Localized Corrosion: Passive Film Breakdown vs. Pit

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Journal of The Electrochemical Society, 165 (11) C762-C770 (2018)

Localized Corrosion: Passive Film Breakdown vs. Pit Growth Stability: Part III. A Unifying Set of Principal Parameters and Criteria for Pit Stabilization and Salt Film Formation Tianshu Li,

1,z

J. R. Scully,

2,∗

and G. S. Frankel

1,∗

1 Fontana 2 Center

Corrosion Center, The Ohio State University, Columbus, Ohio 43210, USA for Electrochemical Science and Engineering, University of Virginia, Charlottesville, Virginia 22904, USA

The framework for pit stability established in Part II of this series is elaborated upon and expanded. The maximum pit dissolution current density, idiff,max , which depends on temperature and maximum potential at the pit surface in a given pit environment, is proposed and compared with the diffusion current density to determine the conditions for pit stability and salt film formation. For an open pit, the criterion for pit stability is that idiss,max must be greater than or equal to the critical diffusion current density required for maintaining a critical local environment for active dissolution, idiff,crit . The critical condition of idiss,max = idiff,crit is associated with three parameters that must be exceeded for stabilization of an open pit: critical temperature Tcrit , critical potential Ecrit and critical pit depth rcrit . Analogously, the criterion for salt film formation is idiss,max must further exceed the diffusion-limited current density ilim , and three parameters associated with salt film formation are: saturation temperature Tsat , saturation potential Esat and saturation pit depth rsat . Unifying criteria for control of pit growth by charge transfer or diffusion are proposed, which provide interrelationships for all principle parameters. The pit stability criteria for covered pits, such as metastable pits, are also described. © The Author(s) 2018. Published by ECS. 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: 10.1149/2.0251811jes] Manuscript submitted May 25, 2018; revised manuscript received July 29, 2018. Published August 16, 2018.

Background The relative importance of passive film breakdown and pit growth stability in pitting corrosion has been debated over the last few decades. Recently, the authors of this paper proposed a new perspective on the critical step of pitting by suggesting that pit growth stability controls pitting corrosion under aggressive conditions, i.e. harsh electrolytes and extreme environments and/or susceptible microstructures, while passive film breakdown is more likely the critical factor in less extreme environments and/or for less susceptible alloys.1 Based on this perspective, a conceptual model for the critical pitting temperature (CPT), which considers both pit growth stability and passive film breakdown, was proposed.2 The model was able to explain various aspects of CPT, including those that could not be explained by previous models, such as why CPT measurements can exhibit a distribution of values and why the CPT of an alloy can depend on a variety of factors such as surface treatments and the chloride concentration of bulk solution. To establish the model, a new framework for pit growth stability was presented by considering both dissolution and diffusion processes inside the pit to maintain a sufficiently aggressive local environment required for active dissolution. The purpose of this paper is to elaborate upon and expand the framework to describe the principal parameters and criteria for pit stabilization, as well as for salt film formation. Passive metals exhibit a low corrosion rate in aqueous environments due to the formation of a thin passivating oxide film on their surface.3–5 Pitting corrosion arises when passivity is lost at localized points on the metal surface, resulting in accelerated dissolution of the underlying metal.3–8 Irrespective of the nucleation event, for a pit to undergo sustained growth, the pit surface must be kept in the active state and prevented from repassivating. Nowadays, the most prevalent viewpoint is that active pit growth requires the maintenance of a local aggressive environment, and several criteria have been proposed to describe pit stability based on this notion.4,9–12 In 1976, Galvele proposed a pit stabilization theory by considering the influence of pit chemistry changes.9 He hypothesized that the hydrogen ion concentration, [H+ ], is the only critical parameter in determining pit stability. Using a one-dimensional (1D) pit model and considering metal cation hydrolysis and diffusion, he calculated the concentration of various ionic species as a function of the product of ∗ Electrochemical Society Fellow. z E-mail: [email protected]

pit depth, x, and current density, i. Galvele proposed that the critical localized acidification for sustained pit growth can only be reached at a critical value of ix.10 The model did not consider the depassivating effect of aggressive anions such as chloride on stabilizing pit growth. However, as low pH and high [Cl− ] must accompany each other in pit solutions, the critical value of ix can still be used to describe the aggressive local environment required for pit stability. In 1992, Pistorius and Burstein proposed a pit stabilization theory based on the consideration of steady-state diffusion control of metal cations out of the pit.4 According to their theory, the local aggressive environment inside the pit can only be maintained if the pit growth is controlled by diffusion of metal cations out from the pit interior into the bulk solution. Similar to the critical value of ix in Galvele’s model, they proposed a critical pit stability product of ir, where r is the depth of a hemispherical pit and i is the pit current density. According to Pistorius and Burstein, both metastable and stable growth of a pit are under diffusion control.4 In 1994, Salinas-Bravo and Newman proposed an hypothesis for pit stabilization in their CPT model by suggesting that a salt film must be present on the pit surface to sustain the stable growth.13 According to their model, a pit can only transition to stability above the CPT, since only then can the pit form a salt film on its surface. However, the crystallographic pit morphologies observed in many cases14–16 cannot be explained by the perspective of diffusion control, and such observations indicate that a salt film might not be required for pit stability. Although many details of pit stabilization remain unclear, it is agreed that a sufficiently aggressive pit environment is required for pit stability, and such an environment must be continuously maintained during the sustained pit growth. The greatest challenge to extend this idea lies in how to consider the competition between the dissolution and diffusion processes inside the pit. However, at steady state, for both metastable or stable pit growth, the actual dissolution rate of metal at the pit surface always equals the actual diffusion rate of metal cations out from the pit interior into the bulk solution. Because they must be equal, it is not beneficial to use the actual dissolution and diffusion rates to consider the competition between the dissolution and diffusion processes. As a result, in most previous theories, only one process, either diffusion or dissolution, was considered. In Part II of this series of papers, the concept of maximum dissolution current density of a pit was proposed to address this issue, and a new framework for pit growth stability was established by comparing this maximum pit dissolution current density with the critical diffusion current density required for maintaining a sufficiently aggressive pit environment for active

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dissolution. In this paper, the framework is further expanded. Principal parameters and criteria for pit growth stability of both metastable and stable pits and salt film formation are proposed based on new ideas, presenting a unified description of various aspects of pit growth. Criteria for control of pit growth by charge transfer or diffusion are also described. Details of the salt film formation and its role in pit growth process will be addressed in a subsequent paper as part of this series. Basic Considerations for Stable Pit Growth Pit stability requires the maintenance of a continuous active dissolution state at the pit surface, which depends on an aggressive pit environment. The most aggressive species inside a pit are considered to be H+ and Cl− . It is well known that local acidification occurs through the hydrolysis of the metal cations, and chloride anions migrate into the pit to balance the charge of the dissolved metal cations and maintain electroneutrality.3 Therefore, the aggressive pit environment is actually derived from the accumulation of metal cations, and thus the concentration of metal cations at the pit surface (Csurf ) can be used to evaluate the aggressiveness of the pit environment. In addition to Csurf , the potential drop across the pit surface/pit solution interface (Esurf ) and the temperature (T) are two other critical parameters that determine the active dissolution of metal at the pit surface. It can be considered that, within a certain range of (Csurf , Esurf , T), if any two of these three parameters are fixed, the remaining one should have a critical value, above which the active dissolution state can be sustained. In this paper, Csurf was selected as the parameter to determine the condition for active dissolution, because maintaining Csurf above a critical value, Ccrit , is associated with both dissolution and diffusion processes, and this facilitates the consideration of the competition between these two processes. Even though Ccrit might be a function of Esurf and T, these dependencies have not been well defined so Ccrit is assumed to be independent of temperature and potential for simplicity in this paper. Additionally, there exists a saturation concentration of metal cations, Csat , associated with the solubility limit of the metal chloride salt. Supersaturation of metal cations is not a stable state,17 and it is not considered here for simplicity. The accumulation of metal cations inside the pit depends on the dissolution of the metal at the pit surface and the transport of metal cations out from the pit interior into the bulk solution. The pit solution is highly concentrated and often well supported so that metal cations carry only a small fraction of the ionic charge, thus electromigration is typically negligible in this transport process. Furthermore, convection is limited in a pit, so only diffusion needs to be considered as a transport process. The focus of this work will be on Fe- or Ni-based alloy systems in which hydrogen evolution at the bottom of pits is not a major factor. Modifications of this framework would be required to account for the convection associated with the copious hydrogen evolution in pits in Al alloys. Assuming that the pit grows as an open hemisphere, the processes of dissolution and diffusion are schematically shown in Fig. 1. For a given pit depth r and temperature T, the diffusion rate of metal cations out of the pit into bulk solution is determined by the concentration of metal cations at the pit surface, Csurf . When Csurf < Ccrit , pit growth cannot be sustained and the pit will repassivate, whereas active dissolution will continue when Csurf ≥ Ccrit . Therefore, there should exist a critical diffusion current density, idiss,crit , associated with the critical condition of Csurf = Ccrit . Moreover, when Csurf = Csat , the diffusion current density will reach its maximum value, which is defined as the diffusion-limited current density, ilim . On the other hand, for a given temperature and Csurf , the metal dissolution rate depends on the actual potential at the pit surface, Esurf , the maximum value of which should be: E max = E app − sol

[1]

where Eapp is the applied potential, and sol is the solution ohmic potential drop between the pit surface and the reference electrode, i.e. the total ohmic potential drop of bulk solution and pit solution.

Figure 1. Schematic view of an open hemispherical pit with radius r. idiss,max represents the maximum dissolution current density at the pit surface. idiff,crit represents the critical diffusion current density. ilim represents the diffusionlimited current density.

Note that ohmic potential drop associated with the salt film is not included in sol , thus Emax represents the maximum potential that can exist at the pit surface for a given applied potential. Emax can also be used to define a maximum pit dissolution current density, idiss,max , which describes the maximum value of dissolution rate at the pit surface. In simple terms, idiss,max can be considered to be the active dissolution current density of the bare metal when its surface potential is anodically polarized to Emax (thus Eapp if sol is negligible) in pitlike solution. The role of sol will be discussed in more detail in On the Stabilization of Pits at Ccrit ≤ Csurf < Csat section. For stabilization of an open pit to occur, the maximum rate of metal dissolution at the pit surface must be sufficiently high to prevent dilution of Csurf to a value lower than Ccrit by diffusion out of the pit. In other words, to maintain Csurf ≥ Ccrit , the maximum dissolution current density of the metal at pit surface must be equal to or larger than the critical current density for diffusion of metal cations out of the pit, otherwise, the pit will repassivate. Therefore, a basic criterion for the stability of an open pit should be: i diss,max ≥ i diff,crit

[2]

Two further situations must be considered here. When idiff,crit ≤ idiss,max < ilim , metal dissolution is the slow step and Csurf will thus adjust to a value below Csat to balance the diffusion current density with idiss,max . The pit growth is therefore under charge transfer (or activation) control with a pit surface that is crystallographic if the pit growth can be maintained under this condition for an extended period. When idiss,max ≥ ilim , diffusion is slow and the pit growth will be under diffusion control. Under this condition, idiss,max will not be reached since the actual dissolution current density will be limited by ilim . A natural consequence of idiss,max > ilim is that the metal cations at pit surface will eventually accumulate to reach the saturation concentration and a salt film will form, resulting in an electropolished pit surface. Note that, in practice, the supersaturation of metal cations might be required for salt film formation, thus idiss,max must be higher than the corresponding diffusion current density associated with the supersaturation concentration to precipitate the salt film. Critical Values of Temperature, Potential and Pit Depth Considering an open hemispherical pit, as shown in Fig. 1, and assuming the concentration of metal cations in the bulk solution is negligible, the current density associated with diffusive transport of metal cations out of the pit, idiff , can be approximated by:4 3n F DCsurf [3] 2πr where n is the average oxidation state of the metal cations, F is the Faraday constant, D is the effective diffusion coefficient of metal cations that takes into account the effect of concentration-dependent diffusivity in the pit and the possible influence of electromigration,17,18 r is the pit depth or the radius of the hemispherical pit, and 2πr/3 is the i diff =


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effective diffusion length4 . Using Ccrit and Csat to replace Csurf results in expressions for idiss,crit and ilim , respectively: i diss,crit = i lim =

3n F DCcrit 2πr

3n F DCsat 2πr

[4]

[5]

On the other hand, idiss,max describes charge transfer controlled dissolution, thus for a given Csurf , idiss,max should be a function of T and Emax , i.e. idiss,max (T, Emax ). If Emax is set at a constant value of Econst , the maximum dissolution current density can be expressed as idiss,max (T, Econst ). Similarly, if the temperature is set at a constant value of Tconst , the maximum dissolution current density can be expressed as idiss,max (Tconst , Emax ). For simplification, idiss,max (T, Econst ) and idiss,max (Tconst , Emax ) are written as idiss,max (T) and idiss,max (Emax ), respectively. Obviously, both idiss,max (T) and idiss,max (Emax ) are increasing functions. As mentioned above, Ccrit is assumed to be independent of potential and temperature for simplicity. D is a weak function of temperature and thus assumed to be independent of temperature, which is supported by the analysis given in Appendix A. Additionally, D should also be independent of potential. Therefore, according to Eq. 4, idiff,crit (T) and idiff,crit (Emax ) can be considered to be constant for a given pit depth r. On the other hand, ilim (T) is an increasing function, but should be a weaker function of T than imax (T), as shown in Appendix A. In addition, ilim (Emax ) should be constant, independent of the potential. For fixed values of Emax and pit depth r, the relationships between idiff,crit (T), ilim (T) and idiss,max (T) are shown in Fig. 2a, and the relationships between idiff,crit (Emax ), ilim (Emax ) and idiss,max (Emax ) for fixed values of T and r are shown in Fig. 2b. According to Fig. 2a, if Emax and r are fixed, there should be a critical temperature that must be exceeded for stabilization of an open pit, Tcrit . At temperatures below Tcrit , the critical concentration of metal cation (Ccrit ) required for active dissolution cannot be maintained because idiss,max < idiff,crit , and open pits will passivate. The measurement of CPT during a potentiostatic temperature-scanning experiment in which a pit initiates and grows so that both r and T increase with time was addressed previously.2 In that situation, idiff,crit for a given pit decreases with T, but only because the pit depth, which determines the idiff,crit , increases with time (and thus with T). Only potential is controlled and the pit stabilization therefore depends on a critical combination of r and T. Fig. 2a describes a situation for fixed r and Emax , so it is an instant in time for a growing pit. It should also be noted that the form of Fig. 2a is similar to the concepts put forth by Newman and coworkers, which describe the CPT as the point where two currents cross.13,19 However, the currents described in Fig. 2a are different than those considered by Newman et al., as has been discussed in our previous paper.2 There also exists a critical temperature for salt film formation, Tsat . At temperatures above Tsat , the metal cations at pit surface will eventually accumulate to reach the saturation concentration (Csat ) because idiss,max > ilim , a salt film will form on the pit surface, and pit growth will transition to diffusion control. Therefore, Tsat is also a critical temperature for diffusion control. If r is fixed at a larger depth, according to Eqs. 4 and 5, both idiss,crit and ilim will be decreased, then both Tcrit and Tsat will decrease accordingly. This indicates that both Tcrit and Tsat will decrease with increasing r. Analogous to the previous discussion, if T and r are given, there should exist a critical Emax for pit stabilization, Ecrit , and a critical Emax for salt film formation and diffusion control, Esat , as shown in Fig. 2b. When Emax ≥ Ecrit , an open pit can be stabilized, and when Emax > Esat , a salt film will form on the pit surface. Both Ecrit and Esat will also decrease with increasing r . Eapp = Ecrit + sol is thus the repassivation potential for a growing pit. It is worth noting that the concept of repassivation potential for a growing pit is different from that of repassivation potential that might be measured for an alloy. Details on this will be discussed in a subsequent paper.

Figure 2. Schematic representations of idiff,crit , ilim and idiss,max dependencies on temperature, maximum potential at pit surface and pit depth, showing that there are two critical values for both of the (a) temperature (Tcrit and Tsat ), (b) maximum potential at pit surface (Ecrit and Esat ) and (c) pit depth (rcrit and rsat ). For simplification, the increasing functions in (a) and (b) are just represented by straight lines.

If T and Emax are both fixed for an open pit, there should exist a critical pit depth for pit stabilization, rcrit , and a critical pit depth for salt film formation and diffusion control, rsat . For the given T and Emax , and considering that idiff,crit and ilim are given by Eqs. 4 and 5, respectively, values of rcrit and rsat are obtained as shown in Fig. 2c. Additionally, when T or Emax is fixed at a larger value, idiss,max will increase, then both rcrit and rsat will decrease accordingly.



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Based on the above analysis, the basic criterion for the stability of an open pit, idiss,max ≥ idiff,crit , can be developed into three specific criteria: T ≥ Tcrit

[6]

E max ≥ E crit

[7]

r ≥ rcrit

[8]

Similarly, the basic criterion for salt film formation and diffusion control of an open pit, idiss,max ≥ ilim , can also be developed into three other specific criteria: T ≥ Tsat

[9]

E max ≥ E sat

[10]

r ≥ rsat

[11]

Note that Tcrit , Ecrit and rcrit are equivalent representations of the same critical condition where idiss,max = idiff,crit , so Eqs. 6, 7 and 8 are just different forms of the same criterion, which means if any one criterion is satisfied, the other two criteria will be satisfied at the same time. Similarly, Eqs. 9, 10 and 11 are also identical criteria for salt film formation and diffusion control. Using the set of parameters (T, Emax , r) to define the state of an open pit, stability will be achieved if (Tcrit , Ecrit , rcrit ) are exceeded, and if (Tsat , Esat , rsat ) are further exceeded, pit growth will transition to diffusion control with the presence of a salt film on the pit surface. When Tcrit ≤ T < Tsat , Ecrit ≤ Emax < Esat or rcrit ≤ r < rsat , pit growth is controlled by charge transfer, and salt film will not precipitate on the pit surface. Under this condition, the actual pit current density, ipit , equals idiss,max and is a function of Emax and T. It is therefore necessary to obtain the values of (Tcrit , Ecrit , rcrit ) and (Tsat , Esat , rsat ). As idiss,max represents the charge transfer controlled dissolution rate of active metal in pit-like solution, idiss,max (T) should obey the Arrhenius equation, while idiss,max (Emax ) should obey the Tafel equation. Therefore, if Emax is given, the dependency of idiss,max on T should be: −G ∗a [12] i diss,max (T ) = Aexp RT where A is the pre-exponential factor, G ∗a is the activation energy of active metal dissolution under a polarized surface potential of Emax in pit-like solution, and R is the gas constant. On the other hand, if T is given, the dependency of idiss,max on Emax should be: E max − E corr [13] i diss,max (E max ) = i corr exp βa where Ecorr , icorr and βa are the corrosion potential, corrosion current density and anodic Tafel slope in pit-like solution, respectively. Therefore, when Emax is given, Tcrit and rcrit can be obtained by applying the critical condition of idiss,max = idiff,crit . Combining Eq. 4 and Eq. 12: Tcrit (r ) = rcrit (T ) =

G ∗a Rln 3n 2πAr F DCcrit

[14]

3n F DCcrit ∗

[15]

2πAexp

−G a RT

Similarly, applying the critical condition of idiss,max = ilim and using Eq. 5 gives Tsat and rsat : Tsat (r ) =

G ∗a Rln 3n2πAr F DCsat

3n F DCsat rsat (T ) = ∗ a 2πAexp −G RT

[16]

Figure 3. Assumed configuration of a covered metastable pit. The pit, with radius r, is covered by a film that contains a centrally located perforation of radius a.

It should be noted that DCcrit in Eq. 14 and DCsat in Eq. 16 are assumed to be independent of temperature to simply the equation. For DCsat , this approximation should be valid within a small temperature range, probably ∼10◦ C as indicated in Appendix A. Analogously, if T is given, Ecrit and rcrit can be obtained by equating Eq. 4 and Eq. 13: E crit (r ) = E corr + βa ln rcrit (E max ) =

3n F DCcrit 2πi corr r

3n F DCcrit corr 2πi corr exp Emaxβ−E a

[19]

Similarly, Esat and rsat can be obtained by equating Eq. 5 and Eq. 13: E sat (r ) = E corr + βa ln rsat (E max ) =

3n F DCsat 2πi corr r

3n F DCsat corr 2πi corr exp Emaxβ−E a

[20] [21]

The pit stability criteria given by Eqs. 6, 7 and 8 and the principal parameters (Tcrit , Ecrit , rcrit ) are valid for open pits that are stabilized by their own depth. Any pits growing under conditions that do not meet these criteria are metastable. Therefore, the criteria of Eqs. 6, 7 and 8 are also the criteria for the transition of a pit from metastable to stable growth. The factors stabilizing metastable pit growth must then be understood. Stability of Metastable Pits Eq. 4 shows that idiff,crit varies inversely with pit radius so the value of idiff,crit for very small pits is very large and the situation where idiss,max < idiff,crit may easily arise. Therefore, an additional diffusion barrier is required for maintaining the critical pit environment for very small pits, which is the situation for metastable pits. According to previous researchers,4,12,20,21 the pit cover is the additional diffusion barrier for metastable pits. It was previously reported that the undermined passive film is the pit cover that creates a diffusion barrier on metastable pits in stainless steel.12 Larger pits may be covered by a so-called lacy pit cover that includes undermined passive film and unreacted metal.22–25 According to Pistorius and Burstein,4 the simplest configuration of a metastable pit with an undermined cover is shown in Fig. 3, where a is the radius of a centrally located perforation in the pit cover. The diffusion current for such a covered hemispherical pit, Idiff,cov , can be approximated by:4 Idiff,cov = can F DCsurf

[17]

[18]

[22]

where c is a dimensionless constant with a value between 2 and 3. Therefore, the critical diffusion current density for a covered pit,


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Table I. Principal parameters and criteria for pit stability, charge transfer control, diffusion control and salt film formation. idiss,max = idiff,crit or idiff,crit,cov

idiss,max = ilim or ilim,cov

Pit Stability

Charge Transfer Control

Diffusion Control and Salt Film Formation

Tcrit Ecrit rcrit acrit

Tsat Esat rsat asat

T ≥ Tcrit Emax ≥ Ecrit r ≥ rcrit a ≤ acrit

Tcrit ≤ T < Tsat Ecrit ≤ Emax < Esat rcrit ≤ r < rsat asat < a ≤ acrit

T ≥ Tsat Emax ≥ Esat r ≥ rsat a ≤ asat

idiff,crit,cov , can be expressed as: can F DCcrit [23] 2πr 2 Similarly, the diffusion-limited current density for a covered pit, ilim,cov , should be: i diff,crit,cov =

can F DCsat [24] 2πr 2 Because of the presence of the pit cover, the maximum potential that can be applied at the pit surface should be Eapp − sol − pc , where pc is the ohmic potential drop across the pit cover. As detailed in Appendix B, if the thickness of pit cover is small but a is not an extremely small value, then pc can be neglected. Therefore, Emax of a covered pit is still equal to that of an open pit. Similar to the basic pit stability criterion for an open pit proposed in Eq. 2, the basic stability criterion for a covered pit should be: i lim,cov =

i diss,max ≥ i diff,crit,cov

[25]

and the basic criterion for salt film formation and diffusion control for a covered pit then should be: i diss,max ≥ i lim,cov

[26]

There should exist a critical perforation radius for stabilization of a covered pit, acrit , above which repassivation occurs because of the loss of the critical pit environment (Ccrit ) required for active dissolution. There should also exist a smaller critical perforation radius for salt film formation and diffusion control, asat . For a perforation radius smaller than asat , the growth of a covered pit is under diffusion control and a salt film will form on the pit surface. The values of acrit and asat can be obtained by applying the critical conditions of idiss,max = idiff,crit,cov and idiss,max = ilim,cov , respectively. If Emax is given, acrit can therefore be obtained by equating Eq. 12 and Eq. 23: −G ∗ 2πAr 2 exp RT a acrit (r, T ) = [27] cn F DCcrit Similarly, asat can be obtained by equating Eq. 12 and Eq. 24: −G ∗ 2πAr 2 exp RT a asat (r, T ) = cn F DCsat

[28]

Analogously, if T is given, acrit can be obtained by equating Eq. 13 and Eq. 23: corr 2πi corr r 2 exp Emaxβ−E a [29] acrit (r, E max ) = cn F DCcrit and asat can be obtained by equating Eq. 13 and Eq. 24: corr 2πi corr r 2 exp Emaxβ−E a asat (r, E max ) = cn F DCsat

The dissolution of metal at the pit surface and the diffusion of metal cations out of the pit occur in sequence. Therefore, when a pit grows at steady state, the actual dissolution current density (idiss,act ) should always equal the actual diffusion current density (idiff,act ). According to the proposed framework, when idiff,crit ≤ idiss,max < ilim (for open pits) or idiff,crit,cov ≤ idiss,max < ilim,cov (for covered pits), pit growth is under charge transfer control with Ccrit ≤ Csurf < Csat . For any given pit, the actual diffusion current density (idiff,act ) can be expressed by: i diff,act =

[31]

and the criterion for salt film formation and diffusion control of metastable pit is: a ≤ asat

On the Stabilization of Pits at Ccrit ≤ Csurf < Csat

[30]

Therefore, the stability criterion for metastable pit growth is: a ≤ acrit

Under the condition of a ≤ asat , the increase in pit current is derived from the rupture events of the pit cover, and discrete rupture events of the pit cover might result in a discontinuous increase in pit current. A current jump might also be observed at the end of the metastable growth because of a large rupture of the pit cover to an extent above acrit . When asat < a ≤ acrit , a salt film will not precipitate on the pit surface and the growth of a covered pit is controlled by charge transfer. Therefore, the actual pit current density ipit equals idiss,max , and thus it should be a function of Emax and T. Under this condition (asat < a ≤ acrit ), the current transient of a metastable pit should increase continuously and smoothly, and a current jump will not be observed at the end of the metastable growth. The principal parameters and criteria for pit stability, charge transfer control, diffusion control and salt film formation are summarized in Table I, which shows that a salt film is not required for pit stabilization, it is just a consequence of a pit achieving diffusion-controlled growth. It also indicates that both metastable and stable pit growth can be controlled either by charge transfer or by diffusion. This perspective is different than the viewpoint held by Pistorius and Burstein,4 who considered that the growth of both metastable and stable pits is under diffusion control. According to the proposed framework, when pit growth is controlled by charge transfer, the pit current density equals idiss,max , while if pit growth is controlled by diffusion, the pit current density equals ilim (for open pits) or ilim,cov (for covered pits). However, although a pit can grow under charge transfer control, such a pit will tend to concentrate the pit solution and transition to diffusion control. This will be discussed in detail below. Interestingly, the principal parameters for stability of open pits, Tcrit , Ecrit and rcrit , can also be deduced from the perspective of metastable growth. In fact, they represent a critical state where acrit = r. Similarly, the principal parameters for salt film formation and diffusion control of open pits, Tsat , Esat and rsat , represent a critical state where asat = r. This is derived in Appendix C. Although the expressions of these principal parameters are derived for hemispherical pits, the concept of these parameters is applicable to any pit geometry. The derivation of the set of parameters (Tcrit , Ecrit , xcrit ) and (Tsat , Esat , xsat ) for a one-dimensional (1D) artificial pit geometry is presented in Appendix D. However, as the 1D pit does not have a pit cover, the concept of metastable pitting is not applicable, so the concepts of acrit and asat are also not applicable to 1D pits.

[32]

n F DCsurf deff

[33]

where deff is the effective diffusion length, which is determined by the pit geometry. On the other hand, for a given temperature and Csurf , idiss,act is determined by the potential drop across the pit surface/pit solution interface Esurf , i.e. the actual potential applied on the pit surface.


Journal of The Electrochemical Society, 165 (11) C762-C770 (2018) Because Csurf is smaller than Csat , a salt film will not form on the pit surface, so there is no ohmic potential drop associated with a salt film and the actual potential at the pit surface is Emax , thus idiss,act equals idiss,max . Two further situations must now be considered. In the first situation, the total ohmic potential drop of bulk and pit solutions (sol ) is assumed to be negligible, hence Emax = Eapp . Therefore, idiss,act can be expressed by: E app − E corr [34] i diss,act = i corr exp βa This equation shows that idiss,act is constant during pit growth for a given Eapp . As idiff,act = idiss,act , and according to Eq. 33, the concentration of metal cations at the pit surface can be expressed as: i diss,act deff [35] nFD Because idiss,act is constant and deff increases during pit growth, Csurf will keep increasing to a supersaturated level and then a salt film will eventually precipitate with Csurf equal to Csat . It can be concluded that, when the solution resistance is negligible, Csurf cannot be maintained at a constant value below Csat during sustained pit growth. However, according to Eq. 35 the time required to reach Csat will depend on pit growth rate, i.e. ddeff /dt, which is actually determined by idiss,act . Note that the effect of Csurf change between Ccrit and Csat on idiss,act is ignored here for simplicity. When Eapp is large, idiss,act will be large and Csurf will reach Csat very quickly so that pit growth will consequently transition to diffusion control with the presence of a salt film. In contrast, if Eapp is small, then it will take a relatively long time for Csurf to reach Csat , during which time the pit will grow under charge transfer control without the presence of a salt film. Therefore, the pit morphology will be crystallographic for a pit growing without a salt film for an extended period, which might occur during immersion at the open circuit potential16 or polarization at a relatively low potential.15 It is also of interest to understand conditions in which Csurf can be maintained at a constant value less than Csat and not changed during the whole pit growth process. To address this issue, the second situation must be considered, in which sol is not negligible. Thus, idiss,act can be expressed by: E app − sol − E corr [36] i diss,act = i corr exp βa Csurf =

When the pit grows in a steady state, idiff,act = idiss,act . Thus, combining Eq. 36 with Eq. 33, sol = E app − E corr − βa ln

n F DCsurf deff i corr

F DCsurf E app − E corr − βa ln dneff (t)i corr n F DCsurf deff (t)

· Apit (t)

R (x) =

surf E app − E corr − βa ln n FxiDC corr

n F DCsurf x

.A1D

[40]

where A1D is the cross-sectional area of the 1D pit. However, the actual solution resistance (Ract ) of a 1D pit increases linearly with the pit depth.17 Therefore, the situation described by Ccrit ≤ Csurf < Csat for a 1D pit is not a steady state condition during sustained pit growth. In fact, for most pits, Ccrit ≤ Csurf < Csat is just a transient state during the sustained pit growth, and Csurf will keep increasing until a salt film is eventually formed on the pit surface. In other words, the pit growth under charge transfer control will tend to spontaneously shift to diffusion control. This perspective is in agreement with the observations by Frankenthal and Pickering,14 who found that the pits on iron are initially crystallographic, but then become electropolished. According to the proposed framework, a high potential or temperature will accelerate the transition from charge transfer control to diffusion control by decreasing rsat (for open pits) or by increasing asat (for covered pits), and also by increasing the pit growth rate. However, in some specific cases, if sol increases rapidly during growth under charge transfer control, i.e. dsol (r ) d E crit (r ) > [41] dr dr rcrit ≤r