Fe interface beneath a thin electrolyte

Electrochimica Acta 51 (2006) 3256–3260

Numerical analysis of galvanic corrosion of Zn/Fe interface beneath a thin electrolyte Jong-Min Lee Faraday Technology, Inc., Clayton, OH 45315, USA Received 13 July 2005; received in revised form 26 August 2005; accepted 12 September 2005 Available online 24 October 2005

Abstract A numerical analysis of galvanic corrosion of a Zn/Fe interface beneath a thin layer electrolyte is presented. Specifically, a circular defect, where the zinc coating has been removed, is considered. It is assumed that both oxygen reduction and iron oxidation can occur on the Fe surface, while only zinc oxidation occurs on the Zn surface. The importance of electrolyte thickness and conductivity and defect radius is considered. It is assumed that the iron and zinc oxidation rates are described by a Tafel relationship. If the kinetic parameters of the oxidation reactions are known, the cathodic protection of Fe is a function of a Wagner number, the ratio of the electrolyte thickness to the defect radius, and the ratio of the radius of the defect to the outer radius of the zinc layer. © 2005 Elsevier Ltd. All rights reserved. Keywords: Galvanic corrosion; Galvanized steel; Boundary element method; Galvanic protection; Zn/Fe interface

1. Introduction Zinc coatings on steels are ubiquitous because of their effective corrosion protection and low cost [1]. Zinc-coated steels are thus the most common application of a galvanic couple. Two major zinc-coating techniques are electroplating and a hotdipped method for industrial applications. Under many environmental conditions, zinc may corrode by a factor of 5–100 times slower than iron [2,3]. Thus galvanized steel having a defect-free coating typically is characterized by a low overall corrosion rate. When a defect is formed in the zinc and the underlying iron is exposed, the corrosion of the neighboring zinc may however accelerate significantly because of the newly established galvanic couple. Nevertheless, the iron surface may be protected depending on the composition of any electrolyte on the surface, the thickness of the electrolyte, and the size of the defect. Previous theoretical studies of a galvanic couple include a study by Wagner [4] who assumed a linear polarization law on both the anode and cathode. Waber [5] used linear corrosion kinetics with equal constant polarization parameters. McCafferty [6] extended the previous models assuming linear corrosion kinetics with unequal polarization parameters. His results show E-mail address: [email protected]. 0013-4686/$ – see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2005.09.026

that the local current densities of both the anode and cathode increase with an increase of the electrolyte thickness. In contrast, Zhang and Valeriote [7] reported that the galvanic current increases with a decrease of electrolyte thickness in experimental measurements. However, they concluded that the galvanic corrosion rate of zinc increases with the area of the exposed steel up to a certain area, then decreases slightly with larger area. The present communication provides a treatment including both an iron oxidation and oxygen reduction on the iron surface and zinc oxidation on the zinc to investigate the influence of electrolyte thickness, conductivity, and defect radius systematically on the iron oxidation current distribution on the iron surface, where the zinc coating has been removed. The novel aspect of this paper is an introduction of a method of predicting protection of iron by estimating the potential difference between the center of the defect and the edge of the defect instead of understanding a detailed understanding of the iron and zinc oxidation kinetics. The predictions are compared to the iron oxidation current distribution calculated for the surface with electrode kinetics. 2. Model A schematic diagram of the galvanic cell is shown in Fig. 1. The cell shown in Fig. 1 contains three characteristic dimensions.

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Fig. 2. Assumed polarization curves for anodic reaction on both zinc and iron.

expression) occur: Fig. 1. Schematic diagram of an axisymmetric galvanic-coupled cell, indicating the coordinate system used for all of the results.

It is assumed that three reactions can occur: Zn → Zn2+ + 2e−

(1)

+

(2)

−

O2 + 4H +4e → 2H2 O Fe → Fe

2+

+ 2e

−

(3)

It is assumed that variations in ion concentration are negligible, although this is clearly a gross approximation, especially at high corrosion rates. Given this assumption, the current density is determined from the local gradient of an electrical potential in solution. i = −κ∇φ

(4)

The potential field is determined by Laplace’s equation, which for the present study can be written in cylindrical coordinates as ∂2 φ 1 ∂φ ∂2 φ + + 2 =0 ∂r 2 r ∂r ∂z

(5)

At the air/electrolyte interface, an insulating boundary condition is imposed. At the zinc/electrolyte interface, the zinc oxidation rate is assumed to be given by 
∂φ i0,Zn αa,Zn nF at =− exp (V − φ − U0,Zn ) ∂z κ RT r z = 0, >1 (6) r0 Along the iron surface, both the mass-transfer limited reduction of oxygen and iron oxidation (assumed to be given by a Tafel

nFcsat DO2 i0,Fe ∂φ = − exp ∂z κδ κ r at z = 0, <1 r0






αa,Fe nF (V − φ − U0,Fe ) RT

(7)

where csat is the saturation concentration of oxygen, δ the thickness of the electrolyte, and DO2 is the diffusion coefficient of oxygen. In the present investigation, the saturation concentration of oxygen is assumed to be 2.5 × 10−4 M and the diffusion coefficient of oxygen is assumed to be 1.9 × 10−5 cm2 /s [8]. In all simulations the following kinetic parameters are assumed: i0,Zn = 10−7 A/cm2 , U0,Zn = −0.75 V, i0,Fe = 10−6 A/cm2 , U0,Fe = −0.25 V, αa,Zn = 0.36, and αa,Fe = 0.3 [9]. By arbitrarily setting the metal potential V = 0, the partial current densities as a function of electrolyte potential φ can be calculated. The curves are shown in Fig. 2, along with one oxygen limiting current density, which is independent of potential. 2.1. Dimensionless forms The equations presented in the previous section can be recast in dimensionless form by defining the following variables: δ∗ =

δ , r0

r∗ =

r , r0

φ∗ =

Fφ RT

(8)

For the potential field, the dimensionless equation can be written as 1 ∂φ∗ ∂2 φ∗ ∂2 φ∗ + ∗ ∗ + ∗2 = 0 ∗2 ∂r r ∂r ∂z

(9)

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Along the iron surface, ∂φ∗ i0,Fe r0 F 1 − = exp ∗ ∂z Wa κRT




αa,Fe nF (V − U0,Fe ) RT

× exp(−αa,Fe nφ∗ )

Table 1 Typical variables and values for simulation



(10)

where the Wagner number is defined here in terms of the oxygen limiting current density: Wa =

κRTδ nr0

F 2c

Values

r0 (cm) rf (cm) δ (cm) κ (S/cm) Wa ψ

10−5 to 0.2 10−4 to 1 10−5 to 0.1 10−3 to 0.4 0.02–50 0–20

(11)

sat,O2 DO2

Along the zinc, the dimensionless equation can be written as
 ∂φ∗ i0,Zn r0 F αa,Zn nF (V − U0,Zn ) =− exp ∂z∗ κRT RT × exp(−αa,Zn nφ∗ )

Variables

at z∗ = 0, r ∗ > 1

(12)

The numerical calculations were performed by a boundary element method that has been previously benchmarked [10–14]. The node density was systematically varied to ensure that the numerical error associated with the grid was less than 1%. Typically, 80 points at which results did not change further were used along the electrode surface in all calculations. 3. Results and discussion Fig. 3 shows the influence of the electrolyte thickness on iron-oxidation rate for κ = 0.0462 S/cm, r0 = 0.01 cm, rf /r0 = 2, and varying electrolyte thickness δ. Typical parameters and values for simulation are shown in Table 1. The limitation of range of the parameters used in the present work was not systematically examined. The results indicate that as electrolyte thickness decreases, oxygen mass transfer effects become important and the current distribution then tends to the more non-uniform distribution. In all cases, the oxidation rate is a maximum at the center of the defect. The iron surface close to the boundary

Fig. 3. Iron-oxidation current distribution on the iron surface as a function of an electrolytic thickness for κ = 0.0462 S/cm, r0 = 10−2 cm, and rf /r0 = 2.

between the iron and zinc is relatively more protected. Presumably, the uneven corrosion rate, with a maximum at the disk center, would lead to the formation of a hemispherical-shaped pit. The influence of iron radius for κ = 0.0462 S/cm and rf /r0 = 2 is shown in Fig. 4. It is assumed that the electrolyte thickness is δ = 10−4 cm. The oxidation rate at r0 = 10 ␮m is zero on the entire iron surface. The influence of increasing iron radius is to increase the importance of the electrode kinetics relative to ohmic resistance, and to increase the potential difference between the center and edge of the iron surface, resulting in non-uniform oxidation current distribution. The results are not in agreement with previous studies [7]. The source of the discrepancy between simulation and experiment is possibly due to the mass-transfer limitation of oxygen on the iron surface. Nevertheless, non-idealities, such as corrosion products [1,15] or solid films [16], may play an important role in dictating current distribution. In addition to the non-idealities under a thin electrolyte, it was indicated that the solubility of the dissolved species is limited [1]. Such a case, the concentrations of the dissolved ions at the electrode surface are dictated by the presence of salt films or corrosion products, which are formed when the solubility of the dissolved ions is exceeded. The reaction rates are then governed

Fig. 4. Iron-oxidation current distribution on the iron surface as a function of an iron radius for κ = 0.0462 S/cm, δ = 10−4 cm, and rf /r0 = 2.

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quent potential difference. A dimensionless potential difference between the center and the edge of the defect is defined as 

  Fφ   Fφ   ψ= −   RT r=0 RT r=r0 

Fig. 5. Iron-oxidation current distribution on the iron surface as a function of electrolyte conductivity for δ = 10−4 cm, r0 = 0.01 cm, and rf /r0 = 2.

by the diffusion of oxygen into the iron surface, by the diffusion of the dissolved iron ions away from the iron surface, and by the diffusion of the dissolved zinc ions away from the zinc surface. The non-idealities cannot be considered in the present simulations. Additional works are necessary for definitive conclusions. The iron-oxidation partial current distribution as a function of electrolyte conductivity for δ = 10−4 cm, r0 = 0.01 cm, and rf /r0 = 2 is shown in Fig. 5. The oxidation rate near the center significantly increases with a decrease of electrolyte conductivity. This results from increasing the influence of ohmic resistance as compared to kinetic effects. However, at extremely low conductivities, current could not flow in practical system because the electrolyte would behave like an insulator [16]. Consequently, the conductivity except the extreme case plays an important role in dictating the current distribution on the iron surface with the electrolyte thickness and the radius of the defect. When a ratio of the radius of the defect to the outer radius of the zinc layer was less than 0.5, zinc area was found not to be an important parameter to influence current distributions on the iron surface in the preliminary results although the results are not shown here. A maximum peak current of zinc oxidation was found on the zinc surface closest to the boundary between the iron and zinc. This is generally true near the boundary, where current density on both the sites is higher in absolute magnitude than elsewhere [17,18]. The question of protection of the defect can be addressed without a detailed understanding of the iron and zinc oxidation kinetics. Instead, it is sufficient to note that the oxidation potential of iron is separated by approximately 200 mV [7]. Thus the potential difference between the center and the edge of the defect may provide a method of predicting protection. This can be estimated by setting the iron-oxidation current to zero, only allowing oxygen reduction on the iron surface, and calculating the subse-

(13)

At room temperature, ψ = 7.79 corresponds to 200 mV that is the most crucial value of investigation to understand the prediction of the defect. Thus, when ψ < 7.79, the defect should be protected. Fig. 6 shows the variation of dimensionless potential difference with (1/Wa)(1 + r0 /δ) for different δ/r0 , assuming rf /r0 = 2. This ratio is large enough to approximate an isolated defect. (1/Wa)(1 + r0 /δ) represents an importance of the ratio of the electrode kinetics to ohmic resistances relative to the dimensionless geometric parameter r0 /δ and is introduced for the design analysis. When δ/r0 > 10, the potential dependence is only a function of 1/Wa, and all curves collapse onto to the δ/r0 = 10 curve. Furthermore, for large electrolyte thicknesses, protection is predicted when 1/Wa < 18. For values of δ/r0 < 1, all curves nearly collapse onto the δ/r0 = 0.1 curve. In such cases, protection is predicted when (1/Wa)/(1 + r0 /δ) < 34. The simulation results shown in Figs. 3–5 are for small δ/r0 , as general cyclic, environmental corrosion is the present focus. The predictions from Fig. 6 are in good agreement with the more complete simulation results of Figs. 3–5. For example, the values of (1/Wa)/(1 + r0 /δ) in Fig. 3 were 62.71 for δ = 0.5 ␮m, 51.37 for δ = 0.55 ␮m, 43.17 for δ = 0.6 ␮m, and 31.65 for δ = 0.7 ␮m, respectively. However, the employment of a Tafel kinetics relationship for the iron-oxidation rate precludes attaining a zero corrosion rate. Thus whenever the iron-oxidation rate was less than 1% of the oxygen reduction rate, the surface was considered to be protected.

Fig. 6. Dimensionless potential differences along the surface of the defect, as estimated by neglecting the iron-oxidation reaction.

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4. Conclusions A numerical analysis of the galvanic corrosion of the Zn/Fe interface under a thin layer electrolyte has been studied. Specifically, a circular defect, where the zinc coating has been removed, was considered. Assuming both oxygen reduction and iron oxidation on the Fe surface and zinc oxidation on the Zn surface, results showing the iron-oxidation current distribution on the iron surface were obtained. The uniformity of the iron-oxidation current distribution depends on the electrolyte thickness and conductivity as well as the defect radius. The cathodic protection of Fe is a function of a Wagner number, the ratio of the electrolyte thickness to the defect radius, and the ratio of the radius of the defect to the outer radius of the zinc layer if the kinetic parameters of the oxidation reactions are known. A method of predicting protection of the defect was provided by calculating the subsequent potential difference between the center and the edge of the defect without a detailed understanding of the iron and zinc oxidation kinetics. Appendix A. Nomenclature

csat D F i i0 n r rf r0 R T U0,Fe U0,Zn V Wa z

saturation concentration of oxygen (mol cm−3 ) diffusion coefficient of oxygen (cm2 s−1 ) Faraday’s constant (96,500 C mol−1 equiv.−1 ) current density (A cm−2 ) exchange current density (A cm−2 ) valence number (equiv./mol) radial coordinate (cm) outer radius of zinc (cm) radius of iron (cm) universal gas constant (8.314 J mol−1 K−1 ) temperature (K) reversible potential of iron (V) reversible potential of zinc (V) metal potential (V) Wagner number in terms of the oxygen limiting current density coordinate normal to the surface (cm)

Greek letters anodic charge transfer coefficient of iron αa,Fe αa,Zn anodic charge transfer coefficient of zinc δ electrolyte thickness (cm) κ electrolyte conductivity (−1 cm−1 ) φ potential in electrolyte (V) ψ dimensionless potential difference between the center and the edge of the defect ∇ gradient operator References [1] G.A. El-Mahdy, A. Nishikata, T. Tsuru, Corros. Sci. 42 (2000) 183. [2] C.J. Slunder, W.K. Boyd, Zinc: Its Corrosion Resistance, 2nd ed., International Lead Zinc Research Organization, Inc., New York, 1986. [3] Metal Corrosion in the atmosphere, STP 435, ASTM, Philadelphia, PA, 1968, p. 360. [4] C. Wagner, J. Electrochem. Soc. 98 (1951) 116. [5] J.T. Waber, J. Electrochem. Soc. 102 (1955) 420. [6] E. McCafferty, J. Electrochem. Soc. 124 (1977) 1869. [7] X.G. Zhang, E.M. Valeriote, Corros. Sci. 34 (1993) 1957. [8] J. Newman, Electrochemical Systems, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1991. [9] D.A. Jones, Principles and Prevention of Corrosion, Macmillan, Englewood Cliffs, NJ, 1992. [10] R.A. Adey, S.M. Niku, ASTM Spec. Tech. Publ., STP 1154 (Comput. Model. Corros.) (1992) 248. [11] R.S. Munn, ASTM Spec. Tech. Publ., STP 1154 (Comput. Model. Corros.) (1992) 215. [12] C.A. Brebbia, S. Walker, Boundary Element in Engineering, Butterworths, 1979. [13] S. Aoki, K. Kishimoto, M. Miyasaka, Corrosion 44 (1988) 926. [14] S. Aoki, K. Kishimoto, Math. Comput. Model. 15 (1991) 11. [15] M. Stratmann, H. Streckel, K.T. Kim, S. Crockett, Corros. Sci. Sci. 30 (1990) 715. [16] G. Prentice, W.H. Smyrl, J. Newman, Mass Transport and Potential Distribution in the Geometries of Localized Corrosion, Perspectives on Corrosion, AIChE Symposium Series No. 278, vol. 86, 1990, p. 1. [17] M.J. Pryor, D.S. Keir, J. Electrochem. Soc. 104 (1957) 269. [18] W.H. Smyrl, J. Newman, J. Electrochem. Soc. 123 (1976) 1423.