On the Critical Factors for Estimating the Pit Stability Product under a Salt Film

The pit cavity includes the following four species: Na⁺, Cl⁻, Fe²⁺, FeCl⁺ (named as components 1–4, respectively). Then, the Nernst-Planck equation is given as:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {c}_{i}}{\partial t}+{\rm{\nabla }}\cdot {{\rm{N}}}_{i}={R}_{i}\end{eqnarray} \tag{ 1 }$$

where c_i, N_i and R_i are the concentration, molar flux and reaction rate of species i, respectively.

The molar flux of species i is defined as,

$$\begin{eqnarray}&&{{\rm{N}}}_{{\rm{i}}}=-{D}_{i}{\rm{\nabla }}{c}_{i}-{z}_{i}{\mu }_{i}F{c}_{i}{\rm{\nabla }}{\rm{\Phi }}\end{eqnarray} \tag{ 2 }$$

where, D_i , z_i and u_i are the diffusion coefficient, charge number and mobility of species i, respectively; F is Faraday's constant; Φ is the electrolyte potential.

The mobility, u_i, is given by the Nernst-Einstein equation.

$$\begin{eqnarray}&&{\mu }_{i}=\displaystyle \frac{{D}_{i}}{RT}\end{eqnarray} \tag{ 3 }$$

where, R is the universal gas constant; T is the temperature in K unit.

Besides considering mass transfer due to diffusion and electro-migration, the electro-neutrality condition is taken into account to ensure a zero net charge at any local point in the pit solution.

$$\begin{eqnarray}&&\displaystyle \sum _{i=1}^{n}{z}_{i}{c}_{i}=0\end{eqnarray} \tag{ 4 }$$

The potential distribution was assumed to be governed by, ³⁰

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot \left(\sigma {\rm{\nabla }}{\rm{\Phi }}\right)+F\displaystyle \sum _{i=1}^{n}{z}_{i}{\rm{\nabla }}\cdot \left({D}_{i}{\rm{\nabla }}{c}_{i}\right)=0\end{eqnarray} \tag{ 5 }$$

where,

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{{F}^{2}}{RT}\displaystyle \sum _{i=1}^{n}{z}_{i}^{2}{D}_{i}{c}_{i}\end{eqnarray} \tag{ 6 }$$

Jun et al., and Srinivasan et al. have demonstrated the importance of solution viscosity on diffusivity, indicating that the diffusion coefficient would decrease with an increase in pit depth. ^8,19,31 Thus, it was necessary to consider a variable diffusion coefficient due to the concentration gradient inside the pit. The diffusion coefficients of Cl⁻, Na⁺, Fe²⁺ and FeCl⁺ in FeCl₂ solutions of different concentrations-obtained with OLI Studio 10.0-are plotted versus chloride concentration, as is shown in Figure 4. The influence of the bulk NaCl concentration (0.6 M NaCl) on ${D}_{F{e}^{2+}}$ is also shown in Figure 4, as indicated by the arrows. The results showed that the function representing the change in ${D}_{F{e}^{2+}}$ with Cl⁻ can be used to estimate the diffusion coefficient inside the pit.

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The anodic reaction is given by:

$$\begin{eqnarray}&&Fe\to F{e}^{2+}+2e\end{eqnarray} \tag{ 7 }$$

The dissolution current density, i, is estimated as

$$\begin{eqnarray}&&i={i}_{0}\left({10}^{\displaystyle \frac{E-{E}_{Fe}}{{b}_{a}}}\right)\end{eqnarray} \tag{ 8 }$$

where, E is the electrode potential at the pit base, E_Fe and i₀ are the equilibrium potential and exchange current density of the oxidation reaction, respectively. In the saturated pit solution, the equilibrium potential was independent of pit depth. The equilibrium potential of the iron oxidation reaction was estimated based on the Nernst equation with a ferrous concentration of 4.2 M. ²⁷ Although the exchange current density (0.25 A m⁻²) was chosen arbitrarily, it did not influence the estimation of the pit stability product if the saturation concentration could be maintained at the pit base.

A salt film precipitates at the dissolving metal interface when the concentration of metal cations reaches the saturation value, and it can accommodate the extra IR drop to main the metal cation at the saturation point at the pit interface. ^14,24,32 The oxidation reaction combined with the salt film deposition was needed to maintain the surface concentration at the saturation point. Due to the uncertainty in defining the precipitation kinetics constant, dissolution kinetics constant, as well as the initial value of the salt film thickness, the dissolution of the salt film was also introduced in our model, increasing its robustness. The salt film would dissolve only when the concentration was lower than the saturation concentration and the salt film thickness was higher than zero.

As indicated by Rayment et al., FeCl₂·4H₂O was found to be the main composition of the salt film formed on 316L SS. ³³ In a recent work by Li et al., they found that the co-precipitated salt film formed on 304SS one-dimensional pit interface contained both Fe²⁺ and Ni²⁺. ³⁴ In this work, the salt film was assumed to be FeCl₂ to simplify the simulation. The reaction order of Fe²⁺ and Cl⁻ was assumed to be one. Thus, the net precipitation rate of the salt film can be expressed as:

$$\begin{eqnarray}&&{R}_{FeC{l}_{2}}=\delta \left(\zeta \right)\cdot {\mathop{k}\limits^{\longrightarrow}}_{FeC{l}_{2}}{c}_{F{e}^{2+}}{c}_{C{l}^{-}}-\theta \left(\zeta \right)\cdot \delta \left(h\right)\cdot {\mathop{k}\limits^{\longleftarrow}}_{FeC{l}_{2}}\end{eqnarray} \tag{ 9 }$$

with

$$\begin{eqnarray}&&\zeta =\displaystyle \frac{{c}_{F{e}^{2+}}}{{c}_{sat}}-1\end{eqnarray} \tag{ 10 }$$

where, h is the thickness of the salt film, ${\mathop{k}\limits^{\longrightarrow}}_{FeC{l}_{2}}$ and ${\mathop{k}\limits^{\longleftarrow}}_{FeC{l}_{2}}$ are the precipitation kinetics constant and dissolution kinetics constant, respectively.

Here, $\theta (\zeta ),\delta (h)$ are step functions, and presented as:

$$\begin{eqnarray}&&\delta \left(x\right)=\left\{\begin{array}{l}0,x\leqslant 0\\ 1,x\gt 0\end{array}\right.\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\theta \left(x\right)=\left\{\begin{array}{l}1,x\leqslant 0\\ 0,x\gt 0\end{array}\right.\end{eqnarray} \tag{ 12 }$$

The Fe²⁺ and Cl⁻ consumption rates at the salt film interface can be expressed as:

$$\begin{eqnarray}&&{R}_{F{e}^{2+}}^{{\prime} }={R}_{FeC{l}_{2}}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{R}_{C{l}^{-}}^{{\prime} }=2{R}_{FeC{l}_{2}}\end{eqnarray} \tag{ 14 }$$

The thickness of salt film can be calculated by the integral of precipitation rate with time.

$$\begin{eqnarray}&&h=\displaystyle \int \displaystyle \frac{{R}_{FeC{l}_{2}}{{\rm{M}}}_{FeC{l}_{2}}}{{\rho }_{FeC{l}_{2}}}dt\end{eqnarray} \tag{ 15 }$$

The salt film is assumed to have a high electric field, which remains unchanged with variations in the salt film thickness. ^14,24,32,34 The potential across the salt film is calculated by:

$$\begin{eqnarray}&&\psi =h\cdot {\psi }_{0}\end{eqnarray} \tag{ 16 }$$

where ψ₀ is the electrical field across the salt film, the value of which is assumed to be 2.1 kV cm⁻¹ in this work. ²⁸

Thus, the electrode potential at the pit base is obtained as:

$$\begin{eqnarray}&&E={E}_{app}-{\rm{\Phi }}-\psi \end{eqnarray} \tag{ 17 }$$

The solution IR drop is not considered in this model, because solution IR drop will not influence the estimation of the pit stability product once the saturation concentration can be maintained at the pit interface.

The equilibrium constant of the Fe²⁺complexation reaction is approximately five orders of magnitudes higher than the equilibrium constant of the hydrolysis reaction ³⁵ ; therefore Fe²⁺ is more likely to react with Cl⁻ instead of hydrolyzing to generate H⁺. The hydrolysis reaction was not considered for pit stability product estimations. The homogeneous complexation reaction considered in the present model is:

$$\begin{eqnarray}&&F{e}^{2+}+C{l}^{-}\underset{\mathop{\unicode{x027F5}}\limits_{}}{\overset{{\mathop{k}\limits^{\longrightarrow}}_{FeC{l}^{+}}}{\longrightarrow }}FeC{l}^{+}\end{eqnarray} \tag{ 18 }$$

Where ${\mathop{k}\limits^{\longrightarrow}}_{FeC{l}^{+}}$ and ${\mathop{k}\limits^{\longleftarrow}}_{FeC{l}^{+}}$ are the forward and reverse kinetics constants, respectively, which are related to the equilibrium constant,

$$\begin{eqnarray}&&{K}_{FeC{l}^{+}}=\displaystyle \frac{{\mathop{k}\limits^{\longrightarrow}}_{FeC{l}^{+}}}{{\mathop{k}\limits^{\longleftarrow}}_{FeC{l}^{+}}}\end{eqnarray} \tag{ 19 }$$

Then, the consumption rates of Fe²⁺, Cl⁻, and FeCl⁺ are,

$$\begin{eqnarray}&&{R}_{F{e}^{2+}}=-{\mathop{k}\limits^{\longrightarrow}}_{FeC{l}^{+}}{c}_{F{e}^{2+}}{c}_{C{l}^{-}}+{\mathop{k}\limits^{\longleftarrow}}_{FeC{l}^{+}}{c}_{FeC{l}^{+}}\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{R}_{C{l}^{-}}=-{\mathop{k}\limits^{\longrightarrow}}_{FeC{l}^{+}}{c}_{F{e}^{2+}}{c}_{C{l}^{-}}+{\mathop{k}\limits^{\longleftarrow}}_{FeC{l}^{+}}{c}_{FeC{l}^{+}}\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&{R}_{FeC{l}^{+}}={\mathop{k}\limits^{\longrightarrow}}_{FeC{l}^{+}}{c}_{F{e}^{2+}}{c}_{C{l}^{-}}-{\mathop{k}\limits^{\longleftarrow}}_{FeC{l}^{+}}{c}_{FeC{l}^{+}}\end{eqnarray} \tag{ 22 }$$

The electrolyte potential and applied potential at the pit mouth have a value of 0 V_SSC and +0.30 V_SSC, respectively.

At the pit surface, the diffusion flux of Fe²⁺ equals its production rate involved in the electrochemical reaction, which could be calculated based on Faraday's law,

$$\begin{eqnarray}&&{{\rm{N}}}_{F{e}^{2+}}\cdot {\rm{n}}=\displaystyle \frac{{i}_{Fe}}{nF}\end{eqnarray} \tag{ 23 }$$

For other species that are not involved in electrochemical reactions the boundary condition at the pit surface is:

$$\begin{eqnarray}&&{{\rm{N}}}_{{\rm{i}}}\cdot {\rm{n}}=0\end{eqnarray} \tag{ 24 }$$

The concentration of species at the pit mouth is assumed to be equal to the bulk solution:

$$\begin{eqnarray}&&{c}_{i}={c}_{i,bulk}\end{eqnarray} \tag{ 25 }$$

The computational geometry was realigned timely with the new position of the pit base via an arbitrary Lagrangian-Eulerian (ALE) method continuously to track pit propagation, which allows moving boundaries without movement of the mesh. ^35,36 If the mesh displacement becomes large and the mesh quality degrades during the calculation, the computational domain would be re-meshed, and all solution variables would be mapped from the old mesh to the new one. For the pit base, the normal component of the pit growth rate can be calculated by the following equation:

$$\begin{eqnarray}&&-{\rm{n}}\cdot {\rm{v}}=\displaystyle \frac{iM}{nF\rho }\end{eqnarray} \tag{ 26 }$$

Lastly, the artificial boundaries are considered to have zero displacements.

A linear concentration gradient was applied inside the pit as the initial condition. The pit depth and the salt film thickness were taken as 100 and 0.6 μm, respectively. A time step Δ t = 10 s was selected for the time-dependent mass transport equations because the stable condition inside the pit was obtained after 10 s. Table II presents the parameters used in the calculations.

Table II. Parameters used in the model.

Parameter	Value	Unit	Source
EFe	−0.45	VSSC	Ref. 27
Eapp	0.30	VSSC	Experiment
iFe,0	0.25	A m−2	Chosen arbitrarily
ba	0.12	V dec−1	Ref. 7
${\mathop{k}\limits^{\longrightarrow}}_{FeC{l}_{2}}$	10−8	m3 (mol s−1)−1	Chosen arbitrarily
${\mathop{k}\limits^{\longleftarrow}}_{FeC{l}_{2}}$	0.1	mol (m−3 s−1)	Chosen arbitrarily
ψ0	2.1	kV cm−1	Ref. 28
${\mathop{k}\limits^{\longleftarrow}}_{FeC{l}^{+}}$	104	1 s−1	Chosen arbitrarily
K	4 × 10−6	m3 mol−1	OLI Studio
${M}_{FeC{l}_{2}}$	0.1988	kg mol−1	OLI Studio
${\rho }_{FeC{l}_{2}}$	1930	kg m−3	OLI Studio
M	0.5574	kg mol−1	Ref. 7
ρ	8030	kg m−3	Ref. 7

Figure 5 shows the current density versus time plot during pit growth until a depth of around 1 mm. It was found that the current density reached the maximum value quickly after pit initiation, followed by a gradual decay of the current density. When the potential decreased from +0.80 to +0.45 V_SSC and further to +0.30 V_SSC, the amplitude of the current density remained stable, indicating that the pit propagation was under mass transport control.

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Figure 6 shows the limiting current density (i_lim ) under a salt film versus the reciprocal of the pit depth (1/d). A linear relationship was observed, the slope of which yielded the pit stability product (x·i), which was 1.00 A m⁻¹ under a salt film and in the range of the results documented in the literature (i.e., 0.8 to 1.1 A m^−17,9,10 ). The linear function indicated that the pit growth was under mass transport control. ^7,9,16 In this regard, a constant pit stability product corresponds to a negligible boundary layer thickness, ^7,9,16 validating the assumption that the concentration at the pit mouth was equal to the bulk solution concentration. However, although the 1D Fick's law of diffusion is commonly used to estimate the pit stability product under a salt film, ^{7–9,14,16,18,21–24} it is still unclear whether the pit growth was under pure diffusion control, which is addressed in the following sections.

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The pit stability product can be solved analytically when the hydrolysis and complexation reactions inside the pit are ignored. During the pit growth, the following equation can be used to calculate the dissolution current density: ¹

$$\begin{eqnarray}&&i=2F\left({D}_{F{e}^{2+}}({c}_{F{e}^{2+}})\displaystyle \frac{d{c}_{F{e}^{2+}}}{dx}+\displaystyle \frac{2F{D}_{F{e}^{2+}}({c}_{F{e}^{2+}})}{RT}{c}_{F{e}^{2+}}\displaystyle \frac{d{\rm{\Phi }}}{dx}\right)\end{eqnarray} \tag{ 27 }$$

where i is the dissolution current density, x is the pit depth; ${D}_{F{e}^{2+}}({c}_{F{e}^{2+}})$ is the diffusion coefficient of ferrous ions, which is assumed to be a function of the ferrous concentration; $\tfrac{d{\rm{\Phi }}}{dx}$ is the electrical field inside the pit.

Once a stable condition is reached, it is reasonable to assume that the net Cl⁻ and Na⁺ flux is zero, ¹ thus,

$$\begin{eqnarray}&&{D}_{C{l}^{-}}\left({c}_{C{l}^{-}}\right)\displaystyle \frac{d{c}_{C{l}^{-}}}{dx}-\displaystyle \frac{F{D}_{C{l}^{-}}\left({c}_{C{l}^{-}}\right)}{RT}{c}_{C{l}^{-}}\displaystyle \frac{d{\rm{\Phi }}}{dx}=0\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{D}_{N{a}^{+}}\left({c}_{N{a}^{+}}\right)\displaystyle \frac{d{c}_{N{a}^{+}}}{dx}+\displaystyle \frac{F{D}_{N{a}^{+}}\left({c}_{N{a}^{+}}\right)}{RT}{c}_{N{a}^{+}}\displaystyle \frac{d{\rm{\Phi }}}{dx}=0\end{eqnarray} \tag{ 29 }$$

where, ${D}_{C{l}^{-}}({c}_{C{l}^{-}})$ and ${D}_{N{a}^{+}}({c}_{N{a}^{+}})$ are the diffusion coefficient of chloride and sodium ions respectively, which are the functions of concentration; ${c}_{C{l}^{-}}$ and ${c}_{N{a}^{+}}$ are the chloride and sodium ion concentrations, respectively.

Then, the electric field or potential gradient inside the pit can be obtained from Eq. 28 as:

$$\begin{eqnarray}&&\displaystyle \frac{d{\rm{\Phi }}}{dx}=\displaystyle \frac{RT}{F{c}_{C{l}^{-}}}\displaystyle \frac{d{c}_{C{l}^{-}}}{dx}\end{eqnarray} \tag{ 30 }$$

By taking into account the following boundary conditions for x = 0 (${c}_{C{l}^{-}}={c}_{N{a}^{+}}={c}_{0};$ and ${\rm{\Phi }}=0$), the concentration of chloride and sodium ions can be obtained by integrating Eqs. 28 and 29, respectively, as:

$$\begin{eqnarray}&&{c}_{C{l}^{-}}={c}_{0}\exp \left(\displaystyle \frac{F{\rm{\Phi }}}{RT}\right)\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&{c}_{N{a}^{+}}={c}_{0}\exp \left(-\displaystyle \frac{F{\rm{\Phi }}}{RT}\right)\end{eqnarray} \tag{ 32 }$$

where c₀ is the concentration of chloride and sodium ions in the bulk solution.

Electroneutrality is considered inside the pit, thus:

$$\begin{eqnarray}&&{c}_{C{l}^{-}}-{c}_{N{a}^{+}}-2{c}_{F{e}^{2+}}=0\end{eqnarray} \tag{ 33 }$$

By solving Eqs. 31–33, the concentration of chloride ions inside the pit is obtained:

$$\begin{eqnarray}&&{c}_{C{l}^{-}}={c}_{F{e}^{2+}}+\sqrt{{c}_{F{e}^{2+}}^{2}+{c}_{0}^{2}}\end{eqnarray} \tag{ 34 }$$

Then, by replacing Eq. 30 into Eq. 27, the dissolution current density can be obtained as:

$$\begin{eqnarray}&&i=2F{D}_{F{e}^{2+}}\left({c}_{F{e}^{2+}}\right)\left(\displaystyle \frac{d{c}_{F{e}^{2+}}}{dx}+\displaystyle \frac{2}{{c}_{C{l}^{-}}}{c}_{F{e}^{2+}}\displaystyle \frac{d{c}_{C{l}^{-}}}{dx}\right)\end{eqnarray} \tag{ 35 }$$

The ${c}_{C{l}^{-}}$ in Eq. 35 can be replaced by Eq. 34. Thus,

$$\begin{eqnarray}&&i=2F{D}_{F{e}^{2+}}\left({c}_{F{e}^{2+}}\right)\left(1+\displaystyle \frac{2{c}_{F{e}^{2+}}}{{c}_{C{l}^{-}}}\left(1+\displaystyle \frac{{c}_{F{e}^{2+}}}{\sqrt{{c}_{F{e}^{2+}}^{2}+{c}_{0}^{2}}}\right)\right)\displaystyle \frac{d{c}_{F{e}^{2+}}}{dx}\end{eqnarray} \tag{ 36 }$$

According to Eq. 34, $\sqrt{{c}_{F{e}^{2+}}^{2}+{c}_{0}^{2}}={c}_{C{l}^{-}}-{c}_{F{e}^{2+}},$ thus,

$$\begin{eqnarray}&&i=2F{D}_{F{e}^{2+}}\left({c}_{F{e}^{2+}}\right)\left(1+\displaystyle \frac{2{c}_{F{e}^{2+}}}{{c}_{C{l}^{-}}-{c}_{F{e}^{2+}}}\right)\displaystyle \frac{d{c}_{F{e}^{2+}}}{dx}\end{eqnarray} \tag{ 37 }$$

By substituting ${c}_{C{l}^{-}}-{c}_{F{e}^{2+}}$ back to Eq. 37, then the dissolution current density is obtained:

$$\begin{eqnarray}&&i=2F{D}_{F{e}^{2+}}\left({c}_{F{e}^{2+}}\right)\left(1+\displaystyle \frac{2{c}_{F{e}^{2+}}}{\sqrt{{c}_{F{e}^{2+}}^{2}+{c}_{0}^{2}}}\right)\displaystyle \frac{d{c}_{F{e}^{2+}}}{dx}\end{eqnarray} \tag{ 38 }$$

The concentration of ${c}_{F{e}^{2+}}$ is taken as zero at the pit mouth. Integrating Eq. 38 yields,

$$\begin{eqnarray}&&\displaystyle {\int }_{0}^{x}idx=\displaystyle {\int }_{0}^{{c}_{F{e}^{2+},s}}2F{D}_{F{e}^{2+}}\left({c}_{F{e}^{2+}}\right)\left(1+\displaystyle \frac{2{c}_{F{e}^{2+}}}{\sqrt{{c}_{F{e}^{2+}}^{2}+{c}_{0}^{2}}}\right)d{c}_{F{e}^{2+}}\end{eqnarray} \tag{ 39 }$$

Based on the mean value theorem for definite integrals, ³⁷ the pit stability product (x·i) is obtained as:

$$\begin{eqnarray}&&x\cdot i=2F{D}_{eq}\left({c}_{F{e}^{2+},s}+2\sqrt{{c}_{F{e}^{2+},s}^{2}+{c}_{0}^{2}}-2{c}_{0}\right)\end{eqnarray} \tag{ 40 }$$

with the equivalent diffusion coefficient, D_eq , defined as:

$$\begin{eqnarray}&&{D}_{eq}={D}_{F{e}^{2+}}\left(\varepsilon \right),\varepsilon \in \left(0,{c}_{F{e}^{2+},s}\right)\end{eqnarray} \tag{ 41 }$$

where, ${c}_{F{e}^{2+},s}$ is the saturation concentration of Fe²⁺ at the pit interface.

The analytical solution showed that the pit stability product (x·i) remained constant even when the variable diffusion coefficient and migration were included. Additionally, the variable diffusion coefficient can be replaced by D_eq during pit growth, which is independent of pit depth. Although it is known that D_eq should lie between the values of dilute and saturated solutions, it is not possible to obtain an accurate value analytically when migration is taken into account. Thus, based on the diffusion coefficient values used in this study (Figure 4), it was found that:

$$\begin{eqnarray}&&0.82A/m\lt x\cdot i\lt 1.45A/m\end{eqnarray} \tag{ 42 }$$

The pit stability product in Eq. 40 is similar to that presented in a recent publication by Nguyen and Newman. ²⁵ Unlike Nguyen and Newman, our model considered variable diffusion coefficients to account for viscosity effects on diffusivity. These variable diffusion coefficients were taken into consideration in mass transport equations, which was not attempted in Ref. 25. The constant diffusion coefficient used by Nguyen and Newman was defined as the equivalent diffusion coefficient (D_eq ) in Eq. 41. Admittedly, using a constant diffusivity value would yield the same solution to Eq. 40. However, our work went beyond a mathematical simplification and provided a physical meaning to the constant diffusion coefficient (i.e., D_eq ), as explained above.

When migration is not considered in the mass transport process, the dissolution current density, i_nm , can be calculated as:

$$\begin{eqnarray}&&{i}_{nm}=2F{D}_{F{e}^{2+}}\left({c}_{F{e}^{2+}}\right)\displaystyle \frac{d{c}_{F{e}^{2+}}}{dx}\end{eqnarray} \tag{ 43 }$$

Based on the mean value theorem for definite integrals, ³⁷

$$\begin{eqnarray}&&{\left(x\cdot i\right)}_{nm}=2F{D}_{eq}^{nm}{c}_{F{e}^{2+},s}\end{eqnarray} \tag{ 44 }$$

Where the equivalent diffusion coefficient without migration, ${D}_{eq}^{nm},$ is

$$\begin{eqnarray}&&{D}_{eq}^{nm}={D}_{F{e}^{2+}}\left(\varepsilon \right)=\displaystyle \frac{1}{{c}_{F{e}^{2+},s}}\displaystyle {\int }_{0}^{{c}_{F{e}^{2+},s}}{D}_{F{e}^{2+}}\left(c\right)dc,\varepsilon \in \left(0,{c}_{F{e}^{2+},s}\right)\end{eqnarray} \tag{ 45 }$$

In contrast to the previous case, ${D}_{eq}^{nm}$ can have an analytical solution and should lie between the values of dilute and saturated solutions as well.

Based on the diffusion coefficient values used in this study (Figure 4), it was found that:

$$\begin{eqnarray}&&0.30A/m\lt {(x\cdot i)}_{nm}\lt 0.53A/m\end{eqnarray} \tag{ 46 }$$

If the pit stability product was evaluated with 1D Fick's law of diffusion, ${D}_{eq}^{nm}$ should be at least 10.12 × 10⁻¹⁰ m² s⁻¹ to reach the lowest (x·i) value in Eq. 42, i.e., where migration is taken into consideration. When FeCl₂ was used as the simplified pit solution, if the 1D Fick's law of diffusion was applied to fit the pit stability product obtained experimentally (Figure 6), a value of ${D}_{eq}^{nm}$ = 12.23 × 10⁻¹⁰ m² s⁻¹ was obtained. Even when a high saturation concentration is considered, i.e., [Fe²⁺] = 5.02 M, for the saturated pit solution and 2.18 is used for charge transfer number, the estimated ${D}_{eq}^{nm}$ needs to be 9.40 × 10⁻¹⁰ m² s⁻¹ to reach the pit stability product. These values are higher than those in an infinitely dilute solution (i.e., 7.1 × 10⁻¹⁰ m² s⁻¹), suggesting that 1D Fick's law of diffusion is not an adequate fit to estimate the pit stability product and will result in the overestimation of the diffusion coefficient and saturation concentration. The analytical solutions shown in Eqs. 40 and 44 indicated that migration could not be ignored during the pit growth and the estimation of pit stability products. Therefore, the pit growth under a salt film is not purely diffusion-controlled, and migration effects should be considered.

The complexation reaction of ferrous ions with Cl⁻ was ignored in this part to validate the analytical results. It is thus necessary to first discuss the reliability of the proposed model. An extensive body of work documents the dependence of IR drop inside the pit and salt film thickness on pit depth. ^{7,14,17,20,32} In this regard, the IR drop inside the pit was found to be independent of pit depth, ^7,14,17,20 whereas the salt film thickness should increase with pit depth. ³²

The IR drop inside the pit and the salt film's thickness during pit growth under a salt film were estimated to validate the model. As seen in Figure 7, in agreement with previous work, ^7,14,17,20 the IR drop inside the pit cavity remained constant. The IR drop value obtained herein was in accord with the values documented in the literature. ^1,17 Similarly, the salt film thickness variation with pit depth was consistent with the findings in the literature. ³² Figure 7 suggested that the salt film thickness had to increase with increasing pit depth to accommodate the extra IR drop during 1D pit potentiostatic anodic polarizations, validating the choice of exchange current density.

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The migration effect on the pit stability product under a salt film was evaluated numerically with the 1D pit model developed herein, while the diffusion coefficient functions shown in Figure 4 were used as inputs. If Fe²⁺ migration was not taken into account, ${D}_{F{e}^{2+}}$ was extracted from the modeling results with migration included as a function of the Fe²⁺ concentration, which incorporates the effect of both NaCl and FeCl₂.

The calculated and experimental mass transport limiting current densities (i_lim) vs the inverse of the pit depth (1/d) with and without migration are presented in Figure 8. It was found that i_lim followed a linear relationship with 1/d, independently of migration inside the pit. However, at the same pit depth, i_lim was almost three times higher when migration was considered, suggesting that migration accounted for ⅔ of the i_lim. Therefore, migration should not be ignored when estimating the (x·i) values under a salt film, consistent with the analytical solutions presented above. The pit stability product with migration was 1.05 A m⁻¹, extracted from the slope of the curve in Figure 8. An error of 5% would be introduced if 4.2 M FeCl₂ is used as the saturated pit solution needed to estimate the pit stability product without considering the complexation reaction.

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Although the analytical solution predicted that an equivalent diffusion coefficient (D_eq ) could replace the variable diffusion coefficient inside the pit, a numerical model was required to obtain D_eq . Figure 9 shows i_lim vs 1/d curves with different D_eq values. The i_lim followed a linear relationship with the 1/d for different D_eq values. The pit stability product varied between 0.81 and 1.38 A m⁻¹, which agreed with the prediction of the analytical solution in Eq. 42. The (x·i) value was between those obtained in dilute and saturated solutions, in accordance with the prediction of the analytical results. When D_eq was 4.83 × 10⁻¹⁰ m² s⁻¹, the results overlapped with those obtained using a variable diffusion coefficient. This D_eq value was close to that of Gaudet et al. (4.98 × 10⁻¹⁰ m² s⁻¹), ¹⁶ and Nguyen and Newman (4.80 × 10⁻¹⁰ m² s⁻¹) ²⁵ when the constant diffusivity with electromigration was considered. Additionally, when D_eq = 4.83 × 10⁻¹⁰ m² s⁻¹ was substituted into Eq. 40, a value of (x·i) = 1.07 A m⁻¹ was obtained under a salt film. This 2% discrepancy with the numerical result could be attributed to the assumptions made in the analytical solutions, i.e., that the net flux of Cl⁻ and Na⁺ was zero. As is shown in Figure 10, with the increase of the pit depth, more chloride migrated into the pit cavity, leading to a non-zero net Cl⁻ flux. These findings confirmed the robustness of the new model to incorporate both continuous pit growth and salt film precipitation.

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As indicated in Eq. 40, the pit stability product is independent of the Cl⁻ diffusion coefficient. Figure 11 illustrates the effect of chloride diffusivity on the pit stability product under a salt film. A linear relationship was found between i_lim and 1/d, the slope of which was independent of the chloride diffusion coefficient. These findings confirmed the analytical results, suggesting that the chloride diffusion coefficient did not influence the (x·i) value under a salt film. In the modeling of pit growth under mass transport control, the viscosity effect on the diffusion coefficient of metal cations needs to be considered, while the viscosity effect on chloride diffusivity could be ignored.

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The pit stability product calculated in Figure 8 was higher than the experimental value by 5%, which could be attributed to the influence of complexation reactions inside the pit that were ignored thus far. In this regard, the charge carried by FeCl⁺ is lower than that of Fe²⁺, which would reduce the flux due to electromigration. Figure 12 compares the experimentally and numerically determined i_lim vs 1/d, which agreed with the expected linear relationship. When a complexation reaction was included, the calculated pit stability product was 1.02 A m⁻¹, in good agreement with the experimental data. The difference between the model and the experimentally determined pit stability product was within 2%. The results indicated that 4.2 M FeCl₂ was an accurate simplification of the saturated solution at the pit. Thus, the estimation of the pit stability product under a salt film under mass transport control does not require considering the complete set of reactions expected in the pit solution based on the 316L SS composition (e.g., hydrolysis and complexation reactions of Cr³⁺, Fe²⁺, Ni²⁺, and Mo³⁺).

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Fe²⁺ and FeCl⁺ were assumed to have the same equivalent diffusion coefficient (D_eq ) inside the pit cavity. Figure 13 shows the simulated i_lim vs 1/d using the proposed D_eq and considering migration inside the pit when the complexation reaction was taken into account. As with the results shown in Figure 9, when D_eq = 4.83 × 10⁻¹⁰ m² s⁻¹, the values overlapped with those obtained with a variable diffusion coefficient, indicating that D_eq can also be used as an approximation even when the complexation reaction was included.

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