On the calculation of corrosion rates of inhomogeneous metal surfaces

Corrosion Science, Vol. 25, No. 2, pp. 99-105, 1985 Printed in Great Britain

ON THE CALCULATION INHOMOGENEOUS V. Yu.

FILINOVSKY,

0010-938)(/85 $3.(~) + 0.0(} © 1985 Pergamon Press Ltd.

OF CORROSION RATES METAL SURFACES

OF

V. I. DMITRIEVand I. V. STRIZHEVSKY

K. D. Pamfilov A c a d e m y of Municipal Economy, Wolokolamski Road 116, Moscow 123373, USSR

A b s t r a c t - - T h e theory of the corrosion processes on i n h o m o g e n e o u s and unequi-accessible metal surfaces is given. Calculation of the corrosion rates in the cases under consideration consists of the simultaneous solution of the Laplace equation and the convective diffusion one. The general solution of the problem is given. The simplest limiting cases are also investigated.

IN THEmajority of cases the surfaces of metallic structures corrode inhomogeneously. Among the reasons for inhomogeneous propagation of corrosion damages one can place physico-chemical particularities of metallic constructions, inhomogeneities or disruptions of protection films, external heat and momentum fluxes and electric or other fields acting on the corroding surfaces. In contrast with local corrosion damages (cracks, pits, etc.) the above-mentioned sources of inhomogeneous distribution of the corrosion process have a macroscopic nature. It is important to take into account the influence of the factors mentioned in the determination of the rates of propagation of corrosion damages as well as in selection of efficient methods of protection against such damage. The existing methods of calculation of the metal corrosion rates 1 are generally based on the use of the metal polarization curves depending on local polarization of the surface. Being obtained by means of special measurements performed in isolated electrochemical cells, such curves describe only very approximately the real pattern of conversions accompanying corrosion processes under practical conditions. Apart from polarization, the corrosion interaction kinetics depends on many other parameters, the evolution of which obeys its own regularities. The problem of estimation of the corrosion rate appears to be closely linked with the problem of heat and mass transfer calculations. In the most general case the rate of corrosion of a metallic surface can be found only by means of solution of coupled problems. It should be noted that the majority of corrosion calculations carried out to the present time have been done in terms of direct problems of mathematical physics. Formulation of such problems enable, using a given distribution of the electrochemical activity over the metal surface, a calculation of the electric state of the medium near the corroding surface, or estimation of the corrosion rates at different points on the surface. For the purposes of substantiating anti-corrosion measures it is much more important to deal with the inverse problem, namely, using data of electric measurements near the corroding surface to evaluate the character of propagation of Manuscript received 16 August 1984. 99

100

V. Yu. FILINOVSKY, V. I. DMITRIEV and I. V. STRIZHEVSKY

corrosion processes along the surface. The information obtained in this case would help in working out efficient methods of anti-corrosion protection. However, the use of model solutions significantly diminishes the value of recommendations following from the theory. Thus, the analysis of the general case of inhomogeneous corrosion of a metallic surface, S, is of primary interest. The factors causing the inhomogeneous distribution of the corrosion process over S can be divided into three groups: (1) the inhomogeneous distribution of the elecrochemical activity over the metal surface; (2) the unequi-accessibility of the surface (or its separate parts) with respect to oxidizer supply (or to corrosion product removal); (3) external electric, thermal, magnetic or other fields causing changes in the corrosion activity of different parts of the surface. In order to take into account the influence of the factors mentioned on the corrosion process, we shall represent the latter as a set of two simultaneously occurring partial reactions. For simplicity we shall suppose that the corrosion process consists of the anodic reaction of metal dissolution and the cathodic reaction of oxidizer (oxygen) reduction. The kinetics of the first reaction can be described by the slow discharge equation ia = i ° exp [fla(E - Ee)],

(1)

whereas the cathodic process kinetics should obey the equation

ic =-i°(Cs/C,)

exp [ - a c ( E - Ee)].

(2)

The surface potential, E, is the potential drop at the metal-medium interface and is equal to the difference between the metal potential, V, and the electric potential of the medium in the vicinity of the corroding surface, ~s(E = V - ~s); the values E~ and E~ are the equilibrium potentials of the corresponding electrochemical processes in a given corroding medium; t•oa and t.oc are the exchange currents of the processes indicated (supposing the values E e and E~ are very different from one another, the process of cathodic reduction of metal and the anodic process of oxidizer discharge will be neglected); the quantities a c and/3 a are the electron-transfer coefficients for the corresponding processes. Equation (2) allows for the fact that due to diffusion limitations, the oxidizer concentration at the corroding surface, Cs, may differ from the bulk value, C.. To find the corrosion process distribution over the surface, S, one needs to determine the function

i(M)

= ia(M) + ic(M).

(3)

Here the argument, M, indicates the local current values at different points on the surface S. It is seen from equations (1) and (2) that the values ia(M) and it(M) are determined by the distributions qb and C~. The potential distribution qb is described by the Laplace equation A~ = O.

(4)

For characterizing the ohmic current which appears in the medium due to electrochemical reactions on the electrodes we shall introduce the parameter

Corrosion rates of inhomogencous metal surfaces

1(}1

/ohm = K R T / F r o .

(5)

where ro is the characteristic linear dimension of the corroding fragment of the surface and K is the electric conductivity of the medium. In the general case an equation analogous to (4) must be written for the electric potential distribution, V ( P ) , inside the metallic construction. In the case of bulky structures their ohmic resistance affects corrosion process distributions rather remarkably. Analogously to equation (5) we shall characterize the ohmic conductance of a corroding material by the parameter i~,~m = K(m)RT/Fro . In further considerations we shall suppose that KIra) >> K, and therefore shall neglect the ohmic resistance contribution of metal. It must be kept in mind that equation (4), and particularly the boundary conditions to this equation, may take into account inhomogeneous electric properties of the aggressive medium (e.g. the contact of media with different electric conductivity) as well as the presence of external electric field sources. The rate of transportation of the oxidizer depends upon the hydrodynamic velocity field, v, and on the molecular diffusivity, D. If the oxidizer local concentration, C, is not too high it can be found from the solution of the convective diffusion equation (vV)C = D A C .

(6)

The mass transfer rate can be characterized by the mean value of the limiting diffusion current 7~d = 7~d(V, D, C,,

ro).

(7)

The particular form of the dependence Of Tcd on the parameters involved in equation (7) is determined by the convective mixing conditions. The coupling between the electric and diffusion parts of the problem is accomplished through the boundary conditions at the corroding fragment surface. According to assumptions made, the electric current, i, arising as a result of corrosion of the metallic surface, S, is the sum of two currents caused by two simultaneously occurring reactions i = ia + ic = i ° exp [/3a(E - E~)] - i ° ( C f l C , ) exp [ - a c ( E - Ee)].

(8)

The corroding fragment of the metal surface appears to be a source of conductance current in the environment, the intensity of the source being described by the equation i = -K(0~/0n)s at S,

(9)

where n is the unit normal vector to S. As is shown by Newman (1973) 2 the latter equation is fulfilled satisfactorily enough for media with an indifferent electrolyte excess. The condition of the material balance of oxidizer on the surface S leads to the equation i c = - n c F D ( O C / O n ) s at S.

(10)

102

V. Y u . FILINOVSKY, V. I. DMITRIEV a n d I. V. STRIZHEVSKY

Equations (4) and (6), together with the boundary conditions (8)-(10), formulate the general problem of calculation of the corrosion process rate along the surface S. Under non-isothermal conditions equation (6) should be substituted by the convective heat transfer equation, and in the polarization characteristic (8), the temperature dependence should be written in an explicit form. Now let us proceed to analysing a general solution of the problem formulated. For any point, P, in an external aggressive medium at some electric current distribution, i (Q), at the metal surface the potential, ~ ( P ) , can be represented in the form

~(P) = ~J J G(P, Q)i(Q)dSo.

(11)

s

G(P,Q)

Here is the Green function of equation (4). Integration of (11) should be performed at all the points Q at S. From (11) one can find the potential value at an arbitrary point M at the surface S itself. In an analogous manner, the solution of the diffusion part of the problem at an arbitrary cathodic current distribution over S can be written as

C(P) =

it(Q) F (P, Q)ic(Q)dSo/ncF,

~

(12)

21 s

where F (P, Q) is the Green function of equation (6). The solution (12) is valid for any point of the external medium including the metal surface. Insertion of (11) and (12) into (1) and (2) results in ia(M)

=

"°exp{fla[V-E~a-~ G(M,Q)i(Q)dSo]},

ta

(13a)

s

Expressions (13a) and (13b) give a general solution of the problem of corrosion on an unequi-accessible metallic surface when combined with condition (3). The equations above should be supplemented by the integral condition

~ i(Q)dSo =

0,

(14)

s

which states the conservation of the total electric charge in the system medium/ corroding metal surface and represents the necessary and sufficient condition for uniqueness of the problem solution. It should be noted that the accepted kinetics of the anodic (1) and the cathodic (2) reactions can change depending on the character of conversions accompanying the corrosion process. The approach suggested above is valid at any functional form of kinetic equations. The general method of corrosion calculations remains principally the same in the

Corrosion rates of i n h o m o g e n e o u s metal surfaces

103

case when the electrochemical activity of the metal changes along the surface S according to the same law. Let us suppose that the fragments with different electrochemical activity have macroscopic dimensions. Furthermore, let us suppose that for any point M at the metal surface there exists its own polarization characteristic

i(M) = i°(M) exp {fla(M)[E(M) - Ee(M)]} -i°(M)[Cs(M)/C,]exp {-eec(M)[E(M) - E~c(M)]}.

(15)

It should be emphasized that according to the approach suggested above, the difference in behaviour of different fragments of the surface is reduced to a difference in the electrochemical parameters of the partial reactions. In the case of inhomogeneous electrochemical activity of the surface, the quantities i °, i °,/3a, ac, Ea~ and Ec~ in equations (13a)-(13b) become variable; they change with changing the position of the point M on the surface. A bulk of computational work is encountered when one attempts to solve the problem for particular cases. 3'4 However, some limiting cases exist for which the problem can be analysed by comparatively simple ways. The formulae above contain four parameters accounting for electrochemical activity, electric and transport properties of the aggressive medium. These • and 1"cd" Firstly we shall consider well-aerated and highly parameters are t•oa, l.oc, /ohm conductive media, for which the relations --: l~d, /ohm ~ l.oa and l-oc are fulfilled. In the absence of diffusion limitations (Cs ~ C,) and at negligible ohmic losses in the medium (~s ~- 0), equations (13) and (15) are significantly simplified and lead to the following equation for the corrosion process distribution:

i(M) = i°(M) exp { f i a ( m ) [ v -

i°(M) exp {-eec(M)[V- Ee(M)]}. (16)

Eae(M)]} -

When considering inhomogeneously corroding surfaces, one must keep in mind that equation (16), which is widely used in the literature, is justified only for aerated and highly conductive media. The stationary potential of the metal structure, Vst, is determined by equation (14). For the case under consideration this equation has the form

~

i°(M) exp {/3a(M)[Vst- Ea~(M)]} dSM =

~I°(M) exp {-ac(M)[Vs,- Ee(M)]} dSM.

(17)

Now we assume that the coefficients a¢ and/3a do not vary along the surface S. Such an assumption is quite justified if the cathodic and the anodic half-reactions occur at different places according to similar mechanisms. For this case we obtain

i°(M) exp [acE~(M)] dSM

V~, = In S

i°(M) exp [-/3aEe(M)] dSM S

(O¢c "1- ~3a).

The value Vst, unlike Vst, corresponds to surfaces with constant cec and/3 a.

(18)

104

V. Yu. FIIJNOVSKY,V. I. DMITRIEVand I. V. STRIZHEVSKY

The polarization characteristic of an inhomogeneous surface can be written as

i°(M._____~) ex__pp[-fl, So i°(M) exp [-/3,E~(M)]dSM exp

i(M) = 7st1( _

i°(M) ~i°(M)

exp

[acE~(M)]So

(/3,,/)

exp ( - a c , / ) ~ "

(19)

J

exp [acEe(M)] dSM

S

The value 7/= V - ~'~t denotes the surface overvoltage under the action of the external current, So being the total area of the surface S; i~t stands for the average corrosion current at the stationary potential. In contrast with the case of homogeneous metals, 5 inhomogeneous activity of metals results in additional weight factors in front of the exponential terms in the polarization characteristic expression. Equation (19) can be used in particular for the calculation of the current of galvanic pairs being formed on a corroding surface in the case of contact of two different metals. The second limiting case is encountered in systems where diffusion transport of the oxidizer is the rate-controlling stage of the cathodic process. For this case the "7" -o >> t -o relations /ohm, lc a and /ca are fulfilled, and the polarization characteristic is described by the equation

i(M) = i°(M)

exp {/3,(M)[V- E~(M)]} - icd(M).

(20)

In deriving (20), we neglected the effect of ohmic losses in the medium and took into account unequi-accessibility of the surface S for the diffusing oxidizer. The distribution icd(M) can be obtained from the solution of the convective diffusion problem. The relationship (20) represents a generalization of the differential aeration pairs equation for the case of the corrosion of unequi-accessible surfaces. The stationary potential Vst is calculated by means of the procedure described above. The result is V,t = In

i°(M) exp

icd(M) dS S

[-/3,El(M)] dS M

13,.

(21)

S

The polarization characteristic takes the form

i(M)

=

{~ i°(M) exp [-/3aEea(M)lSo exp T~t i°(M) exp [-/3aEe(M)] dS M

(/3a*/)

S

--

icd(M)S° ff~ icd(M) dSM

~"

]

(22)

In deriving (21) and (22) we used the condition of constant/3, over S. It is easy to trace from (21) a direct dependence of Vst on the distribution of the limiting diffusion cathodic current over S. In particular, it follows from (21) that the

Corrosion rates of inhomogeneous metal surfaces

105

potential V~I depends on the intensity of motion of the medium near the corroding metal, in accordance with the available experimental data. 5 Equations (19) and (22) are transformed to the well known polarization characteristics in the limiting cases of homogeneous and equi-accessible corroding surfaces. The behaviour of metals with respect to corrosion in media with high enough ohmic losses cannot be described in terms of the stationary potential or deviations of this value, Electric current and potential distributions can be represented as primary, secondary or tertiary 2 depending on the relationship between the parameters mentioned above. Even in the absence of an external polarizing current the polarization of a corroding unequi-accessible metallic surface appears to be different at different points. 6 Evidently, the approach suggested above and the general solution which has been obtained in this paper can also be used for solving the inverse problem of the anti-corrosion protection theory. REFERENCES 1. V. N. OSTAPENKOet aL, Methods of Calculation o f Electric Fields for Electrochemical Anti-Corrosion Protection o f Metallic Constructions. Naukova Dumka, Kiev (1980). 2. J. NEWMAN,Electrochemical Systems. Englewood Cliffs, New Jersey (1973). 3. N. VAHDATand J. NEWMAN,J. electrochem. Soc. 120, 1682 (1973). 4. R. ALKIREand G. NICOLAIDES,J. electrochem. Soc. 121,183 (1974). 5. I. L. ROSENFELD, Corrosion and Metal Protection. Metallurgia, Moscow (1970). 6. V. Ju. FILINOVSKY,[. V. STRIZHEVSKYand I. V. CHEPURINA, Elektrochimia 19, 1592 (1983).