A

Wa

b

c

d

a

ARRAA

KABC

1

adsrHatsfctt

e

SO

h0

Corrosion Science 110 (2016) 157–166

Contents lists available at ScienceDirect

Corrosion Science

j ourna l h omepage: www.elsev ier .com/ locate /corsc i

phase field model for simulating the pitting corrosion

eijie Maia, Soheil Soghrati a,b,c,∗, Rudolph G. Buchheita,d

Department of Materials Science and Engineering, The Ohio State University, USADepartment of Mechanical and Aerospace Engineering, The Ohio State University, USASimulation Innovation and Modeling Center, Columbus, OH, USAFontana Corrosion Center, Columbus, OH, USA

r t i c l e i n f o

rticle history:eceived 22 October 2015eceived in revised form 17 March 2016ccepted 9 April 2016vailable online 13 April 2016

eywords:. Stainless steel. Modeling studies. Pitting corrosion

a b s t r a c t

This manuscript presents the formulation and implementation of a phase field model for simulating theactivation- and diffusion-controlled pitting corrosion phenomena in metallic materials. Introducing anauxiliary phase field variable to represent the physical state of each material point, the free energy of thecorroding metal–electrolyte system can be expressed as a function of field variables and their gradients.The governing equations for the evolution of each field variable are then derived such that the systemfree energy is decreased during the mass transfer and metal dissolution processes. The propagationof the diffuse interface is tracked implicitly by solving the phase field variable over the entire system,which allows the accurate approximation of complex morphological evolutions. A comprehensive study ispresented to verify the accuracy of the proposed model. We show that by calibrating the interface kinetics

parameter with the exchange current density, the proposed model can reproduce different portions of thepolarization curve associated with the activation-controlled, diffusion-controlled, and mixed-controlledcorrosion kinetics. We also demonstrate the application of this model for simulating the electropolishingprocess, interactions between multiple growing pits, and pitting corrosion in a metal matrix compositeand a polycrystalline stainless steel.

© 2016 Elsevier Ltd. All rights reserved.

. Introduction

Corrosion is a destructive electrochemical interaction between susceptible metal and its environment, which leads to the degra-ation of integrity and durability of the material [1]. Most hightrength alloys such as stainless steel and aluminum alloys areesistant against general corrosion by forming a passive film [2].owever, these alloys are vulnerable to pitting corrosion, which is

form of localized corrosion initiated by the partial breakdown ofhe protective film on the metal surface [3,4]. The localized corro-ion attack on the exposed metal can lead to accelerated mechanicalailure of structural components by perforation or forming sites ofrack nucleation [5–7]. Thus, the reliable design and health moni-oring of such metallic structures in corrosive environments require

he ability to accurately predict the evolution of corrosion pits.

Several numerical models are proposed for simulating thelectrochemical phenomena occurring in the electrolyte solution

∗ Corresponding author at: Mechanical and Aerospace Engineering and Materialscience and Engineering, The Ohio State University, 201 W. 19th Avenue, Columbus,H 43210, USA. Tel.: +1 614 292 2768.

E-mail address: soghrati.1@osu.edu (S. Soghrati).

ttp://dx.doi.org/10.1016/j.corsci.2016.04.001010-938X/© 2016 Elsevier Ltd. All rights reserved.

during the corrosion process [8–11]. However, majority of thesemodels are based on the mass transport equation, which is validonly in dilute solutions. For concentrated solutions, the diffusionof dissolved species is driven by the chemical potential gradi-ent instead of the concentration gradient [12]. Moreover, themovement of the corrosion interface is assumed to be negligi-ble, which is valid only when the corrosion rate is sufficientlysmall.

More recently, moving boundary models based on numericaltechniques such as the finite element method (FEM) have beenproposed to simulate the evolving morphology of the pit duringthe metal dissolution [13,14]. In such sharp interface models, themoving interface velocity is incorporated in the boundary condi-tions assigned along the pit interface, which requires an extensiveremeshing to create a conforming (matching) finite element meshat each time step [15]. One can also implement advanced tech-niques such as the arbitrary Lagrangian–Eulerian (ALE) [14,16] andthe level set method (LSM) [17] to track the interface locationand locally evolve the conforming mesh. However, in addition to

increasing the implementation complexity and the computationalcost, adjusting the conforming mesh in such methods leads tobuilding an additional source of error associated with violating theconservation of mass at each time step.

158 W. Mai et al. / Corrosion Scien

Nomenclature

� overpotential� phase field variablec molar concentrationc′ normalized molar concentrationcsolid molar concentration of the solid metalcsat saturation concentration in the solutionc′

Se normalized solid equilibrium concentrationc′

Le normalized solution equilibrium concentration˛c concentration gradient energy coefficient˛� phase field variable gradient energy coefficientf local free energy densityfS free energy density of the solid phasefL free energy density of the solution phasefint interface free energy densityF free energy functional of the systeml thickness of the diffuse interface� interface energyw height of the double well potentialA curvature of the free energy density functionL interface kinetics parameterL0 interface kinetics parameter corresponding to � = 0M diffusion mobilityD diffusion coefficientia anodic current densityi0 exchange current densityn average charge numberF Faraday’s constantv velocity of the corrosion interfaceba Tafel slope� solid-solution interface� solid boundary excluding �

uafaHtit

tituapctciecotibs

i

S� L solution boundary excluding �

Alternatively, some researchers have employed the finite vol-me method (FVM) to simulate the pitting corrosion [18,19]. Thispproach obviates the difficulties associated with creating con-orming meshes by determining the location of the pit interface as

function of the ions concentration in each volume element [20].owever, the accurate representation of the interface location in

he FVM and its application for simulating the pitting phenomenan materials with complex microstructures can be a challengingask.

Meshfree methods (MMs) are also among other numericalechniques that can properly handle moving boundary problemsncluding the pitting corrosion. For example, in the Green’s discreteransformation method (GDTM), the problem domain is discretizedsing a set of un-connected nodes, where only the nodes locatedt the pit surface need to be updated at each time step [21]. In theeridynamics (PD) model proposed in [22,23], the domain is dis-retized using a simple structured grid, where similar to the FVM,he interface location can be identified implicitly based on the ionsoncentration associated with each node. Another approach is tomplement mesh-independent finite element methods such as theXtended/Generalized FEM (X/GFEM) [24,25] and the Hierarchi-al Interface-enriched FEM (HIFEM) [26,27], which allow the usef non-evolving nonconforming structured meshes for discretizinghe domain by introducing appropriate enrichments to capture theons concentration discontinuity along the pit boundary. A com-

ination of the X/GFEM and the LSM has been implemented forimulating the pitting corrosion in [28].

Phase field method is another approach that has received signif-cant attention for simulating problems with evolving geometries,

ce 110 (2016) 157–166

including the solidification of materials [29,30], dendrite growth[31,32], and dislocation interactions [33,34]. In this method, thefree energy of the system is assumed to be a functional of a setof field variables and their gradients, which are assigned to eachmaterial point. The governing equations for the phase field vari-ables are derived in a thermodynamically consistent manner suchthat the free energy of the system is decreased during the dynamicprocess being simulated (e.g., solidification) [35]. Since the mate-rial interface is tracked implicitly by solving the dynamic governingequations in the phase field method, there is no need to assignboundary conditions along the moving interface, which consider-ably facilitates the implementation of this method for simulatingproblems with evolving morphologies. A detailed review of thephase field method and its applications is provided in [36,37].

Despite the extensive application of the phase field methodfor simulating various physical phenomena with complex movingboundaries, only recently it has been formulated for simulatingthe corrosion process in a dual-oxidant environment [38]. Thismodel is capable of simulating the corrosion kinetics by incorpo-rating realistic free energy functions and diffusivity data. A similarmodel is employed to approximate the microstructure evolutionand the distribution of transformation-induced stress as a resultof V2O5 hot corrosion [39]. The corrosion kinetics in such modelsis controlled by the diffusion of reactants (e.g., oxygen or moltensalt) in the solid phase. However, most corrosion phenomenaincluding the pitting corrosion are affected by the diffusion ofdissolved species and reactants in the electrolyte, which mustbe incorporated in the phase field model to simulate the pittingprocess.

The current manuscript aims at developing and implementinga phase field model for approximating the mass transport and sur-face morphology evolution during the pitting corrosion. To achievethis objective, we adopt the Kim–Kim–Suzuki (KKS) phase fieldmodel [40], assuming that every material point in the corrod-ing system is composed of coexisting solid and liquid phases. Theweighted sum of free energies of these phases contributes to thelocal free energy density, which is employed to derive the gov-erning equations by minimizing the free energy during the metaldissolution process. The evolution of the metal–electrolyte inter-face is simulated by approximating the resulting coupled governingequations using the FEM. Adaptive mesh refinement and adaptivetime marching schemes are implemented to assure the accuracyand convergence of the simulation at a reasonable computationalcost. We show that by calibrating the interface kinetics parameterwith the exchange current density before applying the overpo-tential, both the activation- and diffusion-controlled kinetics canproperly be simulated using the proposed phase field model.

The remainder of this manuscript is structured as follows.In Section 2 we present the formulation of the phase field gov-erning equations for approximating the corrosion phenomenon.Section 3 provides several numerical examples to verify/validatethe accuracy of the proposed model. Section 4 demonstrates theapplications of the phase field model for simulating four corro-sion problems: electropolishing, corrosion pits interaction, pittingcorrosion in a particulate composite, and the effect of the crystal-lographic orientation of the stainless steel on the pitting corrosionprocess.

2. Phase field corrosion model

2.1. Problem description

Consider a corrosion system composed of a metallic solid phaseimmersed in NaCl solution, as schematically shown in Fig. 1.Although the metal surface is covered with a protective pas-sive film, which is resistant against general corrosion, the partial

W. Mai et al. / Corrosion Scien

Fig. 1. Schematic of a typical corrosion pit and the initial and boundary conditionsuci

bsrpaptva

[

wdcp

bic

i

wim

v

wa(�

bcrfisftcb

v

TmDat

sed in the phase field model. � S and � L are the solid and liquid boundaries withoordinates xS and xL , respectively. The solid and liquid phases are separated by thenterface � .

reakdown of the film can lead to the formation of a corrosion pitimilar to that depicted in Fig. 1. While it is possible to incorpo-ate a random term in the phase field formulation to simulate theit initiation [41,42], in this work we focus on the propagation ofn existing pit. Assuming that the diffusion of ions in the metallichase is negligible, the Rankine–Hugoniot condition [18] satisfieshe equilibrium between the dissolved metal atoms flux and theelocity of the moving pit boundary � (metal/electrolyte interface)s

D∇c + (c(x, t) − csolid)v] · n = 0 ∀x ∈ �, (1)

here D is the diffusion coefficient, c(x, t) is the concentration ofissolved ions at point (x, t) on the pit boundary, csolid is the con-entration of atoms in the metal, and v is the velocity of the movingit boundary.

The driving force for the atom-ion transformation across the pitoundary is the interface overpotential �. When this driving force

s small, the corrosion rate increases exponentially with �, whichan be determined using the Tafel equation

a = i0exp(

�

ba

), (2)

here i0 is the exchange current density, ba is the Tafel slope, anda is the corrosion current density. The latter can be related to the

oving pit boundary velocity using the Faraday’s second law

n = v · n = ianFcsolid

, (3)

here n is the average charge number, F is the Faraday’s constant,nd n is the unit normal vector to the pit interface. According to2) and (3), the corrosion velocity vn increases exponentially with, which is often referred to as an activation-controlled corrosion.

As the pitting proceeds, metal ions accumulate along the pitoundary until the surface concentration reaches the saturationoncentration csat. At this stage, the saturated dissolved metal ionseact with Cl− ions in the electrolyte and precipitate a chloride saltlm along the pit surface, which prohibits further increase of theurface concentration. After the formation of the salt film, the inter-ace velocity is no longer governed by (2) and instead controlled byhe diffusion of metal ions away from the pit boundary (diffusion-ontrolled corrosion). By replacing c with csat in (1), the moving pitoundary velocity can then be evaluated as

n = D∇c · ncsolid − csat

. (4)

o simulate the pitting corrosion using a sharp interface model, one

ust apply either a Robin boundary condition according to (1) or airichlet boundary condition of c = csat along the pit interface for thectivation-controlled and diffusion-controlled processes, respec-ively. The moving boundary velocities corresponding to each case

ce 110 (2016) 157–166 159

are computed using (3) and (4), respectively. Unlike sharp inter-face models, the phase field approach allows for approximating theinterface evolution implicitly by evaluating the distribution of anauxiliary field variable � over the entire system. Assuming con-stant values in the bulk of each phase with � = 0 for the electrolyteand � = 1 for the metal, the phase field variable � varies contin-uously across the diffuse interface � with a finite thickness l. Itcan be shown that the sharp interface condition is replicated atthe thin interface limit [41,43], which allows using the phase fieldmodel as an alternative approach to obviate the difficulties asso-ciated with assigning boundary conditions along the moving pitboundary. As shown in Fig. 1, the boundary conditions are onlyassigned along the outer edges of the domain, with Neumann con-dition on the solid (metal) boundary � S and Dirichlet conditionon the liquid (electrolyte) boundary � L, respectively. Although theRankine–Hugoniot condition (1) is not directly incorporated in thephase field model, it indicates that the rate of the metal dissolutionis affected by the ions concentration and its gradient along the pitsurface. To incorporate the effect of the ions concentration in thephase field model while satisfying the consistency of units in theresulting formulation, we introduce the normalized concentrationc′ evaluated as

c′ = c

csolid. (5)

Boundary conditions associated with the concentration field in thephase field model are depicted in Fig. 1; noting that assigning aboundary condition along the pit surface is no longer required.

2.2. Phase field governing equations

The driving force for any phase transformation is the reductionof the system’s free energy F. In a phase field model, this free energyconsists of the homogeneous bulk energy Fbulk and the interfaceenergy Fint given by

F = Fbulk + Fint =∫

[f (c′, �) + fint] dV, (6)

where f(c′, �) is the local free energy density, which is a functionof c′ and �, while fint is the excessive energy associated with thediffuse interface. The excessive interface energy arises from theinhomogeneity within the interface region, which can be writtenas a function of the field variable gradients

fint = ˛�

2(∇�)2 + ˛c

2(∇c′)2. (7)

In (7), ˛c and ˛� are the gradient energy coefficients associated withthe concentration and phase fields, respectively.

To assure the reduction of the free energy during the corro-sion process, the phase field governing equations are derived byminimizing F via variational differentiation [39,36], which yields

∂�

∂t(x, t) = −L

ıFı�

= −L

(∂f

∂�− ˛�∇2�

), (8)

∂c′

∂t(x, t) = ∇ · M∇ ıF

ıc′ = ∇ · M

(∇ ∂f

∂c′ + ˛c∇2c′)

, (9)

where L is the interface kinetics parameter and M is the diffusionmobility for mass transport. Eqs. (8) and (9) are referred to as theAllen–Cahn and Cahn–Hilliard equations, respectively. Note that,

in practice, only one of the gradient terms (∇c′ or ∇�) would besufficient to approximate the energy contribution from the diffuseinterface; thus the concentration gradient energy coefficient ˛c isassumed to be 0 for simplicity.

1 Science 110 (2016) 157–166

2

momwte

c

wefFshaf

f

wIo(mtatd

f

f

ItpwTiaf(p(

2

wc

�

l

wiiims

Table 1Values of the phase field corrosion model parameters [18,39].

Parameter Physical interpretation Value

� Interface energy 10 J/m2

l Interface thickness 1 �mD Diffusion coefficient 8.5 × 10−10 m2/sn Average charge number 2.1L Interface kinetics coefficient 2 m3/(J s)A Free energy density curvature 5.35 × 107 J/mol

60 W. Mai et al. / Corrosion

.3. Local free energy density

While (8) and (9) represent the general form of a phase fieldodel, the key difference between various models is the definition

f the local free energy density f(c′, �). In the KKS model [40], eachaterial point is assumed to be a mixture of two or more phasesith different concentrations but similar chemical potentials. In

he present phase field corrosion model, these assumptions can bexpressed as

′ = h(�)c′S + [1 − h(�)]c′

L, (10)

∂fS(c′S)

∂c′S

= ∂fL(c′L)

∂c′L

, (11)

here c′S and c′

L are the normalized concentrations of the co-xisting solid and liquid phases, respectively, while fS(c′

S) andL(c′

L) are the free energy densities corresponding to each phase.urthermore, h(�) is a C∞-continuous interpolation functionuch that h(� = 0) = 0 and h(� = 1) = 1. In this work, we assume(�) =−2�3 + 3�2. Since the concentration at any point is evaluateds the weighted sum of the solid and liquid concentrations, the localree energy density can be computed using a similar approach, i.e.,

(c′, �) = h(�)fS(c′S) + [1 − h(�)]fL(c′

L) + wg(�), (12)

here w is the height of the double well potential g(�) = �2(1 − �)2.t must be noted that in the Cahn–Hilliard equation (9), the diffusionf dissolved species is governed by the chemical potential gradientıf/ıc′) instead of the concentration gradient; thus the proposed

odel is valid in both dilute and concentrated solutions providedhat appropriate free energy density functions are used for fS(c′

S)nd fL(c′

L) [12]. In this manuscript, we mainly focus on simulatinghe pitting corrosion in dilute solutions, for which the free energyensities can reasonably be approximated as [44]

S(c′S) = A(c′

S − c′Se)2, (13)

L(c′L) = A(c′

L − c′Le)2. (14)

n the above equations, c′Se = csolid/csolid = 1 and c′

Le = csat/csolid arehe normalized equilibrium concentrations for the solid and liquidhases, respectively. Also, A is the free energy density curvature,hich is assumed to be similar for both the solid and liquid phases.

he value of A is computed such that the phase transformation driv-ng force in the resulting approximate system is similar to that of thectual thermodynamic system [44,39]. Using an appropriate localree energy density f(c′, �), together with the assumptions made in10) and (11), one can then simulate the propagation of a corrosionit by solving the Allen–Cahn and Cahn–Hilliard equations given in8) and (9), respectively.

.4. Numerical implementation

The gradient energy coefficient ˛� and the height of doubleell potential w in the present phase field corrosion model are

orrelated to the interface energy � and its thickness l as [39]

≈√

16w˛�, (15)

= ˛∗√

2˛�

w, (16)

here ˛* = 2.94 is a constant parameter corresponding to the def-nition of interface region 0.05 < � < 0.95. While decreasing the

nterface thickness l reduces the error associated with the diffusenterface, the computational cost also increases as a more refined

esh has to be generated within the interface. In this study, weelect the interface thickness such that it is significantly smaller

csolid Average concentration of metal 143 mol/Lcsat Average saturation concentration 5.1 mol/L

than characteristic length scales of the problem to assure a reason-able accuracy by allowing the presence of at least six mesh nodesin the interface region. Also, the diffusion mobility M is evaluatedbased on an analogy between the Cahn–Hilliard equation (9) andthe Fick’s second law in the bulk of each phase, which can be writtenas

M = D

2A. (17)

The governing equations are then approximated using the FEMwith six-node triangular Lagrangian elements for discretizing thedomain. Note that the phase field model does not require theconstruction of a conforming mesh that adapts to the evolvingmorphology of the pit boundary at each time step. However, toassure the numerical stability and accuracy of simulations, thereshould be at least 6 nodes within the interface region to properlyapproximate �. To avoid using an excessively refined finite elementmesh everywhere, we implement an adaptive refinement schemeusing the magnitude of | ∇ �| as the refinement criterion to concen-trate this process on elements located within the diffuse interface.Moreover, an adaptive time marching scheme is adopted with atolerance of 10−3 to assure the numerical convergence, while main-taining the computational efficacy. Unless stated otherwise, valuesof the phase field parameters used for performing the simulationspresented in the following sections are selected from Table 1.

3. Validation and verification studies

3.1. Pencil electrode test

As the first example, we implement the phase field model to sim-ulate pit evolution of a pencil electrode and compare the numericalprediction with analytical results. In this numerical experiment,which is schematically illustrated in Fig. 2a, an artificial corrosionpit is created by mounting a metal wire with diameter d = 25 �m inan epoxy coating, which only leaves one end of the wire exposed tothe solution. With a sufficiently high applied potential, the electro-chemical reaction at the metal/electrolyte interface occurs at sucha high rate that a saturated salt film is immediately formed on themetal surface; thus the corrosion can be modeled as a diffusion-controlled process [45,46].

Simulating the pencil electrode test using the phase fieldmethod does not require imposing the boundary condition c = csat

along the corroding pit interface. Instead, Dirichlet conditions c′ = 1,� = 1 and c′ = 0, � = 0 are assigned to the lower edge of the metalwire and on the pit mouth, respectively. Neumann boundary con-ditions (∂c′/∂n) = 0 and (∂�/∂n) = 0 are assigned along the lengthof the wire, which is insulated by the epoxy coating. Fig. 2b illus-trates the phase field approximation of the ion concentration inthe electrolyte at t = 38 s and t = 152 s (l = 5 �m), indicating that the

Dirichlet condition c = csat on the pit interface is implicitly satisfiedin these simulations. Note that since in this test the dissolution ofmetal occurs only in the longitudinal direction of the wire, it can beapproximated as a 1D problem, for which an analytical solution is

W. Mai et al. / Corrosion Science 110 (2016) 157–166 161

F g the

t ion of

avi

3

lbicabeomf

2loapptwpco

F(p

ig. 2. First example problem: (a) schematic setup of a pencil electrode test showin = 38 s and t = 152 s; (c) phase field simulation and analytical solution for the variat

vailable [18]. The analytically computed evolution of the pit depthersus

√t, together with its phase field approximation are depicted

n Fig. 2c, which shows a perfect agreement between both results.

.2. Semi-circular pit growth

In this example, the phase field model is employed to simu-ate the evolution of the semi-circular pit depicted in Fig. 3a. It haseen shown experimentally [47] that when a corrosion pit with an

nitial semi-circular shape propagates under diffusion-controlledonditions, it evolves into a flat shape resembling a semi-ellipse,s corners of the pit have higher limiting current density than itsottom. However, by covering the top edge of a stainless steel foilxposed to an electrolyte by lacquer everywhere except for a smallpening, Ernst and Newman [45] showed that the corrosion pitaintains a semi-circular shape, as diffusion distances are identical

or all the points located on the pit boundary.To numerically simulate this phenomenon, we consider a

00 �m × 400 �m rectangular domain, in which all edges are insu-ated except for a small opening with length 16 �m in the middlef the top edge. Dirichlet boundary conditions c′ = 0 and � = 0 aressigned along the length of this opening. At t = 0 s, a semi-circularit with a diameter of 16 �m is initiated on this opening and thehase field simulation is carried out assuming l = 5 �m. Fig. 3a illus-rates the evolving pit interface at different time steps, together

ith the concentration field at t = 1000 s, showing that the growingit maintains its semi-circular shape throughout the pitting pro-ess. Fig. 3b illustrates the phase field simulation of the variationf the pit depth versus time and its comparison with the XFE-LSM

ig. 3. Second example problem: pitting corrosion in a stainless steel foil coated by lacquea) Evolving pit morphology at different times and the concentration field at t = 1000 s; (b)resented in [28].

boundary conditions. (b) Concentration distribution within the pit solution at timethe pit depth versus

√t.

results presented in [28], indicating that both methods yield simi-lar results and predict the pit depth growth as a parabolic functionof time during this diffusion-controlled process.

3.3. Activation- and diffusion-controlled corrosion

This example demonstrates the predictive capability of theproposed phase field model for simulating both the activation-and diffusion-controlled corrosion phenomena after calibrating theinterface kinetics parameter L with the exchange current densityi0. In the sharp interface model presented in [19] (FVM) and alsoimplemented in [21] (GDTM) and [28] (XFE-LSM), the activation-controlled corrosion rate is evaluated using the Butler–Volmertype equation given in (3). However, as soon as the concentra-tion at a boundary point reaches the saturation level csat (becomesdiffusion-controlled), the moving boundary velocity must imme-diately be evaluated using (4). Although this strategy can simulateboth corrosion modes, the sudden jump in the corrosion rate ofpoints that reach csat is not realistic, as at intermediate potentialsthe corrosion rate is simultaneously affected by both the interfacereaction kinetics and ions diffusion.

In the present phase field model, the rate of the phase transfor-mation (interface reaction) is governed by the Allen-Cahn equationgiven in (8), in which the interface kinetics parameter L controlsthe reaction rate. Therefore, the transition from the activation-

controlled to the diffusion-controlled corrosion can be simulatedby continuously varying L as a function of the overpotential �.The impact of L on the corrosion rate is depicted in Fig. 4a, whichshows the transition from the linear (activation-controlled) to

r everywhere on the top surface except for a small opening of 16 �m in the middle. variation of the pit depth versus time and its comparison with the XFE-LSM results

162 W. Mai et al. / Corrosion Science 110 (2016) 157–166

F differea ased

taabee

�

wTpeatda

L

ivpLct(wsstrirdatmi

4

pm

ig. 4. Third example problem: (a) variation of the corrosion depth versus time for

nd mixed-controlled corrosion regions in the polarization curve by calibrating L0 b

he parabolic (diffusion-controlled) kinetics as L increases. Sinceccording to (8), the corrosion rate is proportional to L in thectivation-controlled stage, there should be a linear relationshipetween the corrosion current density ia and L. Thus, the depend-nce of L on the overpotential has a similar form as the Tafelquation,

= balog(

iai0

)= balog

(L

L0

), (18)

here L0 is the interface kinetics parameter associated with � = 0.o perform the phase field simulation, a similar strategy as thatroposed in [22] is adopted to obtain the value of L0 based onxperimental data. First, a sufficiently small L is implemented tossure that the corrosion is activation-controlled. After evaluatinghe interface velocity v, it is transformed to the corrosion currentensity ia using the Faraday’s second law (3). L0 can then be evalu-ted by rearranging (18) as

0 = Li0ia

. (19)

Using the experimentally measured current density0 = 1.422 mA/cm2 reported in [22] for stainless steel, the L0alue is evaluated as L0 = 1.2 × 10−11 m3/(J s). To reproduce theolarization curve, we first employ (18) to evaluate values of

corresponding to different overpotential values, followed byomputing the interface velocity at t = 20 s and then transforminghat into the current density using (3). The resulting Evans diagram� − log ia variation) is presented in Fig. 4b, which is compatibleith the analytical solution obtained from the Tafel equation for

mall values of � (activation-controlled corrosion). When � isufficiently large, the current density reaches its limiting value dueo the high reaction rate on the pit boundary; thus the corrosionate is limited by the diffusion of dissolved species away from thenterface. In the transition region, where the interface reactionate and the diffusion rate are comparable, the polarization curveeviates from the Tafel equation and the current density graduallypproaches its limiting value as � increases. As shown in Fig. 4b,his behavior is properly simulated by the phase field model. It

ust be noted that the Peridynamics corrosion model presentedn [22] is also capable of producing similar results.

. Application problems

In this section, we demonstrate the application of the proposedhase field model for simulating four corrosion problems withore complex evolving morphologies: (i) electropolishing, (ii)

nt values of L; (b) reconstruction of the activation-controlled, diffusion-controlled,on the exchange current density.

multiple pit interaction, (iii) pitting corrosion in a compositematerial, and (iv) crystallographic orientation-dependent pittingcorrosion in stainless steel.

4.1. Example 1: Electropolishing

In this example, we implement the phase field model tosimulate the electropolishing process, which is widely used inindustry to reduce the surface roughness of metallic components[48]. After immersing a metallic part with rough surface in anappropriate electrolyte solution, the metal surface is subjectedto diffusion-controlled corrosion by applying a sufficiently largeanodic potential. Under such conditions, extruded portions of themetal corrode at a higher rate due to their shorter diffusion dis-tance, which leads to a larger limiting current density. To examinethe predictive capability of the phase field model for simulating thisphenomenon, we consider a 350 �m × 150�m rectangular domainwith the interface thickness l = 5 �m. The boundary conditions aresimilar to those shown in Fig. 1, except that the passive film is elim-inated to allow free corrosion of the entire metal surface (Dirichletboundary conditions c′ = 0 and � = 0 along the top edge).

Fig. 5 illustrates the morphology of the corroding metal at differ-ent times, showing how its initially rough surface is transformedinto a relatively smooth surface due to the electropolishing phe-nomenon. Also, Fig. 6 shows the finite element discretization of therectangular region shown in Fig. 5 at different times, which demon-strates the adaptive mesh refinement in the vicinity of the diffuseinterface as described in Section 2.4.

4.2. Example 2: Multiple pit interaction

While majority of previous computational efforts for simulat-ing the pitting corrosion have focused on studying the growth ofa single pit, in reality corrosion pits can initiate adjacent to oneanother and thus interact with one another during their evolution.Depending on the test set up, both positive (accelerating the ini-tiation of another pit) and negative interactions between adjacentpits are observed in experimental studies [49–51]. In this example,we simulate the growth and interaction of two corrosion pits in arectangular domain with the dimensions and boundary conditionsshown in Fig. 7a. By applying a sufficiently large potential underthe potentiostatic condition, pitting corrosion is assumed to be a

diffusion-controlled process. The initially semicircular-shaped pitshave radii r0 = 0.2 �m and are separated by distances d = 0.5 �m,0.2 �m, and 2 �m in the three simulations carried out in this exam-ple. As shown in Fig. 7a, while in the beginning the corrosion rates

W. Mai et al. / Corrosion Science 110 (2016) 157–166 163

e mor

atswtpatigc

4

mapiscshiot(tt

Fl

Fig. 5. First application problem: evolution of the metal surfac

re similar for all cases, the rate of metal dissolution decreases ashe growing pits approach one another. As this occurs, the dis-olved ions associated with each pit interact with one another,hich increases the concentration in the vicinity of both pits and

hus slows down the diffusion-controlled pitting process. Since thishenomenon occurs earlier for pits that are closer to one anothert t = 0 s (d = 0.5 �m), the overall corrosion rate for this case is lesshan that of pits initially located farther from one another. Fig. 7bllustrates the phase field simulation of the evolving pits morpholo-ies and their interaction in this problem, which also shows theoncentration field associated with each case at t = 1 ms and t = 5 ms.

.3. Example 3: Pitting corrosion in a composite material

This example demonstrates the application of the phase fieldodel for simulating the pitting corrosion in a 200 �m × 140 �m

luminum matrix composite specimen with embedded ceramicarticles, as shown in Fig. 8a. While ceramic particles are embedded

n the relatively softer aluminum matrix to improve its stiffness andtrength, corrosion can considerably deteriorate the integrity of thisomposite material by detaching the particles from the matrix. Ashown in Fig. 8a, the composite specimen studied in this exampleas a protective passive film on its upper edge with a 10 �m open-

ng with Dirichlet boundary conditions c′ = 0 and � = 0. Other edgesf the domain, including portions of the upper edge with the pro-

ective coating has zero flux boundary conditions (∂c′/∂n) = 0 and∂�/∂n) = 0. Since ceramic particles are considerably less reactivehan aluminum, we assume they are non-corrodible. Fig. 8 illus-rates the phase field simulation of the evolving pit morphology at

ig. 6. Adaptively refined finite element mesh within the rectangular region shown in Focation of the diffuse interface. (For interpretation of the references to color in this figur

phology at different times during the electropolishing process.

different times, which shows the ability of this method to handlethe complex shape of the moving pit boundary during the corrosionprocess.

4.4. Example 4: Crystallographic orientation-dependent pittingcorrosion

In this final example problem, we implement the phase fieldmodel to simulate the impact of the stainless steel polycrys-talline microstructure on the propagation of a corrosion pit. It hasbeen shown that crystal orientations can affect the pit growth,as crystallographic planes with higher atomic density corrode ata slower rate during the activation-controlled pitting corrosion[52]. Fig. 9a illustrates the domain geometry and microstruc-ture of the 8 �m × 3 �m stainless steel specimen studied in thisexample. For simplicity, it is assumed that only {101}, {001},and {111} atomic planes are exposed to the electrolyte solution.Using the experimental data reported in [52], we employ thecalibration procedure described in Section 3.3 to evaluate the inter-face kinetic parameters for the three crystallographic orientationsshown in Fig. 9a. Based on the resulting parameters, the {111} crys-tals corrode three times slower than the {101} and {001} grains(L{111} = 10−16 m3/(J s)).

To perform the phase field simulation, a corrosion pit is initiatedat an opening with a length of 0.3 �m in the protective coating on

the top edge of the specimen. The simulated evolving morphologiesof the pit at real corrosion times are depicted in Fig. 9 (l = 0.1 �m),which is obtained by proper normalization of the governingequations to allow using large time steps during the simulation.

ig. 5 at times t = 3 s, t = 9 s, and t = 32 s. The dotted red line shows the approximatee legend, the reader is referred to the web version of this article.)

164 W. Mai et al. / Corrosion Science 110 (2016) 157–166

F issolva

A(mc

FE

ig. 7. Second application problem: (a) domain configuration and variation of the dnother; (b) pits morphologies and ions concentration at t = 1 ms and t = 5 ms.

s shown in Fig. 9b, in the early stages of the pitting process

t = 73 h) the pit propagation is confined to a single crystal and thus

aintains a semi-circular shape during its growth. However, as theorrosion pit interacts with {101} and {001} crystals, the corrosion

ig. 8. Third application problem: pitting corrosion of SiC particle reinforced aluminumvolution of the corrosion pit morphology at different times.

ed mass versus time for two evolving pits initiated at different distances from one

rate increases along certain portions of the pit boundary, which

leads to more complex pit morphologies shown in Fig. 9c–f. Thisexample illustrates the impact of the crystallographic orientationon the activation-controlled pitting corrosion, as also reported in

composite. (a) Microstructure of the composite specimen before corrosion. (b–d)

W. Mai et al. / Corrosion Science 110 (2016) 157–166 165

Fig. 9. Fourth application problem: (a) domain geometry and polycrystalline microstructure of a stainless steel specimen with a 0.3 �m opening on its protective passivefi al oriet

taplts

5

mamw(tafBmatToeetimt

[

lm at t = 0 s. Dark, medium, and light gray represent {001}, {101}, and {111} crysthe pit morphology at different times.

he literature [4]. Note that, although not shown in this example,fter the formation of the salt film on the pit surface at higherotentials the corrosion becomes diffusion-controlled and is no

onger governed by the crystals orientation. At this stage, similaro the electropolishing example presented in Section 4.1, the piturface becomes smoother as it propagates.

. Conclusion

A phase field model was presented for simulating the evolvingorphology of the metal surface during the activation-controlled

nd diffusion-controlled pitting corrosion phenomena. In thisodel, the local free energy density was formulated as theeighted sum of free energies of the liquid (electrolyte) and solid

metal) phases. The Cahn-Hilliard and Allen-Cahn equations werehen implemented to evaluate mass transport in the electrolytend the metal dissolution at its corroding interface such that theree energy of the system is decreased during the pit propagation.y eliminating the need to both create conforming finite elementeshes and assigning boundary conditions along the pit surface

t each time step, the phase field is an appropriate numericalechnique for simulating this complex moving boundary problem.hree benchmark problems are presented to verify the accuracyf the proposed phase field corrosion model versus analytical,xperimental, and sharp interface simulation results. Further, sev-ral examples were presented to demonstrate the application of

he proposed model for simulating the growth of corrosion pitsn materials with complex microstructures such as an aluminum

atrix composite with embedded ceramic particles and polycrys-alline stainless steel.

[

ntations with respect to a fixed coordinate system, respectively. (b–f) Evolution of

Acknowledgements

This work was partially supported by the Ohio State UniversitySimulation Innovation and Modeling Center (SIMCenter) throughsupport from Honda R&D Americas, Inc. and an allocation of com-puting time from the Ohio Supercomputer Center (OSC).

References

[1] E. McCafferty, Introduction to Corrosion Science, Springer Science & BusinessMedia, 2010.

[2] A.J. Sedriks, Corrosion of Stainless Steel, 2nd ed., John Wiley and Sons, Inc,New York, NY, United States, 1996.

[3] Z. Szklarska-Smialowska, Review of literature on pitting corrosion publishedsince 1960, Corrosion 27 (6) (1971) 223–233.

[4] G.S. Frankel, Pitting corrosion of metals a review of the critical factors, J.Electrochem. Soc. 145 (6) (1998) 2186–2198.

[5] E.J. Dolley, B. Lee, R.P. Wei, The effect of pitting corrosion on fatigue life,Fatigue Fract. Eng. Mater. Struct. 23 (7) (2000) 555–560.

[6] K.K. Sankaran, R. Perez, K.V. Jata, Effects of pitting corrosion on the fatiguebehavior of aluminum alloy 7075-T6: modeling and experimental studies,Mater. Sci. Eng. A 297 (1) (2001) 223–229.

[7] L.Y. Yu, K.V. Jata, Physics driven pitting corrosion modeling in 2024-T3aluminum alloys, in: SPIE Smart Structures and Materials+ NondestructiveEvaluation and Health Monitoring, International Society for Optics andPhotonics, 2015, p. 94372E.

[8] B.G. Ateya, H.W. Pickering, Effects of ionic migration on the concentrationsand mass transfer rate in the diffusion layer of dissolving metals, J. Appl.Electrochem. 11 (4) (1981) 453–461.

[9] T.W. Tester, H.S. Isaacs, Diffusional effects in simulated localized corrosion, J.Electrochem. Soc. 122 (11) (1975) 1438–1445.

10] A. Turnbull, J.G.N. Thomas, A model of crack electrochemistry for steels in the

active state based on mass transport by diffusion and ion migration, J.Electrochem. Soc. 129 (7) (1982) 1412–1422.

11] S.M. Sharland, C.P. Jackson, A.J. Diver, A finite element model of thepropagation of corrosion crevices and pits, Corros. Sci. 29 (9) (1989)1149–1166.

1 Scien

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

66 W. Mai et al. / Corrosion

12] S.M. Sharland, A review of the theoretical modelling of crevice and pittingcorrosion, Corros. Sci. 27 (3) (1987) 289–323.

13] J. Xiao, S. Chaudhuri, Predictive modeling of localized corrosion: anapplication to aluminum alloys, Electrochim. Acta 56 (16) (2011)5630–5641.

14] N. Kota, S.M. Qidwai, A.C. Lewis, V.G. DeGiorgi, Microstructure-basednumerical modeling of pitting corrosion in 316 stainless steel, ECS Trans. 50(31) (2013) 155–164.

15] S. Sarkar, J.E. Warner, W. Aquino, A numerical framework for the modeling ofcorrosive dissolution, Corros. Sci. 65 (2012) 502–511.

16] W. Sun, L.D. Wang, T.T. Wu, G.C. Liu, An arbitrary Lagrangian–Eulerian modelfor modelling the time-dependent evolution of crevice corrosion, Corros. Sci.78 (2014) 233–243.

17] J.A. Sethian, A fast marching level set method for monotonically advancingfronts, Proc. Natl. Acad. Sci. U.S.A. 93 (4) (1996) 1591–1595.

18] S. Scheiner, C. Hellmich, Stable pitting corrosion of stainless steel asdiffusion-controlled dissolution process with a sharp moving electrodeboundary, Corros. Sci. 49 (2) (2007) 319–346.

19] S. Scheiner, C. Hellmich, Finite volume model for diffusion- andactivation-controlled pitting corrosion of stainless steel, Comput. MethodsAppl. Mech. Eng. 198 (37) (2009) 2898–2910.

20] Y. Onishi, J. Takiyasu, K. Amaya, H. Yakuwa, K. Hayabusa, Numerical methodfor time-dependent localized corrosion analysis with moving boundaries bycombining the finite volume method and voxel method, Corros. Sci. 63 (2012)210–224.

21] S. Soghrati, W.J. Mai, B.W. Liang, R.G. Buchheit, A boundary collocationmeshfree method for the treatment of Poisson problems with complexmorphologies, J. Comput. Phys. 281 (2015) 225–236.

22] Z.G. Chen, F. Bobaru, Peridynamic modeling of pitting corrosion damage, J.Mech. Phys. Solids 78 (2015) 352–381.

23] Z.G. Chen, G.F. Zhang, F. Bobaru, The influence of passive film damage onpitting corrosion, J. Electrochem. Soc. 163 (2) (2016) C19–C24.

24] T. Belytschko, R. Gracie, G. Ventura, A review of extended/generalized finiteelement methods for material modeling, Model. Simul. Mater. Sci. Eng. 17 (4)(2009) 043001.

25] J. Chessa, P. Smolinski, T. Belytschko, The extended finite element method(XFEM) for solidification problems, Int. J. Numer. Methods Eng. 53 (8) (2002)1959–1977.

26] S. Soghrati, Hierarchical interface-enriched finite element method: Anautomated technique for mesh-independent simulations, J. Comput. Phys.275 (2014) 41–52.

27] S. Soghrati, H. Ahmadian, 3D hierarchical interface-enriched finite elementmethod: Implementation and applications, J. Comput. Phys. 299 (2015) 45–55.

28] R. Duddu, Numerical modeling of corrosion pit propagation using thecombined extended finite element and level set method, Comput. Mech. 54(3) (2014) 613–627.

29] A. Karma, W. Rappel, Phase-field method for computationally efficient

modeling of solidification with arbitrary interface kinetics, Phys. Rev. E 53 (4)(1996) R3017.

30] C. Beckermann, H.J. Diepers, I. Steinbach, A. Karma, X. Tong, Modeling meltconvection in phase-field simulations of solidification, J. Comput. Phys. 154(2) (1999) 468–496.

[

[

ce 110 (2016) 157–166

31] J.A. Warren, W.J. Boettinger, Prediction of dendritic growth andmicrosegregation patterns in a binary alloy using the phase-field method,Acta Metall. Mater. 43 (2) (1995) 689–703.

32] A.A. Wheeler, B.T. Murray, R.J. Schaefer, Computation of dendrites using aphase field model, Physica D 66 (1) (1993) 243–262.

33] D. Rodney, Y.L. Bouar, A. Finel, Phase field methods and dislocations, ActaMater. 51 (1) (2003) 17–30.

34] C. Shen, Y. Wang, Phase field model of dislocation networks, Acta Mater. 51(9) (2003) 2595–2610.

35] A.G. Lamorgese, D. Molin, R. Mauri, Diffuse Interface (DI) Model forMultiphase Flows, Springer, 2012.

36] W.J. Boettinger, J.A. Warren, C. Beckermann, A. Karma, Phase-field simulationof solidification, Annu. Rev. Mater. Res. 32 (1) (2002) 163–194.

37] L.Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater.Res. 32 (1) (2002) 113–140.

38] Y.H. Wen, L.Q. Chen, J.A. Hawk, Phase-field modeling of corrosion kineticsunder dual-oxidants, Model. Simul. Mater. Sci. Eng. 20 (3) (2012) 035013.

39] A. Abubakar, S.S. Akhtar, A.F.M. Arif, Phase field modeling of V2O5 hotcorrosion kinetics in thermal barrier coatings, Comput. Mater. Sci. 99 (2015)105–116.

40] S.G. Kim, W.T. Kim, T. Suzuki, Phase-field model for binary alloys, Phys. Rev. E60 (6) (1999) 7186.

41] N. Provatas, K. Elder, Phase Field Methods in Materials Science andEngineering, John Wiley & Sons, 2011.

42] P. Ståhle, E. Hansen, Phase field modelling of stress corrosion, Eng. Fail. Anal.47 (2015) 241–251.

43] M. Ode, T. Suzuki, S.G. Kim, W.T. Kim, Phase-field model for solidification ofFe–C alloys, Sci. Technol. Adv. Mater. 1 (1) (2000) 43–49.

44] S.Y. Hu, J. Murray, H. Weiland, Z.K. Liu, L.Q. Chen, Thermodynamic descriptionand growth kinetics of stoichiometric precipitates in the phase-fieldapproach, Calphad 31 (2) (2007) 303–312.

45] P. Ernst, R.C. Newman, Pit growth studies in stainless steel foils. I.Introduction and pit growth kinetics, Corros. Sci. 44 (5) (2002) 927–941.

46] P. Ernst, R.C. Newman, Pit growth studies in stainless steel foils. II. Effect oftemperature, chloride concentration and sulphate addition, Corros. Sci. 44 (5)(2002) 943–954.

47] J.N. Harb, R.C. Alkire, The effect of fluid flow on growth of single corrosionpits, Corros. Sci. 29 (1) (1989) 31–43.

48] D. Landolt, P.F. Chauvy, O. Zinger, Electrochemical micromachining, polishingand surface structuring of metals: fundamental aspects and newdevelopments, Electrochim. Acta 48 (20) (2003) 3185–3201.

49] N.D. Budiansky, L. Organ, J.L. Hudson, J.R. Scully, Detection of interactionsamong localized pitting sites on stainless steel using spatial statistics, J.Electrochem. Soc. 152 (4) (2005) B152–B160.

50] N. Laycock, S. White, D. Krouse, Numerical simulation of pitting corrosion:Interactions between pits in potentiostatic conditions, ECS Trans. 1 (16)(2006) 37–45.

51] N.J. Laycock, D.P. Krouse, S.C. Hendy, D.E. Williams, Computer simulation ofpitting corrosion of stainless steels, Electrochem. Soc. Interface 23 (4) (2014)65–71.

52] D. Lindell, R. Pettersson, Crystallographic effects in corrosion of austeniticstainless steel 316L, Mater. Corros. 66 (8) (2015) 727–732.