Multi-physics analysis of the galvanic corrosion of Mg-steel couple under the influence of time-dependent anisotropic deposition film

Ki Wng,Chenpei Li,Ynhui Li,Jinling Lu,Yueshe Wng,?,Xingqi Luo

a State Key Laboratory of Eco-hydraulics in Northwest Arid Region,Xi’an University of Technology,No.5 Jinhua South Road,Xi’an 710048,China

b State Key Laboratory of Multiphase Flow in Power Engineering,Xi’an Jiaotong University,No.28 Xianning West Road,Xi’an 710049,China

Abstract The anisotropic deposit film formed during the galvanic corrosion can impede the mass transfer of the involved species,thereby affecting the electro-chemical behavior and the evolution of galvanic corrosion.The limitations of experimental studies in the spatial-temporal scales restrict a deeper understanding of the corrosion mechanism,which can be complemented by numerical simulation.A multi-physics coupled model is proposed in this work to systematically investigate the temporal and spatial evolution of galvanic corrosion of the Mg-steel couple with the growing anisotropic deposition layer.By utilizing the multi-physics field coupled technique,various coupled physical-chemical processes underlying the corrosion behavior are built into the model,including chemical reactions,ionic mass transfer in the bulk solution and the deposition layer,interfacial reaction,deposition of corrosion products as well as the morphological transitions caused by metal dissolution and deposition.In particular,the anisotropic deposit film is considered to be a porous layer with a porosity varying in time and space as the corrosion evolves.The predicted corrosion morphology by this model is better than the previous models.The coupled relationship between the electrochemical behavior(e.g.,electrode reaction kinetics,current density,surface potential)and the physical processes(e.g.,ionic transport,geometric evolution of metal surface and film interface)is revealed.The results indicate that a porous deposition layer with a denser inner layer and a loose outer layer is generated,leading to more significant inhibition of mass transfer in the inner layer than the outer layer.The anisotropism of the deposition layer results in a non-uniform conductivity distribution and a discontinuous current density distribution in the electrolyte.The current density on the electrode surface is inhibited by the deposition layer and the variation in the cathode/anode area ratio during the corrosion process.The competition between the transport process and the electrochemical reaction determines the spatial-temporal evolution of the ion concentration.? 2021 Chongqing University.Publishing services provided by Elsevier B.V.on behalf of KeAi Communications Co.Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/)Peer review under responsibility of Chongqing University

Keywords:Galvanic corrosion;Corrosion product film;Modeling;Mass transport;Porosity.

1.Introduction

Magnesium has been attracting widespread interest in a variety of technical applications for its unique physical and chemical properties,such as high strength to weight ratio,superb castability,good thermal conductivity,excellent electrical shielding effect,outstanding biocompatibility and non-toxicity to human body and environment,which lead to its great performance in designing lighter engineered systems,reducing energy consumption and developing viable biodegradable materials[1,2].However,its inherent defects like weak corrosion resistance considerably restrict the use of Mg alloys in some new fields[3].Galvanic corrosion,which could decrease service life and even cause catastrophic failures,is recognized as one of the most serious challenges in technical applications of magnesium[4].For the purpose of the reliable design and maintenance of metallic structures,it is of significance to predict the galvanic corrosion susceptibility and the damage evolution of magnesium.

Generally,the higher activeness of magnesium to almost all other structural metals[5]makes it vulnerable to severe galvanic corrosion when in contact with a different metal.This is due to the electrochemical coupling between these two metals with a significant potential difference in a common electrolyte.During the galvanic corrosion,magnesium act as an active metal supporting net anodic current,while a noble metal,such as aluminum,steel and zinc,supports net cathodic current.Meanwhile,this separation of interfacial current is balanced by the ionic currents in the electrolyte that are induced by the motion of charged species under the concentration and potential gradients.In addition,Mg(OH)2,a corrosion product during the reactions,would deposit on the metal surface to form a partially protective deposition layer,and in turn has a significant influence on the subsequent corrosion dynamics.Therefore,galvanic corrosion involves a number of coupled physical and chemical phenomena in different spatial and temporal scales,including electrochemical and chemical reactions,electro-migration,ionic diffusion,the formation of corrosion product layer,and the evolution of corrosion interface.The multiple spatial-temporal scales and dynamic interplay among the heterogeneous interface,transient potential and electrolyte concentration gradients aggravate the complexity of galvanic corrosion.To reveal the internal relationship among these underlying processes and then provide a mechanistic understanding of galvanic corrosion,extensive experimental work has been carried out[6]by using electrochemical techniques(e.g.,galvanostatic and galvanodynamic polarization[7–10]),local electrochemical techniques(e.g.,scanning vibrating electrode technique[11–13],scanning electrochemical microscopy[14–16]and atomic emission spectroelectrochemistry[17–19])and nonelectrochemical techniques(e.g.,weight loss measurements[20]and pH measurements[21]).These experimental studies could provide abundant valuable information that relates corrosion behavior to the solution composition,metal microstructure,surface chemistry and other variables.However,during the experimental research,there remain a number of limitations in the time scale,spatial scale and other aspects.In terms of the time scale,long-term exposure tests[22,23]are significantly time-consuming and may induce relatively inaccurate predictions;accelerated tests[24,25]can be faster,but the test conditions are far away from the practical application.In addition,the corrosion process is multi-scale in time,and the ionic transport is significantly quicker than the evolution of anodic surface,posing a challenge for the experimental work.With regard to the spatial scale,despite a great progress in local electrochemical tests,only spatially and temporally discrete data can be measured,and the details at a fine scale are still unable to be offered[3].In the experimental tests,it is difficult to obtain continuous real-time local electrochemical and chemical information under an irregular porous corrosion film at the micrometer scale.These information,being quite different from that in the bulk solution,is of great significance to link the corrosion behavior to multiple underlying sub-processes including ion transfer,film formation,interface movement and electron motion.Moreover,the severity of galvanic corrosion depends on various coupled factors varying in space and time,such as electrical condition,solution chemistry,corrosion morphology and film microstructure.Yet it is extremely difficult to decouple these factors and clearly distinguish the effect of each individual factor by experimental techniques.

With the development of computational material science,numerical simulation is emerging as a promising way to investigate galvanic corrosion.Firstly,numerical models can provide a powerful means to simulate the evolution of galvanic corrosion in various exposure environments with minimal cost of time and material,thereby expanding the research to longer time scales and larger spatial scales.Secondly,based on the electrochemical kinetics laws and the transport laws composed of diffusion and migration,numerical models can use physical analogs and mathematical equations to deliver the interface evolution and the time-dependent change of physicalchemical variant in the corrosion system.Therefore,the realtime dynamic information at any given location and time,including the evolution of corrosion morphology,potential,species concentration and current density,can be readily captured through numerical simulation.Finally,some complex and harsh service conditions that are difficult to perform in most laboratories can be numerically modeled easily.Numerical simulation provides greater freedom in the control of a wide variety of factors and can easily distinguish the effect of each individual factor.In general,mathematical models,as a valuable complement,can deepen our understanding of the hidden relationship between various parameters in the corrosion process,meanwhile revealing the evolution mechanism of galvanic corrosion in relation with the real-time electrochemical and chemical behavior.In the existing numerical research,there is a good agreement between numerical models and experimental measurements of various galvanic couples in acidic electrolytes and neutral NaCl electrolytes,demonstrating the reliability of these models.There are a number of numerical techniques available for corrosion studies in meso-or continuum scale:boundary element method(BEM)[26–29],finite difference method(FDM)[30,31],finite volume method(FVM)[32,33],peridynamics(PD)[34,35],finite element method(FEM)[36–39]and phase field(PF)method[40].Song et al.[26–28,41]conducted many meaningful numerical studies on the galvanic corrosion of one-dimensional systems based on BEM approach,proposing the theoretical expressions for galvanic potentials and currents and providing a useful insight into galvanic corrosion system.However,due to the neglect of two-dimensional influence,their practical application for predicting the corrosion evolution in the geometrically complex case was restricted.This problem could be overcome by using the finite element method to approximate the current and potential distributions.The finite element method has been widely used to study various physical and chemical phenomena involved in complex corroding systems.Hoche[42]built a 2D finite element model to simulate the corrosion behavior of Mg-Al galvanic couple,with geometrical aspects of the electrolyte introduced.Deshpande[43]established a time-dependent model based on FEM to simulate the galvanic corrosion of AE44-steel couples in neutral NaCl solution.A reasonable agreement was found between the current densities obtained by numerical model and scanning vibrating electrode tests.In addition,Lacroix et al.[44],Trinh et al.[45],Murer et al.[46]and Shi[47]also gave an extended insight into the topic and made a great contribution to the development of numerical simulation of galvanic corrosion.

Despite the current advances in numerical modeling,to the best knowledge of the authors,there remain some issues to be addressed.As is well known,the deposition of corrosion product plays an extremely important role in the evolution of corrosion.The deposition layer,like a physical barrier against the mass transfer of the involved species,can substantially affect the metal surface state,change electric behavior around the metal surface,and thus further impact the development of metal corrosion.Sun et al.[37]proposed a significant model to investigate the influence of the corrosion product precipitation,and assumed that the porosity of the deposition layer is uniform and constant.Based on this assumption,the steric hindrance effect of the deposition layer on the mass transport across the interface was constant during the corrosion evolution process.But actually the porosity of deposits is influenced by numerous factors including electro-chemical reactions,solution composition,deposit structure,ionic concentration field,current density distribution and temperature.Therefore,an anisotropic deposit film with different porosities in space forms on the metal surface and varies significantly as the corrosion propagates.This anisotropic property has a critical influence on the evolution of corrosion behavior.For instance,it may enable the species within the pores of different sizes in the anisotropic film to experience differential mass transfer and reactions,leading to a non-uniform species concentration distribution and thus different corrosion kinetics at different regions of the surface.In turn,it can lead to different growth dynamics of the deposition film as well as various solution chemistries and electrical conditions,which may control the overall corrosion process.This influence induced by the anisotropic film will be clearly revealed in this paper.Another significant challenge that needs to be considered is how to couple electrochemical processes(i.e.,electrode reaction kinetics,evolution of current density,surface potentials,etc.)into physical processes(i.e.,ionic transport,geometric evolution of metal surface and film interface,etc.).The electrochemical processes have a great influence on the evolution of metal surface and film interface,and the physical processes of ionic transport can be changed by the anisotropic film,thus cause the variation of solution conductivity distribution,and then greatly affect the electrochemical processes.The mathematical interrelationship between the time-dependent electrochemical behavior,heterogeneous reactions,mass transport and solution chemistry is essential for a deeper intrinsic understanding of the underlying mechanism,but it has yet to be completely investigated in existing studies.

Considering the anisotropic property of the deposition layer,this paper proposed a multi-physics coupled model to investigate the interaction between these multiple processes and then reveal the evolution mechanism of galvanic corrosion under this anisotropic influence.To comprehensively describe the deposition layer’s actual formation process and corrosion evolution from the perspective of physics,various coupled physical-chemical phenomena underlying the corrosion behavior were introduced into the model by utilizing the multi-physics field coupled technique,including chemical reactions,ionic mass transfer in the bulk solution and deposition layer,interfacial reaction kinetics and the deposition of corrosion products.Moreover,a smooth moving boundary technique was adopted to trace the time-dependent growth of the metal surface interface and the deposition layer.The effect of the anisotropic corrosion product layer on the spatialtemporal evolution of chemical and electrical environment in the complex corrosion system was revealed.The growth dynamics of corrosion product and corrosion morphology was clarified by the multi-physics coupled characteristics of the species concentration distribution,solution conductivity,transient electrical field,current distribution,interface evolution and the porosity and thickness of deposition layer.This paper can capture the physical-electrochemical changes of the corrosion process as the anisotropic deposition film grows,helps in understanding their intrinsic interrelationship,and thus provide an insightful analysis of the underlying mechanism of galvanic corrosion under the time-dependent anisotropic deposition layer.

2.Physical and mathematical model

2.1.Model system

This study focuses on the galvanic corrosion behavior of AE44-mild steel couple exposed to NaCl solution.The model system in this work is shown in Fig.1.The AE44 magnesium alloy acts as the anode to generate Mg2+ions,while the mild steel acts as the cathode on which the reduction reactions take place to generate OH?ions.The mass transfer of the involved ions is driven by the diffusion under concentration gradients and the electro-migration under the induced or spontaneous electric field.The generated Mg2+and OH?move to the bulk solution,and then react with each other to produce Mg(OH)2deposits.Moreover,the continuously growing deposits not only hinder the transport of Mg2+and OH?through the deposition layer,but also change the electric behavior near the metal surface.As for the AE44 magnesium alloy,anodic electrochemical reactions result in the metal dissolution and the sustaining interface evolution.Meanwhile,the species generated and consumed by various chemical/electrochemical reactions can also change the thickness and porosity of the Mg(OH)2deposits and cause the anisotropism of the deposition layer.These key processes are also integrated in this model.In conclusion,the galvanic corrosion of AE44-mild steel couple is an extremely complex phenomenon involving a series of interactional physical and electrochemical processes,such as mass transfer,chemical reactions,interface involution of the electrode surface and the corrosion product layer,as well as the electrochemical reactions at the metal/solution interface.In this work,these complex processes are coupled and incorporated into the proposed model through a multi-physics field coupled technique.

Fig.1.Model system for galvanic corrosion of Mg-steel couple.

Regarding to the geometry design of the computational domain,the length of both the cathode and anode is 10mm,and the origin of the used Cartesian coordinate system is at the junction between the anode and cathode.The height of the computational domain is set large enough to eliminate the disturbance of the electrolyte film thickness.The computational domain is discretized into 13,556 triangular finite element cells.The coarser elements are employed in the region of bulk electrolyte,while the finer elements with a maximum size of 1μm are adopted for the region adjacent to the electrode surface,where the gradients of potential and concentration would be the highest.Especially,the elements with the minimum size of 0.2μm appear near the junction.In addition,a satisfying independent mesh is found until the solution accuracy is unchanged with finer discretization.

2.2.Fully coupled multi-physics model with moving interfaces

The mathematical model is proposed based on the coupled mechanisms of concentration field,electrical field,corrosion interface evolution and the sustaining growth of anisotropic porous deposition layer.The model framework for this complex multi-physics coupled issue is illustrated in Fig.2.The mass transport is ascribed to diffusion and electro-migration.The electrical field has a significant effect on the concentration field by changing the electro-migration of charged species and altering the kinetics of electrochemical reactions,in which some species are consumed and produced.In turn,the concentration field of the involved species can also affect the electrical field due to the change in electroconductivity of the electrolyte.Moreover,the concentration field and the chemical reactions can also affect the growth of deposition layer,thereby changing its thickness and porosity,which will in turn affect the mass transport of ions.As well,the chemical reactions would play an important role in the concentration field and ionic mass transport due to the consumption and production of species in various chemical reactions.Furthermore,the interface evolution due to the metal dissolution and precipitation reaction results in the update of computational domain,and then puts forward the requirement of recalculation on all physics fields at next time step.Therefore,a complex interrelationship exists among these physical-chemical processes,which would be revealed through the multi-physics coupled model in this study.

In this work,the multi-physics coupled mathematical model consists of a mass transfer model estimating species distribution,an electricity model solving electrical field,and an interface evolution model tracking the movement of corrosion interface and simulating the time-dependent physical properties of the deposition layer.These three sub-models are solved simultaneously by using a fully-coupled approach for exploring the coupled physical and electrochemical phenomena in the corrosion system.Moreover,the model is able to predict the spatial-temporal evolution of corrosion morphology,current density,concentration field and the deposition layer’s physical properties including thickness,porosity and interface growth.In particular,the interrelationship between the anisotropic deposition layer and the dynamic corrosion behavior is established.

2.2.1.Mass transfer model

The mass transfer model is established to solve the concentration field of the involved species.Their mass fluxes are attributed to the species motion under the conjunct effects of concentration gradient and electric field,which separately induce diffusion and electro-migration sub-processes.In this study,the flux of speciesiis described by the Nernst-Planck equation[48]:

whereNi,Di,ciandzidenote the mass flux,diffusion coefficient,concentration and charge number of the specific speciesi,respectively.F,RandTare the Faraday constant(96,500 C·mol?1),universal gas constant(8.314 J·mol?1·K?1)and kelvin temperature(298K),respectively.φrepresents the electrostatic potential obtained by solving the electricity model,which will be described in detail in the following section.The first and second terms on the right-hand side of the Nernst-Planck equation denote the diffusion and electro-migration effects,respectively.

Fig.2.Framework of the multi-physics model for galvanic corrosion.

Subsequently,the mass conservation of each species can be expressed as follows[48]:

whereRirepresents the production/consumption rate of speciesidue to the chemical reactions in the electrolyte solution.εis the porosity of the deposition layer,as a means of mathematically permitting access of electrolyte into the interior of deposition layer,andε=1 represents the aqueous solution.εis a key physical characteristic that varies over time and space,which can be solved in the later section.

Moreover,the formation of Mg(OH)2deposits is inevitable when the concentrations of Mg2+and OH?exceed the solubility limit:

The consumption rate of Mg2+and OH?due to the deposition reaction is calculated by the following equations,respectively:

wherekis the rate constant of the precipitation reaction of Mg(OH)2,andkspis the apparent solubility product constant of Mg(OH)2[49].Sis the supersaturation of Mg(OH)2,defined as:

H(S)is a step function expressed as:

The boundary conditions for the mass transfer model are presented in Fig.3.At the electrode interface,the production or consumption of species results from the electrochemical reactions.Therefore,for the mass flux of speciesiparticipating in electrochemical reactions at the electrode interface,there exists the following constrain equation[50]:

wherendenotes the outward normal vector,vi,jis the stoichiometric coefficient of speciesiin reactionj,ijrepresents the current density contributed by electrochemical reactionj,andnjis the number of electrons involved in the electrochemical reactionj.

At the top boundaries,the species concentration should be equal to the bulk concentration,which is expressed as follows:

whereci,bis the bulk concentration of speciesi.

For other boundary,there exits the following constraints expression:

It should be stressed that the deposited film of Mg(OH)2has a significant influence on the corrosion process mainly by posing a significant mass transfer resistance to species.Thus,the diffusion coefficient should be modified to taken into account this influence.A Bruggeman relationship with coefficient of 1.5 is often used to describe the porous layer under the effect of porosity and tortuosity.As a result,the diffusion coefficient for species in this porous film,Di,f,can be estimated by the following equation[38]:

Fig.3.Boundary conditions for the multi-physics coupled model.

whereDiis the diffusion coefficient in film-free electrolyte,?is the film porosity,τis the film tortuosity indicating the interconnection between pores.

2.2.2.Electricity model

Generally,the potential distribution in an electrochemical system occurs on a much shorter time scale than mass transfer.For water system,any small separation of charge would generate a significant potential gradient which tends to rapidly restore the system to the state of electro-neutrality.Therefore,it has been acknowledged that the assumption of local electroneutrality is reasonable.It is a very good approximation and greatly simplifies the mathematical treatment of the system.However,this simplified method ignores the electrostatic interactions among ions.There is a concern that which ion should be selected as a“make-up”ion to enforce electroneutrality at each time step for each spatial element.Improper selection probably leads to a negative concentration,which is definitely unreasonable.Therefore,the Poisson’s equation,as given in Eq.(13),is introduced in this study to overcome these problems[51]:

whereEεis the dielectric constant of the electrolyte.

The boundary conditions for the electricity model are presented in Fig.3.At the anode interface,the boundary condition is given as[50]:

whereσis the electric conductivity of the electrolyte solution,andiarepresents the current density of the electrochemical reactions occurring on the anode interface.i0,ais the exchange current density,andbais the Tafel slope.φsis the corrosion potential,φlis the potential in the solution,andErevis the reversible potential.The conductivityσcan be calculated according to the following equation[51]:

wherezi,uiandciare the charge number,mobility and concentration of speciesi,respectively.

At the cathode interface,the boundary condition is given as[50]:

whereicrepresents the current density of the electrochemical reactions occurring on the cathode interface.These polarization parameters of electrochemical reactions are available in the work of Deshpande[43,52].

Guangling Song et al.[53]proposed that magnesium may be oxidized by the Fe3+ions produced by the cathode steel corrosion,with the Fe3+ions being reduced into iron which deposits on the magnesium surface,which result in a highly active surface and extremely low polarization resistance.The quantitative kinetic data of these electrochemical reactions under the galvanic corrosion environment are not sufficient,and thus this effect is not considered in this model but would be incorporated in the further improved model by obtaining more relevant electrode kinetics data.

At other boundaries,the electrical insulation is applied as follows:I

n addition,the deposited Mg(OH)2film will undoubtedly reduce the conductivity of electrolyte in this porous film by reducing the cross-section areas through which the conductive species can travel.To take into account this effect,the effective conductivity of electrolyte through the deposited film needs to be corrected,as shown in the following equation[38]:

2.2.3.Interface evolution model for electrode surface and deposition layer

The electrode interface evolves constantly due to the metal dissolution in the actual corrosion process.Meanwhile,the continuous deposition growth results in the evolution of the deposition layer interface.As a result,the interested calculation domain is changed inevitably,which has a significant effect on concentration field and corrosion process.In addition,the deposition layer’s physical characteristics(e.g.,thickness and porosity)vary significantly with time and space,and thus have a significant influence on the whole corrosion process.These key factors should be considered into the model for a realistic description and accurate prediction of the corrosion evolution process.It is crucial that the computational geometry and mesh is realigned timely with the new position of the moving interface.In this study,the arbitrary Lagrangian-Eulerian(ALE)method is introduced to track the time-dependent interface of the dissolved metal and deposition layer with a relatively less mesh distortion and a higher resolution.For the ALE method,the combination of the spatial and material coordinate systems are employed,respectively corresponding to Eulerian formulation and Lagrangian formulation.For the Eulerian formulation,the physics equations are formulated in the fixed coordinate system to feature the physical state at fixed points in space.In contrast,a coordinate system printed on a self-defined material zone/surface is employed in the Lagrangian formulation,to follow the material deformation,which exhibits a great capability in handling the moving boundaries.Therefore,the ALE method combines the advantages of both formulations.

As the metal corrosion proceeds,the mesh deformation becomes increasingly larger,and thus the mesh quality degrades obviously.To alleviate this problem,it is necessary to re-mesh the computational domain and map all the physical quantities from the old mesh to the new one.Therefore,the Laplacian smoothing method is incorporated into the numerical modeling to retain the mesh quality.For the transient case in this work,the mesh displacement is governed by the following equations[50]:

wherexandyare the spatial coordinates of the spatial frame,whileXandYare the reference coordinates of the material frame.

The boundary conditions for interface evolution model are schematically represented in Fig.3.At the anode interface,the interface evolution is determined by the metal dissolution process,which results in the variation of corrosion morphology.The normal deformation velocity of the anode surface can be calculated by the following equation based on Faraday’s Law[54].

whereVdisis the normal deformation velocity,nMgis the number of electrons involved in the anodic reaction,andMMgandρMgare the molecular weight and density of the dissolved metal,respectively.

As for the growing deposition layer,the interface velocity for is given as:

whereVdepis the normal deformation velocity caused by metal dissolution and deposition layer growth;MMg(OH)2andρMg(OH)2are the molecular weight and density of Mg(OH)2,respectively.The first and second terms on the right-hand side of this equation denote the velocities of anodic dissolution and deposit layer growth,respectively.

At the top boundary,the following expression is employed:

At other boundaries,the prescribed boundary displacement is given as:

As stated before,the formation of Mg(OH)2deposition on the metal surface is a time-dependent process,and the porosity of the deposits changes with time and space.Therefore,a key factor that should not be neglected is the spatial-temporal porosity,which is solved by the following governing equation[42],indicating that the precipitation of corrosion product is associated with the decrease of porosity:

3.Results and discussion

3.1.Model validation

In order to verify the numerical model in this study,a verification case is conducted under the same conditions in the experiment of Deshpande[52].For a rigorous comparison with the experimental results,the simulation time of one case in this work is also 3 days,and then the corresponding corrosion damage profile predicted by the current model is compared with that obtained from the experiments under the same immersion period(3 days),as shown in Fig.4.The corrosion damage profile calculated by the numerical model shows a predominantly reasonable agreement with the experimental data.The maximum corrosion damage is at the junction in the anodic region and then gradually diminishes beyond the junction.Under the same period of 3 days,compared

Fig.4.Comparison of the predicted corrosion morphology and experimental data.

with the maximum corrosion depth of nearly 2mm by the immersion experiments,the value of 1.745mm predicted by this model is closer to the experimental result than that predicted by Deshpande(1.6mm)[52]and Sun(1.71mm)[37].This is because that the influence of corrosion product deposition on the corrosion evolution was not considered in the model of Deshpande,and the anisotropism of the deposit layer was neglected in the model of Sun,although the effect of deposition was integrated in his model.By comparison,this model takes into account the anisotropism influence of the deposition layer by considering it as a porous film with a porosity varying over time and space as corrosion evolves.The above-mentioned comparison indicates that this model is reliable to explore the influence of the anisotropic deposit layer on corrosion evolution and make a more realistic description of the galvanic corrosion process from the perspective of physics.

3.2.Growth dynamics of corrosion product and corrosion morphology

For visual presentation of the growth dynamics of galvanic corrosion,the evolution of the corrosion interface and corrosion product interface are illustrated in Fig.5,with the red color area corresponding to the corrosion product layer.It is obvious that the corrosion depth increases significantly with the immersion time,and the anodic surface near the junction propagates more quickly with time than the rest anodic area.This is more obvious in Fig.6,which shows that the time evolution of corrosion depth is more pronounced near the junction than that beyond the junction.Moreover,the variation of corrosion depth slows down with the increase in time,especially in the area near the junction,reflecting a decreasing trend of the corrosion rate.All these phenomena are consistent with the spatial-temporal evolution of the corrosion rate along the anodic surface,as shown in Fig.7.Meanwhile,the reduction of corrosion rate with time is more significant near the junction than that beyond the junction,which is associated with the negative influence of deposition layer on the corrosion evolution.This will be explained in detail later.

Fig.5.Spatial and temporal changes of the corrosion interface and corrosion product interface:(a)24h,(b)36h,(c)48h,(d)60h.

Fig.6.Time evolution of the electrode interface.

Fig.7.Time evolution of the corrosion rate along the electrode surface.

Regarding to the time evolution of the corrosion product deposition,Fig.5 shows that the corrosion product deposits both on the anodic area and on the cathodic area due to the mass transport of the involved species in the electrolyte solution.It should be noted that the corrosion product layer on the anodic surface is thicker than that on the cathodic surface,due to the faster mass transfer of OH?(the product of cathodic reaction)than Mg2+(the product of anodic reaction)in the electrolyte solution.For a clear presentation of the time evolution of the corrosion product deposition,Fig.8 shows an obvious growth of corrosion product layer with the increase in time.Moreover,the thickness of the corrosion product layer reaches a maximum near the junction and then gradually decreases away from the junction.This phenomenon indicates the uneven deposition distribution along the electrode surface,which will be discussed in the later section in terms of the spatial-temporal evolution of electrical environment and concentration field.

Fig.8.Time evolution of the corrosion product interface along the electrode surface.

Fig.9 shows the time evolution of porosity distribution of the corrosion product layer,with the color scale representing the porosity of this deposition layer.The area with a porosity value of 1 indicates the aqueous solution without deposition.It can be seen that the lowest porosity occurs near the junction,due to the above-mentioned faster deposition process in this area.As time proceeds,the area with the lowest porosity expands significantly,implying the continuous deposition process.In addition,the corrosion product layer with an obvious porosity gradient appears near the junction,i.e.,a lower porosity at the electrode surface and a higher porosity at the outer layer,implying the porous deposition layer with a denser inner layer and a relatively loose outer layer.For a quantitative description of the time-dependent porosity distribution,the time evolution of the porosity along the electrode surface is shown in Fig.10.The porosity at the anodic surface near the junction is significantly smaller than that at the cathodic area,reflecting the relatively greater deposition rate at the anodic surface.This is consistent with the distribution of deposition layer thickness.Meanwhile,the porosity decreases significantly with time,i.e.,the deposition layer becomes increasingly compact,which implies that the protective effect of the deposition layer on the electrode surface becomes strong gradually,thus leading to the above-mentioned declining trend of the corrosion rate.In addition,the porosity reduction with time is much more pronounced near the junction,while this reduction is almost negligible beyond the junction.Therefore,the compact deposition layer mainly appears near the junction and its protective effect is more and more significant,resulting in an obvious decrease of the corrosion rate near the junction,as shown in Fig.7.

3.3.Effect of corrosion product layer on the spatial-temporal evolution of electrical environment

Due to the anisotropic porous structure of the corrosion product layer,the porous deposition layer has a significant influence on the spatial-temporal evolution of electrical environment,such as the conductivity,potential and current density,which are analyzed in this section.

The distribution of the time-dependent current density along the electrode surface is illustrated in Fig.11,in which the negative values represent the cathodic reduction current,while the positive values represent the anodic metal dissolution current.The maximum anodic current density appears near the junction and then decreases with the increase in time,indicating that the sustained growth of the deposition layer act as a protective barrier to inhibit the corrosion evolution process.This is consistent with the time evolution of corrosion rate.Another reason for the decreasing anodic current density is that the expanded anodic area with time results in a smaller cathode/anode area ratio,which is ignored in many investigations but should be emphasized.Moreover,a close inspection reveals that the change of anodic current density gradually decelerates with time.At the beginning,the appearance of a higher current density is because of a relatively weaker inhibition effect of the deposition layer,and then it causes the rapid expansion of the anodic area.Consequently,the rapidly enlarging anode/cathode area ration significantly reduces the anodic current density in the initial stage.As time goes,the current density becomes lower,and then the expansion of the anodic area slows down.As a result,the reduction of anodic current density with time is not as significant as that in the initial stage.In addition,the time evolution of the cathodic current density shows a similar pattern,whereas the reduction of the cathodic current density with time is obviously smaller than that of the anodic current density,especially at the junction.From Fig.10,it can be seen that the porosity at the cathodic area is greater than that at the anodic area,implying that the protective effect of the deposition layer at the cathodic surface is not very significant.Therefore,the reduction of the cathodic current density is not as significant as the anodic current density.

Fig.9.Distribution of the porosity of the corrosion product:(a)24h,(b)36h,(c)48h,(d)60h.

Fig.10.Time evolution of the porosity along the electrode surface.

Fig.11.Time evolution of the current density along the electrode surface.

Fig.12.Time evolution of the integrated anodic current along the anodic surface.

To quantitatively indicate the influence of time-dependent deposition layer on the total current on the entire anodic surface,the temporal evolution of the integrated anodic current along the anodic surface is shown in Fig.12.It can be seen that the integrated current increases initially,reaches a peak at about 20 h,and then gradually decreases with time.This phenomenon is determined by the competition between the expanding anodic surface and the inhibition effect of the deposition layer.As stated before,the growing deposition layer can inhibit the current density,whereas the enlarging electrode area with time leads to the increase of the integrated current.At the beginning,the thin and loose(high porosity)deposition layer has a negligible protective effect,and thus the expanding electrode area predominates in the competition and makes the integrated current increase with time.As time proceeds,the deposition layer becomes increasingly thick and compact,and forms an effective protective barrier to inhibit the corrosion current density.In this situation,the expansion of the electrode area cannot offset the inhibition effect of the growing deposition layer,resulting in a decreasing trend of integrated current with the increase of time.

The potential distribution is shown in Fig.13,with the color scale representing the value of the potential.It can be seen that the potential varies from?1.27V over the cathodic region to?1.43V over the anodic region after 60 h.A nonuniform potential distribution can be obviously observed.The arrows indicate the electric field induced from the lower potential region(anode)to the higher potential region(cathode).For a better quantitative illustration,Fig.14 shows the distribution of the time-dependent potential along the electrode surface.The potential varies rapidly near the junction,leading to a big potential gradient.The overpotential near the junction is also higher,as shown in Fig.15,providing a great driving force for the investigated galvanic corrosion.According to Eqs.(14)and(21),the overpotential along the electrode surface directly controls the interface current density and the interface evolution velocity,and thus the current density and the metal dissolution rate is the highest near the junction.In addition,the overpotential decreases significantly with time,representing that the driving force for the corrosion process becomes increasingly weaker.The reason is that the ion concentration decreases with time,due to the ion consumption in the formation of corrosion product and the inhibited ion generation by the protective deposition layer.This leads to a decrease in the electrolyte conductivity and a resultant increase in the ohmic potential drop,thereby decelerating the corrosion process.

Fig.13.Potential distribution in the entire domain after 60 h.

Fig.14.Time evolution of the potential along the electrode surface.

Fig.15.Time evolution of the overpotential along the anodic surface.

Fig.16.Distribution of solution conductivity(a)along the electrode surface(b)in the whole domain.

Fig.16(a)shows the time evolution of the solution conductivity distribution along the electrode surface,which is similar to the time-dependent porosity distribution.This also implies that the deposition layer has an effect on the solution conductivity.The mass transport of ions by diffusion and electromigration is largely inhibited by the compact deposition layer,resulting in a significant reduction of the local electrolyte conductivity.The minimum local electrolyte conductivity is 0.98 S/m after 12 h,and then this value drops to 0.04 S/m after 24 h,eventually it reaches the critical value of 0.00025 S/m after 36 h.Meanwhile,as shown in Fig.16(b),the anisotropism of the deposition layer also results in a non-uniform conductivity distribution,which is also similar to the porosity distribution,implying the influence of the porous structure of deposition layer.

The time evolution of the current density distribution in the electrolyte is shown in Fig.17,with the color scale representing the value of the current density.As a consequence of the non-uniform electrolyte conductivity distribution,the current density distribution in the electrolyte is discontinuous,especially near the junction,and the absolute value of the current density also decreases significantly with the increase of time.

3.4.Effect of corrosion product layer on the spatial-temporal evolution of chemical environment

The corrosion product layer can retard the mass transport of the involved ions and inhibit the electrochemical reactions generating a large number of ions.Accordingly,the chemical environment is changed significantly.Thus,the influence of deposition layer on the spatial-temporal evolution of chemical environment is discussed in detail in this part.Fig.18 shows the time evolution of the OH?concentration distribution.As expected,higher OH?concentration appears near the cathodic area,due to the generation of OH?by the reduction reaction on the cathodic surface.Meanwhile,the OH?concentration in the cathodic region decreases gradually with time,indicating the inhibition effect of the deposition layer.For a better illustration,Fig.19 shows the plot of time-dependent OH?concentration along the cathodic surface.It can be seen that the OH?concentration away from the junction decreases significantly with the increase in time.In contrast,this phenomenon is absent near the junction.As time proceeds,the deposition layer grows continuously and thus reduces the current density of cathodic reactions,thereby decelerating the generation rate of OH?on the cathodic surface.However,it cannot be ignored that the dense deposition layer can also retard the outward transport of the generated OH?.This outward transport competes with the generation of OH?,resulting in the change of OH?concentration.At the cathodic region away from the junction,the loose deposition layer with a relatively small thickness and high porosity cannot have a pronounced hindering effect on the ion transport.Thus,the outward transport of the generated OH?is almost unaffected,and then the reduction of OH?concentration beyond the junction is mainly caused by the reduced generation rate of OH?with time.In contrast,the thicker and denser deposition layer near the junction has a pronounced hindering effect on the ion transport and leads to a lower outward transport rate.In this situation,the lower outward transport rate of OH?and the lower generation rate competes,resulting in that the change of OH?concentration near the junction is not in a single direction.Therefore,it can be concluded that the competition between the transport process and the electrochemical reaction influenced by the deposition layer determines the spatial-temporal evolution of the ion concentration.

The time evolution of the Mg2+concentration distribution is shown in Fig.20.As expected,higher Mg2+concentration appears near the anodic area,due to the generation of Mg2+from the Mg dissolution on the anodic surface.For a quantitative illustration,the plot of the time-dependent Mg2+concentration along the anodic surface is shown in Fig.21.It can be seen that the Mg2+concentration near the junction decreases gradually with the increase of time.In contrast,in the region away from the junction,the Mg2+concentration increases gradually with time.This phenomenon is also caused by the competition between the transport process and the electrochemical reaction affected by the deposition layer.The generation rate of Mg2+is determined by the current density of anodic electrochemical reaction,which decreases significantly with time near the junction,due to the inhibition effect of the growing deposition layer.The decreasing Mg2+concentration near the junction indicates that the inhibition effect on the generation rate outweighs the hindering effect on the mass transport.However,the reduction in the anodic current density with time is not obvious away from the junction(shown in Fig.11),which leads to a negligible change of the generation rate of Mg2+.In this situation,the inhibition of the outward transport of Mg2+by the growing deposition layer play a dominant role,leading to a reduced outward transport of Mg2+.Therefore,the Mg2+concentration away from the junction increases gradually with time.

Fig.17.Distribution of the current density in the electrolyte:(a)24h,(b)36h,(c)48h,(d)60h.

Fig.18.Spatial distribution of the OH?concentration:(a)24h,(b)36h,(c)48h,(d)60h.

Fig.19.Time evolution of the OH?concentration along the electrode surface.

Fig.20.Distribution of the Mg2+concentration in the whole domain.

Fig.21.Time evolution of the Mg2+concentration along the electrode surface.

Fig.22 shows the time evolution of the OH?concentration and the porosity along the normal direction from the electrode surface to the electrolyte solution atL=1mm,where theLdenotes the distance to the junction in x axis direction,whileyis the distance to the electrode surface inyaxis direction.As shown in Fig.22(a),the OH?concentration decreases as the distance to the electrode surface increases,and then stabilizes,which implies there is a resistance against outward mass transport within the growing deposition layer.It should be noted that at 48 h and 60 h,the OH?concentration firstly experiences a rapid decrease and then a steady decrease as the distance to the electrode surface increases.In contrast,the decrease rate of OH?concentration with distance is always steady at 12,24 and 36 h.Combined with Fig.22(b),a compact film can be defined as the deposition layer with a porosity less than 0.6,while a loose film is defined as the deposition layer with a porosity more than 0.6.At 48 h and 60 h,the porosity of the deposition layer is less than 0.6 in the inner layer and more than 0.6 in the outer layer,representing a porous deposition layer with a denser inner layer and a relatively looser outer layer.The denser inner layer has a significant inhibition influence on the ion transport and then induces a sharp change of the ion concentration.In this situation,the OH?concentration decreases rapidly in the inner layer and then decreases steadily in the outer layer.In contrast,at 12,24 and 36 h,the porosity is always more than 0.6 in the entire deposition layer,indicating that the deposition layer is entirely loose.In this situation,the OH?concentration decreases steadily in the whole layer.

To reveal this influence at different locations along the electrode surface,Fig.23 shows the OH?concentration and the porosity along the normal direction from the electrode surface to the electrolyte solution at various L values.It can be seen that atL=0.5mm and 1.0mm,the porosity of the deposition layer is less than 0.6 in the inner layer and more than 0.6 in the outer layer,i.e.,a porous deposition layer is formed with a denser inner layer and a looser outer layer.This leads to a plunge of the OH?concentration in the inner layer and a steady decrease with the increase in distance.AtL=1.5mm,the region of the deposition layer with a porosity less than 0.6 is so small that the OH?concentration decreases sharply in the thin inner layer,and then decreases steadily as the distance increases.AtL=2.0mm,the whole deposition layer with a porosity more than 0.6 steadily reduce the OH?concentration in the whole deposition layer as the distance increases.

In summary,when the AE44 Mg alloy is in contact with mild steel,they can form a galvanic corrosion couple due to the difference in the corrosion potential,in which the AE44 Mg alloy with lower corrosion potential acts as an anode and correspondingly suffers serious galvanic corrosion damage.The anodic reaction causes the dissolution of Mg and the production of Mg2+,while the cathodic reaction can generate the OH?.These charged species,driven by the concentration and potential gradients,inevitably move in the electrolyte.As a result of the encountering of OH?and Mg2+,meanwhile the electrolyte around the metal surface becomes more alkaline than the bulk solution,a corrosion product Mg(OH)2would deposit on the metal surface to form a partially protective deposition layer[55].This deposition layer,like a physical barrier against the mass transfer of the involved species,can substantially affect the concentration field,change the chemical-electrochemical processes around the metal surface,and thus further impact the development of metal corrosion.It should be noted that the porosity is one of the important physical properties of the deposition layer,which is influenced by numerous factors including electro-chemical reactions,solution composition,deposit structure,ionic concentration field,and current density distribution.Therefore,an anisotropic deposit film with different porosities in space exists and varies significantly as the corrosion propagates.In the initial stage of galvanic corrosion,the porosity of the deposition layer is high,implying that the entire deposition layer is relatively loose,which results in a steady decrease of the OH?concentration in the whole layer.As time proceeds,a porous deposition layer with a denser inner layer and a relatively looser outer layer is formed,which results in a dramatic decrease in OH?concentration in the inner layer and a steady decrease in the outer layer.Meanwhile,the anisotropism of the deposition layer results in a non-uniform distribution of electrolyte conductivity and electrolyte current density.Moreover,as the deposition layer becomes denser and thicker with time,the solution conductivity decreases gradually due to the restricted mass transfer and the inhibited ion generation,thus leading to a decreasing current density.In addition,the deposition layer is the thickest and densest near the junction due to the ions motion and current density,which results in an obvious decrease of the corrosion rate with time near the junction,and a slight decrease beyond the junction.

Fig.22.Distribution of(a)OH?concentration(b)porosity along the outward direction normal to the electrode surface.

Fig.23.Distribution of(a)OH?concentration(b)porosity along the outward direction normal to the electrode surface at different locations.

4.Conclusions

In this work,a multi-p hysics model is proposed to systematically investigate the galvanic corrosion of Mg alloys under the influence of anisotropic deposition layer.This model can capture the spatial-temporal evolution of corrosion interface,deposition thickness and porosity,species concentration field and current profiles during the growth of the anisotropic deposition layer.The reliability of this model is validated by the correspondingly experimental data.The influence of the continuously growing anisotropic deposition layer on the temporal-spatial evolution of the corrosion behavior is clarified.The growth dynamic of galvanic corrosion behavior is revealed in terms of the multi-physics coupled characteristics of the involved processes.Based on this work,the main conclusions can be drawn as follows:

(1)A porous deposition layer with a denser inner layer and a relatively looser outer layer can be observed.The deposition layer becomes increasingly denser as corrosion evolves,and then it is the thickest and densest near the junction.The uneven physical characteristics of the deposition layer results in an obvious decrease of the corrosion rate with time near the junction,and an almost negligible decrease beyond the junction.

(2)The anisotropism of the deposition layer results in a non-uniform distribution of electrolyte conductivity and electrolyte current density.Moreover,as the deposition layer becomes denser and thicker with time,the solution conductivity decreases gradually due to the restricted mass transfer and the inhibited ion generation,thus leading to a decreasing current density.The deposition layer with a wide range of thickness and porosity in space lead to the different variation trend of the current density on the electrode surface.The current density on the electrode surface is not only inhibited by the growing deposition layer,but also is affected by the change of cathode/anode area ratio caused by the expanded anodic area with time.These two factors combine to result in a nonlinear change of the current density with time.

(3)The competition between the transport process and the electrochemical reaction influenced by the deposition layer determines the spatial-temporal evolution of the ion concentration.In the initial stage of galvanic corrosion,the high porosity of the deposition layer means that the entire deposition layer is relatively loose,which results in a steady decrease of the OH?concentration in the whole layer.In contrast,as time proceeds,a porous deposition layer with a denser inner layer and a relatively looser outer layer is formed,which results in a dramatic decrease in OH?concentration in the inner layer and a steady decrease in the outer layer.This interesting phenomenon is also true for the concentration profiles across the deposition layer at different locations on the electrode surface.

Acknowledgments

This work is supported by the National Natural Science Foundation of China(Grant no.51906200),the Key Project of National Natural Science Foundation of China(Grant no.51839010),the Key Laboratory Foundation of Education Department of Shaanxi(Grant no.19JS045)and the China Postdoctoral Science Foundation(No.2019TQ0248;No.2019M663735).