High-purity Cu and Sn metals (Goodfellow copper rod > 99.999 wt. % and Sn Johnson-Matthey tin slug > 99.9985 wt. %) and Cu-10 wt. % Sn alloy (5.60 at. % Sn) were used in this study. The bronze was prepared from the pure copper and tin through a procedure detailed elsewhere [20] . The working electrodes made from this alloy were embedded into a chemically inert resin with an exposed area (only one face) of 0.33 cm². Before use, the electrodes were mechanically polished with abrasive paper up to 2500 SiC grade, washed with distilled water and dried in a room temperature.

The electrochemical experiments were carried out in a standard three-electrode cell, with a large size graphite sheet as counter electrode and a saturated calomel electrode (SCE) as reference electrode. The electrochemical impedance spectroscopy (EIS) measurements were performed with an Autolab® PGSTAT 20 potentiostat equipped with an impedance analysis module. Software NOVA was used for the instrumentation control, data treatment as well as spectra fitting.

The chloride patina was electrochemically formed at the Cu10Sn bronze using potentiodynamic technique (anodic potential sweep) according to the experimental design described here above.

The full factorial design was used to study the influence of different experimental factors on the chloride patina formation at the bronze substrate. The patina layer was artificially electrodeposited at the Cu10Sn alloy using potentiodynamic technique. After a preliminary investigation, three experimental variables were chosen:

U₁: first factor representing the immersion time in the chloride electrolyte, t.

U₂: second factor representing the potential limit for the anodic sweep, E_lim.

U₃: third factor representing the potential scan rate, n.

Table 1 summarizes the experimental field.

The total number of the experiments to be carried out was 2ⁿ [13] - [19] , where n is the number of experimental variables. Therefore, the experimental design consisted of eight experiments.

The experimental design as well as the experimental matrix is listed in Table 2.

The experimental responses studied were the electrochemical parameters obtained after fitting of the the EIS spectra for the different experiments.

- E_corr: the corrosion potential, the rest potential at which the EIS spectra were taken.

- Re: the electrolyte resistance,

- R_t: the charge transfer resistance,

- CPE_dl: the constant phase element associated to the double layer capacitance,

- k: the effective transfer rate of the chemical reaction (Gerischer type impedance, see below),

- 1/Y₀: the admittance parameter (Gerischer type impedance, see below).

The first order-model for the three variables can be represented by the following equation:

ψ i = b 0 + b 1 X 1 + b 2 X 2 + b 3 X 3 + b 12 X 1 X 2 + b 13 X 1 X 3 + b 23 X 2 X 3 + b 123 X 1 X 2 X 3 (1)

where:

- ψ_i is the experimental response,

- X_i is the coded variable relative to natural variable U_i, which obtained as detailed elsewhere [11] ,

- b₀ is the intercept,

- b_i represents the main effects of the factor i,

- b_ij represents the interaction between the factors i and j,

- b_ijk represents the interaction between the factors i, j and k.

The coefficients (b_i, b_ij and b_ijk) were calculated by the least squares method using [16] :

B = ( X ′ X ) − 1 X ′ Y (2)

where:

- B is the vector of the estimates of the coefficients,

- X is the model matrix,

- X ′ is the transposed matrix,

- ( X ′ X ) is the information matrix,

- ( X ′ X ) − 1 is the dispersion matrix,

- Y is the vector of the experimental design.

EIS analyses of the electrochemically formed interfaces according to the experimental design were performed. In fact, after patina deposition at the bronze substrate according to the experimental design described above, the Nyquist plots of the EIS spectra were recorded for 5 minutes of immersion in 1 g/L Na₂SO₄ electrolyte and the results are shown in Figure 1.

As Figure 1 shows, the Nyquist plots obtained for the different bronze surfaces

are characterized by two badly separately capacitive loops. At high frequency, the depressed semi-circle in the high frequency range could be related to the charge transfer process. Moreover, the low frequency tail at near 45⁰ could be attributed to mass transfer control of the corrosion process. The EIS spectra were analyzed according to the equivalent circuit presented in Figure 2.

The equivalent circuit consists of the electrolyte resistance (R_e), the double layer capacitance (CPE_dl) the charge transfer resistance (R_t) and the Gerischer element (G).

Instead of using capacitance in the equivalent circuit, a constant phase element was introduced (CPE_dl). It represents the deviation from the true capacitor behavior. The impedance of a constant phase element is defined in Equation (3):

Z C P E ( ω ) = Y 0 − 1 ( j ω ) − n (3)

where:

- Y₀ is the admittance representative for the CPE_dl.

- n_d is related to the non-equilibrium current distribution due to the surface roughness and defects.

CPE_dl describes a pure inductor for the case n = −1, an ideal resistor for n = 0 and an ideal capacitance for n = 1. In our case, the CPE_dl could be related to the non-uniform porous patina layer grown at the bronze surface.

The Gerischer impedance was introduced to describe a diffusion type impedance in which the species also participates in a chemical reaction along the diffusion path [18] . This impedance was generally observed in a mixed conducting solid electrolyte systems [2] . The Gerischer impedance is given by Equation (4) [21] :

Z G ( ω ) = Z 0 k + j ω (4)

where, k holds for the effective transfer rate of the chemical reaction, and the admittance element Y₀ = 1/Z₀.

It is interesting to note that the obtained equivalent circuit was different to that found in the literature for bronze corrosion in sulfate containing media. The proposed model consisted of two parallel RC circuits and three parallel RC circuits [22] [23] .

From the EIS spectra depicted in Figure 1, the best fitting parameters corresponding to the equivalent circuit presented in Figure 2 were extracted. The results are summarized in Table 3.

The responses were analyzed by regression analysis according to the proposed mathematical model. The estimated coefficients models parameters were listed in Figure 3.

In order to evaluate the weight of the different coefficients on the electrochemical parameters Pareto analysis was performed [24] . Plots of the contribution of each term are displayed in Figure 3 where the percentage effect P_i of every term i, was calculated through Equation (5):

P i = 100 ( b i 2 ∑ i   b i 2 ) (5)

Analysis of the statistical results showed that:

- The factors b₃, b₂ and b₂₃ could explain about 96% of the corrosion potential variation

- The scan rate is the most influent parameter as P₃ = 48%.

- The third factor was the most significant factor for the electrolyte resistance variation as P₃ = 57.2%.

- b₃, b₁₂₃ and b₂ were the most significant for charge transfer resistance evolution. The main effect was attributed to the limit potential P₂ = 54.1%.

- b₃, b₁₂ and b₁₂₃ could contribute to 93.1% of the double layer capacitance. The third factor was the most important (P₃ = 40.7%).

- 92.4% of the depressed feature coefficient variation was due to b₃, b₁₂ and b₁₂₃, the main effect was linked to the interaction between the second and third factor (P₂₃ = 45.8%).

- The limit potential could present the main effect on the admittance parameter responses 1/Y₀, P₂ = 34%.

- The second factor was the most important parameter for the effective transfer rate of the chemical reaction as P₂ = 66.8%.

It could be concluded from the previous results that the scan rate was the most important factor for the response E_corr, R_e and CPE_dl. This parameter could reflect the kinetic of the patina growth at the Cu10Sn alloy.

For Y₀, R_t and k, we found that the limit potential of anodic polarization curve was the most important. Such a result could be related to the composition and the structure of the chloride patina.

Finally, the interaction between the scan rate and the limit potential was the important factor for n variation. Therefore this parameter could affect the to the non-equilibrium current distribution linked to the surface roughness and patina porosity.