Predictive modeling of localized corrosion: An application to aluminum alloys

Electrochimica Acta 56 (2011) 5630–5641

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Predictive modeling of localized corrosion: An application to aluminum alloys J. Xiao, S. Chaudhuri ∗ ISP/Applied Sciences Laboratory, Washington State University, Spokane, WA 99210-1495, United States

a r t i c l e

i n f o

Article history: Received 21 December 2010 Received in revised form 4 April 2011 Accepted 5 April 2011 Available online 13 April 2011 Keywords: Aluminum alloys Localized corrosion Repassivation Pit stability Corrosion rate Finite element method Numerical simulation

a b s t r a c t Corrosion prevention in light-weight alloys is currently an area of major research for civilian, aerospace, and defense applications. In order to understand thoroughly the complex corrosion system and hence develop effective and “green” corrosion prevention strategies, predictive modeling is believed to be an essential tool. This work presents a new finite element method (FEM)-based corrosion model, specifically tailored for localized pitting corrosion of aluminum alloys. The model distinguishes itself from existing ones by its strong predictive power and high generality. By resorting to this methodology, not only corrosion rate but also pit stability can be quantitatively evaluated for a wide range of systems involving heterogeneous alloy microstructure, complex pit morphology, and versatile solution chemistry. Moreover, the knowledge discovered can shed light on the control of pit repassivation, which will eventually lead to effective corrosion inhibition approaches. A thorough investigation of pitting corrosion of aircraft aluminum alloys is presented in order to demonstrate the efficacy and attractiveness of our method. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Aluminum alloys are light-weight high strength materials that have been extensively used in the aircraft industry. However, they are susceptible to corrosion, especially localized corrosion [1–4]. The various forms of localized corrosion, including pitting corrosion, crevice corrosion, and stress corrosion cracking, are difficult to detect and can often lead to sudden and catastrophic failures. Thus, understanding and prediction of localized corrosion of aluminum alloys have been an area of intense interest for many years [1–27]. The complexity of this problem is primarily due to two factors: material microstructure and corrosion environment, both of which dynamically change over time and demonstrate high levels of heterogeneity. It is recognized that the microstructure of aluminum alloys is very complicated: exhibiting a range of intermetallic particles (IMPs), periphery phases around composite particles and clustering. Many IMPs are “multiphase” in that they contain more than one composition indicating very heterogeneous precipitation processes. For instance, AlCuFeMn IMPs have a variety of compositions including Al7 CuFe2 , Al6 MnFe2 , (Al,Cu)6 Mn, etc. As a result, the composition of IMPs has never been fully characterized [5,6]. Once corrosion has commenced, the situation is further complicated by matrix etching, IMP dealloying and copper redistribution on the alloy surface, leading to significant changes in local electrochemical activity [7–10]. Efforts were identified to study intermetallic phases

∗ Corresponding author. Tel.: +1 509 358 7782; fax: +1 509 358 7627. E-mail address: [email protected] (S. Chaudhuri). 0013-4686/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2011.04.019

using electrochemical cells [11–14]. The reported data on corrosion potentials, pitting potentials and polarization curves greatly improve our understanding of the electrochemical behavior of individual IMPs, such as the ␪-phase (Al2 Cu), the S-phase (Al2 CuMg) and many others. However, investigation of isolated IMPs can only provide a partial picture of the alloy corrosion problem, where clustering and coupling of IMPs play a key role on surface electrochemical activity and pit initiation, propagation and repassivation [1–4,15–17]. For instance, when the ␪-phase is coupled to the Sphase or the aluminum matrix, it is cathodically polarized and its’ dissolution can be further inhibited by the formation of a surface layer of insoluble cuprous salt [17]. The problem becomes more complicated when the heterogeneous alloy surface is exposed to uncertain service environment [18–23]. It is the local corrosion environment that controls all the key interfacial phenomena, such as matrix etching, oxide formation and dissolution, IMP dealloying, copper redistribution, and corrosion product precipitation, etc. However, the local environment within a micrometer-sized pit is hardly measurable. It also shows prominent gradients and could be significantly different from the bulk conditions. Furthermore, a multitude of factors such as alloy microstructure, pit morphology, solution chemistry and electrical condition, which are spaceand time-variant, have coupled effects on the localized corrosion behavior. It is extremely difficult for experimentalists to clearly differentiate the impacts of each individual factor. For example, the distinct roles of the solution chemistry (e.g., pH) and the galvanic coupling (that determines electrostatic potential) on pit initiation, propagation and repassivation around a Cu-rich cathodic IMP are still not well understood [24]. Although characterization of pit mor-

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phology has been attempted [25–27], how the pit shape and surface roughness affect corrosion remains an unsolved problem. Numerical simulation, on the other hand, offers greater freedom in the control of material microstructure and corrosion environment. Dynamic data on localized corrosion conditions can be readily obtained as well. Thus, a good numerical model can greatly improve our fundamental understanding on the localized corrosion behavior. Over the past decades, finite element method (FEM)based modeling efforts have been continuously improving to mimic the complex corrosion system [24,28–42]. However, most efforts in this area are focused on the corrosion of iron or steel [28–38]. Only a few papers studied aluminum [39,40] or aluminum alloys [24,41,42]. Deshpande predicted the corrosion rate of an AE44AA6063 galvanic couple [42]. The developed model assumed that there is no concentration gradient in the electrolyte solution, which is unrealistic for studying localized corrosion. Although the zero gradient assumption was not made by Murer et al. [41] and Oltra et al. [24], the connections between pit morphology, local solution chemistry (e.g., pH and chloride ion concentration) and aluminum dissolution kinetics were not established. Simple homogeneous reactions adopted by those authors [24,41,42] may lead to large errors in the estimation of species concentration as well. Moreover, a majority of developed models cannot be applied to study pit repassivation and corrosion under different bulk conditions. Thus, they have limited predictive power for the corrosion of aluminum alloys, where heterogeneous microstructure and localized environment fluctuations dominate the surface chemistry. Up until now, the interrelationships between aluminum alloy microstructure, pit morphology, bulk and localized corrosion environment, and surface chemistry have not been clearly established. It is the objective of this paper to explore such correlations by modeling multiple physical and (electro)chemical phenomena in a coupled fashion. As an initial step towards the development of more complicated models where coupling between multiple IMPs are taken into account, this work concentrates on pit propagation and repassivation around a cathodic IMP (i.e., ␪-phase). The repassivation studied here is caused by oxide layer formation [20]. Other schemes such as formation of a passive layer from insoluble precipitates [21,22] are not considered. Also note that this work aims at revealing the full anodic polarization behavior by sweeping applied potentials. The special case of free corrosion, where the applied potential equals to the open circuit potential (OCP), is not investigated. In the following text, a generic FEM-based model framework is presented at the outset. After that, the methodology of each individual modeling task as well as their connections is followed by application details for the aluminum alloy system. Finally, a thorough investigation of localized corrosion of 2000 series aluminum alloys is presented to demonstrate the efficacy of our method.

2. Modeling methodology A generic framework has been constructed to guide the development of corrosion models. Fig. 1 illustrates the system investigated in this work and the model framework. Pitting corrosion in the aluminum matrix adjacent to a cathodic IMP is the focus. As shown in Fig. 1a, pit initiation is not investigated. Our main interest is in propagation and repassivation of an existing active pit. The model system consists of four domains, which are respectively, the electrolyte domain ˝e , metal matrix domain ˝ , IMP domain ˝ , and passive film domain ˝p . The electrolyte–metal interface is ∂˝e m = ∂˝e ∩ (∂˝ ∪ ∂˝ ∪ ∂˝p ). Moreover, the model framework contains two layers of modeling blocks, which are bi-directionally coupled and closely connected with a block of experimental data (Fig. 1b). Within the electrolyte domain (i.e., the top layer), mass transfer coupled with homogeneous reactions

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needs to be characterized in order to obtain chemical (i.e., species concentration) and electrical (i.e., electrostatic potential) environment. The corrosive environment then serves as an input for the bottom layer (i.e., the interface models), where surface reactions determine the fluxes of reactive species across the interface as well as the changes of interface location. As a result, they provide boundary location and boundary condition for the electrolyte domain models. In this section, general methods for all modeling tasks listed in Fig. 1b are presented with an emphasis on their unique features. Modeling of aluminum alloys is then delineated in order to facilitate the reader’s understanding on methodological implementation. 2.1. Integrated FEM modeling 2.1.1. Material balance In the electrolyte domain ˝e , the dynamic change of mass for each chemical species, j, is attributed to mass transfer and homogeneous reactions. Mathematically, it can be expressed as: ∂Cj

 j + Rj = −∇ · 

∂t

(1)

 and R are the flux density where C is the concentration (mol/m3 );  (mol/(m2 s)) and reaction rate (mol/(m3 s)), respectively. It is well known that mass transfer of species in an aqueous environment is driven by differences of electrical or chemical potential between the two locations or by motion of a volume element of the electrolyte [36]. Thus, the flux is contributed by diffusion, migration and convection, i.e.:  j = −Dj ∇ Cj − zj F 

Dj Rg T

Cj ∇ ϕ + Cj v

(2)

where F, Rg and T are the Faraday’s constant (96,485.34 C/mol), gas constant (8.314 J/(mol K)) and temperature (K), respectively; z and D are, respectively, the charge number and diffusion coefficient (m2 /s); ϕ is the electrostatic potential (V). In a stagnant electrolyte, the mass transfer due to the convective force can be neglected (i.e., v = 0). The rate of production or consumption of chemical species, j, due to homogeneous reactions can be generalized as [28]: Rj =

⎧ Nr ⎨ 

⎛



−ωjm ⎝kfm

m=1

⎩

(C )

ωm 

− kbm

∀ωm >0 

 ∀ωm <0 

⎞⎫ ⎬ −ωm (C )  ⎠ ⎭

(3)

where Nr is the total number of reactions; ωm is the stoichiometric coefficient for species  in the mth reaction, which is positive for reactants and negative for products; kfm and kbm are, respectively, rate constants for the forward and backward reaction m. Traditionally, all homogeneous reactions are assumed to be in equilibrium at any location and time. This assumption is not used in this work since it may lead to large errors in the estimation of species concentrations at high dissolution current densities [40]. Substituting Eqs. (2) and (3) into Eq. (1) yields: ∂Cj ∂t

+

= Dj ∇ 2 Cj + zj F

⎧ Nr ⎨ 

⎛

Dj Rg T

−ωjm ⎝kfm

m=1

⎩

∇ · (Cj ∇ ϕ)

 ∀ωm >0 

(C )

ωm 

− kbm

 ∀ωm <0 

⎞⎫ ⎬ −ωm (C )  ⎠ ⎭

(4)

The species concentration and the electrostatic potential can be determined by solving Eq. (4) combined with an electroneutrality condition, i.e.:

 j

zj Cj = 0

(5)

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Fig. 1. Model system for pitting corrosion: (a) system domain decomposition and (b) model framework.

Note that Eqs. (2) and (5) are only valid for the dilute solution where the charge density is negligible, which is not the case for the solution in a thin double layer (at nanometer scale) adjacent to the metal surface [28,43]. The effects of the electrical double layer on the local chemical and electrochemical environment and hence on the corrosion dynamics are beyond the scope of this work, and will be thoroughly investigated in an upcoming multiscale modeling paper. 2.1.2. Surface reactions Modeling interfacial phenomena is a most challenging task as it requires an in-depth understanding of the interplay between material and solution chemistry, surface morphology, and interfacial kinetics. 2.1.2.1. Anodic kinetics. In this work, the idea of pit state transition (i.e., active to passive or vice versa) resulting from the competition between formation and dissolution of the passive oxide layer is adopted [44,45]. The oxide coverage fraction  o is given by ia,2 o = ia,2 + ia,p

ia = (ia,1 + ia,2 )(1 − o )

(7)

where ia,1 is the current density of active dissolution. Each component of the total current density is then quantitatively correlated to the applied potential ( ), electrostatic potential in the solution (ϕ), temperature (T), and most importantly, the material and solution chemistry. As suggested by Anderko et al. [44], active dissolution and oxide formation are mediated by the adsorption of aggressive and inhibitive species, respectively. The kinetics of these two types of electrochemical reactions can be written as:





∗ i1,j (Cjs )

ωj

exp

j

ia,2 =

 j

ia,p =

exp

−ϕ



(8)

Rg T

 ω ∗ i2,j (Cjs ) j

˛j F(

0 ) − EM,j

˛j F(

0 − ϕ − EMO,j )

Rg T





ωj

(10)

Corrosion rate Rcorr (kg/(m2 s)) is determined by ia according to:



Rcorr

Mm = zm F

∂˝e a

ia dS

∂˝e a

(11)

dS

where the integration is over the electrolyte–anode interface ∂˝e a = ∂˝a ∩ ∂˝e m . 2.1.2.2. Cathodic kinetics. Similarly, the kinetic expression for the electrochemical reduction reactions on cathodes has the following general format:



ic = −



ω ∗ ic,j (Cjs ) j

exp

−

˛j F(

j

− ϕ − Ej0 ) Rg T



(12)

where the subscript j refers to a dissolved species to be reduced. As usual, a negative scalar quantity is used to quantify the cathodic current density. Note that the anodic domain (˝a ) and the cathodic domain (˝c ) discussed in this sub-section have been differentiated from ˝ and ˝ shown in Fig. 1a. This is because for different corrosion systems, an IMP can be either an anode, or a cathode, or even both. Similarly, it is also possible for the metal matrix to be a cathode. 2.1.3. Interface location evolution The electrolyte–metal interface ∂˝e m moves continuously throughout the corrosion process, which can be described as:  ∂ = ie ∂t

m,m

Mm  ∈ ∂˝e  ∀ n

m zm F

m (t)

(13)

where Mm , m , and zm are, respectively, the molecular weight, den is a position vector; n  is sity and effective valence of the metal; a unit normal vector pointing out of the electrolyte domain. The metal ion involved current density (ie m,m ) across the interface is location and time dependent, i.e.:

(9)

where i* is the concentration-independent part of the exchange current density [46]; Cs is the species concentration near the electrode surface [47]; ω is the reaction order; ˛ is the electrochemical transfer coefficient; E0 is the reversible potential; the subscript j represents an aggressive species in Eq. (8) and an inhibitive species

kp,j (Cjs )

j

(6)

where ia,2 and ia,p are, respectively, the magnitude of current densities (A/m2 ) that lead to oxide formation and dissolution. In the following text, current density is treated as a scalar, unless otherwise stated. The total anodic current density is:

ia,1 =

in Eq. (9). On the other hand, passive dissolution is a chemical process, which strongly depends on the solution chemistry (i.e., pH and specific active ions). The reaction kinetics can be expressed by:

ie

m,m

=

⎧ ⎪ i ⎪ ⎨ a ⎪ ⎪ ⎩

 ∈ ∂˝a (t) ∩ ∂˝e m (t) ∀

ic,m

 ∈ ∂˝c (t) ∩ ∂˝e m (t) ∀

0

 ∈ ∂˝p (t) ∩ ∂˝e m (t) ∀

(14)

As shown in Eqs. (13) and (14), pit propagates into the anode at a speed determined by ia , and there is no material exchange between the electrolyte and passive film. A special term ic,m is introduced,

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Fig. 2. Simulation system geometry: (a) pitting next to ␪-phase and (b) pitting in pure Al.

 t = 0) = ϕ∞ ϕ( ,

Table 1 Homogeneous reactions during dissolution of Al in chloride solution [40]. Hydrolysis

Al3+ + H2 O ⇔ Al(OH)2+ + H+ Al(OH)2+ + H2 O ⇔ Al(OH)2 + + H+ 2Al3+ + 2H2 O ⇔ Al2 (OH)2 4+ + 2H+

Reactions with Cl− ion

Al3+ + Cl− ⇔ AlCl2+ Al(OH)2+ + Cl− ⇔ Al(OH)Cl+

Product formation

AlCl2+ + 2H2 O → Al(OH)2 Cl + 2H+ Al(OH)Cl+ + H2 O → Al(OH)2 Cl + H+

Water dissociation

H2 O ⇔ H+ + OH−

which is the metal-ion-reduction part of the total cathodic current density. It allows our model to tackle some challenging corrosion problems where metal deposition on cathodes is a critical phenomenon (e.g., Cu redistribution for the corrosion of aluminum alloys [1,10]). Another attractive feature of our method is that heterogeneous material microstructure characterized by ˝a , ˝c , and ˝p is an integral part of the model system and the structural information can be dynamically updated throughout the simulation. 2.1.4. Initial and boundary conditions Essentially, the problem to be solved is a mass transfer problem in domain ˝e with moving boundaries. The initial boundary ∂˝e m (t = 0) can have any shape, e.g., a random rough surface, or more rigorously, a surface constructed based on the morphological parameters quantified from experimental measurements (e.g., light scattering experiments [48]). In addition, we have  t = 0) = C ∞ Cj ( , j

 ∈ ˝e ∀

(15)

0.05

Rcorr (kg/m 2 /s)

0.04

(16)

where Cj∞ is the bulk concentration of species j. The electrostatic potential in the bulk solution ϕ∞ is usually set as 0. Moreover, bulk conditions are also applied to the boundaries far from the pit mouth, i.e.:  t > 0) = C ∞ Cj ( , j

 ∈ · ∂˝e /∂˝e ∀

m

(17)

 t > 0) = ϕ∞ ϕ( ,

 ∈ · ∂˝e /∂˝e ∀

m

(18)

The location of the boundary ∂˝e m (t > 0) can be calculated using Eqs. (13) and (14). If some chemical species do not participate in any interfacial reaction, their flux across this boundary will be equal to zero. The flux of other species are given by:  t > 0) = −ie m,j n  j ( ,   zj F

 ∈ ∂˝e ∀

m

2.2. Application to aluminum alloy system The generic methodology introduced above was applied to the study of aluminum alloys immersed in a sodium chloride solution, where pitting took place at the periphery of ␪-phase [1,17]. 2.2.1. Assumptions The following assumptions were made in order to simplify the problem and capture the key phenomena during corrosion. First, the hemispherical shaped ␪-phase is covered by a layer of passive film, which prevents its dealloying. It has been found by experiments that when Al alloy was immersed in a chloride containing solution, ␪-phase can be passivated by a layer of insoluble cuprous

Ionic species

0.02 0.01

10 3

10 2

CCl

10 1

(M)

Fig. 3. Effect of chloride ion concentration on corrosion rate.

10 0

(19)

where ie m,j is the current density due to an electrochemical reaction involving species j and the transfer of zj electrons. The expression of ie m,j has been given in Eq. (14).

Table 2 Comparison of ionic species concentration at the pit bottom.

0.03

0 10 4

 ∈ ˝e ∀

Positively charged Al3+ AlCl2+ Al(OH)Cl+ H+ Na+ Al(OH)2+ Al2 (OH)2 4+ Al(OH)2 + Negatively charged Cl− OH−

Concentration (M) 1.08 7.47 0.09 0.006 0.005 0.004 5.5 × 10−6 1.8 × 10−5 18.3 2.3 × 10−12

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20

CCl

8

(b)

=100 M

=10 4 M

6

12 8

5 4

4 0

CCl

7

16

pH

Cl concentration (M)

(a)

0

10 2 M 10 4 M

1

2

3

4

10 2 M 100 M

3 5

6

2

7

Length along pit surface ( m)

0

1

2

3

4

5

6

7

Length along pit surface ( m)

(a)

-0.55 -0.6

Erp

CCl

(b)

= 0.01 M 0.1 M

-0.7 -0.75

Erp (V)

(V)

-0.65

1M

-0.8 -0.85 -0.9 -0.95 -1

icrit -10

-8

Log (Rcorr

-6

-4

-0.6

1

-0.65

0.8

-0.7

0.6

-0.75

0.4

-0.8

0.2

-0.85

-2

(kg/m2 /s))

-2

-1.5

-1

-0.5

0

Rcorr_crit ( 10 3 kg/m2 /s)

Fig. 4. Local environment within the pit: (a) chloride ion concentration and (b) pH.

0

Log(CCl (M) )

Fig. 5. In silico polarization scan: (a) anodic polarization curves at different chloride ion bulk concentrations and (b) chloride ion concentration effect on repassivation potential and critical current density.

chloride [17]. Second, due to a passive film covering ␪-phase, copper plating on ␪-phase is neglected. Third, the thick passive film (i.e., domain ˝p ) keeps intact throughout the simulation. Fourth, precipitation of corrosion product Al(OH)3 on pit surface is negligible. This is because Al(OH)3 is present in insignificant concentration at a pH below 4 [40], and the electrolyte within an active pit is highly acidic due to hydrolysis reactions [49]. 2.2.2. Simulation domain construction Fig. 2 shows the 2D axially symmetric geometry of the simulation system. It is clear that the aluminum matrix and the ␪-phase are anode and cathode, respectively. A pit propagates into the aluminum matrix, which is covered by a thick layer of passive film composed of Al2 O3 (Fig. 2a). A special case shown in Fig. 2b was also considered, where the ␪-phase has been removed from the metal surface due to severe pitting corrosion. It is then equivalent to pitting corrosion in pure aluminum. 2.2.3. Chemical and electrochemical reactions Homogeneous reactions during pitting corrosion of Al in sodium chloride solution have been investigated extensively [40,50,51]. Comprehensive reactions listed in a very recent article by Guseva et al. [40] were directly adopted here (see Table 1). 2.2.3.1. Anodic reactions. The active dissolution mediated by Cl− ion is a two-step process, i.e., adsorption of species at the metal surface followed by the dissolution of the adsorbed complex: Al + Cl− → AlCl− →

AlCl−

Al3+ + Cl− + 3e−

(20) (21)

Without any inhibitive species, water molecules promote the formation of an oxide layer, which is responsible for the repassivation of an active pit. 2Al + 3H2 O →

Al2 O3 + 6H+ + 6e−

(22)

Due to the acidic environment within a pit, the dissolution of oxide films (i.e., a chemical reaction) is mainly contributed by the proton ions. Al2 O3 + 6H+ → 2Al3+ + 3H2 O

(23)

2.2.3.2. Cathodic reaction. The four-electron O2 reduction was the only cathodic reaction considered here: O2 +2H2 O+4e− → 4OH−

(24)

2.2.4. Material, structural properties and corrosion environment The molecular weight and density of aluminum are, respectively, 26.98 g/mol and 2700 kg/m3 . A total of 12 dissolved species are involved in the above-listed reactions, i.e., Na+ , Cl− , H+ , OH− , Al3+ , Al(OH)2+ , Al(OH)2 + , Al2 (OH)2 4+ , AlCl2+ , Al(OH)Cl+ , Al(OH)2 Cl, and O2 . Their diffusion coefficients and kinetic constants of homogeneous reactions were taken from Refs. [40,50]. At an ambient temperature T = 298.15 K, Al alloy was immersed in a sodium chloride solution, which has a concentration ranging from 10−4 M to 1 M and a pH ranging from 0 to 8. The bulk concentration of dissolved O2 was set as 60 ␮M [52,53] and the applied potential was varied between −1 V and −0.55 V (vs. SCE). Note that, all potentials in this work are relative to saturated calomel electrode (SCE). Moreover, structural properties in terms of the pit size and surface morphology can be implemented straightforwardly. In the present research, coupled PDEs were solved by COMSOL Multiphysics [54], while the moving boundary was implemented

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-0.55

rpit = 5 m 10 m 15 m 20 m

-0.6 -0.65 -0.7

(V)

-0.75 -0.8 -0.85 -0.9 -0.95 -1

-10

-8

-6

-4

-2

Log (Rcorr (kg/m2 /s)) Fig. 8. Anodic polarization curves for systems with different sized pits.

localized corrosion environment, and hence on pit stability and the extent of corrosion were thoroughly investigated. Dynamic data generated from in silico pit propagation were analyzed as well. 3.1. Effect of chloride ion concentration

Fig. 6. pH and potential effect on corrosion rate: (a) pH effect on corrosion rate at different potentials and (b) potential–pH 3D diagram.

using MATLAB. Seamless coupling between COMSOL and MATLAB solutions was accomplished using MATLAB scripts. 3. Results and discussion For the aluminum alloy system shown in Fig. 2, effects of the solution chemistry (i.e., bulk chloride ion concentration and pH) and the pit structure (i.e., size and surface morphology) on the

3.1

10 m

3.05

pH

(b) 3.5

rpit = 5 m 15 m

3

20 m

2.95 2.9 2.85 2.8

5

10

rpit = 20 m

3 2.5

15 m

2

10 m

1.5

5 m

1

pit bottom 0

Cl concentration (M)

(a) 3.15

The first case was designed to investigate the effect of aggressive ions (i.e., Cl− ) on corrosion. The system shown in Fig. 2a was immersed in a neutral NaCl solution and it was under potentiostatic control with an applied potential of −0.55 V. A fixed pit geometry was applied, which corresponds to a specific time instant during pit propagation. The change of corrosion rate as a function of the bulk chloride ion concentration is shown in Fig. 3. When the chloride ion bulk concentration was increased, an increase of corrosion rate was predicted. The same trend was observed in experiments as well [18]. With the help of our model, this increasing trend can be easily explained by examining the local corrosive environment within the pit. The distributions of chloride ion concentration and pH near the electrolyte–aluminum interface ∂˝e a are plotted in Fig. 4. It is shown that a higher Cl− bulk concentration not only drives more Cl− ions towards the pit surface, but also generates a higher acidic environment within the pit. Both factors lead to an increased corrosion rate with increasing Cl− bulk concentration. It is also suggested by Fig. 3 that, with a fixed potential and pH, decreasing the chloride ion bulk concentration can change the status of a pit from active to passive. This feature was also identified by experiments in Ref. [55]. Table 2 gives the concentrations of each ionic species at the bottom of the pit when the bulk Cl− concentration is high (i.e., 1 M). The negative charge of chloride was neutralized mainly by two species, Al3+ and AlCl2+ , whose concentrations reached up to 1.08 M and

15

20

25

30

Length along pit surface ( m)

35

0.5

pit bottom 0

5

10

15

20

25

30

Length along pit surface ( m)

Fig. 7. Local environment within the pit with different sizes: (a) pH and (b) chloride ion concentration.

35

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7.47 M, respectively. According to Beck [56], the saturation concentration of AlCl3 is around 3.11 M. Thus, AlCl3 salt film should be formed in this case although the role of pH will also be needed to be considered. Our current model, which assumes a sufficiently large solubility of Al-containing salts during the initial periods of pit growth, will be improved in future to take into account the complex effect of salt film precipitation and dissolution on corrosion dynamics. These considerations will be especially important for wet-dry cycling that metal alloys exposed to environment presumably undergoes over longer periods of time. In this work we assume that the continuous wet conditions can allow the concentrations of the ionic species can be very high in ways similar to salt fog chambers. In addition to corrosion rate, our methodology allows the quantification of the repassivation potential (Erp ), which is a direct indicator of pit stability. This challenging task was accomplished through in silico polarization scan, where the influence of potential on pitting can be explored computationally. The anodic polarization curves shown in Fig. 5a were obtained by using a series of constant Cl− concentrations and decreasing the applied potential ( ) from values close to Erp to well below Erp at a step size of −10 mV. According to the definition given in Frankel et al. [57], the repassivation potential is the potential at which the pit current density drops below a critical value (icrit ), which is the minimum current density needed to maintain the critical pit environment and prevent repassivation. Then Erp and icrit can be readily identified from a specific polarization curve. For the 0.01 M NaCl case, the values of these two variables have been indicated in Fig. 5a. Note that the anodic current density has been converted to corrosion rate using Eq. (11). It is shown that decreasing from −0.55 V initially led to a linear decrease of log(Rcorr ). At −0.60 V, the current density reached a critical value. Then, further decease of to −0.61 V (i.e., Erp ) caused a drastic decrease of corrosion rate, which indicates the repassivation of the pit. This repassivation behavior was successfully captured by characterizing a sudden growth of an oxide layer on metal surface. It was observed that the oxide layer coverage was increased from ∼1.6% to 100% during this sudden transition. The chloride concentration effects on Erp and Rcorr crit (i.e., corrosion rate at the critical current density) are plotted in Fig. 5b. For a system with a lower Cl− concentration, both Erp and Rcorr crit are higher. It means that pits are less stable when surrounded with less concentrated aggressive ions. The experimentally observed nearly ∞ ) [18,55] was successlinear relationship between Erp and log(CCl − fully predicted by our model (see the solid line in Fig. 5b). 3.2. Effect of pH In this case, the Cl− ion concentration was kept at 0.01 M and the fixed pit geometry shown in Fig. 2a was applied. The bulk pH (pH∞ ) of the solution was adjusted by adding hydrochloric acid (HCl) or sodium hydroxide (NaOH). It has been found in experiments that exposure of aluminum alloys to alkaline solutions usually results in general corrosion rather than pitting corrosion, which is because the uniform thinning of the passive oxide film (by OH− ion attack) overwhelms the pitting corrosion [18]. Since the focus of this work is localized corrosion, acidic or near neutral solutions (i.e., a pH range from 1 to 8) were investigated. Fig. 6a shows how corrosion rate reacts to the change of pH∞ at different applied potentials. Consistent with the experimental observation [58], the decrease of pH∞ led to an increase of corrosion rate. Much more information can be derived from these data. For the case at −0.55 V, there exists a critical pH (see pH∞ crit indicated in Fig. 6a), above which a drastic decrease of corrosion rate was observed. Such prominent decrease indicates a transition of pit status from active to passive. Moreover, corrosion rate was not sensitive to the change of pH within a range between 4 and pH∞ crit .

Fig. 9. Comparison of proton concentration distributions.

A clear increase of corrosion rate was observed only when pH∞ was below 4. Although the curves at other potentials demonstrate very similar features, the change of potential has a clear effect on pH∞ crit . It can be concluded that at a lower potential, a more acidic environment leads to a sustained growth of an active pit. A more informative representation of the data is a 3D potential–pH diagram. Fig. 6b shows such a plot, where the red dots are the data shown in Fig. 6a and the 3D surface is a result of interpolation. From this figure, pit stability and corrosion rate under any corrosive environment (i.e., a combination of pH and applied potential) can be readily obtained. Pourbaix (2D) diagrams, which give the equilibrium phases for specific metal-electrolyte systems at different pH levels and potentials, are widely used for evaluating the tendency of metals to corrode [59]. Our 3D potential–pH diagram extends the concept of the Pourbaix diagram by offering an additional dimension of information, i.e., corrosion rate. This new 3D diagram can potentially be a valuable tool in the field of corrosion science and engineering. 3.3. Effect of pit size In addition to the solution chemistry investigated so far, structural properties of a pit were also taken into account. In this case, hemispherical pits (see Fig. 2b) with different sizes were exposed to a neutral 0.01 M NaCl solution. At a constant of −0.55 V, localized corrosion environments adjacent to the pit surface are given in Fig. 7. Compared with the bulk solution, the solution within the pit was much more aggressive (i.e., lower pH and higher Cl− concentration). Furthermore, a monotonic increase of the solution aggressivity from the pit mouth to the pit bottom was observed. It was also shown that the solution became more aggressive as pit size increased. In silico polarization scans were conducted for systems with different pit sizes. As evidenced in Fig. 8, increasing pit size led to the decrease of both Erp and icrit . It suggests that larger pits are more stable, i.e., harder to repassivate. Buzza and Alkire [60] investigated the growth of an array of pits (on aluminum) with different sizes under galvanostatic control, where the total corrosion rate was lim-

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Fig. 10. Comparison of corrosive environments with the pit: (a and b) contour plot of pH, (c and d) contour plot of chloride ion concentration, and (e and f) contour plot of electrostatic potential.

Fig. 11. In silico polarization scan for systems with different pit geometry: (a) anodic polarization curves and (b) polarization curves at the high potential range.

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2.4

ited by the total cathodic current. It was observed that the smallest pits repassivated first, a result that is consistent with our model prediction. This phenomenon can be mainly attributed to the increase of solution aggressivity with pit size (see Fig. 7).

It has been observed in experiments that a real pit surface has a complex geometry (e.g., a rough surface [26,27]). Due to the technical limitations of experimental methods, until now, it is unclear whether the complex geometry has a significant effect on corrosion and how the pit surface geometry affects corrosion. In order to shed light on fundamental understandings of the geometry effect, two systems were designed. The first system is shown in Fig. 2a, where the electrolyte was 0.01 M neutral NaCl. The second system had the same settings expect that the pit had a rough surface represented by a fractal surface. In order to achieve a fair comparison, pits in two systems had the same volume, i.e., the same amount of mass loss.

2.3

Rcorr ( 10 3 kg/m2 /s)

3.4. Effect of pit morphology

2.35

2.25 2.2 2.15 2.1 2.05 2 1.95

0

1

2

3

4

5

6

t(s) Fig. 12. Dynamics of corrosion rate during pit propagation.

Fig. 13. Corrosive environment in a growing pit at two time instants: (a and b) contour plot of pH, (c and d) contour plot of chloride ion concentration, and (e and f) contour plot of electrostatic potential.

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Fig. 14. 3D half-cut view of the growing pit: (a) at time 0, (b) at the 3rd second, and (c) at the 6th second.

Local solution chemistry within the pit can be obtained for a comparison. Using the axially symmetric data of the 2D simulation system, a pseudo-3D distribution plot has been generated (see Fig. 9). As shown in this figure, at the same applied potential of −0.55 V, the electrolyte in the fractal-shaped pit was more acidic (i.e., higher H+ ion concentration). The same trend was observed for the Cl− ion concentration as well (see Fig. 10). In this case, the fractal-shaped pit has a surface area that is 2.49 times larger than the surface area of the smooth pit. Due to the difference in surface area, a rougher surface led to the generation of a larger amount of Al-containing positively charged species (see the data listed in Table 3), which subsequently attracted more Cl− ions from bulk solution into the pitting area for charge neutralization. Consequently, pit geometry plays an important role in modifying the local chemical environment. A surprising finding is that the corrosion rate of the fractalshaped pit was 2.215 × 10−3 kg/m2 /s, which was lower than that of the smooth pit (i.e., 2.351 × 10−3 kg/m2 /s), although the former pit encountered a more aggressive electrolyte (see Figs. 9 and 10a–d). This phenomenon can only be explained when the coupled electrical environment is also taken into account. As shown in Fig. 10e and f, the solution within the fractal-shaped pit exhibited a higher ϕ, i.e., a higher ohmic potential drop. In summary, the solution aggressivity and the ohmic potential drop both increase as the surface roughness increases, but the former has a positive contribution to corrosion rate and the latter has a negative contribution (see Eqs. (6)–(11)). For the systems shown in Fig. 9, the ohmic effect outweighed the chemistry effect, which led to the overall lower corrosion rate of the fractal-shaped pit.

The geometry effect on pit stability can be unveiled by the in silico polarization scan results, which are given in Fig. 11. The fractal-shaped pit demonstrated lower Erp and icrit . Thus, it was more stable. This is because the ohmic effect is negligible when the applied potential approaches Erp . The more aggressive solution makes the rough pit more stable. Fig. 11 also suggests that the ohmic effect becomes increasingly important as the applied potential increases. There is a critical potential (see the crossing point between two curves in Fig. 11b), at which the chemistry effect balances the ohmic effect and two types of pits have the same corrosion rate. Above this critical potential, the ohmic effect outweighs the chemistry effect.

Table 3 Comparison of the amount of ionic species within the simulation domain. Ionic species

Amount (×10−12 mol) Rough surface case

Positively charged Al3+ AlCl2+ Al(OH)Cl+ H+ Al(OH)2+ Al2 (OH)2 4+ Al(OH)2 + Na+ Negatively charged Cl− OH−

71.8 43.4 2.2 0.85 6.7 0.014 0.77 11.7 331.2 0.000039

Smooth surface case 32.7 10.9 0.82 0.48 3.99 0.004 0.61 12.3 142.4 0.000034

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3.5. Pit growth dynamics Finally, the growth dynamics of a pit around ␪-phase was investigated. The pit shown in Fig. 2a was allowed to propagate in a 0.01 M neutral NaCl electrolyte at an applied potential of −0.55 V. As shown in Fig. 12, corrosion rate decreased with time throughout the complete corrosion process. This deceasing trend was also observed in several experiments [49,60]. Plausible explanations for such phenomenon can be derived by analyzing the evolution of electrolyte–metal interface as well as chemical and electrical environment within the pit. Fig. 13 compares solution chemistry (pH and Cl− ion concentration) and electrostatic potential within the growing pit at two time instants. Note that the initial environment at t = 0 has been given in Fig. 10a, c and e. As time proceeded, the solution became more aggressive (i.e., lower pH and higher Cl− ion concentration) and the ohmic potential drop increased as well. Again, a competition between the chemistry effect and the ohmic effect determined the net trend of the change in corrosion rate. The type of electrochemical reactions at different locations of a pit surface was different and changed dynamically over time due to the moving boundary and the heterogeneous alloy microstructure. As pit grew, more ␪-phase surface was exposed (see Fig. 14). The oxygen reduction reaction took place on ␪-phase (i.e., the cathode) and generated OH− ions that neutralized the acidic solution within the pit. A larger exposed ␪-phase surface led to a higher pH. As a result, the inhibition of the effect of solution aggressivity on corrosion rate increased with time. The increase of pit size and the development of an occluded geometry led by the increasingly exposed Al2 O3 cover (see Fig. 14) contributed to a higher potential drop within the pit. Thus, the ohmic effect on corrosion rate became more prominent as pit grew. It can be summarized that, throughout the pit propagation process, the ohmic effect (enhanced by the occluded geometry) outweighed the chemistry effect (inhibited by the cathodic reaction on ␪-phase) and led to the overall decrease in the corrosion rate.

4. Conclusions Developing predictive models is a promising route towards identifying effective corrosion prevention pathways in aluminum alloys. Different from existing FEM-based modeling efforts that mainly depend on “empirical” parameterization of corrosion process using experimental data, this work aims at a “deterministic” modeling approach that can systematically identify, couple and characterize multiple physical and (electro)chemical phenomena under a modular framework. Through integrated multi-physics simulation, corrosion performance can be quantitatively correlated to the local corrosive environment within the pit that is hardly obtained from experiments, and further to many key dependent factors, including the alloy composition and microstructure, chemistry of the bulk solution, applied potential, and pit geometry, etc. Thus, significantly enhanced predictive capability can be achieved. The attractiveness of our method has been demonstrated through the pitting corrosion simulation of a representative Al alloy system. To the best of our knowledge, this is the first modeling effort in this area that simultaneously allows: (i) quantitative description of pit growth with material composition and microstructure fully taken into account, (ii) establishment of a 3D pH-potential diagram that greatly enhances the capability of relevant areas of a Pourbaix diagram, (iii) investigation of any pit surface morphology so that evaluation of the effect of geometry on corrosion propagation becomes possible, and (iv) insightful analysis of the effect of solution chemistry and pit morphology on local environment and pit stability.

We believe the most powerful predictive model should be derived by resorting to the first-principles linked multiscale modeling. The model will eventually enable in silico design and synthesis of materials so that corrosion performance can be tailored with greater freedom and efficiency. This work built a solid basis for such a multiscale corrosion model. The kinetic data of homogeneous and heterogeneous reactions are expected to be obtained from micro- to nano-scale kinetic Monte Carlo simulations. And alloy element dependence of Gibb’s free energy on alloys surfaces can be calculated from subnano scale density functional theory calculations. Descriptions of a multitude of aggressive species and solvated species in a nanoscale electrode surface double layer are currently being implemented. Moreover, couplings between multiple IMPs and the initiation of localized corrosion need to be thoroughly investigated in the near future. Acknowledgments We acknowledge funding from Office of Naval Research (Grants #N00014-04-1-0688 and N00014-06-1-0315) for the major part of this development work. In addition we thank Boeing Company for funding parts of this study (contract #365923) and especially Joseph Osborne for many insightful discussions on corrosion processes in Al alloys. References [1] A. Boag, R.J. Taylor, T.H. Muster, N. Goodman, D. McCulloch, C. Ryan, B. Rout, D. Jamieson, A.E. Hughes, Corros. Sci. 52 (2010) 90. [2] A. Boag, A.E. Hughes, A.M. Glenn, T.H. Muster, D. McCulloch, Corros. Sci. (2010), doi:10.1016/j.corsci.2010.09.009. [3] A. Boag, A.M. Glenn, D. McCulloch, T.H. Muster, C. Ryan, C. Luo, X. Zhou, G.E. Thompson, A.E. Hughes, Corros. Sci. (2010), doi:10.1016/j.corsci.2010.09.030. [4] A.M. Glenn, T.H. Muster, C. Luo, X. Zhou, G.E. Thompson, A. Boag, A.E. Hughes, Corros. Sci. (2010), doi:10.1016/j.corsci.2010.09.035. [5] A. Boag, A.E. Hughes, N.C. Wilson, A. Torpy, C.M. MacRae, A.M. Glenn, T.H. Muster, Corros. Sci. 51 (2009) 1565. [6] A.E. Hughes, C. MacRae, N. Wilson, A. Torpy, T.H. Muster, A.M. Glenn, Surf. Interface Anal. 42 (2010) 334. [7] B.G. Buchheit, R.P. Grant, P.F. Hlava, B. Mckenzie, G.L. Zender, J. Electrochem. Soc. 144 (1997) 2621. [8] M.B. Vukmirovic, N. Dimitrov, K. Sieradzki, J. Electrochem. Soc. 149 (2002) B428. [9] Y. Yoon, R.G. Buchheit, J. Electrochem. Soc. 153 (2006) B151. [10] H.M. Obispo, L.E. Murr, R.M. Arrowood, E.A. Trillo, J. Mater. Sci. 35 (2000) 3479. [11] R.G. Buchheit, J. Electrochem. Soc. 142 (1995) 3994. [12] N. Birbilis, R.G. Buchheit, J. Electrochem. Soc. 152 (2005) B140. [13] N. Birbilis, R.G. Buchheit, J. Electrochem. Soc. 155 (2008) C117. [14] K.D. Ralston, T.L. Young, R.G. Buchheit, J. Electrochem. Soc. 156 (2009) C135. [15] M.B. Jensen, A. Guerard, D.E. Tallman, G.P. Bierwagen, J. Electrochem. Soc. 155 (2008) C324. [16] P. Leblanc, G.S. Frankel, J. Electrochem. Soc. 149 (2002) B239. [17] R. Grilli, M.A. Baker, J.E. Castle, B. Dunn, J.F. Watts, Corros. Sci. 52 (2010) 2855. [18] B. Zaid, D. Saidi, A. Benzaid, S. Hadji, Corros. Sci. 50 (2008) 1841. [19] T. Suter, R.C. Alkire, J. Electrochem. Soc. 148 (2001) B36. [20] S. Pyun, E. Lee, Electrochim. Acta 40 (1995) 1963. [21] K.A. Yasakau, M.L. Zheludkevich, S.V. Lamaka, M.G.S. Ferreira, J. Phys. Chem. B 110 (2006) 5515. [22] G. Williams, A.J. Coleman, H.N. McMurray, Electrochim. Acta 55 (2010) 5947. [23] M.W. Kendig, R.G. Buchheit, Corrosion 59 (2003) 379. [24] R. Oltra, B. Malki, F. Rechou, Electrochim. Acta 55 (2010) 4536. [25] G.O. Ilevbare, O. Schneider, R.G. Kelly, J.R. Scully, J. Electrochem. Soc. 151 (2004) B453. [26] G.N. Frantziskonis, L.B. Simon, J. Woo, T.E. Matikas, Eur. J. Mech. A: Solids 19 (2000) 309. [27] T. Holten, T. Jossang, P. Meakin, J. Feder, Phys. Rev. E 50 (1994) 754. [28] D.D. Macdonald, G.R. Engelhardt, Shreir’s Corros. 2 (2009) 1630. [29] A. Anderko, Shreir’s Corros. 2 (2009) 1585. [30] S.M. Sharland, Corros. Sci. 27 (1987) 289. [31] S.M. Sharland, Corros. Sci. 28 (1988) 621. [32] S.M. Sharland, C.P. Jackson, A.J. Diver, Corros. Sci. 29 (1989) 1149. [33] J.C. Walton, G. Cragnolino, S.K. Kalandros, Corros. Sci. 38 (1996) 1. [34] G. Engelhardt, M. Urquidi-Macdonald, D.D. Macdonald, Corros. Sci. 39 (1997) 419. [35] B. Malki, T. Souier, B. Baroux, J. Electrochem. Soc. 155 (2008) C583. [36] J. Amri, E. Gulbrandsen, R.P. Nogueira, Corros. Sci. 52 (2010) 1728. [37] N.J. Laycock, S.P. White, J. Electrochem. Soc. 148 (2001) B264. [38] S. Scheiner, C. Hellmich, Comput. Meth. Appl. Mech. Eng. 198 (2009) 2898. [39] M. Verhoff, R. Alkire, J. Electrochem. Soc. 147 (2000) 1349.

J. Xiao, S. Chaudhuri / Electrochimica Acta 56 (2011) 5630–5641 [40] O. Guseva, P. Schmutz, T. Suter, O.V. Trzebiatowski, Electrochim. Acta 54 (2009) 4514. [41] N. Murer, R. Oltra, B. Vuillemin, O. Neel, Corros. Sci. 52 (2010) 130. [42] K.B. Deshpande, Corros. Sci. 52 (2010) 3514. [43] Y. Liu, R. He, Q. Zhang, S. Chen, J. Phys. Chem. C 114 (2010) 10812. [44] A. Anderko, N. Sridhar, D.S. Dunn, Corros. Sci. 46 (2004) 1583. [45] Y.A. Popov, S.N. Sidorenko, Theory of Interaction of Metals and Alloys with a Corrosive Environment, Cambridge International Science Publishing, Cambridge, 1998. [46] A. Anderko, P. McKenzie, R.D. Young, Corrosion 57 (2001) 202. [47] G. Engelhardt, D.D. Macdonald, Corros. Sci. 46 (2004) 1159. [48] Y.P. Zhao, C.F. Cheng, G.C. Wang, T.M. Lu, Appl. Phys. Lett. 73 (1998) 2432. [49] K.P. Wong, R.C. Alkire, J. Electrochem. Soc. 137 (1990) 3010. [50] T.H. Nguyen, R.T. Foley, J. Electrochem. Soc. 129 (1982) 27.

5641

[51] R.T. Foley, T.H. Nguyen, J. Electrochem. Soc. 129 (1982) 464. [52] N. Murer, N. Missert, R. Buchheit, Proceedings of the COMSOL Users Conference, Boston, 2008. [53] M. Sosna, G. Denuault, R.W. Pascal, R.D. Prien, M. Mowlem, Sens. Actuators B 123 (2007) 344. [54] COMSOL AB, COMSOL Multiphysics User Guide and Model Library, Version 3.5a, COMSOL AB, Sweden, 2008. [55] S.T. Pride, J.R. Scully, J.L. Hudson, J. Electrochem. Soc. 141 (1994) 3028. [56] T.R. Beck, Electrochim. Acta 29 (1984) 485. [57] G.S. Frankel, J.R. Scully, C.V. Jahnes, J. Electrochem. Soc. 143 (1996) 1834. [58] V. Branzoi, F. Golgovici, F. Branzoi, Mater. Chem. Phys. 78 (2002) 122. [59] Z. Ahmad, Principles of Corrosion Engineering Corrosion Control, ButterworthHeinemann, Oxford, UK, 2006. [60] D.W. Buzza, R.C. Alkire, J. Electrochem. Soc. 142 (1995) 1104.