Towards multiscale modelling of localised corrosion

Continuum models: rudimentary description of microstructures

Several early single-scale (continuum) corrosion models (Sharland ⁶² ) and many recent works (e.g., Refs. 29, 63 and 64) did not consider the microstructure. However, the modellers who did, assumed the microstructure as a given. The modellers who followed the deterministic path (e.g., Refs. 65–67) have defined the microstructure either in their computational grids ^65,66 or treated them as a regular array and used analytical solution methods. ⁶⁷ The modellers who work in the stochastic realm have used probability density functions (PDFs) (e.g., Ref. 68) to account for the probabilistic nature of the features. Brown and Barnard ^65,66 incorporated the microstructure that defines a certain electrochemical property (e.g., different Tafel dissolution kinetics for individual phases) for each finite difference method (FDM) computational cell. Their latter model ⁶⁵ could predict the distribution of cathodic regions. Jakab et al. ⁶⁷ also developed a simplified deterministic model by treating a heterogeneous AA2024-T3 electrode as a regular array of Cu-rich favoured cathodic sites (partially covered with an inactive aluminium oxide layer) in a benign Al matrix. Furthermore, Zhang et al. ⁶⁸ developed a stochastic model for the same alloy AA2024-T3, which, according to the authors, provided a new approach to the prediction and quantification of localised corrosion kinetics based on the alloy microstructure. Their PDFs for grain dimensions had parameters that were fitted based on the observed three-dimensional grain sizes, and a ‘brick wall model’ was used to model the grains (and the inter-granular regions) in 3D. There are also empirical models that correlate the corrosion consequences with the microstructural aspects. For example, Cavanaugh et al. ⁶⁹ developed an empirical relationship between the accumulated corrosion damage in AA7075-T651 and the physical and electrochemical characteristics of the intermetallic particles, and this model predicted the pit radii (assuming hemispherical pitting) as a function of the immersion time in the 0⋅1 M NaCl. However, such models have limited applicability and cannot be relied upon for any experimental conditions other than those for which they were developed. In summary, the traditional continuum models that incorporate the microstructure have assumed it as a given quantity and have described it in relatively simplistic terms. No work appears to have considered details such as the recently observed ⁷⁰ non-random clustering of buried intermetallic particles in AA2024.

Multiscale models for solidification and prediction of service life: early versions are promising

Microstructure evolution during solidification

The formation of microstructure originates at the atomic scale, where the initial nucleation and the growth of critical nuclei occur. ¹⁰ This formation is followed by the growth of microstructure at the intermediate mesoscopic scale. Rafii-Tabar and Chirazi ¹⁰ have reviewed several deterministic, stochastic, hybrid models and some early MSMs that predict the microstructure evolution; they have also developed a well-documented, validated, generic MSM that spanned the nano-meso-continuum scales for microstructure formation (Fig. 3).

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3.

Multiscale model (MSM) for solidification by Rafii-Tabar and Cherazi ¹⁰

Their MSM is representative of the limited number of available MSMs for microstructure prediction that can be linked to an MSM for localised corrosion. In the Rafii-Tabar and Chirazi model (Fig. 3), MD simulation techniques were used at the atomic scale and linked to a cellular automaton (CA)-based mesoscopic model for microstructures. The Rappaz model ⁷¹ that was used for the microstructure combined a stochastic approach to the nucleation of grains (which was implemented using CA) with a deterministic, diffusion-controlled approach to their growth. The parameters that were used in the Rappaz model were calculated using MD simulations. Finally, the continuum scale simulations were performed based on finite element (FE) and finite volume (FV) techniques. The models at different scales were executed independently and coupled through databases using a serial framework (Fig. 2e ). The procedure was to associate an MD simulation box with a CA volume to couple the nano-meso models, followed by associating a CA volume with a finite volume to couple the meso-macro models. Because only one-way coupling is possible in a serial framework, and because the three scale models were run separately using different codes, it was necessary to run the meso and macro models more than once to achieve the two-way coupling between the scales through iteration. For example, in its first run, the meso model generated the inputs for the macro model, and in the second run, which was performed after the macro model had predicted the temperature evolutions, it predicted the microstructure-based on the temperature histories and the cooling rates.

Recent advances

As the Rafii-Tabar and Chiraz model ¹⁰ applies in its current form only to binary alloys, the important effect of the alloying elements on the corrosion morphology ⁵⁰ cannot yet be modelled. Therefore, the more recently developed and validated MSM models ^74–76 are more attractive for the corrosion modellers who work with engineering alloys, although these models span only the meso to continuum length scales and the associated temporal scales. These latter models are useful because they are capable of handling both ternary ^74,76 and quaternary ⁷⁵ alloys that are industrially more relevant. These 3D models can also account for the influence and the interaction of multiple phases (solid/liquid/gas) during the microstructure development. In addition, some of the latter models (e.g., Wang et al. ⁷⁵ ) have further enhanced their realistic nature by accounting for the effects of the cooling rates and the alloy content in their prediction of the microstructural features that included the shape, the size and the distribution of the intermetallics and the defects such as porosity. The main output of these models is typically a suggested deterministic microstructure. The information flow between the two levels in the Wang et al. ⁷⁵ model was facilitated through the coupling of temperature and pressure variables (see Ref. 76). In addition, the lower-scale model was implemented as a subroutine of the macro model, which was solved using FEM in an embedded multiscale framework (Fig. 2). The mesoscale sub-region formed a total volume fraction of approximately 0⋅001% of the domain volume that was covered by the continuum scale model, which allowed for the mesh sizes in the meso model to be on the order of micro metres. In summary, the microstructure-evolution related MSMs treated the nucleation events stochastically (e.g., with a pre-set nuclei density and a nucleation potential with a Gaussian distribution PDF ⁷⁵ ) and modelled the growth events deterministically using a hybrid strategy. Thus, there is no purely deterministic microstructural model. Regarding MSMs for linking the microstructure to the prediction of service life, validated works ^72,73 by Olson’s teams appear to be the most pre-eminent, and their strategies may be used as a basis to link the microstructure to the in-service corrosion performance in MSMs.

Proposed improvements

Because the existing models on corrosion treat microstructure as a given quantity or have a simplistic description of microstructure, the ability of an MSM (that incorporate such models) to tailor materials is limited. An MSM on microstructure prediction should ideally be linked to an MSM on localised corrosion, so that through a feedback loop, the alloy system may be optimised for corrosion performance. Furthermore, none of the existing models account for the formation of the oxide layer (the Passivating oxide film, its thickness, and its porous nature section below), and they do not provide other corrosion-related information such as the interfacial energies at the grain boundaries and the grain mismatch, which are indicators ⁷⁷ of the propensity of various sites to corrode in some cases. Therefore, in the long term, it is desirable to have the microstructure model also predict the composition and the morphology of the oxide film. Recently, promising efforts at atomistically simulating the precipitation kinetics in the solidification of multicomponent interstitial/substitutional alloys ⁷⁸ have been published. It is also desirable to predict the surface roughness characteristics. However, in the short term, the prescription of the entire final microstructure as a given quantity remains a convenient starting point.

Continuum models: lack resolution

Atomistic models and paucity of data at the atomic scale

The continuum assumption of a uniform passivating layer for the complex heterogeneous structure of the oxide film is quite inadequate. In addition, as noted by Williams et al., ⁹⁹ although the continuum models for migration and accumulation of point defects in the film are appropriate to describe the behaviour of high-purity single-phase metals, they are inefficient for the engineering alloys. Thus, atomic scale models are required. Maurice and Marcus ⁷ reviewed some of the atomistic models and highlighted their usefulness in helping one understand the mechanisms that occur at the lower scales. The ab initio model of Bouzoubaa et al. ¹⁸ predicted that the surface undulations on the oxide film played a major role in its chloride thinning mechanism. Kim et al. ¹⁰⁰ used DFT to understand the origins of the natively n-type characteristics of ZnO by modelling the interaction of a Zn interstitial with an oxygen vacancy. The 3D atomistic model of Diawara et al. ¹⁰¹ successfully modelled the selective dissolution and the passivation of Fe–Cr alloys by simulating the formation of oxide nuclei from chromium-rich clusters on the surface. This model confirmed the experimental observation that Cr preferentially diffused towards the Cr clusters on the surface, whereas iron atoms showed no such preferential diffusion. Their kMC model also showed that passivation occurred within a matter of seconds. The ab initio modelling work by Costa et al. ²⁰ on SSs studied the effect of water coverages in the Cr₂O₃ film on the surface, and the energies of adsorption that it predicted were successfully validated with experimental data. Similarly, Hendy et al. ²¹ calculated the energy barriers that were associated with the transport of cations through the oxide film by combining ab initio simulations with experimental observations at the atomic scale. These investigations reinforce the value in modelling the film at the atomic scale. However, in this connection, Maurice and Marcus ⁷ noted that the development of such models will be limited by the complexity involved in considering the three phases (alloy substrate/oxide/electrolyte), their interfaces, the electric field, the temperature, the orientation of the oxide film, its nanostructural defects and the surface defects. The issue of heterogeneous passive films on engineering alloys makes it critically important to select what is modelled. The examples above are limited in terms of computational load when attempting to model engineering alloys even with a reasonably low concentration (in relation to the number of atoms that can be considered in the ab initio methods) of second-phase particles. This problem is even more challenging when pit initiation is considered. Current ab initio models are rather weak at modelling the defects that exist at second-phase particles in a matrix because the required number of atoms is too high.

The relative scarcity of models at the atomic scale is unsurprising given the computational burden that we discussed above and because the experimentalists could not observe the oxide film in detail at an atomistic level ^28,79 until recently. Nevertheless, there is an increasing amount of literature (e.g., Refs. 79, 99, 102 and 103) that addresses such measurements, slowly filling the present gap in knowledge at such length scales.

Proposed improvements

The oxide film is a classic example of the need for a multiscale approach to corrosion modelling because its atomic scale thickness is coupled with its mesoscopic-scale microstructure, which is punctuated with continuum scale defects. Thus, either a three-scale approach or a less comprehensive two-scale platform (meso/continuum scales) may be chosen to model the film. However, in the latter approach, a generic phenomenological model that adequately describes, in the meso or continuum scales, those mechanisms ^104,105 that actually occur in the nanoscale through derived transport and kinetic parameters will be considered mandatory. The advanced version of a prospective MSM should have the capacity to predict the oxide composition and structure (both geometric and electronic), including its thickness and its porosity distribution, based on a given microstructure of the substrate and the electrochemical environment. Of course, this hypothesis assumes that sufficient experimental data become available for validation in the short to medium term. For a deterministic model to have a strong predictive capability in relation to the electrochemical behaviour of the oxide film, the following must be undertaken: ¹⁰⁶ (a) the kinetic and the transport parameters that are pertinent to different metallic constituents of the film must be determined either based on the available values or using advanced experimental techniques, and (b) the energy heterogeneity of the transport medium must be considered using the distribution functions for the diffusion coefficients of individual species in the compact film and by quantifying the role of the grain boundaries in the transfer of matter and charge through such a film. The model should also account for the interaction between the film and the environment using kinetic data (and thermodynamic data) on the adsorption, the surface complex formation and the re-precipitation reactions of various cations at different pH levels and locations of the oxide. It is also desirable that the atomistic modelling of nano-scale events such as the oxide penetration/thinning mechanism (e.g., Refs. 18 and 28), which leads to the prediction of nucleation sites. In addition, some of the free parameters required for the phenomenological models may be determined from first principles through such simulations where theories are available, rather than being fitted empirically. However, given the enormous effort and time required to achieve all of the above, the early versions of an MSM are likely to incorporate approximations for many of the aforementioned aspects.

Continuum models: lack of universal applicability

Traditional models ^{25,47,62,109–114} have either neglected the influence of pH or Cl⁻ or incorporated their influence in a rudimentary manner. Sharland ⁶² has reviewed some of these models. More recently, as noted by Frankel and Sridhar, ¹¹⁵ a point defect model (PDM) ¹¹⁶ incorporating empirically determined parameters correctly predicted the logarithmic dependency of pit initiation on Cl⁻ concentration, although it could not calculate the often observed bi-logarithmic dependence. The Cl⁻ influence was integrated in the PDM by hypothesising that its concentration altered the generation and the transport of cation vacancies. A simplified PDM for iron, ¹¹⁷ which was based on the high field model (HFM) formulation, also predicted the correct qualitative trend for the dependence of the film thickness on the pH of the electrolyte. Nevertheless, neither PDM nor HFM can be universally applied (they are only valid for crystalline materials) and are limited by the assumptions (e.g., homogeneous oxide) and the need to fit the parameters empirically for each scenario. Lastly, the effect of the inhibitors has not been tackled by continuum models as much as the present authors are aware, which is understandable given the patent lack of resolution of such models in accounting for the surface/liquid interactions at the molecular level. In summary, traditional approaches such as Pourbaix diagrams are inadequate, and the continuum models lack universal applicability and resolution in describing non-homogeneous microstructure. Thus, although some traditional models may have to be used in early versions of an MSM, a truly generic advanced MSM will require more detailed modelling at the lower scales.

Atomistic models: promising, but currently at early stages

The atomistic models of more recent origin have shown promise in this area by incorporating the influence of ions in the aqueous solution from a more fundamental point of view. Molecular or atomistic modelling brings to multiscale modelling a unique tool that can discriminate behaviour on the molecular scale ^118,119 and allow a molecular design. Consider the development of organic corrosion inhibitors. Their performance is strongly controlled by bonding (electron sharing) and the type of pendant functional groups that are attached to aromatic rings. However, the exact molecular structure has a critical impact on the inhibition performance; because conventionally, inhibition is measured by exposure of metal plates in inhibited solution or by electrochemical tests, a direct correlation to the inhibitor’s efficiency does not exist because of the multiple scales that are involved. Hence, incorporating molecular modelling into a multiscale modelling framework offers the possibility of forming a continuous link from the molecular structure to the inhibition of anodic and cathodic activity on the surface, which inhibits corrosion.

Bridging the nano-gap

The design of inhibitors must span the scales from 10⁻¹⁰ to 10⁻¹ m (the scale of a test plate) and cover a wide range of phenomena (see Fig. 4). The properties derived from DFT in vacuum studies may not be relevant to the inhibitor surface-binding and the link between the molecular properties and the mesoscale phenomena such as anode/cathode development, pit initiation, and pit growth is not evident. Thus, a large gap currently exists between the DFT models and the measured inhibitor efficiency in terms of both scale and phenomena. This gap is often ¹³³ covered using a pattern recognition or a neural network approach such as the quantitative structure activity relationships (QSAR). In the near future, the DFT models will likely accurately include the solvation effects and model the competitive reactions (i.e., oxygen reduction v. inhibitor absorption), and local electrochemical techniques such as local EIS can be used to develop refined local parameters and their spatial variation (e.g., coating resistance or diffusional properties), whereas the post-test analytical procedures can examine the structure of the protective layers on the metal surface (i.e., using X-ray photoelectron spectroscopy (XPS), etc.) down to the sub-micron scale. Two additional developments can further close this gap. The DFT studies can be linked to MD studies, which will allow an order-of-magnitude expansion on the molecular scale. ¹²⁵ Lastly, most electrochemical formulations are based on the Butler–Volmer equation, which provides an average free-energy formulation of charge transfer but does not examine the individual processes that are involved in such transfer. Recent phenomenological models ¹⁴⁶ break down the charge transfer in the solution into a number of components and allow each component to be addressed, which effectively refines the electrochemical scale.

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‘Nano-gap’ between density functional theory (DFT) models and measured inhibitor efficiency

At this point, a brief note about the use of ex situ techniques such as XPS is appropriate. In such techniques, the specimens are removed from the experimental solution and placed in vacuum for examination. Such inspection under vacuum may significantly alter some aspect of the surface; for example, dehydration can occur, and the physio-sorption between the inhibitor and the surface will not be maintained, although chemical bonding and chemo sorption in particular will be. Thus, such techniques must be used with care, but they can provide valuable information.

Proposed improvements

Based on atomic modelling, a more fundamental approach can be used for both crystalline oxides ²¹ and the amorphous variety ¹⁴⁷ and is well suited to describe the solid/oxide bonding ¹⁴⁸ and oxide/electrolyte interactions. ^17,18,141 This approach is likely the preferred option for a design-optimisation MSM because of the fine resolution that it provides. There is currently no such exhaustive and generic model, and until one becomes available, empirical correlations that are based on mesoscopic and continuum scale models must be used to build an MSM. In the intervening period, however, it may be possible to explore the use of KSDs to model the influence of pH and aggressive ions at the higher mesoscopic and continuum scales. The current DFT work addresses the accuracy of surface-binding studies and can also examine the competition between inhibitor binding and the cathodic reactions. Hence, the objective of correlating DFT or MD ¹⁴⁹ studies with a coarse parameter such as the experimental inhibition efficiency may have to be modified in the light of data from electrochemical and analytical techniques such as XPS, which probes the surface. Lastly, most electrochemical kinetic models are based on the Butler–Volmer equation, which provides an average free-energy formulation of charge transfer but does not examine the individual processes that are involved in such transfer. The proposed MSM should also address the commonly used but inaccurate assumption that uniform conditions exist in the bulk solution; for example, this assumption often neglects the concentration gradients that extend into the bulk solution from a pit or a crevice.

Continuum models: contain limiting assumptions

The continuum philosophy considers that while the potential difference on the corroding surface is zero, there are anodic partial currents across the metal–electrolyte interface at microscopic surface sites, the total of which is balanced by an equal value of the total cathodic current that is similarly distributed under free corrosion conditions. This balance is achieved at the open circuit potential (OCP) of the ‘homogeneous’ metal substrate, and is determined by the rates of the partial electrochemical reactions ¹⁵⁹ (which are described by either the Tafel or the Butler–Volmer equations). This information is sufficient for modelling the O₂ reduction-driven cathodic tendency only when the influence of the microscopic inclusions and the alloying elements are ignored. There is a limited number of continuum works (e.g., Ref. 160) that address the anode–cathode separation, and they are rudimentary. A model that automatically predicts the anode–cathode separation by solving the mixed potential problem was developed by Venkatraman et al. ⁶⁴ for corrosion under a differentially aerated NaCl droplet, which was deposited on a Zn substrate. The surface was assumed to be homogeneous; thus, no microstructural details were considered. The previously visited model of Brown and Barnard ⁶⁵ could predict the cathodic regions. More recently, researchers have started to address the microstructural inhomogeneity-driven phenomena that are relevant to the anode–cathode separation in alloys, e.g., the previously discussed model of Jakab et al. ⁶⁷ (the Microstructure of the metal substrate: its formation and influence section). In addition, Alodan ¹⁶¹ simulated O₂ reduction using an approximated alloy-like microstructure and employed circular discs for the cathodes (inclusions), which were surrounded by an annulus-like region for the anode (aluminium). Although these latter models represent a step forward in the simulation of engineering alloys, they still suffer from the over-simplification of the microstructure. Vautrin-Ul et al. ^31,162 developed a CA-based mesoscopic model with a random walk of H⁺ and OH⁻ ions, where the spatial separation between the anodic and the cathodic sites could be made smaller than that in the continuum scale; thus, the model could more realistically locate discrete cathodic and anodic sites as interwoven within a propagating pit (i.e., not only on the side walls) based on the solution pH at the mesoscopic vicinity. This model was unlike most continuum models, where the cathodic sites are usually located outside the anodic pit or crevice or simply assumed to be along the side walls of a pit. Although it constituted a step up from the continuum scale models in terms of details, the model still lacked the resolution to handle the microstructural details of engineering alloys and was limited by the approximations that described atomistic phenomena at the mesoscopic scale. Therefore, lower-scale models must be developed for the anode–cathode separation; any continuum treatment that assumes homogeneity in the microstructure is inadequate for design purposes.

Paucity in atomistic models and experimental data

Atomistic models were difficult to find for this topic. However, because characterisation studies (e.g., Ref. 50) have started to generate relevant kinetic data in the form of polarisation diagrams for different alloy phases at the mesoscopic level, such models may soon be developed. An alternative that does not depend on experimental data is an atomistic model that is constructed bottom-up from first principles. No such model appears to exist in the current open domain for metallic corrosion, except a model that is a part of a recent MSM by Yu ¹¹ for glass corrosion. This situation is likely to change with the generation of physico-chemical data at the atomistic level (e.g., energies for surface formation, chemisorption, and adsorption) and the increasing application of high-resolution characterisation techniques for corrosion-related studies (e.g., Refs. 79 and 102). Advancing observation techniques such as the in situ atomic scale studies that were reported by Magnussen et al., ¹⁶³ where the local removal/addition of atoms at atomic kinks at the steps of Cu crystal surfaces during dissolution in 0⋅01 M HCl were observed and quantified, also serve to help accumulate knowledge at lower length scales. However, until a sufficient amount of such data exists, the microstructure-based empirical rules (e.g., for dissolution rates, such as the combination of a chemical rate law (first-order law based on [H⁺]) and an electrochemical law (Butler–Volmer equation) as used by Suter et al. ⁴⁹ for MnS dissolution in NaCl) may be necessary. These rules should also quantify the effects such as the probable catalysis by [Cl⁻] of the MnS dissolution, which was proposed by Williams et al., ¹⁶⁴ and any possible influences of the oxide films.

Proposed improvements

One of the reasons for the lack of atomistic models might be the difficulty inherent in coupling the electron transfer (current flow) between atomistic models of anode and cathode. The problem arises because, at the atomistic level, calculations are typically performed for equilibrium configurations whereas the formation of anodes and cathodes are highly dynamic events. It is suggested therefore that the potential use of fluctuation theorems ¹⁶⁵ be considered for simulating the non-equilibrium mechanisms. The central premise of these theorems revolves around comparing the probability of phase-space trajectories of the system with that of the anti-trajectories (one that the system would traverse if it were moving in the negative time direction), thus introducing the concept of time-irreversibility into the continuum processes. Thus a non-reversible, non-equilibrium process such as corrosion may potentially be modelled using fluctuation theorems at the mesoscopic level, facilitating the prediction of the evolution of favourable phase-space trajectories. Fluctuation theorems are discussed in more detail later, in the Candidates for future MSM and mathematical coupling of scales section. Early versions of an MSM are likely to have varying degrees of approximations to simulate the anode–cathode separation. One of the available avenues for the development of such models is perhaps the extension of models like that of Diawara et al., ¹⁰¹ discussed earlier in the Influence of aggressive ions (pH and Cl-51) and role of inhibitors section , and that of Legrand et al. ¹⁶⁶ These models simulate the selective dissolution of Fe and the passivation of a cluster of Cr atoms (which simulates the Cr₂O₃ layer) in Fe–Cr alloys. Another pathway is available from the previously noted work of Yu, ¹¹ who simulated the corrosion of glass. This MSM had an atomistic model that was coupled to an MC simulation of surface phenomena such as hydroxylation with cations, chemisorptions, adsorption and dissolution. Then, the continuum scale phase field simulations were performed using material properties that were fed from separate MD simulations. Although the MSM contained approximations, educated guesses and at times relied upon empirically obtained parameters, its framework provides a potential platform for the development of an MSM for localised corrosion.

Continuum models: mainly stochastic

Because the triggers for the initiation of metastable pits and their repassivation or transition to the stable pits were poorly understood deterministically, a stochastic model that was proposed by Williams et al. ³⁶ based on a treatment by Shibata ¹⁸⁰ has been frequently used. Experimental data analysis is required to fit this model to a particular system. ³⁶ However, this model simplistically assumed that each micro pit had an equal chance of propagating into a macro pit, which disregarded the role played by the microstructure, the local microenvironment and any aspect of the pit itself, e.g., the narrowness of the mouth or the existence of a pit cover. Nevertheless, it is interesting that the likelihood of interactions between the metastable pits, which were recently observed using in situ techniques at the micrometre scale, ^170,181 were actually predicted by a stochastic model by Wu et al., ³⁷ which pre-dated the observations. By assuming through a ‘memory effect function’ that each pitting event will influence subsequent events and the influence would exponentially decay with time, that models could successfully reproduce the current transients that were recorded during the metastable pitting of an SS alloy and an aluminium alloy (however, it must be noted that the current transients may also be caused by events such as trenching or de-alloying). This temporal model was subsequently extended to a spatiotemporal version by Organ et al. ¹⁸² However, the model remained limited by the assumption that the surface was homogeneous without any preferred sites for nucleation because the microstructural effects were not modelled. A model developed by Walton et al. ¹⁸³ (which is discussed later in the Stable pit initiation section) appears to be one of the first model that accounted for the active/passive transition based on the potentials. The CA models ^184,185 that addressed metastable pits concluded that their growth is controlled by the anodic dissolution probability. Other models of note are by Malki and Baroux ^186,187 and Hoerle. ¹⁸⁸ In summary, the continuum models that accounted for metastable pitting were stochastic in nature and did not consider the influences of microstructure or other factors such as ions (the Cl⁻ ions were neglected). They also did not account for the non-random nature of some features such as the intermetallic particles in some aluminium alloys, which were found in clusters instead of being arbitrarily distributed. ⁷⁰

Proposed improvements

The present authors did not find any atomistic model that addressed metastable pitting. The common drawback among the existing continuum models is that none of these consider the microstructural effects despite their significant influence. ¹⁸⁹ Therefore, an MSM should comprise a mesoscale model that adequately describes the microstructural features, which affect the metastable pitting, including the mechanism of active/passive transition if possible. An MSM should also be able to handle the differences in the behaviour of the current transients between systems, such as SS and aluminium (see Ref. 190), and such events as cooperative spreading. ^169,170 In addition, because the pit nucleation events occur at the nanoscopic scale, ¹⁰ an MSM that incorporates metastable pitting should preferably include a model at this length scale, but a stochastic approach may be used for the initiation phenomena until such model is available.

Conventional theories: insufficient mechanistic detail

Two conventional theories encapsulate the traditional deterministic criteria: (a) critical solution chemistry theory (CSCT) or critical crevice solution theory (CCST) ^194–196 and (b) Ohmic resistance-based IR drop theory (IRDT). ^63,197 According to CSCT/CCST, passive film breakdown occurs when the pH and the Cl⁻ concentration reach a critical state. However, according to IRDT, localised corrosion starts abruptly when the electric potential drop (IR) between the mouth and the interior of a pit or crevice is large enough to activate anodic potentials. Shortcomings of IRDT have been discussed by Frankel and Sridhar ¹¹⁵ and others (e.g., Refs. 198 and 199). Although CSCT/CCST has been supported overwhelmingly in the past, a recent attempt ⁶³ has been made to develop and validate a model that unifies CSCT/CCST and IRDT. However, all of these theories are for pure metals and ignore the seminal role played by microstructural inhomogeneities (including surface roughness) on pitting initiation on engineering alloys. For example, neither of these theories include the preferential dissolution of select alloy phases as a cause of pit initiation.

Continuum models: multiple approaches

As mechanisms relevant to pit initiation take place on the atomic scale, the continuum treatment involves significant approximations and assumptions. In addition, the absence of a consensus view on pit initiation is reflected in the variety of approaches that have been used to model pit nucleation. Many of these studies have been reviewed by Sharland, ⁶² Frankel, ^115,172 Kennell et al., ⁶³ Papavinasam ²⁰⁰ and Anderko, ¹⁵ but a small sample of this research is discussed here. Anderko ¹⁵ reviewed several theories for passivity breakdown and concluded that all of them share a common theoretical result: the passivity breakdown potential varies with the logarithm of the concentration of aggressive ions, which is confirmed by experimental data. He also notes that, while this observation is accepted widely, its generalisation to systems with multiple types of aggressive and inhibitive ions is not obvious. Given the poorly understood nature of pit initiation mechanisms, it was convenient to use a stochastic approach to model pit nucleation (e.g., Refs. 38 and 34–37). Hybrid models (discussed in the Single scale models section) typically take a stochastic approach to model pit initiation (e.g., Refs. 46–48). These formulations calculate quantities such as pit birth rates based on probabilities or assumed statistical distributions rather than physically viable mechanisms. A common feature of stochastic models is that physical mechanisms (e.g., for passive film breakdown and pit initiation and growth) and microstructure are not included explicitly. These considerations limit the applicability of stochastic pit initiation models for alloy design. The pre-eminent deterministic models are the 1D steady-state models by Galvele ^194–196 , the proponent of CSCT/CCST, which influenced deterministic thinking for decades by providing a step change in the ability to rationalise experimental results in various systems. However, as Newman ²⁰¹ notes, the Galvele approach needs minor revisions to include highly concentrated metal salt solutions in pit nuclei. Other significant deterministic models are those of Alkire and Siitari, ²⁰² Sharland, ²⁰³ Laycock and Newman, ²⁰⁴ Cong et al. ²⁰⁵ and Walton et al. ¹⁸³ The transient 1D model by Walton et al. was one of the first that considered the electrode kinetics of both cathodic and anodic reactions with active/passive transitions. An electrode kinetic model developed and validated by Mccafferty ¹¹³ took into account the adsorption of chloride ions on aluminium oxide surfaces, the penetration of chloride ions through oxide films, and the localised dissolution of aluminium at the metal/oxide interface in consecutive one-electron transfer reactions. The 1D pseudo-steady state model of Webb et al. ¹⁹² modelled the influence of MnS inclusions on SS surfaces. However, all these continuum models suffer from the shortcomings discussed in the Conventional theories: insufficient mechanistic detail section.

Atomistic models: showing promise

Decades ago, Williams et al. ²⁰⁶ attempted to develop an atomistic model of pit initiation on random binary Fe–Cr alloys. They added a scheme for passivity breakdown based on CCST to existing atomistic models ^207,208 by postulating that passivity breakdown corresponds to the critical chemistry necessary for the activation of the alloy. Their model identified the most important factors for pit initiation on SS: (a) the dissolution probability of Cr atoms; (b) the alloy composition, which determines the cluster size distribution; and (c) diffusion and migration within the cluster volume. A more recent atomistic work by Bouzoubaa et al. ¹⁸ has proposed tenable mechanisms of passivity breakdown. Currently, experimental observations are being made on the atomistic scale: e.g., the recent observations of aluminium oxide by Zavadil et al. ¹⁰⁵ Rashkeev et al. ²⁰⁹ have performed first-principles quantum-mechanical calculations to provide an atomistic understanding of corrosion initiation on Al under atmospheric conditions. Their results suggest that atomic hydrogen penetrates oxide films and causes structural damage in oxides and at Al/Al₂O₃ interfaces. To summarise, atomistic modelling increases the understanding of pit nucleation by modelling mechanisms on their characteristic scales. Therefore, atomistic models are most suitable for modelling pit initiation.

Proposed improvements

Nucleation events are best modelled on atomic scales, and appropriate models have started to become available. These models, however, are still in their infancy and only explore probable mechanisms for pit nucleation, which remains a controversial topic. More investigations, like the XPS studies and MD simulations of amorphous Al₂O₃ by Chang et al., ²¹⁰ may illuminate the relationship between oxide structures and passivity breakdown. First principle quantum-mechanical calculations such as those carried out by Rashkeev et al. ²⁰⁹ and Scully et al. ¹⁷¹ may provide an alternative to experimentation until better techniques become available for observing pit initiation events. However, until a deterministic model is developed for pit initiation, MSMs on localised corrosion may use stochastic models. It is necessary to prescribe the microstructure to apply the early deterministic models to the selective dissolution of inclusions/second-phase particles at rates based either on empirical polarisation curves (e.g., Ref. 50) or dissolution models (e.g., Refs. 25, 49 and 211). The effect of microstructural roughness can be incorporated by drawing from, for instance, the empirical relationship between surface roughness and pitting potential (e.g., Ref. 212). If continuum deterministic approaches are adopted until models on mesoscopic scales are available, other approximations may be required, for example, Olson et al. ²¹³ They considered that surface energy must affect pit formation and treat pitting corrosion as a nucleation and growth process where surface energy promotes an activation barrier.

Continuum models: the only option available

Because the propagation mechanism is better understood than the nucleation mechanism, more continuum models are available for pit or crevice propagation. Most models, including hybrid versions, follow the deterministic path for propagation. However, a number of stochastic models also exist. Sharland, ⁶² Turnbull, ²¹⁴ Frankel, ¹⁷² Scheiner and Hellmich, ¹⁹⁹ Kennell et al., ⁶³ Papavinasam ²⁰⁰ and Macdonald and Engelhardt ²¹⁵ have reviewed traditional models and formulations for propagation-related mechanisms, and only a selection of these models are reviewed here from a multiscale modelling perspective. Stochastic models typically calculate expected values of quantities such as the induction time or the number of stable pits as functions of defined parameters such as the nucleation frequency, the survival probability and the critical age. Then, through the interpretation of measured current transients, values for the parameters are retrieved. Stochastic models of Williams et al., ³⁶ Valor et al. ³⁵ and Alamilla and Sosa ³³ are noteworthy examples. While none of these models explicitly consider the microstructure, the previously mentioned stochastic model for inter-granular corrosion developed by Zhang et al. ⁶⁸ (the Continuum models: rudimentary description of microstructures section) utilised a brick wall model with a rectangular 2D geometry for aluminium grains. This rather simplistic description appears to be the closest stochastic approach published for modelling alloy microstructure. To summarise, stochastic models have not explicitly accounted for critical factors. Rather, stochastic models have relied on empirical parameters that were used subsequently to estimate various quantities associated with propagation. Deterministic models invariably followed either the CSCT/CCST or the RDT approaches. Often, early models (e.g., Refs. 160, 194–196, 202, 216, and 217) were developed for idealised 1D geometries, supported only the steady state, and contained several simplifying assumptions. Sharland and co-workers ^30,203,218 and Engelhardt et al. ²¹⁹ developed some noteworthy models, but Walton et al. ¹⁸³ appear to be the first to develop and validate a transient model applicable to a wide variety of metals and electrolytes and supported by different kinetic rate equations. Laycock and White ¹⁹⁸ developed a 2D FEM model that successfully recreated the lacy covers observed on pits on SSs. The deterministic model of Turnbull et al. ⁴⁸ contained statistically distributed parameters for the pit growth equation. When Laycock and White applied the previously developed propagation models ¹⁹⁸ in their hybrid version, ⁴⁷ they accounted for different SS alloy compositions by changing the OCP for each alloy and for surface roughness by altering the distance of the initial spherical pit beneath the metal surface in the Tafel equation for anodic dissolution. Additionally, the deterministic propagation equation in the hybrid model of Engelhardt and Macdonald ⁴⁶ for manganese steel in CO₂-acidified seawater and Al in tap water contains empirically fitted parameters rather than values obtained from physical mechanisms. Such empirical modelling has been carried out for Al alloy AA7075-T651 using neural networks by Cavanaugh et al. ⁴¹ Investigating two orientations of metal exposed to varying temperature, pH and [Cl⁻], those workers concluded that pit growth generally followed t ^1/3 kinetics. To summarise, most models used the 1D approach to predict pit size and transfer processes, ²¹⁵ and existing continuum models of pit propagation have not explicitly considered the effects of causal parameters such as alloy microstructure, oxide films or surface roughness on propagation rates. Rather, these parameters have been incorporated using quantities such as the OCP or the exchange current density (in the Butler–Volmer equation, for instance) that were obtained empirically. The lack of generality makes it difficult to use such models in an MSM for designing alloys or corrosion-resistant systems.

Proposed improvements

There are very few atomistic models of pit propagation in the literature. The probable reasons for this situation are (a) the continuum models have tackled this phenomenon reasonably well on the continuum scale and/or (b) the computational cost does not justify applying atomistic simulations to phenomena on mesoscopic scales or continuum expanses. However, it is likely that atomistic methods will be used increasingly to simulate propagation and extend the boundaries of knowledge. There is a need for advancement in the fundamental theoretical understanding of electrode kinetics before reliable atomistic models can be developed. For instance, Macdonald and Engelhardt ²¹⁵ recently noted that while Tafel constants may be calculated ab initio, the exchange current density is almost always measured experimentally because theory is not sufficiently developed to calculate this quantity from first principles. This is because Tafel constants contain relatively little kinetic information and the transfer coefficient is usually assumed to be 0⋅5 corresponding to a presumed symmetric barrier, whereas the exchange current density is derived from a highly kinetic and inherently complex set of mechanisms involving solvent reorganisation, which are still poorly understood. ^220–222 Once an understanding of the charge transfer reactions at the electrode/electrolyte interface and related mechanisms is established on the atomic scale, however, rate equations such as Butler–Volmer or Tafel with empirically determined parameters to describe boundary conditions at the continuum scale will unnecessary. Recent phenomenological models ¹⁴⁶ analyse charge transfer in solution in terms of a number of components and effectively refine the electrochemical scale. Lower-scale models will manage the difficulties with mesh refinement on finer scales to better describe, for instance, small-scale material removal or large concentration gradients near the electrode/electrolyte interface under diffusion control. In a nutshell, pit propagation can be modelled reasonably well on mesoscopic scales in the absence of a lower-scale model. In this case, the incorporation of microstructural details at the electrode/electrolyte interface should be considered in the medium term. However, because the atomistic description gives the best resolution, efforts should be made to incorporate atomistic-scale models in advanced versions of the MSM. Development of such models may remain a challenge until issues with computational loads are resolved through hardware and/or software improvements. Finally, the potential use of mesh-free methods for pit propagation should also be explored because these methods are well suited for modelling moving discontinuities.

Continuum models: purely deterministic

Stochastic modelling of pitting is not commonplace. Sharland ⁶² reviewed some of the early studies and concluded that no single model predicted (qualitatively or quantitatively) all the available experimental observations. The Walton model ¹⁸³ predicted concentration profiles more accurately than previous versions. Evitts ²²⁷ carried out an extensive review of both steady-state and transient models that dealt with changes in chemistry, including the studies by Sharland ^30,203,218 and Walton, ¹⁸³ and concluded that most models assumed isothermal conditions and constant bulk solution chemistry. Macdonald and Engelhardt ²¹⁵ noted that most models are 1D and erroneously neglect the potential drop outside the corrosion cavity, leading to significant errors in the prediction of damage. The White et al. ²²⁸ model relied on the Nernst–Planck equation rather than the electro-neutrality condition for the calculation of potential differences and predicted concentration profiles for an SS system. Laycock and White ¹⁹⁸ calculated the local chemistry in an SS pit and accounted for several relevant criteria including a moving boundary due to propagation. More recently, Kennell et al. ⁶³ developed a combined CCST/CSCT-IRDT model that provided the best agreement yet with the experiments of Alavi and Cottis, ²²⁹ surpassing the models of Sharland, ²¹⁸ Walton, ¹⁸³ White ²²⁸ and Evitts, ²²⁷ and introduced the possibility that such a combined approach may be superior to the CCST/CSCT-only path. Taxen and Persson ²³⁰ considered concentration changes due to the evaporation of aqueous electrolyte and used a moving mesh, but the application of the dilute solution theory was discontinued beyond a certain loss of volume. Finally, Heppner et al., ²³¹ noting that most models assumed a dilute solution, carried out theoretical calculations for non-ideal solutions where ionic interactions could no longer be neglected. Their work, which used the Pitzer model, ²³² simulated crevice corrosion evolution including species concentration changes and delivered better predictions than those models that did not account for ionic interactions. Thus, the use of the Pitzer model in the MSM should be explored, especially for non-ideal solutions. Moreover, in actual solutions, the activities of species can change not only with temperature but also with pH values. ¹⁵ Even for dilute solutions, traditional models relied on estimating activity coefficients from theoretical models (see Refs. 15, 183, and 233), and advanced modelling on this front should be considered. Another common feature in traditional modelling was the lack of data on chemical reaction kinetics. Therefore, several assumptions were made. For instance, Sharland and Tasker ²¹⁸ assumed that the chemical reactions occurred at very high rates compared with other phenomena such as diffusion. While many such assumptions may be justified for continuum models, they are invalid for atomic scale models. To conclude, the existing models that describe chemical changes in a pit all belong to the continuum scale and thus lack detail on the mesoscopic scale for some relevant events discussed below.

Atomistic models: may be required for non-dilute solutions

Lower scale atomistic models that describe changes in solution chemistry are rare in the corrosion domain. This may be because the chemical changes are largely a response to other events rather than being causal events themselves. This means that, if events at electrolyte–electrode and electrolyte–air boundaries were modelled with sufficient accuracy and the important chemical reactions were properly accounted for, the chemical changes in the pit would be adequately modelled at the continuum scale without the need for a computationally intensive atomistic treatment. Nevertheless, as the theoretical work of Heppner et al. ²³¹ showed, ionic interactions may become significant in non-dilute solutions where dilute solution theory would break down. Such cases might benefit from atomistic-level computations, which can provide the necessary resolution to model the interactions in more detail. For instance, in the multiscale framework, an atomistic-scale model can be used to calculate the empirical parameters (e.g., the activity coefficients and the osmotic coefficient) for the Pitzer model ²³² for deployment at a higher scale, although these parameters have been experimentally derived for some systems. ²³⁴ Furthermore, the following events may be better modelled at a scale lower than the continuum: (a) the influences of alloy microstructure on local chemistry (e.g., through space-dependent dissolution) or vice versa (e.g., the local chemistry can preferentially attack metallurgical defects); (b) the passive film or its influence on the chemistry; (c) the microenvironment defined by the microscopic surface roughness; or (d) interactions between pit sites. Such detail is desirable in an MSM that will be used in a predictive role in alloy design. Molecular dynamics simulations of non-ideal solutions have been performed, ^235,236 and perhaps similar studies could be used to explore the concentrated solutions that develop in pits.

Proposed improvements

In non-dilute solutions, phenomena such as ionic interactions within the electrolyte may not be neglected and ideally are modelled on the atomic scale to arrive at an optimum generic approach that could be used widely. Another area that needs to be strengthened is the quantification of reaction rate constants for some reactions. Although these constants could be obtained experimentally for some reactions, other extremely rapid reactions present difficulties in the determination of rates because reliable experimental measurements cannot be made easily using traditional methods. ²³⁷ In the absence of dependable measurements, the application of theories such as ²³⁸ the collision theory or transition state theory may be attempted for estimating reaction rates, and subsequently rate constants (see Ref. 239). Alternatively, these may be obtained from the application of first principles (e.g., Ref. 19). Similarly, it appears that no current document has a comprehensive listing of all relevant parameters such as those above and others such as dissociation constants and solubility products that a modeller could conveniently access, although some handbooks contain samples of these. Therefore, there are opportunities for developing atomistic models capable of estimating these quantities.

Continuum models: severely lacking in detail, but the only type available

As with the changes in pit chemistry (the Changes in electrolyte chemistry in pits section), only deterministic models are relevant for describing precipitation. Models that account for precipitation include those of Gravano and Galvele, ¹⁹⁶ Sharland and Tasker, ²¹⁸ Sharland, ²⁰³ Walton et al. ¹⁸³ Laycock et al. ²⁵ and Laycock and White. ¹⁹⁸ These models provide different mathematical treatments of precipitation. For instance, Gravano and Galvele ¹⁹⁶ treated precipitating solids as colloidal diffusing species in their work that focussed on solution chemistry immediately after film breakdown. The 1D transient model constructed by Sharland ²⁰³ was one of the first models to predict the evolution of solid phases (corrosion products) as a function of time and space in addition to solution chemistry and electric potential. All the above models assumed that the precipitation of the metal hydroxide occurred when its concentration reached the saturation level in the solution; Laycock and White, ¹⁹⁸ however, allowed up to 10% super-saturation to occur owing to their additional assumption that a hydroxide salt film could precipitate only on an existing salt film or on the corroding metal. A slightly different approach was followed by Taxen and Persson, ²³⁰ who assumed that precipitation occurred when the activity coefficient for any solid species in solution exceeded unity. Another aspect of traditional modelling has been the limited use of data for rates of precipitation by many models (e.g., Ref. 218) based on the assumption that rates of precipitation are much higher than other events such as diffusion. It was necessary to make these assumptions in the traditional models due to the general lack of data on chemical kinetics and rates of precipitation, ²¹⁸ which has remained the case to this day. ²¹⁵ Some workers such as Farrow et al. ¹¹⁰ worked around this issue by using theory ²⁴¹ to estimate the rates. Some works such as those by Nesic and Lee, ²⁴² Sun and Nesic, ²⁴³ Anderko ¹⁵ and Taxen and Persson ²³⁰ dealt with the subject in the sphere of general, rather than localised, corrosion. Also worthy of mention is the work of Tidblad et al. ²⁴⁴ who used the well-known GILDES model ²⁴¹ for atmospheric corrosion to predict the formation of corrosion products on copper substrates exposed to aqueous species. The GILDES model solves the mathematically formulated transformations and transitions that occur in the six relevant ‘regimes’: the gas phase, the interface between a gas and a liquid, the liquid phase, the deposition layer, the electrode region near the surface and the solid phase. With the exception of studies such as that of Farrow et al. ¹¹⁰ and the GILDES model, ²⁴¹ none of these models have accounted for the well-known influence of surface pH on precipitation by simulating the dissolution of precipitate layers in certain pH ranges: so these models were limited to pH ranges where the products were stable. None of these models also took into account the effect of microstructure and were unable to provide the details required to model mixed corrosion products, such as different oxides of alloy components or their spatial distribution that depends on microstructural features besides alloy composition. Additionally, none of these models explicitly accounted for the passive films. Because the electrode was assumed to be homogeneous on the macroscopic scale, the available models do not contain the details that a prospective microscopic or lower-scale model would be capable of furnishing. Thus, the available models need fine-tuning from the perspective of incorporating precipitation algorithms into an MSM.

Proposed improvements

The present authors did not find any atomistic-scale models that dealt with corrosion products, although, like oxides films (the Passivating oxide film, its thickness and its porous nature section), the usually ultrathin and porous nature of precipitate layers make them classic examples of why a multiscale approach should be taken for modelling localised corrosion. The atomistic scale provides the ideal framework for modelling transport at not only the interfaces between the products and other components such as the passive film or the metal itself but also through the products themselves. This is also the most suitable method for modelling interactions between the electrolyte and the products including adsorption. In addition, bonding at interfaces may be modelled elegantly using an atomistic approach. Thus, there is scope for building models of corrosion products at the lowest scales in the medium to long term. These models should take into account the microstructural inhomogeneity of the metal substrate and the oxide when predicting the space-dependent structure and thickness of the precipitate layers. It may be possible to develop atomistic precipitation models for corrosion based on precipitation kinetics-related models that are becoming available in other fields, e.g., solidification-atomistic ⁷⁸ and hybrid atomistic-kMC. ²⁴⁵ Additionally, the inverse of precipitation (dissolution) of salt films has been studied recently using MD, ²⁴⁶ and this approach may be examined for developing a basis for precipitation models of corrosion.

Continuum models: inadequate

Traditionally, conditions under which repassivation occurs have been determined by the use of Pourbaix diagrams. ²⁴⁷ However, this phenomenon has been addressed by a handful of empirical and continuum mathematical models as engineering situations deviate significantly from the equilibrium-based scenarios. Examples of empirical models include that of Song et al. ²⁵¹ However, as the empirical models do not explicitly incorporate the influence of, for instance, the environment or material properties, they cannot be utilised as generic tools. Stochastic modelling works have attempted to incorporate the repassivation phenomenon using probabilities. For example, in the model by Williams ³⁶ an assumption regarding the probability that a metastable pit will repassivate was made. However, these stochastic models, like their empirical counterparts, do not explicitly account for mechanisms leading to repassivation. Among deterministic models, the Walton et al. model ¹⁸³ appears to be the first that is capable of simulating repassivation, as it supported the active/passive transition. Laycock et al. ²⁵ assumed that repassivation would occur if the diffusion of metal ions away from the electrode exceeds their production through dissolution, in which case the ion concentration tends to zero. Anderko et al. ^252,253 developed mechanistic models based on statistical thermodynamics. Their models calculated the repassivation potential for selected SS and Ni-base alloys whilst accounting for the influence of solution chemistry (aggressive species, oxyanions and inhibitors), temperature, competitive dissolution, adsorption and oxide formation. In particular, the models predicted the transition from the concentration range where localised corrosion was favoured to the region where inhibition was expected. Some of the parameters in the Anderko models ^252,253 were experimentally obtained. A PDM ^26,104,215 was capable of explaining the experimental observations of Ahn et al. ²⁵⁴ for the repassivation kinetics of Ti. However, as noted by Wu and Celis, ²⁵⁵ HFM is the most widely used explanation of repassivation kinetics, although its validity has been questioned by some. While some experimental works (e.g., Refs. 255–257) have supported the application of HFM, Wu and Celis ²⁵⁵ caution that HFM may not be physically realistic at very short times following depassivation. To summarise, while continuum scale models exist, they are inadequate for modelling repassivation because it is known that lower scale factors influence this phenomenon. For instance, second-phase particles or inclusions in an engineering alloy would exhibit electrochemical potentials different from that of the matrix, and can preferentially passivate (or dissolve). Such resolution at the microscopic level is mandatory for an MSM aimed at alloy design.

Atomistic models and experimental data at this scale are scarce

Using MC simulations, Malki and Baroux ¹⁸⁵ determined that varying the repassivation probability on the atomic scale on the pit walls can control pit growth kinetics (t ⁿ law), but the authors admitted that there was no experimental evidence to suggest that this unexpected prediction was tenable. This model appeared to be the only one in the open literature that dealt with repassivation, and the lack of models at this scale is probably explained by the scarcity of experimental data on this scale. However, atomistic simulations have been performed for the oxidation of metal surfaces, which is a critical step in repassivation, e.g., DFT work by Schröder. ²⁵⁸ Perhaps strategies found in these works can be applied to develop a model for repassivation.

Proposed improvements

More experimental observations on the atomic scale are required to understand the mechanisms that lead to repassivation in situations that do not involve the preferential dissolution of phases or inclusions. It therefore stands to reason that atomistic-scale models will be the most appropriate means to describe such phenomena. However, the existing models that account for repassivation have all been developed on the macroscopic scale. Additionally, no existing model has taken into account the role of the microstructure, for instance in the apparent repassivation of pits formed by selective dissolution. It is also necessary to determine through experimentation whether the presence of different phases or other microstructural features in the local neighbourhood would have any influence on whether a pit repassivates. Such influences may be modelled at the microscopic level and will enrich an MSM designed for alloy design.

Continuum models: hybrid treatment

Only a handful of existing models address this topic, and this is probably due to (a) the lack of experimental data and associated understanding, which is only just beginning to filter into the open literature, and/or (b) the probable perception that such interactions may only be of secondary importance. Available models generally take a hybrid approach treating nucleation and interaction events stochastically but employing deterministic equations for the influence of the environment. One of the first models that accounted for interactions was the previously mentioned stochastic model of Wu et al., ³⁷ in which a memory parameter, M, was incorporated to retain the influence of a given event over time. Lunt et al. ²⁶⁰ extended this model and their simulation results, which agreed with experimental data obtained by the same workers on an array of SS electrodes, showed that surface damage gave the highest M value. Organ et al. ¹⁸² extended the Lunt et al. model to 2D, and this model suggested that interactions among metastable pitting events can lead to the formation of clusters of pits. White et al. ²⁶¹ developed an FEM model incorporating interactions as an extension of a previously discussed model by Laycock et al. ⁴⁷ Their model confirmed that an existing pit would increase the probability of pits nucleating at nearby sites. Harlow and Wei ²⁶² developed a stochastic model incorporating interactions for pit growth in Al alloys. Although not validated with experiments, the authors mentioned that its predictions were in qualitative agreement with observations. Significantly, the authors advocated a deterministic approach for a more realistic analysis of interactions. A mechanism of competitive interaction between pits at the early stage of their development was considered by Popov, ²⁵⁹ who assumed that the interaction occurred due to the hydration of metal ions by the solvent. Popov derived analytical equations and proposed a 1D model of interaction between two linearly connected pits. However, this model was not validated with experimental data.

Proposed improvements

Only a limited number of models have addressed the phenomenon of interactions between pits. All of these models were on the continuum scale, and the electrode surface was modelled as homogeneous, without preferred nucleation sites. Therefore, a stochastic approach was followed for nucleation events, and interactions were modelled by deterministic equations that incorporated the influence of the environment on interactions. Clearly, the assumption of homogeneity reduces the detail required in a model for alloy design. Therefore, a model at the mesoscopic scale should be developed for describing the interactions on their characteristic scales. It would be an advantage to have an atomic scale model to describe nucleation events. However, limitations of atomistic models addressed earlier in the Proposed improvements section with respect to non-equilibrium events must be adequately dealt with before dynamic processes such as interactions can be realistically modelled at that scale. In addition, although recent in situ experiments and modelling indicate that an existing pit would increase the likelihood of pits nucleating at adjacent sites, at least on SS, it would be desirable to have additional observations of different corrosion systems. Lastly, when more experimental data becomes available, replacement of the stochastic treatment by deterministic mechanisms would help interrogate the models and thereby increase our understanding of the controlling factors.