The generalization of the three-point method and its application to the corrosion-rate measurement

Corrosion Science, Vol. 30, No. 1, pp. 9-21, 1990 Printed in Great Britain

0010-938X/90 $3.00 + 0.00 © 1989 Pergamon Press plc

THE GENERALIZATION OF THE THREE-POINT METHOD AND ITS APPLICATION TO THE CORROSION-RATE MEASUREMENT GABRIELE ROCCHINI E N E L - - I t a l i a n Electricity Board, T h e r m a l and Nuclear Research Center, Via Rubattino 54, Milano 20134, Italy

A b s t r a c t - - T h e utility of an approximate analysis of experimental results is briefly discussed and the two formulations of the three-point m e t h o d by Barnartt and Le Roy are introduced. T h e mathematical developments that justify these techniques show that they are a particular case of the more general four-point m e t h o d , obtained by a suitable choice of the points. T h e two criteria that govern this choice were introduced to simplify all the mathematical operations. In this paper, however, the electrochemical parameters were determined by seeking the root of a characteristic function on which the generalization of the two m e t h o d s to a greater n u m b e r of points is based. Although the application relating to the behaviour of A R M C O iron in H 2 S O 4 solutions at various p H values and at 25°C yielded very satisfactory results and presented no anomalies, the two m e t h o d s proved to be rather critical for the A R M C O iron + 1 N HCI or 1 N H2SO 4 + commercial inhibitors systems at 75°C. In this case the values of the electrochemical parameters are d e p e n d e n t on the choice of the three points and in some situations there is no solution to the problem, because the values of the quantities a and fl which were considered without physical m e a n i n g were rejected even if they allowed the corrosion current density to be estimated.

INTRODUCTION

COMPARISON BEXWEENthe values of the corrosion current density of a metal obtained from electrochemical and direct measurements often shows that the two determinations are different, though in some cases the discrepancy tends to disappear. This experimental observation, which generally holds if special precautions are taken in the measurements, gives rise to the question whether it is more convenient to analyse experimental polarization curves using sophisticated mathematical techniques or to limit one's efforts to an approximate calculation of the corrosion current density. In this connection, mention can be made of the characterization of the inhibitor effectiveness of some commercial products for a classification of their merits and of the on-line monitoring of the corrosion rate, if uniform. In both cases the aim is to get a quantitative idea of the corrosion process without going into the mechanisms that govern it. This approach simplifies work considerably because it is sufficient to standardize the electrochemical technique and choose a mathematical method of processing experimental results for the evaluation of the parameters that meet our requirements. Obviously, in this case it is advisable to use simpler methods than those based on the non-linear best-fitting of experimental polarization curves, whose application requires the development of complex calculation codes, also because the information obtained may sometimes be the same.

Manuscript received 10 October 1988; in a m e n d e d form 1 D e c e m b e r 1988. 9

10

G. ROCCHINI

From this point of view, the three-point method, in the two versions by Barnartt a and by Le Roy 2 is ideal because it is based on the non-linearity of the response of the electrochemical system and, in principle, should permit the evaluation of the Tafel slopes and the corrosion current density. Even if the idea was not developed in the original papers, the three-point method affords also the value of the polarization resistance. The main objective of this study is to review the two mathematical techniques in order to introduce a generalization which will make it possible to apply them to wider classes of electrochemical systems without neglecting the physical significance of the parameters on which they work. Of course, this implies a thorough revision of the original notions preserving the two selection criteria but using entirely different equations.

THE UTILITY OF AN APPROXIMATE ANALYSIS The problem of the utility of an approximate evaluation of the anodic and cathodic Tafel slopes can readily be solved by the theory of error propagation. If we consider a function f depending on the independent variable AE and the two parameters a and fl affected by the indeterminations Aa and Aft, the maximum relative error on the quantity f, for an arbitrary value of AE, and a = a, fl =/3 is given by: 3

This relation clearly expresses the concept that the indetermination, by which the quantity f i s affected, remains practically unaltered if instead of considering the true values ~ and fl we use the parameters a' E [ ~ -

Aa, a + Act] and

fl' E [ f l - A f t , ~ + Aft].

It may be useful to remember that the theoretical variation of the ratio IAf/fl as a function of a and fl can be investigated using either the expressions o f f that are given below or an equivalent formulation. In this way the concept of the utility of an approximate evaluation of the quantity considered can be defined more precisely, as explained in the introduction. Obviously, all the values of a and fl that give an unacceptable value of the relative error must be rejected. In our case, the q u a n t i t y f should be identified with the corrosion current density, Ic, and its analytical expression is dependent on the schematization adopted for the calculation. If an electrochemical system is investigated in the linear zone around its mixed potential, recourse can be had to Stern and Geary's well-known formula 4

Ic = 1/[gp(a + fl)l, where

(2)

Rpis the polarization resistance and can be determined using the relation

In this relation ij denotes the experimental current density value, and the potential difference, AEi , referred to the mixed potential, must generally satisfy the condition

]aej[ ~ lOmV.

Generalization of the three-point method

11

Another expression of Ic, which was found by the a u t h o r f is given by I~ =

If 3

L jx~

i ( x ) x dx

1/

[(x~ - x~)(a +/3)]

(3')

and constitutes a generalization of equation (3), based on a different definition of the linear response. Of course, with (3') use must be made of the best-fitting of the experimental polarization curves. It should also be noted that besides (3') there exist other formulas which are a little more complicated but are not subject to limitations of any kind. THEORY General considerations The three-point method was the first attempt to analyse an experimental polarization curve without going into Tafel regions so that it promoted the development of all the following numerical techniques intended to improve data processing. Furthermore this method had the merit of starting a reflection on the experimental procedures employed to perform corrosion rate measurements, because it showed that the values of the electrochemical parameters can be obtained considering the behaviour of the given system over a small overpotential interval around the point AE = 0. We decided to look over these ideas for improving our software developed for corrosion rate monitoring in real time with computerized systems and for optimizing the experimental techniques based on the use of the direct current. The two versions of the three-point method are based on the same concept of the determination of the three electrochemical parameters, a,/3 and I c, by means of a set of non-linear equations. The only difference lies in the criterion that governs the choice of the three points, AE1, A E 2 and AE3, which results in different expressions of the final equation. Consequently, it will be convenient to deal first with the common theory and then with the two diversifications. Like other approaches, this method is based on the assumption that the currentvoltage characteristic of the electrochemical system is i ( A E ) = lc[exp (etAE) - exp (--flAE)],

(4)

where Ic gives the corrosion current density and AE is the potential difference referred to the corrosion potential E c. This law satisfies the requirement that the system should contain only two processes under the exclusive control of the activation energy. The developments that follow apply only to electrochemical reactions that are described by equation (4). Any inadequacy of the theory is to be attributed to a behaviour that is different from the ideal one. If the shape of the response is too different from equation (4) the values obtained for a,/3 and Ic are physically unacceptable. Factorization of the quantity Ic in (4) allows the problem to be reduced to the solution of a set of two non-linear equations in the two unknowns ct and/3. If we consider the three values, il, i2 and i3, of the current density relating to the potential difference AE~, A E 2 and AE3, we can write the set of equations [exp (aAE2) - exp (--/3AE2)]/[exp (aAE1) -- exp (--flAE1)] = i2/il [exp (aAE3) - exp (-flAE3)]/[exp (aAE1) -- exp (--flAE1)] = i3/i~, which represents a particular case of the four-point method 6 when AE1 = AE4.

(5)

12

G. ROCCHINI

It was Barnartt who first inferred from the analysis of the response, equation (4), that this set of equations can be reduced to a very simple expression by a suitable choice of the values of AE1, AE 2 and AE 3. Barnartt' s version The present exposition does not follow the original one, the form having been adapted to the generalization mentioned above. For further details concerning the origin of the method reference can be made to Barnartt's publication. ~Moreover for simplicity the first point AE~ is chosen so that its value is always greater than zero. However, this choice is uninfluential in the mathematical developments. If the three points are such that AE 2 = 2AE 1 and AE 3 = - 2 A E 2 and we set AE 1 = x, equation (5) will become

exp ( a x ) + exp ( - f i x ) = i2/il exp ( - 2 a x ) exp (2flx)[exp ( a x ) + exp ( - f i x ) ] = - i 3 / i l ,

(6)

where according to our choice and equation (4), it follows that i 1 > 0, i 2 > 0 and i 3 < 0. On the basis of the first equation, the second can be written exp (2ax) exp ( - 2 f i x ) -- -i2]i 3

(7)

whence, on eliminating the term exp ( - 2 f i x ) , we obtain the final equation exp ( 2 a x ) [i2[i 1 - exp (ax)]2 = _i2/i3.

(8)

To revert to Barnartt's initial form it is sufficient to set y = exp ( a x ) and extract the square root of the left-hand and right-hand sides of equation (8). Thus we obtain the more familiar equation: y2 _ (i2/il)y + (_i2/i3)1/2 ---- 0.

(9)

Between equations (8) and (9) there is, however, a substantial difference, due to the extraction of the square root, which may result in the presence of complex conjugate roots of equation (9) in practical applications. This is not the case with the formulation (8) because its right-hand member is always greater than zero and we can always find a real value of a that will satisfy it. On the other hand, the value of a may be negative or far too different from the value obtained by other means and consequently not physically acceptable. In any case equation (8) is always an improvement upon the set of equations (6). L e R o y ' s version This second formulation was suggested in 19772 by Le Roy, who described it using a relation proposed by Reeve and Bech-Nielsen. 7 As in the previous case, we shall not follow the original development and shall not, in particular, refer to the formula given in reference, 7 which is a direct consequence of the law (4). In this case the points selected must satisfy the conditions AE2 = 2AE~ and AE 3 = --AE1. If we set AE~ = x, we shall obtain the non-linear set of equations:

exp (o.x) + exp ( - f i x ) = i2/ix exp (-~zx) exp (fix) = - i 3 / i 1.

(10)

Thus the two sets of equations (6) and (10) differ in the form of the second equation, which reflects the different criteria used for the choice of the three points.

Generalization of the three-point method

13

Following the same procedure as in the previous case, we find that the second equation of (10) may be written exp ( a x ) [i2/i 1 - exp ( a x ) ] = - il/i3

(11)

which, on substituting y = exp ( a x ) , becomes y2 _ (i2/il)y - il/i3 = 0.

(12)

This equation is slightly different from the original one ,2 which even if it has the same form refers to the quantity fl, owing to the development used for its derivation. Practical applications of equation (12) may result in complex conjugate roots and negative values. The latter circumstance may occur whenever the quantity - i l / i 3 is less than one. This remark is always valid, even if it is not present in the original paper, because the equality fl = a + [In ( - i 3 / i l ) ] / x holds. Furthermore the two approaches give the same results when they work properly. Henceforward instead of equation (11) the following expression will be used: exp ( 2 a x ) [ i 2 / i l - exp ( a x ) ] 2 = (il/i3) 2

(13)

which has the same characteristics as equation (8). This equation shows that there always exists a real value of a that satisfies it; therefore, its use ensures that no complex conjugate roots will exist. The inadequancy of the two versions to describe the physical reality of the system under examination depends on the presence of negative values or on solutions that are positive but meaningless because they cannot be justified by any hypothesis about the reaction mechanism. D i f f u s i v e c o n t r o l in the limiting c u r r e n t z o n e

The above forms are notably simplified when one of the two electrochemical reactions is under diffusive control in the limiting current zone. If diffusive control is exerted on the cathodic process, fl = 0, (7) and (11) become, respectively exp ( 2 a x ) = -i2/i3

(7')

exp ( a x ) = - i l / i 3 .

(11')

An example illustrating this situation and referring to the study of the corrosion of zinc in a Na2SO4 3 wt% solution at 30°C can be found in Ref. 2. EXTENSION OF THE METHOD As shown in the previous section, the search for a solution to equation (5) may yield values of a and fl that are not physically acceptable. A negative value of fl may be obtained which is in contrast with the experimental observation that the cathodic process attenuates as AE increases. This is exclusively due to the condition which requires that the values of the three parameters, a, fl and Ic, should be determined using only three points. An obvious generalization can be achieved by modifying equations (8) and (13) slightly and comparing the values obtained with those derived by the NOLI method.S In the case of equation (8), the distance G to be minimized is given by: G = ~

j=l

{]iEjli3j I1/2

_

_

exp (axj)[i2j/ilj - exp (axj)]} 2,

(14)

14

G. ROCCHINI

while the function G~ relating to equation (13) is:

G1 = ~ (lilj/i3j] - exp (axj)[i2j/ilj - exp (axj)]) 2.

(15)

j=l

In both formulas the values ilj refer to xj. The significance of equations (14) and (15) is obvious. Contrary to the NOLI method, if we choose suitable groups of three values, we can reduce the problem to the analysis of a function of only one variable, which simplifies the numerical elaboration considerably. This formula was chosen also for its convenience, because the solving equation contains fewer zeros than those obtained using equations (8) and (13) specifically_ Once the value a of a has been determined searching by means of appropriate methods, which are normally found on numerical analysis textbooks, the minimum of the function defined by the equations (14) or (15), fl can be calculated in two different ways. In one case, we solve the first equations of (6) and (10) relating to a given number of AE~ points and take the mean value of the results obtained. Thus we have f l = [i=~1 (In [i2//ilj-ex p (-~xj)])/xj]/n.

(16)

In the other case, we consider the distance Fdefined by

F = ~ [i2j/i,i - exp (axj) - exp (-flx)] 2.

(17)

j=l

In this way the determination offl is given by the search of the minimum of equation (17). This formulation is more satisfactory from a physical point of view because the analysis is extended to a wider region of the experimental curve and the indications it yields are in harmony with the overall behaviour of the phenomenon. EXPERIMENTAL RESULTS Equations (8) and (13), together with the definition, equation (14), were applied in order to investigate the behaviour of A R M C O iron in H 2 S O 4 solutions at various normalities and at 25°C, in 1 N H 2 S O 4 + commercial inhibitors ( M A G N A 385, R O D I N E 92A), and in 1 N HC1 + B O R G P16 and R O D I N E 213 at 75°C. All the details concerning correct application of the original three-point method can be found in Ref. 9 where equations (9) and (12) are used for some of the electrochemical systems under consideration. The experimental conditions adopted for this investigation are described in Ref. 10. It may be useful to remember that all the polarization curves were obtained using the galvanostatic pulse technique and that the electrochemical parameters for the H2SO 4 solution at pH 0.3 refer to a polarization curve mentioned in Ref. 10. In the present case equations (8) and (13) were solved numerically by the method of false position. 3 As the mathematical developments are the same in both cases, all the considerations that follow concern the f u n c t i o n f ( a ) = i2/i3 + exp (2ctx)[(i2/il exp (ctx))] 2 whose qualitative shape is illustrated in Fig. 1. Examination of Fig. 1 shows that the functionf(a) generally presents three roots, two of which, a' and a", should be discarded as physically unacceptable because a' <

Generalization of the three-point method

15

P

,

I

~ _

o,

,, °

?

,,

M /2 /3

FIG. 1. Shape of the characteristic function for Barnartt's formulation. 0 and a" > 1. It may h a p p e n , however, that the relative m a x i m u m of f ( ~ ) , c o r r e s p o n d i n g to the point P, is less than zero. In this case the function f(a) has a single r o o t which is greater than one and consequently does not provide a solution to o u r problem. This situation c o r r e s p o n d s to the presence of complex conjugate roots in B a r n a r t t ' s original formulation. All this shows that a slight change in the formulation of the p r o b l e m is not sufficient to ensure a satisfactory solution, which is strictly c o n n e c t e d with the type of experimental information available. Thus, while it is comparatively easy to r e m o v e the c o m p l e x conjugate roots, it is difficult to i m p r o v e the significance of the m e t h o d . Figure 1 also shows that on passing from equation (8) to (9) the simple calculation of the square root m a y alter the nature of the problem. T h e values of cq and a2, c o r r e s p o n d i n g to the points P and M, can be f o u n d by equating the first d e r i v a t i v e f ' ( a ) to zero and are given by a, = [In (i2/i , - 1)]/x a2 = [In (i2/il)]/x.

(18)

T h e y can serve as starting points of the iterative process i f f ( a l ) > 0 and f ( a 2 ) < 0. Table 1 lists some results o b t a i n e d by B a r n a r t t ' s m e t h o d for the H2SO4 solutions TABLE 1. ARMCO

APPLICATION OF BARNARTT'S METHOD TO THE SYSTEM IRON d- H 2 S O 4 AT VARIOUS

pH VALUES Bc (mY)

AND AT 2 5 ° C

pH

x (mY)

B~ (mY)

lc (~A cm-2)

0.03

4 8 12

87 87 86

90 9(1 90

1254 1246 1244

0.3

4 8 12

77 78 80

108 109 110

605 615 626

1

4 8 12

72 73 74

86 86 87

263 266 269

1.5

4 8 12

66 67 67

89 90 91

272 274 277

16

G. ROCCHINI

at various pH values. The values of the anodic and cathodic slopes, B a and Bc, refer to decimal logarithms. The values of the same quantities obtained by Le Roy's method for the same electrochemical systems are given in Table 2. Table 3 shows the electrochemical parameters Ba, Bc and Ic, for the same systems, derived from the analysis of the experimental polarization curves by the NOLI method and the value of the corrosion current density, 11, calculated by Barnartt's method associated with interpolation with the INTER1 program, n The values of a and fl used for this calculation refer to x = 12 mV. Table 4 compares various determinations of the corrosion current density, Ic, referring respectively to linear polarization in the POLAN program version, m to equations (2) and (3') calculated with the values of a and fl employed in the previous cases. The integral that appears in equation (3') was calculated over the interval [0, 10] mY. The data in Table 5 were obtained through generalization of Barnartt's method represented by equation (14). The x values used for the calculation were the following: x = 4, 6, 8, 10, 12 and 14 mV. Therefore the electrochemical parameters refer to 16 experimental points and the utility of the best-fitting of the experimental polarization curve is evident. The value of Bc was determined by means of equation (16), while Ic was determined by means of equation (4) over the interval [ - 10, 10] mV.

Some results concerning the electrochemical systems at 75°C are listed in Table 6. They all refer to Barnartt's formulation. Le Roy's formulation does not add anything to these data and has therefore been omitted. Table 7 compares the electrochemical parameters calculated with Barnartt's and Le Roy's formulations using the point AE1 = 12 mV, and by the NOLI method in the case of the 1 N HzSO4 solution inhibited with RODINE 92A and MAGNA 385. The same comparison for the 1 N HCI solution inhibited with RODINE 213 and BORG P16 can be found in Table 8.

TABLE 2.

APPLICATION OF LE ROY'S METHOD TO THE

ELECTROCHEMICAL SYSTEM OF TABLE 1

pH

x (mY)

B. (mY)

B c (mY)

~ ( p A cm -2)

0.03

4 8 12

87 85 84

89 88 87

1241 1223 1210

0.3

4 8 12

76 80 82

110 114 119

614 635 657

1

4 8 12

72 73 74

86 87 89

264 267 271

1.5

4 8 12

66 67 67

89 90 91

272 274 277

Generalization of the three-point m e t h o d TABLE 3,

17

ELECTROCHEMICAL PARAMETERS OBTAINED BY

NOLI

ANALYSIS OF THE POLARIZATION CURVES OF THE PREVIOUS SYSTEM

pH

Ba (mV)

B a (mV)

~ ( ~ A cm -2)

I 1 ( ~ A cm -2)

0.03 0.3 1 1.5

93 76 68 64

103 101 89 100

1382 602 252 279

1336 621 266 274

TABLE 4.

COMPARISON AMONG DIFFERENT DETERMINATIONS OF THE CORROSION CURRENT DENSITY

pH

POLAN

Ic (/~A cm 2) Equation (2)

Equation (3')

0.03 0.3 1 1.5

1384 583 262 278

1251 623 272 276

1248 646 274 286

TABLE 5.

ELECTROCHEMICAL PARAMETERS OBTAINED BY THE

GENERALIZED THREE-POINT METHOD BASED ON EQUATION

pH

B a (mV)

Bc (mV)

Ic (/~A cm -2)

0.03 0.3

86 80 74 68

89 114 88 92

1236 636 271 278

1

1.5

TABLE 6.

(14)

ELECTROCHEMICAL PARAMETERS OBTAINED WITH BARNARTT'S THREE-POINT METHOD FOR THE

ARMCO

IRON IN TWO INHIBITED ACID SOLUTIONS

Acid

Inhibitor

x (mY)

B a (mY)

1 N H2SO 4 +

M A G N A 385 0.5 c c l i

4 8 12 4 8 12

237 323 733 118 140 184

185 219 302 161 183 211

140 176 289 69 80 99

4 8 12 4 8 12

142 160 184 210 262 426

309 363 419 2116 ---

93 106 122 83 ---

R O D I N E 92A 0.5 cc 1-1 1 N HCI +

B O R G P16 0.1cc 1 1 R O D I N E 213 1 cc 1-1

Be (mY)

I c ( p A cm -2)

G . ROCCHINI

18

TABLE 7.

COMPARISON AMONG ELECTROCHEMICAL PARAMETERS OBTAINED WITH N O L I , BARNARTT AND LE ROY'S METHODS. A E 1 = 12 m V

1 N H2SO 4 +

B~ ( m V )

inhibitor cc 1-I

NOLI

RODINE 92A 0.1 RODINE 92A 0.5 M A G N A 385 0.5 M A G N A 385

Barnartt

Bc ( m V )

I~ (/~A c m -2)

LeRoy

NOLI

Barnartt

LeRoy

NOLI

Barnartt

LeRoy

121

117

115

113

105

102

697

657

643

120

183

--

108

210

--

56

101

--

379

733

--

190

302

--

174

292

--

152

1313

--

146

908

--

43

317

--

127

143

146

110

125

128

49

56

1 MAGNA 3

385

57

Application of equation (14) has not proved very satisfactory because the values of a and fl generally depend on the n u m b e r of experimental points used and may differ considerably from those obtained by the N O L I method. For example, for the 1 N HCI + 0.1 cc 1-1 B O R G P16 system the results were: Ba = 189 m V , Bc = 540 m V and Ic = 134/~A cm -2, whereas the p a r a m e t e r s obtained by the N O L I method were: B a = 135 mV, B e = 199 m V and Ic = 73ktA cm -2. The results obtained for the 1 N H2SO 4 solution inhibited with 0.5 cc 1 - 1 R O D I N E 92A are analogous, viz.: B a = 196 mV, B c = 281 m V and I c = l l 6 p A cm -2. In this case the N O L I data are: B a = 120 mV, Bc = 108 m V and Ic = 56/~A cm -2. DISCUSSION

Examination of the results concerning the HESO 4 system at various p H values, and particularly of those contained in Table 4, confirms the original assumptions as to the utility of an approximate analysis of the shape of an experimental polarization TABLE 8.

COMPARISON AMONG ELECTROCHEMICAL PARAMETERS OBTAINED WITH N O L I , BARNARTT AND LE ROY'S METHODS.

AE~ =

12

mV

1 N HCI + inhibitor cc 1-1 BORG P16 0.1 BORG P16 0.5 R O D I N E 213 0.1 R O D I N E 213 0.5 R O D I N E 213 1

B, (mV)

Bc ( m V )

lc ( p A c m =)

NOLI

Barnartt

LeRoy

NOLI

Barnartt

LeRoy

135

184

--

199

419

--

149

193

231

231

281

401

205

292

--

357

922

166

162

129

302

141

426

--

238

NOLI

Barnartt

LeRoy

73

124

--

55

69

--

225

381

--

209

143

194

161

120

--

--

38

--

--

88

Generalization of the three-point method

19

curve. From this standpoint the few experimental data examined have shown that in an acid environment and at low temperatures Barnartt's method can provide indications that are in harmony with those obtained by the N O L I method, which is much more laborious and time-consuming. In most cases there is no difference between the indications of equations (9) and (12) and those of equations (8) and (13). If we use the latter, however, we can base the formulation of the problem on the distance G and G1, defined by equations (14) and (15) respectively. In principle, the use of G and Gl should ensure a mean of the a and fl values that refers to a greater number of experimental points and consequently may prove more satisfactory. All this is shown by the analysis of equations (14) and (15). For example, the minimum value of the distance G is found when the contributions of the individual terms that constitute it is the smallest possible. Its failure can therefore be explained by the fact that under certain conditions the minimum value of the distance may coincide with a value of a that differs considerably from the values relating to the individual terms. When applicable, the generalization of the three-point method affords indications that are quite reliable in that they are more conditioned by the shape of the curve. Moreover they start to be of a less stochastic nature. The fact that the application of the generalization to the electrochemical systems at 75°C has not given satisfactory results is not surprising because examination of Table 6 shows that the parameters B~, Bc and Ic are largely dependent on the choice of the point x. A possible explanation for this tendency is that the experimental polarization curves do not exhibit a well-defined trend within the potential difference interval considered. At this point it may be worth recalling that the NOLI method was applied examining the whole of the polarization curve and that, as reported in Ref. 10, the values of the electrochemical parameters may be dependent on the width of the interval used for the analysis. This is confirmed by Tables 7 and 8 which demonstrate that the agreement between the three numerical techniques may depend on the electrochemical system that is being investigated. This observation shows how important the experimental technique is for the success of the three-point method. This could be inferred also from the initial premises specifying that the electrochemical system must present a well-defined response, so that the problem of the determination of the parameters a, fl and Ic will be greatly simplified. A negative aspect of the three-point method is the very restrictive condition that the three values of the current density should satisfy equations (8) and (13). In many cases this requirement may entail the presence of values that satisfy the problem from a mathematical point of view but are not acceptable for a corrosionist. This applies also to the expressions (14) and (15), which may be regarded as the sum of operations of the same kind. Therefore the introduction, on an experimental level, of restrictive conditions, such as the two criteria of choice for the three points, instead of simplifying the determination of the three electrochemical parameters, c~, fl and I~, may result in the problem being physically unsolvable. This is because these conditions require that the electrochemical system should be described very rigorously by the current-voltage characteristic, equation (4), but experimental data may present some oscillations with respect to this ideal behaviour. However, these considerations are not surprising and do not impair the practical application of the three-point method to the corrosion problems because nearly all the mathematical

20

G. ROCCHINI

methods are susceptible to giving an inaccurate analysis in some cases. Their validity cannot be stated in an absolute manner. It was decided to include Barnartt's version of the three-point method, based on equation (8), in our S O F T C O R - D C - P S 112 and SOFTCOR-DC-113 codes which drive the Solartron electrochemical interface model 1286 and the E G & G computerized potentiostat respectively. These determine the value of the polarization resistance over the AE interval (+20 mV) using the potentiostatic configuration and the polynomial best-fitting of the experimental data for evaluating the first derivative of the function i -- i(AE) at AE = 0. At present they are employed to study the thermal stability of some commercial inhibitors for acid solutions and to carry out corrosion rate monitoring on our pilot loops. Their main advantage compared to commercial software is to give the values of the electrochemical parameters in real time without operator intervention and to manage the data storage on files automatically. Our experience referred to a great number of tests relating to different electrochemical systems is quite positive and the results obtained, when the method works properly, are self consistent. CONCLUSIONS On the basis of our experience in the fields of corrosion rate monitoring, the inhibitors characterization and numerical analysis of experimental polarization curves the application and reliability of the three-point method in its two versions can be summarized. The approximate analysis of the geometric shape of an experimental polarization curve in the vicinity of the corrosion potential is quite successful in determining the values of the electrochemical parameters when the kinetics of the process follows the Tafel law even if the results can depend both on the technique adopted for performing the measurements and the choice of the first value of the overvoltage. In some cases this approach gives practically the same results obtained using more complex methods and is a useful tool to study the basic mechanisms which govern a corrosion process. For practical purposes the two methods are very convenient to perform on-line corrosion rate measurements with computerized systems using the polarization resistance technique and the Stern and Geary formula because with this combination it is possible to have a realistic evaluation of the quantity (a + fl) each time which enters into the computation of the corrosion current density. This aspect of the corrosion monitoring problem is very important because many authors use the values quoted in literature which can differ markedly from the true situation. The use of polynomial best-fitting of the experimental data permits the effect of the ohmic drop on the shape of the polarization curve to be removed and the choice criteria of the three points to be properly applied.

1. 2. 3. 4. 5. 6. 7.

REFERENCES S. BARNARTT,Electrochem. Acta 15, 1313(1970). R. L. LE RoY,J. Electrochem. Soc. 124, 1060 (1977). B. DEMIDOVlTCHand I. MARON,Elements De CalculNumdrique, Editions MIR, Moscow(]979). M. STERNand A. L. GEARY,J. Electrochem. Soc. 104, 56 (1957). G. RoccmNI, Corrosion 44, 158 (1988). G. RoccmNi(work in progress). J. C. REEVEand G. BECH-NIELSEN,Corros. Sci. 12,351 (1973).

Generalization of the three-point method

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8. G. ROCCHINI, ENEL-DSR/CRTN, Paper No. G6-1 (1979). 9. M. CASTELLANk G. PERBON1 and G. ROCCHINI, 94th Riun. Annu. AEI, Paper No. A.28, Cagliari (1983). 10. G. PERBONIand G. ROCCHINI, ENEL DSR/CRTN, Paper No. G6/81/02. 11. G. ROCCHINI, Corrosion 43,326 (1987). 12. G. ROCCHIN1, Corrosion 189, Paper No. 619, New Orleans (1989). 13. G. ROCCH1NI, Corrosion Reviews 7~ 273 (1987).