Frumkin correction of corrosion electric field generated by 921A-B10 galvanic couple

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Frumkin correction of corrosion electric field generated by 921A-B10 galvanic couple Qinglin Xu a, Xiangjun Wang a, *, Chong Xu b a b

College of Electrical Engineering, Naval University of Engineering, Wuhan, 430033, PR China College of Education, Central China Normal University, Wuhan, 430079, PR China

A R T I C L E I N F O

A B S T R A C T

Keywords: Corrosion electric field Frumkin correction Electric double layer Diffuse layer potential 921A-B10 galvanic couple Galvanic potential

In order to study the Frumkin correction effect of corrosion electric field generated by 921A-B10 galvanic couple in different concentrations of NaCl solution, the electric double layer (EDL) capacitance of 921A steel and B10 copper alloy at the galvanic potential was acquired by electrochemical impedance spectroscopy (EIS), which was calculated for the experimental value of diffuse layer potential. The theoretical value of diffuse layer potential was got by the theoretical relationship between the diffuse layer potential and the total potential difference of EDL and concentration of supporting electrolyte. According to the Frumkin correction theory, the modified Tafel equation related to diffuse layer potential was obtained. Furthermore, the uncorrected and corrected Tafel equation was used as the boundary condition of the boundary element model (BEM) of corrosion electric field for 921A-B10 galvanic couple, thus the corrosion electric field distribution before and after Frumkin correction was obtained and compared with the experimental results. The results show that the correction effect is 35.6% and 8.8% when the concentration of NaCl is 0.012 M and 0.6 M, respectively. Therefore, it is necessary to correct the Tafel boundary condition during simulation, especially under low electrolyte concentration. The Frumkin correction can greatly reduce the error between results of simulation and experiment.

1. Introduction The current field generated by corrosion and anti-corrosion current of ship is called ship corrosion electric field [1–3]. Relevant research shows that ships can be detected or located by corrosion electric field, therefore, the ship corrosion electric field is a new type of underwater signal source [4,5]. The measurement of corrosion electric field of ship takes a lot of cost and a long time, furthermore, it can only obtain the path distribution of corrosion electric field, and it is difficult to obtain the electric field distribution in a certain plane and three-dimensional space under water [6,7]. However, the boundary element method is a high-precision numerical calculation method, which is very suitable for solving the open-field problem [8]. Therefore, this method has been widely used in the simulation of corrosion protection of marine structures such as oil rigs and ships [9,10]. In recent years, some scholars have begun to apply the boundary element model (BEM) to the simulation study of ship corrosion electric field [11–13]. The boundary condition of the BEM for corrosion electric field of ship is the Butler-Volmer equation describing the dynamics of electrode process. In the derivation of the Butler-Volmer equation, it is assumed that the potential difference of electric double

layer (EDL) is completely distributed in compact layer, that is, the potential of diffuse layer is zero. However, the Stern model of EDL which is closer to the actual situation shows that the EDL is consisted of a compact layer caused by electrostatic action and a diffuse layer caused by thermal motion of molecule [14,15], as shown in Fig. 1(a). Correspondingly, the total potential difference ϕa of EDL includes the compact layer potential difference ϕa
ϕ2 and the diffuse layer potential difference ϕ2 , as shown in Fig. 1(b). Consequently, the Butler-Volmer equation ignores the effect of changes in structure of EDL on the electrochemical reaction rate. In the theory of electrode kinetics, the effect of diffuse layer potential on the charge transfer rate is called Frumkin correction [16–19], namely, Ψ 1 effect or double layer effect [20,21]. The Frumkin correction is a more accurate method of considering the structure of EDL, and it assumes that the electron transfer takes place on the Helmholtz plane, which gives the following two corrections [22–25]: (1) The effective potential difference driving electron transfer is not the total potential difference between metal and solution phase, but the potential difference between electrode surface and the Helmholtz plane;

* Corresponding author. E-mail address: [email protected] (X. Wang). https://doi.org/10.1016/j.jelechem.2019.113599 Received 15 July 2019; Received in revised form 22 October 2019; Accepted 24 October 2019 Available online xxxx 1572-6657/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: Q. Xu et al., Frumkin correction of corrosion electric field generated by 921A-B10 galvanic couple, Journal of Electroanalytical Chemistry, https://doi.org/10.1016/j.jelechem.2019.113599

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Fig. 1. Schematic diagram of the Stern model.

2. Experiment procedure

(2) The concentration of reactive species on the Helmholtz plane is not the concentration of bulk solution, but it is given by the Boltzmann distribution:
 zi e ni ¼ n0i exp
ϕ2 kB T

2.1. Materials The high strength low alloy 921A steel was used for the hull and rudder and the chemical composition (mass fraction, %) is as follows: C ¼ 0.07–0.14; Si ¼ 0.17–0.37; Mn ¼ 0.30–0.60; S  0.015; P  0.020; Ni ¼ 2.60–3.00; Cr ¼ 0.90–1.20; Mo ¼ 0.20–0.27; V ¼ 0.04–0.10 and a Fe balance. The chemical composition of B10 copper alloy with good corrosion resistance that was used for the propeller is as follows (mass fraction, %): Ni ¼ 10.02; Fe ¼ 1.54; Mn ¼ 0.10; P < 0.02; S < 0.02; Zn ¼ 0.13; Pb < 0.2 and a Cu balance [13].

(1)

where ni is the concentration of species i in the Helmholtz plane, n0i is the concentration of species i in bulk solution, zi is the number of charges of species i, e is the charge of elemental charge, ϕ2 is the potential at Helmholtz plane, kB is Boltzmann constant and T is the absolute temperature. The Frumkin correction theory has been applied to many fields such as corrosion [26,27], nanoelectrode [28,29], and chemical battery [24,30] since it was proposed in 1933. Z. Nagy [27] pointed out that the calculation error caused by the neglect of the double layer effect in the electrochemical corrosion rate determination from polarization measurements is more than 10% when the solution concentration is lower than 0.1 M or the corrosion potential is less than 0.25 V respect to the potential of zero charge. In summary, the application of the Butler-Volmer equation as the boundary condition for the BEM of ship corrosion electric field may cause a large error between the results of simulation and measurement. Therefore, two problems were studied mainly based on electrochemical theory and electrochemical testing technology in this paper: one is to explore the necessity of Frumkin correction for the Butler-Volmer boundary condition; the other is to investigate the Frumkin correction effect in 0.012 M and 0.6 M NaCl solution. This is because the conductivity scales down accordingly in the physical scale modeling (PSM) [31, 32], the 0.6 M NaCl solution simulating seawater is correspondingly reduced to 0.012 M if the ratio of the ship prototype to the scaled model is 50:1. The steel hull and copper alloy propeller consist of a circuit of hull-seawater-propeller-shaft-hull in seawater, which constitutes galvanic corrosion. In order to simplify the problem, the commonly used hull material of 921A steel and propeller material of B10 copper alloy were taken as the research objects. The BEM of corrosion electric field of 921A-B10 galvanic couple was established, which lays a foundation for the establishment of the BEM of corrosion electric field of ship.

2.2. Measurement of polarization curves and EIS The potentiodynamic polarization curves and EIS at galvanic potential of 921A and B10 were measured on electrochemical workstation using a three-electrode system, and the medium was NaCl solution with concentration of 0.012 M and 0.6 M. The working electrode was encapsulated by epoxy resin, and the sample is a cylinder with a diameter of 1.1 cm and a height of 1 cm, so the exposed area of metal is 1.0 cm2. Before the tests, the specimens were abraded with wet SiC paper (initially 220, 600 and 1000 grades), rinsed with acetone and ethanol, and finally dried with pure nitrogen gas. The reference electrode was an Ag/AgCl electrode and the counter electrode was a platinum plate electrode with an area of 2 cm2. The working electrode was immersed in the solution for a period of time until the open circuit potential was stabilized, and then started to test. The potential of polarization curve was scanned from
1.1 V to
0.1 V (relative to reference electrode) and the scanning rate was 1.0 mV/s. The 921A electrode and the B10 electrode were shortcircuited, and the surface potential of the two electrodes were tested with time. The stabilized electrode potential was galvanic potential of the corresponding electrode under the 921A-B10 galvanic couple. In the measurement of EIS, the DC potential of 921A and B10 were set to their respective galvanic potential, the AC amplitude was 10 mV and the frequency sweep range was 105–10
2 Hz.

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2.3. Measurement of 921A-B10 galvanic corrosion electric field

8 ϕ ¼ ϕ0 > > > > > > > > q ¼ ∂ϕ ¼ q0 > > < ∂n ja 1 > > q ¼ ¼ fa ðϕa Þ > > σ σ > > > > > jc 1 > : q ¼ ¼ fc ðϕc Þ

The experimental pool is a plastic water tank with size of 1.8 m  1.2 m  1 m and water depth of 0.8 m. The 921A with size of 14 cm  10 cm  0.2 cm and the B10 with size of 10 cm  10 cm  0.2 cm were connected as T-type with insulated glue and short-circuited to form galvanic couple. Moreover, the welded joint between wire and metal plate was sealed with insulated glue. The 921A steel plate was horizontally placed in the simulated seawater and the center of water was selected as the coordinate origin. The Ag/AgCl electrode was employed as electric field sensor and six Ag/AgCl electrodes were fixed on a non-metallic structure frame, as shown in Fig. 2(a). In the figure, 1# and 2# sensors constitute an electrode couple that measures the x-component of electric field, similarly, 3# and 4# sensors constitute y-component electrode couple and 5# and 6# sensors constitute z-component electrode couple. The distance between the two sensors of electrode couple is 10 cm. Three pairs of electrodes are respectively connected to three channels of the multi-channel underwater electric field acquisition system, and each channel records the potential difference of the corresponding component. The potential differences recorded in three channels are expressed as U1, U2, and U3 respectively. Thus electric field is calculated as follows: 8 Ex ¼ U1 =d1 ¼ 10U1 > > > E ¼ U d ¼ 10U < y 2 2 2 3 =d 3 ¼ 10U3 > Ez ¼ Uq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : jEj ¼ ðE Þ2 þ E 2 þ ðE Þ2 x y z

σ

ðAnodic polarization boundaryÞ

(4)

ðCathodic polarization boundaryÞ



 αc e ! E
Eθf j ¼ neKc n0O exp
kB T

(2)

(5)

where Kc is total rate constant of reduction reaction, E is electrode potential and E θf is the formal potential, moreover, the formal potential is defined as [33,34]: Eθf ¼ Eθ


kB T γ R ln ne γ O

(6)

where E θ is the standard potential and γ i is the activity coefficient. According to the Frumkin correction, the total potential difference of EDL and the solution concentration in Eq. (5) are replaced by the potential of compact layer and the reactant concentration at Helmholtz plane, respectively. Thus the electrode kinetics of (5) is corrected to Ref. [25]:
 

 zO e α e ! j ¼ neKc n0O exp
ϕ2 exp
c E
Eθf
ϕ2 kB T kB T    

αc e ðzO
αc Þe ¼ neKc n0O exp
E
Eθf exp
ϕ2 kB T kB T

3. Model of 921A-B10 galvanic corrosion electric field The potential distribution of electrolyte solution and electrode surface is in accordance with the Laplace equation (3) and it is solved using the boundary conditions (4) [8–10]:

∂2 ϕ ∂2 ϕ ∂2 ϕ þ þ ¼0 ∂2 x ∂2 y ∂2 z

ðConstant current boundaryÞ

where ϕ is the potential of electrolyte solution and electrode surface; n is the unit outward normal to the surface; q is the normal derivative of potential (intensity of electric field); ϕ0 and q0 are the constant values of ϕ and q at the boundaries of constant potential and constant current, respectively; σ is conductivity of solution; ja and jc represent the anodic current density and the cathodic current density, respectively; fa ðϕa Þ and fc ðϕc Þ represent the anodic polarization equation and the cathodic polarization equation, respectively. For the multi-electron electrode reaction O þ ne⇌R, it is assumed that αa and αc are the total transfer coefficient of oxidation reaction and reduction reaction, respectively. Thus the absolute speed of the reduction reaction is:

Adjusting the concentration of NaCl solution to 0.012 M and 0.6 M respectively, and the T-type galvanic couple was immersed in the experimental pool for a period time until the corrosion electric field was stabilized before starting to test. The center of the sensor frame was placed at (0,
7,
7) cm, and the T-type galvanic couple was driven by bridge crane through the sensor to obtain passage curve of electric field, which is used to characterize the corrosion electric field distribution around the galvanic couple. Fig. 2(a) is a schematic diagram of the observation path of corrosion electric field, and Fig. 2(b) is a physical diagram.

r2 ϕ ¼

σ

ðConstant potential boundaryÞ

(7)

The reactant of cathodic reaction is electrically neutral during oxygen absorption corrosion, so zO ¼ 0, therefore, the cathodic kinetics at large overpotential is corrected to:

(3)

Fig. 2. The observation path of corrosion electric field shows as (a) schematic diagram and (b) physical diagram. 3

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α e  αc e ! ! c E
Eθf exp jc ¼ j
j  j ¼ neKc n0O exp
ϕ kB T kB T 2

(8)

Similarly, the anodic kinetics is corrected to: 


α e  αa e a E
Eθf exp
ϕ ja ¼ neKa n0R exp kB T kB T 2

(9)

Equations (8) and (9) show that the Frumkin correction factors for

 cathodic and anodic kinetics are exp kαBcTe ϕ2 and exp
kαBaTe ϕ2 , which are consistent with the Frumkin correction factors given by Z. Nagy [27]. The size of seawater domain and galvanic couple, the coordinate system and its origin in the simulation model are consistent with experiment. The BEM of 921A-B10 galvanic corrosion electric field was established by using the current distribution (boundary element) interface in the COMSOL Multiphysics simulation software. There are two simplifications of boundary conditions in simulation. Firstly, the equilibrium potential of oxygen reduction is 0.615 V and that of iron anodic dissolution is
0.817 V, as shown in Table 1. According to the galvanic potential of 921A and B10 in section 4.2, it can be seen that the above two reactions occur in the case of overpotential greater than 200 mV, which belongs to the electrode reaction under large overpotential. Therefore, the Butler-Volmer equation was simplified to the Tafel equation. Secondly, because of the cathodic polarization of B10 after electrical contact with 921A, so the rate of anodic dissolution reaction of B10 is very small. Therefore, the anodic dissolution reaction on the surface of B10 was neglected during the simulation, and only the oxygen reduction reaction on the surface was considered. Based on the above simplifications, the solution domain and boundary conditions of the BEM were set as follows: (1) the infinite space was NaCl solution, and the conductivity was 0.14 S/m and 4.80 S/m respectively (corresponding to 0.012 M and 0.6 M NaCl solution). (2) The conjugated reaction of iron oxidation and oxygen reduction occurred simultaneously on the surface of 921A, and their kinetic were described by the anodic Tafel equation and the cathodic Tafel equation, respectively. (3) Only the oxygen reduction occurred on the surface of B10 and its kinetic was described by the cathodic Tafel equation. The uncorrected and corrected Tafel equation was used as the boundary condition to obtain corrosion electric field of before and after the Frumkin correction respectively. In order to be consistent with the experimental observation path, the path with point (
75,
7,
7) cm and (75,
7,
7) cm as endpoint in the vicinity of the galvanic couple is selected as the observation object.

Fig. 3. Polarization curves of 921A and B10 in 0.012 M and 0.6 M NaCl solution.

the protective oxidation film, as well as the greater the corrosion current density. The electrochemical parameters of iron oxidation and oxygen reduction occurred on the surface of 921A and oxygen reduction occurred on the surface of B10 were obtained by the polarization curve are shown in Table 1. The equilibrium potential of electrode reaction was acquired by the Nernst equation, and the corrosion potential, corrosion current and Tafel slope were obtained by fitting the Tafel region of polarization curve. Furthermore, the exchange current density of electrode reaction can be calculated from above parameters combined with the Tafel equation. Finally, the transfer coefficient was calculated from Tafel slope.

4.2. Electric double layer capacitance at galvanic potential Electrode reaction is carried out at the galvanic potential when 921AB10 constitutes galvanic corrosion. Therefore, galvanic potential of the 921A-B10 galvanic couple should be used in the measurement of EIS and calculation of the theoretical diffuse layer potential. Fig. 4 shows the surface potential of 921A and B10 changing with time after shortcircuited in 0.012 M and 0.6 M NaCl solution, respectively. It can be seen from the figure that at the instant of two electrodes short-circuited, the electrode potential is their respective corrosion potential, then the 921A anodic polarization and its potential shift positively but the B10 cathodic polarization and its potential shift negatively. After 200 s, the potential of 921A and B10 is almost stable, and the stable electrode potential is the galvanic potential of the corresponding electrode under 921A-B10 galvanic couple. The galvanic potentials of 921A and B10 in 0.012 M NaCl are
0.621 V and
0.562 V, and in 0.6 M NaCl are
0.604 V and
0.582 V, respectively. Thus the galvanic potential of the two electrodes are not equal, instead, there is a small potential difference of ΔE. The possible cause of this potential difference is the ohmic voltage

4. Result and discussion 4.1. Corrosion electrochemical parameters Polarization curves of 921A and B10 in 0.012 M and 0.6 M NaCl solution are shown in Fig. 3. The current density of 921A and B10 increases with the increase of NaCl concentration. This is because chloride ions have a destructive effect on the corrosion oxidation film of the metal surface, so the higher concentration of chloride ion, the more damage to

Table 1 Corrosion electrochemical parameters of 921A and B10. Electrode

Concentration (M)

Reaction

Eeq (V)

Ecorr (V)

Icorr (A/cm2)

B (V/decade)

j0 (A/cm2)

α

921A

0.012

Iron oxidation Oxygen reduction Steel corrosion Iron oxidation Oxygen reduction Steel corrosion Oxygen reduction Copper corrosion Oxygen reduction Copper corrosion


0.817 0.615 –
0.817 0.615 – 0.615 – 0.615 –

– –
0.661 – –
0.668 –
0.170 –
0.220

– – 6.28  10
6 – – 8.56  10
6 – 6.74  10
6 – 9.52  10
6

0.116
0.507 – 0.075
0.310 –
1.79 –
1.63 –

2.84  10
7 1.91  10
8 – 8.91  10
8 6.17  10
10 – 2.45  10
6 – 2.93  10
6 –

0.509 0.117 – 0.788 0.191 – 0.033 – 0.036 –

0.6

B10

0.012 0.6

4

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Fig. 4. Variation of electrode potential with time after short-circuited of 921A-B10 galvanic couple in (a) 0.012 M and (b) 0.6 M NaCl.

Cd ¼ Q1=n Rð1
nÞ=n e

drop caused by solution resistance between the two electrodes. Moreover, ΔE in 0.012 M NaCl is greater than ΔE in 0.6 M NaCl, this is because the lower the electrolyte concentration, the larger the solution resistance and the greater the ohmic voltage drop. The EIS of 921A and B10 in 0.012 M and 0.6 M NaCl solution are shown in Fig. 5. Considering the “dispersion effect” of the EIS, it is fitted with a constant phase element (CPE) instead of a capacitor, and the impedance of the constant phase element is defined as: Z¼

1 QðjωÞn

The EIS is fitted to obtain the parameters of each element in equivalent circuit as shown in Table 2. Then the capacitance of EDL at electrode/solution interface is calculated according to formula (12). Re is solution resistance between the Luggin capillary and the surface of working electrode, and it decreases with the increase of NaCl concentration. Rt is the charge transfer resistance of electrode reaction. In the case of the galvanic corrosion of 921A-B10, the anode of 921A is corroded, while the cathode of B10 is protected. The higher the concentration of NaCl, the more serious the corrosion of 921A and the better the protection of B10, therefore, the charge transfer resistance of 921A in 0.6 M solution is smaller than that of 0.012 M, while that of B10 is the opposite, that is, in 0.6 M solution is larger than that of 0.012 M. Cd is capacitance of EDL at the electrode/solution interface and it increases with the increase of NaCl concentration. This is because the effective thickness (Debye length) of diffuse layer decreases with the increase of NaCl concentration, thus the capacitance of diffuse layer increases. Nevertheless, the capacitance of compact layer does not change with the increase of concentration, thus the total capacitance of the EDL increases.

(10)

where, Z and Q are the impedance and admittance of the CPE, ω is the angular frequency of disturbance signal and n ð
1  n  1Þ is the dispersion index. When n ¼ 1, the CPE is equivalent to ideal capacitor, n ¼
1, the CPE is equivalent to inductor, n ¼ 0, the CPE is equivalent to resistor. n ¼ 0:5, the CPE is equivalent to Warburg impedance. The relationship between the capacitance of EDL and the parameters of CPE is [35]:  
1 ðn
1Þ=n Cd ¼ Q1=n R
1 e þ Rt

(12)

(11)

where Cd is the capacitance of EDL, Re is resistance of solution and Rt is charge transfer resistance. When the solution resistance is much smaller than the charge transfer resistance, Eq. (11) is simplified as:

4.3. Diffuse layer potential at galvanic potential Electrode potential E is consist of ion double layer potential caused by residual charge, specific adsorption double layer potential of charged particles, dipole layer potential of polar molecule and metal surface

Fig. 5. The EIS of 921A and B10 in 0.012 M and 0.6 M NaCl solution. 5

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Table 2 Fitting parameters by EIS and capacitance of EDL. Electrode 921A B10

Concentration (M) 0.012 0.6 0.012 0.6

n 0.791 0.824 0.842 0.848

Q=Ω
1 ⋅ cm
2 ⋅ s
n
4

1.98  10 2.90  10
4 1.92  10
4 3.17  10
4

potential formed by short-range force. Only the first case constitutes double layer potential difference ϕa , while the other three cases constitute potential of zero charge (PZC, Ez) of metal. According to the theoretical relationship between the total potential difference of EDL and the diffuse layer potential and the concentration of supporting electrolyte [22]: 
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 eϕ2 8kB T εr ε0 n ⋅ sinh ϕa ¼ ϕ2 þ CH 2kB T


pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eϕ2 8kB T εr ε0 n0 sinh 2kB T

Electrode 921A B10

(13)

  2kB T Cd ⋅ ϕa ⋅ arcsinh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 8kB T εr ε0 n0

Cd =μF ⋅ cm
2

56.3 3.42 61.7 3.12

552 384 9.49  103 3.93  104

60.4 66.2 83.6 91.7

Concentration (M)

ϕa (mV)

Theoretical (mV)

Experimental (mV)

Error (%)

0.012 0.6 0.012 0.6


131
114
342
362


99
46
182
106


129
39
195
103


30.3 15.2
7.1 2.8

from Fig. 6 and the experimental values are shown in Table 3. The absolute values of theoretical and experimental diffuse layer potential decrease with the increase of the concentration of NaCl solution. The absolute diffuse layer potential of B10 is much larger than that of 921A, which is because the total potential difference of EDL of B10 is much greater than that of 921A. The error between experimental and theoretical value may be caused by the following three reasons: Firstly, the total potential difference of EDL of 921A is relatively small and the diffuse layer accounts for a large proportion in the whole double layer structure. Especially at low concentration (0.012 M), the EDL has almost only the diffuse layer, while the compact layer is still considered in theoretical calculation of the diffuse layer potential, which results in a large error of up to 30%. For B10, the total potential difference of EDL is relatively large, the hypothesis of compact layer is reasonable, and the error is within 10%. Secondly, the test time of EIS is relatively short, and there is not enough time to form a stable diffuse layer on the surface of electrode, which results in the experimental value of the absolute diffuse layer potential being smaller than the theoretical value. Finally, when calculating the theoretical value of the diffuse layer potential, it is assumed that the thickness of the compact layer is 0.5 nm and remains unchanged, while the actual thickness of the compact layer is not equal to 0.5 nm, which causes a certain error between the theoretical value and the experimental value.

(14)

Therefore,

ϕ2 ¼

Rt =Ω ⋅ cm2

Table 3 Comparison of theoretical and experimental values of diffuse layer potential.

The variation curves of the diffuse layer potential with the total potential difference of EDL in different electrolyte concentrations can be obtained, as shown in Fig. 6. jϕ2 j increases with the increase of jϕa j, but the increasing speed of the former is less than that of the latter. That is to say, with the increase of jϕa j, the proportion of jϕ2 j in the whole potential of EDL becomes smaller and smaller. When jϕa j increases to a certain extent, the influence of jϕ2 j can be neglected. Furthermore, jϕ2 j decreases with the increase of concentration of NaCl at the same potential of EDL, indicating that the dispersibility of double layer structure decreases with the increase of electrolyte concentration. For the entire double layer structure, qM ¼ Cd ⋅ ϕa and qM is expressed as [18,22]: qM ¼

Re =Ω ⋅ cm2

(15)

where Cd is the electric double layer capacitance measured in section 4.2. The electric double layer capacitance and galvanic potential measured by experiment are brought into Eq (15) to obtain the experimental value of diffuse layer potential. The theoretical diffuse layer potential obtained

4.4. Simulation and experiment of 921A-B10 galvanic corrosion electric field The corrosion electric field before and after Frumkin correction was simulated by using the corrected and uncorrected Tafel equation as boundary conditions, respectively. The distribution of corrosion electric field before and after Frumkin correction on the specified path was compared with experimental result, as shown in Fig. 7 and Fig. 8. Table 4 shows the variation of the amplitude of electric field modulus and current density with the concentration of NaCl solution on the specified path. It can be seen from the figure that the shape of electric field distribution remains unchanged before and after Frumkin correction, but the amplitude of electric field after correction increases compared with that before correction. Especially when the concentration of NaCl is 0.012 M, the amplitude of electric field after correction increases obviously. The simulated results of electric field distribution are in good agreement with experimental results, and the electric field amplitude after Frumkin correction is closer to experimental results. When the concentration of NaCl is 0.6 M, the error between the amplitude of electric field modulus before Frumkin correction and experimental value is 10.9%, while the error is reduced to 3% after Frumkin correction. When the concentration of NaCl is 0.012 M, the error between the amplitude of electric field modulus before Frumkin correction and experimental value is 29.6%,

Fig. 6. The variation curves of the diffuse layer potential with the total potential difference of EDL in different electrolyte concentrations. 6

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Fig. 7. Electric field distribution before and after Frumkin correction is compared with experimental results at 0.012 M NaCl.

4.5. The effect of extended electron transfer (EET) on the Frumkin correction

while the error is reduced to 4.3% after Frumkin correction. The above analysis illustrates that the validity of the BEM of 921A-B10 galvanic corrosion electric field established in this paper, moreover, it is essential to correct the Tafel boundary condition in simulation. Especially when the electrolyte concentration is low, Frumkin correction can greatly reduce the error between simulated and experimental results. The correction effect is defined as: ΔjEj ¼

jEjafter
jEjbefore jEjbefore

The inclusion of distance-dependent electron transfer is able to mitigate the Frumkin correction, especially at higher support ratio where the Debye length is short [30]. Therefore, the impact of distance-dependent electron transfer on the Frumkin correction is related to the concentration of supporting electrolyte. Thus under the present research conditions, does the distance-dependent electron transfer have a perturbation on the Frumkin correction? Our discussion on this issue is as follows: Gavaghan and Feldberg [18] studied the correlation between extended electron transfer(EET, the so-called electron tunnelling effect) and the Frumkin correction, which quantify the effect of EET by comparing the electron transfer rate constants deduced with (k’eff ) and without (keff ) EET, namely:

(16)

where jEjbefore and jEjafter are the amplitude of electric field modulus before and after Frumkin correction respectively. The correction effect is 35.6% and 8.8% when the concentration of NaCl is 0.012 M and 0.6 M, respectively, indicating that the smaller the concentration of electrolyte solution, the more obvious the correction effect. This is because the smaller the concentration of NaCl solution, the larger the proportion of diffuse layer potential in the total potential difference of EDL, that is, the stronger the dispersion of the double layer structure, as well as the more necessary the correction. In addition, the galvanic current density increases with the increase of NaCl concentration, while the corrosion electric field decreases greatly. This is because the conductivity increases with the increase of NaCl concentration, the current density increases from 2.17  10
2 A/m2 to 2.73  10
2 A/m2 (25.8% increase) and the conductivity increases from 0.14 S/m to 4.80 S/m (34 times increase) when NaCl concentration increases from 0.012 M to 0.6 M. It can be seen from E ¼ j=σ that when the concentration of NaCl solution increases, the increase of galvanic current density is much smaller than that of conductivity, which greatly decreases the corrosion electric field.

k ’eff ¼ keff

β κ





4ðzox
αc Þ ejzse jϕ2 exp
arctanh expð
γÞtanh exp
βκ γ dγ γ¼0 jzse j 4kB T   exp
kBeT ϕ2 ðzox
αc Þ

R∞

(17) where k’eff is the effective rate constant with EET; keff is the effective rate constant for the classical Frumkin correction; zox is the charges on the oxidized species, for oxygen reduction, zox ¼ 0, for iron oxidation, zox
αc of the formulation (17) is replaced by zred þ αa , and zred ¼ 0; β is the tunnelling decay constant and has the value of 1.59  1010 m
1 for water [17]; κ is the reciprocal Debye length and is related to the concentration 7

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Journal of Electroanalytical Chemistry xxx (xxxx) xxx

Fig. 8. Electric field distribution before and after Frumkin correction is compared with experimental results at 0.6 M NaCl.

Table 4 The variation of the amplitude of electric field modulus and current density before and after Frumkin correction with NaCl concentration. Electric field modulus(  10-3 V/m)

Concentration (M)

0.012 0.6

Current density modulus(  10-2 A/m2)

Before correction

After correction

Experiment

Before correction

After correction

114 5.22

155 5.69

162 5.86

1.60 2.51

2.17 2.73

of electrolyte; γ ¼ κðx
x2 Þ, x2 is the distance between the electrode and the outer Helmholtz plane, assuming the solution with x  100 mm is bulk solution, the upper limit of integration of formulation (17) is 3.6  107 and 2.5  108 for 0.012 M and 0.6 M NaCl solution, respectively; zse is the charge associated with the 1:1 supporting electrolyte, for NaCl solution, jzse j ¼ 1. The formulation of (17) was solved by the adaptive Simpson quadrature in MATLAB, and the values of k’eff = keff for

Correction effect(%)

35.6 8.8

iron oxidation and oxygen reduction under different conditions were as shown in Table 5. It can be seen from Table 5 that the values of k’eff =keff for oxygen reduction reaction under different conditions are all close to 1, indicating that the impact of EET on the classical Frumkin correction is not obvious in this case, and the impact of EET on Frumkin correction is obviously related to the concentration of supporting electrolyte. The smaller the

Table 5 The values of k’eff =keff for iron oxidation and oxygen reduction under different conditions. Electrode

Concentration (M)

Reaction

921A

0.012

Fe → Fe2þ þ 2e
1=2O2 þ H2 O þ 2e
Fe → Fe2þ þ 2e
1=2O2 þ H2 O þ 2e
1=2O2 þ H2 O þ 2e
1=2O2 þ H2 O þ 2e


0.6 B10

0.012 0.6

→ 2OH
→ 2OH
→ 2OH
→ 2OH


8

zox
αc or zred þ αa

β=κ

e ϕ kB T 2

k’eff keff

0.509
0.117 0.788
0.191
0.033
0.036

44.12


3.86

6.24


1.79

44.12 6.24


7.09
4.13

0.9345 1.0164 0.8193 1.0534 1.0199 1.0308

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Journal of Electroanalytical Chemistry xxx (xxxx) xxx

electrolyte concentration, the closer the value of k’eff =keff is to 1, indicating that the smaller the concentration, the slighter the effect of EET on the Frumkin correction, which is consistent with the conclusions of Dickinson and Compton [30]. The values of k’eff =keff for iron oxidation reaction deviate from 1 to a certain extent, which indicates that the EET has a great impact on the Frumkin correction. Especially when the electrolyte concentration is 0.6 M, the value of k’eff =keff is 0.8193, so the impact of EET on the Frumkin correction cannot be ignored at this time. Therefore, the smaller the concentration of supporting electrolyte is, the slighter the effect of EET on the Frumkin correction. Within the studied conditions in the present paper, EET has little effect on the Frumkin correction of oxygen reduction, but has a greater impact on the Frumkin correction of iron oxidation.

[5]

[6]

[7]

[8]

[9]

[10]

5. Conclusion Based on the Frumkin correction theory, the Tafel boundary of the BEM for 921A-B10 galvanic corrosion electric field was corrected, and the impact of electrolyte concentration on Frumkin correction effect of corrosion electric field was studied. The conclusions are drawn as following:

[11]

[12]

1. When the concentration of NaCl solution increases, the effective thickness (Debye length) of the diffuse layer decreases, so that the capacitance of the diffuse layer increases. Nevertheless, the capacitance of the compact layer does not change with the concentration. Therefore, the total capacitance of EDL increases. 2. The smaller the concentration of electrolyte solution, the greater the dispersion of the EDL. Therefore, the absolute diffuse layer potential increases with the decrease of the concentration of NaCl solution. 3. The amplitude of electric field after Frumkin correction is closer to the experimental value, indicating that it is essential to correct the Tafel boundary condition during the simulation. When the concentration of NaCl is 0.012 M and 0.6 M, the correction effect is 35.6% and 8.8%, respectively. That is, the smaller electrolyte concentration, the greater dispersion of the double layer structure, as well as the more obvious the correction effect. 4. The smaller the concentration of supporting electrolyte, the slighter the effect of EET on the Frumkin correction. Within the studied conditions in the present paper, EET has little effect on the Frumkin correction of oxygen reduction, but has a greater influence on the Frumkin correction of iron oxidation. 5. The electric field distribution obtained by simulation is in good agreement with the experimental results, which verifies the validity of the BEM of galvanic corrosion electric field established in this paper. This provides an effective method for study of corrosion electric field of ship.

[13] [14]

[15]

[16]

[17]

[18] [19] [20]

[21]

[22] [23]

[24]

Acknowledgements [25]

This work is supported by National Natural Science Foundation of China (Grant no. 41476153) and the author wishes to appreciate the technical support of COMSOL software company.

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