*2.4. Electrochemical Test*

Electrochemical measurements were performed using the PARSTAT4000A electrochemical workstation and electrochemical system using the classic three-electrode system with a platinum sheet as the counter electrode, a saturated calomel electrode (SSCE) as the reference electrode, and ductile iron samples as the working electrodes. Electrochemical samples were sealed with high-temperature-resistant epoxy resin. The exposure area was 1 cm2. The open circuit potential (OCP) was measured for at least 30 min until a steady state was reached; then, electrochemical experiments were conducted. Three sets of parallel experiments were set up for each environment to minimize experimental errors [17].

Potentiodynamic polarization curves were obtained by performing kinetic polarization curve measurements with a scan rate of 0.333 mV/s and a scan potential range of −0.5 V/OCP to 0.8 V/OCP.

The test conditions for electrochemical impedance spectroscopy (EIS) in each solution were the same as the working electrode used for the kinetic potential polarization test. The frequency range for EIS was from 100 kHz to 10 mHz with an amplitude of 5 mV (rms) at the open circuit potential.

#### *2.5. Simulation Model Calculations*

The model uses the same solution environment as the autoclave immersion experiment. Finite element simulation was performed using Comsol software [18–21]. Irrespective of insoluble substances, the modeled electrolyte species are H+, OH<sup>−</sup>, Cl<sup>−</sup>, Na+, and Fe2+. Fixed concentrations and electrolyte phase potential were set at the top horizontal electrolyte boundary, facing the air. The iron dissolves at the electrode surface:

$$\text{Fe}^{2+} + 2\text{e}^- \Leftrightarrow \text{Fe} \ (s) \tag{1}$$

Additionally, the kinetics depend on pH (*i*0 proportional to H+):

$$i\_{loc} = i\_0(\exp(\frac{\partial\_a F \eta}{RT}) - \exp(-\frac{\partial\_c F \eta}{RT})) \tag{2}$$

$$i\_{l\alpha} = i\_0(c\_R \exp(\frac{\alpha\_a F \eta\_{ref}}{RT}) - c\_0 \exp(-\frac{-\alpha\_c F \eta\_{ref}}{RT})) \tag{3}$$

Equation (2) is a Butler–Volmer expression [22], where αc denotes the cathodic charge transfer coefficient, *αa* is the anodic charge transfer coefficient, and *i*0 is the exchange current density. The kinetic equation is then coupled with the solution environment. Equation (3) is a Butler–Volmer expression with concentration dependence. This type of expression allows for more freedom in defining the concentration-dependent Butler–Volmer type of expressions, where the anodic and cathodic terms of the current density expression typically depend on the local concentrations of the electroactive species at the electrode's surface. *cR* and *c*0 are dimensionless expressions, describing the dependence on the reduced and oxidized species in the reaction.

*η* is the overpotential given by

$$
\eta = E\_{\rm m} - E\_{\rm exq} \tag{4}
$$

*E*m is the electrode potential and *<sup>E</sup>*eq is the equilibrium potential. The metal potential is set to a fixed value, resulting in a mixed potential not affected by the local pit corrosion. Ions such as Cl− may also be transported in order to maintain electroneutrality. The transport of ions, in combination with the pit shape, determine the local pH. If the iron oxidation reaction is catalyzed by H+, a lower pH within the pit results in faster metal dissolution compared to the metal surface outside the pit. The tertiary current distribution and Nernst–Planck interface defines the mass and ion transport. The water-based charge balance model with electroneutrality defines the H+ concentration and the OH− concentration as built-in variables, and automatically defines the water autoprotolysis equilibrium. The separator node is used to define the pit as a porous structure. Deformed geometry handles the deformation of the pit. The multiphysics nodes couple the electrochemistry to the deformation.

#### **3. Results and Discussion**

#### *3.1. Exploration of the Corrosion Mechanisms of Ductile Iron Shrinkage Holes*
