Multi-state Markov modeling of pitting corrosion in stainless steel exposed to chloride-containing environment

Accepted Manuscript

Multi-state Markov Modeling of Pitting Corrosion in Stainless Steel Exposed to Chloride-containing Environment Yi Xie , Jinsuo Zhang , Tunc Aldemir , Richard Denning PII: DOI: Reference:

S0951-8320(17)30172-2 10.1016/j.ress.2017.12.015 RESS 6035

To appear in:

Reliability Engineering and System Safety

Received date: Revised date: Accepted date:

10 February 2017 2 December 2017 26 December 2017

Please cite this article as: Yi Xie , Jinsuo Zhang , Tunc Aldemir , Richard Denning , Multi-state Markov Modeling of Pitting Corrosion in Stainless Steel Exposed to Chloride-containing Environment, Reliability Engineering and System Safety (2017), doi: 10.1016/j.ress.2017.12.015

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Highlights Multi-state Markov approach to capture the probabilistic aspects of pitting corrosion. Full description of pitting corrosion states based on pitting corrosion mechanisms. Semi-heuristic exhaustive search scheme to determine input parameters. Simulation of the variation of pit depth and density. Comparison of simulation results with experimental measurements.

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Multi-state Markov Modeling of Pitting Corrosion in Stainless Steel Exposed to Chloride-containing Environment Yi Xie, Jinsuo Zhang*, Tunc Aldemir, Richard Denning Nuclear Engineering Program, The Ohio State University, Columbus, OH 43210, United States

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* Corresponding author, Email: [email protected], Phone: (614) 292-5405, Fax: (614) 2923163, Postal address: 201W 19th Ave, Columbus, OH 43210, USA. Abstract

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Although stainless steels (SSs) have excellent general corrosion resistance, they are nevertheless susceptible to pitting corrosion. The variation of pit depth and density is significant for the prediction of likelihood of corrosion damage occurring in service. Among the available pitting corrosion models, it is difficult to identify a specific model capable of characterizing all the pit formation processes observed and one that can be used for estimating the evolution of pit density distribution with time. A physics-based multi-state Markov model giving a full description of pitting corrosion states is presented. The transition rates used in the model are determined by fitting the model to experimental data. The variation of pit depth and density is simulated. The simulation is verified by experimental scenarios of SS exposed to chloride-containing environments.

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1. Introduction

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Stainless steels (SSs) are used in many diverse applications due to their corrosion resistance. Although they have excellent general corrosion resistance, they are nevertheless susceptible to pitting corrosion [1]. Pitting corrosion of metals and alloys is one of the leading causes of structural failure in industrial systems (e.g. [2-5]). Most of the pits are generated at the beginning when first exposed to the environment rather than uniformly in time with constant rates [6]. For SS, the pitting corrosion locations usually occur on the manganese sulfide (MnS) inclusions [7,8]. MnS inclusions are formed during the steel production process: sulfur is added to improve machinability of the steel and Mn is added to precipitate S as MnS in the liquid state during solidification [9]. The density of inclusions correlates well with the density of pitting corrosion sites [10].

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Pitting corrosion is a stochastic process. Frankel [11] indicates that the pitting process contains three stages in chronological order: passive film breakdown, metastable pitting, and stable growth. Breakdown is a rare occurrence that happens extremely rapidly on a very small scale, making direct observation extraordinarily difficult [11]. Thus, the initial state of the surface has no pits and is protected by a passive oxide layer. Probability exists that pits will form and grow through the diffusion of aggressive ions. This probability depends on random fluctuations in local conditions [11,12]. Metastable pits are pits that initiate in micro size (above the level of detectability of the modern microscopy technology) and grow for a limited period before repassivating, or entering a state of stable pit growth [11]. Repassivation occurs when the oxygen diffuses and oxide passive layer regrows at the pit location, returning the surface to a passive condition. Accordingly, three states can be used to describe the metastable pits: growth state, 2

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declining state, and repassivation state. Declining state indicates that the pit depth is decreasing because of the repassivation property of pitting corrosion, which is an oxidation process. Thus, a metastable pit can either enter a state of stable growth or decline to a passive state. In the stable growth stage, the growth rate depends on material composition, pit electrolyte concentration, and pit bottom potential [11]. Note that even what appear to be small stable pits may in fact be metastable [11]. Hence, the actual micro size and limited period of a metastable pit is dependent on the behavior of the pit. The criterion used to determine whether a pit is stable is the achievement of a critical depth beyond which the pit will continue to grow. Below the critical depth, the pits are in the metastable pitting stage in which they may either be growing or declining. If the pit achieves the critical depth; it enters a state of stable growth. The critical depth is empirically determined and depends on the material composition and environment. For example, the critical depth of SS304 at 70 and 80 °C NaCl solution were assessed as 6.4 and 9.4 µm, respectively [13]. Generally, the critical depth of SS304 is recognized as 5-10 µm. With micro-characteristic analysis techniques, all the states can be observed [14]. Some other techniques, such as an electrochemical technique involving analysis of the noise spectrum in the measured current can reveal metastable pit formation [15].

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Pitting corrosion depth and density of pits are used to describe pitting corrosion, quantitatively express the pitting significance, and predict the life of a material. The variation of depth and density can be used to estimate the degree of corrosion attack, and provide input data for the prediction of the change of mechanical properties as a function of material composition and exposure conditions. However, the fundamental knowledge on pitting corrosion modeling is limited and requires further study [16]. Approaches developed for modeling the probabilistic behavior of the pitting corrosion process include the use of Weibull distribution [17], Poisson distribution [18], generalized extreme value distribution [19,20], physically-based differential equations [21], a combination of static and dynamic probabilistic process models [22-25] and the Monte Carlo method [26,27]. Among the available models, it is difficult to identify a specific model capable of characterizing all the pit formation processes observed and estimating the variation of depth and density. In addition, most existing models require initial conditions to be provided as input parameters rather than determining them on a theoretical basis, limiting the applicability of the models. Pitting corrosion is highly sensitive to different environments, such as the temperature, pH and humidity; however, most developed models do not have clearly defined limits of applicability with regard to environment or type of SS.

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In this study, we consider the use of a Markov model [28] to capture the probabilistic aspects of predicting pit depth and density as a function of time. The model uses a semi-heuristic exhaustive search scheme to estimate the transition rates of the Markov model from experiments, and illustrates the applicability and flexibility of the approach by implementation in different environments and for different materials. Section 2 includes a condensed summary of the relevant concepts and methods used for pitting corrosion analyses. In Section 3, we present the Markov approach and the search scheme to determine the transition rates. An important aspect of our Markov approach has the ability to model a non-Markovian physical process as a Markovian process through introduction of auxiliary states (Section 3.2). In Section 4, the model is implemented for SSs exposed to chloride-containing solutions. Two cases are presented to demonstrate the reliability and flexibility of the proposed model. Section 5 presents the conclusions of the study and recommendations for future work. 3

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2. Concepts and methods for analyzing pitting corrosion

2.1 Deterministic and non-deterministic models

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This section is a summary of the important concepts and methods in the analysis of pitting corrosion behavior, which involve deterministic (or mechanistic) and non-deterministic approaches. Section 2.1 describes the deterministic models, which are commonly used to simulate the pit growth rate, and compares it with the models that incorporate the stochastic nature of pitting corrosion. Section 2.2 discusses the static and dynamic stochastic models, which are two distinct non-deterministic methods of pitting corrosion analysis, and indicates the contributions of the multi-state Markov model.

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The wide range of diverse pitting corrosion models that have been proposed can be categorized as two distinct model types: deterministic (or mechanistic) and non-deterministic. The former is often formulated using partial differential equations based on a set of variables and environmental parameters, deploying mathematical representation of various physical and chemical mechanisms to simulate the evolution of pitting corrosion [29-31]. Different deterministic models use diverse variables to characterize the different aspects of corrosion mechanism. Partial differential equations for describing pitting corrosion can be based on either reaction kinetics or electrochemistry [32]. Since there are conditions under which further growth of a stable pit is arrested, the deterministic models consider mechanisms that provide limitations on pit size (e.g. [33]). Under freely corroding conditions, a pit acts as the anode and the surrounding material acts as the cathode; a stable pit can continue to grow only as long as its anodic current is matched by cathodic current from the surrounding material [11, 33]. However, stochastically-based pitting models typically do not account for limitations on pit size. Particularly for long-term pitting corrosion, models are needed that address both the stochastic aspects of pit growth but also include a mechanistic treatment of limitations on pit size.

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Non-deterministic models utilize the probabilistic characteristics of data to interpret or evaluate certain aspects of the evolution of pitting corrosion [26,34,35]. These probabilistic characteristics can be either static or dynamic, which will be further expanded upon in Section 2.2. One example of a dynamic probabilistic model divides the metal surface into a two-dimensional (2D) array of hypothetical cells, then assigns the probabilities for the transitions among pitting states to each cell [27]. In the dynamic analysis, nucleation or destruction of a pit embryo is determined probabilistically by taking random draws from assumed probability density functions for pit formation and growth properties. After a pit embryo grows to a certain stage, it becomes a stable pit and follows the stochastic rules for pitting growth. The environmental impacts enter the model based on a combination of theoretical modeling and observed data. For the initial stages of pit formation, probabilistic models provide a better representation of what is empirically observed than deterministic models because of the random nature of the nucleation process. However the non-deterministic models proposed to date treat pitting mechanisms simplistically and are incapable of predicting pit growth behavior based on material composition and chemical environment. Besides, the non-deterministic models are mostly locally applicable, not incorporating any global behavior (e.g. [26]).

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Mears and Evans stated in 1935 [36] that “from the practical standpoint … it may … be more important to know whether … corrosion is likely to occur at all than to know how quickly it will develop.” Thus to date, deterministic approaches have so far prevailed in corrosion science. Nevertheless, the potential value of non-deterministic analysis has been demonstrated. Stochastic theory has been successfully used to explain pitting corrosion behavior, such as the probability distribution of both the pitting potential and the induction time [37]. Work by Henshall et al. [38] showed that a computational model based on a stochastic approach could describe the pit initiation and growth on SSs, but needs to involve some deterministic elements. To predict the complex corrosion process, we conclude that we need a computational capability that is probabilistic in formulation but includes mechanistic models. 2.2 Static and dynamic stochastic models

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The static non-deterministic approach involves fitting of a probability distribution function or a combination of several probability distribution functions to a set of observed values of the random variables of pit shapes, i.e. pit depth and radius. The majority of researchers agree on the concept that it is impossible to use a single probability distribution function to simulate the complex pitting corrosion process [39]. Although a combination of several probability distribution functions can provide a good fit for the complicated pitting corrosion process, the limitations are still distinct, including the assumption of nominal “homogeneity” in the system (e.g. random distribution of material microstructure) and the inflexibility in dealing with the long-term changes in operating conditions, environment and the pit shape [40]. Furthermore, there is no static approach that has included all the time-dependent factors (e.g. pH and ion concentration in the pit, pit density and shape) for naturally induced pitting corrosion. The longterm changes in behavior should also be taken into account that affect the ultimate severity of pitting damage.

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The DSMs were found to be applicable to pit generation events [26,39]. The pit initiation and repassivation can be described as a birth and death stochastic process [38]. The relation between the distribution of pitting potential and the induction time for pit generation can be stochastically derived [12]. The random current noise generated by pitting corrosion obtained by electrochemical techniques can be modeled as a stochastic process [41,42]. DSMs can be applied that include the effects of inclusions and inhibitors [24].

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Because of the substantial interest for a flexible approach to deal with the long-term changes of pitting corrosion, the dynamic stochastic methods (DSMs) seem to be better suited for events taking place over time than static probabilistic methods. DSMs can also quantitatively include the time-dependent environmental parameters that influence the “birth and death” of corrosion pits. Other advantages of DSMs include:

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A number of potential corrosion growth models were evaluated from a survey in [43]: the linear growth model, time-dependent generalized extreme value distribution (GEVD) model, timeindependent GEVD model, single-value corrosion rate model (National Association of Corrosion Engineers (NACE) model) and Markov model. It was concluded that the Markov model is the best among them for predicting the corrosion rate distribution with time as it considers the ages 5

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and sizes of the corrosion defects as well as the observed dependence of the corrosion defect depth on time [43].

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Markov models that have been used to model pitting corrosion have shown to agree well with experimental results. However, improvements in the method are still needed. For instance, Valor, et al. [40] have treated the pit initiation/generation process as a non-homogeneous Poisson process, and the pit growth process as a non-homogeneous Markov process. Although the model obtains the distribution of maximum pit depths resulting from the combination of the initiation and growth processes for multiple pits, it does not provide an approach for estimating the longterm development of pitting corrosion, nor the resolution of the pit depth distribution with density. Also, most pitting corrosion models are based on a physical description of how a pit is generated and grows on the material surface (e.g. [26,27]) but do not analyze all the pitting corrosion states or interpret all the pit formation processes. The input parameters for the models relative to the environmental factors, such as temperature, pH and humidity for different materials, are assumed and suitable for a very limited range of materials and environments. To address these limitations, we propose the multi-state Markov model to simulate the pitting corrosion states and pit formation process. The model is able to describe the dynamics and determine the input parameters and transition rates accounting for environmental effects (chloride concentration and temperature) for different materials, instead of assuming them. 3. Model development

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This section presents the development of a multi-state pitting corrosion Markov model. It is assumed that the critical depth is 5 µm, and the model is valid only if the equivalent cathodic current for each stable pit is large enough to support the growth. Section 3.1 introduces the pitting corrosion states, and discretization of sub-states based on pit depth. Section 3.2 presents the physical meaning and expression of state transition rates. Section 3.3 illustrates the approach to determine pit density from experimental data. Section 3.4 presents the exhaustive search scheme to determine input parameters to the Markov model. Section 3.5 shows the process to represent the variation of pit depth and density with time. 3.1 Pitting corrosion states and discretization

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Figure 1 shows the pitting corrosion state transition diagram, including the initial state (S), growth state (G), declining state (D), repassivation state (R) and critical state (C).

Figure 1 A physics-based pitting corrosion state transition diagram. S: initial state, G: growth state, D: declining state, R: repassivation state, C: critical state. 6

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These states have been introduced in Section 1. It is assumed that once the pit reaches the critical state, the cathodic current for each stable pit is large enough to support stable growth. Specifically, in State G, the metastable pitting corrosion has pit depth less than the critical depth (i.e. the depth at which pit reaches a stable size and keeps growing at a positive growth rate without declining or repassivation). A well-known characteristic of pitting corrosion rate is that, in addition to its time dependence, it also depends on the pitting corrosion depth [44]. This means that the transition rates f1 and m1 in Figure 1 depend on the residence times in States G and D, respectively, with the mean value and variance of the corrosion rate distribution undergoing changes over time [45,46], and hence make the physical process represented by the States S, G, D, R and C non-Markovian. In the present work, the dependence of f1 and m1 on state residence times is overcome by the introduction of the intermediate States Gi (i = 1,2,…N) and Dj (j = 1,2,…N-1). The States Gi and Dj are obtained by discretizing the critical pit depth using N states of equal depth L (i.e. step depth), i.e. (1)

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Figure 2 illustrates this discretization of pit depth. Figure 3 shows the state transition diagram with sub-states.

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Figure 2 Discretization of depth in a material matrix.

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Figure 3 A physics-based pitting corrosion state transition diagram with sub-states Gi (i = 1,2,…N) and Dj (j = 1,2,…N-1).

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A pit advances through the states as it grows. In Figure 3, Gi (i = 1,2,…N) is the state of a pit that is at depth iL. State G1 represents the nucleation, GN represents a pit whose size exceeds the critical depth and has stable growth. The λi (i = 1,2,…N-1) is the state transition rate for a pit advancing from State Gi to State Gi+1. λN is the state transition rate for a pit advancing from State GN to the critical state C. The metastable pit has a possibility of declining in size, so each substate Gi has a transition rate to the declining state DN+1-i. The sub-transition rate for any Gi to DN+1-i is a constant m1 since the declining states are not affected by pit depth [27,38].

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Similarly, a pit advances through Dj (j = 1,2,…N-1) as it declines. Dj denotes the probability of a pit in declining state j, and γj is the state transition rate for a pit advancing from declining state Dj to state Dj+1. State D1 represents a declining pit at depth (N-1)L, state DN-1 represents a declining pit at initial depth L.

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The Markov model is expressed by

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[

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[

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{

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where are the probabilities of pits to be in States S, Gi (i = ∑ ∑ 1,2,…N), Dj (j = 1,2,…N-1), R and C, respectively, with . At start, . Sections 3.2 and 3.3 describe the process to determine transition rates in Eq.(2) by exhaustive search. 3.2 State transition rates

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The Weibull distribution has been used to characterize the rate from initial state S to growth state G1 [40]. In the present study, the characteristic time for a macroscopic pit being passivated is also assumed to follow the Weibull distribution. The respective transition rates f1(t) and m1(t) in Eq.(2) are expressed as ( )( ) ( )( )

,

(3) ,

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where βi (i = 1,2) are Weibull shape parameters, τi (i = 1,2) are Weibull scale parameters. The parameters are assumed to be chloride concentration dependent and temperature dependent. As indicated earlier, λi (i = 1,2,…N) in Eq.(2) characterizes the rate of moving between growth states Gi (i = 1,2,…N). The simplest model of pit growth rate is to use empirical values for the average corrosion rate according to the corrosiveness of the medium [47]. When no data for the pit growth rate are available, it’s suggested to use a unique value for corrosion rate of 0.4 mm/year [48]. However, pitting corrosion rate changes with time and environment. It is generally accepted that the maximum pit depth Z(t) follows a power function, (5) 9

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where Z(t) is the maximum pit depth, a(t) and b(t) are empirical parameters accounting for dynamic environmental changes and pitting corrosion states, and u is the time at which the pitting corrosion process begins [44,47,49]. The coefficients a(t) and b(t) in Eq.(5) can be obtained from experimental data. Approximating the maximum pit depth Z(t) as (i = 1,2,…N-1) with L as the step depth given in Eq. (1), it can be deduced that (

(i = 1,2,…N-1).

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is the derivative of Z(t) with respect to time. From Eqs.(6,7) we have

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where

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with the prime notation indicating derivative with respect to time. Substituting Eq.(8) into Eq.(7) yields (i = 1,2,…N-1),

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implying that the corrosion process as modeled is non-Markovian because λi depend on the residence time u in state Gi. Substituting Eq.(6) into Eq.(9) yields ]

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(i = 1,2,…N-1).

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Substituting Eq.(5) into Eq.(10) yields *

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From Eq.(5), we also have

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( ) .

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(

To avoid using two time scales t and u, the local pit time (t-u) in Eq.(12) can be substituted into Eq.(11) to relate λi with iL rather than (t-u), yielding [

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( )] (i = 1,2,…N-1).

The transition rate from growth state GN to critical state C is λN (see Eq.(2)).

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In most studies, a(t) and b(t) are assumed as constant. In this study, a(t) and b(t) are obtained from the experimental data in [50,51] (Sections 4.1 and 4.2). Accordingly, Eq.(13) can be expressed as ( ) (

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In a similar manner using Eqs.(5-12), the transition rates γj (j = 1,2,…N-1) can be expressed as )] (j = 1,2,…N-1)

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where c(t) and d(t) are time-dependent parameters accounting for dynamic environmental changes and pitting corrosion states. As indicated in Figure 3, γN-1 characterizes the rate of transition from declining state DN-1 to repassivation state R. Since pit growth and decline are both diffusion-controlled processes as indicated in Section 1, it is assumed that c(t) and d(t) are equal to a(t) and b(t). Accordingly, Eq.(15) is expressed as ) (j = 1,2,…N-1).

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At this point we should indicate that although critical state C is an absorbing state for the Markov model, pits in State C will grow without declining or repassivation following the pit growth law given by Eq.(5). 3.3 Pit density

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The unit surface area is divided into M equal areas (locations) per cm2, each indicating a potential pit location. At each location, a pit evolves as illustrated in Figure 3. The number of detectable pits in the unit area is defined as pit density (PD), calculated as the product of M and the probability of generating a detectable pit. Detectable pits are in states Gi (i = 1,2,…N), Dj (j = 1,2,…N-1) and C. The pit density PD can be expressed as ∑

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with M (cm-2) as the maximum potential pitting locations per unit area in the time period of interest. Estimating the value of M requires estimation of average area of a pit within the time duration of experiment. This area can be experimentally measured by a microstructure analysis instrument, such as by an optical profilometer. Figure 4 shows an example output from a 3D optical profilometer (Bruker, ContourGT-I) to measure pitting corrosion.

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Figure 4 Example 3D morphology profile of pits observed under 3D optical profilometer [50]. The M in Eq.(17) is expressed as ∑

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where K is the total number of time points (k = 1, 2, …K), A(k) is the average area of a pit at time point k. As indicated in Section 1, pitting corrosion is usually nucleated at a MnS inclusion, the MnS inclusion density is about 6×106 cm-2 [7]. In that respect, M should be equal or less than 6×106 cm-2. 3.4 Process of determining optimum input parameters

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The PD(t) in Eq.(17) is used to fit with the pit density by experimental measurements (PDm) and to seek for the optimum input parameters β1, τ1, β2 and τ2 for the best fitting of the density functions for transition rates in Eqs.(3,4). The least squares estimation (S) is the criterion used to determine the best fit to experimental data. √∑

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where K is the total number of time points. A common method to yield PDm is to count the total numbers of pits over the observed area of the test specimen by optical profilometer.

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From Eqs.(2) through (18), it can be seen that Eq.(19) is non-linear with respect to β1, τ1, β2 and τ2. Solving non-linear fitting problems remains a challenge from both mathematical and computational viewpoints [52]. In principle, the best fit can be obtained by Gauss-Newton, gradient descent and the Levenberg-Marquardt method [53-55]. However, for a model with 4 parameters (i.e. β1, τ1, β2 and τ2) to be estimated, these algorithms require 4 partial derivatives with respect to the 4 parameters in addition to the model and an adequate initial guess as a starting point. In terms of Eqs. (2) through (18), these requirements pose a formidable computational challenge. In Eq.(2), the model consists of 2N+2 equations. With 4 parameters to be estimated, the estimation problem consists of 4(2N+2)+(2N+2) = 5(2N+2) equations. If N =10 (see Section 4), the non-linear estimation problem consists of 110 equations. Just as importantly, the fitting process is sensitive to the choice of the initial guess of the vector of parameters to be estimated to assure that the solution represents a global minimum and not a local minimum. Other non-linear estimation techniques pose similar challenges [56]. For these reasons, a semi-

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heuristic exhaustive search scheme is used to identify the parameter combination that minimizes S.

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Step 1 in determining optimum input parameters is to choose the initial input parameters ⃗ to be as close as possible to the optima in order to avoid being trapped in local minima or not converging to a solution. In Step 2, ⃑ is perturbed by observing trend changes to assure that S is not a local minimum. Figure 5 displays the trend changes in the pit density as a function of time for a one at a time perturbation in each of the four input parameters. The last step (Step 3) is to refine ⃑ by exhaustive search. The exhaustive search scheme used in this study considers all the possibilities by generating the candidate parameters within the range of ± 20 % of the parameters ⃑ obtained at the previous step, applying each candidate ⃑ to the Markov chain, computing S and finding the least value of S among the candidates.

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Figure 5 Trend changes of pit density PD for the input parameters (a) β1 (b) τ1 (c) β2 and (d) τ2. 3.5 Variation of pit depth and density

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With the optimum parameters β1, τ1, β2 and τ2 as described in Section 3.4, the transition rates f1(t), m1(t), λi(t) and γj(t) in Eqs.(3,4,14,16) are obtained. Equation (2) is then solved to yield the detectable pit probabilities (i = 1,2,…N), (j = 1,2,…N-1), and . At each depth iL (i = 1,2,…N), the density of pits can be obtained by multiplying these probabilities by M, [

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Eq.(20) yields distributions for pit subpopulation within specific lifetime and depth ranges. Table 1 is a summary of the variables and input parameters of the Markov model.

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Table 1 Summary of the variables and input parameters of the model. Symbols t

Description Global time Pitting corrosion state probabilities

Z(t)

Variables

λi (i = 1,2,…N-1) λN γj (j = 1,2,…N-2) PD(t)

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PDm(t) σm(t) S

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β1, β2

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K L

τ1, τ2

a(t), b(t), c(t), d(t)

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Input parameters

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Transition rates from S to G1 and Gi to Dj, separately Maximum pitting corrosion depth at time t Transition rates from Gi to Gi+1 Transition rate from GN to C Transition rates from Dj to Dj+1 Pit density at time t estimated by fitting Pit density at time t by experimental measurements Standard deviation of PDm(t) (see Tables 2 and 5, Figures 7-8, 10, 15) Square root of sum of squared residuals between simulation results and experimental measurements The number of time points of measurement Step depth Weibull shape parameters for transition rates f1(t), m1(t) estimated by fitting Weibull scale parameters for transition rates f1(t), m1(t) estimated by fitting Time-dependent parameters accounted for dynamic environmental changes and pitting corrosion states (see Section 4) The number of intervals used to discretize critical depth The potential maximum pitting locations per unit area based on measurement

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f1(t), m1(t)

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4. Case studies This section presents two case studies to demonstrate the reliability and flexibility of the proposed model. The cases include two SSs exposed to different concentrations of chloridecontaining solutions at different temperatures. The experimental data are used to empirically determine parameters in the model and to validate the results.

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4.1 Case 1: SS304L exposed to the 6.25 M NaCl solution at 40 °C

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Using Table 2 data with Eq.(18), we have

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The experimental data have been published in [50]. Different coupons were used to test at 2, 5, 10, 20 and 30 days, i.e. 5 time points of measurement with K = 5. Each coupon surface was 2 cm by 2 cm. Table 2 summarizes the experimental data, including measured pit density (PDm), standard deviation of pit density (σm) and pit radius (rm). For each coupon, 10 scan areas with 1.3 mm2 each were randomly selected and analyzed by optical profilometry. Since the pits were highly irregular in shape, the maximum depth was treated as pit depth, and the longest distance from two edges of a pit was treated as pit diameter. The number PDm is the ratio of total number of pits to scan areas (13 mm2), and the standard deviation σm is based on the variation of PDm from 10 scan areas. Since the pitting process is stochastic by nature, PDm for a large sample is expected to converge, while for a small sample size such as the present case, PDm is disperse and thus has a high statistical uncertainty; in other words, σm is large.

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As indicated in Section 3.2, are the parameters of a power function for corrosion depth as a function of time (Eq.(5)), obtained from [50]. The optimum parameters, β1, τ1, β2 and τ2, describing the distributions for transition rates, were obtained by the search scheme.

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Step 1 of the search scheme was to choose a sufficiently large interval for the parameter under consideration and then reduce this interval by a bisection process that checks if the interval covers the observed data (see Section 3.4). The initial inputs are shown in Table 3. In Step 2 of the process, several trials were performed based on the trend changes shown in Figure 6 (e.g. when decreases, the peak increases as can be seen from Figure 6a). The outcome of Step 2 yielded values for β1, τ1, β2 and τ2 of 0.2, 1×1010, 0.9 and 3.4, respectively. Step 3 was performed in two stages: 1) Determine intervals within ± 20 % of β1, τ1, β2 and τ2 obtained in Step 2, i.e. [0.16, 0.24], [0.8×1010, 1.2×1010], [0.72, 0.99], and [2.72, 4.08], respectively. Each interval was evenly divided into 10 equal partitions. Each partition was represented by the starting point. Accordingly, there were 104 possible combinations yielding 104 values for S in Eq.(19). 2) Find the least S and the corresponding β1, τ1, β2 and τ2. Figure 7 shows the values of S and the least number at 0.0703. The corresponding parameters are β1 = 0.192, τ1 = 1.16×1010, β2 = 0.855 and τ2 = 2.72. Table 4 summarizes the empirically-determined input parameters for the model. The value of N in Eq.(1) was selected to be 10 on the basis of numerical accuracy and the computational efficiency. The pitting behavior cannot be resolved if N is too small or step depth L is too large. 15

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For example, in the experimental measurements, step depth was characterized to be 0.5 µm, therefore L must be equal or less than 0.5 µm to characterize the measured pit growth, i.e., N must be equal or larger than 10 for an assumed critical depth of 5 µm. On the other hand, increasing N or decreasing L will improve the numerical accuracy with the penalty of computation cost. Increasing N did not reduce the minimum value of S significantly, however, at substantially larger computational expense. The fitting result is almost independent of N when N ≥ 10.

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Table 2 Experimental data of PDm, σm and rm for Case 1 [50]. t (day) PDm (/cm2) σm (/cm2) rm (μm) 476.92 761.54 692.31 415.38 361.54

231.62 205.13 393.31 221.17 285.64

12.72 14.92 13.39 10.48 15.76

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2 5 10 20 30

Table 3 Initial β1, τ1, β2 and τ2 ranges for the search process. Range [0.1, 0.5] [109, 1010] [0.8, 0.999] [2, 10]

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Figure 6 Trend changes of density PD with the input parameters (a) β1 (b) τ1 (c) β2 and (d) τ2. Error bars denote σm. 16

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Figure 7 Exhaustive search to find optimum parameters β1, τ1, β2 and τ2 for Case 1 that yield the minimum value for S (red point).

Table 4 Case 1 input parameters. Model input parameter M (/cm2) β1 τ1 β2 τ2 a b Critical depth (µm) N L (µm)

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1.76×105 0.1920 1.16×1010 0.8550 2.720 0.22 [50] 0.9 [50] 5 10 0.5

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With the optimum parameters β1, τ1, β2 and τ2, the results for PD are shown in Figure 8. The empirical model shows that pit density peaks at 760 cm-2 within 5 days and drops to 200 cm-2 within 2 months (Figure 8a); it continues decreasing to a minimum of 70 cm-2 in months, then increases to a relatively stable value of about 80 cm-2 (Figure 8b). The trend is consistent with the earlier findings reported in Refs [42,52,57], which indicated that the total number of pits (including metastable and stable pits) in a unit area will increase at the start due to the nucleation, and then decrease after a time as some of the pits are repassivated.

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Figure 8 Comparison of the empirically-based model results PD and the measurement result PDm in (a) 60 days, (b) 1000 days. Error bars denote σm (see Table 1).

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As indicated in Section 3.5, the pit density at each depth, iL (i = 1,2,…N), is obtained through solving Eqs.(2) and (20). Figure 9 shows the distributions for pit subpopulation within specific depth ranges at 2, 10, 20, and 30 days. At 2 days (Figure 9a), the simulation is consistent with measurement. At 10 days (Figure 9b), the simulation slightly deviates from measurement, especially at depth larger than 1 µm; however, the maximum depths of simulation and measurement are similar. At 20 days (Figure 9c), at depth larger than 1 µm, the simulation is larger than measurement, and the maximum depths are not in good agreement. However, based on the measured results, there is clearly some potential for pits with large depths and the disagreement may result from low measurement statistics at large depths. At 30 days (Figure 9d), simulation and measurement at less than 2.5 µm is very similar, but deviate at larger depths. Again the deviation at larger depths could be the result of poor measurement statistics as indicated by the observation of pits at a depth of 5 µm. Thus, in summary the agreement is generally good and the deviations that do exist in Figure 9b-d can be ascribed to the small sample size. The uncertainty in characterizing the distribution is magnified when the population of characteristics is sparse. In addition, because of the metastable property of pitting corrosion, there is substantial statistical uncertainty in pit depth and density, especially at early time.

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Figure 9 Comparison of simulation results (solid bars) and experimental measurements (hollow bars) of variations of depth and density at (a) 2 days, (b) 10 days, (c) 20 days and (d) 30 days. Error bars denote standard deviation of density at the depth.

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Figure 10 illustrates the distribution for pit subpopulation within specific depth ranges at 6, 12, and 36 months. At 6 months (Figure 10a), the maximum depth is approximately 20 μm. There are more metastable pits (< 5 μm) than stable pits. At 12 and 36 months (Figure 10b,c), the maximum depth is approximately 40 and 120 μm, respectively. The density of metastable pits is smaller than the density of stable pits. In Figure 10c, the population of stable pits < 75 μm is less than the population of stable pits > 75 μm. Deep stable pits are generated at the early time period, while stable pits < 75 μm are initiated later. As the oxide layer grows thicker with increasing time, it protects the steel from pitting corrosion. Thus, fewer pits form that can eventually grow to a depth exceeding the critical depth.

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As indicated in Section 1, large pits require large cathodic currents and cathode areas. Limits of cathodic currents can lead to termination of pitting corrosion. This mechanism is not taken into account in the present model. Thus the population of pits with large depths calculated by the model may be larger than the actual population.

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Figure 10 Simulation results of variations of depth and density at (a) 6 months, (b) 12 months and (c) 36 months.

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4.2 Case 2: SS316L exposed to 1.12 M chloride sea-salt solution at 72 °C To examine the range of applicability of the proposed model, we used a data set from Ref [51]. The experimental data include the pit depth and number from 3200 h (133 days) to 8785 h (366 days), each with a sample area of 10 cm2. Pits less than 5 μm were not counted. The experimental data indicate the depth increases with increasing time. The cathodic currents were apparently sufficiently large to support the growth of existing pits. To understand the pitting behavior in the early time, we conducted the same tests as in Ref [51] at the time intervals of 5, 20 and 30 days, using the same analysis method. The experimental data are summarized in Table 5. 19

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Table 5 Experimental data of PDm, σm and r for Case 2 (K=3) (data for 133-366 days were obtained from Ref [51], which only considered pits > 5 μm). PDm (/cm2) 392.3 361.5 146.2 > 4.2 [51] > 5.9 [51] > 6.8 [51] > 8.1 [51]

t (day) 5 20 30 133 223 282 366

σm (/cm2) 133.0 154.7 99.0 unknown unknown unknown unknown

rm (μm) 15.90 21.69 13.95 unknown unknown unknown unknown

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In a similar manner to Case 1, the optimum parameters used in determining distributions for transition rates were obtained by minimizing the least square fit. Table 6 is a summary of the input parameters.

Table 6 Case 2 input parameters.

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Model input parameter M (/cm2) β1 τ1 β2 τ2 a b Critical depth (µm) N L (µm)

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1.04×105 0.205 9×109 0.94 3.5 0.2163 [51] 0.8726 [51] 5 10 0.5

Figure 11 shows the pit density using the optimized parameters. Figure 11a is consistent with Case 1 (Section 4.1) in that the pit density increases in the early exposure period and then decreases. Figure 11b predicts that in the long term the pit density falls below 50 cm-2, which is consistent with the measured data as shown in Table 5. The simulation is also in agreement with the measured data for variations of depth and density.

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Figure 11 Comparison of the fitting result PD and the measurement result PDm in (a) 60 days, (b) 400 days. Error bars denote σm (see Table 1).

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Figure 12 shows pit density for subpopulations within specified depth ranges at 133, 223, 282, and 366 days. The number is for a sample area of 10 cm2. In Figure 12a, maximum measurement is 16.58 µm and maximum simulation is 15 µm, the difference is 9.5 %. In Figure 12b through Figure 12d, the differences are 3.1 %, 12.1 % and 8.4 %, respectively. The differences are attributed possible measurement error (no error bars are available in [51]) and uncertainty in the determination of simulation parameters by search scheme.

Figure 12 Comparison of the simulation and measurement at (a) 133 days, (b) 223 days, (c) 282 days and (d) 366 days. Number is for a sample area of 10 cm2.

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5. Conclusion and future work

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A physics-based multi-state Markov model giving a full description of pitting corrosion states has been developed and tuned with experimental data. The concept and formulation of the model are based on the pitting corrosion mechanism of generation and repassivation (Section 3.1). The continuous-time linear growth Markov process results support the idea that the transition rates in the Markov process are closely related to the pitting corrosion rate (Section 3.2). The input parameters are determined by experimental data using an exhaustive search scheme (Sections 3.3 and 3.4). The distributions for pit subpopulation within specific lifetime and depth ranges are simulated (Section 3.5) The proposed model is a viable and simple alternative to describe the pit formation process under different test conditions (Section 4).

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One of the advantages of the Markov approach over deterministic and other stochastic models for pitting corrosion is that the Markov model is able to capture the dependence of pitting corrosion rate on depth and lifetime. This ability allows for an estimation of not only the probability distribution of the pitting corrosion rate associated with the entire pit population but also such a distribution for a subpopulation within specific lifetime and depth ranges. In other words, it obtains not only the expected pit depth with time, but also the variation of pit depth and pit density at each time.

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Acknowledgement

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The simulation results have been compared with the measured data in Refs [47,48] and show relatively good agreement. However, the experimental data set is sparse resulting in significant statistical uncertainty. Field data (outside of controlled laboratory conditions) used for one of the case studies over the long-term exposures and for metastable pitting corrosion are even sparser. A call for comprehensive experimentation is made to aid continued research in this area. In spite of the supplied laboratory data, obtaining near real-time data will have large benefits. Since the surface to form cathode is limited, there is a limit on the number of large pits that can form. The model will be improved by involving the deterministic elements of cathodic currents in modeling to better emulate the actual corrosion conditions.

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This research has been performed using funding received from US Nuclear Regulatory Commission [grant number NRC-HQ-11-G-38-0036].

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