Application of Laplace Equation-Based Modeling into Corrosion Damage Prediction for Galvanic Coupling Between a Zinc Plate and Stainless Steel Rods Under a Thin Film Electrolyte

Connections between dissimilar materials are frequently encountered in aviation and marine structures, and thus there is an increase in the likelihood of galvanic corrosion when an ionically conductive solution is present. There have been extensive experimental investigations of galvanic corrosion between dissimilar metals or alloys, but many of them used full immersion conditions to simulate an atmospheric corrosion environment. This approach may fail to fully capture the important characteristics of atmospheric exposures. Atmospheric corrosion is a complicated phenomenon affected by multiple external variables¹  (e.g., relatively humidity, solution chemistry, electrolyte layer thickness/droplet size), and when galvanic coupling with a complicated geometry is involved, conventional experimental characterization on galvanic-couple-induced atmospheric corrosion is difficult.

Computational modeling can be combined with experimental characterization to increase the understanding of the dependence of the corrosion susceptibility on external variables. This complementary computational and experimental approach quantifies the important characteristics that control corrosion rate and resultant corrosion morphology. There are several computational modeling approaches available for corrosion studies, but among these, the finite element method (FEM) has been most widely used to study potential and current distributions, as well as transport phenomena in corroding systems.^2-3  There are two prevailing approaches in FEM-based corrosion modeling which are differentiated based on the governing equation used: the Nernst–Planck Equation or the Laplace Equation. The Nernst–Planck Equation takes into account all of the contributions from diffusion, migration, and convection. The solution to Nernst–Planck Equation results in a full transient expression of the distributions of concentration, potential, and current, but at the cost of very long processing time and increases computational complexity. Recent work^4-9  has demonstrated the utility of the use of Laplace’s Equation in the modeling of electrochemical distributions by coupling it with experimentally derived electrochemical kinetics as boundary conditions along with estimations of the electrolyte conductivity. However, the accuracy of numerical solutions by the Laplace Equation needs to be assessed for predicting realistic experimental results.

Stainless steel (SS) is a typical cathode material involved in a number of galvanic coupling systems such as aluminum alloy (AA)/SS,^10-13  zinc (Zn)/SS,^14-15  and magnesium alloy/SS.^16-18  Especially for AA/SS or Zn/SS galvanic couples, the galvanic coupling potential is usually located at the cathodic mass-transfer-limited region of SS based on mixed potential theory assuming a cathode-to-anode area ratio is 1∶1,^4,19  which implies cathodic diffusional kinetics of oxygen reduction reaction (ORR) on SS can further impact the degree of galvanic corrosion if corrosion exposure environment can affect the mass transfer of dissolved oxygen in the electrolyte. Zinc (Zn) is an electrochemically active metal material which generally experiences uniform corrosion when coupled with noble materials in saline solutions, leading to corrosion damage that is more straightforward to assess as compared to materials such as AAs which are prone to localized corrosion. There have been a number of studies focused on exploring the electrochemical, chemical, and corrosion distributions for Zn/noble material couples. Tahara and Kodama²⁰  utilized a Kelvin probe to measure the potential distribution along the Zn/SS couple under a thin layer of electrolyte, and they found that the effective galvanic corrosion distance along the Zn away from the Zn/SS interface was proportional to electrolyte resistance and water layer thickness. Tada, et al.,^21-22  developed both [Zn²⁺] and pH sensors to determine [Zn²⁺] and pH distributions along a Zn/Steel couple in 0.01M NaCl. Simões, etal.,¹⁵  applied both the scanning vibrating electrode technique (SVET) and the scanning electrochemical microscope (SECM) to investigate current distribution along a Fe-Zn cell. As for modeling efforts, Lee²³  used the Laplace Equation to solve for the potential distribution along the Zn/SS galvanic couple interface under a thin layer electrolyte and concluded that the degree of corrosion is affected by electrolyte thickness, conductivity, and the properties of surface.

The objective of this study is to access the application of the Laplace Equation for the prediction of potential, current, and corrosion distributions along a Zn plate with SS rods in the plate under a controlled simulated thin layer electrolyte. The galvanic corrosion as a function of electrolytic conductivity, electrolyte layer thickness, and spacing between two adjacent SS rods has been evaluated numerically. Experimentally determined kinetics of each material serve as boundary conditions. ASTM B117 testing for Zn plate/SS rods galvanic couple followed by optical profilometry was conducted to utilize the numerically predicted results to estimate the electrolyte layer thickness during a modified ASTM B117 testing.

In a dilute electrolyte, the current density can be expressed as²⁴   where c_j is the concentration of species j, D_i is the diffusivity of species i, z_j is the charge number of species j, F is Faraday’s constant, ϕ is the electric potential, u_j is the electrochemical potential of species j, and ν is the fluid velocity.

Three assumptions are typically made in order to simplify the solution to this equation. The first is electroneutrality, which is shown in Equation (2): The second assumption is that the migration term overwhelms diffusion term in Equation (1) and, finally, there is no convection motion in bulk solution. Hence, Equation (1) is simplified to Equation (3): where conductivity (κ)=F²∑_jz_j²u_jc_j. By enforcing the conservation of charge (Equation [4]), the Laplace Equation can be obtained, which is the governing equation used in the modeling framework

A workstation with quad-core processor and 8 GB of RAM was used to perform the mathematical modeling in this work. The Secondary Current Distribution Module at steady state in COMSOL^† Multiphysics software (version 5.3a) was applied to calculate electrochemical distributions in the simulated geometry of the galvanic couple. The modeling domain of interest is the electrolyte only. Two objectives were the focus of the modeling: (1) to investigate the influence of solution parameters (electrolyte layer thickness and solution conductivity) and geometric parameters on electrochemical distributions along the galvanic couple surface; and (2) to compare the modeling results with those from exposure testing to demonstrate the application of the Laplace Equation for corrosion distribution prediction. Two configurations were used in the first modeling category. A single SS rod was set at the center of the Zn plate in Configuration 1, whereas two SS rods was set at the center and side of the plate in Configuration 2. The details of dimensions of two configurations are shown in Figure 1. The electrolyte layer thickness ranged from 5 × 10⁻³ cm (50 μm) to 0.4 cm, and solution conductivity ranged from 0.5 S/m to 5.5 S/m. For spacing between the centers of two SS rods in Configuration 2, three different values were chosen: 1.5 cm (short), 2.5 cm (medium), and 4 cm (long). Electrochemical kinetics of the Zn and SS were mathematically fitted to facilitate the boundary condition implementation. Anodic kinetics of Zn was applied on the Zn plate as the boundary condition, whereas the cathodic kinetics of SS was applied on the SS rod(s) as the boundary condition for those areas.

SS UNS S31600 (composition shown in Table 1) and 99.9 wt% pure Zn were used in this work. They were prepared into 2 cm × 2 cm square coupons and ground to a surface finish of 600 grit with silicon carbide paper. All of the electrochemical measurements were conducted using a Bio-Logic SP200^† potentiostat in a three-electrode cell configuration with a SS (or Zn) coupon as the working electrode (WE, exposure area = 1 cm²), a platinum-niobium mesh as the counter electrode (CE), and a saturated calomel electrode (SCE) as the reference electrode (RE). Potentiodynamic polarization measurements were performed at a scan rate = 0.167 mV/s after 1-h open-circuit potential (OCP) stabilization. For SS, the scan started 100 mV above OCP before reaching the final potential of −1,400 mV_SCE; for Zn, the scan began 100 mV below OCP and ended at 300 mV above OCP. Two types of solutions were tested at 35°C: 0.6 M NaCl and 0.6 M NaCl + 0.04 M K₂S₂O₈. The latter solution with potassium persulfate added was used to accelerate the galvanic corrosion rate in the salt spray testing described in the following subsection.

Zn/SS galvanic couples in Configuration 1 with surface finish equal to 320 grit were tested in the experiment. Optical microscope was used to ensure that no crevice appeared in the interface between SS rod and the Zn surrounding. All of the test samples were horizontally placed in the slots of test rack in a salt fog testing chamber, with a tilt angle equal to 30° with respect to the rack bottom.

A modified salt fog test was utilized in which the testing solution was 0.6 M NaCl with addition of 0.04 M K₂S₂O₈ and the temperature was set at 35°C. A 96-h test with constant salt spray was conducted with a 5-min interruption every 24 h to remove accumulated corrosion products. During this interval, test specimens were sonicated in ammonium persulfate solution (10 wt%) for 1 min to remove the majority of the corrosion products accumulated on the Zn plate. The purpose of this treatment was to minimize the effects of corrosion products on the thin film electrolyte layer on the testing specimens, and to maintain a relatively uniform testing solution composition on the specimen surfaces during the entire testing period. The testing samples were thoroughly ultrasonically cleaned with 10 wt% ammonium persulfate solution for 5 min and rinsed with DI water at the end of the salt fog testing. The corrosion damage was then measured using Zygo NewView™ 7300 Optical Surface Profiler.

Polarization curves of both Zn and SS were mathematically fitted using EC-lab software (Ver.11.01), as shown in Figure 2. For Zn, only the anodic kinetics portion was taken into account from E_corr to −0.95 V_SCE. As for SS, the fitted region started from −0.3 V_SCE down to −1.4 V_SCE. This fitting allowed the removal of the effects of uncompensated IR drop during the generation of the Zn polarization curve at high current density as well as allowing representation of mass transfer limited cathodic kinetics of SS as a function of thin film electrolyte layer thickness. Specifically, the cathodic limiting current density (i_{lim, WL}) was determined via Equation (6), by knowing the value in quiescent solution from the original curve where a reduction in the electrolyte layer thickness on the SS surface increases the pertinent i_{lim, WL}. Actually, i_{lim, WL} is proportional to reciprocal of WL according to Fick’s 1st Law, but 800 μm was chosen for the critical natural convection diffusional layer thickness based on recent work.^1,25  The determination of this value has been discussed elsewhere.⁴  The net result of the natural convection limit is that the dependence of the diffusion limited current density can be described by Fick’s 1st Law as long as WL is smaller than 800 μm, whereas it is a constant regardless of WL thickness above that value. It should be also noted that only mass transfer limited cathodic kinetics of SS in terms of ORR were fitted and extrapolated as the activation region of ORR is not affected by the WL. These fitted and extrapolated electrochemical kinetics of Zn and SS were assigned as anodic (Zn) and cathodic (SS) boundary conditions in the modeling work.

In this subsection, the effects of spacing and the number of SS rods are addressed. For this portion of the work, the WL was set equal to 800 μm and was uniformly distributed across the galvanic couple surface. The conductivity was set to 5.5 S/m. In Configuration 1, the peak current density is at the SS/Zn interface, as expected, with the current density decreasing toward a plateau with increasing distance. This plateau is due to the high conductivity and thickness of the WL. A comparison of dissolution current density distributions away from the center of centric SS rod (center of Zn plate) along the Zn plate, between Configuration 1 and Configuration 2 with three different spacings is shown in Figure 3. When a second SS rod was introduced (Configuration 2), the average dissolution current density between the two SS rods increased relative to Configuration 1. As the spacing between the two SS rods decreased, the average dissolution current density became more intensified, implying the most severe corrosion damage would appear between the two SS rods in the short spacing scenario. Furthermore, the decay in current density away from the Zn/SS interface within the spacing was steeper as the spacing became longer. The peak current density appeared at Zn/SS interface, and did not change significantly when the spacing changed from 4 cm to 2.5 cm, and then rose up when the spacing shrank to 1.5 cm, which was nearly 1.5 times that for a medium or short spacing scenario.

Solution conductivity and electrolyte layer thickness (WL) are the two main electrolyte parameters studied in this subsection. An assumption was made under which electrolytic conductivity is independent of the concentration of solution (fixed at 0.6 M NaCl during the simulation to allow use of the same polarization kinetics). Although it is not physically true, because solution conductivity is a function of solution chemistry, the purpose of this study is to concentrate on the effect of solution conductivity on the potential and current density distributions in the galvanic coupling, making this assumption reasonable for a decent range of solution conductivity which was from 0.5 S/m to 5.5 S/m.

For Configuration 1, the current density distribution as a function of solution conductivity for two different WLs is shown in Figures 4 (4,000 μm) and 5 (50 μm), respectively. When WL = 4,000 μm, the lowest solution conductivity (0.5 S/m) resulted in the highest peak current density at the SS/Zn interface, with the current density decaying sharply with distance from the interface, compared to a more sluggish decay when the conductivity was equal to 5.5 S/m, the peak current density for 0.5 S/m was almost twice that for 5.5 S/m. Whereas when WL = 50 μm (Figure 5), the highest peak current density (0.5 S/m) was more than four times that for 5.5 S/m, a sharper drop in current density occurred, and the large dissolution current density was constrained to a shorter distance away from the interface compared to WL = 4,000 μm, implying that larger ohmic resistance brought by thinner WL electrolyte layer resulted in more severe corrosion attack at the interface before decaying dramatically, and the corrosion attack at the interface became more obvious at lower solution conductivity.

Configuration 2 allows the case of a shorter SS spacing was analyzed. A comparison of current density distributions as a function of WLs for two different solution conductivities is shown in Figures 6(a) and (b). For a solution conductivity of 5.5 S/m, a flat current density distribution between the two SS is observed, and the average current density increased mildly as WL decreased from 4,000 μm to 800 μm. As the WL decreased from 800 μm to 50 μm, a nonuniform current density distribution appeared in which peak current density was at Zn/SS interface and a minimum in current density was at the center of the spacing. Overall, the minimum current density increased as the WL decreased for a conductivity of 5.5 S/cm. However, when conductivity was decreased to 0.5 S/m, although the peak current densities at the Zn/SS interface were all higher than those under 5.5 S/m, the minimum current density decreased with decreasing WL from 200 μm to 50 μm, resulting in a lower minimum for WL = 50 μm than for WL = 4,000 μm. This phenomenon indicates that although lower WL resulted in enhanced cathode delivery capacity of SS rod (due to higher diffusion limited cathodic current densities) which intensified corrosion damage at the Zn/SS interface, the high ohmic resistance due to a thin WL and low solution conductivities prevented the cathode from delivering current far from the interface. This situation can be confirmed from the comparison of current density distributions as a function of solution conductivity for WL = 4,000 μm and 50 μm, respectively (Figures 6[c] and [d]). Both the peak current density and the minimum current density increased as the solution conductivity decreased when WL = 4,000 μm, whereas when WL = 50 μm, the minimum current density as well as average current density decreased as conductivity decreased.

To explore the combined effects of geometric and electrolyte parameters on the galvanic corrosion damage, an example will be discussed in this subsection to demonstrate how spacing between two SS rods and WL affected the average dissolution current density between the two rods in Configuration 2. The definition of average current density is the integral of current density along the length of the Zn spacing divided by the total length of the Zn spacing. The solution conductivity was set at 5.5 S/m. It should be noted that the long, medium, and short spacings were assigned to be 4 cm, 2 cm, 5 cm, and 1.5 cm, as specified previously. A comparison of average current densities as a function of WL for three different spacings is shown in Figure 7. Under the same WL, the average current density increased with reduced spacing due to the more effective interaction between Zn/SS brought by the shortened ohmic resistance distance (spacing), the degree of this increase became more significant as WL decreased below 800 μm, owing to enhanced cathodic kinetics of SS rod brought by enhanced mass transfer of oxygen. For the same spacing, an almost constant average current density was found for the long spacing above 800 μm. It then rose gradually as the WL kept decreasing. The medium spacing experienced a similar change in average current density, except that average current density started to increase once WL was thinner than 2,400 μm. For the short spacing, the average current density increased for all decreased WL, and the degree of increase was more dramatic once WL was below 800 μm. It can be concluded that the lowest electrolytic ohmic resistance from short spacing resulted in more interaction between two SS rods and the Zn portion in between, and the galvanic corrosion became more significant as WL went below 800 μm, thanks to the enhanced mass transfer limited cathodic kinetics from SS cathode to provide more cathode current to maintain the corrosion of Zn.

This subsection focuses on the effect of cathode current availability from SS rod on the galvanic corrosion. Changes in cathode kinetics could occur due to a change in the surface property of SS, or the presence of corrosion products, or an insulating coating. To assess such effects, a comparison of Zn dissolution current map on the plate from the top view is shown in Figure 8, in which the cathode current from SS was reduced from 100% (Figure 8[a]) to 50% (Figure 8[b]), and 10% (Figure 8[c]) of its original value. The WL was set equal to 200 μm, and solution conductivity was set to be 5.5 S/m. It can be seen that, for Configuration 2 with a short spacing between the SS, as the cathode current availability decreased, the interaction between Zn and SS rods within the spacing become disconnected. The reduced cathode current delivery capacity hence made the effective area ratio of cathode to anode smaller than the one with full current delivery capacity, resulting in less corrosion-attacked phenomenon taking place at the Zn/SS interface. Although the IR drop between two cathodes became smaller which would have led to more galvanic corrosion in between because of reduced IR, the effect of the significant reduction in cathodic current availability overwhelmed the reduced IR drop effect which made the area and degree of galvanic corrosion dramatically decreased. This result also implies that a corrosion product accumulation as well as protective coatings on the cathode could actually mitigate the corrosion damage.

In order to validate the application of the Laplace Equation-based modeling work to corrosion damage distribution prediction, the corrosion damage occurring on the Zn surface near the SS rods after 96 h salt fog testing was compared to the modeling results in Configuration 1. The main goal of this comparison is to back calculate the estimated electrolyte layer thickness during a B117 test, due to the technical difficulty of monitoring and measuring the real electrolyte layer thickness during the experimental test.

Modeling work in this section used electrochemical kinetics of Zn and SS in 0.6 M NaCl + 0.04 M K₂S₂O₈ as new boundary conditions which are shown in Figure 9. It can be seen that in a quiescent solution, the mass transfer limited cathodic current density was about 100 times that observed in 0.6 M NaCl solution, which is the main reason why persulfate species was added into NaCl solution as it would be expected to accelerate the galvanic corrosion damage during a 96-h salt spray testing so as to obtain measurable corrosion damage on the Zn plate. In the simulation, a parametric sweep in terms of WL from 4 mm to 50 μm at a fixed conductivity of 5.5 S/m was performed to obtain a series of dissolution current density distributions and then found out under what WL the modeling result matched well with experimental results.

The resultant top view and 3D view of corrosion depth distribution of Zn plate in the vicinity of center SS rod were shown in Figure 10. A comparison of corrosion depth distributions for three samples which were obtained by averaging the depth distributions along the three directions in each sample (shown in Figure 10[a]) was displayed in Figure 11[a]. The average corrosion depth profile based on measurements from these three samples then compared with modeling results, in which current density profiles were converted into corrosion depth profile using Faraday’s Law where i is the current density, M_Zn is the molar mass of Zn (M_Zn = 65.38 g/mol), n is the number of electron transferred (n = 2), t is the exposure time (t = 4 × 24 × 3600 = 345,600 s), ρ_Zn is the density of Zn (ρ_Zn = 7,140 kg/m³), F is Faraday’s constant (F = 96,485 C/mol), and x is the corrosion depth. The comparison is shown in Figure 11(b). From the modeling prediction, the corrosion depth distribution at each WL decreased quickly from the SS/Zn interface until approximately 0.1 cm, and then kept decreasing slowly afterward. A zoom-in of Figure 11(b) focusing on the first 0.1 cm away from the interface is available in Figure 11(c). It is indicated that WL = 3,500 μm would be the appropriate WL formed on the sample surface during salt fog testing as it best matches the profile for the majority of the distribution. Close to the interface it is likely that rapid formation of corrosion products limited the attack.

Laplace Equation-based modeling coupled with mathematically fitted electrochemical kinetics from an experiment as a boundary conditions was applied to simulate galvanic corrosion between Zn plate and SS 316 rods under thin layer electrolyte condition. The effect of geometric, electrolytic parameters on the corrosion distribution had been examined. The modeling results were then compared with experimental results from a 4-d modified B117 test by adding sodium persulfate to predict the thickness of thin film electrolyte during the test from modeling.

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  Considering geometric parameters, increasing the number of cathode rods (SS) results in higher current density (corrosion) on the anode (Zn) due to higher cathode-to-anode ratio; shorter spacing between two adjacent SS rods results in more severe corrosion damage.

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  Considering electrolytic parameters, the high ohmic resistance due to thin WL and low solution conductivities prevented the cathode from delivering a current far away from the interface.

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  For the combined effect of both geometric and electrolytic parameters, the lowest electrolytic ohmic resistance from shortest spacing resulted in more interaction between two SS rods and the Zn portion in between, and the galvanic corrosion became more significant as WL went below 800 μm, due to the enhanced mass transfer limited cathodic kinetics from SS cathode to provide more cathode current to maintain the corrosion of Zn.

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  It was found that an effective WL of 3,500 μm can be used to replicate the damage formed on the sample surface during modified B117 salt fog testing.

The financial support from Office of Naval Research (ONR) via Grants no. N00014-14-1-0012 and no. N00014-17-1-2033, Sea-Based Aviation Program, William Nickerson, Program Manager is gratefully acknowledged.