Further Modeling of Chloride Concentration and Temperature Effects on 1D Pit Growth

The product of diffusion coefﬁcient and saturation concentration of metal ions (D M + · C tot ) is associated with steady state dissolution of metal in corrosion pits. In our previous paper, D M + · C tot was modeled for one dimensional (1D) pit dissolution of super 13Cr stainless steel (S13Cr) by taking into account the common ion effect, as well as viscosity and temperature inﬂuences. However, the modeled values were only half of the experimentally-measured D M + · C tot , implying that D M + and/or C tot were underestimated. To improve the modeling of D M + · C tot , the effects of complexation and electromigration were additionally considered, which led to the deﬁnition of effective diffusion coefﬁcient ( D eff ) applicable for the diffusion of metal ionic species under the combined effect of viscosity, temperature and electric ﬁeld inside a growing 1D pit. The newly modeled D eff · C tot values were very close to the experimental values, validating the modeling approach described in this paper. the

Dissolution of metal under steady state diffusion can be described by the modified Fick's first law, which relates the limiting current density (i lim ) with the product of diffusion coefficient (D M+ ) and concentration difference ( C M+ ) of the metal ion through the diffusion length (L): where F is the Faraday constant, and n is the charge of the metal ion.
For the dissolution of metal in a one dimensional (1D) pit electrode, the concentration of metal ion outside of the pit is commonly assumed to be zero, [1][2][3][4][5][6][7][8] so that C M+ is simply the concentration of metal ion at the pit bottom (C M+ ). It is also usually supposed that L is equal to the pit depth (δ), [9][10][11][12] although the diffusion of metal ions may extend out of pit, forming additional layer of hemispherical diffusion and causing L to be longer than δ. 11 The effect of this additional diffusion layer on i lim was found to diminish if δ of 1D pit with 1 mm diameter was greater than 0.4 mm (aspect ratio of pit depth/mouth: 0.4) for SS 304 11 and super martensitic stainless steel. 10 More recent study on 1D pit growth of SS 316L, however, has suggested that the length of additional diffusion layer (L') is approximately 40% of pit diameter, and δ should be much larger than L' to apply the L ≈ δ condition for the calculation of i lim . 13 In a previous paper, the authors suggested that i lim in 1D pit electrode of super 13Cr stainless steel (S13Cr) can be related to the sum of D M+ · C M+ from each metal component, if the transport of all metal ions is under diffusion control. 14 In this case, i lim can be expressed as: [2] Here each C M+ is replaced by the concentration of metal ions at the pit bottom (C Fe2+/pb , C Cr3+/pb and C Ni2+/pb ). The content of Mo in S13Cr is small (2 wt% or 1.16 at%), so it was omitted for the calculation of i lim . For Eq. 2, three conditions were postulated: 1) The diffusion coefficients of all metal ions are the same (D Fe2+ = D Cr3+ = D Ni2+ = D M+ ), 2) Fe 2+ is the major dissolved species, and the precipitating metal salt at the pit bottom is considered to be FeCl 2 at which point the solution is saturated with the metal ions and supporting anion (Cl − ).
In situ X-ray analysis of the metal salt precipitated on a dissolving stainless steel found that the precipitated salt was mostly composed of Fe and Cl. 12 Based on this, it is reasonable to assume that the precipitating metal salt is FeCl 2 . The concentration of Fe 2+ for the saturated solution in equilibrium with FeCl 2 salt is designated as C sat (C Fe2+/pb = C sat ). 3) Assuming congruent dissolution, the ratios of C sat :C Cr3+/pb :C Ni2+/pb are identical to the mole fractions of Fe, Cr and Ni (X Fe :X Cr :X Ni ) in S13Cr, and C Cr3+/pb and C Ni2+/pb can expressed in terms of C sat using X Fe , X Cr and X Ni : Using Eqs. 3a and 3b, the sum of C sat , C Cr3+/pb and C Ni2+/pb simplifies to: where C tot is the total concentration of metal ionic species at the pit bottom. Note that C tot here defined by C sat /X Fe is only applicable for Fe-based alloys where X Fe is much higher than the fraction of other components (X Cr and X Ni ) so that the congruent dissolution of the alloy would precipitate FeCl 2 upon the saturation of Fe 2+ (C sat ). By applying these conditions and Eq. 4 to Eq. 2, Eq. 5 is obtained. [5] whereñ is the average charge of metal ions dissolving from S13Cr, calculated by adding up the product of charge and mole fraction of metal components (2X Fe + 3X Cr + 2X Ni ). The composition of S13Cr gives X Fe :X Cr :X Ni = 0.81:0.14:0.05 without considering Mo, which yieldsñ = 2.14 and C tot = 1.23C sat . The concentrations of metal ions measured in the saturated pit solution for Fe-18Cr-13 Ni alloy at room temperature resulted in C tot = 1.46C sat , which is the same value determined by assuming congruent dissolution of the alloy. 12 Therefore, the assumption of congruent dissolution, which was also adopted by others, 15,16 seems to be reasonable. Assuming that all of the anodic charge (Q a ) was associated with metal dissolution and a planar electrode assembly, δ can be calculated from: δ = Q a M n FρA [6] where M and ρ are atomic weight and density of S13Cr, and A is surface area of 1D pit electrode. Combining Eqs. 5 and 6 gives Eq. 7, where both i lim and Q a are experimentally measurable and allow for

C824
Journal of The Electrochemical Society, 163 (14) C823-C829 (2016) determination of experimental value of D M+ · C tot : D M+ · C tot (experimental) was found to decrease with increasing concentration of bulk NaCl solution ([NaCl] b ), 9,14,17 indicating that increasing [NaCl] b must decrease D M+ and/or C sat . Increasing the concentration of other chlorides such as LiCl and FeCl 3 were also found to decrease experimental D M+ · C tot of different stainless steels. 18 If Na + ions are present at the pit bottom, the common ion effect could decrease C sat . 3,17,19 This effect would be more pronounced as [Na + ] at the pit bottom ([Na + ] pb ) increased. Meanwhile, D M+ was estimated to decrease with increasing the viscosity of pit solution. 3,17,20 The positive correlation of solution viscosity and concentration implies that D M+ would decrease with increasing concentration of pit solution.
Based on the assumption that increasing [Na + ] pb can cause stronger common ion effect and higher viscosity of pit solution to decrease C sat and D M+ , our previous paper suggested a modeled D M+ · C tot (designated as modeled D M+ · C M+ in the paper) dependent on the pit chemistry and viscosity. 14 The previous study (including the erratum published after the original publication) showed that the modeled D M+ · C tot was about one half of the experimental D M+ · C tot . 21 A reconciliation of the modeled D M+ · C tot with the experimental value is described here, in which the influences of the common ion effect, relative viscosity and electromigration are addressed.

Modeled Concentration Gradients of 1D pit and C sat
To find the contribution of the common ion effect on C sat , [Na + ] pb was assumed to have two extreme cases; [Na + ] pb = 0 (zero [Na + ] pb ) and the maximum [Na + ] pb possible (max. [Na + ] pb ) that neither precipitates NaCl nor surpasses [Na + ] b , i.e. Na + concentration of the bulk solution. The modeled concentration gradients of 1D pits associated with these two conditions are shown in Fig. 1. In the modeled gradients, the total concentration of metal ionic species (C Me/tot ) is the sum of simple and complexed metal ions and it linearly decreases from the saturation level (C tot ) at the pit bottom to zero at the pit mouth. On the other hand, [Na + ] inside the pit decreases from [Na + ] b at the pit mouth to either zero or to the max. [Na + ] pb value at the pit bottom. According to these conditions, the concentration profiles for simple/complexed metal ions and Na + in Fig. 1 can be written as linear functions of distance (x, x = 0 and δ for pit mouth and bottom): The speciation of cations in the pit is crucial to find the total positive charge that determines the gradient of [Cl − ] for electro-neutrality. Hydrolysis and chloride complexation of metal ions produce different types of metal complexes and H + , as follows: where n = 2 and a = 1 or 2 for ferrous and nickel ions, n = 3 and a = 1, 2 or 3 for chromium ion. In Eq. 10a, the net positive charge on both sides is equal to n, indicating hydrolysis would not change the total positive charge inside the pit. Chloride complexation of metal ions in Eq. 10b, however, decreases the positive charge from n to (n-a) and consumes a[Cl − ]. The average charge of ferrous ionic species (n i/Fe ) after chloride complexation is: where X FeCl+ and X FeCl2 are the fractions of [FeCl + ] and [FeCl 2 ] with respect to initial [Fe 2+ ], respectively. The average charge of chromium and nickel ionic species (n i/Cr and n i/Ni ) can also be defined similarly using the fraction of corresponding metal complexes (X CrCl++ , X CrCl2+ , X CrCl3 , X NiCl+ and X NiCl2 ). The fractions of different complexes from the dissolution of each metal chloride, calculated using OLI Stream Analyzer software (version 9.0), are summarized in Table I, showing that X FeCl2 , X CrCl3 and X NiCl2 are insignificant and can be ignored. The OLI software is capable of calculating different chemical properties of complex aqueous systems based on an extensive thermodynamic database, which is described in detail elsewhere. [22][23][24] The effective charge (n eff ) of metal ions and their chloride complexes can be defined by taking X FeCl+ , X NiCl+ , X CrCl++ and X CrCl2+ into account: where 0.81, 0.14 and 0.15 in a 1 × 3 matrix account for the concentration ratio of produced Fe 2+ , Cr 3+ and Ni 2+ from congruent dissolution of S13Cr. Note that X FeCl2 , X CrCl3 and X NiCl2 are omitted for n i/Fe , n i/Cr and n i/Ni in Eq. 12. If metal ions do not form any complexes (i.e. the fractions of all complexes are zero), n eff is identical toñ. More recently, the speciation of pit ions predicted using OLI software showed  Table I

. Calculated fractions of chloride complexes for a) FeCl 2 + H 2 O, b) CrCl 3 + H 2 O and c) NiCl 2 + H 2 O at 85
that the change in pit concentration appeared to influence the fraction of complexation. 7,25 Nonetheless, here the fractions are postulated to be independent of pit concentration, resulting in an invariable n eff along the concentration gradient inside the pit. The calculation of n eff can be further simplified by assuming that the fractions of all metal complexes are equal to X cpx = X FeCl+ = X NiCl+ = X CrCl++ = X CrCl2+ , which modifies Eq. 12 to: n eff = 2.14 − 1.28X cpx [13] The concentration profile of [Cl − ] (C Cl-(x)) in Fig. 1 is then written as: It is also confirmed that the charge neutrality condition where C sat ([Na + ] pb ) is the value of C sat decreased by the presence of [Na + ] pb . The upper limit of [Na + ] pb is predicted to be 0.83 M because larger values resulted in NaCl precipitation according to the OLI software. The maximum common ion effect would then be possible at [Na + ] pb = 0.83 M, which would decrease C sat to 4.47 M. This is in agreement with the analysis of C sat for [NaCl] b = 3 M in the previous paper, where the input of [Na + ] pb = 3 M was estimated to produce 0.83 M of aqueous [Na + ] pb and 1.9 mol of NaCl precipitates with C sat = 4.47 M, which is the same C sat obtained here by inputting [Na + ] pb = 0.83 M. The calculations in Table II show that the concentration of metal ions decreases by less than 10% due to the common ion effect.

Effective Diffusion Coefficient of Metal Ions
The previous analysis 14  The effect of viscosity associated with concentration increase can also be predicted using the Stokes-Einstein equation: 39 where D M+/T (C) and η T (C) are the diffusion coefficient of metal ion and dynamic viscosity in the solution with concentration of C at T, and η r/T is relative viscosity at T defined by η T (C)/η • T . Eq. 17 was previously adopted to calculate D M+/298 for pit diffusion using D • M+/298 and η r/298 estimated from the average concentration of pit solution. 3,11,20 The chemistry of pit solution for varying depth can be characterized as the concentration of aqueous solution containing different amounts of metal chlorides and NaCl. Using Eqs. 8 and 9, the amounts of FeCl 2 , CrCl 3 , NiCl 2 and NaCl for different pit distance (x) were determined as 0.81C Me/tot (x), 0.14C Me/tot (x), 0.05C Me/tot (x) and C Na+ (x), respectively, which were then taken as inputs for OLI software to predict η r/358 . The plots of η r/358 vs. x shown in Fig. 2 were fitted to quadratic equations (also included in Fig. 2) with good agreement, and the equations were determined to be the functions of η r/358 and x (η r/358 (x)). The average pit viscosity at 85 • C (η r/358 ) along the entire pit depth from x = 0 to x = δ can be calculated from: Values of η r/358 for zero and max. [Na + ] pb conditions in 0.1-3 M [NaCl] b are summarized in Table III. Electromigration, another factor that may affect transport, can be addressed by adding the effect of the potential field on the flux of metal ion: 1,41 [19] where C M+ (x) is the function of metal ion concentration and x, R is the gas constant and is the solution potential inside the pit. In a growing 1D pit, Cl − migrates from the bulk solution to the pit bottom to balance the dissolving metal ions, 42 [20] where dC Cl-(x)/dx is obtained by differentiating C Cl-(x) in Eq. 14:  Meanwhile, the total amount of Cl − (m Cl-) in the pit is calculated by multiplying the surface area of 1D pit and integral of Eq. 14 for x = 0 to δ: [22] J Cl-/net is the flux contributing to the increase of m Cl-during pit growth and associated with the time derivative of m Cl-: [23] A unitless coefficient (c r ) is defined to satisfy J Cl-/net = c r · J Cl-/diff . Taking the diffusion flux term of Eq. 20 and using Eqs. 21 and 23, c r can be expended: [24] Laycock and Newman found that dδ/dt in 1D pit growth of SS 304 under steady state diffusion is associated with D M+ and δ: 15 dδ dt where δ 2 /D M+ was defined as characteristic time (τ) for metal ions to diffuse out of pit and c g is a unitless constant associated with the increase of pit depth during τ and was experimentally determined to be 0.03 (3% increase of pit depth) in 1 M NaCl solution at room temperature. The value c g is speculated to depend on experimental conditions including temperature, type of metal and the concentration of bulk electrolyte. Rearranging Eq. 25 in terms of dδ/dt yields: Applying Eq. 26 to dδ/dt of Eq. 24 gives: The self-diffusion coefficient of Cl − (2 × 10 −5 cm 2 /s) is about 2.8 times the predicted D • M+ (0.7 × 10 −5 cm 2 /s) at 25 • C, and assuming this relationship is valid for pit diffusion, D M+ /2D Cl-is approximated to be 0.18. The value of c r calculated for different [Na + ] b and possible c g (0.02 to 0.08) is plotted in Fig. 3, where the largest c r is 0.023 for [Na + ] b = 3 M and c g = 0.08, indicating J Cl-/net is only 2.3% of Eq. 28 can be rearranged in terms of d /dx: dx [29] This electric field (d /dx) acting on Cl − will also drive the migration of metal ions toward the pit mouth so that Eq. 29 can be applied to Eq. 19 along withñ, n eff and C Me/tot , resulting in Eq. 30 which takes the same form to the flux equation derived by Landolt. 43 The underlying assumption made for Eq. 30 is that all simple/complexed metal ions share the same diffusion coefficient (written as D M+ ) so the transport of metal ionic species can be described using single D M+ . The diffusion coefficients of most complexed metal ions are not readily available, but the self-diffusion coefficients of FeCl + , NiCl + CrCl ++ and CrCl 2 + were previously estimated to be 0.8 × 10 −5 cm 2 s −1 at room temperature, 27 which is not effectively different from D • M+ for Fe 2+ , Cr 3+ and Ni 2+ listed earlier in this paper and rationalizes the use of a single D M+ for Eq. 30.
C Me/tot (x)/C Cl-(x) in Eq. 30 is the term assigned for the concentration of simple/complexed metal and chloride ions that are subjected to migration. Using Eqs. 8, 14 and 21, the product of C Me/tot (x)/C Cl-(x) and dC Cl-(x)/dx can be modified to: [31] where k = (n eff C sat /X Fe + [Na + ] pb -[Na + ] b ) and has the units of concentration. The averaged concentration gradient contributing to migration (dC mig /dx) is calculated from Eq. 31 for the entire pit depth (from x = 0 to x = δ): where α is a unitless term. By applying the derivative of C Me/tot (calculated from Eq. 8) and Eq. 32 to Eq. 30, Eq. 33 for i lim is obtained.
Using Eq. 17, D M+ can be replaced with D • M+ /η r , where η r is defined in Eq. 18. Eq. 33 can then be rewritten as: Gaudet et al. used the term 'effective diffusion coefficient (D eff )' which incorporates the contribution of solution viscosity and electromigration on the diffusion-based transport of metal ions. 11 D eff then can be defined from Eq. 34: where the αn eff in the numerator corresponds to the contribution of electromigration on D eff . αn eff is plotted vs. [Na + ] b in Fig. 4 for [Na + ] pb = 0 at 85 • C with varying X cpx conditions, where the relationship of n eff and X cpx is defined in Eq. 13. Electromigration plays a decreasing role as [Na + ] b increases. Landolt described the effect of electromigration on i lim for a diffusion layer where both metal ion and supporting anion are depleted at one boundary (electrodeposition), and the contribution factor of migration in this case was derived to be n+/n-, where n+ and n-are the charge of metal ion and supporting anion, respectively. 43 For 1D pit growth in a bulk solution with near-zero concentration, the pit mouth can be considered as a depletion boundary, so the migration factor could be found using n+/n-, which is n eff for simple/complexed metal and chloride ions. This is in good agreement with the trend that αn eff reduces to n eff as [Na + ] b (also [Cl − ] b ) approaches to zero, predicted by Eq. 32. However, the value of αn eff is close to zero for [Na + ] b higher than 12 M, which is comparable to the concentration of Cl − at the pit bottom To estimate the possible range of n eff , two boundary conditions for X cpx were chosen as X cpx = 0 (no complex formation) and X cpx = 0.1 (10% complexation). The values of n eff , α and D eff/358 for different [NaCl] b , [Na + ] pb and X cpx are summarized in Tables IV and V, where  η r values are taken from Table III. At room temperature (RT), D eff was reported to range from 7 × 10 −6 to 9.2 × 10 −6 cm 2 s −1 for diffusion-limited dissolution of pure Fe 44 and different stainless steels 5,11,18 in acidic and neutral chloride solutions. For the pit dissolution of S13Cr at RT, D eff calculated by Eq. 35 should be close to the reported D eff . To make a fair comparison, the following conditions are considered for the dissolution of S13Cr  (Table IV), can be used for the pit dissolution at RT as well. Applying αn eff = 0.79 and η r = 2.15, the calculated D eff for S13Cr pit dissolution in the considered condition is 8.8 × 10 −6 cm 2 s −1 , which is comparable to 8.24 × 10 −6 cm 2 s −1 reported for SS 304 pit dissolution in 1 M NaCl bulk solution. 11

Comparison of Experimental and Modeled D M+ · C tot
The modeled D M+ · C tot values of S13Cr at 85 • C can now be obtained by the product of C sat /X Fe (C sat values in Table II and X Fe = 0.81) and D eff/358 (in Tables IV and V) as written in Eq. 37.
D M+ · C tot (modeled) = D eff/358 · C sat X Fe = 1.23D eff/358 · C sat [37] Recall that the experimental D M+ · C tot can be determined from the measured i lim and Q a using Eq. 7.  (Fig. 5a). This supports the idea that Na + ions are depleted at the dissolving pit bottom. 17 Then assuming [Na + ] pb = 0, the experimental D M+ · C tot values lie between the modeled values with X cpx = 0 and 0.1 (Fig. 5b), implying that the degree of complexation of metal with Cl − ions in pits is lower than 10%.
In the modeling of D M+ · C tot presented here, the decrease in D M+ (shown in Tables IV and V) is much more pronounced than C tot (in Table II) with increasing [NaCl] b . This approach is clearly distinguished from other modeling of D M+ · C sat , which emphasized the effect of bulk Cl − concentration on decreasing C sat and considered D M+ to be invariant. [17][18][19] Laycock et al. assumed that C sat could further decrease by increasing the fraction of Cl − complexed non-ferrous metal ions which would allow more ingress of Cl − into the saturated pit solution. 19 To fit the [NaCl] b -dependent decline of experimental D M+ · C sat solely by decreasing C sat , they chose a high extent of complexation, where 56% of the non-ferrous metal ions were assumed to be complexed. The C sat in a 1D pit of Fe-18Cr-13Ni alloy in a constant bulk Cl − concentration was measured using in situ X-ray microprobe, 12 but evidence of decreasing C sat with increasing bulk Cl − concentration is hard to find. Tester and Isaacs predicted that a decrease in C sat would be probable for bulk Cl − concentration higher than 5 M but would not be pronounced for bulk Cl − concentration less than 3 M. 3 Considering that [NaCl] b in this modeling was 3 M or lower, C sat of S13Cr may not notably decrease with an increase in bulk Cl − concentration. It is then reasonable to assume that the decrease in D M+ with increasing [NaCl] b would dominate the decrease in D M+ · C tot as discussed here.

Conclusions
An alternative approach to model D M+ · C tot for diffusion-limited dissolution of S13Cr in 1D pit electrode was discussed and applied to fit previously-measured experimental D M+ · C tot for 0.1 -3 M NaCl at 85 • C. With the aid of OLI software, C sat (= X Fe · C tot ) and relative viscosity (η r ) associated with diffusion of metal ions were estimated assuming zero and max. [Na + ] pb conditions for the modeled concentration profiles of pit solution. Effective diffusion coefficient (D eff ) was defined to incorporate the combined effects of temperature, electromigration, average η r of pit solution (η r ) and the fraction of complexed metal ions (X cpx ). The details can be summarized as follows.
1. C sat is predicted to decrease linearly with increasing [Na + ] pb up to 0.83 M, where the maximum common ion effect lowers modeled C sat less than 10% from 4.88 to 4.47 M. 2. The temperature and viscosity dependencies of D M+ were modeled using the Stoke-Einstein equation, where D M+ was inversely proportional to η r and increased by 3.21 times for the temperature increase from 298 to 358 K. 3. The effect of electromigration on the transport of metal ions was derived based on the concept that the [Cl − ] profile inside the pit is maintained by the balance of diffusion and migration on Cl − . The formation of complexed metal ions determines the effective charge of metal ionic species (n eff ) associated with electromigration. The contribution of migration was found to decrease with increasing [Na + ] b . The calculated D eff is comparable to the reported D eff at room temperature. 4. The product of D eff/358 and C sat /X Fe (modeled D M+ · C tot ) for different [Na + ] pb and X cpx was compared to experimental D M+ · C tot . The fitting analysis on modeled and experimental D M+ · C tot suggests that Na + is not likely to exist at the pit bottom during the pit growth. The fitting results also suggest that the degree of complexation of metal and Cl − ions is low (less than 10%). A good fit found between experimental and modeled D M+ · C tot supports the validity of modeling method discussed in this paper.