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2-Dimensional Simulations of Pit Propagation and Multi-Pit Interactions

2-Dimensional Simulations of Pit Propagation and Multi-Pit Interactions Journal of The Electrochemical Society, 2022 169 081503 2-Dimensional Simulations of Pit Propagation and Multi-Pit Interactions 1,z 1, 2 Van Anh Nguyen, Roger C. Newman, and Nicholas J. Laycock Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, ON M5S 3E5, Canada Qatar Shell Research and Technology Centre, Qatar Science and Technology Park, Doha 24750, Qatar This work presents a reaction-transport model for pit propagation coupled with a phase field method to model the moving boundary at the corroding surface. This enables numerical simulations of the simultaneous propagation of multiple pits in close proximity to each other to study the interactions between pits under galvanostatic conditions, with limited applied currents. Results show the formation of lacy covers over pits in stainless steel, which is due to undercutting of the surrounding surface, and reveal the development of other complex morphologies arising from the interaction between neighboring pits; e.g., the growth of “pits within pits” and the evolution of “champion pits.” Such observations are due to the shrinkage of the total active surface to sustain the limited current as pits grow larger. The proposed model can be a valuable tool for studying the evolution of pit morphology in more realistic scenarios when multiple pit initiation sites are present. © 2022 The Author(s). Published on behalf of The Electrochemical Society by IOP Publishing Limited. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is properly cited. [DOI: 10.1149/ 1945-7111/ac8453] Manuscript submitted May 11, 2022; revised manuscript received June 26, 2022. Published August 5, 2022. 1–6 List of symbols artificial pit experiments. Although the 1-D pitting model has 1–9 proved to be valuable in pitting research, to develop predictive b Anodic Tafel slope modeling of pitting corrosion for practical applications, it is −3 C Concentration of species i in solution (mol cm ) necessary to extend the model to 2 and 3-D and to consider multiple −3 C Bulk concentration (mol cm ) bulk propagating pits at the same time. Such modeling also requires n+ −3 C Concentration of Me at saturated condition (mol cm ) sat methods to deal with the moving boundary at the corroding surface, n+ −3 C Critical concentration of Me for pit stability (mol cm ) crit capable of handling the complex morphologies that may develop, 2 −1 10,11 D Diffusion coefficient of species i (cm s ) including the formation of the perforated pit cover, and the 2 −1 D Effective diffusivity of species i (cm s ) ie , ff merging of two or more pits. It is also necessary to consider the E Open-circuit corrosion potential (V) corr interactions between neighboring pits and the competition for −1 F Faraday’s constant (96485 C mol ) resources (such as the available supply of current) that has been −2 12–14 i Current density in the electrolyte (A cm ) shown in some cases to lead to the evolution of a champion pit. −2 i Anodic current density (A cm ) Despite significant developments in numerical models for pitting −2 i Open-circuit corrosion current density (A cm ) corr corrosion in the past few decades, modeling the propagation stage of I Applied current (A) apply corrosion pits is challenging due to the numerical stiffness of the I Critical current for pit stability (A) crit electrochemical system inside a corrosion pit and the moving −2 -1 J Flux of species i (mol cm s ) boundary problem at the corroding interface. Recently, several n Charge of dissolving metal ions pitting models have focused on developing techniques to simulate r Initial pit radius (cm) pit the propagation stage in pitting corrosion. Laycock and White used a −1 −1 R Universal gas constant (8.3143 J mol K ) coupled reaction-transport model and finite element numerical T Temperature (K) techniques for 2-D simulations of single pit propagation that z Charge of species i produced compelling demonstrations of lacy cover formation as a −1 v Velocity at the corroding surface (cm s ) result of dilution effects near the pit mouth. This technique was 3 −1 γ Mobility in phase field model (cm sg ) also used to simulate the simultaneous growth of multiple pits under δ Phase field delta function both potentiostatic and galvanostatic techniques, including their ε Electrolyte volume fraction interactions via potential and concentration effects, but it was not ϵ Parameter controlling the interface thickness (cm) able to handle the physical merger of neighboring pits. Furthermore, η Overpotential (V) the remeshing required at each time step leads to high computational θ 1/2 central angle of the initial pit mouth cost and complexity for these simulations. λ Mixing energy density in phase field method (N) An alternative approach to resolve the moving boundary problem in -1 Interfacial tension in phase field method (N cm ) the computational pitting model is the implementation of the phase field φ Phase variable method to track the moving interface. In this type of model, the ϕ Electrolyte potential (V) interface between two phases is assumed to have a finite thickness and ϕ Electrode potential in galvanostatic pitting (V vs SCE) is governed by a phase field variable. This method can be coupled with ϕ Ohmic potential drop in the electrolytic solution (V) IR the mass transport equations to simulate the evolution of complex −1 17–19 χ Mobility tuning parameter (cm s g ) morphology (e.g., solidification, coalescence). Lately, the phase 3 −1 V Molar volume of dissolving metal ions (cm mol ) Me field method has been adopted to model the growth of pits, incorpor- ating factors such as microstructural variation and the formation of 16,20–23 Since Tester and Isaacs’s pioneering work in the 1970s, many corrosion products. However, because these models do not insights into pitting corrosion have been obtained via the use of one- incorporate any treatment of repassivation behavior, they do not dimensional diffusion-based models for the analysis of results from reproduce the formation of the perforated lacy cover via the under- 10,11 cutting process that is seen in both experiments and Laycock and White’s simulations. Furthermore, little to no information was provided on the interaction mechanisms between pits during the *Electrochemical Society Fellow. E-mail: vananhthi.nguyen@mail.utoronto.ca propagation stage. Journal of The Electrochemical Society, 2022 169 081503 In this paper, a reaction transport model very similar to that of Laycock and White is implemented together with the phase field method as a numerical treatment for the moving boundary. The growth of a single pit and multiple pits is modeled under galvano- static conditions and compared with experimental studies in the literature, focusing on the interactions between neighboring pits. Model Model assumptions.— 1. The proposed model focuses on the metal dissolution in the electrolytic solution (phase Ω in Fig. 1). The pseudo-phase region (phase Ω in Fig. 1) is used to track the corroding surface during the simulation, and thus having little to no physical Figure 1. A schematic view of the 2-D pitting model with the phase field meaning. Here, we assume that the interface has a finite method. thickness and is governed by the dimensionless phase field variable (φ), which takes the values of 1, and −1 in phase Ω and phase Ω , respectively. Within the interface region, 1 2 φ varies from 1 to −1, as shown in Fig. 1. 2. For simplicity, hydrolysis and complexation inside the pit are not included due to their relatively small effects on the dissolution kinetics. However, using the method proposed by Laycock and White, the local pit chemistry and the pH can be approximated from the total dissolved metal concentration. Furthermore, since the concentration of dissolved metal ions is below saturation in most studies under galvanostatic conditions with low applied currents (as considered here), the precipitation of the salt film is also not included. In addition, due to the small effect of the local cathodic reaction inside the pit at a relatively 24–26 high dissolution rate, it is not considered in this model. Thus, the dissolution kinetics at the corroding surface are based on the anodic reaction to form the dissolved metal ions, as shown below n+− Me→+ Me ne where n is the number of electrons transferred in the anodic reaction. 3. In most studies, we choose the initial pit shape to be hemispherical with r = 5m μ and θ=° 30 , as shown in pit Fig. 2. It is noted that the chosen r and θ are within the pit reported range to establish different propagation stages, in- cluding metastable and stable pitting. 4. Since the proposed model focuses on the propagation stage, the Figure 2. Initial pit geometry. critical concentration of dissolved metal ions for pit stability is used as an initial condition for the transient model and based on concentration of NaCl and the electrolyte potential, which is an experimental observations. Thus, the initial chemistry inside electric potential in an electrolyte solution. the pit is assumed to be no less than 70% of the saturated concentration of metal salts (C = 4M), which is slightly System of governing equations with the modified Nernst-Planck sat different from the corresponding values of 100% saturation equation.—The ionic flux of species (J ) in the electrolytic solution assumed in Laycock and White’s model. However, in practice, is described by the Nernst-Planck equation when diffusion and the modeling results have no qualitative difference when the migration are considered. However, since both Ω and Ω are 1 2 lacy cover is formed based on the dilution near the pit mouth. described in the same domain, the Nernst-Planck equation is Here, the initial pit chemistry is also near the marginal regime modified such that no ionic fluxes are sustained in the pseudo-phase when the salt-free initial condition can be reasonably applied for region as described in Eqs. 1–2, which is related to the mass galvanostatic pitting simulation. Furthermore, a diffusive transport in porous media. boundary layer of 10 μm above the pit mouth is employed as a boundary condition, above which the bulk environment is JD =− ∇C − zD C ∇ϕ [1] i i,, eff i i i eff i sustained, as shown below RT −4 C n + = 10 M Me DD=[ ε 2] ie , ff l i +− where C , z , D are the ionic concentration, the charge number, and CC==C Na Cl bulk i i i the self-diffusivity of species i; F, R, and T are the Faraday’s constant, the universal gas constant, and the temperature; D and ε ie , ff l ϕ = 0V are the effective diffusivity and the volume fraction of the electro- where C n +, C +, C − are the concentrations of dissolving metal lyte. Here, the values of ε are calibrated such that it approaches one Me Na Cl ions, sodium, and chloride, respectively; and are the bulk and zero in phase Ω and phase Ω , respectively. The material C ϕ 1 2 bulk l Journal of The Electrochemical Society, 2022 169 081503 is similar to the method proposed by Laycock and White model and 15,27 experimental data recorded by Gaudet et al. Particularly, when the concentration of dissolving metal ions (C n +) falls below the Me critical concentration (C ), which is assumed to be 70% of crit 1–3,15,27 saturation in this paper, the local dissolution is impeded, leading to repassivation. Furthermore, as shown by Gaudet et al., the current density is weakly dependent on the solution chemistry between 70% and 100% of metal ion saturation. Thus, Eq. 7 is remodified with the added sigmoid function f to reflect such behavior, as shown in Eq. 9. η /b ii =( 10 )f [9] a corr Parameter calibration for f is conducted using the 1-D pit propagation model to reproduce the Gaudet’s curve when the pit depth is kept at 0.8 mm, which is close to the values in Gaudet’s experiment. Figure 3 shows the simulated local current density as functions of the surface concentration at different applied potentials and when two sigmoid functions f and f are incorporated. Since a1 a2 f gives a more robust propagation mechanism with a substantially a2 Figure 3. Local current density as functions of the surface concentration at lower local current density in the repassivation regime, f is a2 different applied potentials (vs SCE). Data are obtained from a 1-D pit implemented in most studies. Furthermore, the movement of the propagation model in a 1 M NaCl bulk solution. pit surface is based on the velocity calculated from Faraday’s second law (See Eq. 10) balance of ionic species in the electrolyte is close to that described in Vi the classical electro-transport model with the added electrolyte Me a v=[10] volume fraction parameter ε (Eqs. 3–4). In addition, charge l nF neutrality is included in the mass transport system of ionic species, where v is the velocity of the corroding surface, V is the molar as shown in Eq. 5. Me −1 volume of dissolving metal ions with the molar mass of 56 g mol +− −3 15 For Na , and Cl : ∂(εCt ) /∂ = −∇.J [3] li i and the metal density of 8 g cm . Finally, pit propagation under galvanostatic conditions is inves- n+ tigated in most studies to reflect a more realistic pits’ growth (than For Me : ∂(εδ Ct n +) /∂ = −∇.J +i /(nF) [4] lMe Me a s potentiostatic pitting) when all pits are required to share the limited Charge neutrality: anodic current. Further study on the effect of the initial pit geometry under galvanostatic control will be conducted in future research. zC=[05] ∑ Under galvanostatic growth, the electrode potential, ϕ , from Eq. 8 ii becomes a dependent variable and is evaluated so that the total where δ is the phase field delta function used to impose the current can be sustained as shown in Eq. 11, which is the integration dissolution reaction at the corroding interface. Here, we used of the current density along the corroding pit length in this 2-D the default approximating function given in COMSOL for the phase model. In addition, the total applied current (I ) stated in this apply field method. Furthermore, the current density (c.d.) in the paper is based on the assumption of 1 μm thickness. electrolyte is due to the movement of charged species and calculated as the summation of ionic fluxes, as shown below. idA=[ I 11] ∫ a apply iF=[ zJ 6] ii Implementation of phase field method at the corroding sur- face.—In the proposed model, the dimensionless phase field variable is Dissolution and repassivation kinetics at the corroding sur- used to track the movement of the interface. From this point forwards, face.—The dissolution kinetics at the corroding surface are due to we briefly review the phase field method proposed by Yue et al. and the oxidation reaction of metal, as mentioned previously, and 18,19,29 employed in COMSOL. Extensive analysis has been performed calculated using the anodic Tafel equation (Eqs. 7–8) to compare this phase field model and the classical sharp interface 18,19 η /b model elsewhere. For further reading regarding the numerical a a ii=[ 10 7] a corr 18,19 calculations, readers are referred to the provided citations. In a system with two isothermal immiscible phases separated by a ηϕ=−ϕ − E [8] corr as IR diffusive interface and governed by a phase field variable φ, the total energy density of the two-phase system can be described in the where η , ϕ , and ϕ are the overpotential, the electrode potential a s IR Ginzburg–Landau form, assuming that only the mixing energy with respect to the remote reference electrode in galvanostatic density is considered (See Eq. 12). pitting, and the ohmic potential drop in the electrolytic solution, respectively; E and i are the open-circuit corrosion potential and corr corr λ 1 22 2 f=(φλ − 1) + ∣∇φ∣ [12] the corrosion current density in the pit solution; i and b are the a a 2 4ϵ 2 anodic current density and the anodic Tafel slope, respectively. In all studies, the propagation of corrosion pits in 316 stainless steel is where λ is the magnitude of the mixing energy, ϵ is the capillary modeled with kinetic parameters obtained from the literature (n = 2.2, width, and controls the thickness of the interface. The first term −2 E =−350 mV vs SCE, b = 100 mV, i = 0.1 mA cm ). and second term in Eq. 12 represent the bulk energy density, corr a corr Furthermore, repassivation behavior is also included in this which has a double-well potential form, and the interaction between model and based on the critical chemistry for pit stability, which phases, respectively. Thus, the evolution of the phase variable is Journal of The Electrochemical Society, 2022 169 081503 where γλ describes the relaxation time of the interface. Due to the stiffness of the fourth order partial differential equation (PDE), Eq. 15 is decomposed into two second order PDEs, as shown in Eqs. 16–17, where γ, λ and are critical parameters of the phase field method. ∂φ γλ +∇ u.. φ=∇ ∇ψ [16] ∂t ϵ ψϵ =−∇.1 ∇ϕ +(ϕ − )ϕ [17] It is commonly known that as the thickness of the diffusive interface decreases, the model’s accuracy and the computational cost will increase. Thus, is calibrated to be less than ∼0.4 μm in most studies to sustain a reasonable computational time and accuracy of the model. Furthermore, the magnitude of the mixing energy density λ can be related to the interfacial thickness via the interfacial tension −1 20 ( ), which is assumed to be 10 N m in this model (See Eq. 18). Figure 4. Concentration profiles of ionic species and the electrolyte potential along the centerline (x = 0) from the pit bottom (left side) to the 22 λ σ=[18] diffusive boundary layer above the pit mouth (right side). Single pit growing 3 ϵ under galvanostatic conditions in 1 M NaCl bulk solution at I = 1.15 μA apply t = 2s. and 2 Finally, the mobility parameter equals χϵ , where χ is the mobility tuning parameter. It is noted that χ should be calibrated such that the interfacial thickness remains constant while preventing the overdamping issue to the convective term in Eq. 15. Here, χ is calibrated and dependent on the velocity at the interface in each time step. Numerical setting for the 2-D pitting model.—The coupled governing equations, which include the mass transport of ionic n+ − + species (i.e., Me ,Cl ,Na ) in the electrolyte, the evolution of the phase variable, and the dissolution kinetics, were developed using the commercial code COMSOL Multiphysics® software v6.0. Here, the linear solver PARDISO and Backward Euler methods were implemented to solve the system of differential algebraic −3 equations with a relative tolerance of 10 for each time step. Free triangular elements were used with finer mesh in the area below the diffusive boundary layer. All numerical simulations were conducted on the computer cluster using two Intel(R) Xeon(R) Gold 6130 CPU at 2.10 GHz with 772.42 GB RAM. Data post-processing was performed using in-house codes written in Python. Figure 5. Calculated depth and width of modeled pit and ϕ as functions of Results and Discussion time in 1 M NaCl solution and at I = 1.15 μA. apply In this paper, the proposed model is used to simulate the pit propagation of type 316 stainless steel in chloride environments with modeled using the convective Cahn-Hilliard equation, as shown in 15,27 kinetic parameters acquired from the literature. Here, we choose 18,19 Eqs. 13–14. to simulate the propagation of corrosion pits under galvanostatic conditions because it reflects a more realistic pitting behavior when ∂φ +∇ u.. φγ =∇ ∇G [13] multiple pits are required to share the same limited current. Thus, it ∂t allows us to examine the competition between corrosion pits under various conditions. It is also noted that after checking all the modeling results, the treatment of salt precipitation is not needed δ fd Ω 2 φφ(− 1) ⎡ ⎤ G = = λ −∇ φ [14] in this model because the concentration in these conditions does not ⎢ ⎥ 30 δφ ϵ ⎣ ⎦ reach the levels of supersaturation required for salt precipitation. where u is the velocity field and calculated based on the dissolution Single pit growing under galvanostatic conditions.—A simula- rate at the corroding surface (See Eq. 10), γ is the mobility, G is the tion of a single pit growing in 1 M NaCl solution at 1.15 μA was chemical potential. In Eq. 13, the diffusive term is dependent on the carried out to validate the proposed model. Figure 4 shows the gradient of the chemical potential G, while the convective term, concentrations of ionic species and the electrolyte potential in the which is a kind of analogy in this type of modeling approach, is electrolytic solution along the centerline (x = 0) at t = 2s. Because dependent on the movement of the corroding interface. of migration and diffusion, the concentrations of chloride and Equation 14 is then substituted into Eq. 13 to obtain the transport dissolving metal ions inside the modeled pit are significantly higher equation of the phase field variable (Eq. 15) than in the bulk environment. An increase of the IR drop at the bottom of the pit is also observed, which agrees well with the ∂φ φφ(− 1) ⎡ ⎤ conventional transport model in pitting corrosion and experimental +∇ u.. φγ =∇λ∇ −∇ φ [15] 2 8,15,31–33 ⎢ ⎥ ∂t ϵ data. It is noted that the assumed thickness of the boundary ⎣ ⎦ Journal of The Electrochemical Society, 2022 169 081503 Figure 6. The evolution of the modeled pit at different time steps in 1 M NaCl solution and at I = 1.15 μA. The inner solid line is the original pit shape, and apply the color bar shows the concentration of dissolving metal ions (M). layer in this paper could be an extreme case in actual scenarios. However, the obtained results are close to Harb and Alkire’s works, where the steepest gradients of potential and ionic concentrations occur near the corrosion pit, as shown in Fig. 4. Studies on the thickness of the boundary layer and its effect on the development of pit morphology will be conducted in future work. Furthermore, due to a shorter diffusion length at the area near the pit mouth, the local chemistry is more diluted, leading to repassivation and the emergence of the junction between the active and the passive regions near this area, as described by Ernst et al. Figure 5 and Fig. 6 show the evolving morphology of the modeled pit, the concentration of dissolving metal ions, and the calculated potential (ϕ ) after 10 s of simulation. As shown in Fig. 6, the modeled pit is prone to grow sideways and undercut the nearby metal surface to develop a lacy cover. New holes are formed on both sides of the pit, which gives rise to a locally high outward flux of the dissolving metal ions. Thus, the metal concentration near the holes is decreased to a value below the critical threshold for active dissolution, resulting in repassivation on the neighboring surface. In Fig. 6b, after 10 s of simulation, the modeled pit has a dish-like shape with new lobes formed on both Figure 7. Potential (ϕ ) versus time when two simulated pits are grown in sides, which is in good agreement with the experimental data from 1 M NaCl solution at 1.75 μA. Repassivation behavior is incorporated using Ernst and Newman and the pitting model constructed by Laycock and the calibrated function f . 11,15 a1 White. However, it is also noted that the morphology observed here could transition to a more irregular, tunnel-like shape as the pit grows larger under constant applied current, which is explained in the Coalescence of two pits growing under galvanostatic condi- later section of this manuscript. tions.—The proposed model can also be employed to simulate the The corresponding ϕ versus time is obtained from the proposed coalescence of two pits without any remeshing scheme, while providing kinetic information during the propagation stage. In this model and shown in Fig. 5, which resembles the measured potential 12,13,34 section, two modeled pits are assumed to have the same size and in galvanostatic experiments. As the corrosion pit grows positioned close to each other with similar initial pit chemistry. larger, the required potential to sustain the active pit is stabilized Figure 7 and Fig. 8 show the computed values of ϕ and the to a lower value. After each undercutting event, the calculated s developing morphology of modeled pits growing in 1 M NaCl potential is slightly increased due to the local repassivation, but then solution and at 1.75 μA. Here, the repassivation is described using subsides to a lower value as the pit continues to grow after the function f as defined above. perforation. In general, such a rise in potential will be varied a1 As shown in Fig. 7, the calculated potential decreases as pits depending on the level of repassivation that occurred inside the grow larger. However, a small jump of potential is found during the pit, which will be shown in later sections. Here, a small spike of merging stage, corresponding to a local repassivation event near potential of ∼5 mV is seen during the perforation, but decreases to the merging area. Such an observation is due to a slight increase in 25 mV vs SCE after 10 s of growth (See Fig. 5). It is noted that the the pit openness, which promotes a dilution near the contacting zone, magnitude of the observed potential spikes is smaller than the typical particularly the top part, where diffusion length is the shortest. As values obtained from experiments, which could be due to multiple shown in Fig. 8, two simulated pits after the coalescence continue to factors, including the bulk chemistry, the occludedness of the initial grow as a single pit with a dished-shape and undercutting of the geometry, and the location of each undercutting event. Further analysis on the pit morphology at different applied currents will be metal surface, as discussed previously. Thus, ϕ increases during the discussed in the results section of this paper. undercutting event of the two modeled pits (See Fig. 7). Journal of The Electrochemical Society, 2022 169 081503 Figure 8. Developing morphology of two simulated pits growing in 1 M NaCl solution at 1.75 μA. Repassivation behavior is incorporated using the calibrated function f . The color bar shows the concentration of dissolving metal ions (M), and the inner solid line is the initial pit shape. a1 The critical current for pit stability.—As stated by Suleiman and Newman in 1994, there is a critical applied current (I ) above crit which corrosion pits can sustain the aggressive chemistry for a given pit shape. As the pit grows larger, the total active area needs to decrease so that the local current density remains sufficiently high for continued pit growth. Here, we define I as a minimum crit threshold where the concentration of dissolving metal ions at the bottom of the pit is above C with little to no change from the crit initial pit shape. Figure 9 shows the relationship between the applied current and ϕ when the degree of the pit openness (θ) and the initial pit radius (r ) are varied. After one second of simulation, a pit significant drop of the potential is observed when the applied current is above critical values, suggesting that pits can remain active in this regime. The morphology of pits growing when the applied current is close to I will be discussed in a later section. crit Furthermore, the values of I depend on various factors, crit including but not limited to the initial pit geometry, the bulk chemistry, and the deposition of corrosion products at the pit mouth. As shown in Fig. 9, I decreases as r increases and as θ crit pit Figure 9. The applied current versus the calculated potential of a single pit decreases, which can be explained by the extension of the diffusion in 1 M NaCl solution and at t = 1s. barrier provided in a larger pit and in a more occluded geometry, respectively. Thus, the corrosion pit requires a lower applied current to remain active. Here, a distinct morphology where a new pit is formed at the bottom of the initial pit is observed after six seconds of simulation Pit grows within a pit at low applied current.—The propagation (See Fig. 10a), which could be explained based on the relationship of corrosion pits under galvanostatic conditions is complex since pits between I and C discussed by Suleiman and Newman. Due to crit crit can optimize the total active surface area to sustain the limited the low applied current used in this simulation, the initial pit surface current, particularly at values near I for a given pit shape. crit is split into the active and the passive regions, which in turn reduces Figure 10 presents the simulated data and the evolving morphology the total active surface area and triggers the redistribution of the of a modeled pit growing in 1 M NaCl solution and at local current density inside the pit (See Figs. 10b, 10d). Thus, a I = 0.7 μA, which is close to the I observed in Fig. 9 when apply crit newly developed pit is formed at the bottom of the pit to sustain the r θ 30°, and equal5m μ and respectively. low current in galvanostatic condition, which agrees well with the pit Journal of The Electrochemical Society, 2022 169 081503 Figure 10. Simulated electrochemical data of single pit growing under galvanostatic conditions in 1 M NaCl solution. I == 0.7μμ A, r 5 m,θ= 30°. apply pit experimental observation using the metal foil technique to explore pit, suggesting that the pit propagation is retained near the marginal the 2-D pit morphology of corrosion pits. regime for pit stability. As shown in Fig. 10d, a rise of c.d. at the bottom of the pit is recorded while the remaining surface becomes inactive. The local c. Lateral growth and the transition to the marginal instability −2 d. then decreases and stabilizes to a value of ∼2Acm as the regime.—At moderate to high applied currents and in the absence of newly formed pit becomes larger. The transition between the active the salt film, when most of the corroding surface is active except for and the passive region inside the pit is characterized by the critical the repassivated area near the pit mouth, a corrosion pit is likely to current density for passivation, which is studied extensively in grow sideways and can undercut the nearby surface, as discussed pitting corrosion research and proved to depend on the previously (Fig. 6). However, as the pit grows larger under constant 15,34–36 temperature. Furthermore, the corresponding concentration current, it transitions into the marginal instability regime and of dissolving metal ions, the IR drop, and the surface potential along undergoes a “natural selection” process that favors a certain degree the pit surface are also calculated, as shown in Figs. 10c–10f. In all of occlusion to sustain the critical chemistry for pit stability. time steps, a higher concentration of dissolving metal ions above Specifically, the dissolution at the bottom of the pit is preserved, but below the saturated value is observed at the bottom of the while there is competition for the remaining current between two crit Journal of The Electrochemical Society, 2022 169 081503 Figure 11. Modeled pit grows in 1 M NaCl solution at I = 1.15 μA. In case 1 and case 2, a fillet is added on either side of the pit mouth. In case 3, a step- apply down current of 0.9 μA is applied after 10 s of simulation. The inner solid line is the original pit shape, and the color bar shows the concentration of dissolving metal ions (M). sides of a single pit. Such selective repassivation can occur in natural almost identically in the first 10 s of growth, which is similar to the pit growth conditions due to multiple factors, including the micro- developed morphology shown in Fig. 6. However, as the simulated structure, the surface roughness, and the asymmetry of the actual pit pit grows larger under constant applied current, it would also need to shape. In this paper, the imperfection at the diffusive interface can undergo a selective repassivation to remain active and thus become lead to such observation, but only if the transition to the marginal more sensitive to the initial pit geometry. As a result, the newly instability regime is slow enough. To verify this notion, simulations formed lobe on one side is repassivated while the remaining surface are carried out when a small degree of asymmetry is introduced to stays active, which leads to the observed gradient of the dissolved the initial pit shape. Here, a fillet, which is commonly known as a metal ion concentration along the pit surface, as shown in Fig. 11. rounding arc created at the corner of two intersecting surfaces, is For comparison purposes, another simulation is conducted with added and thus slightly modifies the ionic fluxes between two sides the same initial conditions described in Fig. 6. However, a step- of the modeled pit (case 1 and case 2 in Fig. 11). Due to such small down current of 0.9 μA is applied after 10 s of holding at 1.15 μA differences, the propagation rate and the pit morphology developed (case 3 in Fig. 11), which is similar to the type of simulation of a Journal of The Electrochemical Society, 2022 169 081503 in case 3 when the current is switched to a lower value are observed, suggesting repassivation events could occur inside the corrosion pit during the marginal growth as shown in Fig. 11. The interaction mechanism between pits and the “champion pit” in multi-pit simulations.—Multi-pit simulations are conducted at different applied currents to investigate the interaction mechanism between pits. Here, we simulated the simultaneous growth of four pits with different degrees of pit openness (i.e.,θθ== pit13 pit 30°= ,θθ = 37.5°) in 1 M NaCl solution. As shown in pit24 pit Fig. 13, in the presence of multiple pits, only the more occluded pits (pit 1 and pit 3) remain active, whereas the more opened pits (pit 2 and pit 4) become inactive. Such an observation is due to the competition between pits for the limited current. Specifically, under galvanostatic conditions, propagating pits share the same supplied current and are required to optimize their shape to accommodate such constraints as pits grow larger. Thus, when the applied current is low enough so that all pits cannot propagate concurrently, pits are subjected to the “natural Figure 12. Potential versus time corresponding to the change in the initial selection” process, which favors the more occluded geometry to pit shape (case 1 and case 2) and the applied current (case 3). retain the critical chemistry for propagation. As a result, pits 1 and 3 stay active, while pits 2 and 4 are repassivated, which gives rise to the potential ϕ in Fig. 15. flow disturbance when the boundary layer thickness is reduced. Since the current is reduced to a lower value, both sides of the Furthermore, in a simulation at an even lower applied current but simulated pit are repassivated, whereas the bottom of the pit stays above I , a minimum threshold to sustain the growth of at least one crit active. As a result, a new pit is developed beneath the initial pit, pit in a multi-pit simulation, pit 1 is repassivated after growing to a similar to that mentioned previously. Figure 12 shows the potential certain size, whereas pit 3 remains active and propagates (See as functions of time in three cases. Small humps after 13 s of Fig. 14). A jump of ϕ is observed at ∼1 s corresponding to the simulations in case 1 and case 2, and a sharper increase of potential repassivation of pit 1 in the simulation (See Fig. 15). Although two Figure 13. Multi-pit simulation under galvanostatic conditions in 1 M NaCl solution and at I =2. μA Pit 1 and pit 3 have the same standard shape with apply θ=° 30 , while pit 2 and pit 4 have wider pit mouths with θ=° 37.5 . The color bar shows the concentration of dissolving metal ions (M). Journal of The Electrochemical Society, 2022 169 081503 Figure 14. Multi-pit simulation under galvanostatic conditions in 1 M NaCl solution and at I = 1.75 μA. Pits 1 and 3 have the same standard shape with apply θ=° 30 , while pit 2 and pit 4 have wider pit mouths with θ=° 37.5 . The color bar shows the concentration of dissolving metal ions (M). “champion pit,” it then acquires the remaining current and grows with a faster dissolution rate. As shown in Fig. 14, the concentration of dissolving metal ions inside pit 3 increases significantly upon the repassivation of pit 1. However, as the “champion pit” becomes larger and requires a higher current to remain active, it could also transition into the marginal instability regime, similar to that described in the single pit case. The modeling results of multi-pits growing under galvanostatic conditions agree well with experimental observations when the 2-D morphology of natural pit growth was explored using the metal foil technique and in situ synchrotron X-ray radiography. In the presence of multiple pits, only the larger pit is reported to remain active, while smaller pits are repassivated. Furthermore, the mor- phology of the surviving pit is reported to have a rougher surface and more penetrating than that under potentiostatic conditions. Such an observation from experimental work could be explained based on the competition between pits and the optimization of pits’ shape to accommodate the limited supply of current in galvanostatic condi- tions discussed previously by Gaudet et al. in 1987. The oscillation Figure 15. Potential versus time in multi-pit simulation at different applied behavior of the simulated potential in this model is close to that currents. observed in experimental studies conducted by Suleiman and Newman when the “champion pit” survived for a longer time and pits have identical initial geometry, the arrangement of four pits in obtained a lower potential with a larger size. this study is not entirely symmetrical when pit 1 is placed in the furthest position and pit 3 is placed between pits 2 and 4 in close Comparison with potentiostatic conditions.—Under galvano- proximity. This could result in small variations in the local static conditions, the developed morphology becomes more complex environment near the mouth of each pit and, in turn, induces the than the original pit shape. It is more realistic than potentiostatic repassivation of pit 1 at a lower applied current. Once the pit pitting because the potential has to drop as the pit grows larger, survives the selection process for the limited current and becomes a which is relatively similar to an open-circuit or natural pitting Journal of The Electrochemical Society, 2022 169 081503 experiment. Due to the limited supply of current in galvanostatic Network of Excellence in Nuclear Engineering (UNENE), and the conditions, a corrosion pit can undergo selective repassivation, or Qatar National Research Fund through the National Priorities have a new pit formed beneath the existing pit, which leads to a Research Program (NPRP) under grant number NPRP12S-0209–- narrower and tunnel-like shape with a rounded bottom, as shown 190063. previously. Such complex morphology is reported in the literature ORCID when corrosion pits are growing under current-controlled conditions and in the activation-controlled dissolution without the presence of Van Anh Nguyen https://orcid.org/0000-0001-8299-611X 12,27,34,37,38 the salt film. Furthermore, some open-circuit pits tend to Roger C. Newman https://orcid.org/0000-0002-2422-619X have these shapes and possibly are grown in the activation- Nicholas J. Laycock https://orcid.org/0000-0002-4603-4156 controlled region. In potentiostatic conditions, since the current is allowed to increase References indefinitely as the pit grows into a larger size, there is no competition for 1. J. W. Tester and H. S. Isaacs, J. Electrochem. Soc., 122, 1438 (1975). resources between pits. Thus, pits can propagate until they merge into a 2. R. C. Newman and H. S. Isaacs, J. Electrochem. Soc., 130, 1621 (1983). larger pit, as described in the experimental work. In addition, pits 3. N. J. Laycock and R. C. Newman, Corros. Sci., 39, 1771 (1997). growing under potentiostatic conditions are saucer-shaped, with a 4. J. Srinivasan and R. G. Kelly, J. Electrochem. Soc., 163, C768 (2016). 34,39,40 5. G. S. Frankel, T. Li, and J. R. Scully, J. Electrochem. Soc., 164, C180 (2017). smooth interior surface due to the presence of the salt film. 6. R. M. Katona, J. Carpenter, E. J. Schindelholz, and R. G. Kelly, J. Electrochem. However, at lower applied potentials, the pits can develop a rougher Soc., 166, C3364 (2019). internal surface as the dissolution is transitioned into the activation- 7. V. A. Nguyen, A. G. Carcea, M. Ghaznavi, and R. C. Newman, J. Electrochem. 39,40 controlled growth regime. Furthermore, in some cases, when the Soc., 166, C3297 (2019). 8. V. A. Nguyen and R. C. Newman, Corros. Sci., 186, 109461 (2021). applied potential is dropped to a lower value or in a dilute bulk solution, 9. K. Wang, M. Salasi, and M. Iannuzzi, J. Electrochem. Soc., 168, 121504 (2021). a new lobe is formed at the bottom of the existing pit, and the surface 10. P. Ernst, N. J. Laycock, M. H. Moayed, and R. C. Newman, Corros. Sci., 39, 1133 11,34 near the pit mouth becomes repassivated. This could be due to (1997). various factors, including the dissolution of the salt film when the 11. P. Ernst and R. C. Newman, Corros. Sci., 44, 927 (2002). 12. M. I. Suleiman and R. C. Newman, Corros. Sci., 36, 1657 (1994). potential is reduced, the surface roughness, and the deposition of 13. D. Krouse, P. McGavin, and N. Laycock, Proceedings of 48th Annual Conference corrosion products. Further study on the developing morphology of of the Australasian Corrosion Association 2008: Corrosion and Prevention 2008, p. propagating pit under potentiostatic conditions in various simulated 203, Australasian Corrosion Association, Wellington, New Zealand (2008). environments, including the formation of the salt film and corrosion 14. S. M. Ghahari et al., Corros. Eng. Sci. Tech., 46, 205 (2011). 15. N. J. Laycock and S. P. White, J. Electrochem. Soc., 148, B264 (2001). products, will be conducted in future research to elucidate such 16. S. Jafarzadeh, Z. Chen, and F. Bobaru, Corros. Rev., 37, 419 (2019). observation in experiments. 17. J. W. Cahn and J. E. Hilliard, J. Chem. Phys., 28, 258 (1958). 18. P. Yue, J. Feng, C. Liu, and J. Shen, J. Fluid Mech., 515, 293 (2004). Conclusions 19. P. Yue, C. Zhou, J. J. Feng, C. F. Ollivier-Gooch, and H. H. Hu, J. Comput. Phys., 219, 47 (2006). In this paper, a two-dimensional reaction-transport model of pit 20. W. Mai and S. Soghrati, Electrochim. Acta, 260, 290 (2018). 21. A. F. Chadwick, J. A. Stewart, R. A. Enrique, S. Du, and K. Thornton, stability is coupled with a phase field method to simulate the J. Electrochem. Soc., 165, C633 (2018). propagation of single and multiple interacting pits. The model 22. T. Q. Ansari, J.-L. Luo, and S.-Q. Shi, NPJ Mater. Degrad., 3, 1 (2019). comprises a set of governing equations describing the ionic transport 23. S. Sahu and G. S. Frankel, J. Electrochem. Soc., 169, 020557 (2022). by diffusion and migration in the electrolyte, the dissolution kinetics 24. P. C. Pistorius and G. T. Burstein, Phil. Trans. R. Soc. London, Ser. A, 341, 531 (1992). at the pit surface, and the transport equation of the phase field 25. R. C. Alkire and K. P. Wong, Corros. Sci., 28, 411 (1988). variable to track the moving boundary. 26. N. J. Laycock and R. C. Newman, Corros. Sci., 39, 1771 (1997). Results of the proposed model show the characteristic formation of 27. G. T. Gaudet, W. T. Mo, T. A. Hatton, J. W. Tester, J. Tilly, H. S. Isaacs, and R. the lacy pit cover and the evolution of the pit morphology during the C. Newman, AIChE J., 32, 949 (1986). 28. J. S. Newman and K. E. Thomas-Alyea, Electrochemical systems (Wiley, New coalescence of two propagating pits in a single simulation for the first Jersey, NJ) 3rd ed., p. 517 (2004). time. Furthermore, under galvanostatic conditions, due to a low applied 29. COMSOL Multiphysics® v. 6.0. COMSOL AB, Stockholm, Sweden. www.comsol. current near the repassivation regime, the total active surface is reduced com. to support a larger pit growth at a constant current, which results in the 30. H. S. Isaacs and R. C. Newman, Corrosion and Corrosion Protection, ed. R. P. Frankenthal and F. Mansfeld 120(The Electrochemical Society, Pennington, evolution of complex morphologies (i.e., the formation of a new pit NJ) (1981). inside a pit and the selective repassivations near the pit mouth). 31. J. R. Galvele, J. Electrochem. Soc., 123, 464 (1976). Moreover, in multi-pit simulations, the concept of a “champion pit” 32. S. M. Sharland and P. W. Tasker, Corros. Sci., 28, 603 (1988). 12,34 observed in experimental studies is verified and reproduced using 33. J. N. Harb and R. C. Alkire, J. Electrochem. Soc., 138, 2594 (1991). 34. M. Ghahari, D. Krouse, N. Laycock, T. Rayment, C. Padovani, M. Stampanoni, the proposed model. Due to the competition for the limited current F. Marone, R. Mokso, and A. J. Davenport, Corrosion Sci., 100, 23 (2015). between corrosion pits, surviving pits will have a more occluded 35. V. M. Salinas-Bravo and R. C. Newman, Corros. Sci., 36, 67 (1994). geometry than the repassivated pits. Once the selection process is 36. N. J. Laycock, M. H. Moayed, and R. C. Newman, J. Electrochem. Soc., 145, 2622 completed, the “champion pit” continues to grow with a faster (1998). 37. N. J. Laycock, S. P. White, and W. Kissling, Passivity and localized corrosion: an dissolution rate, but subsides to a slower rate as the pit becomes larger International Symposium in Honor of Professor Norio Sato, ed. M. Seo, and, thus, transitions to the marginal growth at a given constant current. B. Macdougall, H. Takahashi, and R. G. Kelly PV 99-27 (The Electrochemical Society, Pennington, NJ) 541 (1999). Acknowledgments 38. A. Okeremi and M. Simon-Thomas, NACE Conference 2008, paper 08254 (2008). 39. N. Sato, J. Electrochem. Soc., 129, 260 (1982). Portions of this research were funded by the Natural Science and 40. M. P. Ryan, N. J. Laycock, H. S. Isaacs, and R. C. Newman, J. Electrochem. Soc., Engineering Research Council of Canada (NSERC), the University 146, 91 (1999). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the Electrochemical Society IOP Publishing

2-Dimensional Simulations of Pit Propagation and Multi-Pit Interactions

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IOP Publishing
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© 2022 The Author(s). Published on behalf of The Electrochemical Society by IOP Publishing Limited
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0013-4651
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1945-7111
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10.1149/1945-7111/ac8453
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Abstract

Journal of The Electrochemical Society, 2022 169 081503 2-Dimensional Simulations of Pit Propagation and Multi-Pit Interactions 1,z 1, 2 Van Anh Nguyen, Roger C. Newman, and Nicholas J. Laycock Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, ON M5S 3E5, Canada Qatar Shell Research and Technology Centre, Qatar Science and Technology Park, Doha 24750, Qatar This work presents a reaction-transport model for pit propagation coupled with a phase field method to model the moving boundary at the corroding surface. This enables numerical simulations of the simultaneous propagation of multiple pits in close proximity to each other to study the interactions between pits under galvanostatic conditions, with limited applied currents. Results show the formation of lacy covers over pits in stainless steel, which is due to undercutting of the surrounding surface, and reveal the development of other complex morphologies arising from the interaction between neighboring pits; e.g., the growth of “pits within pits” and the evolution of “champion pits.” Such observations are due to the shrinkage of the total active surface to sustain the limited current as pits grow larger. The proposed model can be a valuable tool for studying the evolution of pit morphology in more realistic scenarios when multiple pit initiation sites are present. © 2022 The Author(s). Published on behalf of The Electrochemical Society by IOP Publishing Limited. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is properly cited. [DOI: 10.1149/ 1945-7111/ac8453] Manuscript submitted May 11, 2022; revised manuscript received June 26, 2022. Published August 5, 2022. 1–6 List of symbols artificial pit experiments. Although the 1-D pitting model has 1–9 proved to be valuable in pitting research, to develop predictive b Anodic Tafel slope modeling of pitting corrosion for practical applications, it is −3 C Concentration of species i in solution (mol cm ) necessary to extend the model to 2 and 3-D and to consider multiple −3 C Bulk concentration (mol cm ) bulk propagating pits at the same time. Such modeling also requires n+ −3 C Concentration of Me at saturated condition (mol cm ) sat methods to deal with the moving boundary at the corroding surface, n+ −3 C Critical concentration of Me for pit stability (mol cm ) crit capable of handling the complex morphologies that may develop, 2 −1 10,11 D Diffusion coefficient of species i (cm s ) including the formation of the perforated pit cover, and the 2 −1 D Effective diffusivity of species i (cm s ) ie , ff merging of two or more pits. It is also necessary to consider the E Open-circuit corrosion potential (V) corr interactions between neighboring pits and the competition for −1 F Faraday’s constant (96485 C mol ) resources (such as the available supply of current) that has been −2 12–14 i Current density in the electrolyte (A cm ) shown in some cases to lead to the evolution of a champion pit. −2 i Anodic current density (A cm ) Despite significant developments in numerical models for pitting −2 i Open-circuit corrosion current density (A cm ) corr corrosion in the past few decades, modeling the propagation stage of I Applied current (A) apply corrosion pits is challenging due to the numerical stiffness of the I Critical current for pit stability (A) crit electrochemical system inside a corrosion pit and the moving −2 -1 J Flux of species i (mol cm s ) boundary problem at the corroding interface. Recently, several n Charge of dissolving metal ions pitting models have focused on developing techniques to simulate r Initial pit radius (cm) pit the propagation stage in pitting corrosion. Laycock and White used a −1 −1 R Universal gas constant (8.3143 J mol K ) coupled reaction-transport model and finite element numerical T Temperature (K) techniques for 2-D simulations of single pit propagation that z Charge of species i produced compelling demonstrations of lacy cover formation as a −1 v Velocity at the corroding surface (cm s ) result of dilution effects near the pit mouth. This technique was 3 −1 γ Mobility in phase field model (cm sg ) also used to simulate the simultaneous growth of multiple pits under δ Phase field delta function both potentiostatic and galvanostatic techniques, including their ε Electrolyte volume fraction interactions via potential and concentration effects, but it was not ϵ Parameter controlling the interface thickness (cm) able to handle the physical merger of neighboring pits. Furthermore, η Overpotential (V) the remeshing required at each time step leads to high computational θ 1/2 central angle of the initial pit mouth cost and complexity for these simulations. λ Mixing energy density in phase field method (N) An alternative approach to resolve the moving boundary problem in -1 Interfacial tension in phase field method (N cm ) the computational pitting model is the implementation of the phase field φ Phase variable method to track the moving interface. In this type of model, the ϕ Electrolyte potential (V) interface between two phases is assumed to have a finite thickness and ϕ Electrode potential in galvanostatic pitting (V vs SCE) is governed by a phase field variable. This method can be coupled with ϕ Ohmic potential drop in the electrolytic solution (V) IR the mass transport equations to simulate the evolution of complex −1 17–19 χ Mobility tuning parameter (cm s g ) morphology (e.g., solidification, coalescence). Lately, the phase 3 −1 V Molar volume of dissolving metal ions (cm mol ) Me field method has been adopted to model the growth of pits, incorpor- ating factors such as microstructural variation and the formation of 16,20–23 Since Tester and Isaacs’s pioneering work in the 1970s, many corrosion products. However, because these models do not insights into pitting corrosion have been obtained via the use of one- incorporate any treatment of repassivation behavior, they do not dimensional diffusion-based models for the analysis of results from reproduce the formation of the perforated lacy cover via the under- 10,11 cutting process that is seen in both experiments and Laycock and White’s simulations. Furthermore, little to no information was provided on the interaction mechanisms between pits during the *Electrochemical Society Fellow. E-mail: vananhthi.nguyen@mail.utoronto.ca propagation stage. Journal of The Electrochemical Society, 2022 169 081503 In this paper, a reaction transport model very similar to that of Laycock and White is implemented together with the phase field method as a numerical treatment for the moving boundary. The growth of a single pit and multiple pits is modeled under galvano- static conditions and compared with experimental studies in the literature, focusing on the interactions between neighboring pits. Model Model assumptions.— 1. The proposed model focuses on the metal dissolution in the electrolytic solution (phase Ω in Fig. 1). The pseudo-phase region (phase Ω in Fig. 1) is used to track the corroding surface during the simulation, and thus having little to no physical Figure 1. A schematic view of the 2-D pitting model with the phase field meaning. Here, we assume that the interface has a finite method. thickness and is governed by the dimensionless phase field variable (φ), which takes the values of 1, and −1 in phase Ω and phase Ω , respectively. Within the interface region, 1 2 φ varies from 1 to −1, as shown in Fig. 1. 2. For simplicity, hydrolysis and complexation inside the pit are not included due to their relatively small effects on the dissolution kinetics. However, using the method proposed by Laycock and White, the local pit chemistry and the pH can be approximated from the total dissolved metal concentration. Furthermore, since the concentration of dissolved metal ions is below saturation in most studies under galvanostatic conditions with low applied currents (as considered here), the precipitation of the salt film is also not included. In addition, due to the small effect of the local cathodic reaction inside the pit at a relatively 24–26 high dissolution rate, it is not considered in this model. Thus, the dissolution kinetics at the corroding surface are based on the anodic reaction to form the dissolved metal ions, as shown below n+− Me→+ Me ne where n is the number of electrons transferred in the anodic reaction. 3. In most studies, we choose the initial pit shape to be hemispherical with r = 5m μ and θ=° 30 , as shown in pit Fig. 2. It is noted that the chosen r and θ are within the pit reported range to establish different propagation stages, in- cluding metastable and stable pitting. 4. Since the proposed model focuses on the propagation stage, the Figure 2. Initial pit geometry. critical concentration of dissolved metal ions for pit stability is used as an initial condition for the transient model and based on concentration of NaCl and the electrolyte potential, which is an experimental observations. Thus, the initial chemistry inside electric potential in an electrolyte solution. the pit is assumed to be no less than 70% of the saturated concentration of metal salts (C = 4M), which is slightly System of governing equations with the modified Nernst-Planck sat different from the corresponding values of 100% saturation equation.—The ionic flux of species (J ) in the electrolytic solution assumed in Laycock and White’s model. However, in practice, is described by the Nernst-Planck equation when diffusion and the modeling results have no qualitative difference when the migration are considered. However, since both Ω and Ω are 1 2 lacy cover is formed based on the dilution near the pit mouth. described in the same domain, the Nernst-Planck equation is Here, the initial pit chemistry is also near the marginal regime modified such that no ionic fluxes are sustained in the pseudo-phase when the salt-free initial condition can be reasonably applied for region as described in Eqs. 1–2, which is related to the mass galvanostatic pitting simulation. Furthermore, a diffusive transport in porous media. boundary layer of 10 μm above the pit mouth is employed as a boundary condition, above which the bulk environment is JD =− ∇C − zD C ∇ϕ [1] i i,, eff i i i eff i sustained, as shown below RT −4 C n + = 10 M Me DD=[ ε 2] ie , ff l i +− where C , z , D are the ionic concentration, the charge number, and CC==C Na Cl bulk i i i the self-diffusivity of species i; F, R, and T are the Faraday’s constant, the universal gas constant, and the temperature; D and ε ie , ff l ϕ = 0V are the effective diffusivity and the volume fraction of the electro- where C n +, C +, C − are the concentrations of dissolving metal lyte. Here, the values of ε are calibrated such that it approaches one Me Na Cl ions, sodium, and chloride, respectively; and are the bulk and zero in phase Ω and phase Ω , respectively. The material C ϕ 1 2 bulk l Journal of The Electrochemical Society, 2022 169 081503 is similar to the method proposed by Laycock and White model and 15,27 experimental data recorded by Gaudet et al. Particularly, when the concentration of dissolving metal ions (C n +) falls below the Me critical concentration (C ), which is assumed to be 70% of crit 1–3,15,27 saturation in this paper, the local dissolution is impeded, leading to repassivation. Furthermore, as shown by Gaudet et al., the current density is weakly dependent on the solution chemistry between 70% and 100% of metal ion saturation. Thus, Eq. 7 is remodified with the added sigmoid function f to reflect such behavior, as shown in Eq. 9. η /b ii =( 10 )f [9] a corr Parameter calibration for f is conducted using the 1-D pit propagation model to reproduce the Gaudet’s curve when the pit depth is kept at 0.8 mm, which is close to the values in Gaudet’s experiment. Figure 3 shows the simulated local current density as functions of the surface concentration at different applied potentials and when two sigmoid functions f and f are incorporated. Since a1 a2 f gives a more robust propagation mechanism with a substantially a2 Figure 3. Local current density as functions of the surface concentration at lower local current density in the repassivation regime, f is a2 different applied potentials (vs SCE). Data are obtained from a 1-D pit implemented in most studies. Furthermore, the movement of the propagation model in a 1 M NaCl bulk solution. pit surface is based on the velocity calculated from Faraday’s second law (See Eq. 10) balance of ionic species in the electrolyte is close to that described in Vi the classical electro-transport model with the added electrolyte Me a v=[10] volume fraction parameter ε (Eqs. 3–4). In addition, charge l nF neutrality is included in the mass transport system of ionic species, where v is the velocity of the corroding surface, V is the molar as shown in Eq. 5. Me −1 volume of dissolving metal ions with the molar mass of 56 g mol +− −3 15 For Na , and Cl : ∂(εCt ) /∂ = −∇.J [3] li i and the metal density of 8 g cm . Finally, pit propagation under galvanostatic conditions is inves- n+ tigated in most studies to reflect a more realistic pits’ growth (than For Me : ∂(εδ Ct n +) /∂ = −∇.J +i /(nF) [4] lMe Me a s potentiostatic pitting) when all pits are required to share the limited Charge neutrality: anodic current. Further study on the effect of the initial pit geometry under galvanostatic control will be conducted in future research. zC=[05] ∑ Under galvanostatic growth, the electrode potential, ϕ , from Eq. 8 ii becomes a dependent variable and is evaluated so that the total where δ is the phase field delta function used to impose the current can be sustained as shown in Eq. 11, which is the integration dissolution reaction at the corroding interface. Here, we used of the current density along the corroding pit length in this 2-D the default approximating function given in COMSOL for the phase model. In addition, the total applied current (I ) stated in this apply field method. Furthermore, the current density (c.d.) in the paper is based on the assumption of 1 μm thickness. electrolyte is due to the movement of charged species and calculated as the summation of ionic fluxes, as shown below. idA=[ I 11] ∫ a apply iF=[ zJ 6] ii Implementation of phase field method at the corroding sur- face.—In the proposed model, the dimensionless phase field variable is Dissolution and repassivation kinetics at the corroding sur- used to track the movement of the interface. From this point forwards, face.—The dissolution kinetics at the corroding surface are due to we briefly review the phase field method proposed by Yue et al. and the oxidation reaction of metal, as mentioned previously, and 18,19,29 employed in COMSOL. Extensive analysis has been performed calculated using the anodic Tafel equation (Eqs. 7–8) to compare this phase field model and the classical sharp interface 18,19 η /b model elsewhere. For further reading regarding the numerical a a ii=[ 10 7] a corr 18,19 calculations, readers are referred to the provided citations. In a system with two isothermal immiscible phases separated by a ηϕ=−ϕ − E [8] corr as IR diffusive interface and governed by a phase field variable φ, the total energy density of the two-phase system can be described in the where η , ϕ , and ϕ are the overpotential, the electrode potential a s IR Ginzburg–Landau form, assuming that only the mixing energy with respect to the remote reference electrode in galvanostatic density is considered (See Eq. 12). pitting, and the ohmic potential drop in the electrolytic solution, respectively; E and i are the open-circuit corrosion potential and corr corr λ 1 22 2 f=(φλ − 1) + ∣∇φ∣ [12] the corrosion current density in the pit solution; i and b are the a a 2 4ϵ 2 anodic current density and the anodic Tafel slope, respectively. In all studies, the propagation of corrosion pits in 316 stainless steel is where λ is the magnitude of the mixing energy, ϵ is the capillary modeled with kinetic parameters obtained from the literature (n = 2.2, width, and controls the thickness of the interface. The first term −2 E =−350 mV vs SCE, b = 100 mV, i = 0.1 mA cm ). and second term in Eq. 12 represent the bulk energy density, corr a corr Furthermore, repassivation behavior is also included in this which has a double-well potential form, and the interaction between model and based on the critical chemistry for pit stability, which phases, respectively. Thus, the evolution of the phase variable is Journal of The Electrochemical Society, 2022 169 081503 where γλ describes the relaxation time of the interface. Due to the stiffness of the fourth order partial differential equation (PDE), Eq. 15 is decomposed into two second order PDEs, as shown in Eqs. 16–17, where γ, λ and are critical parameters of the phase field method. ∂φ γλ +∇ u.. φ=∇ ∇ψ [16] ∂t ϵ ψϵ =−∇.1 ∇ϕ +(ϕ − )ϕ [17] It is commonly known that as the thickness of the diffusive interface decreases, the model’s accuracy and the computational cost will increase. Thus, is calibrated to be less than ∼0.4 μm in most studies to sustain a reasonable computational time and accuracy of the model. Furthermore, the magnitude of the mixing energy density λ can be related to the interfacial thickness via the interfacial tension −1 20 ( ), which is assumed to be 10 N m in this model (See Eq. 18). Figure 4. Concentration profiles of ionic species and the electrolyte potential along the centerline (x = 0) from the pit bottom (left side) to the 22 λ σ=[18] diffusive boundary layer above the pit mouth (right side). Single pit growing 3 ϵ under galvanostatic conditions in 1 M NaCl bulk solution at I = 1.15 μA apply t = 2s. and 2 Finally, the mobility parameter equals χϵ , where χ is the mobility tuning parameter. It is noted that χ should be calibrated such that the interfacial thickness remains constant while preventing the overdamping issue to the convective term in Eq. 15. Here, χ is calibrated and dependent on the velocity at the interface in each time step. Numerical setting for the 2-D pitting model.—The coupled governing equations, which include the mass transport of ionic n+ − + species (i.e., Me ,Cl ,Na ) in the electrolyte, the evolution of the phase variable, and the dissolution kinetics, were developed using the commercial code COMSOL Multiphysics® software v6.0. Here, the linear solver PARDISO and Backward Euler methods were implemented to solve the system of differential algebraic −3 equations with a relative tolerance of 10 for each time step. Free triangular elements were used with finer mesh in the area below the diffusive boundary layer. All numerical simulations were conducted on the computer cluster using two Intel(R) Xeon(R) Gold 6130 CPU at 2.10 GHz with 772.42 GB RAM. Data post-processing was performed using in-house codes written in Python. Figure 5. Calculated depth and width of modeled pit and ϕ as functions of Results and Discussion time in 1 M NaCl solution and at I = 1.15 μA. apply In this paper, the proposed model is used to simulate the pit propagation of type 316 stainless steel in chloride environments with modeled using the convective Cahn-Hilliard equation, as shown in 15,27 kinetic parameters acquired from the literature. Here, we choose 18,19 Eqs. 13–14. to simulate the propagation of corrosion pits under galvanostatic conditions because it reflects a more realistic pitting behavior when ∂φ +∇ u.. φγ =∇ ∇G [13] multiple pits are required to share the same limited current. Thus, it ∂t allows us to examine the competition between corrosion pits under various conditions. It is also noted that after checking all the modeling results, the treatment of salt precipitation is not needed δ fd Ω 2 φφ(− 1) ⎡ ⎤ G = = λ −∇ φ [14] in this model because the concentration in these conditions does not ⎢ ⎥ 30 δφ ϵ ⎣ ⎦ reach the levels of supersaturation required for salt precipitation. where u is the velocity field and calculated based on the dissolution Single pit growing under galvanostatic conditions.—A simula- rate at the corroding surface (See Eq. 10), γ is the mobility, G is the tion of a single pit growing in 1 M NaCl solution at 1.15 μA was chemical potential. In Eq. 13, the diffusive term is dependent on the carried out to validate the proposed model. Figure 4 shows the gradient of the chemical potential G, while the convective term, concentrations of ionic species and the electrolyte potential in the which is a kind of analogy in this type of modeling approach, is electrolytic solution along the centerline (x = 0) at t = 2s. Because dependent on the movement of the corroding interface. of migration and diffusion, the concentrations of chloride and Equation 14 is then substituted into Eq. 13 to obtain the transport dissolving metal ions inside the modeled pit are significantly higher equation of the phase field variable (Eq. 15) than in the bulk environment. An increase of the IR drop at the bottom of the pit is also observed, which agrees well with the ∂φ φφ(− 1) ⎡ ⎤ conventional transport model in pitting corrosion and experimental +∇ u.. φγ =∇λ∇ −∇ φ [15] 2 8,15,31–33 ⎢ ⎥ ∂t ϵ data. It is noted that the assumed thickness of the boundary ⎣ ⎦ Journal of The Electrochemical Society, 2022 169 081503 Figure 6. The evolution of the modeled pit at different time steps in 1 M NaCl solution and at I = 1.15 μA. The inner solid line is the original pit shape, and apply the color bar shows the concentration of dissolving metal ions (M). layer in this paper could be an extreme case in actual scenarios. However, the obtained results are close to Harb and Alkire’s works, where the steepest gradients of potential and ionic concentrations occur near the corrosion pit, as shown in Fig. 4. Studies on the thickness of the boundary layer and its effect on the development of pit morphology will be conducted in future work. Furthermore, due to a shorter diffusion length at the area near the pit mouth, the local chemistry is more diluted, leading to repassivation and the emergence of the junction between the active and the passive regions near this area, as described by Ernst et al. Figure 5 and Fig. 6 show the evolving morphology of the modeled pit, the concentration of dissolving metal ions, and the calculated potential (ϕ ) after 10 s of simulation. As shown in Fig. 6, the modeled pit is prone to grow sideways and undercut the nearby metal surface to develop a lacy cover. New holes are formed on both sides of the pit, which gives rise to a locally high outward flux of the dissolving metal ions. Thus, the metal concentration near the holes is decreased to a value below the critical threshold for active dissolution, resulting in repassivation on the neighboring surface. In Fig. 6b, after 10 s of simulation, the modeled pit has a dish-like shape with new lobes formed on both Figure 7. Potential (ϕ ) versus time when two simulated pits are grown in sides, which is in good agreement with the experimental data from 1 M NaCl solution at 1.75 μA. Repassivation behavior is incorporated using Ernst and Newman and the pitting model constructed by Laycock and the calibrated function f . 11,15 a1 White. However, it is also noted that the morphology observed here could transition to a more irregular, tunnel-like shape as the pit grows larger under constant applied current, which is explained in the Coalescence of two pits growing under galvanostatic condi- later section of this manuscript. tions.—The proposed model can also be employed to simulate the The corresponding ϕ versus time is obtained from the proposed coalescence of two pits without any remeshing scheme, while providing kinetic information during the propagation stage. In this model and shown in Fig. 5, which resembles the measured potential 12,13,34 section, two modeled pits are assumed to have the same size and in galvanostatic experiments. As the corrosion pit grows positioned close to each other with similar initial pit chemistry. larger, the required potential to sustain the active pit is stabilized Figure 7 and Fig. 8 show the computed values of ϕ and the to a lower value. After each undercutting event, the calculated s developing morphology of modeled pits growing in 1 M NaCl potential is slightly increased due to the local repassivation, but then solution and at 1.75 μA. Here, the repassivation is described using subsides to a lower value as the pit continues to grow after the function f as defined above. perforation. In general, such a rise in potential will be varied a1 As shown in Fig. 7, the calculated potential decreases as pits depending on the level of repassivation that occurred inside the grow larger. However, a small jump of potential is found during the pit, which will be shown in later sections. Here, a small spike of merging stage, corresponding to a local repassivation event near potential of ∼5 mV is seen during the perforation, but decreases to the merging area. Such an observation is due to a slight increase in 25 mV vs SCE after 10 s of growth (See Fig. 5). It is noted that the the pit openness, which promotes a dilution near the contacting zone, magnitude of the observed potential spikes is smaller than the typical particularly the top part, where diffusion length is the shortest. As values obtained from experiments, which could be due to multiple shown in Fig. 8, two simulated pits after the coalescence continue to factors, including the bulk chemistry, the occludedness of the initial grow as a single pit with a dished-shape and undercutting of the geometry, and the location of each undercutting event. Further analysis on the pit morphology at different applied currents will be metal surface, as discussed previously. Thus, ϕ increases during the discussed in the results section of this paper. undercutting event of the two modeled pits (See Fig. 7). Journal of The Electrochemical Society, 2022 169 081503 Figure 8. Developing morphology of two simulated pits growing in 1 M NaCl solution at 1.75 μA. Repassivation behavior is incorporated using the calibrated function f . The color bar shows the concentration of dissolving metal ions (M), and the inner solid line is the initial pit shape. a1 The critical current for pit stability.—As stated by Suleiman and Newman in 1994, there is a critical applied current (I ) above crit which corrosion pits can sustain the aggressive chemistry for a given pit shape. As the pit grows larger, the total active area needs to decrease so that the local current density remains sufficiently high for continued pit growth. Here, we define I as a minimum crit threshold where the concentration of dissolving metal ions at the bottom of the pit is above C with little to no change from the crit initial pit shape. Figure 9 shows the relationship between the applied current and ϕ when the degree of the pit openness (θ) and the initial pit radius (r ) are varied. After one second of simulation, a pit significant drop of the potential is observed when the applied current is above critical values, suggesting that pits can remain active in this regime. The morphology of pits growing when the applied current is close to I will be discussed in a later section. crit Furthermore, the values of I depend on various factors, crit including but not limited to the initial pit geometry, the bulk chemistry, and the deposition of corrosion products at the pit mouth. As shown in Fig. 9, I decreases as r increases and as θ crit pit Figure 9. The applied current versus the calculated potential of a single pit decreases, which can be explained by the extension of the diffusion in 1 M NaCl solution and at t = 1s. barrier provided in a larger pit and in a more occluded geometry, respectively. Thus, the corrosion pit requires a lower applied current to remain active. Here, a distinct morphology where a new pit is formed at the bottom of the initial pit is observed after six seconds of simulation Pit grows within a pit at low applied current.—The propagation (See Fig. 10a), which could be explained based on the relationship of corrosion pits under galvanostatic conditions is complex since pits between I and C discussed by Suleiman and Newman. Due to crit crit can optimize the total active surface area to sustain the limited the low applied current used in this simulation, the initial pit surface current, particularly at values near I for a given pit shape. crit is split into the active and the passive regions, which in turn reduces Figure 10 presents the simulated data and the evolving morphology the total active surface area and triggers the redistribution of the of a modeled pit growing in 1 M NaCl solution and at local current density inside the pit (See Figs. 10b, 10d). Thus, a I = 0.7 μA, which is close to the I observed in Fig. 9 when apply crit newly developed pit is formed at the bottom of the pit to sustain the r θ 30°, and equal5m μ and respectively. low current in galvanostatic condition, which agrees well with the pit Journal of The Electrochemical Society, 2022 169 081503 Figure 10. Simulated electrochemical data of single pit growing under galvanostatic conditions in 1 M NaCl solution. I == 0.7μμ A, r 5 m,θ= 30°. apply pit experimental observation using the metal foil technique to explore pit, suggesting that the pit propagation is retained near the marginal the 2-D pit morphology of corrosion pits. regime for pit stability. As shown in Fig. 10d, a rise of c.d. at the bottom of the pit is recorded while the remaining surface becomes inactive. The local c. Lateral growth and the transition to the marginal instability −2 d. then decreases and stabilizes to a value of ∼2Acm as the regime.—At moderate to high applied currents and in the absence of newly formed pit becomes larger. The transition between the active the salt film, when most of the corroding surface is active except for and the passive region inside the pit is characterized by the critical the repassivated area near the pit mouth, a corrosion pit is likely to current density for passivation, which is studied extensively in grow sideways and can undercut the nearby surface, as discussed pitting corrosion research and proved to depend on the previously (Fig. 6). However, as the pit grows larger under constant 15,34–36 temperature. Furthermore, the corresponding concentration current, it transitions into the marginal instability regime and of dissolving metal ions, the IR drop, and the surface potential along undergoes a “natural selection” process that favors a certain degree the pit surface are also calculated, as shown in Figs. 10c–10f. In all of occlusion to sustain the critical chemistry for pit stability. time steps, a higher concentration of dissolving metal ions above Specifically, the dissolution at the bottom of the pit is preserved, but below the saturated value is observed at the bottom of the while there is competition for the remaining current between two crit Journal of The Electrochemical Society, 2022 169 081503 Figure 11. Modeled pit grows in 1 M NaCl solution at I = 1.15 μA. In case 1 and case 2, a fillet is added on either side of the pit mouth. In case 3, a step- apply down current of 0.9 μA is applied after 10 s of simulation. The inner solid line is the original pit shape, and the color bar shows the concentration of dissolving metal ions (M). sides of a single pit. Such selective repassivation can occur in natural almost identically in the first 10 s of growth, which is similar to the pit growth conditions due to multiple factors, including the micro- developed morphology shown in Fig. 6. However, as the simulated structure, the surface roughness, and the asymmetry of the actual pit pit grows larger under constant applied current, it would also need to shape. In this paper, the imperfection at the diffusive interface can undergo a selective repassivation to remain active and thus become lead to such observation, but only if the transition to the marginal more sensitive to the initial pit geometry. As a result, the newly instability regime is slow enough. To verify this notion, simulations formed lobe on one side is repassivated while the remaining surface are carried out when a small degree of asymmetry is introduced to stays active, which leads to the observed gradient of the dissolved the initial pit shape. Here, a fillet, which is commonly known as a metal ion concentration along the pit surface, as shown in Fig. 11. rounding arc created at the corner of two intersecting surfaces, is For comparison purposes, another simulation is conducted with added and thus slightly modifies the ionic fluxes between two sides the same initial conditions described in Fig. 6. However, a step- of the modeled pit (case 1 and case 2 in Fig. 11). Due to such small down current of 0.9 μA is applied after 10 s of holding at 1.15 μA differences, the propagation rate and the pit morphology developed (case 3 in Fig. 11), which is similar to the type of simulation of a Journal of The Electrochemical Society, 2022 169 081503 in case 3 when the current is switched to a lower value are observed, suggesting repassivation events could occur inside the corrosion pit during the marginal growth as shown in Fig. 11. The interaction mechanism between pits and the “champion pit” in multi-pit simulations.—Multi-pit simulations are conducted at different applied currents to investigate the interaction mechanism between pits. Here, we simulated the simultaneous growth of four pits with different degrees of pit openness (i.e.,θθ== pit13 pit 30°= ,θθ = 37.5°) in 1 M NaCl solution. As shown in pit24 pit Fig. 13, in the presence of multiple pits, only the more occluded pits (pit 1 and pit 3) remain active, whereas the more opened pits (pit 2 and pit 4) become inactive. Such an observation is due to the competition between pits for the limited current. Specifically, under galvanostatic conditions, propagating pits share the same supplied current and are required to optimize their shape to accommodate such constraints as pits grow larger. Thus, when the applied current is low enough so that all pits cannot propagate concurrently, pits are subjected to the “natural Figure 12. Potential versus time corresponding to the change in the initial selection” process, which favors the more occluded geometry to pit shape (case 1 and case 2) and the applied current (case 3). retain the critical chemistry for propagation. As a result, pits 1 and 3 stay active, while pits 2 and 4 are repassivated, which gives rise to the potential ϕ in Fig. 15. flow disturbance when the boundary layer thickness is reduced. Since the current is reduced to a lower value, both sides of the Furthermore, in a simulation at an even lower applied current but simulated pit are repassivated, whereas the bottom of the pit stays above I , a minimum threshold to sustain the growth of at least one crit active. As a result, a new pit is developed beneath the initial pit, pit in a multi-pit simulation, pit 1 is repassivated after growing to a similar to that mentioned previously. Figure 12 shows the potential certain size, whereas pit 3 remains active and propagates (See as functions of time in three cases. Small humps after 13 s of Fig. 14). A jump of ϕ is observed at ∼1 s corresponding to the simulations in case 1 and case 2, and a sharper increase of potential repassivation of pit 1 in the simulation (See Fig. 15). Although two Figure 13. Multi-pit simulation under galvanostatic conditions in 1 M NaCl solution and at I =2. μA Pit 1 and pit 3 have the same standard shape with apply θ=° 30 , while pit 2 and pit 4 have wider pit mouths with θ=° 37.5 . The color bar shows the concentration of dissolving metal ions (M). Journal of The Electrochemical Society, 2022 169 081503 Figure 14. Multi-pit simulation under galvanostatic conditions in 1 M NaCl solution and at I = 1.75 μA. Pits 1 and 3 have the same standard shape with apply θ=° 30 , while pit 2 and pit 4 have wider pit mouths with θ=° 37.5 . The color bar shows the concentration of dissolving metal ions (M). “champion pit,” it then acquires the remaining current and grows with a faster dissolution rate. As shown in Fig. 14, the concentration of dissolving metal ions inside pit 3 increases significantly upon the repassivation of pit 1. However, as the “champion pit” becomes larger and requires a higher current to remain active, it could also transition into the marginal instability regime, similar to that described in the single pit case. The modeling results of multi-pits growing under galvanostatic conditions agree well with experimental observations when the 2-D morphology of natural pit growth was explored using the metal foil technique and in situ synchrotron X-ray radiography. In the presence of multiple pits, only the larger pit is reported to remain active, while smaller pits are repassivated. Furthermore, the mor- phology of the surviving pit is reported to have a rougher surface and more penetrating than that under potentiostatic conditions. Such an observation from experimental work could be explained based on the competition between pits and the optimization of pits’ shape to accommodate the limited supply of current in galvanostatic condi- tions discussed previously by Gaudet et al. in 1987. The oscillation Figure 15. Potential versus time in multi-pit simulation at different applied behavior of the simulated potential in this model is close to that currents. observed in experimental studies conducted by Suleiman and Newman when the “champion pit” survived for a longer time and pits have identical initial geometry, the arrangement of four pits in obtained a lower potential with a larger size. this study is not entirely symmetrical when pit 1 is placed in the furthest position and pit 3 is placed between pits 2 and 4 in close Comparison with potentiostatic conditions.—Under galvano- proximity. This could result in small variations in the local static conditions, the developed morphology becomes more complex environment near the mouth of each pit and, in turn, induces the than the original pit shape. It is more realistic than potentiostatic repassivation of pit 1 at a lower applied current. Once the pit pitting because the potential has to drop as the pit grows larger, survives the selection process for the limited current and becomes a which is relatively similar to an open-circuit or natural pitting Journal of The Electrochemical Society, 2022 169 081503 experiment. Due to the limited supply of current in galvanostatic Network of Excellence in Nuclear Engineering (UNENE), and the conditions, a corrosion pit can undergo selective repassivation, or Qatar National Research Fund through the National Priorities have a new pit formed beneath the existing pit, which leads to a Research Program (NPRP) under grant number NPRP12S-0209–- narrower and tunnel-like shape with a rounded bottom, as shown 190063. previously. Such complex morphology is reported in the literature ORCID when corrosion pits are growing under current-controlled conditions and in the activation-controlled dissolution without the presence of Van Anh Nguyen https://orcid.org/0000-0001-8299-611X 12,27,34,37,38 the salt film. Furthermore, some open-circuit pits tend to Roger C. Newman https://orcid.org/0000-0002-2422-619X have these shapes and possibly are grown in the activation- Nicholas J. Laycock https://orcid.org/0000-0002-4603-4156 controlled region. In potentiostatic conditions, since the current is allowed to increase References indefinitely as the pit grows into a larger size, there is no competition for 1. J. W. Tester and H. S. Isaacs, J. Electrochem. Soc., 122, 1438 (1975). resources between pits. Thus, pits can propagate until they merge into a 2. R. C. Newman and H. S. Isaacs, J. Electrochem. Soc., 130, 1621 (1983). larger pit, as described in the experimental work. In addition, pits 3. N. J. Laycock and R. C. 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Seo, and, thus, transitions to the marginal growth at a given constant current. B. Macdougall, H. Takahashi, and R. G. Kelly PV 99-27 (The Electrochemical Society, Pennington, NJ) 541 (1999). Acknowledgments 38. A. Okeremi and M. Simon-Thomas, NACE Conference 2008, paper 08254 (2008). 39. N. Sato, J. Electrochem. Soc., 129, 260 (1982). Portions of this research were funded by the Natural Science and 40. M. P. Ryan, N. J. Laycock, H. S. Isaacs, and R. C. Newman, J. Electrochem. Soc., Engineering Research Council of Canada (NSERC), the University 146, 91 (1999).

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