Rafael Amaya-Gómez

Mauricio Sánchez-Silva

Felipe Muñoz

Integrity assessment of corroded pipelines using dynamic segmentation and clustering

Keywords: Corrosion, Dynamic Segmentation, Pipeline Integrity, In-Line Inspections, Clustering

Corrosion defects impact the resistance of hydrocarbon pipelines by increasing the risk of failure and the resulting Loss of Containment (LOC). Different approaches have been proposed to estimate this risk of failure based on empirical approaches (e.g., API 579-1/ASME FFS-1, ASME B318S or API1160), and other focus on probabilistic evaluations. Nevertheless, few works are dealing with spatial variability of corrosion defects and how segmentation (required for reliability evaluation) may affect intervention decisions. In this paper, information obtained from In-Line Inspections (ILI) is used to build a corrosion degradation model under a pressure-stress failure criterion, which is used in turn, to develop a dynamic segmentation strategy. This strategy aims at identifying optimal intervention times and Locations. Results show that existing reliability evaluations using static segmentations are suboptimal and may hide critical zones. This result is illustrated by a real case study in which corrosion evolution is better estimated, and the problems associated with conventional static segmentation are stressed.

Introduction

Pipeline integrity

Pipeline integrity comprises concepts of failure prevention, inspection-repair strategies, and products, practices, and services that help operators maximize assets lifetime. According to [START_REF] Kishawy | Review of pipeline integrity management practices[END_REF], an integrity management program has eight main components out of which four of them are of particular interest in this study: i) a process to identify pipeline segments and their failure modes; ii) repair/replacement criteria ; iii) a continual process of assessment and evaluation to maintain the pipeline integrity; and iv) a process for the review of integrity assessment results. These components are related to pipeline risk management and aim to extend the pipeline lifetime based on integrity evaluation results. To this end, several approaches have been proposed to evaluate pipeline integrity including standards such as API 579-1/ASME FFS-1 (Fitness-For-Service), ASME B318S (Managing System Integrity of Gas Pipelines) or API1160 (Managing system integrity for hazardous liquid pipelines); Finite Elements Modeling (FEM) [START_REF] Berstad | {CO2} pipeline integrity: A new evaluation methodology[END_REF][START_REF] Varga | Continuous integrity evaluation of corroded pipelines using complemented FEA results -Part I: Procedure development[END_REF]; structural evaluations based on a failure assessment diagram [START_REF] Adib-Ramezani | Structural integrity evaluation of X52 gas pipes subjected to external corrosion defects using the SINTAP procedure[END_REF]; and probabilistic/stochastic integrity assessments [START_REF] Amaya-Gómez | Modeling of pipeline corrosion degradation mechanism with a Lévy Process based on ILI (In-Line) Inspections[END_REF], 2016).

Corrosion

Corrosion is one of the main pipeline degradation mechanisms and a top objective of integrity management. It is a progressive degradation process associated with continuous increments of metal loss, which can be determined through In-Line (ILI) inspections. These inspections provide information about defect length, width, and depth to evaluate pipeline integrity at a given time based on phenomenological [START_REF] Tang | Failure analysis and prediction of pipes due to the interaction between multiphase flow and structure[END_REF][START_REF] Zhang | A study of flow accelerated corrosion at elbow of carbon steel pipeline by array electrode and computational fluid dynamics simulation[END_REF], simulation [START_REF] Caleyo | Probability distribution of pitting corrosion depth and rate in underground pipelines: A Monte Carlo study[END_REF][START_REF] Li | Predicting corrosion remaining life of underground pipelines with a mechanically-based probabilistic model[END_REF], or empirical approaches [START_REF] De Waard | Prediction of CO2 corrosion of carbon steel[END_REF]NACE International, 2002;NORSOK, 1998). The main challenge of a corrosion assessment is to make predictions of the pipeline condition between scheduled inspections (i.e., every 2 to 6 years apart). In this direction, some approaches are using random variable adjustments [START_REF] Amaya-Gómez | Pattern recognition techniques implementation on data from In-Line Inspection (ILI)[END_REF][START_REF] Pandey | Estimation of parameters of degradation growth rate distribution from noisy measurement data[END_REF] and stochastic processes [START_REF] Amaya-Gómez | Modeling of pipeline corrosion degradation mechanism with a Lévy Process based on ILI (In-Line) Inspections[END_REF][START_REF] Zhang | Cost-based optimal maintenance decisions for corroding natural gas pipelines based on stochastic degradation models[END_REF] to predict depth increments based on several inspections. For instance, if there are m detected defects and n ILI measurements, the depth increment δ i j of the i-th defect in the j-th inspection could be obtained by:

∆ x = {δ 1 j , . . . , δ m j } ∀ j ∈ 2, 3, . . . , n (1)
∆ x = (x 1 jx 1 j-1 ), . . . , (x m jx m j-1 ) ∀ j ∈ 2, 3, . . . , n, where x i j denotes the depth of the i-th defect at the j-th ILI measurement. These increments are used to fit a given distribution or continuous stochastic process.

Corroded pipeline integrity management

Pipeline integrity management is a performance-based process that handles pipeline serviceability and failure prevention considering the hazardousness of the transported materials. This process includes pipeline inspection, integrity assessment, and pipeline maintenance. Pipeline inspections aim to detect anomalies such as corrosion defects, cracks, or dents through direct assessments, hydrostatic tests or ILI inspections. Integrity assessments seek to evaluate the reliability-based condition of the defects. Finally, maintenance approaches propose intervention policies to assure pipeline serviceability and a lifetime extension. Some available approaches for corroded pipelines are described below.

ILI measurements are the main testing procedure used to follow corrosion evolution. ILI tests provide valuable geometric and localized information of the defects identified along the pipeline using tools such as MFL (Magnetic Flux Leakage) and UT (Ultrasonic). The MFL consists of a magnetic flux induced on the pipe wall, which leaks out when the PIG (Pipeline Inspection Gauge) find an anomaly on the pipe. The UT uses the reflexion time and the angle of ultrasound waves to detect the metal loss. The information obtained from these tools is commonly used to determine the pipeline condition (Wang et al., 2015a;[START_REF] Zhang | Cost-based optimal maintenance decisions for corroding natural gas pipelines based on stochastic degradation models[END_REF].

Based on these ILI inspections, the metal loss can be assessed by several approaches following empirical, numerical or probabilistic models. Among the empirical approaches, McAllister (2014) presents a complete description of methods to address the problem of corrosion and coatings in pipelines. ASME B31G illustrates a manual to determine the remaining strength of a corroded pipeline (ASME, 2009). Moreover, standards such as the API 579-1/ASME FFS-1 and BS 7910 are generic fitness-for-purpose to assess how significant are defects in a range of structures empirically. Numerical approaches to assess pipeline condition usually deal with Finite Elements simulations to determine ductile failure [START_REF] Varga | Continuous integrity evaluation of corroded pipelines using complemented FEA results -Part I: Procedure development[END_REF]. Finally, available probabilistic approaches evaluate plastic collapse, yielding, or leak failure criteria [START_REF] Ahammed | Probabilistic analysis of underground pipelines subject to combined stresses and corrosion[END_REF][START_REF] Amaya-Gómez | Pattern recognition techniques implementation on data from In-Line Inspection (ILI)[END_REF][START_REF] Amirat | Reliability assessment of underground pipelines under the combined effect of active corrosion and residual stress[END_REF][START_REF] Hasan | Probability assessment of burst limit state due to internal corrosion[END_REF][START_REF] Kale | A probabilistic model for internal corrosion of gas pipeline[END_REF]. The plastic collapse is evaluated through burst pressure approaches like ASME B31G, DNV RP-F101, CSA Z662 or the model proposed by Netto [START_REF] Netto | The effect of corrosion defects on the burst pressure of pipelines[END_REF][START_REF] Teixeira | Reliability of pipelines with corrosion defects[END_REF]. Yielding criteria can be described based on longitudinal and circumferential stresses included in a Von Mises approach [START_REF] Amirat | Reliability assessment of underground pipelines under the combined effect of active corrosion and residual stress[END_REF]. A leak failure criterion is based on a critical defect-depth, which is commonly taken as 85% of the pipeline wall thickness [START_REF] Kale | A probabilistic model for internal corrosion of gas pipeline[END_REF]. Overall, these failure criteria are assessed through safety margins -or limit states-using operating or mechanical parameters to estimate pipeline failure probability. Nevertheless, to support maintenance decisions, critical segments have to be identified along the pipeline; hence spatial variability of the pipeline condition or soil properties have to be considered in how these segments are defined [START_REF] Sahraoui | The effects of spatial variability of the aggressiveness of soil on system reliability of corroding underground pipelines[END_REF].

Segmentation is the process of defining pipe sectors with similar characteristics (external or internal) that can be used as units for integrity evaluation. Segmentation can be static -i.e., initially predefined-, or dynamic -i.e., adaptable to mechanical or external conditions. Static segmentations use fixed distances defined arbitrarily (e.g., one mile) or it is defined by specific mechanical elements of particular interest such as valves. In static segmentation, there is considerable variability in the results of risk assessment and may increase intervention costs due to unnecessary evaluations. Furthermore, critical zones can be hidden if risks are weighted throughout long segments. In the dynamic segmentation, the length is not relevant as long the feature on which the segmentation is evaluated remain constant throughout the entire segment [START_REF] Muhlbauer | Pipeline Risk Management Manual: Ideas, Techniques, and Resources[END_REF]. From these two approaches, static segmentations are commonly implemented; some examples of integrity evaluations based on static segmentation can be found in [START_REF] Ahammed | Probabilistic analysis of underground pipelines subject to combined stresses and corrosion[END_REF]; [START_REF] Amirat | Reliability assessment of underground pipelines under the combined effect of active corrosion and residual stress[END_REF]; [START_REF] Hasan | Probability assessment of burst limit state due to internal corrosion[END_REF]; [START_REF] Teixeira | Reliability of pipelines with corrosion defects[END_REF]. However, a dynamic segmentation seems to be more reasonable for corroded pipelines, where localized defects are common along the pipeline.

Finally, the results of an ILI inspection are central to define a condition-based maintenance policy. Many approaches have been proposed in this direction for corroded pipelines [START_REF] Gomes | Optimal inspection and design of onshore pipelines under external corrosion process[END_REF][START_REF] Sahraoui | Maintenance planning under imperfect inspections of corroded pipelines[END_REF][START_REF] Zhang | Cost-based optimal maintenance decisions for corroding natural gas pipelines based on stochastic degradation models[END_REF]. [START_REF] Zhang | Cost-based optimal maintenance decisions for corroding natural gas pipelines based on stochastic degradation models[END_REF] presented a cost-based maintenance policy using costs of inspection, excavation, repair, and failure replacement representative of the typical industry practice in Canada for corroding natural gas pipelines. [START_REF] Gomes | Optimal inspection and design of onshore pipelines under external corrosion process[END_REF] exposed a cost-based model to determine an optimal corrosion thickness, time-to-first inspection, and time between successive inspections. These approaches seek to minimize total expected life-cycle costs, i.e., costs of construction, inspections, repair, and expected loss.

Objective and paper structure

The objective of this paper is to identify critical segments along the pipeline based on modern degradation methods and dynamic segmentation. Dynamic segmentation depends upon the assessment criteria; in this paper, it will be defined for the particular case of corrosion integrity based on information from ILI inspections. The document is structured as follows: Section 2 describes the proposed pipeline integrity evaluation. Section 3 describes the case study, and the results and discussion are shown in Section 4. Finally, conclusions are given in Section 5.

Pipeline integrity evaluation

Overview

The proposed methodology follows three phases. First, a stochastic corrosion degradation process is used based on data collected and processed from ILI measurements. In this paper, the corrosion is modeled by a Mixed-Lévy approach, based on the results from a previous work [START_REF] Amaya-Gómez | Modeling of pipeline corrosion degradation mechanism with a Lévy Process based on ILI (In-Line) Inspections[END_REF]. This approach uses a Gamma Process (GP) and a Compound Poisson process (CPP) with shocks distributed as Delta-Dirac in a linear combination of their constitutive processes. Second, a reliability assessment approach based on pipeline failure probability, dynamic segmentation, and the definition of a critical region is proposed. Finally, critical segments are identified based on recognized threshold evaluations and the results from the critical region.

Reliability assessment approach

In engineering design, the distinction between failure and safe condition is typically defined in terms of a limit state function g(X) [START_REF] Sánchez-Silva | Reliability and life-cycle analysis of deteriorating systems[END_REF]. Several failure mechanisms can be formulated for pipelines such as a plastic collapse, yielding, or leak. A plastic collapse considers a pressure-based limit state (i.e., g P = P b -P op ) between the pipeline burst pressure (P b ), in which the pipe wall will bulge outward and reach a point of instability, and the pipeline operating pressure (P op ). A yielding failure uses a stress limit state (i.e., g S = σ Y -σ Equiv ), which includes longitudinal and circumferential stress loads that can be evaluated with equivalent stresses like Von Mises or Tresca. Finally, leaks are based on the consumption of the wall thickness (i.e., g D = d cd) and predefined critical depth are used for this purpose as resistance. These failure criteria can be implemented separately to support intervention decisions, but in this paper, the pressure and the stress failure criteria are considered jointly for illustrative purposes. The latter with the aim to evaluate a somehow conservative failure criterion by following an isotropic material at which the Von Mises stress is compared to the pipeline ultimate strength, as it was previously implemented in some FEM simulations like [START_REF] Andrade | Structural Evaluation of Corrosion Defects in Pipelines: Comparison of FE Analyses and Assessment Methods[END_REF]. The combined approach is developed based on the models reported by [START_REF] Netto | The effect of corrosion defects on the burst pressure of pipelines[END_REF] and [START_REF] Amirat | Reliability assessment of underground pipelines under the combined effect of active corrosion and residual stress[END_REF], which are described below.

Pressure failure criterion

Netto et al. (2005) proposed a model to calculate the burst pressure of moderate-to-high toughness pipes (API 5L X52 to X77) assuming that the corrosion defects have ideal elliptical shapes. Based on a set of low-scale experiments and non-linear numerical finite elements simulations, both for an intact and a corroded pipeline, they obtained the following limit state:

g P = (1.1σ y )2t D        1 -0.9435 d t 1.6 l D 0.4        -P op , (2)
where σ y is the yield strength, D is the diameter, t is the wall thickness, d is the defect-depth, and l is the defect length. This approach was chosen because it produces less conservative predictions than other recognize models such as ASME B31G, DNV RP-F101, and CSA Z662-07 [START_REF] Amaya-Gómez | Reliability assessments of corroded pipelines based on internal pressure A review[END_REF][START_REF] Amaya-Gómez | Pattern recognition techniques implementation on data from In-Line Inspection (ILI)[END_REF][START_REF] Hasan | Probability assessment of burst limit state due to internal corrosion[END_REF][START_REF] Teixeira | Reliability of pipelines with corrosion defects[END_REF].

Stress failure criterion

Amirat et al. ( 2006) used a model based on longitudinal and circumferential stresses previously described by [START_REF] Ahammed | Probabilistic analysis of underground pipelines subject to combined stresses and corrosion[END_REF]. Longitudinal stresses depend on the load and support conditions throughout the pipeline, whereas circumferential stresses on the transversal propagation of the applied load at the top of the pipeline and the propagation as a reaction at the bottom [START_REF] Ahammed | Probabilistic analysis of underground pipelines subject to combined stresses and corrosion[END_REF]. They used the failure criterion of energy distortion theory, which according to [START_REF] Ahammed | Probabilistic analysis of underground pipelines subject to combined stresses and corrosion[END_REF], has obtained acceptable results compared to other experimental approaches for ductile materials. The limit state function is expressed in terms of longitudinal and circumferential stresses as follows:

g S = σ y -σ Equiv = σ y -σ 2 c -σ c σ l + σ 2 l , (3)
where σ Equiv is the Von Mises equivalent stress.

The circumferential and longitudinal stresses are given by:

σ c = σ Pc + σ S c + σ T c + σ Resc , σ l = σ Pl + σ S l + σ T l + σ Resl , (4)
where σ Pc is the inner pressure stress, σ S c is the circumferential bending stress due to overlaying soil, σ T c is the traffic load, σ Pl is the tensile stress, σ S l is the thermal stress, σ T l is the external bending load, and σ Resc , and σ Resl are the residual stresses. Overall, σ Pc and σ Pl are stresses due to internal pressure; σ S c and σ S l to soil loading; and σ T c and σ T l to bending stresses [START_REF] Amirat | Reliability assessment of underground pipelines under the combined effect of active corrosion and residual stress[END_REF]. Table 1 summarizes these expressions, which are explained in detail by [START_REF] Ahammed | Probabilistic analysis of underground pipelines subject to combined stresses and corrosion[END_REF]. Poisson's ratio (µ), internal pressure (P), internal radius (r), and wall thickness (t).

Soil Loadings

σ S l = α E ∆ θ σ S c = 6k m C d γ B 2 d E t r E t 3 + 24k d P r 3
Thermal expansion coefficient (α), elasticity modulus (E), temperature variation (∆θ), width of the ditch (B d ), coefficient of earth pressure (C d ), bending coefficient (k m ), soil density (γ), and deflection coefficient (k d ).

Bending

σ T l = E r χ σ T c = 6k m I c C t F E t r L e (E t 3 + 24k d P r 3 )
Longitudinal curvature (χ), impact factor (I c ), surface load coefficient (C t ), surface wheel load (F), and effective pipe length (L e ).

This model contemplates residual effects that are commonly found inside materials due to construction, thermal, and mechanical/heat formation processes. Residual stresses are generated in hot lamination processes, which introduce a significant deformation for pipelines [START_REF] Amirat | Residual stress analysis in seamless APIX60 Steel Gas Pipelines[END_REF]. The approach from Amirat and co-workers contemplates the Crampton model, which approximates residual circumferential stresses for thin pipelines by cutting on the complete system and measuring changes of its diameter. These stresses are relaxed in corroded pipelines due to a redistribution throughout the pipeline remaining wall thickness. Thus, this model can be expressed in terms of the corroded layer t c , wall thickness t, and radial coordinate r t as follows [START_REF] Amirat | Reliability assessment of underground pipelines under the combined effect of active corrosion and residual stress[END_REF]:

σ Resc = -70 1 - 2t c t 1 - 2(r t -t c ) t -t c . (5)
The longitudinal residual stress is determined with the Poisson coefficient µ and the circumferential stress σ Resc by σ Resl = µ σ Resc .

Once the limit state function g is defined (e.g., g = g S or g = g P ), the failure probability can be computed based on the so-called generalized reliability equation, which can be expressed as follows:

R = P[g(X) > 0] = • • • g(X)>0 f X (x)dx, (6)
where f X is the joint probability density function of the n-dimensional vector X of basic variables that describe the problem (Sánchez-Silva & Klutke, 2016). Thus, based on the operating conditions and the material properties (i.e., operating pressure and material strength), it is possible to determine the probability that the pipeline is in a safe (g > 0) or in a failure (g ≤ 0) state by computing its reliability R = P[g(X) > 0]. The complexity of this calculation depends on the number of random variables and the form of the limit state function. Then, only in a few cases Eq. 6 has an analytical solution, so alternative solution methods such as FORM/SORM or Monte Carlo simulations should be considered instead. In this paper, Monte Carlo simulations are used following a combined failure criterion described in the next section.

Combined failure probability approach

Although the stress approach includes loads to support the inner fluid pressure, the pressure and stress failure criteria are considered as independent events. The combined failure approach is estimated by the union of these two events, i.e., P (A ∪ B) = P (A) + P (B) -P (A ∩ B). Where A corresponds to the pressure failure event and B is the stress failure event. The failure approach based on the probability of the two events is described in more detail in Algorithm 1.

Algorithm 1 Combined Failure Probability (Monte Carlo Simulation) Input: N (Iterations), T (Evaluating years) and data to estimate both Failure Probabilities (See Sections 2.2.1 and 2.2.2). Procedure: Combined P f 1: for i = 1 to T do 2: Determine the Corrosion rate and defects' depth for the i th year (See Section 1.2). 3:

for j = 1 to N do 4:

Evaluate the Pressure (g P ) and Stress (g S ) limit state functions using the Inverse Sampling Method. 5:

end for 6:

Calculate: I P = #(g P ≤ 0), I S = #(g S ≤ 0), and I C = #(g P ≤ 0 ∧ g S ≤ 0) {Where #(< condition >) is the number of cases that satisfies the condition.} 7: Calculate the combined P f for the i th year for every defect: P f i = I P + I S -I C N 8: end for 9: Determine the mean Failure Probability of the reported defects.

Dynamic Segmentation and Critical region

Consider a fix segment length d seg in which the pipeline length L p can be divided into n = L p /d seg segments. The failure probability of every segment is determined following the combined approach mentioned in Section 2.2.3. Once this task is carried out, segments are shifted a distance ∆d seg < d seg ; i.e., ∆d seg = ζ • d seg , where for example ζ ≈ 0.1, and the failure probability is recalculated. Segments are shifted until they reach the location of an original segment; i.e., k times with k = d seg /∆d seg . Once the segment has been shifted k times and the failure probabilities are calculated, the largest and shortest probabilities are used in a secant interpolation approach to determine upper and lower envelopes (Fig. 1). For this purpose the env secant and smooth1q Matlab®functions developed by [START_REF] Martin | Matlab File Exchange: env secant[END_REF] and [START_REF] Garcia | Matlab File Exchange: SMOOTH1Q[END_REF] Segment length d seg is determined by maximizing the mean difference between the failure probability without segmentation (P f NS ) and both envelopes for a given segment length I:

max I∈S α U N N j=1 E j U (I) -P f NS + α L N N j=1 E j L (I) -P f NS , (7)
where α U = 0.5 and α L = 0.5 are two superposition coefficients and S is the set of possible segment lengths such that max(S) = δ < L p . Considering the entire joint replacement recommendation in case of a corrosion failure (ERCB, 2011), min(S) = δ ≥ υ where υ is the minimum reported joint length of the pipeline. Algorithm 2 illustrates how this segment length is determined per evaluating year. Calculate for each defect (l = 1, . . . , N): Based on these length segments for each evaluating year, a critical region is proposed for a spatial/temporal evaluation. This critical region is obtained from the upper/lower envelopes quartiles, i.e., 25, 50 and 75% of the data. Then, three critical temporal regions are proposed to illustrate the pipeline condition along the entire abscissa.

∆ l U = E l U (d s ) -P f NS and ∆ l L = E l L (d s ) -P f NS . 9: Calculate: dEval j = α U N l=1 ∆ l U + α L N l=1 ∆ l

Pipeline integrity

The pipeline integrity is assessed by identifying critical segments based on a dynamic segmentation, which in turn, is evaluated considering the possibility of observing "corrosion colonies" and how they may cluster depending on the closeness of defects. The cluster-based assessment takes into account the fact that these colonies (which will be named as a group) require lower pressure loads for a burst failure than when evaluated them individually [START_REF] Benjamin | Interaction of corrosion defects in pipelines Part 1: Fundamentals[END_REF].

Grouping criterion

There are several grouping criteria between adjacent corrosion defects on pipelines such as the standards: DNV RP F-101, BS 7910, CSA Z184, ASME B31G or API 579-1/ASME FFS-1. Moreover, grouping criteria based on a limit distance between defects either circumferentially or longitudinally are also commonly used [START_REF] Benjamin | Part 6: Analysis of pipeline containing interacting corrosion defects[END_REF]. Some of the most recognized criteria are shown in Table 2. [START_REF] Benjamin | Part 6: Analysis of pipeline containing interacting corrosion defects[END_REF]. Approach Defect Grouping criterion Longitudinally limit Circumferentially limit

DNV RP-F101 s L ≤ s L lim & s c ≤ s c lim s L lim = 2 √ D t s c lim = π √ t D Kiefner & Vieth s L ≤ s L lim & s c ≤ s c lim s L lim = 25.4mm s c lim = 6t POF s L ≤ s L lim & s c ≤ s c lim s L lim = min(6t, L 1 , L 2 ) s c lim = min(6t, w 1 , w 2 ) ASME B31G s L ≤ s L lim & s c ≤ s c lim s L lim = 3t s c lim = 3t
From the approaches shown in Table 2, DNV RP-F101 and ASME B31G represent attractive alternatives to calculate burst pressure of grouped defects. They obtained errors close to 5% and 23% in experimental results [START_REF] Benjamin | Part 6: Analysis of pipeline containing interacting corrosion defects[END_REF]. Other criteria that have relevant experimental results are the so-called MTI (Mixed-Type Interaction), which is a particular case of the DNV criterion, and the grouping criterion from API 579-1/ASME FFS-1 [START_REF] Benjamin | Part 6: Analysis of pipeline containing interacting corrosion defects[END_REF]. The API 579-1/ASME FFS-1 criterion uses an imaginary box based on the twice of the defect extent (i.e., 2s, 2c) to cluster defects included in this rectangular section (API, 2007). In this work, the grouping criteria from DNV RP-F101, API 579-1/ASME FFS-1, and ASME B31G are compared. Besides, two synthetic learning approaches are also evaluated: i) k-nearest neighbor algorithm (KNN) using a Euclidean distance and k=1 with K-means (KM) with 8085 corrosion groups, and ii) a K-means stand-alone approach with 8000 corrosion groups. Further details about these approaches were described by [START_REF] Amaya-Gómez | Pattern recognition techniques implementation on data from In-Line Inspection (ILI)[END_REF]. Both standards and the synthetic learning criteria aim to generate equivalent dimensions of the grouped defects (i.e., "corrosion colonies") using an enveloping rectangle that covers them as shown in Fig. 2.

Critical segments identification

This approach seeks to determine: i) when should be intervened the pipeline, ii) where these interventions should be addressed, and iii) which corrosion groups have critical failure probabilities. For this purpose, the recognized safety thresholds reported by DNV RP-F101 are considered to determine non-acceptable segments along the pipeline. These thresholds depend on annual failure probability as follows: i) < 10 -5 (High Level), ii) < 10 -4 (Normal Level) and iii) < 10 -3 (Low Level) (DNV, 2010). Fig. 3 illustrates how works the proposed approach. Initially, the time to intervene the pipeline is estimated using the failure region; then, the dynamic segmentation is implemented to determine non-acceptable segments along the pipeline abscissa; and finally, the groups with the greater failure probability are located in those segments. Additionally, a cost evaluation is carried out using the maintenance and failure unit costs reported by [START_REF] Zhang | Cost-based optimal maintenance decisions for corroding natural gas pipelines based on stochastic degradation models[END_REF]. These costs depend on the number of the intervened joints, so this number is estimated using the cumulative non-acceptable segment (according to DNV) of the pipeline and an equivalent distance of the pipe joint. Overall, two main costs are considered: i) A maintenance costs from ILI inspections, pipeline excavations, and coating reinforcements of $72,902.57 USD/Joint, and ii) Direct costs due to a failure costs of $105,221.87 USD/Joint. Algorithm 3 describes the procedure for this method.

Case Study

The case study consists of a carbon steel pipeline grade API5LX52 alloy with the characteristics shown in Table 3. Two corrosion data sets were obtained from ILI measurements 2 years apart. In the first run, 33,466 defects were identified and in the second 59,101 out of which almost 50% were associated with pitting corrosion, and the remaining are distributed mainly in circumferential slotting. The defects are mostly at the inner wall (around 80%) in both measurements.

Algorithm 3 Critical pipeline segments identification and evaluation

Input: T (Evaluating years), data to determine the combined Failure Probability (See Section 2.2), maintenance (m) and failure ( f ) unit costs, a safety acceptable criterion C, and a joint length δ. Procedure: Pipeline critical segments & Maintenance(M)/Failure (F )Costs 1: Determine the corrosion groups and their equivalent dimensions {See Section 2.3.1} 2: L = 0 {L: Non-acceptable length segments} 3: for t = 1 to T do 4: Determine the Combined P f {See the Algorithm 1}. 5:

Determine the Dynamic Segmentation and the Critical Region {See Section 2.2.4}. 6:

Find the segments N such that E L ≥ C {E L the lower envelope} 7:

L ← L + N. 8: end for

9: Calculate M = m L δ and F = f L δ

Results and discussion

Data treatment

Increments obtained from two ILI measurements were used to calibrate the stochastic process assuming independence between the defects and their depths. Because the exact MFL tool used to inspect the pipeline was not known, a circumferential uncertainty of 5 • during the inspection was assumed based on the vendor technical reports as it was described in Amaya- [START_REF] Amaya-Gómez | Pattern recognition techniques implementation on data from In-Line Inspection (ILI)[END_REF]. This uncertainty corresponds to 12 mm or a deviation of 5%, which is used to obtain two different data sets i) defects with a depth change between the two ILI measurements at the same position, and ii) defects located between the ILI measurements with a deviation less or equal than 5%. These sets were used to determine the growth corrosion model based on the defects reported from both ILI measurements at the inner wall; defects with no correspondence between the two ILI measurements were discarded [START_REF] Amaya-Gómez | Modeling of pipeline corrosion degradation mechanism with a Lévy Process based on ILI (In-Line) Inspections[END_REF]. This degradation model was later applied to the defects in the last inspection.

Combined Failure Probability

Some considerations are in place to calculate the pressure and stress failure probabilities. For the energy distortion model (stress failure criterion), the tensile strength was implemented instead of the yield strength to be consistent with the fracture of the material like in the case of the burst pressure. The parameters of each approach were obtained from the ILI measurements of the case study or by the following considerations:

• The operating pressure was assumed to follow a Gumbel distribution following the recommendations of [START_REF] Csa | CSA Z662-07: Limit state equation for burst of large leaks and rupture for corrosion defect[END_REF]. The parameters for this distribution were obtained from the method discussed by [START_REF] Hasan | Probability assessment of burst limit state due to internal corrosion[END_REF].

• Defects depth for the first year were obtained from the last ILI measurement. The remain depths were determined from the Mixed Lévy degradation mechanism with Gamma and CPP-Delta processes (See Amaya-Gómez et al. ( 2018)).

• The trench width in which the pipeline is buried was assumed to be 26 inches (660 mm), following the reported by [START_REF] Mcallister | Pipeline Rules of Thumb Handbook: A Manual of Quick, Accurate Solutions to Everyday Pipeline Engineering Problems[END_REF] for a nominal pipeline of 10 inches.

• The remaining pipeline properties were: i) the elasticity module of API5L X52 grade carbon steel of 210,000 MPa (TMI, 2011), ii) the Poisson coefficient of 0.3 [START_REF] Bokaian | Thermal expansion of pipe-in-pipe systems[END_REF], and iii) the thermal expansion coefficient of 11.7x10 -6 / • C [START_REF] Ahammed | Probabilistic analysis of underground pipelines subject to combined stresses and corrosion[END_REF][START_REF] Bokaian | Thermal expansion of pipe-in-pipe systems[END_REF].

• The soil load was determined using an equivalent volume of the buried pipeline. This volume was calculated using the buried dimensions for a nominal diameter of 10 inches (Gulf Interstate Engineering, 1999). Then, the unitary weight reported by [START_REF] Ahammed | Probabilistic analysis of underground pipelines subject to combined stresses and corrosion[END_REF] was implemented to obtain a soil load of 228 kN.

• The soil properties were taken from [START_REF] Ahammed | Probabilistic analysis of underground pipelines subject to combined stresses and corrosion[END_REF]. Their distributions were assumed to be normal except for C t , k d and k m , which were represented by lognormal distributions. Additionally, L e , B d , C d , E, F, r, α, γ, µ and ∆θ were assumed to be constant, following the suggestions of these authors.

Based on these considerations, the failure probability for the combined, pressure and stress failure probabilities were determined (Fig. 4). For the first 5 years, the differences between the three failure probabilities are imperceptible, but after the 10th year, the pressure failure results differ appreciably from the stress predictions (obtaining a difference near 0.2). Regarding the combined and stress results, their results vary only 1% indicating that practically each failure state in the pressure criterion is also a failure state in the stress criterion.

Dynamic Segmentation and Critical Region

The following parameters for the dynamic segmentation were considered based on the reported joints: δ = 5, δ = 100, and ∆d seg = 1. Fig 5 illustrates the results obtained following Algorithm 2 for 40 years. The y-axis corresponds with the time evaluated, the x-axis with the assessed static segment length within [δ, δ], and the color represents the indicator results from Eq. 7 (see Algorithm 2). The greater results of this indicator were located from 5 to 30 years, whereas for the first 5 years and after the 30th year the results were less than 5%. This result suggests that the failure probabilities of both envelopes are close to the obtained result without segmentation; thus, the required length does not differ drastically with the fixed approach. Interestingly, the greater differences were found in a segment length between 10 to 15 meters; a segment length that matches with the case study joint length range (almost 78% are between 10 and 14 meters). These results would suggest that a great focus of corrosion defects are located near the pipeline joints, which according to [START_REF] Dzioyev | Stress corrosion processes in the metal and welded joints in gas pipelines[END_REF] are commonly found 200 mm from the welding unions. To illustrate how this approach identifies critical zones along the pipeline, a dynamic segmentation at 10 years using d seg = 9 meters was obtained and depicted in Fig. 6. This figure shows the failure probability without segmentation P f NS around 0.04, the upper envelope as the dashed line, and the lower envelope as the solid line. The results for this illustrative example show that significant differences were obtained between the two envelopes with P f NS in 0-5 km and 25-30 km. Note also that the lower envelope surpasses P f NS near 4 km, which demonstrates that the spatial variability is certainly a matter of concern for pipeline reliability analysis.

Finally, the critical region was defined to evaluate pipeline condition over time. Quartiles of the envelopes were used to generate three critical regions depending on the concentration of data: i) 25%, ii) 50%, and iii) 75% (Fig. 7). These critical regions expose an important variability throughout the pipeline obtaining ranges near two orders of magnitude, so their selection is supported against an approach without any segmentation. Additionally, this figure indicates that the main differences between the envelopes with the P f NS are in the range of 5-30 years, as in the case of Fig. 5.

The proposed critical region is an interesting alternative to integrity evaluations that are commonly based on fixed segmentations [START_REF] Dawotola | Applications of Statistics and Probability in Civil Engineering. chapter Estimation of failure rates of crude product pipelines[END_REF][START_REF] Hallen | Probabilistic Condition Assessment of Corroding Pipelines in Mexico[END_REF]. This approach follows a continuous-reliability assessment based on a corrosion degradation process that can be used with acceptability thresholds to support decision-making in future interventions. For instance, if the DNV RP-F101 Low Safety Level (< 10 -3 ) is implemented, an intervention can be recommended in the next 6-7 years because the lower envelope exceeds this threshold. This approach can be used with other spatial evaluations including soil aggressiveness [START_REF] Sahraoui | The effects of spatial variability of the aggressiveness of soil on system reliability of corroding underground pipelines[END_REF] and population density to support a risk-based assessment.

Grouping criteria evaluation

The ILI report provides a list of clusters (potential "corrosion colonies") and a groupable classification of every defect from which these groups are created. Nevertheless, from the 47,663 groupable defects reported on the last ILI measurement, only 18,276 were assigned to one of the 6538 corrosion groups. Then, these corrosion groups were used as a training set and the defects without a group as a test set for supervised learning. Unsupervised learning was developed using all the groupable defects.

The grouping alternatives were compared based on their main capabilities to capture corrosion adjacent defects. Table 4 compares the number of groups obtained, the average number of defects per group, and the total grouped defects. Despite the ASME approach have the highest number of groups, the number of defects per group is not significant in comparison to an individual assessment. Moreover, this approach uses only 44% of the groupable defects on the inner wall, which may not affect the reliability from the case of isolate defects. These results illustrate that the ASME B31G approach does not capture all the possible corrosion groups (colonies); hence, it underestimates the pipeline condition. For the remaining standards criteria, their grouping capabilities follow similar performances. API 579 criterion has a higher number of defect per group based on the mean number of defects and the ratio between the groups with the grouped defects. The latter because 502 records in the last ILI run reported a length greater than 1 meter, which bearing in mind that this criterion is based on the length and the width, would obtain an important amount of clustering defects. Nevertheless, this approach tends to be conservative because the 502 defects do not report a significant depth. DNV RP-F101 approach also has a high amount of defects per group because of its limit distance in the circumferential direction. However, this criterion has a homogeneous behavior on the pipeline that is evidenced in the higher number of groups. The synthetic learning approaches follow a correlated performance, which is evidenced in all the parameters reported in Table 4. For a deeper comparison among these alternatives, corrosion groups distribution were determined for each alternative obtaining the contours depicted in Fig. 8.

This figure confirms that ASME B31G is not an adequate grouping criterion for corrosion defects. This grouping criterion lacks many corrosion defects per cluster in comparison to the other alternatives. For instance, near 18 km this approach report 200 defects and the other grouping approaches a defect density near to 300. These criteria are based exclusively on the location and proximity with their neighbors, so the non-inclusion of a defect with a significant depth may underestimate pipeline risk. Regarding the other alternatives, they did not present significant variations in their groups' distribution although they follow a different construction and evaluation criteria. They implement almost the same proportion of defects, which suggest that these alternatives can represent an interesting possibility in a corrosion colony evaluation.

Finally, the reliability effect of these grouping criteria was compared against isolated defects by their mean failure probabilities (see Algorithm 1). The obtained results -shown in Fig. 9agree with the previous statement about ASME B31G criterion. Its failure probability does not differ significantly from the corresponding for individual defects, which underestimates the pipeline condition in case defects are assessed isolated. The other alternatives show important differences with isolated defects. For instance, the two synthetic approaches increase almost 10% in their failure probability during 12-18 years from the individual defects. The results for these two approaches match almost entirely, except for an initial time-span in which the KNN+KM approach was slightly higher. The results of DNV RP-F101 and API 579 approaches follow expected behaviors based on the discussion mentioned above.

The above results suggest that the standards approaches could represent conservative grouping alternatives in comparison to the synthetic learning techniques. Indeed, ASME B31G grouping criterion is a non-suitable alternative for grouping corrosion defects since it does not describe all possible corrosion groups. API 579 approach is a conservative alternative because of the possibility of corrosion defects with considerable lengths and lower depths. DNV RP-F101 alternative gives conservative results related to their circumferential limit, but it represents an interesting corrosion grouping possibility due to its difference with the synthetic learning approaches.

Pipeline Integrity evaluation approach

To illustrate the proposed approach, the DNV RP-F101 grouping criterion was implemented to evaluate possible clusters along the pipeline. The DNV Low Safety threshold C = 10 -3 was also considered to discriminate non-acceptable segments based on the lower envelope with 75% of data. The results depicted in Fig. 10 show that non-acceptable segments begin around the fifth year. Then, the dynamic segmentation for this evaluating year reveals that the non-acceptable segments would be in the first six kilometers and the range of 40-45 km. Finally, considering the corrosion defects location, two defects with higher failure probability were identified at km 5066.

Regarding the cost evaluation, recall that f = $105, 221.87 USD/ Joint and m = $72, 902.57 USD/ Joint were used as input data in Algorithm 3. For comparison purposes, the three DNV 15 safety thresholds were considered; i.e., C 1 = 10 -1 , C 2 = 10 -2 , and C 3 = 10 -3 . Fig. 11 shows the accumulate non-acceptable segments using these three criteria. The results show that in 10 years the length of the pipeline that requires intervention with a threshold of C 3 is about 40 km, with C 2 is 15 km, and for C 1 less than 8 km. Moreover, for the first 15 years, the pipeline would be compromised since half of its abscissa achieved a failure probability higher than 10 -1 . The corresponding maintenance and failure costs are determined for a system without any intervention using a continuous discounted rate of 5%, a mean joint length of 12 m, and the accumulate non-acceptable segments obtained before with C = 10 -1 (Fig. 12). The results allow decision-makers to evaluate future interventions along the pipeline. For instance, this figure suggests that for the first 5 years the system failure investment would be similar than for required maintenance; however, after the 10th year, the required investment in case of a failure is 20 Million USD greater than if maintenance is implemented. This work proposes an alternative integrity evaluation considering the spatial distribution of defects and the formation of "corrosion colonies" along the pipeline; nevertheless, in this work, we do not contemplate the effect of the surrounding environment (e.g., soil) on the degradation process. However, it has to be recognized the role played by the soil in pipeline degradation (i.e., metal loss) in the form of external corrosion [START_REF] Castaneda | External Corrosion of Pipelines in Soil[END_REF]. A complete integrity evaluation should include a space-dependent degradation process caused by defects located at the outer wall. In this direction, some interesting approaches have been proposed such as the one reported by [START_REF] Sahraoui | The effects of spatial variability of the aggressiveness of soil on system reliability of corroding underground pipelines[END_REF], which used a Gaussian random field based on the Karhunen-Loève expansion depending on the type of soil. Wang and co-workers have developed a framework that cluster defects depending on their corrosion rates, which are later used with Monte Carlo simulations and Markov Chains to estimate a space-dependent corrosion rate probability density [START_REF] Wang | Bayesian Modeling of External Corrosion in Underground Pipelines Based on the Integration of Markov Chain Monte Carlo Techniques and Clustered Inspection Data[END_REF]. This approach uses a Gaussian mixture model (GMM) clustering criterion due to better separability capabilities in comparison to a K-means approach [START_REF] Yajima | A clustering based method to evaluate soil corrosivity for pipeline external integrity management[END_REF].

Conclusions

An integrity evaluation was proposed for corroded pipelines based on a critical region and a dynamic segmentation. The proposed approach can identify critical segments temporally and spatially along the pipeline lifetime that can support future intervention decisions. The dynamic segmentation was developed from upper and lower envelopes from a set of shifted static segmentations. The critical region was determined based on the dynamic segmentation results quartiles. In both cases, a pressure-stress combined failure probability and a grouping evaluation of corrosion defects were considered.

For the corrosion grouping, three limit distance approaches (i.e., API 579, ASME B31G, and DNV RP-F101) and two synthetic learning approaches (i.e., KNN+KM and KM) were compared based on their obtained groups distribution and their failure probability against isolated defects. ASME B31G criterion was a non-suitable alternative because the information captured do not represent the real state of the system. API 579 approach represents a conservative alternative in case defects have important lengths. DNV RP-F101 criterion was conservative because of their limit circumferential distance, but it represents an interesting alternative due to its similarity with the synthetic learning approaches.

The proposed approach allows decision makers to identify critical segments temporally and spatially along the pipeline based on acceptable thresholds as the one reported by DNV RP-F101. Based on a real case study, it was possible to identify that during the first five years interventions should be focus near the fifth kilometer.

Figure 2 :

 2 Figure 2: Equivalent corrosion colony dimensions. Adapted from Amaya-Gómez et al. (2016)

Figure 3 :

 3 Figure 3: Pipeline Integrity Evaluation Approach.

Figure 4 :

 4 Figure 4: Combined Pressure-Stress Failure Probability.

Figure 5 :

 5 Figure 5: Sensitivity Results for the segmentation distance.

Figure 6 :Figure 7 :

 67 Figure 6: Dynamic Segmentation Results for 10 years and d seg = 9.

Figure 8 :

 8 Figure 8: Grouped defects contours based on the grouping criteria.

Figure 9 :

 9 Figure 9: Failure Probability results using grouping criteria.

Figure 10 :

 10 Figure 10: Pipeline Integrity Evaluation Results for the case of study with the DNV grouping criterion.

Figure 11 :

 11 Figure 11: Accumulate non-acceptable pipeline segments.

Figure 12 :

 12 Figure 12: Maintenance and failure discounted costs with C = 10 -1 .

Table 1 :

 1 Longitudinal and Circumferential Stress expressions[START_REF] Ahammed | Probabilistic analysis of underground pipelines subject to combined stresses and corrosion[END_REF]

	Stress cause	Longitudinal		Circumferential	Variables description
	Internal pressure	σ Pl =	µ P r t	σ Pc =	P r t

  are implemented.

	Failure Probability								Upper Envelope
			Lower Envelope
	d	seg	2d	seg	3d	seg	(N-1)d	seg	Nd	seg
					Segmentation
	Figure 1: Dynamic Segmentation Scheme-Upper Envelope

Table 2 :

 2 Corrosion defects-grouping criteria

Table 3

 3

	: Case study main parameters
	Parameter	Value	Units
	Outer diameter	273.1	mm
	Nominal diameter	10	in
	Pipeline length	44	km
	MAOP (Maximum Allowable Operating Pressure)	1500	psig
	SMYS (Specified Minimum Yield Strength)	52,000	psig
	SMTS (Specified minimum tensile strength)	60,000	psig
	Average wall thickness	6.35	mm
	Operating temperature range	303.55-307.05	K
	Operating velocity range	1.7-2.4	m/s

Table 4 :

 4 Summary of the results of the grouping approaches evaluated.

	Grouping criterion	API 579 ASME B31G DNV RP-F101 KNN+KM	KM
	Number of obtained groups	3,106	8,263	5,466	8,085	8,000
	Mean number of defects by group	15	3	9	7	6
	Total grouped defects	46,536	24,176	44,890	48,338	43,399

Acknowledgments

R. Amaya-Gómez thanks the National Department of Science, Technology and Innovation of Colombia for the PhD scholarship (COLCIENCIAS Grant No. 727, 2015).


C Coupeau

D M Kazantsev

M Drouet

V L Alperovich

Kinetics of anticrossing between slip traces and vicinal steps on crystal surfaces

Keywords: Vicinal surface, Slip traces, Anticrossing, Surface atomic diffusion, Au(111) surface

The interaction between vicinal atomic steps and slip traces -straight monatomic steps produced on a crystal surface by the emergence of dislocations -is experimentally investigated and compared to Monte-Carlo simulations. Near the point of apparent crossing between a vicinal step and a slip trace, a checkered three-level surface relief configuration is formed, with two new combinatory steps that borders the opposite highest and lowest terraces. This configuration is unstable with respect to an anticrossing effect which consists in the formation of a nanometer scale bridge that separates the regions with the highest and lowest levels and connects the opposite regions of equal level. It is shown that such an anticrossing effect is a general phenomenon observed on various crystal surfaces, from metals to semiconductors. The anticrossing kinetics was experimentally investigated on the Au(111) surface by scanning tunneling microscopy under ultra-high vacuum. It is observed that the bridge width increases with time according to the power law with exponent β = 0.45 ± 0.01, i.e. significantly smaller than for the single-particle diffusion (β = 0.5). Monte-Carlo simulations were performed in order to clarify the involved atomic diffusion mechanisms. In particular, the competition between two microscopic mechanisms of the bridge formation is discussed, i.e., the adatom diffusion along the combinatory steps versus across the bridge from the uppermost to the lowest terrace.

Introduction

Atomic steps at crystal surfaces are the basic morphological elements that determine the mechanisms of crystal growth, sublimation, wet or gas etching and other technologically important processes [START_REF] Pimpinelli | Physics of crystal growth[END_REF]. Well-prepared vicinal crystal surfaces, which are misoriented from a singular crystallographic face, exhibit regular sets of atomically smooth terraces separated by 'vicinal' steps of monatomic height, with the width of terraces determined by the misorientation angle [START_REF] Jeong | Steps on surfaces: experiment and theory[END_REF][START_REF] Latyshev | Transformations on clean Si(111) stepped surface during sublimation[END_REF]. Another type of atomic steps is caused by the dislocations emerging at the free surface [START_REF] Egberts | Temporal development of indentation plasticity on the atomic scale revealed by force microscopy[END_REF][START_REF] Morgenstern | STM study of step dynamics around a bulk dislocation intersection with a Ag(111) surface[END_REF], which may be inherited from a substrate under epitaxial growth or introduced due to the relaxation of external mechanical stresses. These latter steps are called slip traces as long as their directions are characteristic of the gliding of dislocations in their crystallographic planes [START_REF] Coupeau | Scanning probe microscopy and dislocations[END_REF][START_REF] Engbaek | Atomic structure of screw dislocations intersecting the Au(111) surface: A combined scanning tunneling microscopy and molecular dynamics study[END_REF][START_REF] Egberts | Atomic-scale nanoindentation: detection and identification of single glide events in three dimensions by force microscopy[END_REF][START_REF] Asenjo | Dislocation mechanisms in the first stage of plasticity of nanoindented Au(111) surfaces[END_REF][START_REF] De La Fuente | Dislocation emission around nanoindentations on a (001) fcc metal surface studied by scanning tunneling microscopy and atomistic simulations[END_REF][START_REF] Carrasco | Dislocation cross slip and formation of terraces around nanoindentations in Au(001)[END_REF][START_REF] Douat | Atomicscale insight into non-crystallographic slip traces in body-centred cubic crystals[END_REF].

Various phenomena on crystal surfaces involving each type of these steps were extensively studied and well understood. The formation of step-terraced morphology, step bunching and anti-bunching phenomena induced by the interaction of vicinal steps, electric current or adsorbates were studied on various semiconductor and metal surfaces [START_REF] Jeong | Steps on surfaces: experiment and theory[END_REF][START_REF] Latyshev | Transformations on clean Si(111) stepped surface during sublimation[END_REF][START_REF] Yoshida | Atomically straight steps on vicinal Si(111) surfaces prepared by step-parallel current in the kink-up direction[END_REF][START_REF] Leroy | Step bunching to step-meandering transition induced by electromigration on Si(111) vicinal surface[END_REF][START_REF] Pascale | Self-organization of step bunching instability on vicinal substrate[END_REF]. On the other hand, scanning probe microscopy of dislocation-induced slip trace structures at free surfaces proved to be an effective tool for studying various phenomena associated with dislocation behaviour in crystals, such as partial splitting of screw dislocations [START_REF] Morgenstern | STM study of step dynamics around a bulk dislocation intersection with a Ag(111) surface[END_REF][START_REF] Engbaek | Atomic structure of screw dislocations intersecting the Au(111) surface: A combined scanning tunneling microscopy and molecular dynamics study[END_REF], as well as plastic relaxation processes at nanoscale level under nanoindentation [START_REF] Egberts | Atomic-scale nanoindentation: detection and identification of single glide events in three dimensions by force microscopy[END_REF][START_REF] Asenjo | Dislocation mechanisms in the first stage of plasticity of nanoindented Au(111) surfaces[END_REF][START_REF] De La Fuente | Dislocation emission around nanoindentations on a (001) fcc metal surface studied by scanning tunneling microscopy and atomistic simulations[END_REF][START_REF] Carrasco | Dislocation cross slip and formation of terraces around nanoindentations in Au(001)[END_REF]. Misfit dislocations and dislocation-induced atomic steps are responsible for the formation of the well-known 'cross-hatched' relief patterns of mesoscopic heights on the surfaces of epitaxial films grown on the mismatched substrates, which can be seen by an optical microscope or even with a naked eye [START_REF] Andrews | Development of cross-hatch morphology during growth of lattice mismatched layers[END_REF].

However, vicinal and slip steps of monatomic height were not clearly observed simultaneously until a recent study on a smooth gold surface [START_REF] Coupeau | Slip-trace-induced vicinal step destabilization[END_REF]. This latter study was performed by means of an original experimental apparatus, 'Nanoplast', allowing to examine surface morphology evolution by atomic force microscopy or scanning tunnelling microscopy under ultra-high vacuum environment during in situ straining experiments performed over a large range of temperatures (90 to 600 K) [START_REF] Nahas | An experimental UHV AFM-STM device for characterizing surface nanostructures under stress/strain at variable temperature[END_REF]. The 'Nanoplast' apparatus provides a decisive step forward in studying elementary plastic relaxation processes due to the opportunity to observe directly the in situ evolution of slip traces produced, under increasing controlled strain, by newly born dislocations. In particular, the crystallographic path followed by only one screw dislocation in Nb crystals has been recently studied [START_REF] Douat | Atomicscale insight into non-crystallographic slip traces in body-centred cubic crystals[END_REF]. This apparatus also made it possible to observe for the first time, under external stress, the regular arrays of both vicinal steps and slip traces on Au(111) surface [START_REF] Coupeau | Slip-trace-induced vicinal step destabilization[END_REF].

Simultaneous observations of vicinal steps and slip traces enable us to clarify an old question about the crossing of atomic steps on a crystal surface. According to the theoretical considerations by Nozieres [START_REF] Nozières | Shape and growth of crystals[END_REF], atomic steps cannot cross. Indeed, if one supposes that two atomic steps cross, a checkered three-level relief configuration is created with the highest and the lowest regions being opposite to each other and different in height by two monolayers, while the other two opposite regions are of equal intermediate height. To keep a valid definition of a monatomic step that separates two surface regions which differ in height by one monolayer, two initial intersecting steps should be redefined by exchanging their ends into two new 'combinatory' steps with abrupt turns at the point of contact. This configuration with the touching angles of combinatory steps is unstable because the atomic diffusion along the steps or from the highest region to the opposite lowest region leads to an effective repulsion of the new steps and formation of an 'anticrossing bridge' that connects the two opposite regions with the intermediate height level.

For vicinal crystal surfaces, Nozieres reasoning seems purely theoretical because vicinal steps never cross each other. They can only touch and, thus, form steps of multiple heights or step bunches. In contrast, slip traces result from the dislocation gliding in specific crystallographic planes [START_REF] Duesbery | The dislocation core and plasticity[END_REF]. Slip traces are, thus, aligned along certain crystallographic directions and can cross vicinal steps. An apparent intersection of a monatomic vicinal step and a slip trace makes it possible to confirm the Nozieres theoretical idea and to experimentally explore the anticrossing configuration. In particular, Coupeau et al. observed, for the first time, the formation of a checkered relief structure and an anticrossing bridge resulting from the crossing of a single slip trace and a vicinal step on Au(111) surfaces [START_REF] Coupeau | Slip-trace-induced vicinal step destabilization[END_REF]. A similar interplay between vicinal steps and slip traces was observed on the smooth vicinal surfaces of GaAs/AlGaAs structures bonded to glass, due to the partial plastic relaxation of thermo-mechanical stresses under annealing in quasi-equilibrium conditions [START_REF] Akhundov | Formation and interaction of dislocation-induced and vicinal monatomic steps on a GaAs(001) surface under stress relaxation[END_REF].

Although the coexistence of vicinal and slip steps was experimentally demonstrated and the formation of anticrossing configuration is now qualitatively explained on Au(111) and on GaAs(001), a quantitative description of this phenomenon is still lacking, and a number of basic questions remain open. The main questions are related to the bridge width kinetics and to the involved microscopic mechanisms leading to the formation of anticrossing relief configuration. In particular, the anticrossing configuration can be formed by 'smoothing' abrupt turns of the combinatory steps due to the atomic diffusion along the steps (step diffusion mechanism, SDM). Another, 'recombination' mechanism (RM) consists in the atomic diffusion between the combinatory steps, across the bridge, from the uppermost to the lowest terrace. This diffusion leads to effective step repulsion and to the broadening of the bridge at the intermediate level. The question is, which of these two mechanisms, SDM or RM, plays a dominant role in the formation of anticrossing configuration? This question can be answered by an experimental study of the bridge width kinetics and by comparing the results to a theory that accounts for both mechanisms, but experimental and theoretical data on the kinetics have not been available in literature so far.

As the bridge formation is due to surface diffusion, one may assume that the bridge width w would be proportional to the square root of time w ~ t 1/2 with an 'effective' diffusion coefficient. A more careful analysis should take into account that, in the process of bridge formation, atoms should diffuse to distances which increase with time and step curvature radii. Therefore, similar to the relaxation of macroscopic relief features [START_REF] Pimpinelli | Physics of crystal growth[END_REF][START_REF] Mullins | Theory of thermal grooving[END_REF][START_REF] Jeong | Faceting through the propagation of nucleation[END_REF], a bridge width kinetics with a power smaller than ½ might be expected.

This paper is aimed at answering these questions by means of combined experimental investigations of the anticrossing configuration kinetics on strained Au(111) single crystals by UHV-STM and Monte-Carlo simulations. The results enabled us to clarify the microscopic mechanisms of the step interaction and to obtain the parameters that determine the kinetics of anticrossing bridge width formation on the Au(111) surface, such as diffusion activation and lateral bond energies.

Methods

Experimental

The Au single crystals of nominal section 2x2 mm 2 and 6 mm length were prepared by Surface Preparation Lab [START_REF]Surface Preparation Laboratory[END_REF]. The Au(111) surfaces were first oriented with an accuracy better than 0.1°. After a chemical/mechanical polishing up to a Rms roughness lower than 30 nm, the Au(111) surface was prepared under the UHV environment ( 1010 - < mbar) by repeated cycles of Ar ion bombardments (0.9 kV) and annealing at 850 K for 10 min. As expected, after a few cycles, the surface exhibits atomically flat {111} terraces separated by the vicinal steps of height ℎ = 240 ± 10 pm, close to the theoretical value of 235 pm. The samples were then mechanically strained in UHV conditions at low temperature T = 180 K and heated up to room temperature (RT) using a home-made experimental device allowing us to characterize, by UHV-STM, the in situ evolution of surfaces under increasing controlled strain (see [START_REF] Nahas | An experimental UHV AFM-STM device for characterizing surface nanostructures under stress/strain at variable temperature[END_REF] for details). The uniaxial compression was applied along the 1 10 direction. From crystallographic considerations, the {111}<110> slip systems are expected to be activated in gold single crystals [START_REF] Engbaek | Atomic structure of screw dislocations intersecting the Au(111) surface: A combined scanning tunneling microscopy and molecular dynamics study[END_REF], with slip traces lying along the 011 and 1 01 directions at the Au(111) surface, i.e. at ± 60° from the compression axis. It is recalled that the height of a single trace corresponds to the Burgers vector component perpendicular to the surface. Theoretically, ℎ = . = √ = 0.235 nm, where is the normal to the free surface and a = 0.408 nm is the lattice parameter for gold. It is noticed here that a single trace, i.e. a trace related to the emergence of only one perfect dislocation, has the same height as a vicinal step.

Monte Carlo simulations

The Monte Carlo simulations of the surface smoothing and step-terrace morphology formation were performed on the Kossel crystal in the standard solid-onsolid (SOS) model [START_REF] Brunev | Influence of Schwöbel barriers and surface diffusion anisotropy on surface relief evolution during epitaxial growth: Simulation[END_REF]. The complex processes of mass transfer on a real reconstructed Au(111) surface are replaced by a random walk and attachment/detachment of Au atoms. The atoms are allowed to hop to adjacent free sites on a terrace, to incorporate into a step or to leave a step. The probability P (per second) of each event is determined through the activation energy E of the respective event, = / , where ν0 ~ 10 13

Hz is the Debye frequency, k is the Boltzmann constant, and T is the temperature. The energy E for the atomic surface diffusion is given by:

= + " ($ % -$ ' ))($ % -$ ' ), (1)
where Ed is the adatom diffusion activation energy, Eb is the binding energy, Ni and Nf are the lateral bond numbers of the atom before and after the hop, respectively and Θ(N)

is the step-like function equal to one for Ni > Nf or zero otherwise.

The desorption was neglected in agreement with the relatively low temperature of our experiments (close to or lower than room temperature). The computer program used earlier [START_REF] Kazantsev | Monte Carlo simulation of GaAs(001) surface smoothing in equilibrium conditions[END_REF] was further developed in order to speed up the calculations. In particular, a reverse calculation algorithm was introduced. This algorithm consists in (i) computing the probabilities of all possible events, (ii) randomly selecting the events taking into account their probabilities, and (iii) updating the affected event probabilities and repeating step (ii). The reverse algorithm allowed us to increase the calculation speed by one or two orders of magnitude and, thus, to simulate processes on areas equal to experimental ones within a reasonable time.

In the simulations, the lattice constant of the Kossel crystal a0 was set equal to the monatomic step height at the Au(111) surface, hV = 0.235 nm. The size of the simulated box L was in the range of (240-480) a0, i.e. (56-112) nm. To keep the number of atoms constant, two different boundary conditions were used. For the periodic boundary conditions, each adatom that diffuses outside a simulated box boundary appears at the opposite boundary. For the infinite barrier boundary conditions, the adatom diffusion across the boundaries was forbidden. We proved that the results do not significantly depend on both the box size and the boundary conditions for L ≥ 240 a0. In particular, under the increase of the box size L from 240 a0 to 480 a0, the bridge width increases only by less than 2% for both types of boundary conditions.

Results and discussion

The basic features of the interaction between vicinal steps and slip traces are illustrated in Fig. 1 by the UHV-STM images of the Au(111) sample, which were measured after plastic deformation at low temperature T = 180 K (Fig. 1a) and after a subsequent one-hour anneal at room temperature (Fig. 1b). Prior to deformation, the surface exhibited atomically flat terraces separated by curved vicinal steps of monoatomic height 0.235 nm labelled from V1 to V7. The plastic strain of εp = 0.29% leads to the appearance of straight vertical slip traces labelled T1 and T2, which lie in the 011 * direction and cross the vicinal steps. In this case, the crystal shearing by a dislocation corresponds to the uplift of the material on the right side of the slip trace by an amount of 0.235 nm. Thus, as expected, here the slip traces, which are, essentially, the dislocation-induced monatomic steps, have the same height as vicinal steps. This height corresponds to the Au crystal period in the direction normal to the (111) surface.

At first glance, the sets of curved vicinal steps and straight slip traces in Fig. 1a look like seemingly independent, with each vicinal step and slip trace apparently kept intact. However, a more careful analysis of the resulting relief proves that the introduction of slip traces leads to a radical reorganization of the atomic step system, with the formation of new combinatory steps from the former vicinal steps and slip traces. Indeed, as it is seen in Fig. 1a,a At low temperatures T ≤ 180 K, the checkered surface relief with the touching angles of the new combinatory steps (Fig. 1a) did not significantly evolve for more than one hour. In contrast, at higher (room) temperature, such a structure was shown to be unstable and evolved to the structure that is shown in Fig. 1(b), with curved combinatory steps and a few nanometers wide 'bridge' that separates the lowest and highest terraces and links the opposite terraces with the same intermediate atomic level [START_REF] Coupeau | Slip-trace-induced vicinal step destabilization[END_REF]. This evolution, which is likely caused by the atomic diffusion along the steps and from the highest to the lowest terrace, is called the anticrossing phenomenon in the following. It is clearly seen in Fig. 1b that the anticrossing occurs everywhere on the sample surface. In We showed experimentally that the anticrossing behavior is a universal phenomenon occurring in a large variety of materials, from simple and compound metals (see Fig. 2a, b andc for Au and Ni3Al) to semiconductors (see Fig. 2d for GaAs).

A similar step pattern was experimentally evidenced on rock salt as well [START_REF] Hoche | Combined electron-microscope surfacedecoration and helium-atom-scattering study of the layer-by-layer photonstimulated desorption from NaCl cleavage faces[END_REF]. The height of all steps in Fig. 2d is equal to the period of GaAs in the direction normal to the (001) plane (≈ 0.28 nm), i.e. corresponds to the Ga-As bilayer thickness in this direction [START_REF] Akhundov | Formation and interaction of dislocation-induced and vicinal monatomic steps on a GaAs(001) surface under stress relaxation[END_REF]. Unlike the uniaxial deformation applied to Au(111) and Ni3Al(111) samples [START_REF] Coupeau | Slip-trace-induced vicinal step destabilization[END_REF][START_REF] Nahas | An experimental UHV AFM-STM device for characterizing surface nanostructures under stress/strain at variable temperature[END_REF], the partial relaxation of thermo-mechanical biaxial stress in the The adjacent dislocation steps may have the opposite signs because, in GaAs, the dislocation loops, which are introduced in the (111) and (1 1 1) planes, have opposite signs of the Burger's vector normal component and, thus, produce parallel slip steps of the opposite signs on the GaAs(001) surface. Therefore, on GaAs(001), the dislocation steps form a set of orthogonal stripes with non-monotonically interleaving lower and higher lying terraces [START_REF] Akhundov | Formation and interaction of dislocation-induced and vicinal monatomic steps on a GaAs(001) surface under stress relaxation[END_REF]. Such morphology is in contrast to a fragment of a vicinal surface with a certain sign of the misorientation angle, on which the vicinal step steps have the same sign and, thus, form a monotonically descending (or ascending) staircase.

It should be noted that the lateral AFM resolution was by an order of magnitude lower, compared to the UHV-STM resolution of Au(111) and Ni3Al(111) images shown in Figs. 2a and2c, respectively. Consequently, anticrossing bridges are not clearly seen in Fig. 2d. At higher annealing temperatures, the bridges on the GaAs(001) could be observed [START_REF] Akhundov | Formation and interaction of dislocation-induced and vicinal monatomic steps on a GaAs(001) surface under stress relaxation[END_REF], but still not so clearly compared to the UHV-STM images of Au and Ni3Al.

It should be noted that, although the anticrossing phenomenon is proved to be a universal phenomenon, on semiconductor surfaces it occurs at much higher temperatures than on metal surfaces; in particular, at T ~ 800-900 K on GaAs [START_REF] Akhundov | Formation and interaction of dislocation-induced and vicinal monatomic steps on a GaAs(001) surface under stress relaxation[END_REF] versus T ~ 300 K on Au [START_REF] Coupeau | Slip-trace-induced vicinal step destabilization[END_REF]. A likely explanation consists in relatively small values of metallic bonds and diffusion activation energies that facilitate surface diffusion, in contrast to semiconductor surfaces with higher bond and diffusion activation energies, covalent dangling bonds and unscreened surface potential undulations that hinder diffusion at low temperatures.

The kinetic evolution of a characteristic checkered structure measured by UHV-STM on the Au(111) surface at T = 297 K is shown in Fig. 3. The slip trace (resp. the vicinal step) is labelled T (resp. V). Despite quite low lateral resolution in Fig. 3a induced by the high scanning rate needed to catch the anticrossing kinetics, the sequence of the images, which were taken at various durations after the dislocation emergence, clearly evidences the growth of the bridge. A well-known Au(111) reconstruction, called herringbone or chevron-like pattern [START_REF] Huang | Phase behavior of the Au(111) surface: discommensurations and kinks[END_REF][START_REF] Repain | Reconstruction, step edges and self-organization on the Au(111) surface[END_REF][START_REF] Barth | Scanning tunneling microscopy observations on the reconstructed Au(111) surface -atomic-structure, long-range superstructure, rotational domains, and surface-defects[END_REF][START_REF] Chauraud | Influence of terrace widths on Au(111) reconstruction[END_REF], is seen in Fig. 3c (see the area on the right terrace on which the contrast has been enhanced). It is also seen in Fig. 3c that the segments of the combinatory steps, which originated from slip trace T, remained atomically straight. In contrast, the upper segment of the right-side combinatory step, which originated from the vicinal step, is not straight and contains clear ledges, presumably corresponding to kink bunches.

The bridge width w was extracted from the UHV-STM images by approximating the curved steps by arcs with a constant curvature radius. The time-dependence of w is shown in Fig. 4 in the double log scale for two experiments at T = 297 K (circles) and T = 278 K (triangles). The experimental data at 297 K (resp. at 278 K) correspond to crossing angle 2α = 60° (resp. 2α = 48°). Here, 2α is defined as the angle between the slip trace T and the vicinal step V on the middle terrace of level 1 (see Fig. 3c). The error was estimated by measuring the widths of various bridges observed in the STM images. It is seen that, in both experiments, the bridge width logarithm ln(w) evolves approximately linearly with ln(t), and this evidences a power dependence w(t) ~ t β .

We performed Monte Carlo simulations (MCS) of anticrossing bridge formations in order to elucidate the involved mechanisms. The simulated kinetic evolution of the checkered structure is shown in Fig. 5. The initial checkered configuration resulting from the crossing between a slip trace and a vicinal step at 2a = 60° is shown in Fig. 5a.

Note that the slip trace is perfectly straight due to its singular orientation, while the inclined vicinal step, which is misoriented from the singular direction, contains atomic size kinks. The simulated evolution of the checkered relief after annealing at room temperature T = 297 K is shown in Figs. 5b-d for the increasing annealing time. It is evidenced that the initial checkered configuration with the touching angles of new combinatory steps is unstable, and the anticrossing configuration develops as a result of the annealing. As expected, the bridge width monotonically increases with the annealing time. It can be seen that, at least at the initial stage of the bridge formation, the slip trace remains nearly straight (Fig. 5b) or evolves into a regular array of straight (vertical) segments separated by monatomic kinks (Fig. 5c). In contrast, the vicinal step segments acquire a noticeable roughness. Such a roughness apparently stems from a kink ripening on the vicinal segments, not observed on the slip trace segments due to their initially straight character.

The comparison of experimental UHV-STM images shown in Fig. 3 It is seen in Fig. 4 that the experimental and simulated kinetics are well approximated by a power dependence

β t t w ) (
. The magnitude of the exponent β as a function of Ed is shown in Fig. 6 by triangles. It can be seen that β does not depend on Ed and is equal to β = 0.45 ± 0.01. This value is distinctly lower than β = 0.5, characteristic of a single particle diffusion [START_REF] Pimpinelli | Physics of crystal growth[END_REF]. This lower value of β may be related to the increase with time of the distance at which atoms should diffuse in order to form an anticrossing bridge. Similarly, it is known that the amplitude of macroscopic relief waves evolves towards flat surfaces by massive surface diffusion as t 1/4 [START_REF] Pimpinelli | Physics of crystal growth[END_REF][START_REF] Mullins | Theory of thermal grooving[END_REF]. Thus, in our case of surface diffusion that forms a two-dimensional anticrossing relief configuration, exponent β turns out to have an intermediate value between those characteristic of the single particle diffusion and the relaxation of 3D macroscopic relief features.

The anticrossing bridge can be formed by two distinctly different microscopic mechanisms, which are schematically illustrated in the 3D inset in Fig 7a . The first mechanism consists in the detachment of atoms from the highest terrace, their diffusion across the middle terrace and incorporation into the lowest terrace. In the inset, the direction of the overall diffusion flux corresponding to this mechanism is shown by the double solid arrow. Thus, the bridge is formed due to the recombination of the atoms originating from the upper terrace with the vacant atom positions at the lower terrace; in the following, this mechanism is referred to as 'recombination mechanism' (RM). The second, step diffusion mechanism (SDM) consists in a one-dimensional atomic diffusion along each combinatory step, which also leads to blunting of the initially sharp combinatory step corners (near the former intersection point) and, therefore, to bridge broadening. Specifically, SDM consists in the detachment of atoms from the corner of the upper combinatory step, their diffusion along the step and incorporation into the same step at some distance away from the corner. In the inset, the direction of the overall diffusion flux corresponding to SDM is shown by the dashed arrow. Similarly, at the lower terrace, the diffusion of atoms from the periphery towards the corner (not shown) leads to blunting of the lower combinatory step corner and contributes to the bridge widening. This latter process can also be equivalently described as vacancy diffusion away from the corner.

The simulated bridge width kinetics w(t), that well reproduces the experiment at T = 278 K and takes into account both RM and SDM, is shown in Fig. 7a with the circle dots. The bridge width evolution wSDM(t) simulated for the same model parameters, but for the pure step diffusion mechanism, is shown in Fig. 7a with the triangular dots. In this latter simulation, the recombination mechanism was artificially suppressed with the help of forbidding atom transfer from the middle to the lowerst terrace through the lower combinatory step by setting an infinitely high Schwöbel barrier [START_REF] Benlattar | Ehrlich-Schwöbel barriers and adsorption of Au, Cu and Ag stepped (100) surfaces[END_REF].

The ratio of SDM contribution wSDM(t) to the total bridge width w(t), that takes into account both mechanisms, is shown in Fig. 7b. It is seen that, at t ~ 2 min, the step diffusion mechanism contributes approximately one-third to the total bridge width, and then the SDM contribution monotonically decreases with time down to one-forth at t = 30 min. One would expect that, with the increase of bridge width, the effectiveness of the recombination mechanism, which involves atomic diffusion across the bridge, should decrease. However, this expectation is in contradiction with the observed decrease of the ratio w/wSDM with time (Fig. 7b). This fact is in accordance with the notably smaller exponent β = 0.37 for the SDM kinetics (see Fig. 7a), as compared to the total width kinetics (β = 0.45). An explanation of this apparent contradiction is that the SDM effectiveness also decreases with time due to the increase of curvature radii of the combinatory step corners. In the limiting case of infinitely large radii corresponding to straight steps, the SDM contribution to the bridge width kinetics is zero. Thus, the decrease of the SDM contribution is faster than the decrease of the RM contribution.

Conclusions

The interaction of vicinal steps and dislocation-induced steps (slip traces), which leads to the anticrossing relief configuration, is shown to be a universal phenomenon observed on the surfaces of a large variety of materials, such as smooth step-terraced vicinal surfaces of simple (Au), compound (Ni3Al) metals or semiconductors (GaAs).

110] [011] 0 1 1 2 T 1 T 2 C 2 C 0 (a) V 1 V 2 V 3 V 4 V 5 V 6 V 7 C 2 C 0 (b)

  'checkered' relief is formed near each crossing point. This specific relief is characterized by four terraces around the crossing point, at three sequential atomic levels labelled 0, 1 and 2, as highlighted in the white frame near the intersection between V6 and T1. The lowest and the highest opposite terraces differ in height by two monolayers, while the other two opposite terraces are at the same intermediate level. Despite intuitive considerations of step lines 'continuity', due to the crossing, neither the initial vicinal steps, nor the slip traces can be anymore considered as atomic steps in the conventional definition, which implies that a monatomic step separates two adjacent terraces with the difference in height equal to one monolayer.Instead, new 'combinatory' atomic steps, with sharp angles at crossing points, are formed, by exchanging the ends of the former steps. The segments of two new combinatory steps C0 and C2 near the intersection between V6 and T1 are highlighted by dotted and dashed white lines, respectively. The combinatory step C0 borders the lowest terrace of the checkered relief and consists of the left part of V6 and the upper part of T1, while C2 borders the highest terrace and consists of the right part of V6 and the lower part of T1.

Fig. 1 (

 1 b) the combinatory steps C0 and C2 are highlighted throughout the whole image in order to visualize the formation of staircase of twisted combinatory steps, with each step consisting of interleaving curved segments of former vicinal steps and relatively straight vertical segments of former slip traces.

  Figs. 2a and 2b show a zoomed STM image of an anticrossing bridge formed on the Au(111) surface and a characteristic profile across the bridge along the dashed line. The image and the profile highlight the three sequential atomic levels of the resulting terraces. The image of the anticrossing configuration on the Ni3Al(111) surface shown in Fig. 2c was obtained in the same UHV-STM set-up, in the course of an experiment similar to that performed on Au(111). In Fig. 2c, the heights of both the vertical slip trace T and the inclined vicinal step V are equal to 0.208 nm, i.e. to the period of Ni3Al in the [111] direction.Checkered relief configurations around multiple intersections between monatomic curved vicinal steps (labelled V1-V3) and straight slip traces (T1-T3) are clearly seen in the topographical image of the GaAs(001) surface (Fig.2d). The image was taken by atomic force microscopy (AFM), after the one-hour anneal of a glass-bonded GaAs/AlGaAs heterostructure at relatively high temperature T = 820 K[START_REF] Akhundov | Formation and interaction of dislocation-induced and vicinal monatomic steps on a GaAs(001) surface under stress relaxation[END_REF]. The AFM measurements were done on the equipment of CKP "Nanostructures" (Novosibirsk).

  GaAs/AlGaAs heterostructure led to the formation of a rectangular grid of slip traces oriented in the [110] and 11 * 0 directions. As a result, along with the checkered relief configurations due to the intersections of the slip traces with vicinal steps, checkered configurations are formed due to the intersections of mutually orthogonal dislocationinduced steps with each other; two of such configurations corresponding to the intersections between T3 with T1 and T2 are clearly seen in the right part of Fig. 2d. It is also seen in Fig 2d that, in contrast to the slip traces on Au(111) (see Fig. 1), the signs of two parallel slip traces T1 and T2 are opposite: when going from top to bottom, trace T1 corresponds to the uplift, while T2 corresponds to the lowering of the next terrace.

  with the simulated images in Fig.5proves that the MCS qualitatively describes the kinetics of anticrossing configuration. In order to perform a quantitative comparison, elucidate the relevant microscopic mechanisms and determine the model parameters, we compared the experimental and simulated kinetics of the bridge width w(t). To extract the bridge width from the simulated images, similar to the experimental procedure, the combinatory steps were approximated by arcs in the vicinity of anticrossing, as shown with the dashed lines in Fig.5d. The shortest distance between the arcs, that was averaged over 64 anticrossing simulations with the same parameters, was taken as bridge width w. The experimental kinetics measured at 297 K and 278 K was best-fitted by MCS with the same set of model parameters. The simulated kinetics w(t) are shown in Fig.4with the solid lines. It can be seen that the experimental and simulation kinetics agree reasonably well. The departure of one point of the 297 K kinetics outside the error bar can be hardly treated as a significant deviation of the experiment from the simulation, although, keeping in mind the simplicity of the model, the deviations might be expected. In the double log scale, the kinetics are approximated by straight lines, which correspond to a power dependence of the bridge width on time as +(,)~, . . In the simulations, the diffusion activation energy was first fixed and taken from literature Ed = 0.4 eV[START_REF] Göbel | A study of surface diffusion on gold with an atomic force microscope[END_REF][START_REF] Takano | Motion and conversion energies of adatom and adatom clusters on gold (001) surface[END_REF], while the fitting parameter (lateral bond energy) turned out to be Eb = 0.204 ± 0.001 eV. For the kinetics measured at T = 278 K, additional simulations with Eb deviated from the best fit value by only ± 0.005 eV proved that the resulting simulated kinetics are shifted by about ± 20% from the experimental kinetics. This shift demonstrates the high sensitivity of the fitting procedure to the model parameters.It should be noted, however, that it is not possible to determine definitely both Ed and Eb by fitting the experimental kinetics with MCS. More specifically, the deviation of the simulation curve from the experiment, which is caused by changing the lateral bond energy Eb, can be compensated by an appropriate change in the diffusion activation energy Ed. The description of the bridge width kinetics is, therefore, a oneparameter rather than a two-parameter problem. In other words, the experimental kinetics can be successfully approximated by simulations with various, but strongly correlated pairs of Ed and Eb. The plot of Eb(Ed) corresponding to such pairs calculated for 2α = 60° is shown in Fig.6by circles. It is seen that the dependence Eb vs. Ed is well described with a straight line. Along this line, the variations of the χ 2 criterion were below 2%. In an attempt to determine both Ed and Eb separately, we tried to use experiments with different intersection angles 2α. However, it turned out that the correlation of Eb vs. Ed does not depend on the intersecting angle 2α, since no significant change is observed in the nearly extreme cases of ultimately small 2α = 15° (dotted line) and large 2α = 165° (dashed line) angles.

  The intersection of a vicinal step with a slip trace results in the formation of a checkered relief configuration consisting of four atomic terraces positioned at three different levels around the crossing point. The uppermost and the lowest terraces are bordered by new angular-shaped combinatory steps. Each combinatory step consists of segments of a former vicinal step and a former slip trace. This configuration with the touching angles of combinatory steps is unstable and, due to diffusion processes, an anticrossing bridge grows. The bridge connects the opposite terraces with the same intermediate level and separates the uppermost and lowest terraces. The kinetics of the anticrossing bridge formation was investigated by UHV-STM on the Au(111) surface and compared to the Monte Carlo simulations. It is shown that the bridge width kinetics is described by a power law dependence +(,)~, . with the exponent β = 0.45 ± 0.01, i.e. significantly lower than the value β = 0.5 characteristic of a single particle diffusion. Although the comparison of the experimental results with Monte Carlo simulations does not allow one to determine the diffusion activation energy and lateral bond energy separately, we determined the interdependence between these two model parameters, that describes well the anticrossing bridge width kinetics on the Au(111) surface.We considered two different microscopic mechanisms that may significantly contribute to the anticrossing relief evolution: (1) diffusion of atoms from the uppermost terrace across the bridge and their subsequent incorporation into the lowest terrace (in other words, recombination with vacancies at the lowest terrace), and (2) atomic diffusion along the angular combinatory steps. It turned out that the recombination mechanism dominates over the step diffusion. Moreover, the step diffusion contribution decreases with the increasing bridge width, from approximately one-third at the early stage to one-forth at the mature stage of the observed bridge formation. This unexpected behavior can be explained by the decrease of the step diffusion efficiency induced by the increasing curvature radius of the new combinatory steps. The observed qualitative rearrangement of step-terraced surface morphology due to the interaction between monatomic vicinal steps and slip traces, which leads to the formation of new combinatory steps and anticrossing "bridges", open new opportunities for nanopatterning of surfaces. On the other hand, the anticrossing phenomenon should be taken into account in exploring slip traces for studying elementary mechanisms of plastic relaxation in crystals.

[

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: (a) UHV-STM images of Au(111) surface exhibiting intersections of seven curved vicinal steps (labelled from V 1 to V 7 ) with two straight vertical slip traces (labelled T 1 and T 2 ) induced by uniaxial compression of the single crystal along the ത 110 direction at T = 180 K. (b) UHV-STM images of the same area after annealing at T = 300 K for approximately 60 min. The particle marked with the black circle was used as a reference to position the STM images. The white frame highlights the anticrossing phenomenon at the intersection of V 6 and T 1 . The numbers 0, 1 and 2 in the white frame indicate the different increasing atomic levels of the associated terraces. The dotted and dashed white lines highlight the combinatory steps labelled C 0 and C 2 , respectively.

Figure 4 :Figure 5 :

 45 Figure 4: Experimental (dots) and simulated (lines) kinetics of the bridge width on Au(111). The circles correspond to T = 297 K and 2α = 60°, the triangles corresponds to T = 278 K and 2α = 48. For both temperatures, the solid lines correspond to the Monte-Carlo simulations with E d = 0.4 eV and E b = 0.204 eV.

Figure 6 : 22 Figure 7 :

 6227 Figure 6: Adatom activation energy E d vs. binding energy E b for various intersection angles 2a. Experimental data fit is presented with the black circles. Dotted and dashed lines correspond to the MC simulations for 2a = 15°and 2a = 165°, respectively. The power  of the bridge width kinetics for 2a = 60°i s shown by triangles on the right axis. The dash-dotted horizontal line is drawn as a visual guide for  values.

Acknowledgments

. This work was partially funded by the French Government program Investissements d'Avenir (LABEX INTERACTIFS, reference ANR-11-LABX-0017-01). Monte Carlo simulations are partially supported by the Ministry of Science and Higher Education of the Russian Federation (Project 0306-2019-0003).