Corrosion Science 110 (2016) 157166
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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 metalelectrolyte 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.
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 . Most hightrength alloys such as stainless steel and aluminum alloys areesistant against general corrosion by forming a passive film .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 . 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: email@example.com (S. Soghrati).
ttp://dx.doi.org/10.1016/j.corsci.2016.04.001010-938X/ 2016 Elsevier Ltd. All rights reserved.
during the corrosion process . 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 . 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 . One can also implement advanced tech-niques such as the arbitrary LagrangianEulerian (ALE) [14,16] andthe level set method (LSM)  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
overpotential phase field variablec molar concentrationc normalized molar concentrationcsolid molar concentration of the solid metalcsat saturation concentration in the solutioncSe normalized solid equilibrium concentrationcLe normalized solution equilibrium concentrationc 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 Faradays constantv velocity of the corrosion interfaceba Tafel slope solid-solution interface solid boundary excluding
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 .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 Greens 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 . 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 .
Phase field method is another approach that has received signif-cant attention for simulating problems with evolving geometries,
ce 110 (2016) 157166
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) . 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 . 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 transformati