Point Defect Model (PDM) in Corrosion & Materials

• The Point Defect Model (PDM) is a defect-based framework describing passive oxide film growth and defect thermodynamics in metals and ceramics.
• It integrates electrochemical kinetics, interfacial reactions, and Nernst–Planck transport to model oxide thickness and passive current behavior.
• Recent refinements add explicit electrostatics, current continuity, and multiscale corrections to improve accuracy in corrosion and materials modeling.

Searching arXiv for recent and foundational papers on the Point Defect Model and related point-defect modeling. arXiv search query: "Point Defect Model passivity metals corrosion oxide film arXiv" The Point Defect Model (PDM) denotes a family of defect-centered descriptions in which defects are treated as point-like entities, but the term has acquired distinct technical meanings across subfields. In electrochemistry and corrosion science, PDM is most closely associated with the description of passive oxide films on metals through interfacial reactions, point-defect transport, and electric fields inside the barrier layer. In defect thermodynamics and multiscale materials modeling, the same label is used for dilute or generalized descriptions of vacancies, interstitials, Frenkel pairs, Schottky defects, and their elastic fields. Across these usages, the unifying idea is that defect generation, annihilation, transport, and energetics govern macroscopic observables such as passive current density, barrier-layer thickness, pseudo-phase transitions, or supercell finite-size errors (Bösing et al., 2023, Geng et al., 2013, Varvenne et al., 2013).

1. Terminology and domain of use

In the supplied arXiv literature, “Point Defect Model” is not a single universally fixed formalism but a technical label attached to several related defect-based modeling traditions.

Usage of “PDM”	Core object	Example
Passivity/corrosion PDM	Passive oxide film growth and dissolution via interfacial reactions and defect transport	(Bösing et al., 2023)
Defect-thermodynamics PDM	Dilute or generalized point-defect populations in defective solids	(Geng et al., 2013)
Multiscale point-defect representation	Atomistic defect core plus continuum elastic far field	(Varvenne et al., 2013)

In the electrochemical literature, the PDM is presented as a long-standing framework for describing how passive oxide films form, grow, and dissolve on metals through interfacial reactions plus defect transport inside the oxide, with oxygen vacancies and metal vacancies as central species. In uranium dioxide, by contrast, the PDM is formulated as a practical defect-thermodynamics tool in which defects are treated as point-like objects occupying lattice sites or interstitial positions, first in a simple dilute-limit form and then in a generalized form with an auxiliary constant mean-field to incorporate interactions (Bösing et al., 2023, Geng et al., 2013).

A recurring source of confusion is acronym ambiguity. Other arXiv papers use “PDM” for “position-dependent mass” in nonlinear oscillators and Klein–Gordon systems, or for “Poincaré Discontinuity Mapping” in hybrid impact dynamics; these usages are unrelated to point-defect theory. This suggests that, in technical writing, the expansion of the acronym must be made explicit at first use (Mustafa, 2019, Lima et al., 2024).

2. Classical electrochemical structure of the PDM

The classical passivity-oriented PDM treats the passive film as a defect-mediated semiconductor-like barrier in which growth kinetics are controlled by the creation, annihilation, and migration of point defects such as oxygen vacancies and metal vacancies. Its core statement is that passive oxide growth results from a balance of interfacial reactions at the metal/film and film/solution boundaries and transport of charged defects through the film (Bösing et al., 2023).

Within this formulation, charged-defect transport is written in Nernst–Planck form,

$$J_i = -D_i \frac{\partial c_i}{\partial x} - \frac{z_i F D_i}{RT}\,c_i \frac{\partial \phi}{\partial x},$$

where $c_i$ is defect concentration, $D_i$ the diffusion coefficient, $z_i$ the charge number, and $\phi$ the electric potential. The model is usually organized around a seven-reaction scheme, with nonconservative interfacial reactions controlling net barrier-layer growth and dissolution. In fretting-corrosion applications, the same seven-reaction structure is retained, and reactions 3 and 7 are identified explicitly as the non-conservative reactions that enter the steady-state thickness equation (Bösing et al., 2023, Geringer et al., 2013).

A defining simplification of the classical PDM is the assumption of a constant, potential-independent electric field in the oxide film, often justified by a tunneling-based field-buffering argument with a field on the order of $2$–$5$ MV/cm. A second simplification is a prescribed linear potential drop at the film/solution interface,

$$\phi_{fs} = a \phi_{\text{ext}} + b\,\text{pH} + \phi_{fs}^0.$$

These assumptions made the original model tractable and enabled extensive use in interpreting passivity, film growth, pitting, and related electrochemical behavior, but later work identifies them as physically restrictive rather than generally valid (Bösing et al., 2023, Bösing et al., 2023).

The same defect-reaction logic is used to formulate steady-state growth conditions. In the refined passivity literature, the steady-state film thickness follows from balancing oxide growth and dissolution, for example through

$\frac{dL}{dt}=0=2(k_2-k_5),$

so that the net production of film-forming species at one interface is balanced by dissolution at the other. This steady-state structure is the basis on which later variants introduce more physical electrostatics, more careful current accounting, or additional mechanical wear terms (Bösing et al., 2023).

3. Physically grounded refinements and the variant PDM

A major line of development replaces empirical voltage partitioning and incomplete interfacial bookkeeping with physically explicit electrostatics and current continuity. One variant of the PDM replaces the empirical parameters $\alpha$ and $c_i$0 by two physical parameters, $c_i$1 and $c_i$2, representing respectively the electronic contact resistance at the metal/film interface and the electronic resistivity of the oxide film. In this formulation, the oxide potential drop is represented as an ohmic drop $c_i$3, and the variant is constructed so that metal-vacancy annihilation at the metal/film interface is described correctly and particle and defect volumes are conserved (Emmanuel, 2012).

The volume-conservation step is notable because it causes the Pilling–Bedworth ratio to emerge naturally. For the modified oxide-formation reaction $c_i$4, the model derives

$c_i$5

and the steady-state current reduces to the compact expression

$c_i$6

This links passive current directly to the oxide dissolution or turnover rate $c_i$7, scaled by electrochemical valence and the Pilling–Bedworth ratio, rather than to an empirical voltage partition alone (Emmanuel, 2012).

A second refinement addresses a different inconsistency: the equality of total current across the metal/film and film/solution interfaces. Emmanuel’s analysis argues that the original PDM did not properly enforce the basic electrochemical requirement

$c_i$8

and therefore lacked complete bookkeeping of ion and electron currents. The correction is the introduction of a purely electronic channel at the metal/film interface, treated as a Schottky-like e-channel with

$c_i$9

In this revision, the e-channel is inactive when $D_i$0 and required when $D_i$1, so that interfacial current continuity is restored explicitly rather than assumed implicitly (Emmanuel, 2013).

A third line of refinement removes the constant-field hypothesis by solving the potential profile self-consistently from Poisson’s equation,

$D_i$2

The Refined PDM (R-PDM) introduces two neutral defect layers at the film boundaries, enforces continuity of electric flux through $D_i$3, and includes a compact double layer at the solution side. The resulting calculations show that the electric field is not generally constant, that it varies strongly near the interfaces, and that the potential drop at the film/solution interface is itself a function of external potential. The original PDM assumptions are therefore valid only for very specific parameter combinations of oxide film growth and vacancies transport (Bösing et al., 2023).

4. Chloride, electronic carriers, tunneling, and mechanically assisted breakdown

The most developed breakdown studies extend the PDM beyond neutral passive growth into chloride-induced destabilization, electronic-carrier effects, and fretting-assisted film removal. In the chloride problem, the corrected PDM replaces the original Schottky-pair-mediated coupling with a direct chloride-controlled interfacial mechanism:

$D_i$4

with an equilibrium oxygen-site occupancy

$D_i$5

This chloride occupation pins the oxygen-vacancy-related reaction rate through

$D_i$6

and, through reaction $D_i$7, directly alters metal-vacancy annihilation at the metal/film interface (Emmanuel, 2012).

Under the quasi-steady-state approximation, this variant yields analytical expressions for incubation time, oxide dissolution time, critical pitting potential, passive current transients, and the time evolution of the metal-vacancy concentration. One explicit result is

$D_i$8

so the critical pitting potential decreases with increasing chloride concentration. The same analysis concludes that thicker oxide layers are in general more susceptible to pitting due to the chloride ion, whereas thinner oxide layers destabilize by simple dissolution in the presence of chloride. The paper identifies anion flux-pinning by chloride as sufficient to destabilize the oxide layer, without invoking the Schottky-pair mechanism of the original PDM (Emmanuel, 2012).

Electronic-carrier extensions address another classical assumption: the idea that band-to-band tunneling (BTBT) and electron–hole separation buffer the electric field into a constant value. In the extended R-PDM, the Poisson equation includes electrons and holes,

$D_i$9

carrier transport is described by drift–diffusion,

$z_i$0

and BTBT is modeled by a Kane-type rate

$z_i$1

The minimum field for BTBT across a film of thickness $z_i$2 is written as

$z_i$3

The resulting conclusion is restrictive: BTBT only occurs in very rare cases of narrow band gaps and high electric fields, and electron–hole buffering does not produce either a potential-independent electric field or a uniform electric field (Bösing et al., 2023).

Mechanical wear can be incorporated by augmenting the passive-film thickness balance with a wear term $z_i$4 based on the Archard law. In fretting corrosion of 316L stainless steel, the PDM is used as the microscale electrochemical description of the passive oxide film and then coupled to a friction process through an additive thickness-removal term. The model is fitted with a genetic algorithm over 29 parameters. Reported outcomes include a protective effect of albumin at $z_i$5 mol/L NaCl, little or no protective effect at $z_i$6 mol/L NaCl, an incubation time of about 100 minutes before substantial film loss, and predicted passive-film thicknesses on the order of nanometers with a 95% confidence interval about 2 nm (Geringer et al., 2013).

5. Thermodynamic and multiscale point-defect models in materials science

Outside corrosion science, the PDM is used as a defect-thermodynamics framework for nonstoichiometric solids. In defective uranium dioxide, the simple PDM treats defects as point-like objects on lattice sites or interstitial positions, without explicit spatial extent and without direct interactions. Defect concentrations are controlled by formation Gibbs free energies through relations such as

$z_i$7

$z_i$8

$z_i$9

The stoichiometry deviation is related to defect populations by

$\phi$0

Within this framework, the dominant-defect region is called a pseudo phase, and pseudo-phase transition pressures are used as thermodynamic markers to constrain defect formation energies (Geng et al., 2013).

The same work generalizes the dilute, noninteracting PDM by expanding the configuration free energy in powers of defect concentrations,

$\phi$1

so that interaction effects are absorbed into a concentration-dependent effective formation energy. This generalized simple PDM with an auxiliary constant mean-field is presented as a tool for extrapolating finite-concentration calculations to the dilute limit and for treating nonlinear effects of stoichiometry deviation, reentrant pseudo-transition, multi-defect coexistence, and charged defects (Geng et al., 2013).

A distinct but related multiscale usage appears in defect energetics from atomistic simulation. There, point defects are not treated as truly local objects in a finite periodic supercell because they generate long-range elastic relaxations. A coupled ab initio plus linear elasticity framework writes the raw defective-cell energy as

$\phi$2

with the spurious image interaction given by

$\phi$3

The defect is represented, to first order, by an elastic dipole tensor extracted from residual stress,

$\phi$4

This is explicitly described as distinct from a kinetic corrosion model: the defect core remains atomistic, while the far field is modeled analytically through elasticity so that isolated-defect energetics can be reconstructed from smaller supercells (Varvenne et al., 2013).

6. Computation, diagnostics, and persistent limitations

Recent work casts the passivity-oriented PDM as a coupled moving-boundary PDE/ODE system for defect transport, electrostatics, and film-thickness evolution. In one physics-informed neural network implementation for halide-free iron passivation, the oxide occupies a moving domain $\phi$5, defect transport follows the Nernst–Planck equation, electrostatics follows Poisson’s equation,

$\phi$6

and film growth is governed by

$\phi$7

The implementation uses separate subnetworks for electric potential, cation-vacancy concentration, anion-vacancy concentration, and film thickness, with Swish activation and typically 5 hidden layers with 20 neurons each (Farooqi et al., 3 Oct 2025).

That study identifies four failure modes for PINNs in the PDM setting: imbalanced loss components across different physical processes, numerical instabilities due to variable scale disparities, challenges in enforcing boundary conditions within multiphysics systems, and convergence to mathematically valid but physically meaningless solutions. The mitigation strategy combines nondimensionalization, Neural Tangent Kernel-based adaptive loss balancing, and explicit boundary-condition enforcement. Reported scaling choices include $\phi$8 and $\phi$9; dimensional models became unstable after about 3600 s, whereas nondimensionalization allowed simulation to about 900,000 s, roughly a 250× improvement. A single supervised FEM data point was found sufficient to anchor the PINN to the correct physical branch when a pure PINN drifted toward an incorrect but residual-consistent solution (Farooqi et al., 3 Oct 2025).

The broader literature also defines clear diagnostic and conceptual limits. The variant PDM provides slope diagnostics for $2$0 and $2$1 with respect to pH and potential, intended to distinguish the original Macdonald formulation from resistive and current-consistent variants (Emmanuel, 2012). The refined electrostatic literature argues that constant-field and prescribed interface-drop assumptions are not generally valid (Bösing et al., 2023, Bösing et al., 2023). The defect-thermodynamics literature shows that the simple PDM is a dilute, noninteracting limit rather than a complete description at finite stoichiometry (Geng et al., 2013). The elastic-correction literature likewise emphasizes that linear elasticity and elastic-dipole modeling are reliable only when the defect remains well represented by continuum response and the supercell is not so small that defect images interact through more than elasticity (Varvenne et al., 2013).

Taken together, these strands define the contemporary status of the Point Defect Model: not a single immutable recipe, but a defect-based modeling paradigm that has evolved from empirical electrochemical kinetics toward self-consistent electrostatics, explicit current continuity, interaction-aware thermodynamics, multiscale elasticity corrections, and data-assisted PDE solvers.