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Journal of the Chilean Chemical Society

On-line version ISSN 0717-9707

J. Chil. Chem. Soc. vol.50 no.4 Concepción Dec. 2005

http://dx.doi.org/10.4067/S0717-97072005000400016 

 

J. Chil. Chem. Soc., 50, N° 4 (2005), págs: 739-743

 

OPTIMIZATION OF PHYSICO-CHEMICAL MODELS USING THE GAUSS-NEWTON METHOD

 

SALVADOR BARBATO RAVERA*1 ,MIGUEL ALVAREZ CHAVEZ2 .

1 Department of Chemistry, Sciences Faculty. Universidad de Antofagasta. E-mail: sbarbato@uantof.cl
2. Department of Engineering, Engineering Faculty. Universidad de Antofagasta.


ABSTRACT

This study proposes the use of a numerical calculation method for optimizing non-linear physico-chemical models based on the Gauss-Newton algorithm. This method can be applicable to the Wagner-Traud model used to determine the parameters of corrosion and to adsorption isotherm models, like those of Langmuir and Langmuir-Freundlich, to determine the free energy of adsorption. This method in present work was applicated to experimental data of polarization of the iron electrode in medium of sulfuric acid 0.5 M and experimental values of adsorption of 3-Mercaptopropyltrimethoxisilane on copper surface. For the first case (polarization) the results showed with the proposed method was obtained smaller relative errors than the relative errors obtained by the Polynomial method and the computational method of Betacrunch. In the second case, the results showed that the adsorption parameters agree with the model of isotherm of Langmuir-Freundlich, obtaining a free energy of adsorption of 40 kJ/mol.

Keywords: Gauss-Newton algorithm, isotherm of Langmuir and Langmuir-Freundlich, Wagner-Traud equation, corrosion current, Tafel slopes, free energy of adsorption


INTRODUCTION

The efficiency of interpretation of experimental data requires good determination of the parameters which regulate the physico-chemical model, a condition which is not generally met when the model is non-linear. One example is the use of the non-linear Wagner-Traud (1) model (Eq. 8) for the determination of corrosion currents and Tafel slopes. Due to non-linear characteristics, the usual procedure in information processing of this type is to insert extreme conditions into the model, so that a linear correlation is achieved between the independent and dependent variables at these extremes; this then becomes a problem of partial or incomplete interpretation of the experimental data assimilated in the model2. In order to avoid situations of this type, the present study develops the Gauss-Newton3-8 numerical analysis algorithm for treatment of complete data sets, and we optimize, as examples, two nonlinear physico-chemical models including, (a) the Wagner-Traud corrosion model, and (b) an adsorption isotherm model.

DESCRIPTION OF THE METHOD

We considered that the function which represents the non-linear physico-chemical model to be optimized is given in its general form by equation 1:

where:

y = Dependent variable.
x = Independent variable.
aj = parameters of the model.
n = number of parameters in the mode
l

Thus, if there is a set of experimental values available [yexp, xexp], the parameters aj represents the values to be determined by the model . The squared value of the residual that is generated by each of the experimental data points when comparing the model with the data is given by Eq. 2:

where:

ei = square of the residual for each experimental data point i.
i = 1, m.
m = Number of experimental data points.

If we define the residual according to Eq. 3:

Then the optimal values for aj are obtained when the value of the residual is at a minimum. Each of the residuals of Eq. 2 depends on the values of aj. The matrix of Eq.4 shows the variations which affect the residual of Eq. 3:

Given that the function f (x, aj) which represents the model is not linear, and that the system of equations is over-determined (i.e.:there exist a larger number of data points than equations), to find the solution we use the Gauss-Newton algorithm following Eq. 5:

Equation 5 can be written in vectorial form according to Eq. 6:

Where vector represents the parameters which lead to the solution, and vector represents the variation of the parameters in each iteration and which must converge on values which produce minimal residuals.

RESULTS AND DISCUSSION

Gauss-Newton method for determination of corrosion parameters

Theoretical data were generated using the Wagner-Traud model to test the validity of the model for corrosion currents according to Eq. 9:

where:

Ic = Corrosion current.
ba = Slope of anodic Tafel.
bc = Slope of cathodic Tafel.
h = Overpotential.

The data generated by this equation are included in Table 1, together with the information generated by the Gauss-Newton method.


According to the data of Table 1, complete coincidence is observed between the theoretical model and the optimized model (residual value=0), validating the proposed method.

Experimental polarization data were selected in order to determine the error of the proposed method. Table 2 and Figure 1 show the optimization of the experimental data of polarization of the Fe electrode in 0.5 M H2SO4 at 25 ºC as obtained by J. Jankowski and R. Jchniewczs9.



Figure 1. Polarization curve of Fe in H2SO4 0,5 M. Experimental data are compared with optmized model using the Gauss-Newton algorithm

Based on the data from Table 3, it is observed that the Gauss-Newton method showed a lower percentage of relative error, that is, 0.7 %, compared with the polynomial method2 and the Betacrunch10 computational method, demonstrating that the proposed method was stable and reliable in quantifying the corrosion parameters and that it also permitted working over the entire range of overpotential. This cannot be done with the polynomial method, which does not allow the use of overpotential values near zero. In the case of the Betacrunch computational method, work was done within an overpotential range of ± 100 mV and with an odd number of data points.


Gauss-Newton method for determining DGº of adsorption of organic substances that act as corrosion inhibitors.

The adsorption equilibrium in the electrode / electrolyte interface is described for non-linear models, termed adsorption isotherms, which in their general form are given by the function in Eq.10 as:

f (q X)e-aq = KC000000(10)

where:

q = Molar fraction of the surface covered by the adsorbed molecule.
a = Parameter of the molecular interaction.
X = Relation between the adsorbed molecule and the solvent molecule.
K = Equilibrium constant of the adsorption.
C = Concentration of the substance that is adsorbed in the bulk of the solution.

The constant K is related to the free energy of adsorption by Eq 11:

where:

A = Reciprocal of the solvent concentration.
DG = Free energy of adsorption.

The isotherm most used because of its simplicity is that of Langmuir11, Eqs.12 & 13.

This isotherm has the following restrictions regarding the adsorbed molecules:

o A single molecule per active site.
o Supposition of the formation of a molecular monolayer.
o No lateral interactions between the molecules.

As can be observed from equations 12 and 13 the Langmuir adsorption model is non-linear, however, it is common to transform it to a linear form using logarithmic relations or reciprocal values for the data, which implies the use of transformed values in the optimization of the system. The Langmuir isotherm can be improved by introducing the heterogeneity parameter h , thus obtaining the Langmuir-Freundlich12,13 isotherm given by Eq. 14:

The heterogeneity parameter may assume a range of values between 0< h <1 and it is considered that it is a measure of the distribution of energy of adsorption at the different active sites over the surface14. Figures 2 and 3 show the results of the optimization using the Gauss-Newton algorithm proposed for the Langmuir and Langmuir-Fruendlich isotherms in the adsorption of 3-mercaptopropyltrimethoxysilane in copper15. These results are summarized in Table 3.


 
Figure 2. Langmuir isotherm model. Degree of coverage vs. concentration of 3-mercaptopropyltrimethoxisilane in Cu   Figure 3. Langmuir-Freundlich isotherm model. Degree of coverage vs.concentration of 3-mercaptopropyltrimethoxisilane in Cu

The data from Table 3 (Figures 2 and 3) show that the results of the free energy of adsorption are similar in the three cases.

Nevertheless, the application of Gauss-Newton method to the model of adsorption of Langmuir- Freundlich is obtained the greater coefficient of correlation (0.99). From this result can conclude that the Langmuir-Freundlich isotherm is the best that represent the experimental15 values and therefore, that the free energy of adsorption for this case should be of 40 kJ/mol.


CONCLUSIONS:

Application of the Gauss-Newton model for optimizing non-linear physico-chemical models produces optimal results.

The proposed model has the advantage of being applicable to a complete range of experimental data.

The non-linear model is a good alternative for determining physico-chemical parameters in problems concerning both corrosion and adsorption.

The proposed model can be applied to any physico-chemical system which has non-linear behavior

ALGORITHM CALCULATION:

The calculation programs developed in this paper were programmed in Matlab 6.5.

 

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