit would not

repassivate) could be obtained from the model. It was suggested that this stage

could only occur if the ohmic resistance in the cavity accounted for the major part of

the applied potential. The experiments were carefully designed so that no

adjustable parameters were involved, therefore the good agreement of the theory

indicates that either the assumptions were justified or that any poor assumptions

were inconsequential for this particular system. The latter may well apply in this case

since the electrostatic potential calculated in such a model may be very

different from the true potential distribution. The contribution from the imbalance

between the hydrogen and hydroxyl ions associated with localised corrosion cells will

be significant in most situations. Although the technique of modelling the

migration of a neutral salt is a useful means of neglecting the electromigration in the

pit, the information that may be derived from the results is also somewhat

limited in that only the salt concentration and potential distribution are predicted.

In a later model Alkire and Siitari5 6 attempted to include the effects of potential

variation in the pit without including electromigration directly. A one-dimensional

transport model was developed specifically to investigate the location of the cathodic

reaction in a corroding crevice. It was suggested that anode/cathode

geometry is an important consideration in current-distribution problems but it was not

made clear why these were of particular interest. The influence of the

potential variation was considered only with respect to the electrode reaction rate.

The walls of the crevice were assumed passive and anodic dissolution

occured at the base only. (No hydrolysis was included). The cathodic reaction

the flow of current between the anodic and cathodic regions was included. The

model was formulated by relating the current flow to the potential gradient using

Ohm's Law such that

where i(x ) is the current density and k is the conductivity in the solution per unit

length of cavity. The current varied according to

where j{x ) is the cathodic reaction current related to the potential via a Tafel-type

expression

In this way a second-order differential equation for the potential was derived. An

equation was also derived for the hydrogen ion concentration taking into account

diffusion but not electromigration. These were solved numerically and j(x ) was

evaluated. It was concluded that cathodic processes can occur within localised

corrosion cells as well as on the metal surface outside and when an appreciable

fraction does occur inside, the potential and concentration distributions will be

influenced. However, the degree of influence cannot be correctly assessed without

the effect of electromigration of the charged species being directly included.

No experimental validation of the model was reported. Although the conclusions

could not be tested directly, other more quantitative information could have been

extracted from the same analysis and compared with experiment, for example the pH

in the crevice and the rate of generation of hydrogen gas.

6.2.3 Transport by Diffusion and Electromigration.

Ateya and Pickering5 7 also developed a model to investigate the hydrogen

reduction rate in a system under cathodic-protection conditions (i.e. at low metal

potential) but considering both diffusion and electromigration. Unfortunately, it is not

possible to compare these results directly with those of Alkire and Siitari56 to

assess the effect of the addition of electromigration, since many of the parameters

considered was hydrogen reduction and it was assumed that this took place both on

the walls of the crack and on the metal surface outside the pit. Resistance to

(6.2.11)

(6.2.12)

j(x ) a C(x)exp(—cmF(V—(p)/RT) . (6.2.13)

121

used are quite different. A series of experiments designed to validate the

model was described. The model consisted of a narrow deep slot in the metal filled

with a simple acid electrolyte, HY. The one-dimensional mass-balance

equations for the migration of H + and Y~ ions were constructed. At such low metal

potentials the anodic reaction was considered negligible. Analytic solutions

were obtained for [ / / + ], [Y ~ ] and the potential in various approximation limits

relating the pit dimensions to the diffusion coefficients etc. The calculated solutions

show that both [H +] and [ y - ] decrease with increasing distance into the

slot. This appears to be the opposite of the predictions of other models but it must

be noted that this model is specifically related to low metal potentials at

which the main agent responsible for acidification in pits i.e. metal ion hydrolysis is

relatively ineffective. Comparison of the model with experimental results

showed that the predicted potential was very much less than that observed. This was

attributed to hydrogen gas evolution which has the effect of restricting current

flow and mass transport.P Q

In a later model, Ateya and Pickering concentrated on the effects of ionic

migration in a fairly strong acid electrolyte at higher metal potentials dominated by

anodic processes. The steady-state mass-conservation equations (including

diffusion and migration but no chemical reaction as the acid was assumed strong

enough that the hydrolysis would be negligible) were solved analytically for three

species H + ,Y~ and M ez+ in terms of a dimensionless parameter I = ------ -—zFC°D

where i is the potential-independent anodic-current density, C° the bulk H +concentration and x the distance from the dissolving surface. The application of this

model is particularly interesting since the results are used to assess the

relative importance of diffusion and migration. It is shown that the ratio of the

electromigration term to diffusion term increases with increasing z (charge of the

metal ion) and the limiting value of this ratio is equal to z. Also, ionic migration

becomes negligible for The predicted species concentrations andz

potentials are compared with experimental data and it is found that the predicted

potential gradient is too low. It was suggested that the assumption of dilute-solution

theory in the transport equations was not strictly valid at the high electrolyte

concentrations and ionic interactions should be considered.

In general, it may be concluded that models which neglect ion transport by

electromigration are restricted in their range of application since the potential

distribution in a crevice is strongly dependent on many interrelated factors, such as

the electrolyte concentration, the external electrode potential and indeed the

time elapsed since the corrosion cell is first established. (At short times the

concentration gradients are small and electromigration must dominate).

122

Several authors have produced simple 'diffusion only' models but later replaced

these with more complex formulations which include migration and chemical

oxygen within a crevice . Two crack geometries were considered: a parallel-sided slot

and a crack of variable width, w, characterised by angle 6,

(w = w0 + 2xtan0). An analytic solution to the two-dimensional time-dependent

diffusion equation was obtained for the parallel slot using a Tafel expression

for the oxygen-reduction current density. This solution was compared to a

one-dimensional approximation to the problem which involved averaging the flux to

the crack walls over the crevice width. A range of application of the

approximation was indicated. The tapered-crack problem was reduced to one

dimension in a similar manner and an expression obtained for the steady-state

oxygen concentration.

A more complex 'diffusion only' model was developed by Turnbull and Thomas60

with particular application to cathodically polarised steels in chloride

solutions. This included the anodic dissolution and hydrolysis of ferrous ions and

cathodic reduction of hydrogen ions and water. It was assumed that any oxygen

reduction occurred outside the crevice and the other electrode processes took place

both at the crack tip and on the crack walls. The crevice was modelled as a

parallel-sided slot and the two-dimensional transport equations were reduced to one

dimension by the averaging procedure developed in the earlier study59.

Transport equations were formulated for each aqueous chemical species

(Fe2+, FeOH+ , H + , OH~ and H 20 ) but an improved treatment was developed

in which the solvent concentration was determined by a conservation

equation. The equations were solved numerically to give the solution composition

and electrode potential as functions of bulk pH, crack dimensions and a variety of

other parameters characterising the system. The model included an account of the

effect of metal deposition since in many cases the ferrous ion concentration

attained its equilibrium value associated with

An attempt to include the reverse reaction specifically in the model was unsuccessful

due to numerical difficulties so, as an approximation, as [Fe2+] approached

concentrations. The authors noted that the error introduced by neglecting this effect

reaction. Turnbull5 9 produced a model to evaluate the concentration of dissolved

Fe ^ Fe2+ + 2e . (6.2.14)

the equilibrium value the dissolution rate was determined by the condition [Fe2+] ^

[Fe2+]equU by adjustment of the appropriate input factors. Figure 6.3 shows

the effect of this constraint on the predicted ferrous and ferrous hydroxide

would be small compared with neglecting the effect of solid ferrous hydroxide

123

or magnetite in the crevice. In a subsequent paper61 Turnbull and Thomas included

the effect of migration of species under potential gradients. Some

modifications were made to account for more accurate expressions for the electrode

reaction rates. Numerical solutions were obtainable over a limited range of

potentials 800 mV SCE) so a simpler model which neglected anodic dissolution

was employed. The effects of crack geometry and external electrode potential

were assessed. Comparison of the model's predictions to their experimental data

(measured in a crevice of depth 33mm) showed good agreement with regard to the

variation of potential drop with external electrode potential but the pH was

less accurately assessed with the predicted pH higher than the experimental results

(as shown in figure 6.4). This was attributed to the formation of magnetite or

ferrous hydroxide, which was not included in the model. They suggested that

inclusion of a precipitation reaction would lower the pH and the ferrous ion

concentration. It was also suggested that the pH could be limited because of the

reversibility of the hydrogen electrode although the influence of the magnetite was

likely to be greater.

This series of papers forms one of the most significant contributions to the field of

localised corrosion modelling, although the range of application is limited

somewhat to conditions of cathodic protection i.e. less positive metal potentials. The

work falls into the class of the more general, predictive modelling described

earlier. The models have been tested extensively for sensitivity to the many input

parameters of the system. Their work has since been extended further to include the

effects of mechanical yawning of stress-corrosion cracks.

In a similar sequence of papers, Galvele et al. 62 63 64 constructed a fairly complexfsO

model of pitting corrosion. In the first of these , steady-state diffusion was

combined with anodic dissolution and hydrolysis of metal ions in a parallel-sided slot

with passive walls. Boundary conditions used were constant concentrations at

the pit mouth and flux conditions at the pit base. Analytic solutions were obtained for

the concentrations of 5 species (Men+, Me(OH)^n~x+ , H +, OH~ and

H 20 ) in terms of the product of the corrosion current density, /, and the pit depth,

x. The potential at the base of the pit is effectively defined by the current

density i.e. i = f(E ) but explicit treatment of the potential is avoided by considering

only one electrochemical reaction. This simplifies the mathematics of the

problem considerably but effectively limits the model's application to systems

polarised to reasonably anodic potentials, when cathodic reaction rates are very

much smaller than the anodic ones. The results were used to estimate the minimum

length of crevice in which localised corrosion could be sustained i.e. in which

a critical H + concentration could be exceeded. This was done using experimentally

determined values for this critical pH and the current density within a pit. The model

was extended to include the effect of an aggressive anion salt (NaCl). An

electromigration term was included in the steady-state transport equations (for

and corrosion current density. Again, only one electrochemical reaction is

considered. This model effectively yields the metal potential as a function of anodic

current density whereas Turnbull's models, for example, use the metal

potential as an input parameter and obtain the current density etc from this. The

latter approach is more flexible and probably has more practical applications.

In a subsequent publication63, Galvele treated the hydrolysis of metal ions in more

detail and included the extra reaction

potential at which the critical value of * . / was reached) on the pH, consistent with

experimental observations. The previous formulation was unable to do this. The

appropriate steady-state mass-balance equations without the electromigration terms

were derived and analytic solutions were obtained for the concentrations of 6

species in terms of the parameter*./. This series of papers is one of the few to

consider the effects of the precipitation of solid M e(OH)2. In these calculations, the

[M e(OH)2\aq exceeded the equilibrium value associated with the

precipitation reaction. The mass-balance equations were re-derived treating the solid

as a diffusing species. This formulation represents the ionic distribution at the

very initial stage of precipitation and assumes that the hydroxide is precipitated as

colloidal particles in the early initial of the process. The author argued that if

the precipitation process is much slower than the hydrolysis reaction then the ion

distribution in the crevice will be somewhere between that predicted by the

all-soluble treatment and that predicted by the precipitate treatment. The diffusion

coefficient of the solid, D, was set equal to that of the metal ions. Galvele

points out that changing this value will only shift the concentration curve, C = /(* ./) ,

curve along the C-axis. The results from this improved model were again

used to identify critical * . / values for which pitting could be sustained in various

corroding metals and electrolytes. The pitting potential of bivalent metals was

predicted to be pH-independent up to a pH value of about 10, but pH-dependent

above this.

In the most recent paper in this series, Gravano and Galvele6 4 have extended the

Men+, N a+ and C/~ only, so no chemical reactions were included) and an analytic

expression obtained for the potential in terms of the bulk chloride

concentration and the critical parameter*./, the product of distance along the crevice

Men+ + n O H -^M e (O H )n{aq) . (62.15)

This model predicted a dependence of the pitting potential of a system (i.e. the

125

model by including electromigration of all the species i.e. by no longer

assuming that the aggressive anion solution acts as a supporting electrolyte. They

also include a buffering acid. This extension of the modelling was considered

necessary to study the passivity breakdown of zinc in diluted solutions of aggressive

anions plus a borate buffer. In particular, the previous treatments of the

problem could not predict observed relationships between the pitting potential and

buffer concentration. The steady-state transport, chemical equilibrium and

local electroneutrality equations were solved numerically but the results are given as

functions of x .i for comparison with their previous work, i was assigned an

experimentally determined value of 1 A cm . Precipitation was included as before

and once again the results were analysed with respect to passivity breakdown.

The predicted concentrations were tested against those made by the second model

of this series for iron in a pH 10 solution. The results gave a coincidence for

values of x .i < 1 0 “ 4 but the electromigration effects became important in the

redistribution of chemical species at higher values. The greatest deviations were

those in the Cl~ and N a+ profiles. This value at which migration becomes important

is many orders of magnitude larger than that predicted by Ateya and

Pickering58. The more sophisticated model was then used to predict the effect of a

buffer solution on the pitting potential of zinc in diluted NaCl. The results

showed that the buffer inhibits pitting by increasing the critical x .i value and this

effect was enhanced at low concentrations of the aggressive anion. Some predictions

from the model have been tested against experimental data with good results.

The localised acidification mechanisms assumed in these models yield the following

expression for the pitting potential

Ep = Ec + r] + A</>, (6.2.16)

where Ec is the corrosion potential of metal in the pit-like solution. Ec was

determined experimentally by exposing a smooth metal surface to a solution whose

composition was fixed according to the model's prediction of the chemistry at

the base of a corroding cavity. The open-circuit potential of the system was then

measured, rj is the polarization required to reach the critical x .i value and by

assuming polarization curves (e.g. Tafel kinetics), changes in x .i can be converted to

changes in rj. A</> is the electrical potential drop in the solution when ion

migration is taken into account i.e. in the absence of a supporting electrolyte. The

model yields A 0 at the pitting potential. The expression for the pitting potential

could then be tested against experimentally measured values. The coincidence was

found to be very good for a number of different solution compositions. The

deviations were found to be greater at higher borate concentrations for which the

assumption of dilute-solution theory was less correct. If we take a more direct test of

the accuracy of the model and relate the corrosion current at the base of the

pit to the polarisation of the metal using the predicted potential drop down the cavity

( A <p) and by assuming a Tafel polarization curve, we find that the predicted

metal potentials for the various solutions are lower than one would reasonably expect

for such a cavity propagation rate. They also disagree with the reported

theoretical pitting potentials. This model really predicts the potential drop in the pit

as a function of the current density via the resistivity of the solution (which is

itself a function of the solution chemistry). Although the technique of comparing the

predicted pitting potentials with experiment gave some good correlations, the

results lack the self-consistency between the current density and the actual potential

of the metal necessary for a complete theoretical description of the process.

This suggests that the model has over-simplified some of the basic physics of the

problem.

The modelling of Alkire et al. follows a similar series of stages, although the

necessity for the inclusion of the extra terms is not laid out as clearly as in the

Galvele series. This series is developed to describe different systems whereas the

Galvele papers seek to explain additional phenomena of the same system. The

simple diffusion-only model5 5 was used in relation to the chemistry and

electrochemistry of artificial pits as described earlier. An extension of this was

developed with particular relevance to the location of the cathodic reaction in a

corroding crevice and included some account of potential variations56. Hebert and

Alkire6 5 produced a more sophisticated model of transport mechanisms in

crevices relating the study to initiation of crevice corrosion in aluminium. The model

was based on the hypothesis that for aluminium immersed in a dilute sodium

chloride solution, initiation occurs when the concentration of dissolved metal

exceeds a certain minimum critical value, determined experimentally. The model

consisted of anodic dissolution of aluminium, hydrogen ion reduction and oxygen

reduction within a cylindrical crevice (with variations in the radial direction only). The

hydrolysis of aluminium ions was included. The potential and concentration

of species at the edge of the crevice were assumed equal to the uniform corrosion

potential and bulk concentrations respectively. This boundary condition for

the potential allowed direct comparison with a set of experiments reported. Only the

period prior to passivity-breakdown was considered -after this many of the

assumptions made were no longer valid. The oxygen- and hydrogen- reduction

current densities were assumed to be of Tafel form and linear kinetics were assumed

for the aluminium dissolution since the potential range before initiation was

shown to be within a few mV of the pitting potential, tj>Rl i.e.

j = A{(p - <pRl) for <p><pm (62.17)

j = 0 for <p<(f>mwith A determined from experimental polarization curves. Time-dependent transport

equations were constructed for 6 species including diffusion and migration.

Electromigration of the species involved in the electrode reactions was neglected.

This was presumably considered insignificant compared with the migration of the

ions of the supporting electrolyte. An extra term proportional to the appropriate

electrode current density which reflected the rate of production or depletion at the

base of the cylindrical pit was added to the mass- balance equations of these

ions. The set of equations was completed with the hydrolysis equilibrium reaction

and the electroneutrality equation. At time zero the concentrations were fixed (by

experimental conditions) and the potential set equal to the uniform corrosion

potential (calculated from the condition of no net electrode current density). The

system was solved numerically using a technique which involved linearising the

non-linear terms about a trial solution and iterating until convergence was reached.

Results indicate that the current density and potential are highest at the edge

of the crevice and decrease monotonically towards the centre. The predicted

breakdown times were reported to compare favourably with experimental data so as

to suggest that the mechanism of dissolved metal species, on which the

model is based, may be valid.

6.2.4 Transport by Convection.

A number of studies have considered the effect of convection of ions associated

with either a fast-growing pit or a fluid flow outside the cavity. A full

formulation of the transport equations including diffusion, electromigration,

convection and chemical reactions is difficult to solve for laminar flow and virtually

impossible in the case of turbulence. There have been several approaches to

solving the laminar flow case. A number of models derive and solve approximations

to this problem, neglecting for example electromigration, whereas others treat

the equations in a generally less rigorous manner, and aim to assess the relative

importance of the various transport processes and surface reactions by identifying

various dimensionless groups of parameters. These groups can provide guidance for

experimental design and corrosion prediction even though the equations from

which they are derived cannot be solved.

Silverman6 6 considered such dimensionless groups. He considered fluid flow past

the end of a rectangular crevice. This flow creates a momentum and, in the

presence of corrosion, a mass-transfer boundary layer. The two-dimensional

ionic-transport equation was reduced to one dimension by averaging the flux to and

from the walls across the crevice width. The equation was then cast into

dimensionless form by identifying various scaling factors of the system. These were a

characteristic length of the cavity, L, a saturation concentration of one of the

species, C°, the pure component mobility at relevant ionic strength, U °t a constituent diffusion coefficient in a pure state, D °, the maximum potential

free stream velocity (laminar flow) or the frictional velocity (turbulent flow), V. Several criteria were established from the resulting dimensionless groups. Migration

was negligible compared to diffusion when

This latter criterion was approximate and depended on the particular system

being modelled. To assign a criterion to the relative importance of reaction rate and

diffusion, account was taken of the order of magnitude of the dimensionless

diffusion contribution and the quantity /(C)/C, where f(C) is related to the reaction

rate at the walls, R, by R = kf(C)exp(aF<p/RT). The condition for neglect of

the reaction term was given as

dimensions of the crevice). The implications of these results were discussed with

relation to the hypotheses of other authors. For example Alkire, et al. in reference 55

asserted that acceleration of crevice dissolution will only occur if the ohmic

resistance in the cavity accounted for the major part of the applied potential. Analysis

of the appropriate dimensionless groups gave a consistent conclusion. Also,

the criterion for the domination of diffusion over migration as the important transport

process was shown to be consistent with that predicted by Ateya and

Pickering58.

Shuck and Swedlow6 7 constructed a more rigorous model of ion transport in

crevices or pits to assess the effect of both transport mode and crack geometry on

the solution composition of the cavity. A single reaction of oxide film

formation was considered

No cathodic reduction was included. It was assumed that the oxide film had zero

difference between the external surface and the base of the cavity, (p° , and either the

zF<p°/RT « 1 (62.18)

and convection was insignificant when

V L/D ° « 10" 2 (6.2.19)

( V L /D q is the modified Peclet number (Reynold's number x Schmidt number)).

kiL y ^ ^ f exp(^<t>/RT)lw D° c° « constant (6220)

_2where the constant in this system is of the order 1 0 . (/ and w are the linear

M + 2 H 20 ^ M 0 2 + 4H+ + 4e~ (62.21)

129

electronic conductivity and all the current passing through it was conducted by

migration of ions or vacancies. Initially transport by diffusion only was considered.

The rate of oxide reaction was assumed constant to provide constant flux

boundary conditions. Three geometries were considered, 1) rectangular, 2) hyperbolic

and 3) wedge-shaped. The time-dependent diffusion equations were reduced

to one dimension by averaging the contribution from the walls of the crack across the

width in each case and analytic solutions were obtained. At short times there

were quite significant differences between the concentration profiles but there was

little difference in the steady state. A two-dimensional formulation of the

equations was developed for the rectangular crack and a comparison made with the

one-dimensional approximation. The results indicated that the

two-dimensional transport effects were only significant at short times or at points

near the end of the crack. Similar results were reported for hyperbolic and

wedge-shaped cracks. The model was extended further to consider the effect of

solvent motion associated with the yawning motion of a stress- corrosion crack.

Equations for the fluid velocity inside the crack were given. The transport equations

were solved and it was concluded that the effect was to reduce the magnitude

of the concentration gradient at every point throughout the crack and thereby reduce

the concentrations everywhere. The influence on the gradient was greatest at

the crevice mouth. However, the overall effect was considered small as compared

with the effects of crack geometry and transport by simple diffusion. Transport of

charged species by electromigration was next considered. A one-dimensional

formulation was used to compare the results with those obtained for diffusion only. It

was concluded that little influence is exerted on the concentration of product

species, but the principal effect is to alter the concentration of salt ions in

accordance with the requirement of electroneutrality. This seems consistent with the

comparisons of Galvele et al. between their simple diffusion model and the

more complex diffusion and migration version, (the current density and crevice length

in this paper satisfy the criterion for negligible influence of electromigration,

x .i < 1 0 " 4).

Smyrl and Newman considered the effect of a moving electrolyte in two papers.

In the first the hydrodynamics were analysed and velocity profiles describing

the flow in a wedge-shaped crack were obtained. In a later paper, the transport of

ions in this geometry was considered, with particular reference to the mass transfer of

dissolved water in a molten LiCI-KCI eutectic. A coordinate system

propagating with the crack was chosen and a steady-state mass-transport equation

for a minor species was derived in polar coordinates. A diffusion and

convection term was included but electromigration was considered unnecessary

since the contribution of any minor species would be small compared to that of K+,

130

Li and Cl- . Approximations were made to the equations for various regimes

of r and 0 and analytic and numerical solutions were obtained using infinite

boundary conditions. The potential distribution in the crack was investigated using

V*/ = 0 and = 1d. , assuming variations were mostly radial. It was- drconcluded that minor components being consumed in a crack at rates limited by

mass transfer have vanishingly small concentrations near the crack tip. In addition

the flux of these is extremely small in this region. The analysis yields regions

within the wedge in which various combinations of the transport processes

dominated.

Alkire and Cangellari6 9 developed a fairly detailed model to investigate the

formation of salt films during metal dissolution in the presence of fluid flow. The

presence of salt films can either aid or discourage repassivation depending on the

alloy and its environment. The model considered steady-state anodic

dissolution and aimed to predict the range of fluid velocities over which metal salt

films occur on the dissolving surface. The system consisted of an electrolytic

solution flowing between two widely spaced parallel plates. On one of these a small

anodic region of bare metal was exposed to the solution while the other

served as a cathode. The fluid velocity profile was assumed to be fully developed.

The mass-transfer boundary layer on the active surface was assumed thin and the

Schmidt number (v/D ) large. Electromigration was considered negligible due to the

presence of a supporting electrolyte but the authors point out that this

assumption may not be valid since large metal-ion concentrations may arise in the

diffusion layer. Inclusion of migration was estimated to increase the mass flux

of divalent metal ions and cations by a factor of up to 2. The potential of the working

electrode was resolved into three components; the concentration

overpotential in the diffusion layer, the surface overpotential and the potential in the

solution just outside the boundary layer. The latter was obtained by solving

Laplace's equation in the bulk electrolyte region. The steady-state

convective-diffusive equation for laminar flow was derived for the concentration of

metal ions in the boundary layer, assuming that the fluid velocity varied

linearly with distance from the wall. The current, potential and concentration

distributions along the dissolving surface were obtained numerically for the system of

equations using a 'guess and iterate' technique. An equation was also derived

for turbulent flow, involving a parameter describing the eddy diffusivity. This second

system of equations was also solved numerically. Results for both laminar and

turbulent flow indicated that the current distribution on the plate was not uniform,

with the largest densities found at the leading and trailing edges. This may be

attributed to the ohmic resistance of the solution. The concentration profiles

131

indicated that the saturation of the solution occurs first at the trailing edge where the

boundary layer is thickest and the current density high. The concentration at

the leading edge also tended to increase rapidly since the current density was high

but the increase was limited by the thinness of the boundary layer in this

region. For some slow flow conditions the surface concentration was found to lie

below the saturation concentration and for others the concentration was above this

value. An intermediate region in which part of the surface was saturated and

part not was specified in terms of various parameters of the system e.g. fluid velocity

etc. The critical Reynold's number for salt film formation (i.e. if the flow

exceeds this number saturation of the fluid adjacent to the surface would not occur)

was identified by considering a simple system of a small circular disc

embedded in the walls of a flow channel. Four cases were considered- laminar and

turbulent flow and for each of these ohmic control conditions and

charge-transfer control conditions. An expression for the average current density from

the disc was derived using Ohm's law and this was used to obtain an

expression for the critical Reynolds number. For flows with Reynold's number below

this value precipitation of a salt film is possible but supersaturation

phenomena may impede its occurrence. Comparison of the theoretical predictions

with experimental data was fairly encouraging. It was found that repassivation

occurred at flow conditions which permitted formation of a salt film on at least a

portion of the dissolving surface.

6.2.5 Moving—Boundary Formulations.

There are a few models that consider the effects of the changing dimensions and

geometry of propagating corrosion cavities. None of these, however, involve a

rigorous analysis of the moving-boundary corrections to the equations. Beck and

Grens70 developed a fairly complex model of mass transport within a stress-corrosion

crack in titanium which included a coordinate system propagating with the

crack. This approach is an approximation to a full formulation of a coordinate system

stretching as the cavity expands. Their analysis has some applications to

pitting and crevice corrosion. A wedge-shaped crack was divided into three regions;

the tip zone where the cleavage process occurred, the monolayer zone where

the first monolayer of oxide is formed and the outer layer where the oxide layer

increased in thickness. In the tip zone the reaction

Ti + mXT -* T iX m + me" (6.2.22)

was assumed to occur, where X" is a halide ion. In the other two zones the reaction

Ti + 2H20 -* T i0 2 + 4 /T + 4e" (6.2.23)

was assumed but the kinetic behaviour in the two zones was treated differently.

Hydrogen ion reduction was assumed in the monolayer zone but not further out

(since the potential was considered sufficiently anodic to neglect the reaction).

One-dimensional mass-transport equations were derived for anions, cations and

hydrogen ions with the flux boundary conditions (relating the flux of ions to the

electrode currents associated with (6222) and (6.2.23)) Numerical solutions were

obtained for the ion concentrations and potential as functions of position in

the crack and other parameters. Some correlation with experimental results was

reported but several limitations were suggested. These included doubts that

continuum models such as this were applicable to the crack-tip zones and on the

accuracy of the kinetic data employed.

6.3 Summary

Each model discussed in this survey goes some way towards describing the very

complex system of physical and chemical processes that characterise pitting

and crevice corrosion. The models that neglect ionic diffusion under concentration

gradients are only valid at short times since any build up in potential implies

some imbalance in ionic concentrations and subsequent diffusion. The models that

neglect migration under potential gradients also tend to be rather restricted in

their application; they are either specific to some experimental system in which the

potential gradients have been shown to be small over the relevant timescales

or they consider only the transport of neutral species. Several of the models indicate

ranges of certain parameters in which each transport process dominates. The

more general models are those which consider diffusion and electromigration (and

convection for fluid-flow situations and stress-corrosion cracking). These

generally give better quantitative agreement with experimental data over a wider

range of parameters. They also predict a wider range of qualitative phemonena even

if quantitative comparisons are not so satisfactory.

However, there is no single model that includes all the phenomena necessary for a

complete description of cavity propagation or can predict both qualitatively

and quantitatively all the experimental observations made in this field. Even the more

rigorously derived ones do not agree with all aspects of the real corrosion

situations. One of the most common inaccuracies seems to be in the prediction of

the cavity propagation rates, i.e. the corrosion current within the crevice or pit.

For example, a typical current from Turnbull's model61 is considerably higher than

one would realistically expect from a metal polarized to such a cathodic potential or

a typical metal potential evaluated from Galvele's model6 4 (using the input

current density, the calculated potential drop and a reasonable expression for the

anodic reaction kinetics) seems too low for such a dissolution rate. Turnbull suggests

in his paper that the inaccuracy is partly attributable to the formation of

magnetite within the crevice, which the model does not account for. Galvele,

however, includes precipitation of a solid corrosion product, but his model assumes

passive cavity walls which would tend to make the predicted current high

anyway. Ateya and Pickering58 also predict a potential gradient smaller than

experiment in their model which they attribute to ionic interactions at high electrolytic

concentrations. There are also a number of other effects that have not been

investigated in much detail and may make important contributions to a more

physically realistic model. These include

(1 ) the effect of the solid precipitate on the migration of ionic species within the

restrictive geometries,

(2 ) the effect of the moving boundaries of a propagating cavity and

(3) the effect of some restriction on the quantity and movement of cathodic charge

generated outside the crevice.

It seems likely that a combination of these inaccurate approximations leads to the

general lack of agreement of the propagation models with experimental data.

The degree of influence of some of these will be investigated in the next two

chapters of this thesis using a series of models of a propagating crevice or pit of

increasing complexity.

Figure 6.1 Illustration of the crack shape predicted by Bignold53

135

OJOn

' , . . r • . • ; * • . > •

I Bulk. Solution

Stagnant ;v-Diffusion layer

/‘ VV* ‘ *; • \ •• ,• \ ‘ ! .*•*: . ; . ' r :S

• V«'

Figure 6.2 Schematic representation of the experimental system used by Tester and Isaacs54.

fSh

teri

trd

tio

n

(mo

les

dm

Figure 6.3 Variation of the concentration of ferrous and ferrous hydroxide ions with distance from crack tip (*=0) as predicted by the model of Turnbull and Thomas60.

137

Me

tal

Po

ten

tia

l,

mV

S

CE

7 8 9 10 11 12 13pH

Figure 6.4 Variation In pH In an artificial crevice with external potential at distance x from the crevice tip as measured by Turnbull and Thomas60.

7. A Mathematical Model of the Steady-State Propagation of Localised Corrosion Cavities.

7.1 Introduction.

The aim of Part II of this research project is to develop a mathematical model of

the cavity-propagation stage of pitting and crevice corrosion, which is entirely

self-consistent and predictive, and reproduces experimental data both accurately and

consistently. In particular, cavity growth rates and the solution chemistry

within a cavity should be reproduced. The models will also allow assessments of the

distribution of solid corrosion products in the cavities and the evolution of

their shape. The need for a more realistic mathematical model has been outlined in

the literature review in Chapter 6 . The mathematical models developed in

Chapters 7 and 8 will improve on some of the more common approximations used in

these models.

In this chapter, the first stages of such a model are presented. This model predicts

the steady-state solution chemistry and electrochemistry (and hence metal

penetration rates) within a cavity as functions of the many parameters on which these

depend. These input parameters describe the physical and chemical

environment of the system, for example metal potential, pH of the electrolyte etc.

The accuracy of the model's predictions will be assessed fairly generally in this

chapter, by considering the order of magnitude of the predicted cavity propagation

rates and qualitative variation in the predicted quantities with the parameters

of the system. Further validations will be carried out in Chapter 8 , in which a more

sophisticated version of the steady-state model is presented. The model is

developed in a number of distinct stages, each considering an additional physical

process or chemical equilibrium reaction. This approach allows an assessment of the

importance of each process to the evolution of the corroding cavity. Input

data appropriate to the corrosion of carbon steel will be used and the predictions at

each stage compared with empirical data for this metal. (Carbon steel has

been shown to passivate in a wide range of solutions containing chloride ions5).

However, the model is suitable for application to other metals.

7.2 Description of the Preliminary Steady-State Model.

The pit or crevice is modelled as a parallel-sided slot of length /, width w and

through-thickness d (Figure 7.1). The crevice is assumed filled with a dilute

aggressive solution, in this case sodium chloride. Initially the following assumptions

are made:

(1 ) The metal surface outside the crevice is covered with a passive film and there is

139

sufficient generation of cathodic charge on the outer surface to drive the

localised corrosion;

(2) The important effects are genuinely local; any changes or local variations in the

chemical and electrochemical conditions within the crevice do not affect

the potential of the whole specimen;

(3) Transport in the through thickness (z direction) may be neglected i.e. only

variations along the depth and across the width of the crevice need be

considered and the equations may be reduced to two dimensions or sometimes

to a single dimension;

(4) The electrolyte is static and no fluid-flow effects need be included ;

(5) Cavity propagation is slow compared with the ionic migration rates and both

moving-boundary effects and any induced electrolyte motion may be ignored.

This may not be valid in some cases, for example at high metal

potentials. The effects of the moving boundary are considered in section 7.6.1;

(6 ) The crevice is anaerobic;

(7) Dilute solution theory may be used throughout, and so the activity of water is

not considered specifically.

Several of these assumptions are not always strictly valid and will require further

investigation.

7.2.1 Electrochemical Reactions and Data.

Six aqueous chemical species considered in this model are

Fe2+, FeOH+, Na+, Cl~, H + and O FT. (This is the minimum number necessary to

simulate the behaviour of a real localised corrosion site). Three

electrochemical reactions are included; the dissolution of iron, the reduction of water

and the reduction of hydrogen. The environment is assumed deoxygenated.

This part of the model requires some empirical data in the form of corrosion reaction

rates. The particular data used in the first stage of the model have been

obtained over a range of pH consistent with that found in corroding cavities, at a

temperature of 25° C. For the oxidation of iron

Fe2+ + 2 e ~ -+ F e , (7.2.1)

Turnbull and Gardiner71 found that between pH 3 and 8.5, an expression of the form

i = /0 xexp ( FE/R T) (7.22)

with /01 = 1.96xlO n i4m 2 and arj = 1.0 describes the anodic current quite

was found to show a first-order dependence on hydrogen ion concentration

w ith /03 = — 2.5x10 4A dm 3m 2mol 1 and a3 = — 0.561.

The electrode potential E in each of these expressions which controls the rate of

the electrode reactions needs a little clarification for localised corrosion. <pM is defined as the electrical potential of the corroding metal relative to some standard

electrode in the bulk solution and <p(x) as the potential drop in the solution in

the cavity i.e. the difference in the potential at the top of the crevice and the potential

at position x, just outside the electrical double layer on the metal surface. The

potential driving the corrosion reactions, E, is then

that the chloride and sodium ions do not interact chemically with any of the

others but contribute to the transfer of current within the crevice. Two chemical

reactions are included initially:

This simplification is quite reasonable since the chemical reactions occur on a

several parallel reaction schemes can be collected into one overall scheme: it

is only the equilibrium constant which really matters, and this is determined by the

accurately. E is the electrode potential. The reduction of water

H 20 + e~ H + OH~ ( 7 2 . 3 )

was found to follow the relationship

i = iQ2 exp (oc2 FE/R T) , ( 7 2 . 4 )

with <02 = — 8.0x10 10 Am 2 and a2 = — 0.561. The hydrogen discharge reaction

2 H + + 2e~ H 2 (7.2.5)

' = i<alH+]exP(<XiFE/RT) ’ ( 7 2 . 6 )

E = <Pm ~ ( 7 2 . 7 )

7.2.2 Solution Chemistry.

As a first stage, a very simplified reaction scheme is considered and it is assumed

Fe2+ + H20 ^ Fe(OH)+ + H *

H * + OH~ ^ H 20 .

(7.2.8)

(7.2.9)

62The first reaction is simplified from the many-stage hydrolysis that is likely to occur .

much faster timescale than the migration processes so that the detailed kinetics of

141

free energies of the species. Thus any simplifications here are likely to have

less effect on the results than simplifications made in the electrode kinetics. The

equilibrium constants of the reactions K x and K2 are calculated using

R T ln ^ = Gf (Fe1+) + Gf (H 2Q) - Gf (FeOH+) - Gf (H +) (72.10)

and

RTlnK2 = Gf (H +) + Gf (O H ~) - Gf (H 2Q), (7.2.11)

where Gj(S) is the Gibb's free energy of formation of species S. The individual

forward and backward reaction rates, denoted k lF,k lB and k2F,k2B, are related to

the equilibrium constants by the following,

K i =IF

IB K > = kIF

2 B

7.2.3 Governing Mass—Transport Equations.

In dilute-solution theory, the transport of aqueous species i is governed by the

mass-balance equation describing diffusion under concentration gradients,

electromigration under potential gradients and chemical reaction:

^ = D,V2C, + z M F V t C M ) + R, (72.12)

where Ri represents the rate of production or depletion of species / by chemical

reaction and Tf is the mobility, given by the expression

< 7 - 2 - 1 3 >

Assuming passive cavity walls, the steady-state transport equation for species i is

D. d Ci + h E J L 4 - ( c A + r ,, = odx‘ RT dx 1 dx

(7.2.14)

The six equations are specified fully in Appendix 3. The electrostatic potential of the

system, (p, is governed by Poisson's equation

V 2<t> = ■£ (7.2.15)eo

where p is the charge density. For water e0 is 80, and the magnitude of Eo 1 is

sufficiently large that any departure of the system from electroneutrality results in a

142

very large electrical restoring force. This force tends to remove charge

gradients on a much faster timescale than those associated with the diffusion

processes. As an approximation therefore, Poisson's equation may be replaced by

the equation of local charge neutrality

is proportional to the corresponding current at the crevice tip. The flux of the

other species at the crevice tip is zero.

These conditions are specified fully in Appendix 3.

The equations describing corrosion in two directions i.e. with the electrode

processes occurring on the cavity wails in addition to at the crevice tip are derived in

a similar manner. An approximation employed by Turnbull5 9 is used to convert

the equations from two dimensions into one. This technique assumes that the

concentration and potential profiles across the width of the cavity are uniform. The

contributions from the various electrode processes at the walls may then be

averaged across the width and added to the appropriate mass-balance equations as

terms independent of the transverse coordinate. This effectively adds extra

terms involving the potential- (and hence position-) dependent currents and cavity

width. The parameters used in this study fall within the range of validity of the

approximation. The extra terms are specified fully in Appendix 3. The boundary

conditions for this set of coupled differential equations are as before.

7.2.4 Solution of Mass—Transport Equations.

The two sets of equations are highly non-linear in nature and an analytic solution

would be extremely difficult to obtain. However, by casting the equations into

a dimensionless form, rearranging them and integrating once, a form suitable for

numerical integration may be derived. The details of these methods are given in

Appendix 4. The method of solution involves making a reasonable estimate for the

pH and potential profiles, calculating the various parameters dependent on

these, solving the coupled equations and comparing the resultant pH and potential

distributions with the estimates. If these do not coincide then the calculated

profiles are used for the next iteration. This process is continued until convergence

and hence a self-consistent solution is reached. Numerical integration' is. by

Gear's method72, which adopts backwards differences and chooses automatically the

= o (7.2.16)

for all x in the crevice. The boundary conditions of the problem are as follows;

(1) The concentrations of the species are fixed at the cavity mouth and are equal to

the values in the bulk solution outside the corrosion site.

(2) The flux of species involved in the electrode processes (i.e. Fe , H and O H ~)

143

stepsize and order of the integration formula. It is especially well suited to this

problem since the variables show rapid variation over a very short distance near the

pit mouth and a large number of evaluations are required here. Several

techniques involving a variety of mathematical approximations which reduce

computing time have been developed for solution of this system of equations.

7.3 Results from Preliminary Model.

The parameters used in the calculation are given in Table 7.1. Initially a crevice

with passive walls is considered. Figure 7.2 shows details of the solution chemistry

for a metal potential of -0 -2 V SCE. The pH decreases rapidly in a small

region near the cavity mouth and then decreases more slowly towards the tip. The

Fe2+ and Cl- concentrations also show a marked change near the crevice

mouth and increase towards the crevice tip. The model indicates that the deviations

of concentrations in the crevice from the bulk values increase with increasing

metal potential. Figure 7.3 shows a comparison of the potential drop along the

crevice length (i.e. the difference in potential between the solution at the pit mouth

and the potential in solution at the pit base just outside the electrical double

layer) for the metals at potentials — 04, — 0-2 and 0 0 V. the calculated corrosion

currents associated with these electrode potentials are extremely high under these

hypothetical conditions. It will be shown that inclusion of corrosion on the pit

walls reduces the corrosion-current densities considerably. Figure 7.4 shows the

variation of corrosion current with crevice depth (with each calculation carried out

independently in a static geometry -i.e. no moving boundaries are involved here) for

the case (pM = -0 -2 V. Although the magnitude of the current is

unrealistically large ( for reasons which will be suggested later), the general trend of

decreasing current with increasing diffusion length is significant. The cathodic

current at the pit base is calculated as being very small at such anodic potentials and

it is concluded that the significant portion of cathodic activity occurs on the

metal surface outside the crevice.

Thermodynamic Data:

Equilibrium constants (25° C) 15

Fe2+ + H 20 Fe(OH)+ + H + log £ = -6 .7 8

FeOH+ + H 20 ^ ± Fe(OH)2 + H + log £ = -4 .8 8

/ / + + OH~ ^ H 20 log £= 14

Physical and Chemical Parameters

Passive wall calculations

Crevice length = 2 mm

Metal potential = -0 .4 to 0.0 V SCE

Bulk chloride concentration (no precipitation) = 10~ 3 M

Active wall calculations

Crevice width=10 /zm

Diffusion coefficients

All species except H + and O H ~ , D = 10“ 9 m2 s' 1

H +, D = 9.3X10 " 9 m2 s '1.

O H ~ ,D = 5 .3 x l0 ~ 9 m2 s '1.

Table 7.1 Parameters used in runs of preliminary steady-state model of

corrosion-cavity propagation.

Figure 7.5 show the solution chemistry in a crevice with corroding walls and base,

at a potential of -0 -2 V. The same basic trends as for the passive-wall case

are observed, i.e. increasing chloride and ferrous ion concentration and decreasing

pH towards the crevice tip. However, the magnitude of the deviations from

the bulk values are slightly larger with corroding walls and the profiles show more

rapid variation near the crevice mouth. A comparison of the potential drops for the

active and passive walls (figure 7.6) indicates that the corrosion currents are

reduced when the metal dissolution occurs over a larger area. This trend is consistent

with experimental observations73. In this case the corrosion current density at

the pit base is approximately 2 X 102 A m “ 2 compared with 3 x 103 A m“ 2 for the

passive walls. However, these currents are physically unrealistic and a number of

other phenomena need to be considered. These include the precipitation of solid

ferrous hydroxide, since in all the examples the solubility limit is exceeded.

(with ferrous hydroxide) = 2 x 1 0 2 M

145

Passivation of the Crevice Walls.

The potential drop along the crevice length can be used to calculate the current

distribution over the internal metal surface. Figure 7.7 shows the corrosion

current as a function of position in a cavity of length 2mm at a potential -0 2 V

(SCE). The current at the cavity mouth is extremely large, but the model does not

account for any passivation of the metal. In reality, it seems likely that this region

would repassivate under conditions of higher pH and lower potential drop in this

region. Although the predicted currents in the rest of the cavity are unrealistically

high, the distribution gives an indication of the future shape of the p it- the rapid

broadening of the width close to the mouth suggests an eventual bottle-shaped

cavity. Most of the current distributions generated for this range of parameters

display this general behaviour.

7.4 Precipitation of Ferrous Hydroxide.

In all the examples given in earlier sections, the ratio of the concentrations of

hydrogen and ferrous hydroxide ions has exceeded the solubility limit of ferrous

hydroxide. As a next stage in the model therefore, the reaction

Fe(OH)+ + H 20 ^ Fe(OH)2 + f t (7.4.1)

is considered specifically. The equilibrium constant of this reaction is calculated from

the free energies of the individual species as before. In a recent paper,

Gravano and Galvele6 4 treat the solid as a diffusing species. They include a term dCD ~ - directly in their mass-balance equations and solve for this solid ax

'concentration', C, in the same way as for the other concentrations. The diffusion

coefficient, D, is set to 1 x 10- 9 m2 s-1, the same as for the other diffusing ions

(except H+ and OH- ). This describes the situation in a crevice a few milliseconds

after a flaw in the passive film occurs i.e. before the solid in the pit changes the

diffusion coefficients for the mobile species. In the model presented in this chapter,

the solid concentration is not solved for directly, thus avoiding the necessity

to assign a diffusion coefficient to a non-mobile species. Instead the mass-balance

equations are modified, so that the solubility limit of Fe(OH)2 is not

exceeded.

There are little data available on the rate of precipitation but it is assumed that the

precipitation rate is fast everywhere compared to the diffusion rates. The

governing mass-transport equation are derived as before but with the additional

constraint fixing the ratio of the H+ and Fe(OH)+ concentrations. The boundary

conditions at the crevice mouth are modified slightly to ensure charge neutrality and

chemical equilibria. These equations are given in Appendix 3 and are solved

using the iteration techniques described earlier. Figure 7.8 shows the predicted

solution chemistry in a pit with non-corroding walls. The parameters are the same as

in the previous calculations except that the bulk chloride concentration is

assumed 2 X 10- 2 M. Comparison of figure 7.8 with figure 7.5, in which the solubility

limit is exceeded, shows a much lesser degree of acidification in the crevice,

about pH 5 at the tip compared to pH 3. However, the potential distributions are very

similar and the corrosion-current densities are only reduced by a few per cent

by constraining the pH and [ FeOH+ ] in this way.

7.5 Comparison of the Preliminary Model with Experiment.

The model has been tested against data from several experimental systems. The

first of these is part of a project to evaluate localised corrosion of carbon steel

with specific reference to canisters containing heat-generating nuclear waste5. Part of

the data is in the form of measurements of maximum pit depth against time

of samples polarized to — 0*2 and 0*0 V SCE, exposed to a corrosive environment

(0-1 M NaH C 03 + 1000 ppm Cl~ at pH 8.4) at 90 0 C. Statistical analysis of this

data yielded an expression of the form

Pmax = 8.35 Tm0.46 (7.5.1)

where Pm9X is the maximum pit depth in mm and Tm the time in years. This is plotted

in figure 7.9. The relationship yields a corrosion-current density of 17.5 A m

(equivalent to a penetration rate of about 19 mm year-1) for a crevice of depth 2 mm.

The model predicts a current density of approximately 3 x 102 A m-2,

assuming active walls and the precipitation of ferrous hydroxide. Although these

values are clearly too high, the model does predict a similar rate of decrease in

current with increasing crevice length.

The second test of the model is to compare the predictions of the pH within the

crevice with some experimental data obtained by Turnbull and Gardiner71.

They measured the pH distribution within an artificial cavity in BS 4360 50D steel.

This cavity contained 3.5% NaCl (pH 6 ) and was 33mm deep and 150 [im wide. Figure 7.10 shows the measured pH at distance 4 mm from the crevice tip. Also

shown is the predicted pH from Turnbull's mathematical model using the

above parameters60. The present model give slightly better agreement over lower

potentials. However, Turnbull suggested that the formation of magnetite in the cavity

would account for most of the discrepancy under these conditions which his

model also neglects. This comparison suggests that the form of the solid corrosion

products in the cavity will have a strong effect on the solution chemistry and

the overall behaviour of the pit or crevice and should be more carefully considered in

the formulation of the model.

147

7.6 Sensitivity Tests

As discussed in the previous section, the model gives some reasonable qualitative

agreement with certain aspects of the chemistry and electrochemistry within

the crevice but the quantitative agreement is less encouraging. These inaccuracies,

particularly the overestimation of the corrosion currents, seem consistent with

those from other models in the literature. There are a number of approximations and

assumptions used in the construction of the model that require further

investigation. These include

1 ) The effect of the moving boundaries of a propagating crevice;

2) The chemical form of the precipitates and

3) The effect of the solid precipitates on the migration of the ionic species within

the restrictive geometries.

7.6.1 Addition of a Moving Boundary Representation.

A preliminary attempt has been made to include an account of the effects of the

changing geometry of the crevice with time. Initially, a crevice propagating in

one direction only is considered (the x direction where 0 < x < l and / is the crevice

length). For this calculation, the dimensionless variable X = - is introduced where

0<Ar< l . Transforming equation (72.14) into this coordinate system yields

a 2c fA (— t +

' a * 2

^ ^ ( C , f £ ) ) + IvX + l 2R, = R T 3 X y ‘ 3 X ” 3X ' 31

(7.6.1)

dl SCwhere v = — . The equations are solved for the quasi-steady state i.e. — = 0 by

casting then into dimensionless form and rearranging as before (Appendix 5).

In order to integrate the additional term in each equation (to produce a form suitable

for numerical integration ) it is assumed that each concentration is nearly

constant along the pit length i.e. the integral is approximated as follows,

xs v X ~ d X = vC (l) - v C ( l) ( l ~ X ) - vXC{X) a X

( 7 . 6 2 )

= v X (C (l) - C(X))

The resulting system of equations and description of the solution method is given in

Appendix 5. The original set of parameters with passive walls were used

initially (i.e. as in table 7.1) for comparison with previous results. The precipitation of

ferrous hydroxide was included but the diffusion coefficients were not scaled

148

down, again with a view to direct comparison. It was found that the method of

solution was only successful for values of the metal potential less than about -0-1 V.

For values above this, the solution method produced some very interesting

behaviour which is described in Appendix 6 . For metal potentials that do have a

convergent solution by this method, the addition of the moving-boundary term has

the effect of increasing the potential drop (and so decreasing the current in

the crevice). Figure 7.11 shows a comparison of the potential profiles for a

static-geometry run and a moving-boundary run at a metal potential of — 0*1 V. Figure

7.12 shows the concentration profiles for the same set of parameters. For the

moving-boundary calculation there is a lesser degree of acidification in the crevice

but the [Fe1+] and [Cl~] are higher. A calculation of the corrosion-current

density at the crevice tip for the static- and moving-boundary cases shows that

inclusion of this extra term reduces the current by about a factor of 1 0 to to M x 1 0 4

A m and thus brings the models predictions to a more realistic values.

For metal potentials above 0-1 V, a convergent solution cannot be found by the

current solver method (Appendix 7). An alternative iteration scheme is necessary.

7.6.2 Addition of Ferrous Chloride

The predicted solution compositions in many of the examples given in this chapter

are supersaturated with respect to ferrous chloride. This suggests that the

equilibrium reaction for the precipitation of this solid should be added to the mass

balance equations. However, an attempt to add the reaction

Appendix 6 , i.e. non-convergent, bifurcating solutions for the potential and pH. This

again suggests that a more stable numerical method is necessary for solving

corroding cavity and this is presented in Chapter 8 .

7.6.3 Sensitivity to Diffusion Coefficients in Cavity.

The intrinsic diffusion coefficient is a property of a three-component system: the

diffusing species, the solvent and the porous medium. It is related to the

free-water coefficient, D0, by

where W, the diffusibility, is a property of the porous medium and is generally

independent of the diffusing species. V depends on various geometric factors,

Fe2+ + 2C l’ ^ FeCl2 (7.6.3)

to the steady-state model led to numerical problems similar to those described in

the mass-balance and chemical-equilibria equations describing the solution within a

An = (7.6.4)

149

(7.6.5)W =

where f i is the porosity, <5 the constrictivity and x the tortuosity74. There are little

available empirical data regarding the diffusion coefficients of aqueous species

through saturated solid corrosion products. However, by scaling down the diffusion

coefficients over a range of factors to represent the change of porosity in a

crevice once precipitation occurs, the sensitivity of the predictions to this parameter

may be assessed. Figure 7.13 shows the predicted steady-state corrosion

current density at the base of a cavity with active walls against the diffusion

coefficient of the aqueous species. The metal potential in these calculations is -0.2 V

and the cavity length is 2mm. Clearly, restricting the migration rates of the

aqueous species increases the potential within the cavity and so slows down the

metal dissolution rate. However, it is unlikely that in reality solid corrosion product

would become so tightly packed within the crevice that the diffusion

coefficient would be reduced by four orders of magnitude (which yields a realistic

cavity propagation rate). It must therefore be concluded that the inaccuracies of the

model's predictions are not solely the result of the inaccurate value of this

parameter.

7.7 Summary.

A preliminary steady-state mathematical model of the propagation of localised

corrosion cavities has been developed. This model aims to be self-consistent and use

the minimum empirical data for maximum applicability and flexibility. It is

constructed in a series of stages in order to assess the importance of each aspect of

the physical system to the overall corrosion process. The model predicts the

solution chemistry and electrochemistry within the corrosion cavities as functions of

the many parameters on which these depend. Conclusions that may be drawn

from the results so far are as follows.

Although the preliminary steady-state model makes some reasonable qualitative

predictions of the chemistry and electrochemistry within a localised

corrosion site, the quantitative comparisons with experimental data are less

encouraging. In particular, the corrosion rates of the cavities are predicted as

much as several orders of magnitude too high.

2) Corrosion on the cavity walls in addition to at the base produces lower corrosion

current densities but higher acidity within the crevice.

In most cases, part of the crevice walls near the cavity mouth will be passivated

by a solid film of corrosion product. This suggests that a rectangular

crevice will eventually evolve to a bottle-shaped cavity — with a narrow region

150

close to the mouth and a more even broadening over the rest of the

depth.

4) Inclusion of the precipitation of ferrous hydroxide, where the solubility limits are

exceeded (for metal potentials ^ 0-6 V) reduces the acidity in the crevice

but has little effect on the predicted corrosion-current density.

5) The predicted solutions in many cases are strongly supersaturated with respect

to ferrous chloride. Numerical problems with the current solving method

prevent direct inclusion of this reaction.

6 ) The corrosion-current density is strongly dependent on the diffusion coefficients

within the cavity. There is little empirical data on these, but decreasing

the diffusion coefficient has the effect of reducing the predicted currents.

7) Specific consideration of the effects of the moving crevice walls and base

reduces the predicted currents by an order of magnitude, although this method

has only been so far successful over a limited range of potentials ( 0 - 1

V).

Numerical restrictions with the model have limited extension and improvement to

the accuracy of the model's predictions. In Chapter 8 , a new solving

technique is described and the effects outlined above are added. The results from the

improved model are more extensively tested against empirical data.

Figure 7.1 Schematic illustration of the processes included in the cavity propagation

model.

152

log1

0 co

nce

ntr

ati

on

.

2 r

c/>M = - 0 * 2 V , l = 2 m m t [ C f ] Bulk=10 ' 3 M

Figure 7.2 Concentration profiles along the cavity length for a crevice with passive

walls.

Po

ten

tia

l d

rop

a

lon

g

pit

le

ng

th,

( K r

Figure 7.3 Electrostatic potential drop along the cavity length for a crevice with

passive walls at various metal potentials.

Figure 7.4 Variation of corrosion current with crevice length for a crevice with

passive walls.

log1

0 co

nce

nt

rat

ion

,

2-5

0

-2-5 — 1.4

FeOH*

H'-■ -I

- 5 0

-7-5

Na+

1 0 °0------------- CK3------------0^6------------<k------------ V2------------ 1*5Distance from c a v i t y t ip, mm

<*>M = -0'2V,l = 2mm,tCriBulk = 10*3 M

10 2 0

Figure 7.5 Concentration profiles along the cavity length tor a crevice with active

walls.

Pot

enti

al

drop

al

ong

pit

len

gth

.

0-35

Figure 7.6 Comparison of the potential drops along the crevice length for corroding

and non-corroding walls.

corr

osi

on

c

urr

en

t .

Am

6*5

cvi6 0 -

5 5 -

5 0 -

/»• 5-

4-0-

3-5-

3 0 -0

^ M = - 0 - 2 V . l = 2 m m . [ c n Bu|ks 2 x 1 0 “ 2 M

Figure 7.7 Corrosion current against cavity length for a crevice with- active walls.

Iog1

q

con

cen

tra

t io

n .

<#>M = - 0 - 2 V , l = 2 m m . [ c r ] eulk= 2 * 1 0 "2 M

Figure 7.8 Concentration profiles along the cavity length for a crevice with active

walls (with a precipitation reaction included but no change to diffusion coefficients).

60

8

£< 50

I

c<u

3 AO u

20

10

\\\\\\\\\

\

0 I--------------------------- L-00 2-0

P= 8-35T0-46

P= Pit depth, mm T=Time, years

AO 6-0 8-0 100Crevice length.mm

Figure 7.9 Experimental variation of corrosion current density with crevice length5.

t

Figure 7.10 Comparison of Turnbull and Thomas's experimental and theoretical

results of the variation of pH within a crevice with metal potential and the

predictions of the present model.

Potent ial d r o p in c r e v i c e , V

0 M=-O- 1 V, T= 25 °C . I = 2 mm • [ C n = 2 x 1 0 “ 2 M

Figure 7.11 Comparison of potential drops along crevice length for calculations

carried out in a static and moving geometry.

(co

nce

ntr

ati

on

M

)

0 1o

- 5 0

7-5

H* moving

H+ sialic

i

-10 ol—0 0

I0-3

i___________ i--------------- 1--------------------- <—0-6 0-9 1*2 1-5

Distance from crack t ip , mm

1 -8 ' 2-0

0 M=-O-1V T = 25°C , l = 2mm , [Cl"]= 2 x10-2 M

Figure 7.12 Comparison of predicted concentration profiles in a crevice with a static

and non-static geometry.

Co

rro

sio

n

curr

ent

x 10

A

m

Figure 7.13 Corrosion current against diffusion coefficient of Fe2+, FeOH+ , N a+

and C /“ for a crevice with active walls.

8. A Finite—Element Model of the Propagation of Localised Corrosion Cavities.

8.1 Introduction.

In Chapter 7, a mathematical model of the steady-state propagation of pits and

crevices was developed. This model gave some reasonable qualitative agreements to

experimental data but less accurate quantitative results. The extension to a

wider range of environmental regimes was limited by analytical and numerical

restrictions in the mathematical solving method. In this chapter, a new method of

modelling corrosion-cavity propagation is developed. This method involves dividing

the cavity into a series of finite elements and solving the equations governing

ionic migration and chemical reaction using this grid. It uses the Harwell

finite-element subroutine library TGSL75. This approach allows a more complex

description of the solution chemistry in the crevice and may be applied over a wider

range of physical and chemical conditions than the previous model. It also

allows a relatively straightforward extension to both a two-dimensional geometry (i.e.

a cavity with active walls) and to time-dependent solutions which enables

prediction of the temporal as well as spatial variation of the crevice chemistry.

In section 82, the method of setting up a model with the TGSL subroutine library

is outlined and the model of a propagating cavity is described. In section 8.3,

this model is applied to a number of situations and used to continue the assessment

of the sensitivity of the predictions to the approximations made in this method

of modelling cavity propagation. The finite-element model is further tested against a

number of experimental data.

8.2 A Finite—Element Model of Corrosion-Cavity Propagation.

8.2.1 The TGSL Subroutine Library.

In this section, the structure of a program using the TGSL finite-element

subroutine library is described. The mathematical method of the finite-element

technique is outlined in Appendix 8 . The subroutine library offers considerable

flexibility in constructing packages for solving particular classes of equations. There

are many different facilities for input, calculation of results and processing of

output, each facility requiring different input data. A special input language for TGSL,

TGIN, has been developed76. At any given point in the construction of a

program using TGIN there will be several alternative options available, and these can

be invoked by appropriate commands. Each option may then require data

(different from that of any other option). A particular command may have further

options requiring subcommands. Thus, there is a logical tree-like structure in the

organisation of the TGIN input data. There are four distinct phases in the

construction of a program using TGIN. Briefly these are as follows:

1 . Selecting a package i.e a specific collection of subroutines for solving a

particular type of equation;

2. Defining the model i.e. the geometry, boundary conditions and information

about the size and type of grid;

3. Invoking the solver i.e. to specify the way in which the problem is to be solved

and specifying the physical equations and parameters;

4. Processing the results i.e. specifying the type and form of output required.

8.2.2 Model of Cavity Propagation using TGIN

For the models of cavity propagation, a collection of subroutines from the TGSL

finite-element library called CAMLE (Corrosion And Migration in Localised

Environments) has been constructed. This package solves the mass-transport and

chemical-reaction equations describing the environment within an active crevice. The

cavity is represented as a rectangle of rectangular elements. Initially, a grid 30

elements long by 1 element wide is generated. A grading factor on the size of these

elements is imposed so they are narrowest at the cavity mouth and increase

in width towards the corroding end. This is necessary since the original model in

Chapter 7 suggests that in the steady state the concentrations of the various ionic

species and the electrostatic potential vary rapidly over a small distance close to the

crevice mouth, and change more slowly along the rest of the length. The

precise degree of grading was chosen by comparing the results with those from a run

of the steady-state model with identical input parameters and obtaining the

best agreement.

The boundary conditions for the crevice with passive walls (i.e. the

one-dimensional problem) consist of fluxes of the various ions involved in the

electrode reactions at the corroding end of the grid (for example, the equations for

the boundary conditions given in Appendix 3). These fluxes are specified

directly to the program in terms of the potential drop down the cavity. The numerical

solving-method used in this package is able to iterate for this quantity in a

much more efficient and stable way than the 'guess and iterate' technique developed

in Chapter 7. The boundary conditions at the other end of the grid

representing the cavity mouth are set as fixed concentrations equal to the bulk

values. For the full two-dimensional problem (i.e. the crevice with active walls), a flux

of species along both sides of the grid is included, in addition to the flux at

one end.

The third stage in setting up this finite-element model involves writing two

subroutines that specify the mass conservation equations. In the first of these, the

sets of equations for the migration and reaction of each ionic species and the

charge-neutrality equation (such as those given in Appendix 3) are supplied. The

second subroutine specifies the Jacobian of the solver matrix; the Jacobian is derived

by analytic differentiation of each mass-conservation equation with respect to

each variables. Initially, the program is set up to obtain a steady-state solution for the

crevice chemistry, but in some applications of the program reported in section

8.3 an alternative solving technique is chosen which yields the time evolution of the

chemistry and electrochemistry.

The output from the model consists of profiles of ionic concentrations and

electrostatic potential along the cavity.

8.3 Applications of the Finite-Element Model.

8.3.1 Addition of Ferrous Chloride.

The predicted solution compositions in many of the examples given in Chapter 7

were supersaturated with respect to ferrous chloride. The original steady-state

model could not be extended to include the precipitation reaction,

Fe2+ + 2Cl~ ^ FeCl2

due to restrictions with the numerical solving technique. However, this reaction has

been added to the finite-element model and is now used to investigate the

effect on the predicted solution chemistry and corrosion currents. The equations

governing the steady-state chemistry and electrochemistry within the crevice with

precipitation of both ferrous hydroxide and ferrous chloride are given in Appendix 9.

The assumptions in the calculation are as follows:

Passive walls

Metal potential

Bulk pH

Bulk Cl"

Crevice length

Solids

-0.2 V

6.13

0.57 M

2 - 1 0 mm

Ferrous hydroxide

Ferrous chloride

Results of calculations with and without precipitation of ferrous chloride are

compared at different lengths of cavity. (The diffusion coefficients in these examples

are not scaled down to account for the volume occupied by the solids). The

167

model indicates a lower chloride ion concentration in the cavity when precipitation is

included, and a higher pH; for a crevice of length 2mm, the pH at the tip is

increased from 4.2 to 6.4 and the chloride ion concentration is reduced from 425 x

103 M to a much more realistic 6.23 M. There is also a dramatic effect on the

predicted corrosion current density; figure 8 . 1 shows a reduction of about two orders

of magnitude. Precipitating ferrous chloride in the model effectively reduces

the amount of chloride ions in solution, thus increasing the solution resistivity and

the potential drop down the cavity. The comparison suggests that a realistic

description of the solution chemistry within a corroding crevice is very important for

the accuracy of the predictions, and in particular that the precipitation of

ferrous chloride has a dramatic effect in reducing the propagation rate of the cavity.

8.3.2 Comparison of Finite-Element Model with Experiment.

The model is first tested against the experimental data for pit propagation rates in

carbon steel given in section 7.5s. This experiment gave the following

relationship between pit depth and time

P = 8.35 7J46,

with P measured in millimetres and Tm in years. The original model, with only

ferrous hydroxide present as a solid phase in the cavity, overestimated the

propagation rates by several orders of magnitude. However, when ferrous chloride is

added (in the steady-state, active-wall model), the corrosion currents are

reduced to within an order of magnitude of the experimental values, as shown in the

plot of corrosion current density against pit length in figure 8.2. The

parameters used in this comparison are as follows,

Active walls

Metal potential

Bulk pH

Bulk Cl"

Crevice length

Crevice width

Solids

-0.2 V

6.13

0.57 M

2 - 1 0 mm

2 x 1 0 ‘ 5m

Ferrous hydroxide

Ferrous chloride

In this case, the predicted current is actually less than the measured value. This is

probably because the experiment has been carried out at 90° C whereas the

thermodynamic data within the model is only strictly applicable to 25° C. This

comparison again underlines the point that an adequate description of the solution

168

The steady-state finite-element model is next tested against some experimental

measurements of pH and potential in a simulated crevice in carbon steel by

Alvali and Cottis77. The experiment consisted of a crevice formed between a plate of

carbon steel and an acrylic block. The crevice was filled with a 0.6M NaCI

solution. One set of tests were performed under potentiostatic control with the metal

polarised to a potential -0.5 V SCE. The pH and potential were measured at

different points of the crevice over 70 hours. The measured potential showed a rapid

drop of up to 150 mV in the first few hours and then gradually increased about

20 mV over the remaining time (i.e. the potential drop in the crevice decreased with

time). The potential drop along the crevice at 70 hours is summarised in

figure 8.3. The parameters in the steady-state model were assigned the following

values:

Active walls

Metal potential -0 .5 V

chemistry is essential to accurate predictions with this model. Also, the rate of

metal dissolution inside a cavity with corroding walls is considerably less than that in

a passive-wall crevice or pit.

The predicted potential drop along the pit is also shown in figure 8.3. The general

shape of the profile is very similar to the experimental curve, but the potential

drop is a little low. This may perhaps partly be accounted for by the fact that the

experimental crevice solution had not reached its steady state at 70 hours.

In the experiment, the measurements of pH in the artificial crevice showed some

interesting behaviour; the pH reached a maximum approximately 20 mm into

the crevice and subsequently fell at greater distances from the mouth. At 70 hours,

the pH varied between about 4.9 and 6.5 along the length. At steady state, the

model predicts that ferrous chloride will be present along the entire crevice. However,

the initial solution in the experiment is not supersaturated, so during the

evolution of the crevice solution both ferrous hydroxide and ferrous chloride may

precipitate at various stages. The non-uniform pH profile at 70 hours may then

perhaps be accounted for by an uneven distribution of solid corrosion products within

the crevice.

Bulk pH

Bulk Cl"

8.0 0.6 M

8 mmCrevice length

Crevice width

Solids

10~4 m

Ferrous hydroxide

Ferrous chloride

The final test of the finite-element model involves a time-dependent description of

the solution chemistry and electrochemistry. This model is tested against

some experimental data of Beavers and Thompson73 which demonstrated the effect

of active cavity walls on pit propagation rates in carbon steel. In his

experiments, two simulated pits were constructed of identical shapes, but one with

electrically insulated walls. The 'pits' were prepacked with an acidified paste

of Fe30 4 and 0.1 M H C l to simulate the conditions found in an established pit in

carbon steel exposed to a basaltic groundwater. The currents and potentials at the

base of the pits were measured in both cases as a function of time and it was

found that the dissolution rate of the cavity with passive walls was about two orders

of magnitude greater than the rate for reactive walls. It was also noticed that

in the latter case high rates of attack were confined to the region near the pit mouth.

Figure 8.4 shows the time variation of the corrosion current density in both

cases.

The steady-state finite-element code was run first assuming passive walls to the

crevice and then with active walls, with the following parameters:

Metal potential -0.58 V

The cylindrical artificial pit constructed for the experiment is approximated in the

model as a rectangular slot of width equal to half the diameter. Also, the pH outside

the pit is taken as that of the basaltic groundwater. (It can be shown that the

and is essentially independent of the external pH). The potential over the bulk

metal varied between about -0.56 and -0.62 V during the experiment on the pit with

passive walls. In the case of the pit with active walls, the potential remains

essentially constant at -0.58 V, so for the comparison this value is used. At the bulk

chloride levels used in the experiment, the pit solution should not become

saturated with respect to ferrous chloride, so the model which precipitates ferrous

hydroxide only is used.

In Table 8.1, the experimental current densities at 140 hours are compared with the

steady-state values predicted by the model. The predicted steady-state value

for the passive wall current is about 20 times higher than the measured value at 140

Bulk pH 9.8

0.1 MBulk chloride

Crevice length

Crevice width

Solids

225 x 10'2 m 225 x 10"3 m

Ferrous hydroxide

pH inside the pit very quickly becomes dominated by the solid phases within the pit

170

Experimental

140 Hours

Predicted

Steady-state

Predicted

Steady-state

Predicted

140 Hours

D = 0 ( 1 0 '9 )m 2 s“ ' •D=0 ( 1 (T 10) z>=o(io-10)

Passive 0.4 8.5 2.7 0 . 6 8

Active 0.004 2.4 1 .1 -

Table 8.1 Comparison of experimental values for corrosion current densities (in_O T J

A m ) in artificial pits by Beavers at al. and values from the finite element model.

hours. However, in the experiment the artificial crevice had been packed with

a paste of Fe30 4 which would tend to restrict the diffusion and migration of ions in

solution. The sensitivity of the predictions to this phenomena is tested by

scaling the diffusion coefficients of the aqueous species down by an order of

magnitude. This has the effect of reducing the steady-state current by a factor of

about 3, as shown in Table 8.1. In the case of the crevice with active walls, the

predictions are less accurate for both diffusion coefficients.

The predictions of the time-dependent model of the crevice chemistry are next

compared with the data in figure 8.4 which shows that the current density within a pit

with passive walls decreases to a minimum at about 25 hours and then

steadily increases. Figure 8.5 shows the predicted evolution of the current at the pit

base with time for diffusion coefficients of the order 1 0 - 9 and 1 0 - 1 0 m2 s-1.

Generally, the theoretical values for the current density show a similar shape to the

experimental ones with a rapid drop in current followed by a slow rise. The

figures also indicate that with a lower diffusion coefficient in the crevice, steady-state

conditions take longer to establish and at 140 hours, the value of the

corrosion current is 0.68 A m - 2 compared with 0.4 A m- 2 in the experiment. (Scaling

the diffusion coefficients down by one order of magnitude seems reasonable

as the pit is packed with oxide paste). Thus, the predictions of the time-dependent

model agree well with measurements of a cavity with passive walls.

Observations within a cavity with active walls have been matched only with

steady-state predictions; it may be that chemistry within the cavity is still evolving at

140 hours, leading to the overestimate of the current density.

8.4 Summary

A finite-element model of the solution chemistry and electrochemistry within a

corroding cavity has been developed. This method allows a more complete

description of the chemical equilibria within the crevice or pit than the original model

developed in Chapter 7. The evolution of the corrosion current and of the

concentrations of significant species can be simulated with this model.

The model has been used to test the sensitivity of the predictions to specific

approximations and assumptions. Results have also been compared with

experimental data. The conclusions are:

1) Calculated cavity propagation rates are most sensitive to which chemical

equilibria are included in the model, and to any obstruction to the movement of

species in solution;

2) Cavity propagation rates in good agreement with observed values are obtained

with a model that allows precipitation of both ferrous hydroxide and

ferrous chloride;

Predictions of the evolution of the concentrations of significant species within a

cavity with passive walls, under conditions such that only ferrous

hydroxide precipitates, also agree well with experimental data.

172

log

1 Q (

Co

rro

sio

n

cu

rre

nt

de

ns

ity

) A

m

6.0

CM Without fe r rou s ch lo r id e

5.0

M e ta l p o te n t ia l « - 0 . 2 V

Bulk pH » 6.13

Bulk (C l" ) o 0.67

Figure 8.1 Comparison of variation in predicted corrosion current density with crevice length with and without ferrous chloride present as a solid phase in the

crevice.

0

0

0

0

0

0

Metal potential = - 0-2V Bulk pH = 6-13 Active wallsFerrous chloride and ferrous hydroxide precipitated.

Experiment (P(mm) = 8*35 T0’*6 (years))

P

Theoretical

0 U * 0 6*0Pit depth, mm

8*0 10*0

re 8 . 2 Comparison of predicted corrosion current density with crevice length

finite-element model with ferrous chloride and experimental data of Marsh et al.5.

Pot

entia

l dr

op,

mV

Figure 8.3 Comparison of predicted steady-state potential drop in a crevice with

experiment of Alvali and Cottis77.

-40 \

Time L. hours

Figure 8.4 Experimentally determined conosion current densities at base of an

artificial crevice with passive and active walls by Beavers and Thompson .

log

10 (

co

rro

sio

n

curr

ent

de

nsi

ty)

Am

CNII

Figure 8.5 Comparison of predicted corrosion current density from time-dependent

finite-element model and data of Beavers.

9. Conclusions and Future Work.

The aim of this thesis has been to develop a number of models of the various

stages of pitting and crevice corrosion, both to improve the understanding of the

physical mechanisms underlying these phenomena and as an aid to predicting the

occurrence and extent of these forms of localised corrosion. These models

are of three types;

Initiation models for predicting when localised corrosion will occur in a

particular environment of the metal;

Initiation models which from a more microscopic perspective aim to investigate

the underlying dynamics of the initiation phase;

Propagation models which seek to predict the rate of enlargement of localised

corrosion cavities.

In this thesis, these models have principally been applied to localised corrosion in

iron and steel. However, most of the methods are with appropriate data

applicable to other materials.

(D

• (2)

(3)

The conclusions from each of these sections are summarised below.

(1) In Chapter 3, some fundamental environmental limitations to the occurence of

localised corrosion were identified and a number of models of various

changes in solution chemistry surrounding a corroding metal were constructed.

These changes were related to subsequent behaviour. Two aspects of

the environment were considered: the concentration of oxygen at the metal

surface (critical to the initiation and persistence of both pitting and crevice

corrosion), and the solution chemistry within a passive corroding crevice (the

key to the breakdown of the passive film and the onset of crevice

corrosion).

The model of the oxygen supply to a metal embedded in a porous medium

predicts the period in which localised corrosion is possible on the metal

and how this period depends on the oxygen content of the system and the

transport properties and thickness of the medium. The model is based on

simple mechanistic arguments. The model must still be validated against

experimental data, but the predictions are potentially very useful especially

when it is necessary to provide a corrosion allowance in deciding metal

thickness.

The model of the solution chemistry within a passively corroding crevice in a

metal ultimately aims to predict when a 'critical crevice solution'

(determined experimentally) is achieved. Given a sufficient supply of oxygen to

178

the bulk metal surface, crevice corrosion may then begin. This model

has generally improved agreement with various experimental data over several

models of similar systems in the literature. However, any such results

rely heavily on empirical data in the form of equilibrium constants for chemical

reactions, so predictions can only ever be as accurate as this data.

(2) Chapter 4 summarises various published models and theories associating

microscopic mechanisms with the onset of localised corrosion. These models,

which aim to interpret the generally unpredictable nature of the initiation

phase, attempt either mechanistic interpretations or data-fitting which generally

involves statistical theory. There is, however, a hint of controversy in the

literature whether the initiation is truly a 'rare event' in the statistical sense. In

Chapter 5, this assumption has been investigated. Recent developments

in the theory of non-linear dynamics have demonstrated the existence of very

complex (so-called chaotic) solutions to relatively simple deterministic

differential equations. In the case of dissipative systems, the solution curves

(i.e. the trajectories) eventually remain confined to a subset of the phase space.

This subset is known as an attractor. Chaotic behaviour in the solutions

occurs when there exists a 'strange' attractor (i.e. one having non-integer

dimension). The apparent randomness arises because trajectories with very

slightly different starting points rapidly diverge.

Seemingly random current records from corroding stainless steel have been

analysed by a novel method by Grassberger and Proccacia46. This

analysis suggests that the process of pit initiation is deterministic, i.e. can be

described by a set of well-behaved differential equations. Further, the

analysis suggests that the dynamical system exhibits a strange attractor

(leading to the chaotic behaviour of the direct measures of the system); and

that this attractor is embedded in a phase space of small dimension i.e. the

system could be modelled by a small set of differential equations. Preliminary

results fix the dimension of the attractor at around 1.9 and the minimum

number of variables needed to model the system at 4, but further applications

of the method are necessary to confirm the accuracy of these values.

The method does not unfortunately identify the variables governing the system,

but by exploiting physical insight, it may be possible to construct a

deterministic model of pit initiation.

The system of equations describing the depassivation and repassivation of a

metal will, however, be considerably more complex than a chaotic

system such as the Lorenz equations, described in Chapter 5. Mass transport of

ions in solution adds spatial variations to the temporal variations

resulting from chemical reaction and dissolution. Simulation therefore requires

partial differential equations rather than ordinary differential equations,

but as yet chaotic behaviour in systems of partial differential equations has not

been examined systematically. It seems likely, however, that the

non-linearities producing chaotic behaviour during pitting corrosion arise from

the dependence of electrode reaction kinetics on the solution chemistry

rather than from mass transport, which will nevertheless clearly complicate the

problem.

In Chapter 6 , a number of published models of the propagation stage of pitting

and crevice corrosion are described and discussed. No single model

includes all the phenomena necessary for a complete description of cavity

propagation or accounts quantitatively or even qualitatively for all relevant

experimental observations. Even the more comprehensive models cannot cope

with all aspects of real corrosion. One of the most common deficiencies

seems to be the poor prediction of cavity propagation rates, i.e. of the corrosion

current within the crevice or pit. It seems likely that these models

incorporate several inadequate approximations. The impact of these

approximations has been investigated in Chapters 7 and 8 using a series of

increasingly complex models of a propagating crevice or pit.

Chapter 7 describes the development of a preliminary steady-state model that

includes ionic transport within the crevice, potential- and

pH-dependent electrode reaction rates, and a limited set of chemical reactions.

The model starts from basic phenomena and relies upon a minimum of

empirical data; it is therefore flexible and of wide application. It is constructed

in stages in order to assess the importance of each separate

phenomena to the overall corrosion process. The model predicts the solution

composition and metal dissolution rate within corrosion cavities, and

how they depend on the various physical parameters. The qualitative

predictions of the chemistry and electrochemistry within a localised corrosion

site are reasonable, but the quantitative comparisons with experimental

data are less satisfactory. In particular, the predicted corrosion rates of the

cavities are as much as several orders of magnitude too high. Preliminary

sensitivity studies suggested the need for accurate specification of the diffusion

coefficients within a cavity containing a solid corrosion product, and that

the description of the solution chemistry within the cavity was inadequate. The

model was enhanced to simulate the effects of a moving crevice base

(for a cavity with passive walls); this change generally reduced the predicted

currents. However, the numerical method employed to solve the moving-base

180

problem could cope with only a limited range of environmental

conditions. This difficulty precluded any extensive sensitivity analysis or any

conclusion as to whether a moving base generally yielded better results.

Chapter 8 describes a more sophisticated method of modelling an active

crevice or pit. This model uses the method of finite elements and allows a more

complex description of the solution chemistry within the crevice, over a

wider range of physical and chemical conditions than the original model.

Time-dependent simulations are also straight forward. The predictions of the

finite-element model have been tested against various experimental

data, and for sensitivity to the various approximations and assumptions. The

results of these investigations may be summarised as follows:

a) The cavity propagation rates are most sensitive to the chemical composition

of the solution within the cavity and to any retention of aqueous species

within the pores and interstices of solid corrosion products such as ferrous

hydroxide and ferrous chloride. These products can markedly influence the

composition of the solution and ionic migration. A realistic description of the

solution chemistry in the crevice is essential for accurate predictions,

especially to the cavity propagation rates.

b) Corrosion of the cavity walls in addition to at the base results in lower

corrosion current densities and hence reduced cavity propagation rates.

c) In most cases, part of the crevice walls near the cavity mouth will be

passivated by a solid film of corrosion product. This observation suggests that a

rectangular crevice will eventually evolve to a bottle-shaped cavity —

with a narrow region close to the mouth and a more even broadening over the

rest of the depth.

d) The predictions of the steady-state finite-element model with precipitation of

both ferrous chloride and hydroxide agrees reasonably well with

measured cavity propagation rates.

e) The time-dependent finite-element model has been tested in a situation in

which ferrous hydroxide only is precipitated and the cavity walls are

passive. The predicted evolution of the chemical composition within the cavity

and of the corrosion currents agrees reasonably well with experimental

data.

One possible further extension of the finite-element model is a

time-dependent simulation of the moving boundary. An enhancement of this

kind certainly improved the steady-state model, as described in Chapter

7. The evolution of the shape of the pit could also be investigated with such a

model.

182

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NomenclatureUnit

a Ionic activity

b Concrete thickness m

C Ionic concentration M

c ° Concentration in bulk solution (outside cavity) M

C(r) Correlation function —

D Fractal dimensionality —

Di Diffusion coefficient of aqueous species i m2 s- 1

Eact Activation potential V

Ecorr Potential governing corrosion cell V

E eq Equilibrium potential for electrochemical reaction V

E° Standard electrode potential V

Ep Pitting potential V

Epass Passivation potential V

f Dimensionless constant —

F Faradays constant of electrolysis C mol- 1

Fi Fraction of leakage current from production of ions of metal i —

8 Dimensionless constant —

i Current density A m- 2

k Leakage current density A m“ 2

h Exchange current density A m - 2

k Flux of aqueous species i m mol dm

k Chemical rate constant

K Chemical equilibrium constant

/

Mi

n

Q

R

R:

s

T

u

U

• v,

v

Yi

V

w

W

•

Drag coefficient

Length of model crevice or pit

Molar fraction of metal i in alloy

Embedding dimension of attractor governing a dynamical

system

Dimensionless concentration of hydroxyl ions

Number of points in time series of variable from a

from a dynamical system

Dimensionless concentration of ferrous ions

Rate of production of metal ion i from alloy dissolution

Dimensionless concentration of ferrous hydroxide

Molar gas constant

Rate of production or depletion of species i by

chemical reaction

Dimensionless concentration of hydrogen ions

Temperature

Fluid velocity

Dimensionless concentration of chloride ions

Ionic mobility

Speed of propagating crack

Velocity of species i

Dimensionless concentration of sodium ions

Width of model crevice or pit

Dimensionless concentration of ferrous chloride

Dimensionless equilibrium constant for reaction k

kg m 3 s 1

m

mol m 3 s 1

J mol- 1 K" 1

mol dm - 3 s" 1

K

m s

m mol J s"

- im s

m s

m

188

<t>

P

P

P

P/

V

Charge number

Capacity factor

Electrochemical partition coefficient

Porosity of porous medium

Constrictivity of porous medium

Tortuosity of porous medium

Overpotential V

Proportion of passive metal surface covered by film -

Electrostatic potential V

Electrostatic potential of corroding metal V

_2Charge density C m

Chemical potential J moP

Electrochemical potential J moP

Control parameter in set of non-linear differential equations -

Dimension of attractor in dynamical system -

Dimensionless potential -

Diffusibility of porous medium -

Pitting induction time s

189

Appendix 1. Calculation of Oxygen Concentration Within a Porous Medium Surrounding a Corroding Metal.

Assuming there is no fluid flow through the porous medium and the system may

be regarded as one-dimensional (figure 3.1), the concentration of oxygen in

the pores, C(x,t), is governed by the diffusion equation,

D d 2C(x,t) _ a s>£(xj l 0 * =s b, (A1a * 2 3 /

where D is the intrinsic diffusion coefficient of oxygen within the medium and a is

the capacity factor. This parameter reflects both the physical retention of the

diffusing ion in the porosity and the bulk chemical retention due to equilibrium

sorption. The thickness of the medium is given by b. The boundary conditions to the

problem may be expressed mathematically as follows:

(1) At time zero (the time at which the corrosion starts), a uniform distribution of

oxygen exists through the concrete pore structure:

C(jc, 0) = C0 (A1.2)

(2) At the metal surface, the flux of oxygen is proportional to the leakage current,

*L:

a c ( M =dx 4 F

(A1.3)

(3) The concentration of oxygen in the pore water at the other end of the concrete

surface remains constant at value.

C ( M = C0 (A1.4)

To solve (A1.1) and boundary conditions (A1.2)-(A1.4), a Laplace transform technique

is used. The Laplace-transform of a function f( t) is defined as

L (f(0 ) = }(P) = J/W e-*<fc0

so

190

u r (o) = p u a o) - m

U A ) = f

where A is a constant. The function /( /) may be recovered from its Laplace transform

by integration:

oo

M = jf(p)epldt

Taking the Laplace Transform of equation (A1.1) gives

= apCiX'P) - C0 (A1.5)OX

and of boundary conditions (A1.3) and (A1.4)

ac(o,P) _ ‘L (A1.6)dx 4Fp

CnC(b,p) = j (A1.7)

The solution of (A1.5) is

C = Ae!X + + -2P

where s = (pa /D )0 5 and A and B are constants. Satisfying boundary conditions

(A1.6) and (A1.7) yields

sinh(s(b—x)) Co AFDps coshsb p

The Laplace transform of the concentration can be inverted numerically by a method

which evaluates the inversion integral along the steepest-descent contour of

the function g(t) = l / f78.

Appendix 2. The Computer Programs PHREEQE and CHEQMATE.

A2.1 The PHREEQE program.

The PHREEQE code17 predicts the equilibrium established on mixing aqueous

solutions or on titrating one solution against another, or the change that results from

the addition of further reactants to an equilibrium solution. The calculated

concentrations of different chemical species are in equilibrium with respect to

specified solid phases. In order to predict equilibrium, the program solves sets of

coupled equations that describe the chemical reactions and impose mass

conservation and electrical neutrality. These equations are solved using a

combination of two techniques, a continued-fraction approach is used for the

mass-balance equations and a modified Newton-Raphson technique for all the other

equations, to yield the following quantities:

The pH;

The pe;

The total concentrations of elements;

The amounts of minerals (or other phases) transferred into or out of the

aqueous phase;

5. The concentration of each aqueous species;

6 . The saturation state of the aqueous phase with respect to specified mineral

phases.

The main assumptions used in setting up these equations are as follows:

1. The total mass of each element of the system (including the amount

dissolving or precipitating as a solid phase) must be conserved;

2. The number of electrons in the system must be conserved. This assumption

enables redox reactions to be included in PHREEQE;

3. The system must be electrically neutral or maintain a constant deviation from

electroneutrality;

4. The amount of solid dissolving or precipitating accords with the solubility

product for that phase.

In setting up these equations the program needs to draw on a large amount of

thermodynamic data relating to the elements which may be present in this system,

192

and the aqueous species and mineral phases which these elements may form.

PHREEQE calculates the equilibrium distribution of aqueous species in a particular

solution using the concept of 'master species'. For each element in the

system, one aqueous species is selected as the 'master species' of that element. All

other aqueous species are then described in terms of mass action involving

the master species and the associated equilibrium constant for that reaction. Taking

sulphur as an example, sulphate S 0 2~ has been chosen as the master

species for this element. Hence the concentration of sulphide S2~ is determined by

the equilibrium constant for the reaction that involves sulphate ions, protons

and electrons:

the equilibrium constants for reactions leading to master species. The choice

maintain consistency throughout the database. The thermodynamic data is exterior to

the code and is stored in a user-supplied database which is read in at the

beginning of each PHREEQE or CHEQMATE run. The database can include up to 27

elements, 250 aqueous species of those elements and 40 minerals. A full

description of the methods used in the PHREEQE code may be found in reference

17.

Correction for Ionic Stength in PHREEQE

The formation constants in the thermodynamic database are appropriate at zero

ionic strength and 25° C. The PHREEQE code converts these into a new set

at the ionic strength of a particular solution by calculating an activity coefficient using

one of three standard approximations: the Davies equation, the

Debye-Huckel equation and the WATEQ Debye-Huckel. These are expressed in

Table A2.1. The latter approximation is used in preference to the standard Debye-

Huckel for only a limited number of ions in the code (Ca2+, M g2+, N a+ , K + ,

Sr2+, C l~ , C 0 2~ , S 0 2~). The Davies equation is most commonly used since it is

applicable over the widest range of ionic strength.

Correction for Temperature.

The equilibrium constants given for the species association reactions and the

mineral dissociation reactions in the database are appropriate at 25 0 C. PHREEQE

includes two choices for calculation of corrections to these constants for

temperature; the Van't Hoff expression or an analytic expression of the form

S O + 8 H + + Se~ ^ S2~ + 4H 20 log ^=20.753

Amounts of minerals are likewise determined by the values of solubility products and

of master species for a particular element is arbitrary. However, it is essential to

193

Approxi- Equation Range of

mation applicability

Davies Io g y= - A z 2( 1+j0S - 0 .3 / ) <0.5 M

/4 = 1 .8 2x l06(£ 7 )°66667

where e is the dielectric constant

Debye- lo g y = - A z 2( i+ B a /0 5 ) <0.1 M

Huckel fl= 5 0 .3 (e 7 )-°5

where a is an ion size parameter

WATEQX° % y = - AZ\ + B a I ^ + b I)

<0.1 M

Debye- b is an additional paremeter which accounts

Huckel for dilution of solvent

y is the single-ion activity coefficient, z is the charge on the ion and / the ionic

strength given by / = 0.5 J Q z ?iTable A2.1 Analytic formulae for activity corrections within PHREEQE code

Log(K) = /4j + A 2 T + A ^/T + A 4logT + A$/T^.

For most species in the database only data for the Van't Hoff correction is available.

A2.2 The CHEQMATE Program-Basic Structure.

The CHEQMATE program simulates one-dimensional diffusion and

electromigration of aqueous species while maintaining chemical equilibrium . Within

CHEQMATE, the PHREEQE program repeatedly recalculates the equilibrium

composition of the solution and any dissolution or precipitation of mineral phases16.

The ionic transport part of the code involves solution of the set of

mass-conservation equations for each aqueous species i. This is given by the

following:

where C, is the concentration of species /, is the apparent diffusion coefficient, z,

is the charge number and (p is the electrostatic potential in solution. is the

mobility given by the expression

The first term on the right-hand side of (A2.1) describes the rate of transport of ions

by diffusion under concentration gradients. The second term represents the

migration of charged species in response to electrostatic potential gradients present

in the system and represents the rate of production or depletion of species

/ by chemical reaction. However, it is assumed in the model that chemical

equilibration occurs much faster than ionic migration. Most chemical reactions, with

the exception of certain mineral precipitation reactions, will achieve

equilibrium in times much less than those characteristic of ionic migration. The

migration part of the mass-balance equations are therefore solved without the

chemical reaction terms, but at each timestep the solution is re-equilibrated. To solve

the set of migration equations, the system is first divided into discrete cells. At

present these cells are of equal length, but it would be possible to use an uneven

grading, for example for regions of differing transport properties. The concentration

of each species is stored in a two-dimensional array, C( /, / ) , with I denoting

the PHREEQE species identification number and J the cell number. The initial

equilibrium chemistry with particular mineral systems specified is determined by one

or more calls to PHREEQE. The main program assigns the equilibrated

solutions to the approbate parts of the grid. In reality, diffusion and electromigration

of ionic species act to smooth out any subsequent chemical discontinuities.

The program models this by effectively 'opening' the boundaries between each cell

in the grid during each timestep and allowing an amount of each species,

calculated from the finite-difference representation of the migration equation, to flow

across the boundary. The total changes in concentration of each species in

each cell from movement across the boundaries are stored in an array DC{ /, J).The chemistry of each cell is perturbed from equilibrium by the mass-transport step

and the main program then calls PHREEQE as a subroutine to re-equilibrate

(A2.1)

(A2.2)

the solution in each cell. This equilibration involves the option in PHREEQE that

allows the addition of further reactants to an equilibrium solution. The C( /, J)

reactants and the equilibrated C( /, J) values are the starting solution for the

next timestep. Migration and equiibration steps are repeated in turn.

The electrostatic potential, satisfies Poisson's equation,

where p is the charge density and e the permittivity of the electrolyte. However,

CHEQMATE maintains local electroneutrality because any imbalances in charge level

themselves much faster than variations in concentration. This condition is

imposed by fixing the net current flowing into each cell. In most systems this net

current is zero but the code can be applied where there is an overall current flowing,

for example in the electrolyte within a localised corrosion cavity in a metal.

However, charge neutrality is still maintained over the timescales of ion migration.

Hence, the local potential gradients across the boundaries between the cells

are calculated such that the electric current from the migration and diffusion exactly

balances the current through the system. The electric current may be written

as,

Thus at each timestep, the potential gradient is calculated using the set of ionic

concentrations from the last PHREEQE equilibration, this gradient is then adopted in

the finite-difference representation of the migration equations for the transport

step.

A2.3 Boundary Conditions.

The concentrations at the ends of the grid may be kept constant to represent some

bulk solution that is unaffected by the evolving chemistry within the grid.

Alternatively, sources and sinks of ions may be included at the ends or indeed at any

part of the grid, to represent electrochemical reactions, for example.

(Electrochemical reactions are discussed further in section A2.6) At the start of each

timestep, appropriate amounts of ions are added to or removed from

values describe the initial solution chemistry, the values of DC( /, J) the additional

(A2.3)

(A2.4)

Rearranging for the potential gradient,

v<p = ( - / - RT^Z^VC,) / F^zjD.C,. (A2.5)

196

particular cells. The local chemistry may thus be perturbed far from equilibrium so an

additional equilibration is required here or otherwise ionic fluxes calculated in

the migration step may be unrealistically high. This step is particularly important

when precipitation restricts to low levels the concentrations of ions supplied by

electrochemical reactions. This additional call to PHREEQE for equilibration before

the migration step reemphasises that chemical reactions occur much faster

than migration. The migration step then begins with the calculation of the potential

gradient across each cell boundary, using the concentrations returned from

this additional call to PHREEQE. The basic program structure is illustrated in the flow

chart in figure 3.7.

A2.4 Maintainence of Electroneutrality in CHEQMATE.

It is important to stress again how the program maintains electroneutrality in the

system. Charge neutrality is ensured at two points in the program. Firstly, the

solution in each cell at the end of each transport step is electrically neutral (through

the process described in the previous section), although no longer in chemical

equilibrium. Secondly, charge neutrality is maintained during the equilibration step in

PHREEQE.

A2.5 Minerals accounting.

As mentioned previously, the program PHREEQE can calculate the composition of

an aqueous solution in equilibrium with solid phases and also yields the

amounts of solid precipitating or dissolving in order to achieve this equilibrium.

However, a limitation in the original code is the assumption that all mineral phases

are present in infinite quantities and thus a dissolving mineral is never

exhausted. Similarly, during the course of a reaction the solution may become

supersaturated with respect to a certain mineral and PHREEQE does not

automatically allow precipitation. These details have been amended in the program

CHEQMATE to provide a facility for monitoring both the precipitation and

dissolution of specified mineral phases throughout the grid. At each timestep, the

program checks for supersaturation of the solution with respect to a list of specified

minerals. If this test indicates that the solution is supersaturated with respect

to one (or more) of these, the equilibrium composition is recalculated, including

those minerals as solid phases. The amounts of all solid phases are finite, specified

by the amounts present at the beginning of the run (if any) modified by the

amounts precipitating or dissolving during previous timesteps. If, during the course of

any equilibration, the amounts of a given solid dissolving are such that the

supply would be exhausted, this equilibration step is recommenced, adding the

residual amount of that mineral as additional aqueous species.

197

A2.6 Electrochemical Reactions.

There are two basic types of electrochemical reactions; those that generate

electrons (anodic or oxidation reactions) and those that consume electrons (cathodic

or reduction reactions). In the case of corrosion of a metal, the dominant

anodic reaction is taken as

If the corrosion is uniform, then all the electrons generated by the metal dissolution

reaction are consumed by a number of cathodic reactions which occur at

essentially the same places, and there is no net current flow. However, in the case of

localised corrosion (where corrosion occurs in isolated points on the surface)

the sites of anodic and cathodic reaction are separated and a current flows between

them. For example, in the case of a corroding crevice in a metal, the anodic

reactions occur on the walls and at the base of the cavity and the cathodic reduction

reactions occur principally on the metal surface outside. A current then flows

between the two sites.

In ordinary chemical kinetics, the rate of an endothermic reaction is related to the

height of the energy barrier, Ea:

The rate of an electrochemical reaction requires an additional term to reflect the

additional electrostatic barrier:

where / is a function of the potential, V, of the charge number, rt, and of Faraday's

constant, F. In CHEQMATE, electrochemical reactions serve as sources and

Me ^ Mez+ + ze . (A2.6)

k = A e x p (-E jR T ). (A2.7)

k = A e xp (-(E a + f(VnF))/RT), (A2.8)

sinks of the various ions involved in the corrosion reactions (e.g. ferrous and hydroxyl

ions) at appropriate parts of the grid. Empirical values for the rates of

production of these ions are used.

198

Appendix 3. Mass—Transport Equations and Boundary Conditions for Cavity-Propagation Model.

In dilute-solution theory, the transport of aqueous species i is governed by the

mass-balance equation describing diffusion under concentration gradients,

electromigration under potential gradients and chemical reaction:

a t D,V2C, + ZjUiF V(CjV<p) + Ri (A3.1)

where C, represents the concentration of species /, /?, represents the rate of

production or depletion of species / by chemical reaction and £7 is the mobility,

given by the expression

U = — - .1 RT

V(j> is the potential gradient in the crevice. The concentrations of the species are

denoted as follows

# [Fe2+] = C ,, [FeOH + ] = C2, [ C T ] = C3, [Na + ] = C4, [ t f+] = C5 and

[O H '] = C6.

A3.1 Passive Crevice Walls with no Precipitation.

Assuming passive crevice walls, the steady-state mass-transport equations (from

equation (7.2.14)) for each species are,

D x(d 2Cx

dx2(A3.2)

d2C2

dx2 + + * lfC | “ k 'BC2CS = 0(A3.3)

D _ F d d(/>\\ _ q dx2 R T d x( C id x ” 0

(A3.4)

D |- ^ ^ ( c _ o ° * { dx2 R T d x ^ d x ” U

(A3.5)

199

05 ( ^ 2 ' d x ^ 5~dx^ ^ ^ l F ■*■ k2F ^2B ^sQ — ®

(A3.6)

- S r a (c < S » + * * - i =*c >c - - 0 IA3JI

where k 1F,klB,k2F,k2B are the forward and backward rate constants. The seventh

equation is the equation of charge neutrality

Z i 'C 'ix ) = 0 (A3.8)

for all x in the crevice.

The boundary conditions of the problem are as follows;

(1) The concentrations of the species are fixed at the cavity mouth and are equal to

the values in the bulk solution outside the corrosion site.

c, = C2 = 0 , C3 = C f, C4 = q \ C5 = C5X, C6 = k jC f (A3.9>

where C f — CT + C? — k2/C$ = 0 for charge neutrality.

(2) The flux of species involved in the electrode processes (i.e. Fe ,H and OH ) is

proportional to the corresponding current at the cavity tip. The flux of the

other species at the cavity tip is zero.

D (— ?■ + ——(C — )) = 0 U l( dx + R T(C2d x )] U

D i — F (C = 0 U i( dx R T ^ d x ’ ’ °

D ■ < £ + A<C>S» - 0

D ( ^ S l + J L (C d±X) = ,• s( dx + R T ^ d x " 03 F

exp{a2F(<j>M - <t>)/RT)

D „ ( ^ ~ J f ( C6 ^ ) ) = ^fexp(o-2 F ( ^ - (A3.10)

Rearrangement of Equations and Boundary Conditions.

Equations (A3.2) - (A3.7) are cast into dimensionless form using the following

variables, following Turnbull60

(pM is the metal potential and (p the potential at the crevice tip, where the electrode

reactions are taking place.

D , C , d 7 c ? C 5 d 6 c 6p - 1 \ Q - , S ■, N

D5Cs DSC? r x D5CSD,C ,

u - 3 3 , v = D* c\ =F<p

X = 7-d 5c ; D 5 C f R T ’ l

Adding and subtracting various combinations of (A3.2) - (A3.7) to remove the

reactions terms gives

d X i

d2pd X 2 ^ ^ + 2 ) g > - 0

(A3.11)

d 2QdX 2 - 0 t S « s + " - e ) 2 l | - 0

(A3.12)

*|<

5

c: II o (A3.13)

X X + = 0

ÂT2 d X }(A3.14)

§0. — w p Wi

(A3.15)

Co II (A3.16)

aP + bQ + cS + dN + eU + JV = 0 (A3.17)

where

w, =K t D3

, W, ='2^5

k 2d 6

'2^5

2D, Dca = b = — , c = 1 , d =

D i D,z B iD.

, e = A / -D, Da

The boundary conditions, (A3.9) and (A3.10) transform to the following.

At the crevice mouth, X = 0,

(A3.18)P = Q = 0, U = U ° ° , V = 5 = 1 , N = W 2, I// = 0

At the crevice tip, AT = 1,

§ + + ( 2 p + Q I x ) = * i exP( - * i * ’)

7 j c ~ d X ~ 7 ^ + (S + N ~ = «2exP(_Q2 + S3êxp(-a3V>)

dU _ rjd(pdX dX

dV d£dX dX

0

0

(A3.19)

where

gi = f'o 1 M exp (Q-, T)/2FDS C f

g2 = i02lAexp{a2F<pM/RT)/FDs C*

S3 = i03lAexP (a3F(t>JR T )/FDs

with / l = 1 0 ~ 3 to convert concentrations to moles m .

Integration of (A3.11)—(A3.14) and application of boundary conditions (A3.19) yields

P' + Q' + (2 P + Q W = gi«xp(-ar,v»(l)) (A3.20)

S’ - Q' - N' + (S + N - Q W = g2exp(-02^(l)) + g35(l)cA:p(-a'3^(l))

(A3.21)

U' - U q ' = 0 (A3.22)

V' + Vi//' = 0 (A3.23)

where ' represents dX

Active Cavity Walls with no Precipitation

The electrode processes are assumed to occur both on the crevice walls and at the

crevice tip. An approximation used by Turnbull59 which averages the

contributions from the walls across the width, assuming a uniform transverse profile

is employed here. The potential profile across the width of the cavity is also

assumed uniform. Equations (A3.2), (A3.6) and (A3.7) are replaced by

202

£>.(d2Ci

dx2+

R T d x K 1 d x )} k \F ^ l "*■ ^■lfl^'2^'5 — exp(otxF(<pM - <t>/RT)

(A3.24)

„ , d lc $ + F d (C5% ) + kR Tdx 3 dx IF C\ k lB C2 C5 + k2F k1B C5 Ce

2*03wF

C5exp(a3F((pM - (j>/RT) (A3.25)

A>(0 JC2

F d R T dx

(C6 g ) ) + k2F - k2Bc 5Q

= -^p‘ 02exP(a2F(4>M ~ <I>)/RT) (A3.26)

Casting these equations into dimensionless form, rearranging and integrating as

before gives

i

P' + Q ’ + (2 P + Q W = g \e xp (-a x\l){ 1)) + J / 1 e * p ( - t f 1 T/'(A'))dAr (A3.27)*

5' - Q ' - N ' + (S + N - Q )y ' = g2exp(-(xzxp(l)) + g3S (l)e x p (-a 3y ( l) )

i i

+ { f 2exP ( - ° i2 'P(x ))d x + f f }S(X)exp(-tr}t/>(X))dX (A328)X X

where

f i = 'o i PAexpia, F4>jRT)/wFDs C 5”

/2 = 2i02l2AexP(a2F<t>M/R T)/wFD5C5

h = 2j0 3 l 2Aexp(a} F<f>M/RT)/FD5

Passive Cavity Walls with Precipitation of Ferrous Hydroxide.

The mass-balance equations for a system with passive walls and the additional

reaction of the precipitation of ferrous hydroxide are now derived. This effectively

adds the constraint

c2 = k 3c 5 (A3.29)

to the system where K 3 is the equilibrium constant of the reaction

FeOH+ + H 2O ^F e(O H )2 + H +.

Transforming the seven coupled equations to dimensionless coordinates, rearranging

and integrating as before gives

S' + Q ' + 2P' - N ' + (S + Q + 4P + N)ty' = -2 g xexp(-ocx V>(1))

+ g2 exp(-afeV'O)) + g3S (\)e xp {-a 3t^ ( l)) (A3.30)

LV - Uv>' = 0 (A3.31)

V + v y = 0 (A3.32)

(A3.33)

SN = W2 (A3.34)

Q = W3S (A3.35)

aP + bQ + cS + dT + eU + jV = 0 (A3.36)

Active Cavity Walls with Precipitation

The mass-transport equations are the same as (A3.30)-(A3.31) but in this case the

right hand side of (A3.30) is replaced by

-2 g ie x p (-a riV (l) ) + g2 exp(-ar2 *//(!)) + g} S(l)exp(-<x3f ( l ) )

1

+ J [ - 2 / i ^ / 7 (-a r 1 t/;(l)) + f 2e x p (-a 2\p(l)) + f 3S (X )exp(-a3\p(\))\dX (A3.37)

The boundary conditions for both the passive and active wall cases are

At the crevice mouth, X = 0

S = 1 , N = W2, Q = W3, P = W j/W j, U = U x

V = (aP+bQ +cS+dN+eU)/f, y> = 0

thus ensuring chemical equilibrium and charge neutrality at this point.

(A3.38)

204

Appendix 4. Solution of the Mass—Transport Equations.

The various systems of coupled differential equations decribing the chemistry and

electrochemistry within a corrosion crevice with either active or passive walls

are highly non-linear in nature and an analytic solution would be extremely difficult to

obtain. However, a form suitable for numerical integration may be derived by

rearranging the equations so that

P' =

Q ’ = F2 ( P ,e ,5 ,N , f / ,V ^ X ^ ( l) ,5 ( l ) ) etc.

where Fx ,F2 etc are algebraic functions of the variables P, Q , 5, N , (/, V and ip. These functions also involve the electrostatic potential, t//(l), and pH, 5(1), at

the crevice base. Since the values of ip( 1) and 5(1) are unknown, the equations are

solved initially by making reasonable estimates at these quantities, after

calculating the various parameters dependent on them. If resultant pH and potential

distributions differ from the estimates, then the calculated tp(l) and 5(1) ( or

the average of the new and old values for faster convergence ) are used as the

starting point at the next iteration. This process is repeated until convergence is

reached subject to some specified error limit. Numerical integration is by Gear's

method, which adopts a variable-order backward-differentiation formula. The stepsize

is chosen automatically, the order of integration selected to best effect in the

range 1 to 5. Gear's method is particularly well suited to this problem since the

variables vary rapidly over a very short distance near the pit mouth and the functions

must be evaluated vary many times.

The system of equations describing the case of active crevice walls is treated in a

similar way except that an initial guess for ip and 5 has to be made over the

whole range 0«sA=sl, since the derivatives P ', Q' etc. are now functions of integrals

of functions of ip and 5. The program iterates for ip and 5 in the same way as

before, but at each step has to make use of their entire profiles rather than just the

values at the crevice tip. Such profiles are defined by numerical interpolation

based upon 10 recorded values of ip and 5 at points graded toward the crevice

mouth where these quantities vary most significantly. The interpolation provides

values of ip and 5 at the large number of points needed for numerical evaluation of

those integrals of functions of ip and 5 that appear in the definitions of P ', Q' etc. Since each iteration involves extensive numerical interpolation and integration,

computations of this kind are much more demanding: some use over 50 times

more computer time than calculations that involve passive walls. The convergence

time is reduced somewhat by improving the initial estimate of ip(X) and

205

S(X), although at high metal potentials 0.4 V) not particularly by improving the

5(A") guess. (This is due to the cathodic current generated inside the pit

being very small in these cases, so there is little contribution from the term involving

The computer time is reduced considerably using a method which approximates

the integrals on the right-hand side of (A3.25) and (A3.26). This makes use of

the observation that the potential and pH profiles are extremely flat over most of the

crevice length, with the values changing rapidly to the bulk values in a narrow

region near the cavity mouth. The integrals may be approximated as follows:

The runs converge as quickly as the initial passive-wall calculations and agree quite

well with the full solution. Figure A4.1 shows the solution chemistry obtained

using this approximation. The parameters used are a metal potential of —02 V,

from the full solution indicates a very close agreement near the crevice tip with the

largest deviations occurring near A!' = 0 (the crevice mouth). However these

discrepancies are only of the order of a few per cent. Figure A4.2 shows a

comparison of the potential profiles from the full and the approximate solutions. At

the crevice tip the difference is less than 1 %, but results in a difference in

corrosion currents of about 1 1 %. This loss of accuracy must be weighed against

faster convergence speed. The approximate solutions for ip and S are also useful as

initial iteration estimates in the full calculation, thus reducing the run time by

over 50 %.

The convergence criteria used for the ip and S iterations are, for passive wall runs,

convergence if

S(X)).

J e x p ^ -a ^ X ^ d X - exp(-ax y ( l) ) (1—AT>; (A4.1)x

J S (X )exp (-ax ip(X))dX = 5 ( l ) « p ( - a , t/>(l)) (1— AT). (A42)X

_ocrevice length 2mm, crevice width 10 /xm, bulk chloride concentration 10 M, bulk pH

7 and a temperature of 25° C. Comparison with figure 7.5 showing results

and

IV/h -iGO - fk(J)\ < 10 >*(/)

&+.W - sk(.r>I < w - ' 2sk( j ) (A4.4)

(A4.3)

for 7=1 to 1 0 , and for active walls

IVk+iCO “ VkCOl < 10 3% V ) (A4.5)

206

and \SM V) - sk(J)| < 10~4s k(j) (A4.6)

where tyk(J) is the value of t// at the 7th grid point evaluated at the kth iteration. The

convergence conditions are not so stringent for the active wall runs since the

computation time is so much greater.

207

conc

entr

atio

n .

0

2 1

FeOH4

- 2 5 — ,, 4HN

5 0

O)o i

- 7 5

Na4

- 1000 0 3 0- 6 0 - 9 1-2 1-5

Distance from crack t ip , mm1 8 2*0

«#»M = -0-2V,l=2mm.[Cnaulka10"s3 MFigure A4.1 Concentration profiles along the cavity length for a crevice with

corroding walls calculated using an approximation technique.

Pot

entia

l dr

op

alon

g p

it,

0 15

0 10 -

---------Approximate solution0 0 5 - --------- pun solution

0 _________ I_________ I_________ I______ :__ I__________L-0 0-3 0-6 0 9 1*2 1*5

Distance from crack t ip , mm

4>m = -0-2V,U mm,[Cr]BuL!< = 1CT3 MFigure A4.2 Comparison of potential drops along cavity length for a crevice with

conoding walls using the approximation technique and the full solution.

I1-8 20

Appendix 5. Moving Boundary Equations.

A preliminary attempt has been made to include an account of the effects of the

changing geometry of the crevice with time. We have considered a crevice

propagating in one direction only. The precipitation of solid ferrous hydroxide is

included. The crevice length is parameterised by the variable x where 0< x < l. For

this calculation we introduce the dimensionless variable X = - where 0<Ar< l .

Transforming equation (7.2.14) into this coordinate system yields

D^ — r + ^ A ( c , | ^ ) ) + iv x + i 2r , =SX2 R T d X '' ’ d X ” dX ' 31

dl 9C;where v = — . We aim to solve for the quasi-steady state i.e. — — = 0.

dt dt

(A5.1)

Using the dimensionless versions of the concentrations of the aqueous species

defined in Appendix 3 and rearranging the mass-balance equations to remove the

chemical reaction terms, the steady-state equations become

AT + ((5 + Q - 4P + N ) y y

iw Q' ^ S' 2 P' N \ n + + D, - D, - D . ' - 0

(A5.2)

i r - ( i / y 'y + vl* u ' = o (A5.3)

v + (Vxpy + = o (A5.4)

QS = WXP (A5.5)

S1

IIto (A5.6)

e = w35 (A5.7)

aP + bQ + cS + dN + eU + JV = 0 (A5.8)

where " denotes — d X 2

In order to integrate (A5.2),(A5.3) and (A5.4) analytically to obtain a form suitable

for the numerical integration program developed for the static-boundary

calculations, we make the following approximation:

dX = vC (l) - v C ( l) ( l - X ) - vXC(X). (A5.9)

210

In doing so, we are assuming that the concentrations are approximately constant

along the crevice length. From the previous calculations in a static geometry, for

example figure 7.8, this is a reasonable approximation, and it is unlikely that any

error incurred will obscure the qualitative effects of including the moving boundary.

We also use of the following approximation related to the values of the

variables at the crevice tip:

P ( l) = ^ » G( 1), 5(1) and N( 1) (A5.10)

and t / ( l ) » K ( l ) . (A5.11)

Further we assume

t / ( i ) ■ 5(1) 2

W,W3(A5.12)

from the condition of charge neutrality. Thus integrating (A5.2)-(A5.4) using these

approximations, we obtain

S' + Q ' - IP ' - N ' + {S + Q - 4P + N)ty'

+ vIX {d i + ~ f - w 6 > = ~ 2 g > “ I * - * *

+ g2 exp(-ar2y j( l)) + g3 5(1) e x p (-« 3ip (l)) + 2 V 1 (A5>13)

^ _ w + v i x u = _ U ( l ) v W - X )D A

v + v y + v l 0 V = 0

(A5.14)

(A5.15)

and differentiating (A5.5) and substituting from (A5.5), (A5.6) and (A5.7) we obtain

(2 aS

w; w3 w 3+ — + c -

dW,)S' + eU' + J V ' = 0. (A5.16)

8 i» 82 an< 83 are the dimensionless constants defined in Appendix 3, as are

a , b , c, d, e and/.

Boundary Conditions.

The boundary conditions for the solution of this set of equations are as follows. At

the crevice mouth, X = 0,

211

5 = 1 , N = W29 Q = W3, P = W3/W l y U = t T , (A5.17)

V = - ( :2 P + Q + S -N -U ), y> = 0,

where £/* is the dimensionless bulk chloride concentration.

212

Appendix 6. Solution to Moving-Boundary Problem at High Metal Potentials.

Calculations carried out with metal potentials above — 0*1 V failed to converge in

the first iterated variable i.e. the potential at the crevice tip. Instead, for a

potential very close to this, the crevice-tip potential eventually settled to 2 values,

alternating between each at successive iterations. As the metal potential was raised

further the iterations settled to 4 distinct values and then to 8 at a higher

potential again and eventually the iterate became random and no clear pattern

emerged. This set of period-doubling bifurcations, although very interesting in itself,

is a feature of the mathematical method rather than an indication of any

oscillatory behaviour in the physical system. To be more precise it arises from the

iteration scheme employed.

In the static-boundary, passive-walls calculations an initial guess is made for the

potential at the crevice tip. The parameters dependent on this are calculated

and the equations are solved to yield the concentration and potential profiles along

the crevice length. The calculated crevice-tip potential is then compared to

the original guess and if it does not coincide then the average of the old and new

values is calculated and adopted as the initial guess on the next iteration. This

averaging procedure was devised as a way of increasing convergence speed but for

the moving boundary calculations at high metal potential it produces

problems.

Figure A6.1 shows a schematic illustration of the guessed crevice-tip potential

against the calculated potential for two values of <pM, one of which has a single

stable solution and the other produces period 2 cycle of the iterate described above.

The solution of the set of differential equations can be regarded as a mapping

function from the set of guessed potentials to the calculated potentials. It is the

shape of this mapping function that is important to the solution of the equations. The

solution crevice-tip potential is represented on this figure as the point where

the line (pnew = meets the mapping curve— a fixed point of the the map. The

theory of the dynamics of single humped maps and bifurcating systems is outlined

briefly in Appendix 7. The mapping corresponding to the moving-boundary

problem is slightly different in that the hump is inverted. However the metal potential

acts in the same way as the parameter A, since both serve as tuning

parameters that determine the shape of the mapping curve. The behaviour of the

iterate is determined by the gradient of the mapping function at the fixed point.

Changes in the tuning parameter sweep the gradient through critical values that mark

the onset of successive period doubling.

It is the method of averaging of the guessed and calculated values of xp(l) at each

iteration that leads to the single trough in the mapping function. If the

calculated value of tp(l) is fed back into the system at each iteration, then the new

mapping function is monotonic and the period-doubling phenomenon is lost.

However such an iteration scheme will only converge if the initial guess is already

very close to the solution.

0 N

ew

. ca

lcul

ated

pot

entia

l dr

op a

t cre

vice

tip

, V

0 o ld , 9uessed potential a t crevice t i p , V

Figure A6.1 A comparison of the relation between the guessed crevice tip potential and the calculated value for a convergent case, = —0.11 V and a bifurcating system, = —0.09 V.

Appendix 7. The Dynamics of Simple Single-Humped Mapping Functions.

First-order difference equations arise in many contexts. There are many examples

of such equations that exhibit an interesting array of dynamical behaviour

despite being simple and deterministic. Their behaviour can range from stable points

to a bifurcating hierachy of stable cycles, to apparent random fluctuations. Let

us consider an example of a single-humped mapping function and investigate its

properties using a simple graphical analysis. The chosen mapping function,/(jc), is

to be defined

f(x ) = 4 k x ( l- x ) (A7.1)

where A is a variable parameter. A fixed point of the mapping, jt*, is defined by

* * = /( * * ) . (A7.2)

Figure A7.1a shows the mapping as the curve y = f(x ) together with the line y — x. Where these lines intersect are the fixed points. This single-humped mapping

has 2 fixed points, x* — 0 and x* = 1 — 1/4A. The stability of a fixed point is

determined by the angle between the tangent to the curve and the y = x line. If this

angle is acute then the fixed point is locally stable and attracts all trajectories

in its neighbourhood. In this case

s ‘ (jc*) = = 2 - 4A (A7.3)ax

so the fixed point will be stable if 0.25<A<0.75. If A is increased to some value

greater than 0.75 then x* will become unstable. For A<0.75 consider application of

the mapping function to a starting point jc0. As the function is successively

applied to generate x { x2 etc. (figure A7.1a) the iterate approaches and eventually

reaches the stable point. To investigate the behaviour of the system when x* becomes unstable it is useful to consider a second application of the same map i.e.

to consider f( f(x )) ( or f 2(x)). Clearly the fixed points of f(x ) are also fixed

points of f 2(x). Figure A7.1b shows f 2(x) plotted with A<0.75 showing its relation to

f(x). The slope of the map at this point, s2(x*), is equal to the square of the

slope at the corresponding point of the single map graph. Now if the fixed point, Jt*,

becomes unstable, i.e. A>0.75 and 1 (jrs|e) |> l , then |s2 ( * * ) |> l, and the

slope of f 2 steepens as shown in figure A7.2b. As this happens the curve f 2(x) must

develop a 'loop' and two new fixed points appear.

In short, as the function/(jt) becomes more steeply humped, the fixed point x* becomes unstable. At exactly this stage two new and initially stable fixed points of

period 2 are born and the system oscillates between them in a stable cycle of

period 2 as shown in figure A7.2b. As before, the stability of the 2 new points

depends on the slope of the curve f 2(x) at each of them. As the parameter A is

216

increased further these 2 points will become unstable and bifurcate to give an initially

stable cycle of period 4. This in turn gives way to an 8 cycle and then a 16

cycle and so on. Figure A7.3 illustrates this bifurcation process. It must be noted that

this phenomenon is characteristic of most functions with a hump of tunable

steepness.

Although an infinite sequence of period-doubling bifurcations is produced, the

range of parameter values in which any cycle is stable becomes progressively smaller

and the sequence is bounded by some critical value. For the particular

mapping in this example, this value is Ac = 0.8925.... Beyond this 'accumulation

point' there are an infinite number of fixed points with different periodicities and an

infinite number of periodic cycles. This regime is generally known as the

'chaotic' regime and is characterised by dynamical trajectories that are

indistinguishable from some stocastic process. The chaotic region is not totally

structureless however. There can exist 'windows' of the tuning parameter above the

accumulation point in which unstable cycles of odd period exist.

Such non-linear systems exhibit remarkable universal behaviour. For example if A„

is the value of the tuning parameter at which the period doubles for the n th

time, and if

(A7.4)

then it has been shown 79 that 6n quickly approaches a constant value <5 which is the

same for all bifurcating systems, with 6 = 4.6692016....

217

f (X )

Figure A7.1 a) A plot of the single-humped mapping function/(jc) = 4Ax(l — x ) with

A = 0.7, showing iterations to a fixed point.

A7.1 b) A plot ot'the mapping function applied tw ice,/2 (x)

218

f (X)

Figure A7.2 a) A plot of the single-humped mapping function, f(x ) with A = 0.785, showing the iterations oscillating in a stable cycle of period 2.

A7.2 b) A plot of this mapping function applied twice.

219

So

luti

on

• • • • •

Tuning param eter

Figure A7.3 A schematic illustration of a hierachy of unstable cycles in a bifurcating system and the onset of chaos.

Appendix 8. A Finite-Element Method for Solving Non-Linear Differential Equations.

The application of the finite-element methods to non-linear partial differentialQA

equations has been discussed in detail by many authors , and so only a brief

description is given here. In particular, we discuss the Galerkin method adopted in

the HARWELL finite-element subroutine library TGSL75. This approach

involves solving the weak form of the equations and is one of the more common

techniques associated with finite element modelling. We illustrate the Galerkin

method by considering the equation

l-a (u ) + b(u) • Vm + V (c(u) Vu) = 0 (A8.1)at ~ ~

in a region Q of space, where fl,6 ,c are known functions of the field variable m ; the

boundary conditions imposed are

u = f on Cp

n.c(u)Vu = g on C2,

(A8.2)

where n is the unit normal on C2 and Cx + C2 is the boundary 3Q of Q ;/, g are

known functions of space. The weak form of (A8.1) and (A8.2) is derived by

multiplying (A8.1) by an arbitrary smooth function 0 , which is zero on C j; then

integrating over Q, integrating by parts the term V (cV m ):

f 0 (^-a + b • Vm) — f V0 • cVm + f 0 g = 0. (A8.3)J q a t J q J c2

Now if (A8.3) holds for sufficiently many choices of 0 which satisfy the condition

(p = 0 on C i , then it is equivalent to equation (A8.1) together with the boundary

condition on C2, for sufficiently smooth solutions u.

The spatial discretization of the weak equations is achieved in the finite-element

method by approximating Q by the union Q ' of a finite number of 'elements',

which are usually of simple geometric shape. The field is then approximated on Q '

by a function u which is defined on each element by a small number of

variables, which are commonly called freedoms to distinguish them from the field

variables. These freedoms are usually values of the field variable or its derivative at

special points called nodes. Equations for the freedoms are obtained by

substituting u into (A8.3) for a number of test functions 0, equal to the number of

freedoms. Expressing the integral over Q as a sum over elements,

221

^ elements [ f </>,(|a(«) + b(u).Vu) - f V^.-c(u)V«] + f < p ,g = 0.(A8.4)J element & J element J c2

A steady-state problem is then reduced to a set of coupled non-linear algebraic

equations, and a transient problem to a set of coupled non-linear ordinary differential

equations in time. The fundamental point of the finite-element method is the

choice of test functions that yields sparse equations i.e. the equation for one freedom

only involves relatively few other freedoms. Such test functions are usually

low-order polynomials that are centred at a particular node but extend only over

adjoining elements. The finite-element equations (A8.4) show why; the equation for

one freedom only involves other freedoms on the same or adjacent elements.

Thus powerful direct methods developed for solution of sparse linear systems can be

applied once the system of equations has been linearised in some way. It is

important to note with this method, that if a solution varies rapidly in some region of

the space, it is necessary to have relatively more elements in that region to

represent the solution correctly.

222

Appendix 9. Mass—Transport Equations with Ferrous Chloride Precipitation.

In this appendix, ionic mass-transport and chemical-equilibrium equations are

derived that describe the crevice solution chemistry and electrochemistry with ferrous

chloride precipitation. Thus one extra ionic species (FeCl+ ) and two extra

chemical equilibria equations must be added to the scheme in Appendix 3 which

involves the precipitation of ferrous hydroxide only. The additional reactions are

Fe2+ + C /+ ^ FeCl+ (A9.1)

FeCl* + Cl~ ^ FeCl2 (A9.2)

with equilibrium constants £ 4 (=10 0 5) and t f 5(=10 1,5) respectively. Adopting the

same notation for the dimensionless concentrations and potentials as used in

Appendix 3, with [F e C r ] = C7 and W =d 7c 7

C5, the seven coupled,

time-dependent mass-balance equations for the passive wall crevice rearrange to the

following:

Q" + S" + 2F ' - N ' + W' - IT + ((Q + S + 4P + N + W + U)ip')' =

d(Q + S + 2 P ^ - N + W - U )(A n 3)

V” + (Vip’Y = ~ (A9.4)

Q S = Wt P

SN = W2

q = w3s

w = W4PU

1w u

aP + bQ + cS + dN + eU + j V + gW = 0

(A9.5)

(A9.6)

(A9.7)

(A9.8)

(A9.10)

(A9.11)

d 5c :where W4 = - —f - K 4,

a 2

a * 2 '

(D5Cs )d 3d 7 ’

T* =tD2— — and ” denotes L 2

223

The boundary conditions at the crevice mouth are that the the concentrations of

the species are fixed and must satisfy the equilibria constraints (A9.5-A9.10)

and the charge-neutrality condition (A9.11). The fluxes of ions involved in the

electrode reactions at the walls and base of the cavity are the same as those given in

Appendix 3 (A3.10)

For the active wall crevice equations (A9.3) and (A9.4) are replaced by

B2(Q + S + 2P — N + W - U ) B2(Q + S + 2P - N + W - U) d X 2 BY2

^ ( ( < 2 + S + 4P + N + W + C 0 | | ) + ^ ( ( Q + S + 4P + N + W + U) | | )

B(Q + S + 2 P - N + W - U BT*

(A9.12)

+ ± ( v ^ + ± ( v * t = ^BX2 BY2 aX K BX BYk BY BT*

(A9.13)

where Y = L 2— y-

1

224