Evaluation of the electrochemical parameters by means of series expansion

Corrosion Science. Vol. 36, No. 8, pp. 1347-1361. 1994

t ~

Pergamon

Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0010-938X/94 $7.00 + 0.00

0010.938X(94)E0029-5

EVALUATION OF THE ELECTROCHEMICAL PARAMETERS BY MEANS OF SERIES EXPANSION G. ROCCHINI E N E L - Environment and Materials Research Center, Via Rubattino 54, 20134 Milano, Italy

A b s t r a c t - - A new mathematical method for computing the corrosion current density and Tafel slopes from experimental data is presented. The method is of global type because the theory is based on the Maclaurin's power series expansion of the function i ( A E ) and the calculation of three definite integrals over AE intervals of various width. Its application provides valid information even if the trend of the experimental polarization curve over a suitable interval is approximated in mean by a function of the type i(AE) = l c [ e x p ( a A E ) - exp(-flAE)] with a satisfactory accuracy. A basic feature of this method is that it is not based on any iterative calculations, which sometimes may give wrong results because of convergence problems of the numerical sequences. Some applications to ideal cases have shown its validity when the series expansion is approximated by a polynomial of suitable degrec. The experimental applications concerned the behaviour of Armco iron in some uninhibited and inhibited acid solutions, which was studied by performing current-transient m e a s u r e m e n t s , and afforded satisfactory determinations for the electrochemical parameters 1c, B~. and B c. It is also shown that, if suitable terns ( A E 1 , A E 2, AF<~) are chosen, practically the same values of the electrochemical parameters are obtained. The calculation of the definite integrals is of analytical type being based on the best-fitting of the experimental data with a polynomial of the fourth degree. The use of the best-fitting technique was necessary because the distribution of the N pairs (AE,,, i,,) were not regular.

INTRODUCTION

EXPERIENCE shows that the electrochemical behaviour of many metals and alloys in several environments is described by a simI'le law even if from a microscopic point of view the overall process contains a lot of elementary steps which can run in parallel or series. This observation has a great importance because the resistance to corrosion of metals and alloys can be evaluated by performing simple measurements and processing collected data properly. Some discrepancies may be observed between electrochemical (Ie) and direct (Id) determinations of the corrosion rate. The previous observation is confirmed by the data reported in Fig. 1, drawn from Ref. 1, which illustrates the relationship between the two determinations of the corrosion current density. The experimental data refer to the behaviour of Armco iron immersed in 0.5 m H2SO4 solutions containing some primary aliphatic amines. Examination of Fig. 1 shows that most points, excluding those rather close to zero, fit well to a straight line having the equation I e = 0.905Id. Corrosion current density is a useful quantity which, for instance, as demonstrated by Kelly2 while studying the behaviour of pure iron in H2SO4 solutions saturated with hydrogen and at different pH, can be used to determine the reaction order of the hydrogen evolution reaction. From a physical point of view the use of this quantity as a parameter for ranking the resistance to corrosion of different Manuscript received 13 December 1993; in a m e n d e d form 25 January 1994. 1347

1348

G. ROCCHINI

Ie [mAcm -2] .22

/

.18 Q / A

.14 -

.10

-

QA

.06

•n

• n-octyLamlne .

.02 .00

• n-dodecylamine I

I

I

I

I

I

.04

.08

.12

.16

.20

.24

.

|g [rnA cm -2] FIG. I. Comparison of the direct, Id, and electrochemical, le, determinations of the corrosion current density for Armco iron in H2504 solutions containing some primary aliphatic amines.

materials in a given environment is legitimate only when they are subjected to attack of a uniform type. The importance of the previous remark is due to the fact that it is possible to find in the literature concerning corrosion monitoring many applications of polarisation resistance even if samples experience localised attack. The behaviour of steel embedded in concrete, for example, represents a situation where the use of the polarisation resistance must be evaluated carefully. The main effect of some aggressive species, such as chloride ions, is to destroy the passive state of the steel surface, and cause local attack. Many numerical methods of linear or non-linear type have been suggested for analysing polarisation curves describing electrochemical systems that exhibit the characteristic Tafel behaviour when the AE values, referred to the mixed potential Era, defined by i(E) = 0, are rather far from AE = 0. A good review of some numerical techniques has been made by Jensen and Britz, 3 who compared the reliability of the three point method of Barnartt, 4 the two four point methods of Jankowski and Juchniewicz 5 and Bandy, ° the C O R F I T method of Mansfeld, 7 the B E T A C R U N C H method of Greene and Gandhi 8 and the non-iterative method of Feliu and Feliu. 9 Some of the previous methods may have serious limits as concerns a satisfactory analysis of experimental data, because their mathematical models introduce approximations for searching for the roots of the resolving equations. The importance of this observation has been confirmed by Rocchini 1° who demonstrated that the validity of the Mansfeld method depends on the scheme adopted for computing the polarisation resistance, ~ even if the changes of the values of the electrochemical parameters are not so important as to invalidate the numerical analysis. The method of Barnartt, which is based on an equation of the second degree, is of the local type and has the drawback of having to select the three points

Electrochemicalparameter evaluation by series expansion

1349

rather close to the mixed potential. This choice can give rise to bad determinations of the electrochemical parameters because of the possible contribution of minor reactions to the two dominant processes. The presence of a small perturbation is very important because the values of the current density close to E m are not usually high enough to allow other contributions to be neglected. Experience shows that in some cases the roots of the solving equation have no physical meaning because they are complex conjugate. To remove this difficulty, Rocchini 12 suggested a suitable extension of the previous method which, in principle, is able to provide information of the global type. This extension cancels the simplicity of the original version and requires the use of suitable software for determining the value of the anodic Tafel slope. The foregoing remarks justify other studies in the field of numerical analysis of experimental data aimed at developing more reliable methods which are easier than those already existing. From a mathematical standpoint, the reliability of a given method means that its application is independent of the ability of any user so that different corrosion scientists will find equivalent results when processing the same experimental data. This point expresses the need that the theory of the proposed method must be rigorous and any approximation must concern only the search for the roots of the solving equations set. A method which belongs to the previous class is NOLI1.13 This method is of non-linear type, analyses experimental data over intervals having a reduced width and is based on the functionf(AE) = ln]exp(aAE) - exp(-flAE)] which depends on the two parameters, ct and ft. Another important aspect of the numerical analysis of the polarisation curves concerns the on-line monitoring of a corrosion process using computerised systems. In such problems the success of data processing plays an important role as regards the usefulness and reliability of the application. Moreover, any method different from the polarisation resistance introduced by Stern and Geary 14 has the great advantage of providing more reliable information. Examination of high AE values makes it possible to neglect the influence of random fluctuations which may be important when the specimen is kept close to its free corrosion state. MATHEMATICAL DEVELOPMENTS A key point of any numerical technique is to define its application field. Usually, the main aim of a given method is to provide significant evaluations of the unknown parameters. This end is reached when the mathematical model is ~ble to separate the elementary contributions from the overall reaction. If the method fits only a polarisation curve, the probability that the information is meaningless is not negligible. This remark means that a corrosion scientist must know in advance the kind of response likely to be obtained from the examined system before applying any numerical technique to process experimental data. The basic assumptions are that the system contains one anodic reaction and one cathodic reaction and that they follow the Tafel law when the electrode potential E is quite far from the mixed potential. The first assumption states the possibility of neglecting all the minor contributions to the overall process. This hypothesis is not very restrictive when the electrode potential is rather different from Em. Under these hypotheses the current-voltage characteristic takes the following expression

1350

G. RocCHINI i( A E) = Ic[exp( a A E ) - exp(-/3AE)],

(1)

where Ic indicates the corrosion density and AE gives the potential difference referred to Em. Furthermore, the parameters a and/3 are linked to the usual Tafel slopes B a and B~ by the relationship: B a = a -1 In 10 and B~ = /3-1 In 10. It is interesting to stress that, according to the law (1), the quantities a and/3 take always positive values which are less than 1. This observation is useful because it provides a criterion for establishing whether the method works properly when processing experimental data. The derivation of the expression (1) has been discussed by Gellings 15 who examined the conditions that are necessary to add other contributions to the main anodic and cathodic reactions. Besides simplicity, the importance of the kinetic law (1) is that it is possible to establish some equations that link the parameters Ic, a and/3 to the first, second and third derivatives of i ( A E ) computed at AE = 0. In particular, it is interesting to examine the possibility of reducing the number of unknown parameters so that the solution of the problem is easier. Computing the first three derivatives of the function (1) at z~E = 0, the following equations are obtained: i'(O) = lc(a +/3)

(2)

i"(O) = I~(a 2 - / 3 2 )

(3)

i'"(0) = Ic (a 3 +/33 ).

(4)

Now considering the well-known identity a 3 -b-/33 = (0~2 q._ f12 _ O~/3)(a --{--/3),

(5)

and introducing the ratios A = i"(O)/i'(O) and B = i'"(O)/i'(O), two equations are attained, a -/3 = A

(6)

a 2 + t32 - a/3 = B,

(7)

which allow the values of a and/3 to be determined by means of simple algebraic calculations. In fact, considering the equation (6), the relationship (7) becomes a 2 - A a + A 2 - B = 0.

(8)

The choice of the proper root of the equation (8) is immediate if the equality is considered, a =

A + X/4B - 3A 3 2

(9)

It becomes an identity when A and B are substituted with their expressions obtained from the relationships (6) and (7). Thus, from the equation (6) X/4B - 3A 2 - A /3 =

SO

2

'

(10)

Electrochemicalparameter evaluationby series expansion

1351

a + f i = X / 4 B - 3A 2,

(11)

which shows that the quantity (a + fl) can be expressed in terms of the geometric characteristic of the law (1) at AE = 0. The equation (11 ) permits to compute I c using the expression (2) by means of the formula i'(0) (12) Ic - X/4B - 3A 2 or, remembering that the polarisation resistance is given by Rp = 1/i'(O), the relationship 1

I~ = R~ x / 4 e - 3A 2" (13) The last expression represents a generallsatlon of the theory of Stern and Geary for the determination of the corrosion rate. From this standpoint the equation (13) indicates that there is no constraint on the width of AE intervals containing AE = 0. Moreover, to obtain a better evaluation of the first three derivatives it is convenient to perform measurements over AE intervals greater then [-10, 10] mV. P'

,

•

Approximation ofi(AE) by a polynomial From the previous developments it is evident that the main problem of the present method concerns the approximate evaluation of the quantities i' (0), i"(0) and 7'(0). This problem can be tackled by introducing the Maclaurin's series expansion 16 of the function i(AE). Considering the first n terms of this expansion and using the integral representation of the remainder, Rn(AE), i : ( a + f l ) ~ + (xa

2 - f l -~) ~x. 2+ . . + [ a " + ( - f l ) n ]

1 0 i("+0 (t)(x - t)"dt, (14) +~.T

where for simplicity it has been set x = AE. An important aspect of the series expansion (14) concerns the definition of the interval [0, AE] where the function i = i(AE) is represented faithfully by a polynomial of the n degree. To this end it is useful to recall that the remainder R,(AE), in accordance with the formula of Lagrange 16 is equal to i0'+I)(~) R , ( A E ) = AE ~"+') (n + 1)!'

(15)

where, according to the theorem of the mean value, ~ indicates a point inside the interval [0, AE]. Thus, remembering that i(m(AE) = Ic[a" exp(aAE) - (-/3)" exp(-flAE)], many [0, AE] intervals are found where the remainder Rn(AE ) can be neglected when n is >> 1. Assuming a > fl, considering the anodic interval [0, bE] and setting M = Ic exp(aAE), the inequality is obtained R,(x) < (aAE) n+l

M (n + 1)!'

(16)

which is always valid when 0 -< x -< AE. The inequality (16) provides a tool for verifying the validity of the approximation of i(AE) with a polynomial P,(AE) of the n degree.

1352

G. ROCCHINI

To render this argument clear it is useful to give some numerical results. To this end Table 1 reports the values of the dimensionless quantity e=

i(AE) - P,(AE) × 100 i(AE)

(17)

as a function of AE in the case of the three polynomials of the 4th, 5th and 6th degrees. This example refers to B a = 40 mV and Bc = 120 mV. It is not necessary to give the value of I c because e, according to its definition, is independent of the corrosion current density. Examination of this table shows that a polynomial of the 6th degree provides a satisfactory representation of the current-voltage characteristic over the AE interval [ - 5 0 , 50] mV, the maximum value of the percent deviation being less than 10. Thus, giving this example, the previous statement is confirmed because the faithfulness of a polynomial representation is independent of the values of a and ft. On the other hand this result can be regarded from a mathematical standpoint as an extension of the linear response theory. The same consideration applies for the polynomials of the 4th and 5th degrees even if a reduction of the amplitude of the AE interval, where they provide an accurate approximation, is observed. A characteristic common to the previous approximations is that the goodness of the representation in the anodic zone differs from that exhibited in the cathodic region where the difference between i(AE) and Pn(AE) increases as the absolute value of AE rises.

Calculation of the first three derivatives The calculation of the three quantities i'(0), i"(0) aAd i'"(0) represents a key point of this theory, even if this problem can be solved in several different ways. From the previous developments, which prove the validity of the representation of i(AE) with a suitable polynomial Pn(AE), it is evident that a correct approach for solving this problem is based on the use of the polynomial best-fitting. In this case an attempt is made to represent a polarisation curve, described with a great accuracy by the function (1), over a given interval [AE1, AE2] with the polynomial TABLE 1.

VARIATION OF ~" AS A FUNCTION OF T H E POLYNOMIAL D E G R E E

AE (mV)

4th

5th

6th

-50 -40 -30 -20 20 30 40 50

43.12 18.86 6.20 1.23 0.84 3.49 8.83 16.87

-21.60 -7.50 -1.83 -0.24 0.16 0.95 3.16 7.37

9.28 2.56 0.47 0.04 0.02 0.23 O.99 2.84

Electrochemical parameter evaluation by series expansion

1353

r/

Pn(x) = ~" akx k,

(18)

k=l

where the coefficients a n are determined by solving a set of n equations which are easily obtained from the theory of the linear best-fitting ~7 or other mathematical methods for interpolating a given function. 18 For the sake of simplicity only the linear best-fitting of the ideal law (1) based on a polynomial of the 4th degree is examined by considering some numerical applications. The use of polynomials of higher degree does not present additional difficulties because there are several numerical methods for searching for the roots of sets having a large number of equations. 18 The main aim of the examination of some numerical examples is to demonstrate the validity of the polynomial approximation for computing the electrochemical parameters. To this end, it is not essential to select properly the degree of the polynomial because we can choose a convenient interval where the approximate evaluations of the first three derivatives of i(AE) at the point AE = 0 are very close to the true values. From this mathematical standpoint the choice of a polynomial of the 4th degree has the advantage of allowing an easy analytical solution of the equations set. The values of the parameters Ba, Bc and Ic, which refer to the function examined, are given in Table 2, where the first column singles out the artificial systems. The approximate values of the electrochemical parameters, obtained using the formulae (9), (10) and (12) and performing the best-fitting over the interval [-40, 20] mV, are reported in Table 3. Calculations were carried out using single precision and considering 50 points at regular intervals of 1.2 inV. The determinant of the 4 x 4 TABLE 2.

VALUES OF THE ELECTROCHEMICAL PARAMETERS FOR SOME IDEAL CASES

System

Ba (mV)

B c (mV)

I c ( m A cm -2)

A B C D E F

50 60 40 70 45 55

140 90 120 130 100 85

2.(I 1.0 (1.1 (1.8 0.3 0.5

TABLE 3.

COMPUTED VALUES OF THE ELECTROCHEMICAL PARAMETERS

System

B a (mV)

Bc (mV)

I c ( m A cm -?)

A B C D E F

49.77 59.75 39.66 69.81 44.66 54.72

145.00 90.67 127.15 130.69 102.51 85.82

2.010 1.001 0.101 0.800 0.301 0.501

1354

G. ROCCHINI

matrices was computed by means of Laplace rule. 19 The use of Laplace rule is quite easy for computing the determinant of a 5 x 5 or 6 × 6 matrix even if the algebraic operations are more laborious. The values of B a and B c are given with two decimal digits in order to evaluate the goodness of this approach. Examination of Table 3 shows very clearly that the use of a polynomial of the 4th degree is a valid tool for computing the values of B a and Ie. The approximate evaluations of these quantities are practically coincident with their true values. However, the calculation of the cathodic slope shows in some case a significant difference even if the percent error is rather small. The ratios of the actual and approximate values of i'(0), i"(0) and i"(0) were also examined. In all the cases considered the values of these ratios are very close to 1. This observation provides a further confirmation of the validity of the linear bestfitting. The previous results, which from the point of view of a corrosion scientist must be considered very good, are explained by the data of Table 1 which show that a polynomial of the 4th degree can significantly differ from i(AE) over the interval [ - 4 0 , - 3 0 ] mV. Furthermore, some calculations proved that the difference between the true and approximate values of Baincreases as the amplitude of the anodic region exceeds 20 mV.

Integral formulation Another interesting way for obtaining an approximate evaluation of the unknown quantities i'(0), i"(0) and i"(O) from experimental data is to solve a set of three equations obtained by a suitable rearrangement of the Maclaurin's formula. To this end, if the experimental current density ie (AE) is introduced and equated to the ideal characteristic i( A E),

iAe, ie(x)dx = i'(0) (AE 2 - AE~) + i"(0) 'AE 3

AE3o)+

i ' " ( 0 ) / A t74

AE 4)

(19)

/'"(0)/A174 -- AEB) 4! ~ "-2

(20)

JAE{}

aEzie(X)dX --- Ti'(0) (AE2 -- AE°2) + i"(0) (AE

~aE,,

3!

- AE{ ) +

I a& &(x)dx = i'(0) (AE32 _ AE2) + t'" (0) tAE3 _ AEB) + ~"m/0~ (AE T-, 3 ./AE{~ " "

43 _ AE4) '

(21)

which permit an approximate evaluation of the unknown quantities. The values of the first three derivatives at AE = 0 can be determined, for instance, by setting AE 0 = 0. Obviously the upper bounds AE1, AE 2 and AE 3 should fall inside the interval where the series expansion of i(AE), truncated at the term of the third degree, keeps its validity. However, the equations (19), (20) and (21) provide only a useful suggestion which defines the main lines for setting a set of equations. A good evaluation of the third derivative at AE = 0 is essential for a correct application of the present theory. The last observation is rather interesting because the choice of the number of equations usually could depend on the problem under investigation. Moreover, it is not possible to state a priori if a scheme based on a fixed equations set is suitable for any electrochemical system. Thus it is opportune to have improved versions of the main scheme which are based on a larger number of equations. Furthermore, it is

Electrochemicalparameter evaluation by series expansion TABLE 4.

VALVES OF THE E L E C T R O C H E M I C A L

1355

PARAMETERS

OBTAINED USING THE INTEGRAL FORMULATION

System

Ba (mV)

Bc (mV)

Ic (mA cm 2)

A B C D E F

49.38 59.61 39.19 69.63 44.38 54.57

135.2 89.1 113.1 128.7 97.0 84.0

1.960 0.990 0.097 0.794 (t.294 0.495

useful to stress that the correct application of the equations (9) and (10) requires that the inequality 4B

-

3 A 2 >- 0

(22)

holds because the slopes of the anodic and cathodic Tafel straight lines must be always real numbers. Usually the relationship (22) is an inequality, B~, and B c being different when we are dealing with a corrosion process. The validity of the scheme based on the equations (19), (20) and (21) has been verified by considering the previous examples and truncating the series expansion of i(AE) at the term of the fourth degree. The upper bounds of the four integrals were chosen as follows: AE1 = - 15 mV, AE 2 = - 10 mV, AE 3 = 10 rnV and A E 4 = 15 mV. Calculations were performed using single precision. The choice of a polynomial of the fourth degree was dictated by the need of comparing the goodness of the schemes based on the best-fitting and integral formulation. The results are given in Table 4. Examination of Table 4 shows that the integral formulation provides a good evaluation of the anodic slope and corrosion current density. The comparison of Tables 3 and 4 stresses that the two approaches are essentially equivalent. This observation is also confirmed by the difference existing between the true and the approximate values of the cathodic slope. Furthermore, the ratios of the actual and approximate values of i'(0), i"(0) and i"'(0) are rather close to 1, even if the determination of the third derivative is lightly different from that obtained using the best-fitting method. Notwithstanding the fact that the analysis of the previous functions does not provide the exact values of the electrochemical parameters, however, this scheme seems to be a valid tool for analysing experimental polarisation curves close to AE = 0 when their shape approximates the ideal behaviour quite faithfully. EXPERIMENTAL APPLICATIONS The validity of this new method, based on the equations (19), (20) and (21), has been checked by examining the polarisation curves of various electrochemical systems concerning mainly the behaviour of Armco iron in 5% by weight HCl uninhibited and inhibited solutions at different temperatures. The polarisation curves were of galvanostatic pulse type and were performed using a polarisation time of 100 ms, whereas the simultaneous readings of the electrode potential and current intensity were made 40 ms after the application of the current transient with an integration time of 20 ms. The determination of the solution

1356

G. ROCCHINI

resistance between the working and reference electrodes was based on the use of alternating current having a frequency of about 104 Hz. Usually the polarisation curves were performed, at regular time intervals, over the AE interval [ - 9 0 , 60] mV and the values of the electrode potential were numerically corrected for accounting for the contribution of the ohmic drop. During the performance of the polarisation curves it was carefully verified that the shift of the free corrosion potential was negligible. The systems considered were: (G) 5% HC1 at 25°C; (H) 5% HC1 at 40°C; (I) 5% HC1 at 55°C; (L) 5% HCI at 65°C; (M) 5% HC1 and 0.001 g 1-1 inhibitor U at 75°C; (N) 5% HC1 and 0.01 g 1-1 inhibitor U at 75°C; (O) 5% HC1 and 0.1 g 1-1 inhibitor U at 75°C; (P) 5% HC1 and 0.01 g 1-1 inhibitor V at 75°C; (Q) 5% HC1 and 0.1 g 1-~ inhibitor V at 75°C; and (R) 5% HCI and 1 g 1-~ inhibitor V at 75°C. To compute the first three derivatives of i(AE) at AE = 0 the upper bounds, expressed in mV and indicated by the two terns S =- ( - 2 5 , - 1 5 , 35) and T --- ( - 3 5 , 15, 25), were used. In principle, the choice of a tern is arbitrary on the condition that the inequality (22) holds. The last observation is very important because close to AE = 0 the contribution of minor reactions to the overall process could distort the shape of i(AE), even if for values of AE rather far from the origin it follows the law (1) with a high accuracy. To this end the accuracy and reliability of the present method was verified by considering many other terns having points in the anodic and cathodic zones which were selected inside the AE interval [ - 5 0 , 50] mV. This step was also decided in order to establish a procedure applicable to a wide class of electrochemical systems. Comparison of several evaluations of the first three derivatives of i(AE) showed that the choice of the tern does not affect the calculation of the electrochemical parameters, their values being practically equivalent. At any rate, the evaluation of the electrochemical parameters by considering various terns is a very easy operation which is not time consuming when suitable software is used. For the sake of simplicity it was decided to report only the values of the electrochemical parameters concerning the S or T tern. Experience shows that such a choice is appropriate for many other electrochemical systems. Another application concerns the behaviour of some carbon and low alloy steels in solutions containing 100 g I- 1of E D T A and having different pH in the range of 6-9 at room temperature. In this case the tests were of dynamic type, the solution velocities being 0.5, 1.0 and 1.5 ms -1, and the potentiostatic polarisation curves were performed over the AE interval [-100, 100] mV by 3 mV steps and using a polarisation time of 1 s. It is important to stress that this application was made in order to value the goodness and reliability of the corrosion rate monitoring based on the present approach. To this end the SOFTCOR-DC-PS2 program, which drives the Solartron electrochemical interface mod. 1286, was used. EXPERIMENTAL RESULTS AND DISCUSSION An important aspect concerning the application of the method based on the equations (19), (20) and (21) regards the calculation of the three definite integrals which can be done in several ways. 18 In the present case, for sake of simplicity, a choice was made to compute the definite integrals using the best-fitting of the experimental data with the polynomial obtained from the equation (18) with n equal

Electrochemical parameter evaluation by series expansion

1357

AE [mY]

Experm i ental vatues / - •

,, tom P

/ ¢

.F

.,t /

0

-40 , .....

I

10-3

FIG.2.

........

I

;

10-2 i [ m A c m -2]

,

,~llll

II

,-

10-}

Comparison of the experimental data and their representation with a polynomial of the fourth degree for some electrochemical systems.

to 4. To this end the program INTER1 was developed. This program is not specific for the present application, but has been used in other cases, as, for instance, it is shown in Ref. 20, successfully. Thus, on the basis of a previous experience, it was decided to adopt a polynomial of the fourth degree. The validity of this choice is confirmed, for instance, by Fig. 2 which compares the experimental points with their analytical representation, obtained from the polynomial best-fitting, for the systems O, P and R. Examination of this figure, which reports the behaviour of i(AE) over the interval [-50, 50] mV, shows that the chosen polynomial r~presents all the experimental data very faithfully. Furthermore, this result was not accidental because a good agreement was observed for many other systems. This observation is very significant because it justifies the use of the analytical calculation, based on a polynomial representation, of the definite integrals. The utilisation of a valid numerical technique should provide nearly the same result. This happens because experimental and analytical points are very close. The polynomial best-fitting of experimental data provides a reliable and easy tool for the calculation of the integrals of the left-hand sides of equations (19), (20) and (21) with reference to S and T terns. From this standpoint, the polarisation curve resulting from the bestfitting of the experimental data can be considered as a true curve because usually a lack of reproducibility of the experimental points is observed when several measurements are performed during the same test. Besides the change in the metal reactivity, the lack of superposition can be also determined by the shift of the free corrosion potential. In any case the best-fitting technique becomes necessary when the distribution of the experimental data is so irregular as to jeopardise the numerical calculation of the definite integrals. Another interesting aspect of this approach is the verification if the values of i' (0), i"(0) and i'"(0) depend in a significant way on the choice of the upper bounds. This point was investigated by considering mainly the two terns S and T. The need of this

1358

G. ROCCHINI

TABLE 5.

VALUES OF THE FIRST THREE DERIVATIVESOF i(AE) AT AE = 0 FOR SOME SYSTEMS

System

Tern

i'(0) x 103 mAcm-2mV 1

i"(0) x 10 6 mAcm 2mV-2

i"(0) x 107 mAcm-2mV 3

G

T S T S T S T S

11.570 11.610 70.160 71).200 11.660 11.660 2.622 2.628

-2.490 -2.490 -134.6 00 -134.6 00 -63.210 -63.210 -12.470 -12.470

80.580 79.030 512.000 510.400 39.910 39.720 5.840 5.592

H N P

verification is dictated by the fact that an experimental polarisation curve usually differs from the ideal case, as far as local information and global content of a given AE interval are concerned. This happens because the response of the electrochemical system contains, in principle, the contribution of some minor species as long as their presence is not negligible with respect to the two predominant reactions. Some results of this investigation concerning the systems G, H, N and P are shown in Table 5. Examination of Table 5 shows that, in the case under discussion, the two determinations of the first three derivatives are very close. Owing to the arbitrariness in the choice of a tern, it is not possible to generalise the previous results because the information contained in an experimental polarisation curve depends on the amplitude of the potential interval. It is useful to point out that experience has shown that the choice of the upper limits cannot be completely arbitrary. In the case of the S and T terns the points in the anodic zone were selected concurrently with those in the cathodic region because it was observed that the determination of the electrochemical parameters was in good agreement with that obtained using the NOLI method. 21 Some preliminary calculations had revealed the existence of discrepancies between the two determinations when the three upper limits were selected inside the anodic zone. NOLI method was developed in order to analyse a polarisation curve containing anodic and cathodic points. The value of this method was stated by means of the comparison of the values of the electrochemical parameters with those obtained using the NOLI method, when analysing the curves generated with the polynomial representation of the experimental data. The results of the analysis with the NOLI method are reported in Table 6. The data reported in Table 6 are very instructive because they show that in the region examined the experimental polarisation curves can be analysed using the law (1). From this standpoint that is a reliable reference in order to assess the validity of the proposed method. Lastly, Table 7 lists the values of Ie, Ba and B e obtained using the new method. The tern, S or T, was selected so as to have the best agreement with the NOLI method. The agreement between the data of Tables 6 and 7 can be considered very satisfactory even if it is based on a suitable choice of the S and T terns. This observation does not impair the validity of the present approach. In fact, for the

Electrochemical parameter evaluation by series expansion TABLE 6.

1359

VALUES OF THE ELECTROCHEMICAL PARAMETERS OBTAINED USING THE N O L I METHOD

System

Ic (mA cm -2)

Ba (mY)

Bc (mY)

G H l L M N O P Q R

0.231 1.368 4.754 26.405 0.108 (I.338 0.068 (}.246 (I.060 0.019

90.87 92.00 101.36 113.83 215.12 154.88 250.40 174.46 371.56 193.97

92.22 86.79 82.13 76.85 169.60 116.62 278.08 130.88 254.78 196.18

s y s t e m G the tern T gives the following results: Ic = 0.219 m A c m -2, Ba = 87.63 m V a n d B c = 86.92 m V . F o r the system L, which p r e s e n t s the highest v a l u e of the c o r r o s i o n r a t e , the S t e r n gives: Ic = 25.076 m A cm -2, B a = 109.34 m V a n d Bc = 73.15 inV. It can b e s t a t e d that w h e n the terns are c h o s e n in a c c o r d a n c e with the c r i t e r i o n a d o p t e d for S a n d T the d e t e r m i n a t i o n of the e l e c t r o c h e m i c a l p a r a m e t e r s c a n n o t be c o n t r a d i c t o r y . This result was quite e x p e c t e d b e c a u s e the choice o f terns like S a n d T e n h a n c e s the c o n d i t i o n i n g of the e q u a t i o n s (19), (20) a n d (21) a n d p e r m i t s p r o p e r i n f o r m a t i o n to be o b t a i n e d f r o m an e x p e r i m e n t a l p o l a r i s a t i o n curve. T h e a g r e e m e n t b e t w e e n the two m e t h o d s is not f o r t u i t o u s , d e p e n d i n g m a i n l y on t h e a c c u r a c y o f t h e p e r f o r m a n c e o f the e x p e r i m e n t a l p o l a r i s a t i o n curves. C o n c e r n i n g the use of the S O F T C O R - D C - P S 2 p r o g r a m for s t u d y i n g t h e b e h a v i o u r o f s o m e c a r b o n steels in E D T A solutions, it is n e c e s s a r y to p o i n t o u t t h a t the success of the n u m e r i c a l analysis o f e x p e r i m e n t a l d a t a , with r e f e r e n c e to t h e a m p l i t u d e of t h e p o t e n t i a l interval p r e f i x e d by s o f t w a r e , d e p e n d s on the i n t e n s i t y o f t h e c o r r o s i o n r a t e . This o b s e r v a t i o n is c o n f i r m e d , for e x a m p l e , by the fact t h a t t h e r e was no p r o b l e m in the case o f low alloy steel S A 213 g r a d e T9 which exhibits a v e r y

TABLE 7.

VALUES OF

Ic, Ba AND B c OBTAINED WITH THE EQUATIONS (19), (20) AND (21)

System

Tern

Ic (mA cm -2)

Ba (mY)

Bc (mY)

G H 1 L M N O P Q R

S T T T T S T T T T

0.222 1.301 4.508 25.919 0.108 0.327 0.067 0.243 0.060 0.019

88.61 88.61 98.48 112.80 212.99 152.12 245.05 172.97 363.21 195.69

87.88 82.88 77.30 74.85 170.65 112.01 278.10 128.79 253.93 193.09

136o

G. ROCCHINI

small corrosion rate in the pH range from 6 to 9. Similar behaviour was observed for SA 106 grade B carbon steel at pH 8, where it exhibits the best resistance to corrosion. In our opinion, such a situation is mainly ascribed to the solution resistance between the working and reference electrodes which had a value of about 1 1~. The area of the exposed surface of the specimens was about 60 cm 2. The question of the electrochemical measurements being or not being a valid tool for a faithful representation of the true behaviour of a corrosion process is still open. A careful answer to this question is not very easy, because it depends on the nature of the electrochemical system, and is beyond the scope of the present work which was intended to prove the validity of the scheme based on the equations (19), (20) and (21).

CONCLUSIONS Validity of thc numerical method which uses the scheme based on the cquations (19), (20) and (21) has bccn carefully checked by examining some numerical examples and experimental polarisation curves concerning a wide class of electrochemical systcms exhibiting a markcd difference in thc corrosion currcnt density. Examination of ideal cases had the aim of getting valid information about a suitablc choicc of the degree of the polynomial to be used for an approximate rcprcscntation of the kinetic law. The experimental cascs cxamincd have pointed out that the application of this method isIcss criticalthan the analysis of the ideal cascs. This resultcan bc cxplaincd by considering that the contribution of thc term a4AE 4 of thc best-fittingpolynomial ispracticallyncgligiblc whcn intervals having a small amplitude arc considered. This observation explains how the two methods provide very close dctcrminations of the clcctrochcmical parameters. Thc proposed method is of easy application and has thc great advantage of not rcquiring a dccp knowledge of numerical analysis in order to develop a program for processing cxpcrimcntal data. From this standpoint a good suggestion, when thc aim of thc application concerns thc cvaluation of the magnitudc order of the electrochemical paramctcrs, is to usc a polynomial of the fourth dcgrcc. The main convenience of the present technique with rcspcct to the N O L I method is its simplicity and thc fact of requiring a shorter proccssing time since it docs not use iterativc calculations. Its main constraint is represented by the prcrcquisitc that the inequality (22) must bc always vcrificd. This observation is very significant bccausc it points out the importance of the clcctrochcmical technique uscd to pcfform a polarisation curve. A careful use of the polynomial best-fittingrcquircs the selection of a potcntial interval whcrc from a graphic standpoint the Tafcl bchaviour is not wcll-defined. It is necessary to reduce the influcncc of any factor which has nothing to do with the ovcrall proccss on the shape of the polarisation curve. Finally it is useful to underline the point that cvcn if the determination of the slopcs of the Tafcl straight lincs is not very significant,equation (13) providcs a tool for a qualitative evaluation of thc corrosion rate and aids many problems whcrc the essential objective isto estimate the magnitude order of the intensity of the corrosive attack. From this standpoint cquation (13) represents an cxtcnsion of Stern and Geary's formula and can bc used for computing the corrosion current density without thc nccd to have preliminary knowlcdgc of the values of the Tafcl slopcs.

Electrochemical parameter evaluation by series expansion

1361

REFERENCES 1. G. PERBONIand G. ROCCHINI, Proc. 6th European Symposium on Corrosion Inhibitors, Vol. l, p. 509, Ferrara, Italy (September 1985). 2. E.J. KELLY, J. electrochem. Soc. 112, 124 (1965). 3. M. JENSEN and D. BRrrz, Corros. Sci. 32,285 (1991). 4. S. BARNARTr, Electrochim. Acta 15, 1313 (1970). 5. J. JANKOWSKIand R. JUCHNIEW1CZ,Corros. Sci. 20, 841 (1980). 6. R. BANDY, Corros. Sci. 20, 1017 (1980). 7. F. MANSFELD, Corrosion 29,397 (1973). 8. N. D. GREENE and R. H. GANDHi, Mater. Perform. 26, 52 (1987). 9. V. FEL1Uand S. FELIU, Corrosion 42,151 (1986). 10. G. ROCCHINI, Corros. Sci. 36,567 (1994). 11. K. F. BONH6FFERand W. JENA, Z. Elektrochem. 55, 151 (1951). 12. G. ROCEHINI, Corros. Sci. 30, 9 (1990). 13. G. ROCCmNI, Corros. Sci. 34, 583 (1993). 14. M. SIERY and A. L. GEARV,J. electrochem. Soc. 104, 56 (1957). 15. P. J. GELLtNGS, Proc. 3~ Congrds de la F(d(ration Europ(enne de la Corrosion, Section 1, p. 1, Brussels, Belgium (1964). 16. V. 1. SMmNOV,A Course of Higher Mathematics, Vol. 1, Pergamon Press, London-New York (1964). 17. N. R. DRAPER and H. SMIlH, Applied Regression Analysis, John Wiley and Sons, Inc., New York (1981). 18. B. D~MIDOVI1"CHand I. MARON, Elements de Calcul Num(rique, Mir Publishers, Moscow (1979). 19. G. SCORZADRAGONI, Elementi di Analisi Matematica, Vol. 1, Cedam--Casa Editrice Dott. Antonio Milani, Padua (1963). 20. G. ROECHINI, Corros. Sci. 36, 113 (1994). 21. G. ROCCHtNI, ENEL-DSR/CRTN, Internal report No. G6-1 (December 1979).