Corrosion Science, Vol. 30, No, 8/9, pp. 915-928, 1990 0010--938X/90 $3.00 + 0.00 Printed in Great Britain Pergamon Press plc

M A T H E M A T I C A L M O D E L I N G O F M A S S T R A N S P O R T A N D C H E M I C A L R E A C T I O N I N C R E V I C E A N D P I T T I N G

C O R R O S I O N

JOHN C. WALTON

Idaho National Engineering Laboratory, Geosciences Unit, P.O. Box 1625, Idaho Falls, ID 83415, U.S.A.

Abstract--A theoretical model is developed to evaluate transport processes and chemical reaction in crevices and pits. The model is general in form, allowing application to a variety of systems. The model is applied to recent experimental data on crevice corrosion in iron. Two different electrolytes are analyzed: acetate and sulfuric acid. These experimental systems are of interest because active crevice corrosion is obtained in the absence of localized acidification and aggressive anions. Excellent agreement is found between model predictions and experimental results.

I N T R O D U C T I O N

MATHEMATICAL models are one tool which can be utilized to improve the understand- ing of corrosion processes such as crevice and pitting corrosion. Models are useful for making comparisons between theory, which can be included in the governing equations of the model, and actual experimental systems which are more complex. Numerical modeling is especially important in the area of localized corrosion where the governing processes are described by systems of tightly coupled, non-linear differential equations.

In this work a generalized model is described which is capable of modeling mass transport, interphase mass transfer, and chemical reaction in crevices and pits. The model is applied to recent experimental results on the crevice corrosion of iron. A general overview of theoretical modeling of localized corrosion is contained in the review by Sharland. 1

C O N C E P T U A L M O D E L OF L O C A L I Z E D C O R R O S I O N

Mass transport The equation for transport of dissolved electrolytes in dilute solutions, subject to

diffusion and electromigration is: 2

nziDiF C V J i = - n D i V C i ~-~ i ~s (1)

where Ji is the flux of species i in solution (moles dm-2 s- a), n is the liquid porosity in the corrosion cavity, Di is the diffusivity of species i (dm -2 s-a), V is the vector gradient operator, Ciis the concentration of species i (moles dm-3), zi is the charge of species i, F is the Faraday constant, R is the gas law constant, T is the absolute temperature and ~0 s is the potential in solution (volt).

Manuscript received 23 April 1989; in amended form 31 August 1989.

915

916 J.C. WALTON

A porosity term (n) is included in the equations to allow for the situation where the corrosion cavity is partially filled with gas or solid phase corrosion products such as hydrogen gas, metal oxides or salts. The liquid porosity is the proportion of the cavity volume which is filled with the liquid phase. When corrosion products are not present in the cavity, the porosity is equal to one.

C h e m i c a l react ion

Two types of chemical reactions are of interest in crevices and pits: (1) electro- chemical reactions at the metal-solution interface, and (2) reactions in the crevice solution (e.g. hydrolysis). Examples of reactions at the metal-solution interface are the hydrogen evolution reaction and dissolution of the metal:

H+(aq) + e- ¢:> 0.5Hz(g) (2)

Me(s) <=> Me n+ + ne- . (3)

The current density resulting from an electrochemical reaction at the metal- solution interface will generally be expressed as a function of the corrosion potential and the concentrations of some aqueous species (e.g. Butler-Volmer kinetics). The mathematical formulation of the model allows the use of any functional form of the current density expressions, including tabulated coefficients.

ij = f(~0 M - ~ s , C1, C2 . . . . , Cn) (4)

where q5 M is the potential of metal (volt) and ij is the current density from reaction j (A dm-2).

The rate of chemical reaction/mass transfer for each aqueous species at the metal-solution interface (N/) is obtained from the current density of the electro- chemical reactions using Faraday's law:

Jmax - = ~ v~// Ni

j= 1 n.~..F (5)

where vq is the stoichiometric coefficient of species i in reaction j, N~ is the flux of species i at metal-solution interface (moles dm -2 s -1) and nj is the stoichiometric coefficient of electrons in reaction ].

Rates of reaction of species in the crevice/pit solution can be expressed in terms of kinetic expressions (e.g. first order kinetics) or adjusted arbitrarily to keep the rections at equilibrium. In general, acid/base hydrolysis reactions are rapid, justify- ing the assumption of local equilibrium. An example of a reaction in solution is:

Me 2+ + HzO ¢:> Me(OH) + + H +. (6)

If it is to be assumed that a reaction in solution remains at equilibrium, several distinct approaches can be used to formulate the system of equations. The most common method is to write out all the governing equations in rate format 3'a'5 then add and subtract equations to eliminate the arbitrary rate expressions. The new set of governing equations are then supplemented by the equilibrium constants for the reactions remaining at equilibrium. This methodology is most clearly illustrated in the work by Sharland.5 The disadvantages are that the governing equations generally must be rederived each time a new system is modeled or a new reaction is considered.

Crevice and pitting corrosion 917

Also these models tend to assume implicitly that any solid corrosion product included in the formulation will be present throughout the corrosion cavity.

An alternative methodology for modelling the equilibrium state in solution is to keep the governing equations in rate format but make the reaction rates very fast. As long as reaction rates are fast, relative to rates of mass transport, the reactions will remain at equilibrium and the solution is independent of the kinetics assumed. Any kinetic expression which is physically consistent (i.e. is stoichiometrically true, gives zero net reaction rate at equilibrium, changes sign as the equilibrium point is crossed, and gives fast reaction rates) 6 can be used. The choice becomes a matter of numerical stability and convenience. In this work the following expression, obtained from transition state theory for reactions near equilibrium, is used:

i .... 1 ] Ri = z...a In (7)

k= 1 Kk

where R i is the total rate of reaction of species i in solution (moles dm-3 s- a), rk is the adjustable numerical rate parameter for reaction k (moles dm -3 s -a) and K k is the equilibrium constant for reaction k.

The above equation tends to generate a zero reaction rate contribution from any reaction in solution which is at equilibrium. When a reaction deviates even insignifi- cantly from equilibrium, large, physically consistent reaction rates are generated for all the species in the particular reaction. The total reaction rate for any species is the summation over all reactions containing the particular species. Keeping the equations in rate format allows the system of equations for the localized corrosion cell to be solved without requiring a priori algebraic manipulation of the governing equations. This greatly facilitates adaptation of the model to different electrolytes and metals.

Governing equations As a simplifying assumption, concentrations of species in the solution are

assumed uniform in the cross-width directions. This assumption will be valid when the length of the cavity is significantly greater than the width (i.e. for crevices and relatively deep pits). A balance on a control volume taken as a slice of the model domain assuming steady state gives for each species:

° C i - o = ~ + ziDiF cg (Ci~xS) Ox n--wNi + Ri (8)

where w is the width of crevice (dm) and x is the distance from mouth of crevice (dm). The governing equation for a deep cylindrical pit is similar to that for crevices:

OCi-oot = L)i- ~ 02Ci + ziDiF O Ox + 2Ni + (9)

where r is the radius of pit (dm). Equation 8 (or 9) is written for each of the aqueous species being considered in

the model system. Because the potential is also an unknown variable, another

918 J.C. WALTON

governing equation is required. The requirement is satisfied by the electroneutrality equation:

i

Ciz i : O. (10)

The boundary conditions at the cavity mouth are fixed concentrations and potential. At the base of the cavity the boundary conditions are Ni = - J i and electroneutrality. In the situation where the bottom of the cavity is non-metallic (e.g. many crevices) a boundary condition of zero gradient for all species is imposed.

To obtain a solution, the derivatives are approximated by finite difference equations. The equations are then solved by relaxation. 7 An initial guess for all values is made at the grid points, then the numerical solution improved through iteration. To reduce the number of grid points needed for an accurate solution a variably-spaced grid is used with a greater concentration of nodes near the mouth where the highest gradients are present.

The numerical constant r k is adjusted automatically in the computer code to keep all reaction rates rapid enough to maintain equilibrium but not so fast as to make the numerical solution unstable. As an option rk can be specified to be zero depending upon the direction of reaction k. This allows the equilibrium state for a single reaction to be maintained without change. For example, one might require that precipitation of a solid occurs when the solution becomes supersaturated without arbitrarily assuming that the solid is always present throughout the cavity and available for dissolution. The only physical significance of rk is that it is either large enough to keep the reaction close to equilibrium, or set to zero to eliminate the reaction from consideration in a simulation.

In the numerical code (a) the solution error from the transport equation of each aqueous species, (b) the deviation of each reaction from equilibrium, and (c) the deviation of the solution from electroneutrality, are monitored relative to tolerances at each grid node in the model domain. Iterations continue until the solution of the governing equations, within the allowable tolerances, is reached.

DIMENSIONAL ANALYSIS Prior to numerical solution, it is useful to analyse the governing equations in

terms of important dimensionless groups. Silverman 8 discusses the use of dimension- less groups in the modeling and prediction of corrosion processes. Useful definitions are:

X ' = x/! N'~ = N / N o

D'i = Di/Do R~ = R i l / N 0

C'~ =Ci/Co ~P = ¢psF/RT

where I is the length (depth) of crevice, Do is the characteristic diffusion coefficient, Co is the characteristic concentration, No is the characteristic reaction rate at metal-solution interface, X' is the dimensionless length, D' is the dimensionless diffusivity, C' is the dimensionless concentration, N' is the dimensionless reaction rate at metal-solution interface and R' is the dimensionless reaction rate in solution.

In terms of dimensionless variables, the steady state transport equation becomes:

Crevice and pitting corrosion 919

O= D ~ + ziD'i cl (c" d7 ) t+ N'i+ R" d X d X ' \ dX'} ~ "

(11)

Boundary conditions at the mouth are:

, Ci. ~bsF (12) C i - co, ~P - R T

Boundary conditions at the base are:

D" d X + ziD'C" ' - DoG} Ni. (13)

It is instructive to rearrange equation (11) such that the transport terms are on the opposite side of the equation from the reaction terms.

/ / ~,h" dX ' 2 - ~' + zi D i - ~ [Ci -~d~P _ CoDolN{} N'i + R~ . (14)

The equations can now be viewed as a competition between chemical reaction and mass transport rates. The terms on the left hand side of equation (14) representing mass transport by diffusion and electromigration must balance with the terms on the right hand side representing mass transfer/chemical reaction at the metal-solution interface and chemical reaction in solution. If mass transport is rapid, relative to reaction rates, the concentration and potential gradients will be small. Conversely, if mass transport is slow in relation to reaction rates, the cell will be strong.

Two dimensionless groups multiply the mass transfer rate. The dimensionless groups are influenced by both shape or aspect ratio (l/nw) and scale (lNo/CoDo). Thus aspect ratio alone (usually the only parameter that is reported in the experimental literature) is inadequate to explain the influence of system geometry on crevice and pitting corrosion. Rather the equations suggest an additional effect of scale. All else being equal, larger systems will be predicted to form stronger cells than small systems.

The influence of system geometry on the equations appears as (12/nw) in the metal-solution interface reaction rate, as (l 2) on the reaction rate in solution, and as (l) in the boundary conditions. If the time scale of the reactions in solution is fast relative to transport times (i.e. the reactions can be assumed to be at equilibrium) then the R' i terms can be eliminated from the governing equations by substitution. Likewise if the cavity is deep or non-metallic at the base then the boundary conditions at the base become unimportant. In these situations the influence of system geometry on the solution appears only as (12/nw). The same observation has been made by Turnbull. 3 The factor (12/nw) is referred to as the geometry factor in this work.

The equations above assume a crevice with metal on one side. A two sided (i.e. metal-metal) crevice has a geometry factor of (212/nw) and a cylindrical pit has a geometry factor of (212/nr) where r is the radius. The simplifying assumptions leading to these governing equations (i.e. one dimensional transport) will only be valid for crevices and relatively deep pits. Within this realm, numerical solutions will be identical for each of these systems as long as the geometry factor is held constant.

920 J,C. WALTON

MODEL APPLICATION The model is applied to recent experimental data obtained by Valdes. 9 The

experimental systems were for crevice corrosion of iron in electrolyte solutions of acetate buffer and sulfuric acid at 25°C. The experimental apparatus 9'1° consisted of a 5 mm long, 0.5 mm wide, and 10 mm deep crevice with iron metal on one side. Corrosion products were not allowed to build up extensively in the experiments considered herein, giving a porosity of 1.0. This gives a geometry factor of 2 dm (12/nw = {10 mm}2/{1.0 × 0.5 ram}). A perpendicular section of metal measuring 5 × 20 mm was included on the surface. The systems were anodically polarized and the potential measured as a function of distance inside the crevice by insertion of a Luggin capillary.

The systems are of some interest 1° because active crevice corrosion was found in electrolytes which were designed to exclude aggressive anions and minimize local- ized acidity. Potential drops of over 1 V were measured from the bulk solution to the base of the crevice.

Acetate electrolyte The acetate solution consisted of a mixture of equal amounts of 0.5 M acetic acid

solution and 0.5 M sodium acetate solution. This forms a buffer solution of pH 4.8. The reactions considered in modeling the system with equilibrium constants n'12'13 at 25°C are:

H20 q- Fe2+(aq) ¢:> FeOH+(aq) + H+(aq) K1 = 1.63 x 10 -7

Fe(OH)2(s) + 2H+(aq) ~ FeZ+(aq) + 2H20 K2 = 4.79 x l0 ll

HzO ¢:> H+(aq) + O H - ( a q ) K3 = 1.014 x 10 -14

CH3COOH(aq) ¢:> H+(aq) + CH3COO-(aq) K = 1 . 7 5 × 1 0 -5

C H z C O O - ( a q ) + FeZ+(aq) ¢:> CH3COOFe+(aq) K = 25.1.

(15)

(16)

(17)

(18)

(19)

These reactions, with the exception of reaction (16), are assumed to remain at equilibrium. Reaction (16) only becomes active if the solution becomes supersatur- ated with ferrous hydroxide. Eight aqueous species and the potential are considered in the model. They are listed below along with their diffusion coefficients (units: dm 2 s-l):

1. Fe 2+ 0.7 x 10 -7 6. CH3COOH 1.1 × 10 -7 2. H + 9.3 × 10 -7 7. CH 3 CO O - 1.1 × 10 -7 3. O H - 5.3 × 10 -7 8. CH3COOFe + 1.1 x 10 -7 4. F e O H + 1 × 10 -7 9. q~ 5. Na + 1.3 × 10 -7

In the acetate experiments Valdes 9 noted that the viscosity of the acetate

Crevice and pitting corrosion 921

FIG. 1.

10

0~ o o

" ~ 0 . 0 0 1 e - <

o , o o o l - - i _ _ I J _ _ i - 0 . 5 0 O. 5 1 1 , 5

Potential (SHE)

Anodic polarization curve for an uncreviced iron specimen in acetic acid solution.

solutions appeared to be much greater than would be expected in pure water. Based upon rough estimates of bubble rise Valdes estimated that the viscosity could be as much as 500 times that of pure water. The diffusivity of aqueous species is inversely related to the viscosity of the solution according to the Stokes Einstein relation.14 The diffusivities in pure water given above are thus adjusted by:

Dsolution = Do r}° (20) Y]solution

where ~/is the viscosity. Although pure acetic acid has a viscosity only slightly greater than water,

mixtures can have a viscosity significantly greater than the pure components, 15 although nowhere near a factor of 500. In the modeling a factor of 4 increase in viscosity relative to water is assumed, consistent with actual viscosity measurements in acetate solutions.16 The importance of this area of uncertainty in model input is discussed further below.

Electrochemical kinetics in the occluded cell are described by polarization data for an uncreviced sample of iron in the same solution (Fig. 1). The polarization curve is discretized into a series of (current, potential) data pairs for use in the code. Intermediate values are estimated assuming a linear relation between the log of the current density and the potential. The system always remains above the rest potential for the active metal. Thus only the anodic polarization curve is required. The current density and interfacial flux of ferrous ions (NFe2*) at any point inside the crevice are thus given as:

Fe(s) ¢:~ Fe 2+ + 2e- (21)

iFe ---- f(~bM -- ~bS) = a recognised tabulated value (22)

_ lVe ( 2 3 )

The governing equations for the acetate system consist of eight versions of equation (8), one for each aqueous species, and electroneutrality (equation 10). The

922 J .C. WALTON

rates of reaction in solution (Ri) are obtained from equation (7) assuming local equilibrium for reactions (15-19). Flux at the metal-solution interface (Ni) is given by equation (23) for the ferrous ion and is assumed zero for all other ions. Boundary conditions are fixed concentrations and potential at the crevice mouth and zero gradient in concentrations and potential at the crevice base. Concentrations at the crevice mouth are assumed to be equal to concentrations in the bulk solution. An arbitrarily small concentration of ferrous ion is assumed to be present in the bulk solution. Concentrations of the iron complexes at the boundary are estimated assuming equilibrium with the ferrous ion levels. In order to preserve exact electroneutrality, sodium concentrations at the boundary are estimated from other species concentrations assuming electroneutrality. The potential at the crevice mouth is the experimentally measured potential at the mouth.

In the experiments, significant potential drops were found in moving through the external solution to the crevice mouth. The potential drops in the bulk solution, outside the occluded cell, are not within the scope of the model presented herein but are evaluated for the sake of completeness. Potential drops in the bulk solution, where concentration gradients are expected to be small, can be estimated using Ohm's law. Application of Ohm's law leads to Laplace's equation for the potential. 2

V2(#S = 0. (24)

Boundary conditions are the measured potential at the crevice mouth, the applied potential at distance in the solution, and Odp/OX = 0 on the exposed metal surface. Laplace's equation is solved for the two dimensional geometry outside the crevice using the computer code PORFLO, 16 which was designed for groundwater appli- cations.

EXPERIMENTAL RESULTS The model predicts the potential, current density, and electrolyte composition as

a function of distance inside the occluded cell. A comparison of measured and predicted potentials for the system polarized to 0.84 V(SHE) is given in Fig. 2. The agreement is quite good. The predicted anodic current density inside the crevice is

1

u I

° ~

Eo Q.

-o2

-0A

nmdek~

[ ]

D

I k I i I I I "4 "2 0 2 4 6 8 10

Position Inside Crevice (mm)

FIG. 2. Predicted and measured potentials from crevice in acetate solution polarized at 0.844 V(SHE). The negative distances refer to measurements outside the crevice mouth.

The data points represent the results from several experiments.

Crevice and pitting corrosion 923

i 0.5

°f 0.001 0.N~ 0.01 0~3 0.1 0.3 1 3 10

P o s i t i o n I n s i d e C r e v i c e ( m m )

FiG. 3. Predicted distribution of anodic current density inside crevice in acetate solution polarized at 0.844 V(SHE). The active/passive transition occurs inside the crevice near the mouth. The dissolution rate is most rapid just below the active/passive transition and

declines rapidly with depth.

shown in Fig. 3. The active/passive transition occurs inside the crevice with the most rapid corrosion rate occurring just below this point. The predicted distribution of anodic current is consistent with the observed corrosion rate of the specimens and the observed location of the active/passive transition. 9,t°

The predicted composition of the solution is shown in Fig. 4. Most of the iron is tied up in the acetate complex. This minimizes the amount of iron hydrolysis. In the absence of significant metal hydrolysis, the hydrogen ions move rapidly out of the crevice as a result of the large potential gradients. The system never reaches saturation with respect to ferrous hydroxide. Figure 4 also illustrates that the numerical code correctly keeps all reactions at equilibrium and maintains electro- neutrality throughout the crevice domain.

Another set of experiments was performed at a polarization of 1.24 V(SHE). The comparison between modeled and measured potentials is given in Fig. 5. Polariz- ation at a variety of other potentials indicated that the potential at the base of the

FIG. 4.

CH3COOH( ) . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . : ~ -; ._-- -_-- '- -2Z2~ . . . . . . . . . . . . .

. . . . . C ~

' ~ o+m ' ~ ~ ................... ~ Fe(2+)

. . . . . . . . FeOH(+) ................. \ " " " " H ( + ) \ N a ( + )

1 E - 0 6 ¢ . . . . . . . . . + I

0.001 0.01 01 1 10

Position Inside Crevice (ram)

Predicted solution composition inside crevice in acetate solution polarized at 0.844 V(SHE).

924 J .C. WALTON

-1-

-j

o

o.

o I O D D D

[ I I I [ I I • , 4 - 2 0 ~1 4 iS I I

Po,mon Imad, Crovk:. Imm)

Fro. 5. Predicted and measured potentials from crevice in acetate solution polarized at 1.24 V(SHE). The negative distances refer to measurements outside the crevice mouth. The

data points represent the results from several experiments.

crevice varied very little. Irrespective of the external anodic polarization, the base of the crevice tended toward the same limiting potential (Elim). The model also predicts that potentials at the base are nearly independent of the mouth potential.

The uncertainty concerning the viscosity of the acetate solutions can be evaluated in light of the dimensional analysis. Since viscosity is inversely proportional to diffusivity, the impact of viscosity on the system is contained in the dimensionless groups controlling the development of the occluded cell [(lNo/CoDo) and (l/nw) in equation (14)]. An increase in viscosity (i.e. lowering of Do) is analogous to narrowing (or lengthening) the crevice. Thus as viscosity increases the system will more quickly reach the limiting potential. Since the solutions reach the limiting potential quickly in any case, the uncertainty concerning viscosity of the solution turns out to be relatively unimportant for modeling this particular electrolyte.

Sulfuric acid electrolyte The sulfuric acid solution consisted of 0.001 M sulfuric acid. The reactions

considered in modeling the system with equilibrium constants are: 11,12,13

Ha0 + Fe2+(aq)¢:~ FeOH+(aq) + H+(aq) (25) K = 1.63 x 10 -7

H20 ¢:~ H+(aq) + OH-(aq) (26) K = 1.014 x 10 -14

FeSO4-7Hz0(S ) ~ FeZ+(aq) + SO42-(aq) + 7H20 (27) K = 0.027.

These reactions, with the exception of reaction (27), are assumed to remain at equilibrium. Reaction (27) only becomes active when the solution becomes supersa- turated with ferrous sulfate. Five aqueous species and the potential are considered in the model. They are listed below along with their diffusion coefficients (Units: dm 2 s - - l ) :

1. Fe 2+ 0.7 × 10 -7 4. FeOH + 1 × 10 -7 2. H + 9.3 × 10 - 7 5. S O 2 - 1.1 X 10 - 7

3. O H - 5.3 x 10 -7 6. ~b

Crevice and pitting corrosion 925

O.S i

0.6

v

-~0~ 0 o ,

-0.2

-O-4

4 I [ I I I -2 0 2 4 $ 8 10

Position Inside Crevice (mm)

FIG. 6. Predicted and measured potentials from crevice in sulfuric acid solution polarized at 0.844 V(SHE). The negative distances refer to measurements outside the crevice mouth.

As with the acetate solution, the boundary conditions are fixed concentrations and potentials at the mouth and zero gradient at the base. The kinetics were obtained by discretization of the anodic polarization curve obtained from an uncreviced sample in the same solution. 9 This curve is virtually identical to the one obtained in acetate solution (Fig. 1).

The model predicts the potential and electrolyte composition as a function of distance inside the occluded cell. A comparison of measured and predicted poten- tials is given in Fig. 6. The predicted composition of the solution is shown in Fig. 7. Ferrous sulfate is predicted to precipitate at the base of the crevice. The region of ferrous sulfate precipitation is indicated by the flat portions in the Fe z+ and SOl - curves.

The potential at the crevice base in sulfuric acid solutions also reached a limiting value at a depth which was independent of the degree of anodic polarization in the experiments. This is examined in the model through variation of the boundary

0003 S04(2-) ..... ~:'~ ~ . . . . . . . . . . . . . . . . . . . . .

~ OOOO3

~ 00001

F~DH(+)

o

3E-06 ., H(+) ,7"

1E-06 ' ," I I 0.001 0.01 O. 1 1 10

Oist~w~ Inside Grevk~ (men)

FIG. 7. Predicted solution composition inside crevice in acetate solution polarized at 0.844 V(SHE). Ferrous sulfate is predicted to precipitate throughout most of the crevice.

926 J.C. WALTON

I 1 !

.! T W ~ 0 . 4

L •

-11.4

-ID.6

1 . 0 V

0.6 V

-0.036 V ",,, ....

• ,, ..~..

limiting potential

O.OQ01 O.G01 0.01 0.1 1 10

Position Inside Crevice (ram)

FIG. 8. Predicted potentials inside crevice in sulfate solution polarized at different applied potentials. In every case the potential asymptotes to a limiting value (E~im). Note that potentials at the crevice mouth will be less than the applied potential due to IR losses in the external solution. The potential assumed at the mouth of the cavity for each simulation is

placed above each line.

potential at the mouth of the crevice (Fig. 8). The model predicts the same limiting potentials at the base as were found in the experimental data.

DISCUSSION The results of the modeling analysis need to be viewed in the context of the

experimental setting. In the more general case of localized corrosion, factors such as the presence of aggressive anions, localized acidity, blockage of the cavity by gaseous or solid corrosion products, and salt film formation may be important. In the interest of focusing on the role of the potential in localized corrosion, the referenced experiments of Valdes were designed to eliminate the above factors as controlling aspects of the study systems. Elimination of the aspects of localized corrosion which are most difficult to quantify greatly simplifies the task of modeling the corrosion system. This is perhaps one of the major reasons for the evident agreement between theoretical analysis and experimental results. Current understanding of mass trans- port and thermodynamics appears to be reasonably adequate to describe chemically simple systems.

The explanation for the large observed and modeled potential drops lies in the balance between chemical reaction and mass transport processes embodied in the governing equations (equation 14). Since no single factor dominates the system, and the processes are highly coupled, a qualitative explanation of the observed and modeled phenomena is difficult. One very important aspect of these systems which promotes large potential drops is the chemical reaction/complexation of the anions. In both electrolytes, the anions present in the bulk solution were consumed by reactions at the base of the crevice. In the acetate systems, an iron acetate complex was formed. In the sulfate systems, salt precipitation occurred (ferrous sulfate). The chemical reaction of the anions limits their ability to migrate into the crevice and serve as a supporting electrolyte. At relatively low levels of a supporting electrolyte, ion movement is driven predominantly by potential gradients.

The importance of chemical reaction/complexation in the crevice solution to

Crevice and pitting corrosion 927

overall system behavior can be easily examined with the model. For example, if the ferrous acetate complex or the ferrous sulfate precipitate are (erroneously) assumed not to form, then the anions present (acetate or sulfate) act as supporting electro- lytes. This causes the model to (a) underpredict the potential drops in both systems, (b) predict formation of localized acidity at the crevice base, and (c) predict greater current density in the crevice. This exercise suggests that the chemical behavior of the electrolyte species in the corrosion cavity is very important to the development of the corrosion cell---irrespective of (or in addition to) any specific interactions with the passive layer. The systems chosen for study by Valdes may be fairly unique in giving large potential drops with only moderate current density and no localized acidity. Further experimental research in carefully controlled corrosion cavities is needed to elucidate the exact role of the electrolyte and many other aspects, besides the potential, which are important in localized corrosion.

The limiting potentials observed in these systems result from a sharp decline in anodic current density at lower potentials in conjunction with the concentrated electrolyte solutions at the crevice base. The anodic current density at the base for both systems is in the vicinity of -0 .05 A dm -2 as opposed to - 6 A dm -2 near the active/passive transition (Figs 1 and 3). The total current carried at any point in the crevice solution is the integral of the current density from that point to the base of the crevice. As the potential drops inside the crevice in the active region, the total current carried in the solution drops exponentially with potential. Low currents can be carried in the concentrated solutions with insignificant potential drops. Thus the potential naturally declines asymptotically to a limiting value with depth in the crevice as the equilibrium potential is approached. 10,17

Even though the local net current density in the model and experimental systems always remained in the region of anodic dominance (Figs 1, 2, 5, 6), evolution of hydrogen gas was observed in the experiments. These observations appear inconsist- ent, but are not. Even though the cathodic reactions do not dominate the net current density anywhere in the cavity, they nonetheless occur throughout the system. Because of the low solubility and uncharged nature of hydrogen gas, transport rates in aqueous solution are limited. Experience with other model systems suggests that localized corrosion cavities quickly become supersaturated with hydrogen gas, even at very low cathodic current densities. 18 Cathodic kinetics (e.g. hydrogen evolution reaction) were not included in the modeling analyses presented because (a) they are quantitatively unimportant in determining the current/potential distribution, and (b) cathodic polarization curves from uncreviced specimens were not included in Valdes' thesis (i.e. the data were not available).

Based upon a dimensional analysis of the governing equations, a scale effect is predicted for the corrosion cells. All other parameters being equal (including aspect ratio), larger crevices will be expected to develop stronger cells than smaller crevices. The impact of system geometry on many corrosion cells is expected to depend upon the product length squared divided by width (12/nw). This conclusion contrasts sharply with much of the corrosion literature where aspect ratio (length/width) is assumed to be controlling.

In summary, a conceptual, mathematical, and numerical model of the electro- chemistry in crevices and pits has been developed. The model is general in form, precluding the need to manipulate the governing equations by hand prior to modeling different systems. Excellent agreement was obtained between theoretical

928 J.C. WALTON

predic t ions and exper imen ta l results in two i ron systems. The ag reemen t be tween expe r imen ta l and theoret ical results gives s t rong and c o m p l e m e n t a r y evidence that large po ten t ia l drops can occur in some localized corros ion systems even in the absence of cavity b lockage by bu i ldup of corros ion products . The general ized mode l ing approach should be appl icable to a wide var ie ty of o the r systems.

Acknowledgements--The assistance of Drs S. A. Rawson, M. R. Sharpsten, R. L. Miller, and G. A. Giesbrecht with execution of this work and obtaining thermodynamic data for the modeled systems is gratefully acknowledged. Work partially funded under the auspices of the U.S. Department of Energy, DOE Contract No. DE-AC07-76IDO1570.

REFERENCES 1. S. M. SHARLAND, Corros. Sci. 27,289 (1987). 2. J. NEWMAN, Electrochemical Systems. Prentice Hall, New Jersey (1973). 3. A. TURNBULL and J. G. N. THOMAS, J. electrochem. Soe. 129, 1412 (1982). 4. S. M. GRAVANO and J. R. GALVELE, Corros. Sci. 24, 517 (1984). 5. S. M. SHARLAND, Corros. Sci. 28,621 (1988). 6. A. C. LASA~A and R. J. KIRKPATR1CK, Kinetics of Geochemical Processes. Mineralogical Society of

America, Washington (1981). 7. W. H. PRESS, B. P. FLANNERY, S. A. TEUKOLSKY and W. T. VETTERLING, Numerical Recipes--The Art

of Scientific Computing. Cambridge University Press, New York (1985). 8. D. C. S1LVERMAN, Corrosion 41,679 (1985). 9. A. VALDES-MOULDON, PhD Thesis, The Pennsylvania State University (1987).

10. H. W. PICKERING, Corros. Sci. 29,325 (1989). 11. R. A. Rosin, B. S. HEMINGWAY and J. R. FISHER, Thermodynamic Properties of Minerals and Related

Substances at 298.15 K and 1 Bar Pressure and at Higher Temperatures. USGS Bulletin 1452, U.S. Government Printing Office, Washington (1978).

12. A. E. MARTELL and R. M. S~IITH, Critical Stability Constants, Volume 3: Other Organic Ligands. Plenum Press, New York (1977).

13. C. F. BAES and R. E. MESMER, The Hydrolysis of Cations. Wiley, New York (1976). 14. A. LEHMAN, Geochemical Processes in Water and Sediment Environments. Wiley, New York (1979). 15. R. H. PERRY and C. H. CH1LTON, Chemical Engineers' Handbook. McGraw-Hill, New York (1973). 16. A. RUNCHAL, B. SAGAR, R. G. BACA and N. W. KLINE, PORFLO--A Continuum Model for Fluid

Flow, Heat Transfer, and Mass Transport in Porous Media: Model Theory, Numerical Methods, and Computational Tests. Rockwell Hanford Operations, RHO-BW-CR-150-P (1985).

17. H. W. P1CKERING, Corrosion 42,125 (1986). 18. J. C. WALTON, Nuel. Chem. Waste Management 8, 143 (1988).

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