Electro-Chemical Impedance Spectral ( EIS ) Study of Patinated Bronze Corrosion in Sulfate Media : Experimental Design Approach

The aim of the present investigation was to model the experimental conditions of tin bronze patination using full factorial experimental design. In this sense, a full factorial design approach was developed to model the corrosion behavior of patinated tin bronze alloy in sulfate electrolyte. Three experimental factors (the immersion time in the chloride electrolyte, the potential limit for the anodic sweep Elim, and the potential scan rate) were chosen to identify the significant factor on the patina growth process at the bronze substrate. The experimental responses were the kinetic parameters extracted from the electro-chemical spectra (EIS) for eight different experiments. An equivalent electrical circuit containing an electrolyte resistance (Re), a double layer capacitance (CPEdl), a charge transfer resistance (Rt) and Gerischer element (G), was developed to model the patinated bronze corrosion process. The electro-chemical spectra (EIS) show that the corrosion process of the patinated bronze alloy occurred from a chemical reaction is followed by an electrochemical one. Analysis of the experimental responses showed that while the scan rate is the most influent factor for the corrosion potential (Ecorr), the electrolyte resistance (Re), and the double layer capacitance CPEdl variation, the potential limit is the significant factor for charge transfer resistance Rt, reciprocal of the admittance parameter Y0 and the effective transfer rate of the chemical reaction k variation.


Introduction
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 [1] [2] [3].Electrochemical investigations were also undertaken [4]- [8].We used the cyclic voltammetry technique to explore archaeological Punic bronze corrosion behavior in chloride electrolyte [9] and we compared also modern and archaeological materials voltammetric behaviors [10] recently, we showed [11] 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 2 CuCl − 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 [12]- [18].Among them, only de Lago et al. [19] 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.

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 [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 cm 2 .Before use, the electrodes were mechanically polished Journal of Materials Science and Chemical Engineering 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.

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: U 1 : first factor representing the immersion time in the chloride electrolyte, t.U 2 : second factor representing the potential limit for the anodic sweep, E lim .U 3 : third factor representing the potential scan rate, ν.The total number of the experiments to be carried out was 2 n [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/Y 0 : the admittance parameter (Gerischer type impedance, see below).
The first order-model for the three variables can be represented by the following equation: 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], -b 0 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]: ( ) where: -B is the vector of the estimates of the coefficients, -X is the model matrix, -Y is the vector of the experimental design.

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 Na 2 SO 4 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 45 0 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): ( ) ( ) where: -Y 0 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]: where, k holds for the effective transfer rate of the chemical reaction, and the admittance element Y 0 = 1/Z 0 .
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].
Figure 2. The equivalent electrical circuits used for fitting of the impedance spectra.

Experimental Design Study
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): Analysis of the statistical results showed that: - -92.4% of the depressed feature coefficient variation was due to b 3 , b 12 and b 123 , the main effect was linked to the interaction between the second and third factor (P 23 = 45.8%).
-The limit potential could present the main effect on the admittance parameter responses 1/Y 0 , P 2 = 34%.
-The second factor was the most important parameter for the effective transfer rate of the chemical reaction as P 2 = 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 Y 0 , 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.

Conclusions
The aim of the present work was to study the corrosion behavior of chloride patinated bronze in 1 g/l Na 2 SO 4 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 E corr , R e and CPE dl variation.However, the limit potential was the significant factor for R t , Y 0 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.

Figure 1 .
Figure 1.Electrochemical impedance spectra obtained for the patinated bronze immersed in 1 g•L −1 Na 2 SO 4 solution in different condition.

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The factors b 3 , b 2 and b 23 could explain about 96% of the corrosion potential variation -The scan rate is the most influent parameter as P 3 = 48%.The third factor was the most significant factor for the electrolyte resistance variation as P 3 = 57.2%.-b 3 , b 123 and b 2 were the most significant for charge transfer resistance evolution.The main effect was attributed to the limit potential P 2 = 54.1%.-b 3 , b 12 and b 123 could contribute to 93.1% of the double layer capacitance.The third factor was the most important (P 3 = 40.7%).

Figure 3 .
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.

Table 1
summarizes the experimental field.

Table 3 .
Experimental responses extracted from the EIS spectra.