NUMERICAL STRATEGIES FOR CORROSION MANAGEMENT:
SPATIAL STATISTICS AND FINITE ELEMENT SIMULATION
NUMERICAL STRATEGIES FOR CORROSION MANAGEMENT:
SPATIAL STATISTICS AND FINITE ELEMENT SIMULATION
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus Prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 1 december 2008 om 12.30 uur
door
Juliana Mar´ıa L ´
OPEZ DE LA CRUZ
Wiskundig ingenieur geboren te Bogot´a, Colombia
Prof. dr. ir. R. de Borst Prof. dr. ir. M.A. Guti´errez
Samenstelling promotiecommissie:
Rector Magnificus, Voorzitter
Prof. dr. ir. R. de Borst, Technische Universiteit Delft, promotor Prof. dr. ir. M.A. Guti´errez, Technische Universiteit Delft, promotor Prof. dr. ir. R. Benedictus, Technische Universiteit Delft
Prof. dr. R.M. Cooke, Technische Universiteit Delft Prof. dr. C. Guedes Soares, Technical University of Lisbon Prof. ir. A.C.W.M. Vrouwenvelder, Technische Universiteit Delft
Dr. R. Cottis, University of Manchester
Cover Design: Hugh Geoghegan
Copyright c°2008 by J.M. L´opez De La Cruz Printed in the Netherlands by PrintPartner Ipskamp ISBN-13: 978-90-79488-31-5
The work presented in this thesis has been supported by the technology foundation STW and it has been developed at the Aerospace Faculty at Delft University of Technology.
I wish to express my gratitude to my promotor, Rene de Borst, for believing in me and for offering me the opportunity to make my PhD in his group. I would also like to thank my supervisor, Miguel Guti´errez, for his helpful comments in the last four years. In the same manner, I would like to thank the members of my committee for their helpful comments.
The help of Carla Roovers and Harold Thung have made the whole process a more pleas-ant time. Therefore, I am very grateful to them. I would like to thank my colleagues: Dr. Steven Hulshoff, Dr. Harald van Brummelen, Dr. Sergio Turteltaub, Dr. Akke Suiker, Dr. Christian Michler, Dr. Edwin Munts, Dr. Doo Bo Chung, Dr. Denny Tjahjanto, Clemens Verhoosel, Jingyi Shi, Kris van der Zee, Ido Akkerman, Gertjan van Zwieten, Andr´e Vaders and Thomas Hille, for our very enjoyable discussions. Special thanks should be given to Marcela and Wijnand for their friendship through these years. To my friends Astrid and Ana, thank you very much for being there for me.
I wish to thank my sister Adriana and Hugh for their permanent help and encouragement at all times. The constant love and support of my parents made this work possible therefore I wish to express my sincere gratitude to them. Finally, I wish to thank Theodore, who has always supported me, bearing the load by my side encouraging me to continue pursuing my dreams. Thank you!
Juliana L´opez De La Cruz Delft, August 2008
Contents
Contents vii
List of Symbols xi
1 INTRODUCTION 1
1.1 Aims of the study . . . 2
1.2 Notation and Study outline . . . 3
2 BASICS OF ELECTROCHEMICAL CORROSION 5 2.1 Corrosion Terminology . . . 6
2.2 Corrosion Mechanism . . . 7
2.3 Electrochemical Equations . . . 8
2.4 Corrosion stages and localized corrosion . . . 10
3 SPATIAL STATISTICS OF POINT PATTERNS 13 3.1 Complete Spatial Randomness (CSR) . . . 14
3.2 Traditional CSR Tests . . . 14
3.2.1 Inter-Event distances method . . . 15
3.2.3 Ripley’s methods . . . 18
3.2.4 Quadrat Counts method . . . 19
3.3 Spatial Point Processes . . . 20
3.3.1 The homogeneous Poisson point process . . . 21
3.3.2 The nonhomogeneous Poisson point process . . . 21
3.3.3 The Neyman-Scott process . . . 22
3.4 Goodness of Fit Tests . . . 23
3.4.1 χ2 goodness of fit test . . . 23
3.4.2 Kolmogorov-Smirnov goodness of fit test . . . 24
4 APPLICATIONS OF SPATIAL STATISTICS TO CORROSION 25 4.1 Inter-Event Distances Method Applied To Corrosion Patterning Analysis . . 26
4.2 Quadrat Counts Method For The Analysis Of Corrosion Patterns . . . 31
4.2.1 Empirical pattern analysis and quadrat size dependence . . . 32
4.2.2 The binomial index of dispersion in corrosion analysis . . . 33
4.3 Goodness of fit to a non-homogeneous Poisson process: Distance-based sta-tistics and quadrat analysis . . . 34
4.3.1 Goodness of fit to a spatially non-homogeneous Poisson process . . 35
4.3.2 Empirical data analysis by means of the non-homogeneous Poisson process . . . 46
4.3.3 Optimum quadrat size and scale dependence . . . 54
5 SIMULATION OF PITTING CORROSION AND PASSIVATION IN 2D 57 5.1 Problem definition . . . 58
5.1.1 Geometry and reactions . . . 58
5.1.2 Model parameters and reaction kinetics . . . 60
5.2 Simulation Implementation . . . 64
5.3 Simulation of pit passivation . . . 72
6 SIMULATION OF PITTING CORROSION AND PASSIVATION IN 3D 75 6.1 Artificially generated cluster process . . . 75
6.2 Electrochemical pit cluster simulation . . . 76
CONTENTS 7 CONCLUSIONS 89 Bibliography 93 Summary 101 Samenvatting 105 Resumen 109 Curriculum vitae 113
LIST OF SYMBOLS
Symbol units Description
M - Metal X - Reduced species m - metal ion o - ion e - electron E Vm−1 Electric field
Ni mol m−2s−1 Mass flux vector of species i
Di m2s−1 Diffusion coefficient of species i
R J mol−1K−1 Ideal gas constant
T K Temperature
ci mol m−3 Concentration of species i
zi - Charge of ionic species i
ui s Ionic mobility of species i
F C/mol Faraday’s constant
φ V/SCE Potential in solution
u m s−1 Velocity vector
t s Time
R kmol m3/s Production of homogeneous chemical reactions
i A/m2 Current density
i0 A/m2 Exchange current density for Fe oxidation α - Apparent transfer coefficient
φm V/NHE External potential of the metal
M - Atomic mass
K - Equilibrium constant
k m3· s/kmol Kinetic constant β V/NHE Tafel slope
Symbol units Description
W - Planar region
AW m2 Area of planar region W
λ m−2 Intensity
xi - Point i in a plain
n - Total number of points in W
D - Random variable of the distance among points
FD - Inter-Event theoretical distribution
d - Realization of the variable D
b
FD - Inter-Event empirical distribution
Ndij - Number of points whose distance from i to j is ≤ to d
U (d) - Upper simulation envelope
L(d) - Lower simulation envelope
s - Number of simulations to generate the envelopes
Cr - Circular region of radius r
Mr - Number of events in Cr
Ri m Distance of the i-th closest point to the origin
FN N - Nearest-Neighbour theoretical distribution
b
FN N - Nearest-Neighbour empirical distribution
Ndi - Number of points whose distance diis ≤ to d
b
K - Ripley’s estimator
M - Total number of quadrats (cells)
m - A quadrat (one cell)
N (B) - Number of points in a region of area B
k - Partitions of W
ni - Points contained in quadrat i
¯
n - Sample mean of ni
I - Index of dispersion
χ2 - Chi square hypothesis test α - Significance level
pi - Events ordered from the smallest to the largest
F (pi) - Theoretical cumulative distribution
BI - Binomial index of dispersion
{ζi, ηi} - Cartesian coordinates of point xi
Ξ - Process composed by the points in W
Lδ - Cells of side δ
λ(x) m−2 Intensity function of L δ
x m Vector containing the points in Lδ
a m−1 Gradient of λ(x)
CONTENTS
Symbol units Description
p(x) m−2 Density of a point in Lδ
δ m Side of a cell
U - A realization from the uniform distribution
T(nm) - Distance-based statistics
Tobserved - Scalar value found with the distance-based statistics
D(Tδ(n)) - Set of points in Lδseparated ≤ Tobserved
b
H - Weighted mean statistics
wm - Weigh of quadrat m
µ - Biased asymptotic mean
σ2 - Asymptotic variance
pvalue - Probability of the test statistic being at least as extreme as the one observed
given that the null hypothesis is true.
A small pvalueis an indication that the null hypothesis is false.
CHAPTER
1
INTRODUCTION
Annually, millions are invested in corrosion management around the world. In the United States corrosion is estimated to cost about $276 billion dollars per year [3]. Figure 1.1 shows a pie chart of the total direct cost of corrosion for different sectors for the year 1998 in the United States. As in Figure 1.1, several industries around world are also investing massive capital in an attempt to control the damage caused by corrosion. In Europe, several companies have turned to research trying to understand the nature of the corrosion problem as new models and techniques can prevent the loss of billions.
Corrosion studies have been mainly attached to the electrochemical field as the govern-ing equations describgovern-ing the problem belong to this discipline. In the last century, several electrochemical researchers have identified different aspects of the corrosion mechanism de-veloping new methods to battle corrosion. In the process, alternative methods outside the electrochemical field have been studied. Statistical analysis has been incorporated to differ-ent fields providing a powerful tool to treat the uncertainties of differdiffer-ent disciplines. Cor-rosion scientists have managed to model the potential profiles, the induction time and the failure probability as function of time by means of stochastic methods.
Corrosion research can benefit from different numerical techniques when multidiscipli-nary models are combined (e.g. probabilistic modelling of uncertainty in experimental data, electrochemical corrosion simulation, etc.). In particular, spatial statistics methods and
fi-$ 47.9 Billion Utilities 34.7 % $ 17.6 Billion Production & Manufacturing 12.8 % $ 20.1 Billion Government 14.6 % $ 22.6 Billion Infrastructure 16.4 % $ 29.7 Billion Transportation 21.5 %
Figure 1.1: Total direct cost of corrosion in analyzed sectors: $137.9 billion/year (1998) [3]
nite element simulations are used in this document as a tool in the study of pit interaction. Electrochemical studies suggest that pits appear in clusters and that there exists interaction among them [22][39]. As consequence, the existence of one pit can enhance the growth or the appearance of new pits.
Spatial statistics techniques are used to detect the spatial distribution of corrosion spots in space. The analysis of empirical patterns by this mean can provide new insights into the mechanisms that govern the appearance and evolution of pits. Finite elements simulations are used as a coupled tool to the spatial statistics techniques in the analysis of pit interaction. The results of an electrochemical corrosion simulation can either support or refute the spatial statistics findings. In this way, reliable information about the nature of corrosion can be obtained.
1.1
Aims of the study
Corrosion has traditionally been viewed as a process governed by the electrochemical the-ory. Innovative techniques from different disciplines can provide new insights into an area based on experimentation and field expertise. Life threatening failures and inestimable cap-ital losses can be avoided when the corrosion management grounds are improved by new
1.2. NOTATION AND STUDY OUTLINE
techniques. The latter constitutes the main focus of this study.
The techniques developed and applied through this document are focused on assisting corrosion management independently of the industrial sector. Therefore, the models through this document are formulated for a general cases and only small variations are necessary in order to make them suitable for a particular analysis.
Spatial statistics models are used here to detect the distribution of different corrosion patterns. These models are advantageous in that they can be applied to several corrosion sys-tems since they are independent of material type and environmental conditions. In addition, the spatial statistics techniques can be applied to perform smart maintenance which targets places that exhibit clusters of pits because in these regions the risk of failure is higher. Sev-eral spatial statistics methods have been proposed. This work focuses on finding the most appropriate spatial statistics method to analyze corrosion patterns as not all methods exhibit good performance.
Expensive and time consuming experimentation can be improved when electrochemical simulations are properly combined with reliable experimental data. A robust electrochemical model that can represent the corrosion process at different stages constitutes an asset for corrosion management. This thesis focuses on the formulation and implementation of a finite element model that can represent the electrochemical behavior of a pit. The finite element model has to support 3D simulations in order to represent more accurate practice. Moreover, it has to offer the opportunity to implement transitions in corrosion stages since only in this way the corrosion process can be represented by a model.
1.2
Notation and Study outline
In this study, the notation used for the electrochemical formulation is completely independent of the symbols and variables used in the spatial statistics models. Therefore, the notation should be read and understood in the context of each chapter. The list of symbols provided at the beginning of this thesis can be used as an aid to follow the notation of each Chapter.
In Chapter 2 the basics of the electrochemical corrosion system are explained. The equa-tions and models described in Chapter 2 are implemented in Chapter 5 and 6 where an alternative numerical technique to the spatial statistics methods is applied.
Chapter 3 presents a brief introduction to spatial statistics and the techniques used in this document. In Chapter 4, different empirical corrosion patterns are analyzed by means of the
methods described in Chapter 3. The robustness of the selected methods is calibrated with different artificially generated patterns. Since controversial results are found after analyzing the same empirical pattern with different spatial techniques, a reliable method based on an optimization routine is implemented.
The grounds of the goodness of fit to a non-homogeneous Poisson process method are explained in Chapter 4. The method is based on the combination of different spatial statis-tics techniques. This method is applied to different empirical corrosion patterns and to the analysis of a pit pattern in an oil pipeline section.
In Chapter 5, the finite elements methods is applied to the simulation of pitting corrosion and passivation for a single iron pit in a 2-dimensional domain. Chapter 6, presents an ex-tension of the same electrochemical model in a 3-dimensional domain. In the 3-dimensional model the bias induced by the simplification of the domain and the applied electrochemi-cal input is reduced. Therefore, the simulation results represent more accurate the values measured in practice.
In Chapter 6, pit interaction is also studied in a cluster generated with the Neyman-Scott process. The interaction among pits is analyzed in the 3-dimensional electrochemical model for a cluster composed by three pits. The passivation phenomenon is simulated for one of the pits that seems to passivate spontaneously due to interaction with its neighboring pits. Thus, the information provided by the electrochemical studies, the spatial statistics and the finite elements simulations can be combined and conclusions regarding the corrosion nature and pit interaction can be done in a reliable way.
CHAPTER
2
BASICS OF ELECTROCHEMICAL
CORROSION
Everybody has observed once the damage produced by a corrosion attack. Corrosion can be defined as the degradation of a material caused by its interaction with the environment. Researchers have spent years of investigation trying to understand the mixed mechanisms that drive corrosion to occur. Essentially, corrosion takes place at the surface that acts as a common boundary between the metal and the environment. This surface is known as interface. Thus, the transport of ions, molecules or atoms at the interface of a material is what controls the corrosion process.
In the corrosion field different forms of corrosion are encountered. Uniform and local-ized corrosion are two of them and they can be easily recognlocal-ized by their appearance on a surface. Uniform corrosion causes 30% of the structural corrosion induced failures. It oc-curs evenly over a metal surface and it has a predictable rate of appearance. Conversely, in localized corrosion specific spots over a metal are damaged. The damaged locations are usu-ally associated with material imperfections or metal composition. Its rate of appearance is greater than that associated with uniform corrosion. Therefore, 70% of all corrosion induced failures are related to this kind of corrosion [4].
Pitting and crevice corrosion are two similar types of localized corrosion. Pitting devel-ops as a collection of small cavities that at early stages are imperceivable to the human eye. Crevice corrosion is observed when adjacent surfaces create an obstructed region enhancing the corrosion process. Since the growth mechanisms of pits and crevices are very similar, the electrochemical equations described in the following sections can be employed to study both corrosion types. Other forms of localized corrosion such as stress corrosion cracking and corrosion fatigue cracking are not treated here. However, similar techniques as those explained in this document can be further developed to treat other corrosion forms.
2.1
Corrosion Terminology
Corrosion phenomena have been widely studied in the last decades. The process itself is related to a specific area of the chemical sciences known as electrochemistry. The formulas and equations describing the corrosion systems are grounded in the electrochemical theory. Important terminology and relevant definitions are given in this section.
Corrosion can take place in different environments. This work is particularly focused on corrosion occurring in aqueous solutions. An aqueous environment means that the surround-ings of the metal contain in some proportion water. When a metal corrodes in an aqueous environment, atoms at the surface of the metal enter the solution as metal ions. The metal site where this occurs is called anode. Then, at the the metal or electrode, electrons migrate to a site called cathode. At the anode is where oxidation or corrosion occurs. An environment capable of conducting electricity is called electrolyte. In the electrolyte, current flows from the anodic to the cathodic site.
The electrolyte is characterized by its mobile species. The term species is used to refer to ions as well as to neutral molecular components that do not associate [5]. Ions are electrically charged atoms (e.g. Na+, Cl−) or group of atoms as radicals (e.g. NH+4) [7]. The amount of work required to move these electrically charged particles inside the electrolyte is known as Potential. The difference in potential between the anodic and cathodic site is what causes the current to flow. The passing of current generates a shift in the potential which is referred as overpotential [1]. The overpotential is a very important measure in corrosion as it increases in magnitude proportionally to the current density. The current density is the rate of flow of positive charge per unit area [5] expressed as amperes per square meter [ A · m−2] or amperes per square centimeter [A · cm−2]. The latter relates directly to the corrosion intensity (CI)
2.2. CORROSION MECHANISM
defined as mass loss per unit time per unit area or to the corrosion penetration rate (CPR) defined as mass loss per unit time [1].
2.2
Corrosion Mechanism
The process of electrochemical corrosion is characterized by the mass transport across a metal interface. The main electrochemical mechanism is when a metal corrodes in an aque-ous solution where ions from the metal (m+) are transferred into the solution. Then, the electrons that migrate to the cathodic site are consumed by other species (Xo+) to produce new ones (X) [1]. Figure 2.1 illustrates the process.
ANODE CATHODE e -m+ o+ X OXIDATION REDUCTION METAL X
Figure 2.1: Schematic representation of the corrosion process in a metal
In the anodic part of the metal oxidation takes place. The anodic reaction involves the dissolution of metal ions into the environment. This phenomenon causes the migration of the electrons inside the metal towards the cathodic part where they are consumed by other species at the metal interface. The reactions befalling at the cathode are known as reduction reactions.
The rate of corrosion depends on the kinetics of the processes influencing the transport of ions from the metal to the environment. Thus, species reacting with the metal ions in solution decrease the concentration of free ions. The latter favors the metal dissolution which increases the corrosion rate. Conversely, protective layers formed by precipitates at the metal
interface decrease the corrosion rate [1] [5]. These protective layers are commonly referred to rust and they resemble a thin film in the metal surface.
2.3
Electrochemical Equations
As mentioned in Section 2.2, metal oxidation is produced by the combination of various electrochemical reactions. An electrochemical reaction is defined as a chemical reaction involving the transfer of electrons [7] which entails the oxidation and reduction of the species involved . The process occurring at the anode can be generalized by the expression
M → Mm+
+ me
Similarly, the process taking place at the cathode is generalized by
Xo+
+ oe → X
The driving mechanism for these reactions is the potential difference. At the electrode, electrons flow because of the difference in the electric field E which is related to the potential gradient by E = −∇φm. In the electrolyte, ions move in response to the electric field (migration), the concentration gradients (diffusion) and the fluid motion (convection). Thus, the net flux of an ion in the solution is given by the sum of the three terms,
Ni = −Di∇ci− ziuiF ci∇φ + ciu (2.1)
where Ni denotes the mass flux vector of species i, ci the concentration , Di its diffusion coefficient, u is the ionic mobility, F is the Faraday’s constant, φ is the potential in solution, z is the charge of the ionic species and u is the velocity vector. The convection term ciu, is used in cases where the solution is moving at certain velocity. The corrosion caused by the flow of water in a pipe illustrates this case. Hence, the convection term is dropped from equation (2.1) in the models here implemented as the electrolyte is considered to be steady.
The mobility u and the diffusion coefficient Di are connected by the Nernst-Einstein relation [5],
ui = Di
2.3. ELECTROCHEMICAL EQUATIONS
where R is the universal gas constant, 8.3143 [J/mol · K] and T the temperature [K]. The change in concentration of species i as a function of time is described by the Nernst-Planck equation,
∂ci
∂t = −∇ · Ni+ Ri (2.3)
where Riis the production of homogeneous chemical reactions in the solution.
The reactions rates are coupled by the principles of conservation of charge and elec-troneutrality. In the electrolyte, the electroneutrality equation is expressed as,
X
zici = 0 (2.4)
Equation 2.4 states that a balance must exist between the amount of anions and cations [5]. Additionally, eletctroneutrality is related to the Laplace’s equation ∇φ = 0, for a medium with no free charges [6]. Equation 2.4 is coupled with equation 2.3 in order to achieve a consistent system as the latter posses more unknowns than equations.
The electric field E in the solution produces an ionic current density i that is related to the net flux of charged species by,
i =
X
i
ziF Ni (2.5)
Moreover, the current density i is related to the potential in solution φ by the Tafel equa-tion,
i = ioexp{
αF
RT(φm− φ)} (2.6)
where φm is the metal potential, io is the exchange current density which depends on the species concentrations and the nature of the electrochemical system. The parameter α is known as the apparent transfer coefficient and its sign changes from positive to negative depending on the reaction. Thus, for the anodic reaction the parameter is positive and for the cathodic reaction is negative.
Once the current density i is computed by means of the previous equations, the corrosion intensity (CI) can be computed as
CI[g/m2 · year] = 0.327Mi
zFe
(2.7) where M is the atomic mass.
The equations described in this Section represent the basics of the electrochemical cor-rosion process. In Chapter 5, iron corcor-rosion is simulated with a specific electrochemical system.
2.4
Corrosion stages and localized corrosion
In general, all corrosion systems tend to behave in a similar way [1]. In Figure 2.2 an schematic representation of the corrosion process in a pit is illustrated.
Figure 2.2: Current-potential relation for the corrosion stages in a pit that exhibits passivation
In the active region the current density increases with the potential. Then, for a critical potential, a protective anodic oxide film appears in the metal surface. This phenomenon is called passivation. Passivity is defined as the loss of chemical reactivity exhibited by certain metals and alloys under specific environmental conditions [7]. At this stage, a decay in the current density is observed as the potential increases. Thus, a potential is reached where the
2.4. CORROSION STAGES AND LOCALIZED CORROSION
protective film disappears and the corrosion rate becomes very large. This stage is known as transpassive region. The models in this study are focused on the corrosion process at the active region together with the breakpoint to a passive stage. These models can be extended to the transpassive region since the same electrochemical mechanisms continue to apply.
As mentioned, pitting is a form of localized corrosion that occurs in different parts of the metal surface. Figure 2.3 illustrates two examples of pitting corrosion [1]. Pitting can lead to costly maintenance and it is the cause of more unexpected corrosion losses than any other type of corrosion [7]. Pitting is associated to intergranular corrosion where intergran-ular cracks grow into the metal from the pit cavity. A pit may undergo four different phases [7]: Initiation, propagation, passivation and reinitiation. Several mechanisms can be asso-ciated to pit initiation. Pits appearing in a mechanically damaged film is one example. In the propagation phase, the corrosion rate increases due to changes in the environment (i.e. acidification). A pit can passivate when it is filled by corrosion products such as rust. It reinitiates when the protective film formed by the corrosion products disappears enhancing dissolution again.
(a) (b)
Pit
Figure 2.3: (a) Pitting corrosion of inner wall of boiler tube. (b) Cross section of pipe wall showing distribution of pits [1]
Pits at the propagation and passivation phases are of special interest in this study. Pits at the propagation phase that do not exhibit the passivation stage are called stable pits. Pits that passivate but in the process become active again are called Metastable pits.
Crevice corrosion, illustrated in Figure 2.4 [1], is often encountered in metals protected by films. It is believed to initiate as the result of a differential aeration mechanism when the environment enters into the crevice. Inside the crevice, the concentration of metal ions in-creases and the oxygen concentration dein-creases. It is observed usually in cracks or crevices
formed between mating surfaces of metal assemblies. The present study focuses only in pit-ting corrosion. However, the models developed here can be applied to both types of localized corrosion.
Metal
Epoxy
CHAPTER
3
SPATIAL STATISTICS OF POINT
PATTERNS
Statistical analysis applied to the corrosion field is relatively new. Research has been focused on developing stochastic models that can predict the growth of a pit and the time to failure. Melchers [8–14] has developed several probabilistic models through the years. His research has been mostly focused on implementing maximum pit depth distributions and the esti-mation of the failure probability. Shibata [15–18] elaborated significant stochastic models to predict the maximum pit depth applying extreme value analysis [19]. His birth and death sto-chastic process is also applied in the computation of the statistical distribution of the pitting potential and the distribution of the induction time for pit generation. Hong [20] developed a combined model for pit generation and pit depth growth applying the nonhomogeneous Poisson process and a nonhomogeneous Markov process. Engelhardt and Macdonald [21] and Laycock et al. [22] utilized extreme value statistics to predict the development of lo-calized corrosion damage and to model maximum pit depths respectively. Cooke and Jager [23] constructed a model to compute the failure frequency of underground pipelines applying expert judgement.
All these authors have successfully applied statistical models to analyze different corro-sion phenomena based on the effect of time in the corrocorro-sion evolution. Thus, such models compute the failure probability in terms of the elapsed time or the growth of a pit as function
of time. Hence, the influence of the spatial parameter has not been yet considered and its influence in the process has been overlooked.
The spatial location of corroded spots and their influence in the corrosion process is studied in Chapter 4. Spatial models applied to statistical corrosion analysis are discussed in Chapter 4 as well. The preliminary concepts and methods used in corrosion patterning analysis are explained in the following Sections.
3.1
Complete Spatial Randomness (CSR)
The Complete Spatial Randomness (CSR) test is a hypothesis test that divides patterns be-tween regular or aggregated [24]. In a regular pattern, the events are homogeneously distrib-uted per unit area whereas in a clustered pattern the events exhibit aggregation. Addition-ally, a random pattern is a pattern where no structure or arrangement of points is perceived [24][25].
A pattern is denoted as completely random when its events (points) do not interact among themselves. The latter follows from the basic principles of the CSR definition:
1. Consider a planar region W with area AW. The number n of events xi, i = 1, ..., n in W follows a Poisson distribution with mean λAW,
2. Any set of events {xi}ni=1 in region W is a random sample of independent uniformly distributed events on W .
where λ is the intensity defined as the mean number of events per unit area.
The first statement conditions the intensity to remain constant in the planar region W . The second statement prevents any interaction among the events in W . Otherwise, the CSR hypothesis is rejected and aggregation of the points contained in W is considered.
3.2
Traditional CSR Tests
The CSR test is one of the most commonly used tests in spatial statistics. Its success relies on the fact that many methods have been developed to test pattern deviation from complete randomness. In this section only three of these methods are presented as they will be used in Chapter 4 to analyze different empirical corrosion patterns. Ripley’s method is briefly
3.2. TRADITIONAL CSR TESTS
explained because some of the empirical patterns have been previously analyzed by means of this method but it is not used as a statistical test in this study.
3.2.1
Inter-Event distances method
The theoretical distribution of the distance between two events depends on the size and shape of the studied region. The theoretical distribution of the distance D between two events independently and uniformed distributed in a region W is denoted by FD. The cumulative distribution function (CDF) of D on a unit side square region is given by [24],
FD(d) = πd2− 8d3 3 + d4 2 , for 0 ≤ d ≤ 1 (3.1) FD(d) = 13 − 2d2− d 4 2 + 4√(d2−1)(2d2+1) 3 + 2d2sin−1 ¡ 2 d2 − 1 ¢ , for 1 < d ≤√2 (3.2)
For a circular region with unit radius, the theoretical distribution is given by [28] FD(d) = 1 + 1π µ 2(d2− 1)cos−1¡d 2 ¢ − d ³ 1 + d2 2 ´ q 1 −d2 4 ¶ , for 0 ≤ d ≤ 2 (3.3)
The method to test CSR for a given pattern is explained for two different region shapes as the test implementation varies depending on the region shape to analyze.
Inter-event distances estimator for a square region
The empirical inter-event distribution, bFD(d), is obtained by counting the inter-event dis-tances for which the events are at most a distance d separated. Let n be the total number of events (points), then the total number ofpossibleinter-event distances is given by
n−1
X
i=1 i = 1
2n(n − 1) (3.4)
The empirical inter-event distribution is then computed as
b FD(d) = Ndij≤d 1 2n(n − 1) (3.5)
where Ndij represents the number of points for which the distance from point i to point j is
less or equal to d. The plot of FD(d) versus bFD(d) should be roughly linear in case the point pattern follows a random distribution.
In order to assess the significance of departures from linearity, upper and lower simula-tion envelopes are computed as
U(d) = maxi=1,...,s( bFDi(d)) (3.6)
L(d) = mini=1,...,s( bFDi(d)) (3.7)
where s random samples of n events are generated from the uniform distribution for a unit side square region to compute bFDi(d) for each run.
Figure 3.1-right presents the results for the Empirical Distribution Function (EDF) af-ter the generated random point pataf-tern in Figure 3.1-left is analyzed with the Inaf-ter-Event distances method. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z y 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
inter event distance
Cumulative pr obability Upper envelope Lower envelope Analytical Empirical
Figure 3.1: (Left) artificial random point pattern. (Right) simulation results for the inter-event distance distribution with s=1000
From the graph in the right it is possible to observe that the EDF remains inside the envelopes that are computed after one thousand simulations (s=1000). Therefore, it can be concluded that the generated pattern is randomly distributed.
Figure 3.2 presents the results for the EDF after an artificially clustered pattern is ana-lyzed. The EDF falls outside the simulation envelopes. Hence, a random distribution in the pattern is rejected.
Inter-event distances estimator for a circular region
The simulation envelopes for a circular region are computed in a different way than for a square region. The random pattern generated inside the circle does not follow a uniform
3.2. TRADITIONAL CSR TESTS 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z y 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
inter event distance
Cumulative pr obability Upper envelope Lower envelope Analytical Empirical
Figure 3.2: (Left) artificial cluster point pattern. (Right) simulation results for the inter-event distance distribution with s=1000
distribution. Denote with Cr the circular region of radius r, centered at the origin. Let Mr = N(Cr), be the number of events in this circular region. Since Cr has area πr2, Mris Poisson distributed with parameter λπr2. Let Ri denote the distance of the i-th closest point
to the origin, then
Ri ≤ r if and only if Mr ≥ i, (3.8)
which means that if the radius of the i-th closest event is less than or equal to r, there are at least i events in the circular region with radius r. In particular, with i = 1 and r =√a,
P (R2
1 ≤ a) = P (R1 ≤
√
a) = P (M√
a> 0) = 1 − exp−λπa. (3.9)
In other words: R21 is exp(λπ) distributed [28]. For a general i a similar equation holds,
P (R2 i ≤ a) = 1 − exp−λπa i−1 X j=0 (λπa)j j! , (3.10)
which means that R2i has a Γ(i, λπ) distribution. Consequently, the simulation envelopes for a circular region are simulated by sampling from this Gamma distribution.
3.2.2
Nearest-Neighbour distances method
This method resembles very much the Inter-Event distances method as both methods are based on computing an empirical distribution function for the distances among points in a given region.
Let didenote the distance from the ith event to the nearest other event in a region W con-taining n points. The di are called the nearest-neighbour distances [24][26]. The empirical distribution function bFN N is computed as,
b
FN N(d) = Ndi≤d
n (3.11)
where Ndi is the number points for which the distance di is less than d.
The theoretical distribution FN N under CSR is not expressible in closed form because of complicated edge effects [24]. However, some approximations can be done by ignoring this edge effects,
FN N(d) = 1 − (1 − πd2|AW|−1)n−1 (3.12)
The upper and lower simulation envelopes from the sampled bFN Ni: i = 2, ..., s are
generated exactly as in section 3.2.1. Moreover, the CSR hypothesis is tested is the same way as for the Inter-event distances method. Therefore, specific examples are not developed to illustrated this method since only the empirical distribution function bFNN is used as test parameter in Chapter 4.
3.2.3
Ripley’s methods
Ripley’s methods are a spatial statistics technique widely used in ecology where the spatial distribution of organisms is assessed [27]. Ripley’s methods have been adapted and applied to localized corrosion to study the factors that control the formation of different patterns [39]. These methods are used to determine deviations from CSR by applying a normalized estimator bL of the mean number of events within a circle of radius h defined as [41],
b L(h) = q b K(h) π − h where bK(h) is defined as,
b K(h) = λ−1Pn i=1 Pn j=1 i6=j 1(dij ≤h) wijn
3.2. TRADITIONAL CSR TESTS
where the proportion of the circle within the study area centered at the i-th event with radius equal to the distance dij between the i-th and the j-th event is wij, 1(dij≤h) is the indicator
function, which is 1 if the argument is true and 0 otherwise and λ is the intensity or mean number of events per unit area of the process.
Ripley’s estimator bK is approximately unbiased for sufficiently small h. However, bias is introduced when the radial distance becomes very large [24], possibly rendering an un-bounded estimator that would be meaningless [43].
3.2.4
Quadrat Counts method
The Quadrat Counts method is an alternative approach to the distance based methods previ-ously presented. It is a spatial statistics technique widely used in Biostatistics. Recently, it has been applied to engineering when the distribution of components or material composition is needed [48]. The method’s name derives from its application to ecology where a group of rectangular plots (quadrats) of land are used in the study of plants or animals distribution.
In the Quadrat Counts method a partition of a given domain W into M sub-regions (quadrats) of equal area is necessary. Then, the Complete Spatial Randomness (CSR) test is carried out in the counts of number of events in m. Figure 3.3 illustrates the process.
W W
M
1
Figure 3.3: The domain of analysis W is partitioned into m quadrats of equal area
Under the hypothesis of CSR, the number N (B) of events in any region with area B fol-lows a Poisson distribution with mean λB, where λ is the intensity. Explicitly, the probability distribution of N (B) is
Pn(B) = exp(−λB)
(λB)n
n! n = 0, 1, 2, ... (3.13)
The available data is then comprised of independent counts n1, n2, ..., nM in M quadrants each with area B [24]. More specifically, let us assume that W is the unit square and is
partitioned into a regular k × k grid of square sub-regions, so that M = k2. Let ni, i =
1, ..., M, be the quadrat counts which result from the partitioning of W . Then ¯n = ntotal/M is the sample mean of ni.
The hypothesis that the niare an independent random sample from a Poisson distribution is tested statistically computing Pearson’s criterion or index of dispersion ,
I = M X i=1 (ni− ¯n)2 (M − 1)¯n (3.14)
Under CSR, the sampling distribution of (M − 1)I is χ2M −1. The performance of the test increases with M and it is generally effective checking aggregated alternatives to CSR [29]. In order to apply the index of dispersion test to empirical data, it is necessary to define the significance level α for which the null hypothesis will be rejected. It is relevant to recall that the postulate that the niare an independent random sample from a Poisson distribution is the null hypothesis in the Quadrat Counts method. The significance level chosen is α = 0.05, which means that the alternative hypothesis is accepted when the results of the statistical test fall within the interval (0.95, 1].
Advantages of the method are its straight forward implementation and its efficiency in detecting clustered patterns. As drawbacks, it can be mentioned that the results are affected by the quadrat size [24] [47] and that it performs poorly in detecting regular patterns.
The χ2 goodness of fit test is explained later in this Chapter as it will be used in Chapter 4.
3.3
Spatial Point Processes
A spatial point process is a stochastic mechanism that generates countable sets of events xi in a plane [24]. Generally, a distinction must be done between the spatial processes as they can be classified as stationary or non-stationary. A process is stationary if its probability properties remain invariant under an arbitrary translation in the plane. Conversely, a process is non-stationary if its probability properties depend on the spatial location.
In this Section, three spatial processes are explained. Each process describes a specific distribution of a pattern in space. In Chapter 4, these processes are used to test the perfor-mance of different spatial statistics methods.
3.3. SPATIAL POINT PROCESSES
3.3.1
The homogeneous Poisson point process
The homogeneous Poisson point process (HPPP) is a stationary process and it is considered as the simplest mechanism to generate spatial point patterns [24]. A realization from the ho-mogeneous Poisson point process (HPPP) is illustrated in Figure 3.4. The formal definition of the HPPP is given by the following postulates,
Figure 3.4: Point pattern corresponding to a realization of the homogeneous Poisson point process
1. Consider a planar region W with area AW. The number n of events xi, i = 1, ..., n in W follows a Poisson distribution with mean λAW,
2. Any set of events {xi}ni=1 in region W is a random sample of independent uniformly distributed events on W .
Postulates 1. and 2. correspond exactly to the definition of complete spatial randomness given in Section 3.1 . The homogeneous Poisson process is used as an idealized standard of this test since it gives a helpful approximation to some observed patterns despite the fact that this kind of pattern is seldom encountered in practice [24].
3.3.2
The nonhomogeneous Poisson point process
The nonhomogeneous Poisson point process (NHPPP) is a non-stationary process as its in-tensity function λ(x) changes through the plane. Figure 3.5 presents a realization of this process. The nonhomogeneous Poisson process is defined with the following postulates,
Figure 3.5: Point pattern corresponding to a realization of the nonhomogeneous Poisson point process
1. Consider a planar region W . The number of events xnin W follows a Poisson distri-bution with meanRW λ(x)dx,
2. Any set of events {xi}ni=1in region W form an independent random sample from the distribution on W with probability density function (PDF) proportional to λ(x). This kind of pattern is more often encountered in practice. Hence, in Chapter 4 this process is applied to improve the analysis of empirical corrosion patterns.
3.3.3
The Neyman-Scott process
This process is also known as the Poisson cluster process. It was developed by Neyman and Scott [31] to model aggregated spatial point patterns. Figure 3.6 illustrates a realization of this process. The Neyman-Scott process is defined by the following postulates [26],
1. The parent events form a stationary Poisson process with intensity λp
2. Each parent produces a random number of offsprings which are scattered indepen-dently and with an identical distribution around the parent
The pattern produced by this process is mostly composed by the offsprings unless stated otherwise. The Neyman-Scott process is used in Chapter 4 to generate a cluster pattern in order to test the performance of a method based on the non-homogeneous Poisson process
3.4. GOODNESS OF FIT TESTS
Figure 3.6: Point pattern corresponding to a realization of the Neyman-Scott process
3.4
Goodness of Fit Tests
A goodness of fit test is a statistical hypothesis test. It measures the discrepancy between the observed data and the data whose distribution (under hypothesis) is known [32]. In this way it can be checked wether two samples are drawn from identical distributions or whether the computed frequencies follow a specified distribution.
The significance level is a common term when testing hypothesis. The hypothesis to test is rejected at the α significance level if the value of the test statistics calculated form the data is in the upper α of the distribution. Through this document the α used in the computations is equal to 5%.
In this section only two statistical hypothesis tests are explained as they will be used later in Chapter 4.
3.4.1
χ
2goodness of fit test
The χ2 test is used to find out whether a sample comes from a population with a specific distribution. It is applied to data divided into groups and it is defined by the null and alter-native hypotheses. The null hypothesis states that the events follow a specified distribution. Conversely, the alternative hypothesis states that the events do not follow the specified dis-tribution.
In the general χ2 goodness of fit test the null hypothesis is that the relative frequency of observed events follow a specified frequency distribution. The events are assumed to be
independent and to share the same distribution. Then, the χ2 computation for M groups can be described as χ2M −1 = M X i=1 (Oi− Ei)2 Ei (3.15)
where Oi is the observed number of elements of group i and Ei is the expected number of elements for group i. The number of degrees of freedom in this test is equal to the number of possible outcomes minus one (M − 1). The method performance increases with the number of groups.
The Quadrat Counts method uses this hypothesis test to assess departures from CSR where M is the total amount of quadrats in which an area is divided.
3.4.2
Kolmogorov-Smirnov goodness of fit test
The Kolmogorov-Smirnov (K-S) test is based on measuring the distance between the empir-ical distribution function and the hypothesized distribution function [32]. In a similar way as with the χ2 test, this test is used to decided whether a sample comes from a specific distrib-ution [33]. Therefore, the null and alternative hypotheses are exactly the same as in the χ2 test.
The K-S test statistics for n events is defined as KS = max 1≤i≤n{F (pi) − i − 1 n , i n − F (pi)} (3.16)
where pi are the events ordered from the smallest to the largest value and F is the theoretical cumulative distribution of the distribution being tested which must be continuous. If F is continuous then the KS test converges [30].
Almost all mathematical software has built in a K-S statistical hypothesis test to analyze sets of data. Therefore, manual implementation is seldom necessary.
CHAPTER
4
APPLICATIONS OF SPATIAL
STATISTICS TO CORROSION
As mentioned in Chapter 3, the statistical models developed to assess the failure probability and the pit growth do not consider the influence of the spatial location in their formulation. The scarcity of models involving the spatio-temporal relation is due to the fact that studies regarding the influence of the spatial variable are rather recent. It is important to clarify that the spatial location of an event in a plane is what is considered in this document as the spatial variable. Any other spatial parameter such as the pit depth in terms of the spatial growth is not accounted for in the models here formulated.
Significant work focused on the analysis of corrosion patterns by means of spatial tech-niques has been carried out in the last decade. J. L´opez De La Cruz et al. [64] applied a Bernoulli lattice process [26], a simple spatial point process, to generate samples of pitting spots in iron. Harlow and Pei [34] and Cawley and Harlow [35] implemented the Nearest-Neighbour method to study the arrangement of constituent particles in an aluminum alloy and its influence in the nucleation of corrosion pits. Punckt et al. [36] investigated the nucleation and evolution of individual pits experimentally. Additionally, they applied the Nearest-Neighbour method to their empirical data and concluded that individual pits
en-hance the probability of appearance of further pits at defect sites. Feng et al. [38] studied the pitting process on a copper electrode. Their findings show that early pitting proceeds in favor of nucleation of pit clusters after the Nearest-Neighbour method is applied. Budiansky et al. [39] employed spatial statistics to characterize the spatial patterns of pitting sites on AISI 316 stainless steel. By applying the method’s of Ripley [41], a spatial statistics technique, the authors found evidence of pit interaction in their experiments. Organ et al. [40] applied the bL function, a Ripley’s method [41][42], to analyze metastable pitting in a homogeneous metal surface. Their results show that pits exhibit significant clustering. Therefore, they concluded that the pits were interacting among each other.
In this chapter several spatial statistics techniques are applied to published empirical data. In some cases, the findings of the methods contradict each other. Consequently, a new method based on the combination of distance-based statistics and quadrat analysis is used to overcome the drawbacks of the methods employed until now.
4.1
Inter-Event Distances Method Applied To Corrosion
Patterning Analysis
The basics of the Inter-Event distances method is explained in Chapter 3. In this section the Inter-Event distances method is applied to published empirical corrosion data in order to assess the distribution of pits in each empirical pattern. J. L´opez De La Cruz et al. [43] presented the outcome of this analysis where the results obtained are compared with the results by other authors for the same corrosion pattern.
Punckt et al. [36] collected the data of the onset of pitting corrosion on stainless steel in 50 mol/m3 NaCl at 22 ◦C. The data is gathered in situ visualizations directly in the elec-trolyte using two different techniques: ellipsomicroscopy for surface imaging (EMSI) and a specially adapted high-resolution contrast-enhanced optical microscopy. In both techniques the current density is monitored in parallel. In this way, stable and metastable pits are recog-nized. The ellipsometry method (EMSI) permits real time observations of ultra thin layers in the metal surface. But since the spatial resolution of the EMSI is limited to about 12 µm, it exhibits some problems capturing individual pits. Therefore, the need of a high-resolution optical microscopy. The data of this study is presented in a movie format from where a snap-shot of size 0.4 × 0.4 mm2 at time 44 s is shown in Figure 4.1. The authors modified the
4.1. INTER-EVENT DISTANCES METHOD APPLIED TO CORROSION PATTERNING ANALYSIS 0 0.4 0.4 x (mm) y (mm)
Figure 4.1: Sudden onset of pitting corrosion for stainless steel in 50 mol/m3 NaCl, Punckt et al. [36]. The green spots are metastable pits whereas the red spots are stable pits
movie format adding green dots to the pits that they identified as metastable and red dots to the stable pits.
The Inter-Event distance method is applied to the pattern in Figure 4.1 after the coordi-nate pair of each pit kind (stable/metastable) is obtained. The x and y coordicoordi-nates of each pit are measured with a CAD program where the spatial location of the cursor centered in a pit is delivered by the program.
0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Upper envelope Lower envelope Analytical Empirical
Inter-Event distances method
Cu m ula tiv e pr obabilit y 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Upper envelope Lower envelope Analytical Empirical
Inter-Event distances method
Cu m ula tiv e pr oba bili ty
Figure 4.2: Stainless steel in 50 mol/m3NaCl after 44 s. (left) Inter-Event distances method for stable pits. (right) Inter-Event distances method for metastable pits
Figure 4.2 presents the results of the Inter-Event distances method after the empirical pattern in Figure 4.1 is analyzed. The image shows 9 stable pits and 36 metastable pits. The other small red spots in Figure 4.1 passivated immediately, therefore they are not accounted.
The upper and lower envelopes of the Inter-Event distances method are computed after 1000 simulations (s=1000) are performed. The results with more simulation coincide with the results obtained after 1000 simulations. The left graph in Figure 4.2 shows the result of the Inter-Event distances method for the stable pits. The right graph shows the result for the metastable pits. Both satisfy the CSR hypothesis, thus interaction among the pits cannot be established.
Figure 4.3: Spatial distribution of pits in 2% (weight) NaCl, Hashimoto et al. [45]
0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Inter-Event distances Cumulative pro bability Upper envelope Lower envelope Analytical Empirical
Figure 4.4: Inter-Event distances method for the spatial distribution of pits in 2% (weight) NaCl
In evidence of this result, the hypothesis proposed by Punckt et al. [36] regarding the existence of interaction among pits is not validated by the Inter-Event distances method. Analysis of the patterns at different times yielded no rejection of the CSR hypothesis.
4.1. INTER-EVENT DISTANCES METHOD APPLIED TO CORROSION PATTERNING ANALYSIS
In Figure 4.3 a spatial distribution of pits is presented. This pattern corresponds to an iron specimen (exposed area of 1 × 10−4m2) after immersion for 292 h in a 2% (weight) NaCl solution [45]. The photograph is taken with a 100× magnification from where the authors computed the pits center coordinates. The pattern analysis is performed for 49 pits. The graph in Figure 4.4 shows that the pits in this case are randomly distributed since the Inter-Event distances empirical distribution falls inside the simulated envelopes.
Figure 4.5 presents the location of pit site centroids (x − y coordinates) of a 316 stainless steel in neutral NaCl solution [39]. This image is taken by the authors using a Nikon Epiphot optical microscope fitted with a Nikon DXM 1200 digital camera which allows a magnifica-tion of 200×. The image is convert into black and white using a software for image analysis from where the x-y coordinates of the pits centroides are computed.
Figure 4.6, shows the result of the inter-event distances distribution for the pattern in Fig-ure 4.5. The pattern is composed by 190 pits. The EDF falls inside the simulation envelopes, thus the CSR-hypothesis is not rejected. Moreover, it is possible to observe that the EDF is very close to the lower simulation envelope which means that the probability of finding small distances among pits is high but the hypothesis of randomness cannot be discarded.
0 50 100 -50 -100 100 50 0 -50 -100 x (µm) y (µ m)
Figure 4.5: Location of pit site centroids for 316 stainless steel in neutral NaCl solution, Budiansky et al. [39]
Budiansky et al. [39] obtained a different result in their research by using the methods of Ripley. This difference might be due to the way in which the random patterns are generated on the circle. They generated their pitting sites with a random number generator at discrete x − y locations. However, the x and y location of a random pattern on a circle are not
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Inter-Event distances Cumulative pro bability Upper envelope Lower envelope Analytical Empirical
Figure 4.6: Inter-Event distances method for 316 stainless steel in neutral NaCl solution
independent. Although, in polar coordinates the radius and the angle of a random process are independent [28], where the radius follows a Gamma distribution Γ(i, λπ) and the angle a Uniform distribution U (0, 2π).
Figure 4.7: Electron micrograph of 2024-T3 aluminium alloy after exposure to 500 mol/m3 NaCl solution for 72h at 40◦C [35]
Cawley and Harlow [35] analyzed the pattern in Figure 4.7 by means of the methods of Ripley and the Nearest-Neighbour method. The pattern, corresponding to a 2024-T3 aluminium alloy after exposure to 500 mol/m3 NaCl solution for 72 h at 40 ◦C, is taken with an optical microscope. The pits coordinates are measured by the authors to perform the spatial analysis of the pits. The same values defined by them are used here for the Inter-Event distances method.
Figure 4.8 illustrates the result of the Inter-Event distances method for the pattern in Figure 4.7. The analysis is performed for 149 pits. The EDF falls inside the simulation
en-4.2. QUADRAT COUNTS METHOD FOR THE ANALYSIS OF CORROSION PATTERNS 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Empirical Upper envelope Lower envelope Inter-Event distances Cu m ula tiv e pr obabilit y
Figure 4.8: Inter-Event distances method for 2024-T3 aluminium alloy after exposure to 500
mol/m3 NaCl solution for 72h at 40◦C
velopes, therefore the CSR-hypothesis is not rejected. Consequently, the findings of the Inter-Event distances methods are in agreement with Ripley’s methods and the Nearest-Neighbour method for the same corrosion pattern.
From the spatial statistics techniques, the distance-based approaches are commonly used in the corrosion field. However, some of these methods are very sensitive to record inaccu-racies in the data [24]. For this reason, it is advisable to perform pattern analysis by means of different tests since each test examines different aspects of a pattern Cox [44].
An alternative method to the distance-based approaches is the Quadrat Counts method explained in Chapter 3. The following Section presents this method applied to corrosion pattern analysis.
4.2
Quadrat Counts Method For The Analysis Of
Corro-sion Patterns
As mentioned in Chapter 3, this method is applied in mathematical ecology where the in-terplant dispersion of a disease is measured by counting the amount of plants infected in a region [46] [47]. Recently, Cunningham et al. [48] applied also the Quadrat Counts method to identify the spatial pattern of visual defects on wafers used in the assembling of integrated-circuits and transistors.
In corrosion, the Quadrat Counts method offers a suitable alternative to analyze pit pat-terns. L´opez De la Cruz et al. [43] showed that two widely used distance-based methods,
Inter-Event distances and Ripley’s methods, failed to agree in their results for the same cor-rosion pattern. In such cases, an alternative method like the Quadrat Counts can increase the outreach of pattern analysis [49].
4.2.1
Empirical pattern analysis and quadrat size dependence
The patterns in Figures 4.5 and 4.7 are analyzed once again with the Quadrat Counts method [49]. This method presents a good alternative to methods previously applied as the distribu-tion of distances among pitting events does not make part of the analysis as it is the case for the methods previously selected by Budiansky et al. [39] and L´opez De La Cruz et al. [43].
Figure 4.9, shows the grid sizes employed to test the CSR hypothesis applying the Quadrat Counts method for the pattern in Figure 4.5. The pit centroid coordinates are nor-malized to fit a square [−1, 1]2. In this fashion, each quadrat has the same area.
−1 −0.8 −0.6 −0.34 −0.1 0 0.1 0.33 0.6 0.8 1 −1 −0.8 −0.6 −0.34 −0.1 0 0.1 0.33 0.6 0.8 1 x y −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x y
Figure 4.9: Quadrat Counts test results for the 316 stainless steel pattern in neutral NaCl solution after the pits are normalized to fit the square [−1, 1]2. (left) 3 × 3 grid, (right) 4 × 4 grid
Figure 4.9 (left), shows the grid m = 9 for which the χ2m−1 test result is equal to 0.9999. Therefore, a random distribution of pits is rejected. Figure 4.9 (right), illustrates the grid m = 16. For this case the χ2
m−1 test result is equal to 1. Again, the null hypothesis of pits randomly distributed is rejected. For the case of Figure 4.9 (right), it is possible to observe that the inclusion of an empty quadrat in the analysis cannot be avoided and bias is introduced in the results.
4.2. QUADRAT COUNTS METHOD FOR THE ANALYSIS OF CORROSION PATTERNS 0 100 266.6300 400 533.3 600 700 800 0 100 266.6 300 400 533.3 600 700 800 x [µm] y [µ m] 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 y x [µm] [µ m]
Figure 4.10: Quadrat Counts test results for the 2024-T3 aluminium alloy pattern after ex-posure to 500 mol/m3 NaCl. (left) 3 × 3 grid, (right) 4 × 4 grid
Figure 4.10 shows the grid sizes used to test the CSR hypothesis applying the Quadrat Counts method for the pattern in Figure 4.7. After the χ2m−1test is applied to the lattice m =
9, the CSR hypothesis is not rejected. The statistical test gives a result of 0.70. Therefore, a
random distribution in the pit pattern is not rejected.
Figure 4.10 (right), shows the result of the χ2m−1 test for the lattice m = 16. The results yielded not rejection of the CSR hypothesis. In this case, a random distribution of the pitting sites is assumed after the statistical test gives a result of 0.4596.
The effect of quadrat size can be observed in the results. The change in the test values is noteworthy when going from 9 quadrats to 16. Moreover, the restriction that all the quadrats should share the same area limits the analysis to rectangular or square domains. The next section presents another approach to the index of dispersion (Equation 3.14) employed by the Quadrat Counts method. L´opez De La Cruz and Guti´errez [49] discussed the results of this study and its application to corrosion.
4.2.2
The binomial index of dispersion in corrosion analysis
The influence of the quadrat size in the results delivered by Quadrat Counts method has been the focus of interest for many researchers. In the past, many authors have tried to overcome this issue [53] [50] but the proposed techniques make the results difficult to interpret. Fer-randino’s work [51] is worth mentioning as he implemented a modified binomial index of dispersion which is independent of quadrat size. However, after the technique of Ferrandino
is tested with corrosion patterns, the results show that it performs poorly. Therefore, this technique is briefly explained.
Ferrandino [51] propose to employ a quadrat size independent binomial index of disper-sion (IB) which only depends on the amount of pits in the stable or metastable state. Thus, the analysis of a pattern distribution is based on the amount of pits in the study. That is, if many pits are present, unavoidably the analysis will deliver a cluster pattern. Conversely, few pits will stand for a random pattern.
An alternative method to the Quadrat Counts method and the distance-based methods that does not introduce bias to the analysis is necessary. The next Section presents an innovative method that manages to bring together the distance-based statistics with quadrat analysis to perform reliable pattern studies.
4.3
Goodness of fit to a non-homogeneous Poisson process:
Distance-based statistics and quadrat analysis
In corrosion research the spatial location of pits plays a fundamental role when maintenance models (preventive or corrective) must be formulated. Scarf and Laycock [56] state that larger pitting events are found next to smaller neighbors whose growth is suppressed by the former. Budiansky et al. [39] and Punck et al. [36] show that pits interact among each other after analyzing the results of potentiodynamic experiments in different micrographs of electrodes surfaces. The micrographs illustrate that the dominant pits are surrounded by secondary pitting events highly clustered. The latter suggests that more attention should be given to the distribution of pits in space as in places where clusters of pits are found the risk of corrosion induced failure might be higher.
In this section, a statistical method to assess pit pattern distributions is implemented. The method, developed by Brix et al. [58], has the advantage that it can be applied to different domain shapes and it is very powerful detecting random, regular or clustered patterns. This method is an extension of the classical complete spatial randomness test to non-stationary Poisson point process. So far, the methods discussed assume that the pit patterns are station-ary. In the corrosion field the observed pitting events are frequently non-stationary and pit clusters are usually found.
