Pergamon Corrosion Science, Vol. 38, No. 1, pp. I-18, 1996 Elsevier Science Ltd Printed in Great Britain 0010-938X/96 $15.00+0.00 0010-938X(96)00107-7 A NUMERICAL MODEL OF CREVICE CORROSION FOR PASSIVE AND ACTIVE METALS J.C. WALTON,* G. CRAGNOLINO and S.K. KALANDROST Center for Nuclear Waste Regulatory Analyses, Southwest Research Institute, San Antonio TX 78238-5166, U.S.A. Abstract-A transient mathematical and numerical model for crevice corrosion is presented. The model is general in format and applicable to a variety of metallic and electrolyte systems. It allows consideration of electrode kinetics including both cathodic and anodic reactions with an active/passive transition. Chemical reactions in solution are generalized to facilitate simulation of a variety of electrolytes. The model is applied to experimental data on crevice corrosion initiation in stainless steel and to experiments on active crevice corrosion of iron. Keywords: A. steel, A. stainless steel, B. polarization, C. crevice corrosion, C. pitting corrosion. INTRODUCTION Crevice corrosion is one of the predominant forms of localized corrosion limiting the service life of engineered structures. Mechanistic models of corrosion processes are one of the tools available to assist with interpretation and extrapolation of short term experimental data to long time periods. The goal of this work was to develop a crevice corrosion model that would be sufficiently general to be useful in a variety of applications. Such a model can be useful, when used in conjunction with experimental data, in determining when localized corrosion is likely to occur, maximum crevice or pit penetration depths theoretically possible under anticipated environmental conditions, and the relationship between electrolyte composition and propagation of crevice corrosion. This work presents a new mathematical model for crevice and pitting corrosion and compares the model with several sets of experimental data on crevice corrosion. Models previously developed for crevice corrosion have been reviewed in detail by Sharland. Since the time of the review, a number of newer models have been developed and published. In general, the newer models tend to offer increasingly more rigorous solutions to the coupled transport and chemical reaction equations used to describe electrochemical systems quantitatively. Watson and Watson and Postlewaite3 developed a model for initiation of crevice corrosion. The model considers time dependent development of the chemistry inside a crevice assuming a passivated metal surface corroding with constant passive anodic current density and multiple reactions in solution. Mass transport by diffusion and electromigration is solved with the finite difference method in one dimension. The model uses dilute solution theory for reactions and mass transport (i.e. activity coefficients equal to one), although an *Present address: Department of Civil Engineering, University of Texas at El Paso, El Paso, TX 79968, U.S.A. TPresent address: Department of Electrical Engineering, University of Wisconsin, Madison, WI, U.S.A. Manuscript received 12 June 1994; in amended form 6 March 1995.
empirical correction is applied when considering the critical ph for crevice corrosion initiation. The model was not compared to experimental data. Sharland4 developed a model. CHEQMATE, for coupled mass transport and chemical reaction and applied the model to several transient crevice corrosion experimental data sets with fully passivated crevices. This model considers transport by advection, diffusion, and electromigration in one dimension coupled to chemical reaction. The transport solution is by finite difference. A constant. passive anodic current density is assumed at the metal/solution interface allowing simulation of the time period prior to the initiation of crevice corrosion. Activity coefficients are considered in the chemical reactions with the Davies equation but not in the mass transport equations. The CHEQMATE code. unlike most other codes for crevice corrosion, is written in a general format allowing multiple reactions to be considered. This type of generality is impossible when the chemical reactions under consideration are incorporated directly in the governing equations and thus hard wired directly in the associated computer code. In both models,,4 reactions in the aqueous solution are assumed to be at equilibrium. Furthermore both models have no potential dependent reactions, thus neither active corrosion nor transitions between active and passive corrosion states can be modeled. Several newer models have been developed for active crevice corrosion. Sharland et al. developed a two dimensional, steady-state, finite element model for active crevice corrosion. The model considered a fixed number of chemical reactions appropriate for steel corrosion. Walton developed a steady-state one dimensional finite difference model for active and passive crevice corrosion. The model is capable of modeling multiple cathodic and anodic reactions including heterogeneous equilibria. Both of these models assume dilute solution theory with activity coefficients equal to unity. The published models have a number of similarities in simplifying assumptions related to the difficulty in modeling the systems. All the transient models contain the simplifying assumption of constant anodic current density and thus cannot model active corrosion or active/passive transitions. The active corrosion case is significantly more difficult to model because of: (a) the higher current density: (b) higher concentrations of ions; (c) greater potential drops; and (d) the nonlinear relation between potential, ion concentrations, and current density at the metal/solution interface. As a result of these difficulties, models for active corrosion have all been steady-state. eliminating the derivative in time. Unfortunately, none of the above models can be used for modeling the situation where an experimental crevice moves from a fully passive state into active corrosion (e.g. when polarized at higher potentials) or repassivation of a crevice when the external corrosion potential is lowered. The model presented in this work is capable of simulating processes such as crevice corrosion initiation, propagation rate of crevices (i.e. spatially dependent corrosion rate), and crevice repassivation as a function of external electrolyte composition and potential. The model can also simulate pitting corrosion growth. The model is applied to several experimental data sets including crevice corrosion initiation on stainless steel and active corrosion of iron in several electrolytes. In the corrosion area, the model breaks new ground in three areas: (1) equations for moderately concentrated solutions including individual ion activity coefficients and transport by chemical potential gradients are used rather than equations for dilute solutions; (2) the model is capable of handling passive corrosion, active corrosion, and active/passive transitions in a transient system; (3) the code is general in format, greatly facilitating evaluation of the importance of different species, chemical reactions, metals, and types of kinetics at the metal/solution interface. The generality allows
Crevice corrosion for passive and active metals 3 rapid evaluation of what/if questions such as: how important is a particular aqueous species or individual chemical reaction in controlling the electrochemistry in a localized corrosion cavity? Alternative model evaluations can rapidly be performed with and without the species or reaction of concern. GOVERNING EQUATIONS The model considers mass transport by diffusion and electromigration. The flux equation for diffusion and electromigration of aqueous species is: where the symbols are explained in the Appendix. The chemical potential is given by (1) CL,, = P: + RTlna,, (2) where P,~ is the standard chemical potential of species n and the activity of each aqueous species is given by a, = yncn. (3) Equation (1) can be expanded using equations (2) and (3) giving + N,, = -+%v$$ -+vc,, The flux of ions results in a current in the solution: C,yV,). Combining equation (4) and (5) and solving for the electrostatic potential in solution yields where VA = -(is + C,,(z,FD,llv,l)(n,VC~ + W%J) (6) K K=RT F2 c n z;d,c,. In order to specify the current in solution for a particular problem, one must consider conservation of charge. In the case of electroneutrality, there can be no storage of charge or capacitance in the solution. The current at any point of a crevice assuming electroneutrality and unidimensional transport is.x s 1 L; = - AC L i, Pdx, (8) where i, is the current density at the metal/solution interface resulting from electrochemical dissolution of the metal or alloy, A, is the cross-sectional area, and P is the metal covered
J.C. Walton et al. nodai interface Ax node L Fig. 1. Schematic of model geometry for a crevice. Ion fluxes and current are calculated at nodal interfaces; concentrations and potential are calculated at node locations. g and As may yary with.y. perimeter of the crevice as shown in Fig. 1. Substituting equation (8) into equation (6) gives an expression for the potential gradient at any point in the crevice as a function of solution composition, gradients in activity coefficients, concentration gradients, and the current density across the metal/solution interface. The current density at the metal/solution interface, i,, can be a function of distance into the crevice (s), potential, temperature, and concentrations of the aqueous species present: ie =,f (s, E, T, C,, C2.., C,,). (9) The potential, E, is defined as the potential of the metal relative to the potential in solution: E=&-$,, (10) where &, is the potential of the metal. The electrochemical reactions define the flux of ions at the metal/solution interface: (11) By definition, % points in a direction perpendicular to the surface representing the metal/solution interface. The weighting factor a,, can be based on alloy molar composition. assuming that congruent dissolution occurs, or be specified by the user by an appropriate expression in the case of selective dissolution. Assuming the flux at the metal/solution interface is averaged over the crevice, the material balance at any point is given by where R,, is the change in concentration of species from chemical reactions in the crevice solution.
Crevice corrosion for passive and active metals NUMERICAL SOLUTION Solution of the model for a particular problem requires specification of species to be considered, reactions in solution, thermodynamic data, grid geometry, and kinetics at the metal/solution interface. The numerical solution is described below in the order steps occur in the model. Different numerical methods are used in separate portions of the solution. The simulation domain is broken into a set of calculational nodes (Fig. 1). Nodes can be non-uniformly spaced to increase accuracy where gradients are highest. Equations (8) and (6) are discretized as (13) @s(i) = @s(l) + ~W,,))A.u,,. 2 The summations occur from where either current in solution (at the base of the cavity) or potential (at the mouth) are known to the other end of the corrosion cavity. Equation (13) is solved at the beginning of each time step to estimate the potential distribution in the corrosion cavity. The current density (i,) can be calculated as Tafel kinetics, Butler-Volmer kinetics, or tabular values as required for the problem being solved. V$,, is calculated by application of the finite difference form of equations (4) and (6) at adjacent nodes. In the case of an active/passive transition within the cavity, iteration is required to obtain the correct potential and current distribution since the two variables (i, and V&) are coupled. The system of equations is solved by relaxation (weighted averaging of old and new values at each node) until convergence is obtained. Knowledge of the current and potential distribution in the cavity allows calculation of the mass transfer of species at the metal/solution interface (N,,,n). The Nc,~,~J are treated by direct incorporation in the transport equations and thus do not appear as boundary conditions or reactions in solution. The finite difference form of equation (12) is solved for each species at each node thereby forming a system of (n x j) ordinary differential equations: (14) The in and Out superscripts refer to the nodal interfaces on either side of the node. Fluxes are calculated at nodal interfaces using the finite difference form of equation (4). Boundary conditions are specified as fixed concentration at the mouth and either no flux or corrosion at the base. The interface between nodes is defined to be at the midpoint between adjacent nodes, allowing for variable node spacing as specified by the user. Cross-sectional area, metal surface area, and volume are calculated at each node based upon user input of nodal spacing and crevice gap (or pit radius). When the crevice gap (g) or pit radius (Y) are nonuniform, A,, A,, and V are estimated using linear interpolation of the three dimensional shape between nodes. The electrochemical reactions at the metal/solution interface (Nc,,.jj) appear in the second term on the right hand side of equation (15) where the reaction rate per unit corroding surface area is multiplied by the corroding surface at the node and divided by (15)
6 J.C. Walton el al. the nodal volume. This effectively averages the electrochemical reactions over the node, allowing a one dimensional solution. The interfacial concentration of each species appearing in the finite difference form of equation (4) is normally estimated by central difference (i.e. averaging the concentration at adjacent nodes). However, when high potential gradients are present, the central differencing scheme can become unstable. A dimensionless group describing the ratio of transport by electromigration relative to diffusional transport (analogous to a Peclet number) is defined as MD, V& 0, z,faxv& -_= RT I Ax RT. If the absolute value of the dimensionless group is greater than two, upwinding is used. In the case where z,v&> 0, the interfacial concentration is assumed to be equal to the concentration at the next higher node. When z,v#, < 0, the concentration at the lower node is used. The assumption involved with upwinding is that, when electromigration is strong relative to diffusion, the best estimate of concentration at the nodal interface is the concentration at the up gradient node. The upwinding greatly enhances the ability to model numerically difficult situations such as active/passive transitions. The equations are next integrated for one time increment using one of three specified ordinary differential equation (ODE) solvers with monitored accuracy and step size control. If the desired accuracy is not achieved, then the step size is reduced automatically. The three solvers are an Adams method;* the Bulirsch-Stoer method; and the fifth-order Runge- Kutta method. The equations for chemical reactions in solution at each node (R c,n) (e.g. speciation and precipitation of solid corrosion product phases) are solved separately (operator splitting). The assumption is made that the characteristic times of chemical reactions in aqueous solution are much shorter than those of the mass transport or corrosion processes. Thus, the aqueous solution is assumed to be at equilibrium. At the end of each time step, the resulting aqueous solution composition at each node is solved to equilibrium by a call to the equilibrium solver. Equilibrium of the solution is obtained by stoichiometric Gibbs free energy minimization. lo Three options are provided in the code for determining activity coefficients: (1) ideal solution (activity coefficients equal to 1); (2) B-dot Debye-Htickel equation; and (3) Davies equation. * The B-dot Debye-Htickel equation is given by: log(yj = -AZ; (16) JI + iti (17) I + a;s%/t The Davies equationi is similar to the B-dot equation with ai B= 1 and B= 0.3. The Debye-Htickel and B-dot parameters can be calculated as a function of temperature using correlations valid over the range 0-250 C. Uncharged species, water, and solids are assumed to have activity coefficients of 1. An example of the adequacy of the approximations provide3 by the activity coefficient expressions can be obtained by comparison with experimental data. Figure 2 demonstrates how the activity coefficient of calcium chloride varies with concentration. The dotted line represents experimental data collected by Staples and Nuttal. 3 The dashed line represents the Davies equation, whereas the solid line is the B-dot equation. At low molal concentrations (co.5 molal), both equations accurately predict the activity coefficient. At
Crevice corrosion for passive and active metals 3.5. 3. 2.5 r 2. / F 51.5. s 1' l- / / / Davies Staples and Nutta13 J 2 4 6 8 10 molal concentration Fig. 2. Comparison of calculated activity coefficients for calcium chloride with experimental data higher concentrations, the Davies equation deviates markedly. The B-dot equation exhibits divergence from the experimental data as ionic strength increases with approximate agreement below 2 M and good agreement below 1 M. The series of steps given above is repeated until the final simulation time is reached with system variables written to files at specified intervals. COMPARISON OF MODEL PREDICTIONS WITH EXPERIMENTAL DATA In the following sections, the results of several experiments described in the corrosion literature are modeled. The first case represents crevice corrosion initiation in stainless steel. The last two simulations represent active crevice corrosion of iron. Alavi and Cottis experiment 4 Alavi and Cottis14 conducted an experiment on crevice corrosion of stainless steel. The electrolyte was a 0.6 M NaCl solution at 25 C with a ph of 6.0. A plate of type 304 stainless steel (74% Fe, 18% Cr, 8%Ni) and a perspex electrode holder formed an experimental crevice measuring 8 cm deep and 2.5 cm wide with a 90 pm gap. The steel was coupled to a larger piece of the same steel which was exposed to free-corrosion conditions in the aerated bulk electrolyte. Thirteen reactions were assumed to occur in the system as listed below: H20(1) + H+(aq) + OH-(aq); Cr3+(aq) + H20(1) + Cr(OH> +(aq) + H+(aq); Cr3+(aq) + 2H20(1) ++ Cr(OH)2+(aq)+2H+(aq); (18) (19) (20) Cr3+(aq) + 3H20(1) * Cr(OH),(aq) Cr3+(aq) + 4H20(1) + Cr(OH);(aq) + 3H+(aq); + 4Hf(aq); (21) (22)
J.C. Walton et al. 2Cr3+(aq) + 2H20(1) ($ Cr2(0H)T(aq) + 2H+(aq); (23) 3Cr3+(aq) + 4H#(l) + Cr3(0H)p(aq) + 4H+(aq); (24) Cr3+(aq) + Cl- (aq) * CrC12+(aq); (25) Cr3+(aq) + 2Cl-(aq) * CrC12+(aq); (26) Fe2+(aq) + Cl-(aq) e FeCl+(aq); (27) Fe2+(aq) + 2Cl-(aq) + FeCll(aq); (28) Fe2+(aq) + 4Cl-(aq) + FeCl%(aq); (29) Ni +(aq) + Cl-(aq) ($ NiCl+(aq). (30) These reactions involve 18 aqueous species in addition to water. The Na+ ion from the sodium chloride solution must also be included in the input file for the sake of preserving electroneutrality. The thermodynamic and transport properties of the 20 species are listed, among others, in Table 1. The reactions assumed to occur at the metal/solution interface are: Fe(s) + Fe +(aq) + 2e-; (31) Cr(s) * Cr3+(aq) + 3e-; (32) Ni(s) e Ni2+(aq) + 2e-. (33) These reactions are assumed to occur at relative rates proportional to the atomic composition of the steel; thus, the anodic current was assumed to consist of 0.74& of Fe2+, O.l%, of Cr3+ and O.OSi, of Ni*+ The boundary conditions at the mouth of the crevice are 0.6 molal Na+,0.6 molal Cl- and low6 molal H+. The crevice geometry was modeled with 32 finite difference nodes with closer nodal spacing near the crevice mouth. The simulation was run for 90 h using the Adams solver with an error tolerance of 10e4. The B-dot equation was used to determine activity coefficients. The anodic current density was not reported in the experiments, thus modeling the system requires an assumption concerning the anodic current density. The simplest assumption was a constant current density as might be expected on a passivated metal. Three simulations were conducted assuming passive corrosion with three different constant anodic currents: 10e4, 3 x 10e5 and 10V5 A/dm*. Quasi-stationary polarization curves of type 304 stainless steel in chloride solutions of similar concentration exhibited passive current densities of about 1O-4 A/dm*. Even lower current densities can be expected under steady-state conditions. Results were output at times of 2, 18, 24,43 and 90 h, corresponding to the times at which measurements were taken in the Alavi and Cottis experiment. The results for ph of the solution versus position at 90 h are shown in Fig. 3. In the
Crevice corrosion for passive and active metals Table 1. Properties of aqueous species Species AGr D (Jim4 (dm2/s) H@(l) -237,200 - H (aq) 0 9.3 x lo- (a) OH-(aq) - 157,300 5.3 x IO- (a) Na (aq) -261,900 1.3x 10- (a) Cl-(aq) -131,300 2.0 x IO- (a) C?+(aq) - 195,100 0.6 x lo- (a) Cr(OH)*+(aq) -409,400 0.7 x lo- (b) Cr(OH) +(aq) -614,100 0.8 x 10- (b) WOW3(4-803,900 1.0 x 10- (b) CrWh-@q) - 987,400 0.8 x IO- (b) CrAOI-b4 (aq) - 789,600 0.5 x IO- (b) Cr3KWt5+(aq) - 1,487.400 0.4 x IO- (b) CrCl* + (aq) - 325,500 0.7 x 10- (b) CrC12 + (aq) -458,600 0.8 x lo- (b) Fe +(aq) -91,500 0.7 x lo- (a) FeCl+(aq) -221,900 0.8 x IO- (b) FeClW) - 340,100 1.0x10- (b) FeC& (aq) - 605,800 1.1 x 10- (b) FeOH+(aq) -277,400 0.8 x IO- (b) Ni +(aq) -45,600 0.7 x lo- (a) NiCI+(aq) - 171,200 0.8 x 10- (b) CH#ZOOH(aq) - 396,500 1.2x lo- (c) CHXCOO-(aq) - 369.300 I.1 x lo- (c) CHsCOOFe+(aq) -468,200 1.1 x IO- (b) Fe(CH@QMaq) - 844.300 1.2 x IO- (b) S04*-(aq) -44,500 1.1 x IO- (a) FeSWaq) -848,500 1.1 x 10- (b) FeSO&) - 820,800-0 - & - 9.0 3.0 3.0 5.0 4.5 5.5 6.0 4.5 6.0-4.5 3.0 3.0 0.0 Thermodynamic data were taken from the EQ3/6 (Wolery et a1. 5) database (Version: datao.com.rl2). Diffusion coefficients are from: (a) Li and Gregory;16 (b) estimated; and (c) Vitagliano and Lyons. 2.5 t -. ---._. -.-.~.-.-.~.~.~.~.~.~.~.~.~.-.~.-.~.~.~.~... position (cm) Fig. 3. Experimental and predicted ph as a function of position in the crevice after 90 h using different constant anodic current densities (A/dm ).
IO J.C. Walton et al. Alavi and CottiS 4 20 40 60 60 time (hr) Fig. 4. Experimental and predicted ph values 7.5 cm into the crevice as a function of time using different constant anodic current densities (A/dm*). experimental data the ph versus depth curve passes through a minimum then increases at the base of the crevice. Alavi and Cottis suggest that the minimum in ph may correspond to a peak in anodic current density. Since current density is assumed to be constant in the model, the ph minimum is not reproduced. It would be possible to make the anodic current density a function of distance inside the crevice and obtain better agreement with data (i.e. use anodic current density as a fit parameter), however, little would be accomplished thereby. Figure 4 shows the results for solution ph 7.5 cm into the crevice versus time. Most of the experimental ph values lie near the predicted values at a current density of 10e4 A/dm2. Figure 5 shows the results for potential drop in the solution versus position at 90 h. The potential drop agrees qualitatively with model predictions assuming a current density between 1O-4 and lo- A/dm*. Figure 6 shows the results for potential drop in the solution 0 s E - -10 Q) P E -20 5 i -30 5 P -40 _..iand~~~~~ 1 0 2 4 6 position (cm) Fig. 5. Experimental and predicted potential drop after 90 h using different constant anodic current densities (A/dm ).
s g -10 I Crevice corrosion for passive and active metals 11. I 01 1 Q) -- - s P c -20.. 0 3-30.... E g40. _._.-.--- *_._._._._.-.----._._._.-.-.-._...---.-.- Alavi and Cottis 4 0 20 40 60 60 time (hr) Fig. 6. Experimental and predicted potential drop 7.5 cm into the crevice using constant anodic current densities (A/dm ). 7.5 cm into the crevice versus time. The experimental values lie intermediate between the 10e4 and lop5 A/dm* predicted values. The model predicts that, at constant anodic current density, the potential drop will decrease with time as the crevice solution becomes more concentrated. However, the experimental values exhibited a faster decrease of the potential drop than those predicted by the model. It is possible that the passive current density decreased with time during the course of the experiment, leading to smaller potential drops with increasing time. Perhaps the most significant discrepancy between experimental and modeled data is that different assumed current densities are required to even approximately match the observed trends in ph and potential. High current densities provide better agreement with observed ph and lower current densities provide better agreement with observed potential drops. Valdes-Mouldon experiments 9-ac.etate system Valdes-Mouldon conducted several experiments involving crevice corrosion of iron in electrolyte solutions. The experimental crevice was 10 mm deep, 5 mm wide, with a gap of 0.5 mm and iron metal on one side. A perpendicular section of the iron measuring 5 mm by 20mm at the surface of the crevice was anodically polarized to different values. The experiments were carried out with an acetate buffer solution and a sulfuric acid solution. The Valdes-Mouldon experiments are more difficult to simulate numerically because they deal with systems having strong active/passive transitions and the potential drops are very high. These aspects of the experiments place greater emphasis on the non-linear portions of the governing equations. The acetate buffer solution consisted of equal parts 0.5 M acetic acid and 0.5 M sodium acetate solution at 25 C. This formed a buffer solution with ph 4.8. The five principal reactions occurring in the system are listed below: Fe*+(aq) + HzO(1) + FeOH+(aq) + H+(aq); (34) CHsCOOH(aq) + H+(aq) + CHsCOO-(aq); (35)
12 J.C. Walton et al. CHKOO-(aq) + Fe2+(aq) + CHsCOOFe+(aq); (36) CHsCOO-(aq) + 2Fe2+(aq) w Fe(CHsCOO),(aq), (37) along with equation (18). These reactions involve seven aqueous species as well as water. In addition, the Nat ion from the sodium acetate solution must also be included in the input file for the sake of preserving electroneutrality. The properties of these 10 species are also listed in Table 1. The predicted solution composition ranges between 1 and 2 M in acetate species. Vitagliano and Lyons indicate the viscosity of acetate solutions of 1 and 2 M are 1.11 and 1.23 times that of pure water, respectively. An intermediate correction factor for viscosity of 1.17 was applied to the model input diffusion coefficients as listed in Table 1 for the acetate system to reflect viscosity effects. The only reaction assumed to occur at the metal/solution interface is anodic oxidation of iron to ferrous iron. The boundary conditions at the mouth of the crevice are 0.5 molal Nat, 0.5 molal CH$ZOO-, and 0.5 molal CH$OOH. The simulation was run for 30 min using the Bulirsch-Stoer solver with an input tolerance of 10P3. Two runs were made using different methods for determining the activity coefficients: one assuming the ideal case (activity coefficients all set to 1) and one using the B-dot equation. A third run was made using the B-dot equation and omitting equation (35)-(37) from the input file; that is, no complexation of Fe2+ ions was assumed to occur. The purpose of the third simulation was to examine the sensitivity of the results to chemical reactions in solution. The crevice was discretized into 100 nodes with finer nodal spacing near the crevice mouth where most variation in concentration and potential occurs. The finer grid resolution was required to accurately resolve the very sharp active/passive transition characteristic of iron. Valdes conducted the experiment with the system anodically polarized to two different values. The simulations were run for external applied potentials of 0.844 and 1.24 V (SHE). Since the model is only valid inside the crevice, experimental values for the corrosion potential at the mouth of the crevice were used as the fixed boundary conditions; these values were 0.68 and 0.92 V (SHE), respectively. A relaxation factor of 0.2 was used for the prediction of the potential in solution. Electrode kinetics were obtained by discretization of polarization curves of uncreviced specimens immersed in acetate and sulfate solutions. The polarization curve for acetate solution is shown in Fig. 7. The polarization curve for sulfate solution (not shown) is similar. The model is only valid for the inside of the crevice. However, Valdes observed significant potential drops in the bulk solution outside the crevice mouth. Since concentration gradients in the bulk solution should be small, Ohm s law is used to estimate the potential drops outside the crevice mouth. Unlike the Alavi and Cottis simulations where the current density was unknown, the Valdes data set is complete. All parameters in the Valdes simulations represent data obtained with uncreviced specimens. Thus the Valdes simulations can be viewed as model predictions rather than simple fits to data. Figure 8 shows a comparison of experimental and predicted results for the B-dot and dilute solution activity coefficient methods using the 0.68 V boundary potential (external polarization of the system to 0.844V). Both simulations lie within the scatter of the experimental data. The experimental data points represent data from several experiments. In each
Crevice corrosion for passive and active metals 13 Valdes-Mouldon g 1 1 iron in: E 2 3 0.01 F/ I I I I I I I -0.25 0 0.25 0.5 0.75 1 1.25 potential (V SHE) Fig. 7. Anodic polarization curve for an uncreviced specimen in acetate solution (used for electrode kinetics in the model). 3 experiment the data points lie on a smooth curve, however when the data is lumped on one plot, data scatter appears. The scatter in experimental data is greatest near the mouth of the crevice. This may reflect experimental difficulties in obtaining precise measurements where the gradients are highest, since small variations in location or disruption of the fluid by introduction of the Luggin capillary may lead to greater potential variations in the presence of strong gradients. Deeper inside the crevice, where the limiting potential is reached, less scatter in the data is observed. The movement from dilute solution theory to a more rigorous treatment with activity coefficients and mass transport by chemical potential gradients has relatively little influence on the model predictions for this system. Figure 8 also compares the results with and without the complexation reactions (equations (36) and (37)). The complexation reactions clearly have a large influence on the predicted potential in the solution. In the absence of complex formation, the ion concentrations in the crevice cavity build up to higher levels, allowing current transmission with lower ZR drops (under * _ ; I -4 Valdes-Mouldott g acetate solution 4.21 iron in Fig. 8. Experimental and predicted potentials for 0.844 V potentials.
14 J.C. Walton et al. If..I..l.l 1,I..l.l III,,*. I, 0.01 0.05 0.1 0.5 1 5 10. position (mm) Fig. 9. Species concentration in acetate solution for 0.844 V polarization. prediction of potential drops). Note that the two complex formation equations result in combination of cations and anions leading to lower charge density in solution. The lower charge density solution requires greater potential drops to carry the current. Figure 9 illustrates the concentrations of various species at the end of the 30 min period. The ferrous ions that have entered the solution are mainly present as the (CH3C00)2Fe complex. Note that the solution is not yet at steady-state. The concentrations at the base reflect the initial conditions. The influence of flux at the metal/solution interface on concentrations in the system and potential drops can be understood by viewing Figs 7-9 simultaneously. Most of the action is in the crevice, and the highest corrosion rate occurs near the mouth. Valdes observed that the experimental crevices corroded most rapidly just inside the mouth, consistent with model predictions. Figure 10 represents the experimental and predicted potentials inside the crevice at 1.244 V of applied potential. Figure 11 shows the predicted anodic current density after 30min at the same potential. The active/passive transition occurs just inside the crevice mouth, The transition from passive to active corrosion occurs somewhat deeper into the crevice than the 0.844 polarization case because of the higher boundary potential. The Valdes-Mouldon acetate solution iron in 1 Exp. Data ( \.. \. -. \ I * pyition (mm) I I I -4-2 1. 4 6 - -6---.-.-$0... -0.25. t Fig. 10. Experimental and predicted potentials for 1.244 V polarization
Crevice corrosion for passive and active metals 15 5 f 5 l_ s OS5 : 2 g 0.1 : E s! 0.05 - Valdes-Mouldon 9 iron in acetate solution 0.01 I 0.01 0.05 0.1 0.5 1 5 10. position (mm) Fig. Il. Predicted anodic current density at metal/solution interface for the 1.244 V polarization corrosion rate again peaks a short distance after this transition and then drops off sharply deeper into the crevice. The predicted crevice solution composition (not shown) is virtually identical to the lower polarization case shown in Fig. 9. Vakdes experiments 9-suljiiric acid system Valdes also performed experiments with suifuric acid electrolyte in the same machined crevice. The sulfuric acid solution was a 0.001 M sulfuric acid at 25 C. The reactions involved are: Fe*+(aq) + SO*-(aq) + FeSOd(aq); 4 (38) Fe*+(aq) + SO*-(aq) e FeSOd(s); 4 (39) and equation (18). These reactions involve five aqueous species as well as water. The only reaction assumed to occur at the metal/solution interface is anodic oxidation of iron to ferrous ion. The boundary conditions at the mouth of the crevice are 0.002 molal H+ and 0.001 molal S04*-. The simulation was run for 30min using the Bulirsch-Stoer solver with an input tolerance of 10m4. Two runs were made using different methods for determining the activity coefficients: one assuming the ideal case (activity coefficients all set to 1) and one using the B- dot equation. A third run was made using the B-dot equation and omitting the ferrous sulfate complexation reaction from the input file. This is referred to in the figures as the complexes removed case. Discretization into finite difference nodes was over 100 nodes as in the acetate system. The simulations were run for a corrosion potential of 0.844 V (SHE). Since the model is only valid for inside the crevice, an experimental value for the potential at the mouth of the crevice was used as the fixed boundary condition; this value was -O.O36V(SHE). A relaxation factor of 0.2 was used for the prediction of the potential in solution. Valdes measured the anodic current density for uncreviced specimens of iron as a function of potential for both of the electrolyte solutions he studied; the results for the sulfuric acid
16 J.C. Walton et al. Valdes-Mouldon 9 iron in sulfuric acid -.- Complexes. Removed - Laplace Eq. Fig. 12. Experimental and predicted potentials for sulfuric acid system. solution (not shown but similar to results for the acetate system) were discretized to represent kinetics at the metal/solution interface. Figure 12 compares the experimental results and the predicted results with and without the aqueous complexation reaction. A third simulation used the ideal assumption for activity coefficients. The simulated anodic current density (not shown) predicts a transition from passive to active corrosion just outside of the crevice. The data represent one experiment and most values are near the limiting potential, thus less scatter appears in the data and the agreement between experimental and modeled potentials is very good. CONCLUSIONS A reactive transport model optimized for crevice corrosion is described. The computer code implementing the model is general in format, allowing specification of species and reactions considered at time of execution. Comparison of modeled and experimental data for three different systems with iron and stainless steel gives agreement ranging from approximate (Alavi and Cottis data) to very good (Valdes data). This likely represents ranges in the complexity of the experimental systems and the completeness of the experimental data set required for modeling the system. The Alavi and Cottis data does not contain electrochemical kinetic data required by the model, leading to imprecise predictions probably associated with changes in the passive current density with time during the course of the experiment. In contrast, the Valdes data for the chemically simpler iron metal, is sufficiently complete to allow simulations based only upon data obtained external to the crevice. The electrode kinetics were obtained from polarization curves drawn on uncreviced specimens, the initial and boundary conditions were based upon initial conditions and boundary potential measured in the experiments, and the crevice geometry was carefully controlled in the experiments. Transport and thermodynamic data were obtained from the literature. The resulting model predictions exhibit very good agreement with the experimental data. The Valdes data was used to test the importance of activity coefficient treatment on modeling results. For the system modeled, the activity coefficients are relatively unimportant. Instead, full description of chemical reactions in solution (complex
Crevice corrosion for passive and active metals 17 formation and hydrolysis) and the electrochemical reactions at the metal/solution interface are the most important factors required to accurately describe the chemical environment inside the crevice and the potential distribution. Acknowledgemenrs-This paper was prepared to document work performed by the Center for Nuclear Waste Regulatory Analyses (CNWRA) for the Nuclear Regulatory Commission under Contract No. NRC-02-93-005. The activities reported here were performed on behalf of the NRC Office of Nuclear Material Safety and Safeguards, Division of Waste Management. The paper is an independent product of the CNWRA and the University of Texas at El Paso and does not necessarily reflect the views or regulatory position of the NRC. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. REFERENCES S.M. Sharland. Corros. Sci. 27,289 (1987). M.K. Watson, Ph.D. Thesis, University of Saskatchewan (1989). M.K.Watson and J. Postlewaite, Corrosion 46, 522 (1990). S.M. Sharland, Corros. Sci. 33, 183 (1992). S.M. Sharland. C.P. Jackson and A.J. Diver, Corros. Sci. 29, 1149 (1989). J. Walton, Corros. Sci. 30, 915, (1990). J. O M Bockris and M.D.N. Reddy, Modern Elecrrochemisfry, Plenum Press, New York (1977). A.C. Hindmarsh, Scienrific Computing (ed. R.S. Steplemen et al.), p. 55. North-Holland, Amsterdam (1983). W.H. Press. B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge (1986). J.C. Walton and SK. Kalandros, Mariana-A Simple Equilibrium Solver, CNWRA 92-020, Southwest Research Institute, San Antonio, TX (1992). H.C. Helgeson, Amer. J. Sci. 729 (1969). W. Stumm and J.J. Morgan, Aquatic Chemistry, An Introduction Emphasizing Chemical Equilibria in Natural Waters, John Wiley and Sons, New York (1981). B.R. Staples and R.L. Nuttal, J. Phys. Chem. Ref. Data, 385 (1977). A. Alavi and R.A. Cottis, Corros. Sci. 27, 443 (1987). T.J. Wolery, K.J. Jackson, W.L. Bourcier, C.J. Bruton, B.E. Viani, K.G. Knauss and J.M. Delany, Chemical Modeling ofaqueous Systems II(ed. D.C. Melchior and R.L. Bassett), ACS Symposium Series 416. American Chemical Society. Washington, DC (1990). 16. Y.-H. Li, and S. Gregory, Geochim. Cosmochim. Acra, 38, 703 (1974). 17. V. Vitagliano and P.A. Lyons, J. Amer. Chem. Sot. 78, 4538 (1956). 18. Z. Szklarska-Smialowska, Corrosion, 27, 223, (1971). 19. A. Valdes-Mouldon, Ph.D. Thesis. The Pennsylvania State University (1987). 0 a, A AC AS cl D, F g k z N,, APPENDIX ion size parameter for species n coefficient of Debye-Huckel limiting law cross-sectional area of crevice or pit cavity (nr2 for a pit, wg for a crevice) corroding metal surface area concentration of species n in aqueous solution. diffusion coefficient of species n Faraday constant crevice gap current density at any point in the crevice solution. current density at the metal/solution interface the length (i.e. depth) of the crevice or pit flux of species n in solution
J.C. Walton et al. the flux of species n into solution at the metal/solution interface the corroded perimeter (27rr for a pit, 2~ for a crevice) pit radius gas law constant the rate at which species n is produced through chemical reactions per unit volume in solution absolute temperature volume of aqueous solution at the node crevice width (assumed as one unit of length with cr>>g) the position in the crevice measured from the mouth distance between nodes charge of ionic species n weighting factor indicating the relative rate of production of species n activity coefficient of species n electrochemical potential of species n chemical potential of species n electrostatic potential in solution electrostatic potential of the metal Subscripts in, out nodal interface in the negative and positive x directions m final node at base of cavity n species index j node index
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