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arxiv: 2606.22640 · v1 · pith:WVXCBGNBnew · submitted 2026-06-21 · 💻 cs.CE · cond-mat.mtrl-sci· physics.app-ph· physics.chem-ph

A phase-field model for microbiologically influenced corrosion

S. Kovacevic , E. Mart\'inez-Pa\~neda This is my paper

Pith reviewed 2026-06-26 09:25 UTC · model grok-4.3

classification 💻 cs.CE cond-mat.mtrl-sciphysics.app-phphysics.chem-ph
keywords phase-field modelmicrobiologically influenced corrosionsulfate-reducing bacteriapitting corrosionmechano-chemical couplingcathodic protectionmicrostructure simulation
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The pith

A phase-field reaction-diffusion model couples microbial sulfate reduction to mechanical fields to predict MIC pitting kinetics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a phase-field model that integrates microbial activity, chemical transport, electrochemical reactions, dissolution, and mechanical stress effects into a single reaction-diffusion framework for microbiologically influenced corrosion. Microbial sulfate consumption is handled with a Monod-type rate law while mechanical influence enters through an enhanced mobility that alters local corrosion kinetics. The formulation is calibrated to experimental pitting data under sulfate-reducing bacteria and then applied to microstructure and structural-scale problems, including coupling to a cathodic protection model. This enables simulation of how grain size modulates pitting severity versus crack growth rates and how sacrificial anodes slow defect evolution in large components such as offshore monopiles. The approach supplies a computationally efficient way to forecast long-term damage in microbial environments.

Core claim

The central claim is that a phase-field-based reaction-diffusion corrosion model, incorporating a Monod expression for microbial sulfate reduction, sulfate transport, electrochemical kinetics, material dissolution, and mechano-chemical coupling through an enhanced mobility relation, reproduces experimental pitting kinetics under SRB activity and captures both MIC-induced pitting and stress-assisted corrosion across length scales from microstructure to engineering structures.

What carries the argument

phase-field reaction-diffusion model with Monod-type microbial kinetics and enhanced mobility for mechano-chemical coupling

If this is right

  • Finer grain sizes reduce overall pitting severity while accelerating defect propagation under mechanical loading.
  • Coupling to a cathodic protection model shows that CP delays pitting and suppresses cracking, with effectiveness declining as sacrificial anodes degrade.
  • Sensitivity analyses identify the relative influence of microbial kinetics, transport rates, and thermodynamic driving forces on corrosion behaviour.
  • The model reproduces both MIC pitting and stress-assisted corrosion in microstructure-sensitive and structural-scale simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same framework could be used to explore trade-offs between grain refinement for corrosion resistance and the resulting increase in crack growth risk under load.
  • Structural-scale predictions might inform inspection intervals or anode replacement schedules for assets such as offshore wind foundations exposed to SRB.
  • Extension to time-dependent microbial community changes or varying environmental conditions would require only updates to the Monod parameters and transport fields.

Load-bearing premise

The chosen functional form and parameters of the enhanced mobility relation correctly convert mechanical fields into modified corrosion rates, and the calibration to the cited experiments extends to the microstructures and structural geometries examined.

What would settle it

New experimental measurements of pit depth evolution or crack growth rates under controlled SRB exposure and applied stress that deviate systematically from the model's calibrated predictions.

Figures

Figures reproduced from arXiv: 2606.22640 by E. Mart\'inez-Pa\~neda, S. Kovacevic.

Figure 1
Figure 1. Figure 1: Microbially influenced corrosion mechanism and diffuse interface representation of the corrosive environment (bacteria-rich environment with biofilm ϕ = 0) and metal (ϕ = 1) phases. The evolution of the corroding interface is tracked by a continuous phase-field variable ϕ(x, t). The variable takes the value ϕ = 1 in the uncorroded metal and ϕ = 0 in the corrosive environment. The thin diffuse interface reg… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic illustration of the experimental setup from Ref. [20] and the corresponding one-dimensional computational domain. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between experimental measurements [20] and phase-field predictions for pitting depth as a function of immersion time. may be interpreted as the time required for biofilm formation and SRB acclimation to their new environment. Subsequently, corrosion progressed, leading to more severe pitting. The same trend in results is reproduced by the present framework, except for the initial immersion stage… view at source ↗
Figure 4
Figure 4. Figure 4: Sensitivity analysis. Dependence of pitting depth on variations in the (a) Monod half-velocity Km and maintenance-respiration coefficients b, (b) true yield of bacterial mass Y and maximum specific rate of metal substrate utilisation by SRB q, (c) bacteria density N and effective energy Ψ, and (d) bacterial density in the biofilm cb as a function of immersion time. Reference values for K0 m, b 0 , Y 0 , q … view at source ↗
Figure 5
Figure 5. Figure 5: Application case study. (a) Schematic illustration of an offshore wind monopile foundation with sacrificial anodes. Computational domains with initial and boundary conditions for (b) microstructure-sensitive simulations of corrosion in the buried zone and (c) structural scale simulations at three representative points. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Sensitivity of corrosion kinetics on microstructure. (a) Dependence of corrosion current density on crystallographic orientation. (b) Corrosion evolution in the buried zone in the absence of CP for microstructures with an average grain size of 30 µm, 40 µm, and 60 µm as a function of immersion time. The red lines correspond to uniform corrosion. The initial surrounding corrosive environment is not shown in… view at source ↗
Figure 7
Figure 7. Figure 7: Corrosion evolution in the buried zone in the absence of CP. Pitting corrosion for 3D microstructures with an average grain size of 40 µm as a function of immersion time. Pitting intensity is defined as the difference in corrosion depth between pitting and uniform corrosion. by deeper and more irregular pit morphologies along the corroding interface. Large grains with orientations that are more susceptible… view at source ↗
Figure 8
Figure 8. Figure 8: Corrosion evolution in the buried zone in the absence of CP for microstructures with an average grain size of 60 µm subjected to remote deformation ε∞. The red lines correspond to uniform corrosion. The initial surrounding corrosive environment is not shown in the plots. The scale bar for all plots is 200 µm [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Corrosion evolution in the buried zone in the absence of CP for microstructures with an average grain size of 40 µm subjected to remote deformation ε∞. The red lines correspond to uniform corrosion. The initial surrounding corrosive environment is not shown in the plots. The scale bar for all plots is 200 µm. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Corrosion evolution in the buried zone in the absence of CP for microstructures with an average grain size of 30 µm subjected to remote deformation ε∞. The red lines correspond to uniform corrosion. The initial surrounding corrosive environment is not shown in the plots. The scale bar for all plots is 200 µm. hot spots for crack initiation. The locations of these hot spots with amplified mechanical fields… view at source ↗
Figure 11
Figure 11. Figure 11: Quantifying the interplay between corrosion degradation, mechanics and local heterogeneity. (a) Normalised mass loss and (b) defect depth as a function of immersion time for microstructures with an average grain size of 30 µm, 40 µm, and 60 µm subjected to various remote tensile deformations ε∞. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Evolution of current density (A/m2 ) for the offshore monopile foundation as a function of service time in the presence of cathodic protection. The grey bars indicate sacrificial anodes. Corrosion evolution is evaluated at the three locations along the monopile, see [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Evolution of current density at the three representative points along the monopile foundation as a function of service time in the presence of cathodic protection. 4.4.2. Phase-field simulations of defect growth The corrosion current density obtained from the CP model in the previous section is used to inform the phase-field mobility parameter L0 via Eq. (B.11). Corrosion evolution at each location is mod… view at source ↗
Figure 14
Figure 14. Figure 14: Defect depth as a function of service time at (a) point A and (b) points B and C. The shaded regions indicate the different stages of cathodic protection performance: active protection (full CP), degradation (partial CP), and loss of protection (no CP). The temporal evolution of corrosion damage, quantified as defect depth, in the presence and absence of CP at the three representative points is depicted i… view at source ↗
Figure 15
Figure 15. Figure 15: Evolution of pitting (ε∞ = 0) and stress-assisted (ε∞ = 0.4εy) corrosion as a function of service time at point A. The initial surrounding corrosive environment is not shown in the plots. The scale bar for all plots is 500 µm. Since the phase-field mobility parameter L0 is proportional to the current density (Appendix B), the application of CP increases the characteristic interface time τϕ, thereby reduci… view at source ↗
read the original abstract

A phase-field-based reaction-diffusion corrosion model is developed to predict microbially influenced corrosion (MIC) in metal alloys, with a focus on anaerobic conditions and sulfate-reducing bacteria (SRB). The formulation couples microbial sulfate reduction, sulfate transport, electrochemical kinetics, material dissolution, and mechanical effects. Microbial activity is modelled using a Monod-type expression for sulfate consumption, whereas the mechano-chemical coupling is incorporated through an enhanced mobility relationship that captures the influence of mechanical fields on corrosion kinetics. The model is calibrated against experiments and shows strong agreement in predicting pitting kinetics under SRB activity. Sensitivity analyses quantify the competing roles of microbial kinetics, transport, and thermodynamic driving forces in governing corrosion behaviour. The capability of the formulation to capture both MIC-induced pitting and stress-assisted corrosion across multiple length scales is demonstrated through case studies that include microstructure-sensitive simulations and structural-scale coupling with a cathodic protection (CP) model. Results show that finer grain sizes reduce pitting severity but promote faster defect propagation under mechanical loading. At the structural scale, coupling with the CP model enables predictions of defect growth under varying electrochemical conditions and over engineering-relevant length scales, as exemplified with the analysis of an offshore wind turbine monopile. CP delays pitting and suppresses crack propagation, although its effectiveness diminishes as sacrificial anodes degrade. The framework provides a predictive and computationally efficient tool for assessing MIC-induced damage over extended times, with potential applications in the integrity and life assessment of metallic structures operating in aggressive microbial environments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a phase-field reaction-diffusion model for microbiologically influenced corrosion (MIC) under anaerobic SRB conditions. It couples Monod-type microbial sulfate reduction, sulfate transport, electrochemical kinetics, material dissolution, and mechanical effects via an enhanced mobility relation. The model is calibrated to experiments with reported strong agreement on pitting kinetics, includes sensitivity analyses on microbial kinetics, transport, and driving forces, and demonstrates multi-scale case studies: microstructure-sensitive pitting (grain size effects on severity vs. propagation) and structural-scale coupling to a cathodic protection (CP) model for an offshore wind turbine monopile (CP delays pitting until anode degradation).

Significance. If the calibration generalizes, the work provides a multi-physics phase-field framework that integrates microbial activity with mechano-electrochemical corrosion across scales, which is valuable for integrity assessment of structures in aggressive environments. Explicit strengths include the sensitivity analyses quantifying competing mechanisms and the structural-scale CP demonstration; these go beyond single-scale pitting models.

major comments (2)
  1. [Calibration and case-study sections] The central claim of predictive capability at structural scale (monopile with CP) rests on transfer of calibrated Monod parameters and enhanced mobility relation from the pitting experiments. No independent validation set, hold-out data, or quantitative error metrics (e.g., RMSE on pit depth or growth rate) are reported for the CP-coupled or microstructure cases, so the generalization assumption is untested and load-bearing.
  2. [Model formulation] § on mechano-chemical coupling: the enhanced mobility functional form is introduced to encode stress effects on kinetics, but its parameters appear fitted to the same experiments used for overall calibration; this risks circularity when claiming the model captures both MIC-induced pitting and stress-assisted corrosion without additional verification.
minor comments (2)
  1. Notation for the Monod half-saturation constants and the mobility enhancement factor should be defined consistently in the text and equations to avoid ambiguity when comparing sensitivity results.
  2. Figure legends for the monopile CP simulations could explicitly state the anode degradation timeline and boundary conditions used, improving reproducibility of the structural-scale results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We respond point-by-point to the major comments below, acknowledging limitations where they exist and indicating planned revisions to the manuscript.

read point-by-point responses

  1. Referee: [Calibration and case-study sections] The central claim of predictive capability at structural scale (monopile with CP) rests on transfer of calibrated Monod parameters and enhanced mobility relation from the pitting experiments. No independent validation set, hold-out data, or quantitative error metrics (e.g., RMSE on pit depth or growth rate) are reported for the CP-coupled or microstructure cases, so the generalization assumption is untested and load-bearing.

    Authors: We agree that the microstructure-sensitive and CP-coupled simulations transfer parameters calibrated solely from the SRB pitting experiments and do not include an independent validation dataset or quantitative error metrics (such as RMSE) for those cases. These simulations are presented as demonstrations of the model's multi-scale capabilities and coupling features rather than as validated predictions. We will revise the manuscript to explicitly qualify the scope of these case studies, add a limitations discussion on the need for future independent validation at structural scales, and ensure that calibration error metrics for the primary experiments are reported with greater prominence. revision: partial

  2. Referee: [Model formulation] § on mechano-chemical coupling: the enhanced mobility functional form is introduced to encode stress effects on kinetics, but its parameters appear fitted to the same experiments used for overall calibration; this risks circularity when claiming the model captures both MIC-induced pitting and stress-assisted corrosion without additional verification.

    Authors: The enhanced mobility relation is constructed from established mechano-chemical principles in the stress-corrosion literature, with its functional parameters adjusted to reproduce the observed pitting kinetics in the SRB experiments. The Monod microbial kinetics are calibrated independently of the mobility enhancement term. Nevertheless, we acknowledge the potential interdependence when the same dataset informs both components. We will revise the model formulation section to clarify the separation of terms, cite the literature basis for the mobility form more explicitly, and add a note on the calibration strategy to address concerns of circularity. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper formulates a phase-field reaction-diffusion model from standard electrochemical, microbial Monod kinetics, transport, and mechanical principles, then calibrates parameters to experimental pitting data before demonstrating application to microstructure and structural-scale cases. No quoted equations or steps in the provided text reduce the central claims (e.g., pitting kinetics predictions or CP coupling) to the calibration inputs by construction, nor do self-citations or ansatzes serve as load-bearing uniqueness theorems. The derivation chain remains independent of the fitted values, with calibration functioning as external validation rather than definitional equivalence.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

Abstract-only review limits the ledger to explicitly stated model ingredients. The formulation rests on standard phase-field and reaction-diffusion assumptions plus domain-specific microbial and electrochemical kinetics; several parameters are calibrated to data.

free parameters (2)
  • Monod kinetic parameters
    Rate constants and half-saturation constants for sulfate consumption by SRB, calibrated to experiments.
  • enhanced mobility parameters
    Coefficients linking mechanical fields to corrosion kinetics via the mobility term.
axioms (3)
  • domain assumption Phase-field evolution equations can represent moving corrosion interfaces
    Core modeling choice for material dissolution.
  • domain assumption Monod expression accurately captures SRB sulfate reduction rate
    Used to model microbial activity under anaerobic conditions.
  • ad hoc to paper Enhanced mobility relation correctly encodes mechano-chemical coupling
    Introduced to incorporate mechanical effects on corrosion kinetics.

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Reference graph

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