Cu-Sn alloys were used since antiquity to produce sculptures, coins and artefacts. These materials exhibit good mechanical and esthetical properties.
Numerous analytical techniques were used to study tin bronze corrosion mechanisms    . Electrochemical investigations were also undertaken  -  . We used the cyclic voltammetry technique to explore archaeological Punic bronze corrosion behavior in chloride electrolyte  and we compared also modern and archaeological materials voltammetric behaviors  recently, we showed  that the bronze corrosion reaction order with respect to chloride ions varied as the halide content changed. In fact, for [Cl−] < 0.5 M, the reaction order was about 0.22 suggesting that the Cu10Sn bronze alloy dissolution was not strongly dependent with Cl− ions which could act as corrosion initiator. For [Cl−] > 0.5 M the bronze mechanism alloy was controlled by copper oxidation. Two determining steps were evidenced where the cuprous chloride formation was followed by complex.
We used also electro-chemical impedance spectroscopy to characterize the corrosion behavior of archaeological bronze in 0.1 M chloride medium interface. Indeed, a simple electrical equivalent circuit was used to explain the material reactivity.
As the corrosion ability of materials depends on various conditions, then, the large number of experimental factors to consider remained the major obstacle for the experimenter to understand the alteration mechanisms. Nowadays, the chemometric approach is considered as powerful tool for studying the corrosion and protection process. Many works introduced the use of experimental designs for understand metals corrosion and inhibition  -  . Among them, only de Lago et al.  used the experimental design to study the effect of 2-amino- 5-mercapto-1,3,4-thiadiazole (AMT) as inhibitor for bronze protection in artificial rainwater.
However, the experimental design use for studying bronze corrosion, to the best of our knowledge, was not yet investigated.
The aim of the present investigation was to model the experimental conditions of tin bronze patination using full factorial experimental design. The main interests for application of an artificial patina are rebuilding of historical artifacts, works of arts and for the purpose of scientific research.
2.1. Electrochemical Test
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  . 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.
2.2. Experimental Design Approach
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, Elim.
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  -  , 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.
- Ecorr: the corrosion potential, the rest potential at which the EIS spectra were taken.
- Re: the electrolyte resistance,
- Rt: the charge transfer resistance,
- CPEdl: 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).
Table 1. Experimental field.
Table 2. Experimental matrix-Experimental design.
The first order-model for the three variables can be represented by the following equation:
- ψi is the experimental response,
- Xi is the coded variable relative to natural variable Ui, which obtained as detailed elsewhere  ,
- b0 is the intercept,
- bi represents the main effects of the factor i,
- bij represents the interaction between the factors i and j,
- bijk represents the interaction between the factors i, j and k.
The coefficients (bi, bij and bijk) were calculated by the least squares method using  :
- B is the vector of the estimates of the coefficients,
- X is the model matrix,
- is the transposed matrix,
- is the information matrix,
- is the dispersion matrix,
- Y is the vector of the experimental design.
3. Results and Discussion
3.1. EIS Characterization
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
Figure 1. Electrochemical impedance spectra obtained for the patinated bronze immersed in 1 g・L−1 Na2SO4 solution in different condition.
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 (Re), the double layer capacitance (CPEdl) the charge transfer resistance (Rt) and the Gerischer element (G).
Instead of using capacitance in the equivalent circuit, a constant phase element was introduced (CPEdl). It represents the deviation from the true capacitor behavior. The impedance of a constant phase element is defined in Equation (3):
- Y0 is the admittance representative for the CPEdl.
- nd is related to the non-equilibrium current distribution due to the surface roughness and defects.
CPEdl 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 CPEdl 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  . This impedance was generally observed in a mixed conducting solid electrolyte systems  . The Gerischer impedance is given by Equation (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   .
Figure 2. The equivalent electrical circuits used for fitting of the impedance spectra.
3.2. Experimental Design Study
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  . Plots of the contribution of each term are displayed in Figure 3 where the percentage effect Pi of every term i, was calculated through Equation (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%.
Table 3. Experimental responses extracted from the EIS spectra.
Figure 3. Pareto analysis for the experimental responses (a) The estimated model coefficients, (b) Pareto charts and (c) cumulative Pareto charts for the different electrochemical responses.
It could be concluded from the previous results that the scan rate was the most important factor for the response Ecorr, Re and CPEdl. This parameter could reflect the kinetic of the patina growth at the Cu10Sn alloy.
For Y0, Rt 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.
The aim of the present work was to study the corrosion behavior of chloride patinated bronze in 1 g/l Na2SO4 electrolyte using EIS technique. A full factorial design was chosen to study the effect of the immersion time, the limit potential and the scan rate on the patina growth at the bronze substrate. The experimental responses were the electrochemical parameters obtained after fitting the EIS spectra. We found that the most suitable equivalent circuit to describe the electrochemical behavior of patinated bronze is a Randles type containing an additional Gerischer impedance element. When analyzing the experimental responses, we found that the scan rate was the most influent factor for Ecorr, Re and CPEdl variation. However, the limit potential was the significant factor for Rt, Y0 and k variation.
We hope to investigate the effects of plant extracts on the corrosion of bronze covered with chloride patina by potentiodynamic polarization measurement, electrochemical impedance spectroscopy and SEM/EDX methods.