Catalyst Behavior Analyzed via General Regression Model with the Parameters Depending on a Covariate

In this work, the catalytic activity of modified glassy carbon electrodes with xPd–yLaNi0.5Fe0.5O3–chitosan as an anodic catalyst for the polymeric fuel cell was investigated with cyclic voltammetry and controlled potential coulometry techniques; x and y are the mass loading of noble metal and mixed oxide, respectively. For the first time, the statistical regression mixed models were used to compare the electrocatalytic ability of nanocomposites in a fuel cell. The nonlinear regression model of yi,j = f(xi, (sj)) + εi was considered and simulated, where Xi is a random variable, sj is a covariate value, εi is a normal random error variable, and θ is a P-dimensional vector of parameters of the mentioned model. A strategy to make a mixed model was proposed by using the maximum likelihood or mean square error methods. Then, the appropriate linear and nonlinear models were applied to the electrochemical results. The equations of current density vs time were obtained via the fitting and simulation of experimental data at different potentials and mass loadings of components. The amounts of transferred charge during the methanol oxidation were calculated vs time through the integration of mentioned equations at different potentials and mass loadings of components.


■ INTRODUCTION
For the last few decades, a significant percentage of research on fuel cells was focused on noble-metal nanostructures as suitable electrocatalysts toward the oxidation of liquid fuels with remarkable features, including quantum, structure, size, and surface effects. 1−3 Up to now, platinum and ruthenium are regarded as effective electrocatalysts toward liquid fuel oxidation. 4,5 In spite of this, some main factors restricting the performance of such elements as electrocatalyst are the minor availability, too much high price, susceptibility to poisoning by the oxidation intermediates, and difficulty in durability due to sintering and dissolution that may reduce the active area of the electrocatalyst surface. 6,7 To overcome this bottleneck, remarkable efforts have been made on introducing strong substitutes for catalysts containing platinum and ruthenium to serve as the anode electrode in the fuel cells. 8−10 As an appropriate alternative, palladium is considered due to its comparatively low price and lower poisoning. 11−14 In the following discussion are the investigations conducted for the incorporation of mixed oxide-containing transition metals (M: La, Sr, Mn, Fe,...) with a large number of oxygen vacancies in noble-metal catalysts so that such oxides can be good candidates for replacing noble metals in direct alcohol fuel cells as anodic electrodes. 15,16 The studies showed the decrease in the onset potential related to an upgrade in the kinetics of alcohol electrooxidation. The adjacency of metal particles can remove the intermediate poisoning on the surface of the noble-metal particles. This proved that the selectivity and catalytic efficiency are extremely dependent on both morphology and size of the catalytic material. Therefore, the synthesis of noble metal (Pt or Pd) and mixed oxide on nanoscale may be the fundamental parameter affecting the performance of the noble-metal catalyst. 17 In our previous work, 18 we introduced a novel anodic electrocatalyst for methanol oxidation in direct methanol fuel cell as a type of polymeric fuel cell. The mentioned electrocatalyst was the nanoscale LaNi 0.5 Fe 0.5 O 3 particles incorporated on Pd nanoparticles dispersed into chitosan (CH) polymer.
In the present work, we prepared the mentioned nanocomposite at different loadings of noble metal and mixed oxide dispersed into chitosan polymer. The catalytic activity of glassy carbon (GC) electrodes modified with xPd−yLaNi 0.5 Fe 0.5 O 3 − chitosan was investigated with cyclic voltammetry and controlled potential coulometry techniques at different potentials; x and y are the mass loadings of noble metal and mixed oxide, respectively. In the following discussion, for the first time, use of the nonlinear regression model to analyze the behavior of prepared catalysts is described as for i = 1, 2,..., n and j = 1, 2,..., J; where X i is a random variable, s j is a covariate value, ε i is a normal random error variable with and θ is a P-dimensional vector of parameters of the mentioned model and it can be modeled as for l = 1, 2,..., P and j = 1, 2,..., J as another regression model, where γ is the L-dimensional vector of the parameters and e is a normal random error variable with We proposed a strategy to make a mixed model by using the maximum likelihood or mean square error (MSE) methods. 19,20 For this, first, a regression model of θ l on s j , where j = 1, 2,..., J, was estimated. Then, the new models were taken instead of each related parameter. To illustrate the idea, studies were conducted by using the simulation method 21 and this strategy was applied to the electrochemical results. To do this, assume according to the lth parameter of the main regression model, a model of θ l on s j in the form for j = 1, 2,..., J and l = 1, 2,..., P is possible to estimate. Therefore, one can estimate the latter mentioned model as follows for l = 1,2,..., P or for l = 1, 2,..., P and k ≥ 1.

■ EXPERIMENTAL SECTION
All chemicals were purchased from Merck and employed without further purification. Chitosan with medium molecular weight (400 000 Da) was purchased from Fluka. All solutions were prepared in distilled water.

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On the basis of our previous work, 18 Table 1 offers the list of all modified electrodes. The electrochemical behavior of modified electrodes was investigated by SAMA Electroanalyser (Isfahan, Iran) by using cyclic voltammetry and controlled potential coulometry techniques in a three-electrode cell at room temperature (T = 301 K), for 500 s and different potentials. In this way, the working, counter, and reference

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Article electrodes were the modified GC, platinum, and Hg/HgO electrodes, respectively. A mixture of potassium hydroxide solution with a known methanol concentration (1 M KOH + 1.54 M methanol) was considered as the electrolyte.

■ RESULTS AND DISCUSSION
Cyclic Voltammetry Investigations. The cyclic voltammograms of methanol oxidation on different modified electrodes were recorded at three potential ranges ( Figure 1). On the basis of Figure 1a, for the GC/0A−4B−CH electrode, the peak of methanol oxidation appeared at 1.28 V. By increasing the contribution of A to B component in the catalyst, the current density was increased so the peak at 1.28 V was detected with difficulty. As seen from Figure 1b, the peak comparison for GC/ 4A−0B−CH and GC/3.2A−0.8B−CH electrodes proved that the presence of B components shifted the potential and current to more negative and higher values, respectively. By decreasing the contribution of A to B component in the catalyst, the peak methanol oxidation disappeared. Figure 1c shows the cyclic voltammograms of methanol oxidation on different modified electrodes at a wider potential range. According to this, there are two peak methanol oxidations (1.28 V for B component and 0.43 V for A component, individually) at forward sweep for modified electrodes. To include both peaks of methanol oxidation on such electrodes with different contributions of A and B components, the potential 1.2 V was selected for the controlled potential coulometry technique.
Simulation Study. To study this approach, the model is defined as is a random variable, the sample size n = 100, the iteration number N = 1000, the main random variable x = seq (0, 10, length = n), the fixed values of D = seq (1.8, 2.8, length = 11), and the covariate values s = (2, 25, 50, 75, 100, 125, 150, 175, 200, 225, and 250). We assume that there is a relation between the parameters and the covariate given by the equation The mixed model is extracted as Not only is this a generalized model but it can also estimate the response variable according to optional values of the covariate,

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Article the simulation are subsequently stated. The estimated parameters are displayed in Table 2 where The curves of estimated parameters are shown in Figures 2 and 3.

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Article Real Data. For real data, we consider the following two conditions. First case: For constant potential at +1.2 V, the proposed model was considered for experimental data of different percentages of A and B to find a general model. The general model can predict the value of the response variable (output) for each percentage of A and B. Second case: For constant percentage of A and B, the proposed model was considered for experimental data of different values of the potential to find a general model. The general model can predict the value of the response variable (output) for each applied potential. According to the percentages of A and B components, several nonlinear regressions of current density were chosen in J/(mA cm −2 ) for these data.

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The summary of the model equivalent to P a = 1.00, P b = 0.00 follows.
Formula: J/(mA cm −2 ) ∼a + bT + cT 2 parameters estimate std. error t-value P r (>|t|) By combining the parameter estimators from these models Examples of regression models estimating the current density are reported in Table 3. By combining the A and B levels such as P A × A and (1 − P A ) × B, the resulting robust model is To have an estimation of exchanged charge over time, we consider the values for t 1 and t 2 as displayed in Table 4. Then, the fitting of the related graphs and calculation of the exchanged charge over time intervals (Figure 4) can be possible.
These values are shown in the following Tables 5−8. It is noteworthy that we considered the logarithm of the response variable with a base of 10, as log[J/(mA cm −2 )t/(s) ]; hence, it is easy to calculate the original response variable, J/(mA cm −2 ), to use the last mixed robust model by putting a new p A value and the time values to estimate the related response variable. Figure   Figure 5.  α̂= − + (21) and putting them in the main model, the mixed regression model was obtained. Finally, this nonlinear regression model can easily estimate the response value, the current density, J/(mA cm −2 ), on the basis of the participation potential level and the time.
Examples of regression models to estimate the current density are reported in Tables 9 and 10. By combining the fixed mentioned levels of A and B and potential levels, the resulting robust model is Hence, the fitting of the related graphs and calculation of the exchanged charge over time intervals (Figure 6 and 7) can be possible.

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Article ■ CONCLUSIONS In this work, the catalytic activity of modified glassy carbon electrodes with xPd−yLaNi 0.5 Fe 0.5 O 3 −chitosan as an anodic catalyst toward methanol oxidation was investigated with cyclic voltammetry and controlled potential coulometry techniques; x and y are the mass loading of noble metal and mixed oxide, respectively. The results of the simulation method to extract a suitable mixed model and robust model show that on the basis of the skill of the statistician, this strategy usually works well. It would be better, before starting the related trials, to consult a statistician, and the points of the covariate values should be more. For example, it is impossible to repeat the trials for all arbitrary percentages of the A and B components without spending time and chemicals. But with the use of this method, the same results can be obtained with less expense and time.

Notes
The authors declare no competing financial interest.