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Reaction distribution in a corroding pit
Electrochimica AC%&Vol. 38. No. 12, pp. 1619~1659. Printed in Great Britain.
REACTION
1993 0
DISTRIBUTION
IN A CORRODING
56.00 + wJo 00s4686/93 1993. Pergamoa Prws Ltd.
PIT
MARK W. VERBRUGGE,*DANIELR. BAKERYand JOHN NE\KMAN~ Physical Chemistry Department, General Motors Research, Warren, MI 48090-9055, U.S.A. 7 Mathematics Department, General Motors Research, Warren, MI 480909055, U.S.A. $ Materials Sciences Division, Lawrence Berkeley Laboratory, and Department of Chemical Engineering, University of California, Berkeley, CA 94720, U.S.A. l
(Received 14 October 1992; in revisedform 8 February
1993)
Abstract-An analysis of the reaction distribution over a corroding pit is presented. If the liquid-phase conductivity and the interfacial rate constants for the metal-dissolution reaction are known, the results of this secondary current distribution problem can be used to estimate the maximum rate of pit growth. The approach taken involves the confonnal mapping of the pit geometry onto a new coordinate system; the finished transformation is shown to remove the singularity at the pit edge and provide a bounded region over which only a limited number of mesh points are required to perform accurate numerical analysis. Numerical calculations for the pit dissolution rate subject to linear and Tafel polarization are fitted to simple expressions; as a consequence, the results of this work can be used without having to resort to numerical methods. For the linear polarization analysis, the results are generalized in a manner that allows one to consider an arbitrary number of reactions on the pit surface. Key words: conformal mapping, pitting corrosion,
f” E ET
F i
'0 i G n
Iv
pit radius,
modeling, current distribution.
NOMENCLATURE
r
cm
#
FfRT, v-l ext4 - 5)
Tafel correlation function Faraday’s constant, 96487 C/equivalent current density normal to the pit surface, Arm-2 current density averaged over the pit surface, AlXl-2 exchange current density, A ctn2 linear polarization dimensionless group, tan(j) = J = (a, + a&I0 a/K reaction 8 unit normal vector pointing into solution total number of electrochemical reactions on pit surface radial coordinate, cm gas constant, 8.3143 J mol-’ K-l dimensionless solution resistance dimensionless electric potential [equations (8) and (911 temperature, K electrode potential, V rla cylindrical coordinate, cm zla anodic transfer coefficient cathodic transfer coefficient dimensionless average current, a* ~5~~~ a/K coordinate (Fig. 1) spherical coordinate (Fig. 1) conductivity, S cm- 1 unbounded coordinate resulting from conforma1 transformation coordinate (Fig. 1)
Tafel-polarization dimensionless group, tan(T) = a._& exda.JV)a/~ electric potential, V
Subscripts linear polarization L on the pit surface Tafel polarization ;
INTRODUCI’ION It has been speculated that as much as four per cent of the gross national product in a number of industrialized nations is balanced by costs attributable to corrosion[l]. Although it is well known that passivating metals commonly used as corrosion-resistant coatings generally undergo localized dissolution from hemispherical pits, referred to as pitting corrosion[2,3] rather than by uniform surface dissolution, no specific analytical tools exist that allow one to determine the reaction distribution over the pit surface, which could be used to determine the dissolution rate and the failure time of a corrosionresistant coating. Szklarska-Smialowska’s[2] text is helpful in organizing the literature associated with pitting corrosion. A number of theoretical studies of pitting corrosion have appeared in the literature. The most commonly employed geometry used for simulating the corroding pit corresponds to a cylinder of uniform cross section and a central axis perpendicular to the metal’s surface[4-7-J. The metal surface and the walls of the cylinder are taken to be insulating; the eleu trode surface undergoing corrosion is located at the bottom of the cylinder, parallel to the metal surface. The electrolyte immediately outside the cylinder is
1649
M. W. Vaaaauoo~ et al.
1650
8
=o
I 4z Bulk solution
Fig. 1. Pit geometry. The problem is axisymmetric. Left schematic-pit problem and cylindrical coordinates r and z. For spherical coordinates, the coordinate 0 is als_ogiven. Right schematicpit problem in new coordinates q and (.
taken to be well mixed and at uniform potential. This assumed geometry yields a one-dimensional analysis and has proven useful for gaining qualitative understanding of pitting phenomena. Twodimensional analyses of corroding hemispherical pits have been completed by Newman et ~I.[83 and by Harb and Alkire[9]. The conformal transformation presented in this work, and described briefly in the last paragraph of the Introduction, makes it possible to perform more accurate two-dimensional analyses with reduced computational expense. In this study, we provide a theoretical analysis for a hemispherical pit in contact with a well-mixed electrolyte, representing a secondary current distribution analysis. Since the electrolyte is assumed to be wellmixed, the calculated corrosion rates represent an upper boundary for most corrosion processes, as the additional transport resistances not included in the model equations would slow the rate of dissolved metal ions leaving the pit. The current distribution results show that for very small pits, uniform pit dissolution takes place and the hemispherical geometry is maintained. For larger pits, however, the current density at the pit edge can be much larger than that of the pit center. Thus the analysis allows one to determine when nonuniformities in the current distribution are expected to be significant, in which case the pit geometry would begin to change from its original hemispherical shape, as has been observed experimentally[9]. The electric potential and reaction distribution over the pit surface are given in terms of a single dimensionless parameter; the resulting plots can be used to determine the local rate of corrosion over the pit surface if all the electrochemical reactions are described by linear current-potential relations, as is usually the case with corrosion resistant materials that are designed to minimize the metal dissolution rate. For both linear and Tafel polarization, analytic
expressions are given for the average current density, electrode potential relationship, which should be useful for the interpretation of accelerated corrosion test data and experiments designed to determine the pitting potential[lO-121 by means of current passage. The mathematical problem governing the secondary current distribution involves the solution to Laplace’s equation for the electric potential along with a current-potential relation at the electrode surface and appropriate insulator and bulk-solution conditions for the other boundaries. For linear or Tafel polarization, the results &aii be given in terms of one dimensionless group. Thus.the dimensionless average current densities for the two problems are given in tabulated form, removing the need for future researchers to perform numerical calculations in order to use the solutions presented in this work. A polynomial fit to the tabulated results is also provided. It should be noted that the secondary current distribution problem subject to a linear polarization boundary condition shares a one-to-one correspondence with the mathematical description of: (1) steady-state mass-transport problems characterized by Fickian diffusion and linear boundary conditions with constant coefficients, such as is observed typically in the analysis of ionic diffusion to an electrode that is imbedded in an otherwise insulating wall (the dilute solution containing an excess of inert electrolyte); and (2) steady-state heat-transport problems characterized by Fourier’s law of heat conduction and linear boundary conditions with constant coefftcients, such as typically observed in the analysis of heat transfer through solids to a surface imbedded in an otherwise insulating wall, as described on pp. 18 and 19 of Carslaw and Jaeger[13]. More generally, all systems described by Laplace’s equation for the dependent variable and a boundary condition that is linear in the dependent variable and contains
Reaction distribution in a corroding pit constant coefficients, as is the case for the secondary current distribution problem subject to linear polarization, are mathematically equivalent; the mathematical results can be used interchangeably. Because of the equivalent nature of the secondary current distribution problem to ionic diffusion in the presence of excess inert electrolyte, it is possible, for example, to apply analytical methods derived for the analysis of the secondary current distribution associated with a disk electrode[14, 151 subject to linear polarization to treat the equivalent problem concerning ionic diffusion to a microdisk electrode[ 16, 173. The approach we use for treating the pit problem is to develop a conformal transformation to recast the problem as shown in the left schematic of Fig. 1, with a singularity at the electrode-insulator edge, to a half-infinite strip devoid of singularities; a subsequent transformation is then used to collapse the unbounded coordinate into a new coordinate, resulting in a recasting of the problem to a box geometry, which is illustrated by the right schematic in Fig. 1. The use of conformal mapping for the generation of a new coordinate system more amenable to numerical analysis is a well-established practice[ 181. Conformal and closely related transformations used to remove electrodeinsulator singularities have been shown to simplify greatly subsequent numerical analyses in a number of electrochemical studies. (For examples, see [ 19-221.)
1651
trode potential relative to a reference electrode of the same kind located in the bulk solution (a constant);f refers to F/RT, where F is Faraday’s constant, R is the gas constant and T is the absolute temperature; and the kinetic constants iO, a, and a, refer to the exchange current density and the anodic and cathodic transfer coefficients, respectively. After solving the above equations, the average current density iavp can be obtained by integrating the normal current density over the pit surface, 1
=r”’
27ca2
i dA,
(7)
where dA is the differential surface area associated with i. The remainder of this section details the recasting of-a dimensionless form of equations (l)-(7) in the rl- [ coordinate system shown schematically in Fig. 1. Nondimensionalization
Using the following definitions allows us to write the governing equations in a compact form and identify important dimensionless groups: tanCj) = J = (a, + a,)fifioa/K, t&W
= a,fiO
cvb, fV)a/~,
0 = aV, X = r/a,
GOVERNING COORDINATE
Z = z/a.
EQUATIONS AND TRANSFORMATION
Governing equations For secondary current distribution analyses, Laplace’s equation is used to solve for the electric potential in solution @, V% = 0.
(1)
The potential is taken to be zero in the bulk solution far from the surface of the hemispherical pit, lim @ = 0, r+r+m
for
z = 0 and r > a,
-KVcD.ll,
(4)
= i,(a, + aJf(V
- CD,)(linear polarization),
= i, exp[a.f(V
- @6)] (Tafel polarization).
a
(linear polarization),
V sin(j)
or
(5) (6)
The component of the current density normal to the pit surface is represented by i; the solution conductivity is given by K; @,, denotes the electric potential evaluated at the pit surface (a variable); V is the elec-
(8)
and for Tafel polarization, P = a,@
(Tafel polarization).
(9)
Equations (l)-(3) can be rewritten by replacing Cp and V with P and 6, respectively. The boundary conditions given by equations (5) and (6) become 1 - sin@P, + cos(j)QP . 0 = 0
(linear polarization),
sin(T)e-PO + cos(Y)OP * q = 0
(Tafel polarization).
(3)
where II is the pit radius. The unit normal vector at the pit surface, pointing into solution, is given by II. We consider two boundary conditions on the pit surface, corresponding to linear and Tafel polarization[23] : i=
p=-
(2)
where the cylindrical coordinates r and z are shown in Fig. 1. On the passive surface outside the pit, which acts as an electronic insulator, the normal gradient of the potential is zero: V@ . II = 0
The dimensionless groups j and Y fultil the role of J and 6, respectively, employed by Newman[23, 143 for the analysis of the secondary current distribution of a disk electrode; i and Y, however, range between 0 and n/2. For linear polarization, we define the dimensionless potential P as
(10)
(11) Because of the definitions used for P, j and T, the coefficients in equations (10) and (11) range between 0 and 1, which is convenient for subsequent numerical analysis. We define the dimensionless average current densities as QvJf KV
= -sin(j) linearpoLrintioa
(12)
M. W. V~RBRU~OE et al.
1652
a.hvsa K
= 6= Tafcl pdarintion
’ VP * q sin(e) de, I =I2
(13)
where 0 is the angle to the z-axis (see Fig. 1) and 6 is commonly used to identify the dimension group a. jiavs a/u in secondary current distribution analyses subject to Tafel polarization[23]. For both linear and Tafel polarization, the dimensionless local current densities can be represented as
The gradient VP in equation (12)-(14) is evaluated at the pit surface, defined by X2 + Z2 = 1 and rr/ 2sesz. It is important to note that for the case of linear polarization, the dimensionless average current and the current distribution expressed in equation (14) are dependent solely on the dimensionless parameter j (or J). Similarly, for Tafel polarization, the dimensionless average current 6 and the current distribution are dependent solely on the dimensionless parameter Y. Thus, in the following Results section, the integrals required for calculating these quantities are tabulated over a range of j and Y values. It is possible to generalize the linear polarization model to account for multiple reactions. For each reaction d, ic = i&a,
+ ad)f(V
and its inverse
where i = fl. These relations can be used to map circles onto lines. Now suppose that w = X + iZ and C = < + iq are two complex variables satisfying the relation
(15) = exp(30 + 3 exdt;) 3exp(2r)+l *
- Uf - mO) (linear polarization),
where LJFcan be viewed as an equilibrium for reaction d. In addition,
potential
N
i= Cic, C=l
-Kv@-D.
=
ly near the electrode edge before heading outward along the insulator surface. (Typical behavior of this type is illustrated by the coordinate lines perpendicular to the electrode, shown in Fig. 2.) It is thus advantageous to solve Laplace’s equation using coordinates whose coordinate lines display the same singular behavior at the pit edge that arise in the flux lines, and we will derive such a coordinate system using a wnformal mapping. Conformal mappings are constructed by considering complex analytical functions and then taking their real and imaginary parts. (See Zwillinger’s text[18] and references given there for further information.) The wnformal mapping that we need can be constructed from the complex valued function
Thus the definitions of j and P can be generalized to
W-9
Equation (16) expresses the overall result: the mapping of the entire electrolyte region, including the pit, onto an infinite strip. Note that equations (17) and (18) below give the result in a form more suitable to immediate application, as the cylindrical coordinates X and Z are given in terms of the conformal coordinates [ and q. Figures 2 and 3 show what the coordinate lines for the system t-q look like when plotted in the X-Z coordinates. As mentioned above, the lines of constant q display the same
cp
tan(j) ’ = V sin(j) z= i (1 - LJ,O/V)J, (linear polarization), where the dimensionless group Jt is given by Jc =
i&b
1.2
+ a&f K
’
The expressions for the dimensionless average current i,,,a/rcV and the current distribution i/& remain unchanged. Conformal transjbmation to the c-q system As noted in the introduction, potential functions satisfying either of the boundary conditions 10 or 11 will have singularities at the edge where the electrode meets the insulator. These arise because flux-vectors on the electrode surface start out pointing inward, toward the Z-axis, but then change direction abrupt-
f=
0.9
0.6
3"
0
0.3
0
Fig. 2. Lines of constant r and q plotted on the cylindrical coordinate system.
Reaction distribution in a corroding pit
1653
where : f&E) = E9 + 36E’ + 81E f,(E) = 3Es + 18E6 - 36E4 - 54E2 + 27 Av = ai200 0.001
f,(E) = 12E’ + 108E’
AC - 0.015
f,(E) = -6E6
t
+ 27E4 + 18E2
I
f4(E) = 18E’
0
rl=o
f5(E) = 3E4.
Fig. 3. An expanded view of the pit edge showing lines of constant r and q plotted on the cylindrical coordinate system.
behavior at the pit edge that is exhibited by the flux lines associated with solutions of Laplace’s equation, and this is the main advantage of this coordinate system. Indeed, the function r is a solution to Laplace’s equation with a constant potential condition on the electrode if the electrode is in the shape of an infinite, straight trough of semicircular cross section as depicted in Fig. 1. Both the hemispherical and the trough-shaped electrodes exhibit the same local behavior near the electrode edge, with the consequence that the t-q system allows accurate numerical determination of potential functions in either geometry. For the hemispherical electrode, the transformed version of Laplace’s equation into the [-rj system is obtained by considering the real and imaginary parts of the transformation (16). These are written as Re{w} = X =
E2 cos(3~) + [3 + 9E2 + 3E4] cos(r~)
Rotational coordinate systems generated by conformal mappings are described in chapter 2 of Moon and Spencer[24], where formulas for computing the Laplacian are also given. Using these relations, we can write Laplace’s equation for the dimensionless potential P as o=x($+$)+$~+~$. We note that the right side of this equation actually equal to
times the Laplacian, and this factor must be included when solving inhomogeneous equations. The boundary conditions expressed in equations 2 and 3 can be written as IimP = 0, e-m g=O atl
[l - sin(j)PJ.P(~)
E2 sin(3tf) + [3 - 9E’ + 3E4] sin(q)
6E3 cos(2~) + 9E + Es
’
E = exp(-0.
One also needs the following expressions for partial derivatives: COWI) +fs(E) cos(3~) +f,(E) cos(rt) J+(E) cos(4tl) +f2(E) cos(2tt) + J,(E)
conditions
at the pit
ap
= 0 + cos(i) at c=0 (linear polarization),
(24)
(Tafel polarization).
(25)
The function 9r(r1) is given by
+ 273 cos(3rt) - 429 cos(e) . t26j
WV)= - 9 cos(7rI) + 195 co@q)
+ 1153 cos(3rI) + 2739 cos(rf)
and
aq
(23)
9 cos(7rf) + 147 cos(StJ)
(19)
ax -=-
forq=Oandrf=i,
= 0 [sin(T)e-po]4F(q) + co@) E at +e
where
h(E)
(22)
respectively. The boundary surface become
and
ax -= at
is
6E3 cos(2~) + 9E + Es (17)
Im{w} = 2 =
(21)
h(E) sin(W+f3(E) sin(%) +ME) Wrt) f40
cos(4rl)
+f,(E) cos(2rt) + f,(E)
’
Governing equations in the E-11system In writing equations (21~(26), we have mapped conformally the pit problem into a half-infinite strip
1654
M. W. VBRBRUGGE et al.
half-infinite strip that arises when the < coordinate is used. It is useful at this point to refer to Fig. 1 and recap the overall transformation: l
l
l
l
0
%I*
1
Fig. 4. Surface contour plot of the potential P for the primary current distribution, corresponding to P, = 1.
defined by 0 < q d n/2 and 0 < { < co. For numerical calculations it is convenient to replace the r coordinate with the bounded coordinate <:
g = tanh(c/2)
with
c = In
and
E = 5;.
(27)
Hence, g= 0
when
5 =O,
c=l
when
{=Q),
and numerical calculations using i can be done on the finite rectangle pictured in Fig. 1, instead of the
The pit surface from A to B maps to the bottom boundary of the rectangle. The insulating surface maps to the left boundary of the rectangle. The line of symmetry maps to the right boundary of the rectangle. The bulk solution infinitely far from the pit maps to the upper boundary of the rectangle.
The use of < also ensures that the dimensionless potential P varies linearly with 5 as 5 approaches unity, ie as { tends to 00 (see Figs 4 and 5). This linear behavior can be deduced from the fact that @ is proportional to l/&S? for large values of ,/n; for large X and Z (a~ equivalently for small E),
&T7
+3E
and
[+I-2E,
so
Figure 4 shows a plot of the potential P(& 11) satisfying the constant potential condition P = 1 on the electrod_e surface. Clearly P is almost a linear function of c, although the same function has infinite normal derivatives at the electrode edge when plotted in the X-Z coordinates, and its gradient in the X-Z coordinates tends to zero at large distances from the electrode. The constant potential lines are almost independent of q, reflecting the similarity between the straight trough geometry, for which r is itself a solution to Laplace’s equation, and the hemispherical geometry. Figure 5 shows a plot of the potential P(& q) satisfying constant flux conditions on the electrode. Again, the relatively smooth behavior of the solution in these coordinates implies that numerical solutions can obtain a high degree of accuracy with relatively large mesh spacings. To do the above calculations, one needs to know Laplace’s equation in terms of the coordinate f, which takes the form:
1 +g-x< ;,I-E’,$+g$, 1
o=x+u
a9 -p+Typ
-z 2 2~
[
(28)
(
>
g=O
Fig. 5. Surface contour plot of the potential P for the case of constant flux over the pit surface (VP. II = -1 on the pit surface).
Ph 1) = 4
(29)
forq=Oorq=q,
(30)
0 = [l - sin@PJp(q)
+ co&J A % 2 at2=o (linear polarization),
(31)
1655
Reaction distributionin a corrodingpit 0 = sin(T)e-po4F(q) + cos(T) 1 E (Tafel polarization),
kinetics
Linear
(32)
where X and 8X/a{ must be evaluated as functions of e using equations (17), (19) and (27). The dimensionless average current densities can be expressed in the r-q system as &f
KV
s =I2 i
= -sin(j]
0
ap
I
u_-
X(rl, 0) drt, 2 at pco
--
o
linear polariution
@hi O.*_..0.6 -_!!A-..__ !---
0 0
0.4
0.2
0.6
I
0.6
Position (2/71)-q
(34)
Fig. 7. Potential distribution P,,(q) over the pit surface for linear polarization. The primary current distribution corresponds to j = n/2 and P,, = 1.
For both linear and Tafel polarization, the dimensionless local current densities can be represented as i
--
ap
&=&-
(35)
:r
Wlj
i
O_ 0.2__
EL.._ 0.5__ &!I......... 1
RESULTS Current and potential distributions The flux aP/af and potential P, distributions over the pit surface 0 < 2q/n < 1 are shown in Figs 6 and 7, respectively, for the linear-polariz.ation case. Analogous plots are given in Figs 8 and 9 for the dimenUto &, a sionless current i/iavs and potential distributions, respectively, vs. the spherical coordi-
0.6
0.5
0.7
cl.8
0.9
I
Angular position O/n
Fig 8. Current distribution @7)/i,, for linear polarization vs. the spherical coordinate 6. The pit lip is at 0/x = 0.5 and the pit bottom is at B/n = 1. The primary current distribution corresponds to j = 42, in which case the current density is intkdte at the electrode-insulator edge, in contrast to the results shown in Fig. 6.
51
Linear kinetics
__ aWdi
/’
2
E
2t
/
:w,
,,
/’
0 O&2__ Q.4..._ 0.6__
/’_,...‘--
Es?-_ -
--_/-__._-.-.--
,/./“,_./,_/-------_ . ~___,,_.___..____.__.........................
T-------’
/.
I ...______
___, --;-;
~z.:,:
/.
0 (!...-....
Wn)j O-
0.2 ---
Linear kinetics
0.95 ~ 0.99
o.r..._ Q&L._ P:!..
0
0.2
0.4
0.6
Position (2/7r)q
0.6
1
L._._ 0.6
0.1
04
Angularposition
0.9
S/n
1
Fig 6. Flux distribution aP/8EIF_,, over the pit mface for Fig 9. Dimensionless potential a0 u/i_,, a distribution over and various j values. The primary the pit surface for linear polarization vs. the spherical coorcurrent distribu$on corresponds to j = n/2, @ which case dinate e. the flux aP/at Ike is nearly U&OIIIIin the {-PJS~S~CXII.
linear polarization
hi. W. VxruutvaCe et al.
1656
nate 0 shown in Fig. 1. The dimensionless potentials used in this work are related by
(linear polarization).
F = P, sin01 5 .“s a
Hence both the current and potential distributions o-f Figs 8 and 9 are functions ofj (or .I). The flux cYP/at over the surface is bounded for all values of q. As the parameter j approaches x/2, which in physical terms denotes the dominance of ohmic resistance over that associated with the interfacial charge-transfer reaction, the flux aP/ac over the surface becomes nearly uniform; the dimensionless current i/i,,, in spherical (or cylindrical) coordinates, however, tends to intinity at the pit lip. The contrasting behavior for the two coordinate systems illustrates clearly why the conformal transformation facilitates the numerical analysis. The current i/& is constant over the pit surface for the case of j = 0; the corresponding surface potential shown in Fig. 9 for j = 0 was found to differ by less than 1 per cent from the values given by Newman et aI.[8] in their constant surface flux analysis of the same problem. The dimensionless current i/iasr and potential Co,@_a distributions shown in Figs 10 and 11, respectively, are for the Tafel polarization boundary condition. For the Tafel case, cD,~/i,,~a = P&5, and both the current and potential distributions of Figs 10 and 11 are functions of Y only. For small values of the parameters j and T, the mathematical behavior of the linear and Tafel problems is similar, as would be expected in view of the linearization of the exponential for small T values, resulting in a linear boundary condition.
Dimensionless average current densities In Table 1 the dimensionless average current densities for the linear polarization case (i,,,a/Vrc vs. 5) and for the Tafel polarization case (a, fi.,, a/K = 6 vs. r) are provided. The value of &a/VK = 0.5396 at j = n/2 (J = co) corresponds to the primary current
‘Mel
kinetica
(W-f D o-3-_
or...o&L_ O:~.. 0.99
1
Fig. 11. Dimensionless potential 8,rc/i,,,a distribution over the pit surface for Tafel polarization versus the spherical coordinate 6.
distribution result, which for the disk electrode is 4/ n[23]. Thus, the primary resistance to a disk is about 60 per cent of that to a pit of equal electrode surface area. To obtain the four digits given for the dimensionless average current densities of Table 1, calculations performed with a 51 x 51 and 61 x 61 mesh were extrapolated to zero mesh spacing, assuming an error order of h’, where h is the mesh sp+ng. For the symmetric mesh of this work, h = A{ = 2Aq/x, where A< is the mesh spacing in [ and Aq is the mesh spacing in q. The plot shown in Fig. 12 spans the entire range of j values and was obtained using grid sixes of 31 x 31,41 x 41, 51 x 51 and 61 x 61. The method of least squares was used to fit lines to
(2hdi 0 0
1.001
.
0.204062
A
0.406163
A
0.591037
0
0.796916
l
1
1
(2/7dT 0
0.990
0.997
-
0
0.0006
0.0012
square of mesh spacing, (AT)’ = @Au/r)’
Fig. 10. Current distribution $0)/i.,, for Tafel polarization in spherical coordinates. The primary current distribution corresponds to T = x/2, in which case the current density is infinite at the ekctrodoinsulator edge.
Fig. 12. Extrapolation to xero spacing for i,,a/Vrc. The nunmmr dvSa/VK Llcuhrmdrefers to the calculated value for the indicated mesh spacing, and the denominator %g= sM~Sj-O C, U/VK refers to the value of i,,,a/VK extrapolated to zero mesh spacing using the results from 31 x 31.41 x 41,51 x 51 and 61 x 61 meshes.
Reaction distribution in a cormding pit
1657
the data in Fig. 12. The plot indicates a leading error order of h2, consistent with the error discussion given above. The average current density calculations given in Table 1 for the linear polarization case were fit to the following polynomial: k=6
‘w&f VU
= linear polariution
L
sint_i? c
near
kinetics
(36)
akjL,
k=O
where the coefficients a, are given in Table 2. Shown in Fig. 13 are the average current density results for the linear polarization case in the form of a dimensionless solution resistance R,: 1 --. t.vl a une~rpolarintion J
R,$
To correlate
the Tafel polarization calculations, we make use of a treatment that has been used suc-
Fig. 13. Dimensionless solution resistance R, for linear polarization. Note that J = tan(j) = (K, + u,)ji&c.
cessfully for the disk electrode[25, 26). The Tafel correlation formula is written as Table 2. Coefficients for the fit polynomial of equation (36) k
ak
0 1 2 3 4 5 6
0.9996 - 3.6706 128543 - 30.8668 45.6364 - 36.0201 11.5982
. = I.,*
4
exp[a, f( V - tie)],
T
where w. = i,,,a/(O.S396a), the solution potential drop under primary current distribution conditions. Substituting in the definitions of T and S, we can write l/E, as 1 E,=
6 exp(6/0.5396) tan(r) ’
Since
the dimensionless average current density 6 and the quantity Y are related by equation (13), ET
Table 1. Dimensionless average current densities for linear [equation (12)] and Tafel [quation (13)] polarization (2H.j
cr r)
0 0.02041 0.04082 0.06122 0.08163 0.1020 0.1224 0.1429 0.1633 0.1837 0.2041 0.2245 0.2449 0.2653 0.2857 0.3061 0.3265 0.3469 0.3673 0.3878 0.4082 0.4286 0.4490 0.4694 0.4898
i, dlvk 0 0.02978 0.05565 0.07839 0.09860 0.1167 0.1331 0.1480 0.1617 0.1744 0.1861 0.1970 0.2073 0.2170 0.2261 0.2348 0.243 1 0.2511 0.2587 0.2660 0.2732 0.2801 0.2868 0.2934 0.2998
cr,&,
a/K
0 0.02985 0.05614 0.07975 0.1013 0.1212 0.1397 0.1572 0.1738 0.1895 0.2047 0.2193 0.2335 0.2472 0.2607 0.2740 0.2870 0.2999 0.3127 0.3255 0.3382 0.3510 0.3639 0.3769 0.3900
(2Mi
or r)
0.5102 0.5306 0.5510 0.5714 0.5918 0.6122 0.6327 0.6531 0.6735 0.6939 0.7143 0.7347 0.7551 0.7755 0.7959 0.8163 0.8367 0.8571 0.8776 0.8980 0.9184 0.9388 0.9592 0.9796 1
L, WK 0.3061 0.3124 0.3186 0.3247 0.3308 0.3369 0.3430 0.3491 0.3553 0.3616 0.3679 0.3744 0.3811 0.3880 0.3952 0.4026 0.4105 0.4189 0.4279 0.4377 0.4487 0.4612 0.4763 0.4963 0.53%
a. A., a/k 0.5616 0.5634 0.5661 0.5698 0.5744 0.5800 0.5868 0.5947 0.6039 0.6145 0.6268 0.6408 0.6569 0.6754 0.6968 0.7216 0.7506 0.785 1 0.8267 0.8780 0.9437 1.032 1.164 1.402 co
M. W. V~RBRUOOE et al.
1658
0
Dlms/m/ess
0.2
0.4 average
0.6 cLrrent
0.8 6/(1+6)
Fig. 14. Tafel correlation function l/E,. The dimensionless average current density used in formulating the abscissa is given by b = a&a/K. The correlation formula is given @$= where by i, = (i&%) expCa./(v - 4W 1,a/(O.5j961~), the solution potential drop under primary current distribution conditions.
can be taken to be a function of either 6 or Y. A plot of l/E, vs. S/(1 + 6) is given in Fig. 14. The plot is nearly linear over the range of average currents shown, and the following equation was fit to the data shown in Fig. 14:
sities for linear polarization [equation (12)] and Tafel polarization [equation (13)] are given in terms of the appropriate dimensionless groups. The numerical calculations for the dimensionless average current densities are fitted to simple expressions; as a consequence, future researchers may use the results of this work without having to resort to numerical methods. The approach we have taken involves mapping conformally the pit geometry into a half-infinite strip and collapsing the unbounded coordinate so that it ranges from 0 to 1. The result is a new coordinate system that removes the singularity at the electrodeinsulator interface and provides a bounded region over which a limited number of mesh points is required to perform accurate numerical analysis. The major work in taking this approach consists of calculating the transformed equations. These calculations are presented in this paper. While the transformed equations are more complicated to write than the governing equations in the original coordinates, they are not significantly more difficult to incorporate in a computer code. The transformed coordinates should be useful for solving the more complete problem involving diffusion, salt-film precipitation and other corrosion-related phenomena.
REFERENCES (Editor) Corrosion Processes, Applied Science Publishers, New York (1982). 2. Z. Szklarska-Smialowska, Pitting Corrosion of Metals, 1. R. N. Darkin
+ = 0.9983 - 0.5291 T 3.
(37) This correlation for E, and the data shown in Fig. 14 treat the range 0 < 6/(1 + 6) < 0.8, or 0 < 6 6 4. For larger values of 6, a singular perturbation analysis analogous to that completed by Smyrl and Newman[ZS] for the disk electrode subject to high 6 should be undertaken to ensure accuracy in the calculated results. In a similar vein, a singular perturbation analysis can be used to determine more accurately the value of i,,,a/Vx as j approaches rr/ 2[15]. For both the linear correlation [i,, a/VK as a function of j, equation (36)] and the Tafef correlation [l/E, as a function of 6, equation (37)], the correlation formulas produced results that were indistinguishable from the numerical calculations when plotted as correlated.
4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
14.
SUMMARY The secondary current distribution for a corroding pit is analyzed for both linear and Tafel polarization. One dimensionless group can be identified for secondary current distribution analyses that employ linear and Tafel polarization laws and contain only one characteristic dimension (here the pit radius). In this work, the dimensionless average current den-
15. 16. 17. 18.
National Association of Corrosion Engineers, Houston (1986). P. A. Schweitzer (Editor) Corrosion and Corrosion Protection Handbook, second edition, Marcel Dekker, New York (1989). See p. 164 for a discussion of pitting corrosion and passivating metals. H. W. Pickering and R. P. Frankenthal, J. electrochem. Sot. US,1297 (1972). J. R. Galvele, J. electrochem. Sot. 123,464 (1976). R. Alkire, D. Ernsberger and D. Damon, J. electrochem. Sot. 123,458 (1976). K. Nisaucioglu and H. Holtan, Electrochim. Acta 23, 251 (1978). J. Newman, D. N. Hanson and K. Vetter, EIectrochim. Acta 22,829 (1977). J. N. Harb and R. C. Alkire, J. electrochem. Sot. 138, 2594 (1991). H. P. Leckie and H. H. Uhlig, J. electrochem. Sot. 113, 1262 (1966). H. Biihni and H. H. Uhlig, J. elecrrochem. Sot. 116, 906 (1969). E. McCafferty, J. electrochem. Sot. 131,373l (1990). H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, second edition, Oxford University Press, New York (1959). J. Newman, J. electrochem. Sot. 113, 1235 (1966). [See also J. Newman, J. electrochem. Sot. 114,239 (1967).] K. Nisancioglu and J. Newman, J. electrochem. Sot. 120,1339 (1973). A. M. Bond, K. B. Oldham and C. G. Zoski, J. electroanal. Chem. 245,71(1988). D. R. Baker and M. W. Verbrugge, J. electrochem. Sot. 137,3836 (1990). D. Zwillinger, Handbook of Diffuential Equations, Academic Press, New York (1989). See p. 335, comment 5, for the use of conformal representations and numerical simulation. Related references are given on p. 336.
Reaction distribution in a corroding pit 19. W. H. Smyrl and J. Newman, .I. electrochem. Sot. 123, 1423 (1976). 20. C. Amatore, in Ultramicroelectrodes (Edited by M. Fleschmann, S. Pons, D. R. Rolison and P. P. Schmidt), pp. 169-183. Datatech Systems, Morgantown, NC (1987). 21. A. C. Michael, R. M. Wightman and C. A. Amatore, J. electroad. Chem. 267,33 (1989). 22. M. W. Verbrugge and D. R. Baker, J. phys. Chem. %, 4572 (1992).
1659
23. J. Newman, Electrochemical Systems, second edition, Prentice-Hall, Englewood Clitfs, New Jersey (1991). 24. P. Moon and D. E. Spencer, Field Theory Handbook, second edition, Springer Verlag, Berlin (1971). 25. W. H. Smyrl and J. Newman~J. electrod&. Sot. 136, 132 (1989). (See Fin. A-l for the Tafel correlation ulot analogous to Fig. 14 of this work.) 26. A. C. West and J. Newman, J. electrockem. Sot. 136, 139 (1989). (See Fig. 2 for the Tafel correlation plot similar to Fig 14 of this work.)
REACTION
1993 0
DISTRIBUTION
IN A CORRODING
56.00 + wJo 00s4686/93 1993. Pergamoa Prws Ltd.
PIT
MARK W. VERBRUGGE,*DANIELR. BAKERYand JOHN NE\KMAN~ Physical Chemistry Department, General Motors Research, Warren, MI 48090-9055, U.S.A. 7 Mathematics Department, General Motors Research, Warren, MI 480909055, U.S.A. $ Materials Sciences Division, Lawrence Berkeley Laboratory, and Department of Chemical Engineering, University of California, Berkeley, CA 94720, U.S.A. l
(Received 14 October 1992; in revisedform 8 February
1993)
Abstract-An analysis of the reaction distribution over a corroding pit is presented. If the liquid-phase conductivity and the interfacial rate constants for the metal-dissolution reaction are known, the results of this secondary current distribution problem can be used to estimate the maximum rate of pit growth. The approach taken involves the confonnal mapping of the pit geometry onto a new coordinate system; the finished transformation is shown to remove the singularity at the pit edge and provide a bounded region over which only a limited number of mesh points are required to perform accurate numerical analysis. Numerical calculations for the pit dissolution rate subject to linear and Tafel polarization are fitted to simple expressions; as a consequence, the results of this work can be used without having to resort to numerical methods. For the linear polarization analysis, the results are generalized in a manner that allows one to consider an arbitrary number of reactions on the pit surface. Key words: conformal mapping, pitting corrosion,
f” E ET
F i
'0 i G n
Iv
pit radius,
modeling, current distribution.
NOMENCLATURE
r
cm
#
FfRT, v-l ext4 - 5)
Tafel correlation function Faraday’s constant, 96487 C/equivalent current density normal to the pit surface, Arm-2 current density averaged over the pit surface, AlXl-2 exchange current density, A ctn2 linear polarization dimensionless group, tan(j) = J = (a, + a&I0 a/K reaction 8 unit normal vector pointing into solution total number of electrochemical reactions on pit surface radial coordinate, cm gas constant, 8.3143 J mol-’ K-l dimensionless solution resistance dimensionless electric potential [equations (8) and (911 temperature, K electrode potential, V rla cylindrical coordinate, cm zla anodic transfer coefficient cathodic transfer coefficient dimensionless average current, a* ~5~~~ a/K coordinate (Fig. 1) spherical coordinate (Fig. 1) conductivity, S cm- 1 unbounded coordinate resulting from conforma1 transformation coordinate (Fig. 1)
Tafel-polarization dimensionless group, tan(T) = a._& exda.JV)a/~ electric potential, V
Subscripts linear polarization L on the pit surface Tafel polarization ;
INTRODUCI’ION It has been speculated that as much as four per cent of the gross national product in a number of industrialized nations is balanced by costs attributable to corrosion[l]. Although it is well known that passivating metals commonly used as corrosion-resistant coatings generally undergo localized dissolution from hemispherical pits, referred to as pitting corrosion[2,3] rather than by uniform surface dissolution, no specific analytical tools exist that allow one to determine the reaction distribution over the pit surface, which could be used to determine the dissolution rate and the failure time of a corrosionresistant coating. Szklarska-Smialowska’s[2] text is helpful in organizing the literature associated with pitting corrosion. A number of theoretical studies of pitting corrosion have appeared in the literature. The most commonly employed geometry used for simulating the corroding pit corresponds to a cylinder of uniform cross section and a central axis perpendicular to the metal’s surface[4-7-J. The metal surface and the walls of the cylinder are taken to be insulating; the eleu trode surface undergoing corrosion is located at the bottom of the cylinder, parallel to the metal surface. The electrolyte immediately outside the cylinder is
1649
M. W. Vaaaauoo~ et al.
1650
8
=o
I 4z Bulk solution
Fig. 1. Pit geometry. The problem is axisymmetric. Left schematic-pit problem and cylindrical coordinates r and z. For spherical coordinates, the coordinate 0 is als_ogiven. Right schematicpit problem in new coordinates q and (.
taken to be well mixed and at uniform potential. This assumed geometry yields a one-dimensional analysis and has proven useful for gaining qualitative understanding of pitting phenomena. Twodimensional analyses of corroding hemispherical pits have been completed by Newman et ~I.[83 and by Harb and Alkire[9]. The conformal transformation presented in this work, and described briefly in the last paragraph of the Introduction, makes it possible to perform more accurate two-dimensional analyses with reduced computational expense. In this study, we provide a theoretical analysis for a hemispherical pit in contact with a well-mixed electrolyte, representing a secondary current distribution analysis. Since the electrolyte is assumed to be wellmixed, the calculated corrosion rates represent an upper boundary for most corrosion processes, as the additional transport resistances not included in the model equations would slow the rate of dissolved metal ions leaving the pit. The current distribution results show that for very small pits, uniform pit dissolution takes place and the hemispherical geometry is maintained. For larger pits, however, the current density at the pit edge can be much larger than that of the pit center. Thus the analysis allows one to determine when nonuniformities in the current distribution are expected to be significant, in which case the pit geometry would begin to change from its original hemispherical shape, as has been observed experimentally[9]. The electric potential and reaction distribution over the pit surface are given in terms of a single dimensionless parameter; the resulting plots can be used to determine the local rate of corrosion over the pit surface if all the electrochemical reactions are described by linear current-potential relations, as is usually the case with corrosion resistant materials that are designed to minimize the metal dissolution rate. For both linear and Tafel polarization, analytic
expressions are given for the average current density, electrode potential relationship, which should be useful for the interpretation of accelerated corrosion test data and experiments designed to determine the pitting potential[lO-121 by means of current passage. The mathematical problem governing the secondary current distribution involves the solution to Laplace’s equation for the electric potential along with a current-potential relation at the electrode surface and appropriate insulator and bulk-solution conditions for the other boundaries. For linear or Tafel polarization, the results &aii be given in terms of one dimensionless group. Thus.the dimensionless average current densities for the two problems are given in tabulated form, removing the need for future researchers to perform numerical calculations in order to use the solutions presented in this work. A polynomial fit to the tabulated results is also provided. It should be noted that the secondary current distribution problem subject to a linear polarization boundary condition shares a one-to-one correspondence with the mathematical description of: (1) steady-state mass-transport problems characterized by Fickian diffusion and linear boundary conditions with constant coefficients, such as is observed typically in the analysis of ionic diffusion to an electrode that is imbedded in an otherwise insulating wall (the dilute solution containing an excess of inert electrolyte); and (2) steady-state heat-transport problems characterized by Fourier’s law of heat conduction and linear boundary conditions with constant coefftcients, such as typically observed in the analysis of heat transfer through solids to a surface imbedded in an otherwise insulating wall, as described on pp. 18 and 19 of Carslaw and Jaeger[13]. More generally, all systems described by Laplace’s equation for the dependent variable and a boundary condition that is linear in the dependent variable and contains
Reaction distribution in a corroding pit constant coefficients, as is the case for the secondary current distribution problem subject to linear polarization, are mathematically equivalent; the mathematical results can be used interchangeably. Because of the equivalent nature of the secondary current distribution problem to ionic diffusion in the presence of excess inert electrolyte, it is possible, for example, to apply analytical methods derived for the analysis of the secondary current distribution associated with a disk electrode[14, 151 subject to linear polarization to treat the equivalent problem concerning ionic diffusion to a microdisk electrode[ 16, 173. The approach we use for treating the pit problem is to develop a conformal transformation to recast the problem as shown in the left schematic of Fig. 1, with a singularity at the electrode-insulator edge, to a half-infinite strip devoid of singularities; a subsequent transformation is then used to collapse the unbounded coordinate into a new coordinate, resulting in a recasting of the problem to a box geometry, which is illustrated by the right schematic in Fig. 1. The use of conformal mapping for the generation of a new coordinate system more amenable to numerical analysis is a well-established practice[ 181. Conformal and closely related transformations used to remove electrodeinsulator singularities have been shown to simplify greatly subsequent numerical analyses in a number of electrochemical studies. (For examples, see [ 19-221.)
1651
trode potential relative to a reference electrode of the same kind located in the bulk solution (a constant);f refers to F/RT, where F is Faraday’s constant, R is the gas constant and T is the absolute temperature; and the kinetic constants iO, a, and a, refer to the exchange current density and the anodic and cathodic transfer coefficients, respectively. After solving the above equations, the average current density iavp can be obtained by integrating the normal current density over the pit surface, 1
=r”’
27ca2
i dA,
(7)
where dA is the differential surface area associated with i. The remainder of this section details the recasting of-a dimensionless form of equations (l)-(7) in the rl- [ coordinate system shown schematically in Fig. 1. Nondimensionalization
Using the following definitions allows us to write the governing equations in a compact form and identify important dimensionless groups: tanCj) = J = (a, + a,)fifioa/K, t&W
= a,fiO
cvb, fV)a/~,
0 = aV, X = r/a,
GOVERNING COORDINATE
Z = z/a.
EQUATIONS AND TRANSFORMATION
Governing equations For secondary current distribution analyses, Laplace’s equation is used to solve for the electric potential in solution @, V% = 0.
(1)
The potential is taken to be zero in the bulk solution far from the surface of the hemispherical pit, lim @ = 0, r+r+m
for
z = 0 and r > a,
-KVcD.ll,
(4)
= i,(a, + aJf(V
- CD,)(linear polarization),
= i, exp[a.f(V
- @6)] (Tafel polarization).
a
(linear polarization),
V sin(j)
or
(5) (6)
The component of the current density normal to the pit surface is represented by i; the solution conductivity is given by K; @,, denotes the electric potential evaluated at the pit surface (a variable); V is the elec-
(8)
and for Tafel polarization, P = a,@
(Tafel polarization).
(9)
Equations (l)-(3) can be rewritten by replacing Cp and V with P and 6, respectively. The boundary conditions given by equations (5) and (6) become 1 - sin@P, + cos(j)QP . 0 = 0
(linear polarization),
sin(T)e-PO + cos(Y)OP * q = 0
(Tafel polarization).
(3)
where II is the pit radius. The unit normal vector at the pit surface, pointing into solution, is given by II. We consider two boundary conditions on the pit surface, corresponding to linear and Tafel polarization[23] : i=
p=-
(2)
where the cylindrical coordinates r and z are shown in Fig. 1. On the passive surface outside the pit, which acts as an electronic insulator, the normal gradient of the potential is zero: V@ . II = 0
The dimensionless groups j and Y fultil the role of J and 6, respectively, employed by Newman[23, 143 for the analysis of the secondary current distribution of a disk electrode; i and Y, however, range between 0 and n/2. For linear polarization, we define the dimensionless potential P as
(10)
(11) Because of the definitions used for P, j and T, the coefficients in equations (10) and (11) range between 0 and 1, which is convenient for subsequent numerical analysis. We define the dimensionless average current densities as QvJf KV
= -sin(j) linearpoLrintioa
(12)
M. W. V~RBRU~OE et al.
1652
a.hvsa K
= 6= Tafcl pdarintion
’ VP * q sin(e) de, I =I2
(13)
where 0 is the angle to the z-axis (see Fig. 1) and 6 is commonly used to identify the dimension group a. jiavs a/u in secondary current distribution analyses subject to Tafel polarization[23]. For both linear and Tafel polarization, the dimensionless local current densities can be represented as
The gradient VP in equation (12)-(14) is evaluated at the pit surface, defined by X2 + Z2 = 1 and rr/ 2sesz. It is important to note that for the case of linear polarization, the dimensionless average current and the current distribution expressed in equation (14) are dependent solely on the dimensionless parameter j (or J). Similarly, for Tafel polarization, the dimensionless average current 6 and the current distribution are dependent solely on the dimensionless parameter Y. Thus, in the following Results section, the integrals required for calculating these quantities are tabulated over a range of j and Y values. It is possible to generalize the linear polarization model to account for multiple reactions. For each reaction d, ic = i&a,
+ ad)f(V
and its inverse
where i = fl. These relations can be used to map circles onto lines. Now suppose that w = X + iZ and C = < + iq are two complex variables satisfying the relation
(15) = exp(30 + 3 exdt;) 3exp(2r)+l *
- Uf - mO) (linear polarization),
where LJFcan be viewed as an equilibrium for reaction d. In addition,
potential
N
i= Cic, C=l
-Kv@-D.
=
ly near the electrode edge before heading outward along the insulator surface. (Typical behavior of this type is illustrated by the coordinate lines perpendicular to the electrode, shown in Fig. 2.) It is thus advantageous to solve Laplace’s equation using coordinates whose coordinate lines display the same singular behavior at the pit edge that arise in the flux lines, and we will derive such a coordinate system using a wnformal mapping. Conformal mappings are constructed by considering complex analytical functions and then taking their real and imaginary parts. (See Zwillinger’s text[18] and references given there for further information.) The wnformal mapping that we need can be constructed from the complex valued function
Thus the definitions of j and P can be generalized to
W-9
Equation (16) expresses the overall result: the mapping of the entire electrolyte region, including the pit, onto an infinite strip. Note that equations (17) and (18) below give the result in a form more suitable to immediate application, as the cylindrical coordinates X and Z are given in terms of the conformal coordinates [ and q. Figures 2 and 3 show what the coordinate lines for the system t-q look like when plotted in the X-Z coordinates. As mentioned above, the lines of constant q display the same
cp
tan(j) ’ = V sin(j) z= i (1 - LJ,O/V)J, (linear polarization), where the dimensionless group Jt is given by Jc =
i&b
1.2
+ a&f K
’
The expressions for the dimensionless average current i,,,a/rcV and the current distribution i/& remain unchanged. Conformal transjbmation to the c-q system As noted in the introduction, potential functions satisfying either of the boundary conditions 10 or 11 will have singularities at the edge where the electrode meets the insulator. These arise because flux-vectors on the electrode surface start out pointing inward, toward the Z-axis, but then change direction abrupt-
f=
0.9
0.6
3"
0
0.3
0
Fig. 2. Lines of constant r and q plotted on the cylindrical coordinate system.
Reaction distribution in a corroding pit
1653
where : f&E) = E9 + 36E’ + 81E f,(E) = 3Es + 18E6 - 36E4 - 54E2 + 27 Av = ai200 0.001
f,(E) = 12E’ + 108E’
AC - 0.015
f,(E) = -6E6
t
+ 27E4 + 18E2
I
f4(E) = 18E’
0
rl=o
f5(E) = 3E4.
Fig. 3. An expanded view of the pit edge showing lines of constant r and q plotted on the cylindrical coordinate system.
behavior at the pit edge that is exhibited by the flux lines associated with solutions of Laplace’s equation, and this is the main advantage of this coordinate system. Indeed, the function r is a solution to Laplace’s equation with a constant potential condition on the electrode if the electrode is in the shape of an infinite, straight trough of semicircular cross section as depicted in Fig. 1. Both the hemispherical and the trough-shaped electrodes exhibit the same local behavior near the electrode edge, with the consequence that the t-q system allows accurate numerical determination of potential functions in either geometry. For the hemispherical electrode, the transformed version of Laplace’s equation into the [-rj system is obtained by considering the real and imaginary parts of the transformation (16). These are written as Re{w} = X =
E2 cos(3~) + [3 + 9E2 + 3E4] cos(r~)
Rotational coordinate systems generated by conformal mappings are described in chapter 2 of Moon and Spencer[24], where formulas for computing the Laplacian are also given. Using these relations, we can write Laplace’s equation for the dimensionless potential P as o=x($+$)+$~+~$. We note that the right side of this equation actually equal to
times the Laplacian, and this factor must be included when solving inhomogeneous equations. The boundary conditions expressed in equations 2 and 3 can be written as IimP = 0, e-m g=O atl
[l - sin(j)PJ.P(~)
E2 sin(3tf) + [3 - 9E’ + 3E4] sin(q)
6E3 cos(2~) + 9E + Es
’
E = exp(-0.
One also needs the following expressions for partial derivatives: COWI) +fs(E) cos(3~) +f,(E) cos(rt) J+(E) cos(4tl) +f2(E) cos(2tt) + J,(E)
conditions
at the pit
ap
= 0 + cos(i) at c=0 (linear polarization),
(24)
(Tafel polarization).
(25)
The function 9r(r1) is given by
+ 273 cos(3rt) - 429 cos(e) . t26j
WV)= - 9 cos(7rI) + 195 co@q)
+ 1153 cos(3rI) + 2739 cos(rf)
and
aq
(23)
9 cos(7rf) + 147 cos(StJ)
(19)
ax -=-
forq=Oandrf=i,
= 0 [sin(T)e-po]4F(q) + co@) E at +e
where
h(E)
(22)
respectively. The boundary surface become
and
ax -= at
is
6E3 cos(2~) + 9E + Es (17)
Im{w} = 2 =
(21)
h(E) sin(W+f3(E) sin(%) +ME) Wrt) f40
cos(4rl)
+f,(E) cos(2rt) + f,(E)
’
Governing equations in the E-11system In writing equations (21~(26), we have mapped conformally the pit problem into a half-infinite strip
1654
M. W. VBRBRUGGE et al.
half-infinite strip that arises when the < coordinate is used. It is useful at this point to refer to Fig. 1 and recap the overall transformation: l
l
l
l
0
%I*
1
Fig. 4. Surface contour plot of the potential P for the primary current distribution, corresponding to P, = 1.
defined by 0 < q d n/2 and 0 < { < co. For numerical calculations it is convenient to replace the r coordinate with the bounded coordinate <:
g = tanh(c/2)
with
c = In
and
E = 5;.
(27)
Hence, g= 0
when
5 =O,
c=l
when
{=Q),
and numerical calculations using i can be done on the finite rectangle pictured in Fig. 1, instead of the
The pit surface from A to B maps to the bottom boundary of the rectangle. The insulating surface maps to the left boundary of the rectangle. The line of symmetry maps to the right boundary of the rectangle. The bulk solution infinitely far from the pit maps to the upper boundary of the rectangle.
The use of < also ensures that the dimensionless potential P varies linearly with 5 as 5 approaches unity, ie as { tends to 00 (see Figs 4 and 5). This linear behavior can be deduced from the fact that @ is proportional to l/&S? for large values of ,/n; for large X and Z (a~ equivalently for small E),
&T7
+3E
and
[+I-2E,
so
Figure 4 shows a plot of the potential P(& 11) satisfying the constant potential condition P = 1 on the electrod_e surface. Clearly P is almost a linear function of c, although the same function has infinite normal derivatives at the electrode edge when plotted in the X-Z coordinates, and its gradient in the X-Z coordinates tends to zero at large distances from the electrode. The constant potential lines are almost independent of q, reflecting the similarity between the straight trough geometry, for which r is itself a solution to Laplace’s equation, and the hemispherical geometry. Figure 5 shows a plot of the potential P(& q) satisfying constant flux conditions on the electrode. Again, the relatively smooth behavior of the solution in these coordinates implies that numerical solutions can obtain a high degree of accuracy with relatively large mesh spacings. To do the above calculations, one needs to know Laplace’s equation in terms of the coordinate f, which takes the form:
1 +g-x< ;,I-E’,$+g$, 1
o=x+u
a9 -p+Typ
-z 2 2~
[
(28)
(
>
g=O
Fig. 5. Surface contour plot of the potential P for the case of constant flux over the pit surface (VP. II = -1 on the pit surface).
Ph 1) = 4
(29)
forq=Oorq=q,
(30)
0 = [l - sin@PJp(q)
+ co&J A % 2 at2=o (linear polarization),
(31)
1655
Reaction distributionin a corrodingpit 0 = sin(T)e-po4F(q) + cos(T) 1 E (Tafel polarization),
kinetics
Linear
(32)
where X and 8X/a{ must be evaluated as functions of e using equations (17), (19) and (27). The dimensionless average current densities can be expressed in the r-q system as &f
KV
s =I2 i
= -sin(j]
0
ap
I
u_-
X(rl, 0) drt, 2 at pco
--
o
linear polariution
@hi O.*_..0.6 -_!!A-..__ !---
0 0
0.4
0.2
0.6
I
0.6
Position (2/71)-q
(34)
Fig. 7. Potential distribution P,,(q) over the pit surface for linear polarization. The primary current distribution corresponds to j = n/2 and P,, = 1.
For both linear and Tafel polarization, the dimensionless local current densities can be represented as i
--
ap
&=&-
(35)
:r
Wlj
i
O_ 0.2__
EL.._ 0.5__ &!I......... 1
RESULTS Current and potential distributions The flux aP/af and potential P, distributions over the pit surface 0 < 2q/n < 1 are shown in Figs 6 and 7, respectively, for the linear-polariz.ation case. Analogous plots are given in Figs 8 and 9 for the dimenUto &, a sionless current i/iavs and potential distributions, respectively, vs. the spherical coordi-
0.6
0.5
0.7
cl.8
0.9
I
Angular position O/n
Fig 8. Current distribution @7)/i,, for linear polarization vs. the spherical coordinate 6. The pit lip is at 0/x = 0.5 and the pit bottom is at B/n = 1. The primary current distribution corresponds to j = 42, in which case the current density is intkdte at the electrode-insulator edge, in contrast to the results shown in Fig. 6.
51
Linear kinetics
__ aWdi
/’
2
E
2t
/
:w,
,,
/’
0 O&2__ Q.4..._ 0.6__
/’_,...‘--
Es?-_ -
--_/-__._-.-.--
,/./“,_./,_/-------_ . ~___,,_.___..____.__.........................
T-------’
/.
I ...______
___, --;-;
~z.:,:
/.
0 (!...-....
Wn)j O-
0.2 ---
Linear kinetics
0.95 ~ 0.99
o.r..._ Q&L._ P:!..
0
0.2
0.4
0.6
Position (2/7r)q
0.6
1
L._._ 0.6
0.1
04
Angularposition
0.9
S/n
1
Fig 6. Flux distribution aP/8EIF_,, over the pit mface for Fig 9. Dimensionless potential a0 u/i_,, a distribution over and various j values. The primary the pit surface for linear polarization vs. the spherical coorcurrent distribu$on corresponds to j = n/2, @ which case dinate e. the flux aP/at Ike is nearly U&OIIIIin the {-PJS~S~CXII.
linear polarization
hi. W. VxruutvaCe et al.
1656
nate 0 shown in Fig. 1. The dimensionless potentials used in this work are related by
(linear polarization).
F = P, sin01 5 .“s a
Hence both the current and potential distributions o-f Figs 8 and 9 are functions ofj (or .I). The flux cYP/at over the surface is bounded for all values of q. As the parameter j approaches x/2, which in physical terms denotes the dominance of ohmic resistance over that associated with the interfacial charge-transfer reaction, the flux aP/ac over the surface becomes nearly uniform; the dimensionless current i/i,,, in spherical (or cylindrical) coordinates, however, tends to intinity at the pit lip. The contrasting behavior for the two coordinate systems illustrates clearly why the conformal transformation facilitates the numerical analysis. The current i/& is constant over the pit surface for the case of j = 0; the corresponding surface potential shown in Fig. 9 for j = 0 was found to differ by less than 1 per cent from the values given by Newman et aI.[8] in their constant surface flux analysis of the same problem. The dimensionless current i/iasr and potential Co,@_a distributions shown in Figs 10 and 11, respectively, are for the Tafel polarization boundary condition. For the Tafel case, cD,~/i,,~a = P&5, and both the current and potential distributions of Figs 10 and 11 are functions of Y only. For small values of the parameters j and T, the mathematical behavior of the linear and Tafel problems is similar, as would be expected in view of the linearization of the exponential for small T values, resulting in a linear boundary condition.
Dimensionless average current densities In Table 1 the dimensionless average current densities for the linear polarization case (i,,,a/Vrc vs. 5) and for the Tafel polarization case (a, fi.,, a/K = 6 vs. r) are provided. The value of &a/VK = 0.5396 at j = n/2 (J = co) corresponds to the primary current
‘Mel
kinetica
(W-f D o-3-_
or...o&L_ O:~.. 0.99
1
Fig. 11. Dimensionless potential 8,rc/i,,,a distribution over the pit surface for Tafel polarization versus the spherical coordinate 6.
distribution result, which for the disk electrode is 4/ n[23]. Thus, the primary resistance to a disk is about 60 per cent of that to a pit of equal electrode surface area. To obtain the four digits given for the dimensionless average current densities of Table 1, calculations performed with a 51 x 51 and 61 x 61 mesh were extrapolated to zero mesh spacing, assuming an error order of h’, where h is the mesh sp+ng. For the symmetric mesh of this work, h = A{ = 2Aq/x, where A< is the mesh spacing in [ and Aq is the mesh spacing in q. The plot shown in Fig. 12 spans the entire range of j values and was obtained using grid sixes of 31 x 31,41 x 41, 51 x 51 and 61 x 61. The method of least squares was used to fit lines to
(2hdi 0 0
1.001
.
0.204062
A
0.406163
A
0.591037
0
0.796916
l
1
1
(2/7dT 0
0.990
0.997
-
0
0.0006
0.0012
square of mesh spacing, (AT)’ = @Au/r)’
Fig. 10. Current distribution $0)/i.,, for Tafel polarization in spherical coordinates. The primary current distribution corresponds to T = x/2, in which case the current density is infinite at the ekctrodoinsulator edge.
Fig. 12. Extrapolation to xero spacing for i,,a/Vrc. The nunmmr dvSa/VK Llcuhrmdrefers to the calculated value for the indicated mesh spacing, and the denominator %g= sM~Sj-O C, U/VK refers to the value of i,,,a/VK extrapolated to zero mesh spacing using the results from 31 x 31.41 x 41,51 x 51 and 61 x 61 meshes.
Reaction distribution in a cormding pit
1657
the data in Fig. 12. The plot indicates a leading error order of h2, consistent with the error discussion given above. The average current density calculations given in Table 1 for the linear polarization case were fit to the following polynomial: k=6
‘w&f VU
= linear polariution
L
sint_i? c
near
kinetics
(36)
akjL,
k=O
where the coefficients a, are given in Table 2. Shown in Fig. 13 are the average current density results for the linear polarization case in the form of a dimensionless solution resistance R,: 1 --. t.vl a une~rpolarintion J
R,$
To correlate
the Tafel polarization calculations, we make use of a treatment that has been used suc-
Fig. 13. Dimensionless solution resistance R, for linear polarization. Note that J = tan(j) = (K, + u,)ji&c.
cessfully for the disk electrode[25, 26). The Tafel correlation formula is written as Table 2. Coefficients for the fit polynomial of equation (36) k
ak
0 1 2 3 4 5 6
0.9996 - 3.6706 128543 - 30.8668 45.6364 - 36.0201 11.5982
. = I.,*
4
exp[a, f( V - tie)],
T
where w. = i,,,a/(O.S396a), the solution potential drop under primary current distribution conditions. Substituting in the definitions of T and S, we can write l/E, as 1 E,=
6 exp(6/0.5396) tan(r) ’
Since
the dimensionless average current density 6 and the quantity Y are related by equation (13), ET
Table 1. Dimensionless average current densities for linear [equation (12)] and Tafel [quation (13)] polarization (2H.j
cr r)
0 0.02041 0.04082 0.06122 0.08163 0.1020 0.1224 0.1429 0.1633 0.1837 0.2041 0.2245 0.2449 0.2653 0.2857 0.3061 0.3265 0.3469 0.3673 0.3878 0.4082 0.4286 0.4490 0.4694 0.4898
i, dlvk 0 0.02978 0.05565 0.07839 0.09860 0.1167 0.1331 0.1480 0.1617 0.1744 0.1861 0.1970 0.2073 0.2170 0.2261 0.2348 0.243 1 0.2511 0.2587 0.2660 0.2732 0.2801 0.2868 0.2934 0.2998
cr,&,
a/K
0 0.02985 0.05614 0.07975 0.1013 0.1212 0.1397 0.1572 0.1738 0.1895 0.2047 0.2193 0.2335 0.2472 0.2607 0.2740 0.2870 0.2999 0.3127 0.3255 0.3382 0.3510 0.3639 0.3769 0.3900
(2Mi
or r)
0.5102 0.5306 0.5510 0.5714 0.5918 0.6122 0.6327 0.6531 0.6735 0.6939 0.7143 0.7347 0.7551 0.7755 0.7959 0.8163 0.8367 0.8571 0.8776 0.8980 0.9184 0.9388 0.9592 0.9796 1
L, WK 0.3061 0.3124 0.3186 0.3247 0.3308 0.3369 0.3430 0.3491 0.3553 0.3616 0.3679 0.3744 0.3811 0.3880 0.3952 0.4026 0.4105 0.4189 0.4279 0.4377 0.4487 0.4612 0.4763 0.4963 0.53%
a. A., a/k 0.5616 0.5634 0.5661 0.5698 0.5744 0.5800 0.5868 0.5947 0.6039 0.6145 0.6268 0.6408 0.6569 0.6754 0.6968 0.7216 0.7506 0.785 1 0.8267 0.8780 0.9437 1.032 1.164 1.402 co
M. W. V~RBRUOOE et al.
1658
0
Dlms/m/ess
0.2
0.4 average
0.6 cLrrent
0.8 6/(1+6)
Fig. 14. Tafel correlation function l/E,. The dimensionless average current density used in formulating the abscissa is given by b = a&a/K. The correlation formula is given @$= where by i, = (i&%) expCa./(v - 4W 1,a/(O.5j961~), the solution potential drop under primary current distribution conditions.
can be taken to be a function of either 6 or Y. A plot of l/E, vs. S/(1 + 6) is given in Fig. 14. The plot is nearly linear over the range of average currents shown, and the following equation was fit to the data shown in Fig. 14:
sities for linear polarization [equation (12)] and Tafel polarization [equation (13)] are given in terms of the appropriate dimensionless groups. The numerical calculations for the dimensionless average current densities are fitted to simple expressions; as a consequence, future researchers may use the results of this work without having to resort to numerical methods. The approach we have taken involves mapping conformally the pit geometry into a half-infinite strip and collapsing the unbounded coordinate so that it ranges from 0 to 1. The result is a new coordinate system that removes the singularity at the electrodeinsulator interface and provides a bounded region over which a limited number of mesh points is required to perform accurate numerical analysis. The major work in taking this approach consists of calculating the transformed equations. These calculations are presented in this paper. While the transformed equations are more complicated to write than the governing equations in the original coordinates, they are not significantly more difficult to incorporate in a computer code. The transformed coordinates should be useful for solving the more complete problem involving diffusion, salt-film precipitation and other corrosion-related phenomena.
REFERENCES (Editor) Corrosion Processes, Applied Science Publishers, New York (1982). 2. Z. Szklarska-Smialowska, Pitting Corrosion of Metals, 1. R. N. Darkin
+ = 0.9983 - 0.5291 T 3.
(37) This correlation for E, and the data shown in Fig. 14 treat the range 0 < 6/(1 + 6) < 0.8, or 0 < 6 6 4. For larger values of 6, a singular perturbation analysis analogous to that completed by Smyrl and Newman[ZS] for the disk electrode subject to high 6 should be undertaken to ensure accuracy in the calculated results. In a similar vein, a singular perturbation analysis can be used to determine more accurately the value of i,,,a/Vx as j approaches rr/ 2[15]. For both the linear correlation [i,, a/VK as a function of j, equation (36)] and the Tafef correlation [l/E, as a function of 6, equation (37)], the correlation formulas produced results that were indistinguishable from the numerical calculations when plotted as correlated.
4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
14.
SUMMARY The secondary current distribution for a corroding pit is analyzed for both linear and Tafel polarization. One dimensionless group can be identified for secondary current distribution analyses that employ linear and Tafel polarization laws and contain only one characteristic dimension (here the pit radius). In this work, the dimensionless average current den-
15. 16. 17. 18.
National Association of Corrosion Engineers, Houston (1986). P. A. Schweitzer (Editor) Corrosion and Corrosion Protection Handbook, second edition, Marcel Dekker, New York (1989). See p. 164 for a discussion of pitting corrosion and passivating metals. H. W. Pickering and R. P. Frankenthal, J. electrochem. Sot. US,1297 (1972). J. R. Galvele, J. electrochem. Sot. 123,464 (1976). R. Alkire, D. Ernsberger and D. Damon, J. electrochem. Sot. 123,458 (1976). K. Nisaucioglu and H. Holtan, Electrochim. Acta 23, 251 (1978). J. Newman, D. N. Hanson and K. Vetter, EIectrochim. Acta 22,829 (1977). J. N. Harb and R. C. Alkire, J. electrochem. Sot. 138, 2594 (1991). H. P. Leckie and H. H. Uhlig, J. electrochem. Sot. 113, 1262 (1966). H. Biihni and H. H. Uhlig, J. elecrrochem. Sot. 116, 906 (1969). E. McCafferty, J. electrochem. Sot. 131,373l (1990). H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, second edition, Oxford University Press, New York (1959). J. Newman, J. electrochem. Sot. 113, 1235 (1966). [See also J. Newman, J. electrochem. Sot. 114,239 (1967).] K. Nisancioglu and J. Newman, J. electrochem. Sot. 120,1339 (1973). A. M. Bond, K. B. Oldham and C. G. Zoski, J. electroanal. Chem. 245,71(1988). D. R. Baker and M. W. Verbrugge, J. electrochem. Sot. 137,3836 (1990). D. Zwillinger, Handbook of Diffuential Equations, Academic Press, New York (1989). See p. 335, comment 5, for the use of conformal representations and numerical simulation. Related references are given on p. 336.
Reaction distribution in a corroding pit 19. W. H. Smyrl and J. Newman, .I. electrochem. Sot. 123, 1423 (1976). 20. C. Amatore, in Ultramicroelectrodes (Edited by M. Fleschmann, S. Pons, D. R. Rolison and P. P. Schmidt), pp. 169-183. Datatech Systems, Morgantown, NC (1987). 21. A. C. Michael, R. M. Wightman and C. A. Amatore, J. electroad. Chem. 267,33 (1989). 22. M. W. Verbrugge and D. R. Baker, J. phys. Chem. %, 4572 (1992).
1659
23. J. Newman, Electrochemical Systems, second edition, Prentice-Hall, Englewood Clitfs, New Jersey (1991). 24. P. Moon and D. E. Spencer, Field Theory Handbook, second edition, Springer Verlag, Berlin (1971). 25. W. H. Smyrl and J. Newman~J. electrod&. Sot. 136, 132 (1989). (See Fin. A-l for the Tafel correlation ulot analogous to Fig. 14 of this work.) 26. A. C. West and J. Newman, J. electrockem. Sot. 136, 139 (1989). (See Fig. 2 for the Tafel correlation plot similar to Fig 14 of this work.)