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The determination of the shape of developing corrosion pits
Corrosion Science, Vol. 36, No. 10, pp. 1711-1725, 1994
Pergamon
Copyright ~) 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved (KII0-938X/94 $7.00+0.00
O010-938X(94)OOO62-X
THE
DETERMINATION OF THE DEVELOPING CORROSION
SHAPE PITS
OF
G. E N G E L H A R D T and H.-H. S T R E H B L O W lnstitut fiir Physikalische Chemie und Electrochemie, Heinrich-Heine-Universit~it, D-40225 Dfisseldorf, Germany
Abstract--A simple numerical method was developed to calculate the shape and size of growing corrosion pits. It was shown that the surface of a closed pit, growing under mass-transfer control in the absence of convection, can be described with the help of a general expression which depends only on the initial geometry of the system but not on its physical properties. For sufficiently large times they can be approximated by the equation by an ellipsoid of revolution. The growth of open and closed corrosion pits on iron in 1 M NaCI was simulated in case the current density at their surface is determined by a Tafel kinetic expression. The shape of these pits depends on their initial form and on a possible supersaturation of the system. Numerical calculations are compared with published experimental data.
INTRODUCTION The kinetics of localized corrosion is often decisively influenced by the transport of ions within the electrolyte. In most theoretical works the mathematical analysis of the transfer processes within corrosion pits assumes a pit shape constant with time. 1 For simplicity many authors investigate what they describe as one dimensional pits in their experimental and theoretical investigations. They assume cylindrical channels with a depth much bigger than the radius which are submitted to metal dissolution at their bottom only. 2-7 The ionic concentrations and the potential are assumed not to change outside of the pit. For these simple cases an analytical solution is achieved for the transport phenomena. In several papers the pit is assumed to have the shape of a hemisphere, a spherical section, a cylinder with a dissolving bottom and a lateral surface, a narrow crevice etc. 8-12 It seems obvious that mathematical models describing the pit growth should start with the assumption that the pit form and the size are not known a priori but should be found during the solution of the appropriate equations. Such calculations allow a prediction of the shape and size of a pit as a function of the parameters controlling the growth processes at any given time. They also allow an evaluation of the error when the form of the pit is assumed to be constant. In this paper a simple numerical method for the determination of the form and the size of a corrosion pit is described with different assumptions for the boundary conditions of the dissolution process.
Manuscript received 8 March 1994. 1711
1712
G. Engelhardtand H.-H. Strehblow MATHEMATICAL M O D E L
From the mathematical point of view of the problem of determination of the shape and size of a developing pit belongs to the class of Stefan problems. 13 These problems are reduced to the solution of a system of differential equations of parabolic or elliptic type with unknown moving boundary. The velocity of this boundary is connected to the sought solution by some differential relationship. The implicit equation qb = ~(t, x1, x2, X3) = 0 shall describe the pit surface with time relative to the Cartesian coordinate system Xl, x2, x3. It fulfils the relation d~ = (O~/Ot + V~ • Vs) dt = 0, where Vs(dXl/dt, dx2ldt, dx3/dt ) is the velocity of the dissolving metal surface. In this paper, subscript "s" refers to the electrode surface and subscript "oo" to the bulk of the solution. Let n = Vqb/lVqbI be the unit vector pointing normally from the solution to the metal surface. According to Faraday's law Vs = Kvin, where i is the current density of metal dissolution and Kv is the electrochemical equivalent volume, i.e. the volume of dissolving metal when one Faraday of charge passes through the interface. Introducing the correlation for Vs into the expression for the differential d ~ = (a~/~t + VsV~)dt = 0 yields equation (1) describing the change of the electrode surface
(1)
a~ + gvislV~[ = 0
Ot
Equation (1) is frequently applied to the description of some important processes in electrochemical technology, e.g. electrochemical machining of metals or electrochemical shape-formation. 14,15 For simplicity in the present paper an axially symmetrical pit shape is assumed for any time of the dissolution process (Fig. 1). For the case of a closed pit (Fig. la) it is supposed that a round defect with radius ro is formed on the plane passive surface which stays constant for the time of the dissolution process, and thus yields a remaining passive layer when the metal is dissolved. The consideration of these specific systems is essential for various technological problems as e.g. device technology, 16 photo-electrochemical milling, 17 electrochemical machining, etc. In the case of an open pit (Fig. lb), one assumes an extremely thin protective passive layer of only some few molecular layers, ~8 which is destroyed immediately even for an insignificant subetching.
(b)
(a) X
I Metal
•
"{" / Passive film
~r0~
"r
t,,\\\\',<
Passive film~
LroJ
I= Electrolyte Fig. I. Schematicdiagramof discussedpit geometries:(a) closedpit; and (b) open pit.
The determination of the shape of developing corrosion pits
1713
In both cases the position of the metal/electrolyte interface is denoted in the motionless stationary system of polar coordinates (Fig. 1) by x = 6(r, t), i.e. • = x 6(r, t). Introducing this into equation (1) yields
06 _ at
Kvis~l + (0612\ Or/
Kvi~ cos (n, x)'
(2)
where cos(n, x) is cosine of the angle between the normal of the dissolving pit surface and the x-axis. Besides the information about the boundary position in the initial period 6(0, r) the integration of this equation requires information about the dissolution current density i~, at the metal surface, i.e. equation (1) or (2) should be solved with a set of unsteady equations of ion transfer. However, due to the applied experimental conditions the velocity of the interface is so slow, that the quasi-steady-state approximation can be used for the description of ion transfer processes for the anodic dissolution of metals. 1"~9 This means that in any moment the distributions of all species in the solution as well as electric potential coincide with the stationary profiles corresponding to the position of the pit surface. For this paper the following set of equations is applied describing the steady-state ion transfer in dilute solutions V'Jk=0;
k=l,2 .....
N
-- ZkUkFCk~Tt~;
Jk : - - D k V C k
(3) k = 1,2 . . . . .
EZkCk = 0
N
(4)
(5)
k
Here Ck is the concentration of species k, Jk its flux density, D k is the diffusion coefficient, related to the ion mobility Uk by the Nernst-Einstein relation D k : ukRT; D k depends, in the general case, on the concentration of each ion component. Zk is the charge, Tis the temperature, R is the gas constant, Fis the Faraday constant and is the electrical potential. The index "k" denotes the ionic species in the solution, with a total of N species. It is assumed that convection has no essential influence on the mass-transfer. No homogeneous chemical reaction shall occur within the solution preceding or following the electrochemical reaction. The following boundary conditions were chosen for the set of transfer equations (3)-(5). At points far from the mouth of the corrosion pit the concentrations and potential have their bulk values, i.e. Ck=Ck~;
q~=tPo~=0.
(6)
At the passivated surface, it is assumed that the normal flux of each species disappears, i.e.: Jkn = 0;
k=l,2 ..... N
(7)
At the active electrode surface the simple anodic metal dissolution occurs according to Equation (8) Me ~ Me n+ + n e -
(8)
1714
G. EngelhardtandH.-H.Strehblow
For this process two types of boundary conditions shall hold: (1) At the electrode surface the saturation concentration of a product of an electrochemical reaction is achieved (limiting current conditions) and the normal fluxes of all other ionic species different from the ions of the dissolving metal equal to zero, i.e. F(Cls, C2s . . . . .
C3s) = 0;
Jkn = 0;
k = 2, 3 . . . . .
N
(9)
Here the index "1" refers to the metal ions and F = 0 is the equation for saturation. For example, if in the solution only Fe 2+, Na ÷ and C1- ions are present, the following linear expression will hold approximately for saturation concentrations at the metal surface: 2° Cls = a - bC2s
(10)
Here C1 and (72 are molar concentrations of Fe and Na, a = C l , s a t , b = Cl,sat/f2,sat; C l , s a t = 4.2 M and C2.sat = 5.3 M are the solubilities of FeC12 and NaCI in water, respectively. The saturation condition (9) is frequently valid for the late stage of pit growth. 2~ (2) The rate of dissolution is determined by the kinetic equation is = ZlFJln
=f(Cks , V- q~s);
Jk n = 0,
k = 2, 3 . . . . , N,
(11)
where V is the electrode potential of the metal electrode relative to the reference electrode of the same kind as the working electrode located in the bulk of the solution through which only a negligible current is passed and f i s the known function. For a more general case it is possible that conditions (9) and (11) will hold simultaneously at the different parts of the electrode surface. It has been shown 22 that the solution of the system of ion transfer equations (2)-(5) satisfying boundary condition (6) can be reduced to the nonlinear differential equation V . [D(C1)VCI]
= O,
(12)
if only one electrochemical reaction takes place at the electrode surface. Here D ( C 1) is an effective diffusion coefficient. Its value is given by the following equation during metal dissolution by reaction (8) in a symmetric electrolyte gz+Az-:23 D=DI
I 1+2-n2~/( ~ n )2 2C1
1
(13)
c, + c2~c3~ The subscript "2" refers to the cation K z+ and "3" to the anion A z-. The concentrations C2 and (73 and potential ~ are expressed by CI due to the following equations:
c3=
~z Cl +c2.c3~,
(72 = (73 - _n C1,
z
(14)
The determination of the shape of developing corrosion pits
1715
R T In C 3 . (/) = z--e C3~
Equation (12) can be simplified by the use of the quasi-concentration determined by equation (15) 1 I c, D(C1) dC 1 C = D--~ c,,~
(15)
Function C satisfies Laplace's equation and is equal to zero for the bulk solution. Its normal derivative for the passivated surface part is equal to zero too. If the saturation is achieved for the reaction products at the electrode surface, the quasi-concentration at the surface C~ reaches a constant value which can be expressed according to equations (9), (14) and (15). Taking into account that
(16) the equation of the movement of the metal surface (2) can be rewritten as -Or -
k°X
~
tan (x, n) =
[ Ox
Or OrJ
(17)
In order to reduce the number of the parameters from which the solution may depend when saturation is achieved at the pit surface the following dimensionless variables and parameters are introduced: H = h/ro, L = 1/r,,, C* = C/C~, X = x / r o, R = r/ro, A = 6/ro, "9 d* = d / r o, r = t/to, to = ~ / ( z 1 F D = C ~ K v ) = ~ p m / ( D = C ~ M I ) ,
where Pm and M 1 a r e the specific density and molecular mass of the metal Me respectively. Here Kv = M l ( z l F p m ) was considered according to equation (8). The dimensionless concentration C* and the dimensionless position of the interface A can be found as a solution of the following boundary problem: V2C * = 0,
(19a)
C* = 0 for the bulk solution; 6 C * / 6 n = 0 at the passive surface; and C* = 1 at the active part of the surface of the electrode (at X = A(R,r)), OA Or
OC*
- - -
OX
OC* OA
+ - -
OR OR
A = 0 at r = 0.
(19b)
It is obvious that the solution of the boundary problem should be C* = and A = A ( R , r , d * ) . If the thickness of the passive layer or an insulating surface film becomes vanishingly small, i.e. when d* --> 0, the solution of the problem is described by the universal dependence C* = C*(X, Y, r) and A = C*(X,R,r,d*)
A(R, The form of the corrosion pit does not depend either on the dissolving metal or on the composition of the solution when the saturation concentration is achieved at the
1716
G. Engelhardt and H.-H. Strehblow Table 1. Valuesof the surface concentration Fe 2+ (Cls), Na+ (Czs), the potential drop in the solution ~s, surface quasi-concentration Cs and the value of the effective bulk diffusion coefficient D= for iron dissolution in NaC1 with bulk concentration C2= C2~ (M)
Cls (M)
Czs (M)
~s (mV)
C~ (M)
D~ × 10-5 (cm2 s-1)
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
4.176 4.105 3.984 3.809 3.577 3.277 2.896 2.410 1.774 0.869
0.029 0.120 0.273 0.493 0.787 1.166 1.647 2.260 3.062 4.206
72.8 54.8 44.0 36.2 29.9 24.4 19.5 14.7 9.9 4.4
5.880 5.449 5.066 4.700 4.326 3.918 3.445 2.860 2.084 0.907
0.70 0.64 0.60 0.56 0.53 0.51 0.48 0.46 0.43 0.41
electrode surface. In other words, such non-linear factors as migration and variation of diffusion coefficients with concentration do not influence the shape of the pit. However, these factors influence the time to achieve a given pit size by the quantities D~ and Cs which enter in the definition of the characteristic time to. In Table 1 values of Doo and Cs are presented for Fe dissolution in NaCI solutions of different concentrations. The quasi-concentration Cs was calculated according to equations (10) and (14). The values of D~ that coincide with DI~ according to equation (13) were calculated by Walden's rule (D1/~ = constant) based on known viscosity values/~ of aqueous NaCI solutions. 2° If the boundary condition (11) will hold at the electrode surface the transition to dimensionless variables leads to a large number of dimensionless parameters. In this case the calculations were performed for a specific system: dissolution of Fe in 1 M NaC1. For this situation equations (2) and (12) were solved with boundary conditions (6), (7) and (11). Equation (11) is approximated by a Tafel kinetic expression is = io exp
(13zlF(V
-
~bs)/RT),
(20)
where io = 10 -8 A cm -2 and fl = 0.4 are assumed. 24
NUMERICAL METHOD The solution of the problems formulated above was obtained by the method of finite differences using a rectangular non-uniform grid spacing (Fig. 2). The spatial intervals close to the pit mouth in direction x and r, &x and Ar, were usually chosen from the condition &x = Ar = ro/5. However, for the model with an infinitely thin passive layer the corresponding step along the x-axis &x = d = ro/500 was chosen. The grid spacing steps were enlarged sharply far from the pit mouth outside the pit in order to save computer time. We consider that the position of the metal/electrolyte interface of a corrosion pit is known for a given time t (for example, curve I, Fig. 2). The position of this boundary is defined by the coordinates of its crossing points, marked by numbers 110, with the coordinate axes. Methods for the solution of Laplace's equation and for
The determination of the shape of developing corrosion pits
,._,6
Ax' r"
1717
\
l " - ~ " ~ ~.~
:,0 I" 10~, ,
xT I~ Fig. 2.
12
"/iV/, "//I/.//
r0
3_d 2-
A v]
Cross section through the axis of symmetry of a pit with defined grid system.
the solution of the more general equation (12) for the arbitrary two-dimensional region are well known from literature. 25 For the solution of these equations a locally one-dimensional method was applied for this work. 26 When the solution is found, the current density at the metal surface can be calculated numerically using one of the relations (16). The value of the angle between the surface normal n and the x-axis can be found for point i of the border using the relation tan (n, x) = (Xi+l - xi)(ri - r i - l ) 2 + (Xi - x i - 1 ) ( r i + l - ri)2 ( r i - ri-l)(ri+l - ri)(ri+l - ri-l)
(21)
For the right point of the phase boundary (i = 10, Fig. 2), tan(n,x) = (xi x i ) / ( r i - 1 - ri) will hold. As it follows from equation (2), the coordinates of points lying on an arbitrary grid line parallel to the x-axis (points 1,2, 3, 4, 6, 7), when the phase boundary shifts from position I to II for time t + At, can be found from the relation
X i - l)/(ri -- r i - l), and for the left point (i = 1), tan(n,x) = (xi + 1 -
Xu
= xl -
Kvi~At/cos(n,
x)
(22)
The position of the points of the new phase boundary which are located on grid lines parallel to the r axis can be found in principle with the help of interpolation. The determination of a new position of the interface corresponding to the solution to equation (2) can lead to a decrease in accuracy for the region where cos(n, x) ~ 0, i.e. in the region where the boundary is parallel or almost parallel to the x axis. This situation occurs close to the point of undercutting of the isolating passive layer (point 10) for limiting current conditions. For this region it is convenient to shift all points which lay on the grid lines parallel to r according to equation (23) which is analogous to equation (22) ru = rI -
KvisAt/cos(n,
r)
(23)
1718
G. Engelhardt and H.-H. Strehblow
It is convenient to use equation (22) in the region where the angle between the xaxis and the surface normal n is less or equal ~r/4 (tan(n,x) -< 1), but equation (23) in the region where this angle is larger than :r/4. Thus on Fig. 2 points 1, 2, 3, 4, 6 are shifted along x-axis by equation (22), but points 8, 9, 10 along the r-axis by equation (23). The position of points 5, 7, 8, 9 on the new boundary are found by linear interpolation using the known coordinates of the rest of the boundary points. After determination of the new position of the interface the boundary problem is solved for the new region. The further changes are calculated with the same procedure. When this algorithm is applied, it is useful to keep in mind the following two details. The solution of the boundary problem which is found for time t can serve as a good initial approximation for the solution of time t + At, i.e. for the new position of the pit surface. The concentration values at the grid's points which belong for time t to the metallic phase but for t + At to the solution can be obtained by interpolation. These points are marked by * on Fig. 2. The distance Ax* (or ArT) may be very small. This situation may lead to a significant error for the calculated flux which is proportional to 1/Ax*. In this situation it is reasonable to change AX*oby a small quantity Axo if Ax* < &,Co.For the present work AXo = ro/250 was chosen.
THE RESULTS OF N U M E R I C A L CALCULATIONS Figure 3 presents the change of the metal/electrolyte interface of a closed pit during anodic dissolution, obtained from the numerical solution of the boundary problem (19), for the case of an infinitely thin barrier layer. According to these calculations the surface form can be approximated by an ellipsoid of revolution
100 ~
._o ~
N
-- 50
2
1
I 0
1
2
4
R (dimensionless) Fig. 3. The shape of a closed pit with normalized coordinates at different dimensionless times. Dashed curve represents results of experiments.16 Initial form of the pit is a plane disk.
The determination of the shape of developing corrosion pits
1719
around the x-axis for the related values of dimensionless time r > 5, i.e. the equation for any cross section through the x-axis can be approximated by the following equation of an ellipse X 2 / H 2 + R2/L 2 = 1
(24)
The time dependence of the axes of this ellipse for r > 5 can be approximated with an accuracy better than 10% by H = brl/3; L = a + b r 1/3,
(25)
where a = 0.5 and b = 0.75. From equation (25) follows that the surface of the dissolving pit can be approximated for r ~ o0 by a sphere. For r > 150 the axes H and L differ less than 10%. Figure 3 presents the experimental results for the shape of a cavity found during diffusion controlled dissolution of GaAs. 16 At time r = 13 the depth H of a pit calculated according to equation (25) coincides with the experimental value. The agreement of calculated and experimental results for the pit shape and size confirms the applicability of equations (24) and (25) for the determination of the shape of closed pits during electrochemical or chemical dissolution with diffusion-controlled conditions. For the majority of electrochemical experiments the dependence of the characteristic pit size (depth h, mouth radius rp) on time t can be expressed by simple relation h (or rp) -- tg
(26)
Registered values of g equal approximately 1/3, 1/2, 2/3 and 1.0.11"27-30 It follows from equation (25) that the power g equals to 1/3 during dissolution of a closed pit for current limited conditions. In the case of one-dimensional pits g = 1/2 is obtained for this condition. 31 According to equation (25) the transport limited current l,m through a closed pit with half-ellipsoid shape is described by equation (27): I,m = (dVpit/dt)/Kv = /lim.~ (1 nt- 0 . 8 8 8 / T 1/3 "{,- 0 . 1 4 8 / r 2 / 3 ) ,
(27)
where Vpit = 2:rhl2/3 is the volume of the pit and Ilim,~ -- 0.884r3o/gvto the current flowing through it at t ~ ~. Relation (27) can be compared with experimental results. For a one-dimensional pit one obtains I ~ 1~kit. Calculations have shown that in the region 0 -< d* -< 1 the form of the pit practically does not depend on the passive layer thickness. The shape of the cavity can be described by equations (24) and (25), but the parameter b becomes a linear function of dimensionless isolation d*: b = 0.75 - 0.12d*;0-< d* -< 1
(28)
Although the pit shape depends on its initial form, this influence is smoothed at
1720
G. Engelhardt and H.-H. Strehblow 5O
_o
2
tO
"5 t-
X
0
1
2
3
R (dimensionless) Fig. 4.
The shape of a closed pit with normalized coordinates at different dimensionless times. Initial form of the pit is a hemisphere.
t ~ ~. Figure 4 shows the growth of a closed, initially hemispherical pit which changes to an ellipsoidal form which then transfers to spherical form at t ~ ~. If the metal dissolution is charge transfer controlled and does not depend on transport phenomena, the local current density is constant with time. For this situation it is convenient to introduce the dimensionless time parameter t* = and rewrite equation (2) in the following dimensionless form:
(-isKv/ro)t
aA ~
b-r-
(aAI2
(29)
1 + \0R/
According to equation (29) the pit shape is determined only by the initial conditions. For example, if the initial pit shape is a hemisphere with radius R = 1, one obtains the following solution of equation (29) A=N/(I+t*) 2-R2;
R-
(30)
i.e. the pit keeps its hemispherical form for its further growth. If the initial pit has the form of a cone inclined to the passivated surface with an angle a (Fig. 5) it will remain a cone for any later stage of growth described by the linear function A = t*/cos a + (1 - R) tan (a);
R -< 1 +
t*/sin(a)
(31)
X
Metal
v r
Passive film / Electrolyte Fig. 5.
Schematic diagram of a pit with cone geometry.
The determination of the shape of developing corrosion pits 3
-
2-
.~
1721
1000 s
X 0
I
~
I
1
2
3
\i 4
IN
J
5
6
R (dimensionless) Fig. 6. The shape of closed pit with normalized coordinates for Fe dissolution in 1 M NaCI at different times. Initial form of the pit is a plane disk with radius ro = 30pm. V = 0.55 V.
If we assume the growth of a plane defect within a barrier layer one has to follow the limiting transition a ~ 0 for equation (31) (see Fig. 5). For this case with undercutting of a passivated surface by AL = L - 1 = t*/sin a--~ oo with a--~ 0 a stable pit growth is evidently not possible. In the opposite case, when the initially plane defect growths under mass-transfer control, AL has a finite value for any time. Accordingly, one can expect that in the intermediate case of mixed kinetics the sub-etching under the barrier layer Al will be larger than in the case of transport-limited current densities for the growth a pit with the same depth. This suggestion is confirmed by numerical calculations carried out with the above mentioned method for Fe dissolution in 1 M NaCI at V = 0.55 V. (Fig. 6). This can be also explained by the use of equation (23). For diffusion-controlled conditions the current density where an electrode meets an insulator is either infinite (at a > 90°) or zero (at a < 90 °) unless a right angle is formed 32 (see Fig. 1). Thus, for these conditions, the motion of the interface boundary occurs so that the right angle (a = 90 °) is established by the etching process. For mixed kinetics a is not rectangular, and cos(n,r) becomes less than 1, which leads to the increase of the subetching Al. Figure 7 illustrates the influence of the initial shape of an open pit on its further
3 o o
"0
~ M
2
1
I 0
1
2
3
4
5
R (dimensionless)
Fig. 7. The shape of an open pit with normalized coordinates for Fe dissolution in 1 M NaCI. Initial form of the pit is a hemisphere (1) and a plane disk (2). r o = 10pm. V = 0.55 V.
1722
G. Engelhardt and H.-H. Strehblow
2 o e-
.o
X 0.625 v o
1
2
3
R (dimensionless) Fig. 8.
The shape of an open pit with normalized coordinates for Fe dissolution in 1 M NaCI at different applied potential. Of value ro = 30/Lm.
development. For the initial shape a hemisphere or a plane disk are assumed, both with the same radius r o -- 10 ~tm. In both cases corrosion pits tend to become more shallow as they increase in size, which shows that the dissolution rate is higher at the sides of the pit than at its bottom. When the pit depth h gets three times larger than the initial radius r o the ratio of the pit radius r to its depth r/h equals 1.15 for an initial hemisphere and 1.6 for an initial disk. In Ref. 10 is has been shown experimentally that for pitting of Ni in 0.5 M NaCl this ratio does not exceed 1.25 for pits up to 100 /~m diameter if saturation is not reached in the cavity. Our calculations show that if saturation is not reached in the cavity and the initial shape of pits was a hemisphere this ratio does not exceed 1.2 for Fe dissolution in 1 M NaCl for pits up to 60 ~tm diameter. This result shows that the assumption of an initially hemispherical pit shape is more realistic compared to a planar form. Figure 8 shows the influence of the electrode potential V on the form an initially hemispherical pit when its depth has increased twice of its initial value. The calculated form and size of a pit depend strongly on a possible supersaturation. For example, in Ref. 10 it is suggested that some supersaturation occurs for pitting of Ni. However, it is reasonable to assume for the general case a saturation limit. For this work the calculations were performed in the following way. In any point of the pit surface it was verified if the surface concentration of FeCl 2 exceeds the saturation level, i.e. 4.105 M for the case of Fe dissolution in 1 M NaC1 (see the table). If so, the boundary condition (9) is used for this point, if not, the boundary condition (11) is applied. For V -- 0.625 V (Fig. 8) saturation is reached on the electrode surface for R -1.1. From Figs 9 and 10, one may deduce that supersaturation is an important factor for the calculation of the current density and the shape of the pit. For example, the current density at the bottom of the pit may decrease with the increase of the electrode potential V if the concentration does not exceed uniformly the saturation value at the whole pit surface. This may be explained the following way. The increase of V leads to an increase of the metal ion flux at the sides of the pit wall where the saturation limit is not yet reached. This effect will cause a general increase of the metal concentration inside the pit. As the surface concentration at the bottom of the pit has reached saturation and therefore is constant, the concentration gradient at the bottom will decrease together with the current density at this point. Of course, the
The determination of the shape of developing corrosion pits
. . . . . . . . . °°... °°... -. 0.65 V "'.... 0.65 V'"-.
4
0.625
r..) --
0.6
0
v
V
I
I
I
I
25
50
75
100
Angle (degrees) Fig. 9. Local concentration of Fe 2+ along the bottom of a hemispherical pit at different applied potentials (bottom 0°, edge 90°). Dashed curve was calculated with the assumption of a supersaturation. Of value r o = 30/zm.
1.5 -0.65 v
1.0
"'"""
-
E
<
o,,
0
o/
25
50
laov
75
100
Angle (degrees) Fig. 10. Local current density of the bottom of a hemispherical pit for different applied potentials (bottom 0°, edge 90°). Dashed curve was calculated with the assumption of a supersaturation. Of value ro = 30 pm.
1723
1724
G. Engelhardt and H.-H. Strehblow
~
l
0
1
2
3
4
5
R (dimensionless) Fig. 11.
The shape of a partially closed pit with normalized coordinates. Initial form of the pit is a hemisphere.
total current flow within the pit must increase with the electrode potential V till saturation is reached all over the dissolving pit surface. Dissolution within a pit for limiting current conditions and with zero sub-etching of the barrier layer (Fig. lb) should not occur for simple mathematical reasons. This is a consequence of the properties of the solution of Laplace's equation with a local current density i --~ ~ at the pit edge. One possibility to describe the boundary problem for this condition is to assume that the pit is partially open because of a remaining part of the barrier layer. For example, it can be supposed that the passive layer is mechanically destroyed or dissolved if the sub-etching exceeds some critical value A/1 with a remaining subetching of Al2. Figure 11 shows the shape of a partially open, initially hemispherical pit when its radius increases to three and to five times of its initial value. The calculations were performed with Al 1 = 0.4r o and Al2 = 0.2r o. If, for example, the pit radius increases from r o -- 10/~m to r -- 50~tm, the ratio of radius of the pit to its depth equals 2.6. Ratios of 2.75 were observed for shallow pits growing under masstransport control on stainless steel. 10 For this condition the value of Al 1 equals only 4 /~m which is a small fraction of the pit radius. It seems reasonable that the remaining passive layer is dissolved by the aggressive pit electrolyte. High C1- concentrations favor fast dissolution of Fe-oxides. 18
CONCLUSIONS A simple method was developed to calculate the shape and the size of growing corrosion pits and to demonstrate the concentration distribution of metal ions and the electrical potential within the cavities. It was shown that the surface of a closed pit growing under mass-transfer control in the absence of convection can be described with the help of a general expression which depends only on the initial geometry of the system but not on its physical properties as, for example, diffusion coefficients or charges of species. These expressions were determined for the cases of initially plane or hemispherical pits. For sufficiently large times they can be approximated by the equation by an ellipsoid of revolution. The method was used to simulate the growth of open and closed corrosion pits on iron in 1 M NaCI in the case that the current density at the surface is determined by a Tafel kinetic expression. It was shown that the shape of these pits depends on their initial form and on a possible supersaturation of the system. The calculation of the
The determination of the shape of developing corrosion pits
1725
shape of open pits with limiting current conditions requires assumptions for the limits of the dissolution rates at the contact of the active metal and the passivated surface. The finite difference techniques for the displacement of the boundary of a pit described in this paper is absolutely independent of mass-transport problems. It can be used jointly with an arbitrary finite difference method describing transport phenomena in systems with the fixed boundary including chemical reactions and non-steady-state processes. Acknowledgements--The support of Dr G. Engelhardt by the Deutsche Forschungsgemeinschaft during his research stay at the Heinrich-Heine-Universit~it Diisseldorf is gratefully acknowledged.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.
S. M. Sharland, Corros. Sci. 27,289 (1987). J. R. Galvele, J. electrochem. Soc. 123,464 (1976). H. S. Isaaks, J. electrochem. Soc. 120, 1456 (1973). J. V. Tester and H. S. Isaaks, J. electrochem. Soc. 122, 1438 (1975). R. Alkire, D. Ernsberger and D. Damon, J. electrochem. Soc. 123,458 (1976). H. W. Pickering and R. P. Frankenthal, J. electrochem. Soc. 119, 1297 (1972). D. G. Atea and H. W. Pickering, J. Appl. Electrochem. 11,453 (1981). J. Newman, D. N. Hanson and K. Vetter, Electrochim. Acta 829, 22 (1977). M. W. Verbrugge, M. W. Baker and J. Newman, J. electrochem. Soc. 140, 2530 (1993). J. N. Harb and R. C. Alkire. J. electrochem. Soc. 138, 2594 (1991). S. Szklarska-Smialowska, In Localized corrosion, p. 312. NACE (1974). T. R. Beck and E. A. Grens, J. Electrochem. Soc. 116, 117 (1969). H.S. Karslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edn, Oxford Press, London (1959). J. A. McGeough, Principles o f Electrochemical Machining, Chapman and Hall, London (1974). A. D. Davydov and J. Kozak, High-Rate Electrochemical Shaping, Nauka, Moscow (1990). H. K. Kuiken, J. J. Kelly and P. H. L. Nottin, J. electrochem. Soc. 133, 1217 (1986). B. Chatterjee, Precision Engineering 8, 131 (1986). K. J. Vetter and H.-H. Strehblow, In Localized Corrosion, p. 240. NACE (1974). G. R. Engelhardt, A. D. Davydov and V. S. Krylov, Soviet Electrochemistry 13, 1103 (1979). G. R. Engelhardt, G. N. Zaidman, G. S. Suvorova, G. S. Domente and E. A. Vysokova, Soviet Surface Engineering and Applied Electrochemistry, No. 6, p. 1 (1987). H.-H. Strehblow, Werk. Korros. 35,437 (1984). B. P. Pillay and J. Newman, J. electrochem. Soc. 140,414 (1993). G. R. Engelhardt and H.-H. Strehblow, J. electroan. Chem. 365, 7 (1994). K. J. Vetter, Electrochemical Kinetics, Academic Press, New York (1967). S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington (1980). A. R. Mitchel and D. D. Griffits, The Finite Difference Method in Partial Difference Equations, John Wiley & Sons, Chichester (1980). F. Forchamer and H. J. Engel, Werk. Korros. 20, 1 (1969). N. Sato, T. Nakayawa, K. Kudo and M. Sakashita, In Localized Corrosion, p. 447. NACE (1974). J. Tougek, Werk. Korros. 25,496 (1974). J. Tougek, Corrosion 33,193 (1977). G. R. Engelhardt, A. D. Davydov, T. B. Zhukova and H.-H. Strehblow, Soviet Electrochemistry 28, 199 (1992). J. Newman, Electrochemical Systems, Prentice-Hall, Inc., Englewood Cliffs, NJ (1973).
Pergamon
Copyright ~) 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved (KII0-938X/94 $7.00+0.00
O010-938X(94)OOO62-X
THE
DETERMINATION OF THE DEVELOPING CORROSION
SHAPE PITS
OF
G. E N G E L H A R D T and H.-H. S T R E H B L O W lnstitut fiir Physikalische Chemie und Electrochemie, Heinrich-Heine-Universit~it, D-40225 Dfisseldorf, Germany
Abstract--A simple numerical method was developed to calculate the shape and size of growing corrosion pits. It was shown that the surface of a closed pit, growing under mass-transfer control in the absence of convection, can be described with the help of a general expression which depends only on the initial geometry of the system but not on its physical properties. For sufficiently large times they can be approximated by the equation by an ellipsoid of revolution. The growth of open and closed corrosion pits on iron in 1 M NaCI was simulated in case the current density at their surface is determined by a Tafel kinetic expression. The shape of these pits depends on their initial form and on a possible supersaturation of the system. Numerical calculations are compared with published experimental data.
INTRODUCTION The kinetics of localized corrosion is often decisively influenced by the transport of ions within the electrolyte. In most theoretical works the mathematical analysis of the transfer processes within corrosion pits assumes a pit shape constant with time. 1 For simplicity many authors investigate what they describe as one dimensional pits in their experimental and theoretical investigations. They assume cylindrical channels with a depth much bigger than the radius which are submitted to metal dissolution at their bottom only. 2-7 The ionic concentrations and the potential are assumed not to change outside of the pit. For these simple cases an analytical solution is achieved for the transport phenomena. In several papers the pit is assumed to have the shape of a hemisphere, a spherical section, a cylinder with a dissolving bottom and a lateral surface, a narrow crevice etc. 8-12 It seems obvious that mathematical models describing the pit growth should start with the assumption that the pit form and the size are not known a priori but should be found during the solution of the appropriate equations. Such calculations allow a prediction of the shape and size of a pit as a function of the parameters controlling the growth processes at any given time. They also allow an evaluation of the error when the form of the pit is assumed to be constant. In this paper a simple numerical method for the determination of the form and the size of a corrosion pit is described with different assumptions for the boundary conditions of the dissolution process.
Manuscript received 8 March 1994. 1711
1712
G. Engelhardtand H.-H. Strehblow MATHEMATICAL M O D E L
From the mathematical point of view of the problem of determination of the shape and size of a developing pit belongs to the class of Stefan problems. 13 These problems are reduced to the solution of a system of differential equations of parabolic or elliptic type with unknown moving boundary. The velocity of this boundary is connected to the sought solution by some differential relationship. The implicit equation qb = ~(t, x1, x2, X3) = 0 shall describe the pit surface with time relative to the Cartesian coordinate system Xl, x2, x3. It fulfils the relation d~ = (O~/Ot + V~ • Vs) dt = 0, where Vs(dXl/dt, dx2ldt, dx3/dt ) is the velocity of the dissolving metal surface. In this paper, subscript "s" refers to the electrode surface and subscript "oo" to the bulk of the solution. Let n = Vqb/lVqbI be the unit vector pointing normally from the solution to the metal surface. According to Faraday's law Vs = Kvin, where i is the current density of metal dissolution and Kv is the electrochemical equivalent volume, i.e. the volume of dissolving metal when one Faraday of charge passes through the interface. Introducing the correlation for Vs into the expression for the differential d ~ = (a~/~t + VsV~)dt = 0 yields equation (1) describing the change of the electrode surface
(1)
a~ + gvislV~[ = 0
Ot
Equation (1) is frequently applied to the description of some important processes in electrochemical technology, e.g. electrochemical machining of metals or electrochemical shape-formation. 14,15 For simplicity in the present paper an axially symmetrical pit shape is assumed for any time of the dissolution process (Fig. 1). For the case of a closed pit (Fig. la) it is supposed that a round defect with radius ro is formed on the plane passive surface which stays constant for the time of the dissolution process, and thus yields a remaining passive layer when the metal is dissolved. The consideration of these specific systems is essential for various technological problems as e.g. device technology, 16 photo-electrochemical milling, 17 electrochemical machining, etc. In the case of an open pit (Fig. lb), one assumes an extremely thin protective passive layer of only some few molecular layers, ~8 which is destroyed immediately even for an insignificant subetching.
(b)
(a) X
I Metal
•
"{" / Passive film
~r0~
"r
t,,\\\\',<
Passive film~
LroJ
I= Electrolyte Fig. I. Schematicdiagramof discussedpit geometries:(a) closedpit; and (b) open pit.
The determination of the shape of developing corrosion pits
1713
In both cases the position of the metal/electrolyte interface is denoted in the motionless stationary system of polar coordinates (Fig. 1) by x = 6(r, t), i.e. • = x 6(r, t). Introducing this into equation (1) yields
06 _ at
Kvis~l + (0612\ Or/
Kvi~ cos (n, x)'
(2)
where cos(n, x) is cosine of the angle between the normal of the dissolving pit surface and the x-axis. Besides the information about the boundary position in the initial period 6(0, r) the integration of this equation requires information about the dissolution current density i~, at the metal surface, i.e. equation (1) or (2) should be solved with a set of unsteady equations of ion transfer. However, due to the applied experimental conditions the velocity of the interface is so slow, that the quasi-steady-state approximation can be used for the description of ion transfer processes for the anodic dissolution of metals. 1"~9 This means that in any moment the distributions of all species in the solution as well as electric potential coincide with the stationary profiles corresponding to the position of the pit surface. For this paper the following set of equations is applied describing the steady-state ion transfer in dilute solutions V'Jk=0;
k=l,2 .....
N
-- ZkUkFCk~Tt~;
Jk : - - D k V C k
(3) k = 1,2 . . . . .
EZkCk = 0
N
(4)
(5)
k
Here Ck is the concentration of species k, Jk its flux density, D k is the diffusion coefficient, related to the ion mobility Uk by the Nernst-Einstein relation D k : ukRT; D k depends, in the general case, on the concentration of each ion component. Zk is the charge, Tis the temperature, R is the gas constant, Fis the Faraday constant and is the electrical potential. The index "k" denotes the ionic species in the solution, with a total of N species. It is assumed that convection has no essential influence on the mass-transfer. No homogeneous chemical reaction shall occur within the solution preceding or following the electrochemical reaction. The following boundary conditions were chosen for the set of transfer equations (3)-(5). At points far from the mouth of the corrosion pit the concentrations and potential have their bulk values, i.e. Ck=Ck~;
q~=tPo~=0.
(6)
At the passivated surface, it is assumed that the normal flux of each species disappears, i.e.: Jkn = 0;
k=l,2 ..... N
(7)
At the active electrode surface the simple anodic metal dissolution occurs according to Equation (8) Me ~ Me n+ + n e -
(8)
1714
G. EngelhardtandH.-H.Strehblow
For this process two types of boundary conditions shall hold: (1) At the electrode surface the saturation concentration of a product of an electrochemical reaction is achieved (limiting current conditions) and the normal fluxes of all other ionic species different from the ions of the dissolving metal equal to zero, i.e. F(Cls, C2s . . . . .
C3s) = 0;
Jkn = 0;
k = 2, 3 . . . . .
N
(9)
Here the index "1" refers to the metal ions and F = 0 is the equation for saturation. For example, if in the solution only Fe 2+, Na ÷ and C1- ions are present, the following linear expression will hold approximately for saturation concentrations at the metal surface: 2° Cls = a - bC2s
(10)
Here C1 and (72 are molar concentrations of Fe and Na, a = C l , s a t , b = Cl,sat/f2,sat; C l , s a t = 4.2 M and C2.sat = 5.3 M are the solubilities of FeC12 and NaCI in water, respectively. The saturation condition (9) is frequently valid for the late stage of pit growth. 2~ (2) The rate of dissolution is determined by the kinetic equation is = ZlFJln
=f(Cks , V- q~s);
Jk n = 0,
k = 2, 3 . . . . , N,
(11)
where V is the electrode potential of the metal electrode relative to the reference electrode of the same kind as the working electrode located in the bulk of the solution through which only a negligible current is passed and f i s the known function. For a more general case it is possible that conditions (9) and (11) will hold simultaneously at the different parts of the electrode surface. It has been shown 22 that the solution of the system of ion transfer equations (2)-(5) satisfying boundary condition (6) can be reduced to the nonlinear differential equation V . [D(C1)VCI]
= O,
(12)
if only one electrochemical reaction takes place at the electrode surface. Here D ( C 1) is an effective diffusion coefficient. Its value is given by the following equation during metal dissolution by reaction (8) in a symmetric electrolyte gz+Az-:23 D=DI
I 1+2-n2~/( ~ n )2 2C1
1
(13)
c, + c2~c3~ The subscript "2" refers to the cation K z+ and "3" to the anion A z-. The concentrations C2 and (73 and potential ~ are expressed by CI due to the following equations:
c3=
~z Cl +c2.c3~,
(72 = (73 - _n C1,
z
(14)
The determination of the shape of developing corrosion pits
1715
R T In C 3 . (/) = z--e C3~
Equation (12) can be simplified by the use of the quasi-concentration determined by equation (15) 1 I c, D(C1) dC 1 C = D--~ c,,~
(15)
Function C satisfies Laplace's equation and is equal to zero for the bulk solution. Its normal derivative for the passivated surface part is equal to zero too. If the saturation is achieved for the reaction products at the electrode surface, the quasi-concentration at the surface C~ reaches a constant value which can be expressed according to equations (9), (14) and (15). Taking into account that
(16) the equation of the movement of the metal surface (2) can be rewritten as -Or -
k°X
~
tan (x, n) =
[ Ox
Or OrJ
(17)
In order to reduce the number of the parameters from which the solution may depend when saturation is achieved at the pit surface the following dimensionless variables and parameters are introduced: H = h/ro, L = 1/r,,, C* = C/C~, X = x / r o, R = r/ro, A = 6/ro, "9 d* = d / r o, r = t/to, to = ~ / ( z 1 F D = C ~ K v ) = ~ p m / ( D = C ~ M I ) ,
where Pm and M 1 a r e the specific density and molecular mass of the metal Me respectively. Here Kv = M l ( z l F p m ) was considered according to equation (8). The dimensionless concentration C* and the dimensionless position of the interface A can be found as a solution of the following boundary problem: V2C * = 0,
(19a)
C* = 0 for the bulk solution; 6 C * / 6 n = 0 at the passive surface; and C* = 1 at the active part of the surface of the electrode (at X = A(R,r)), OA Or
OC*
- - -
OX
OC* OA
+ - -
OR OR
A = 0 at r = 0.
(19b)
It is obvious that the solution of the boundary problem should be C* = and A = A ( R , r , d * ) . If the thickness of the passive layer or an insulating surface film becomes vanishingly small, i.e. when d* --> 0, the solution of the problem is described by the universal dependence C* = C*(X, Y, r) and A = C*(X,R,r,d*)
A(R, The form of the corrosion pit does not depend either on the dissolving metal or on the composition of the solution when the saturation concentration is achieved at the
1716
G. Engelhardt and H.-H. Strehblow Table 1. Valuesof the surface concentration Fe 2+ (Cls), Na+ (Czs), the potential drop in the solution ~s, surface quasi-concentration Cs and the value of the effective bulk diffusion coefficient D= for iron dissolution in NaC1 with bulk concentration C2= C2~ (M)
Cls (M)
Czs (M)
~s (mV)
C~ (M)
D~ × 10-5 (cm2 s-1)
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
4.176 4.105 3.984 3.809 3.577 3.277 2.896 2.410 1.774 0.869
0.029 0.120 0.273 0.493 0.787 1.166 1.647 2.260 3.062 4.206
72.8 54.8 44.0 36.2 29.9 24.4 19.5 14.7 9.9 4.4
5.880 5.449 5.066 4.700 4.326 3.918 3.445 2.860 2.084 0.907
0.70 0.64 0.60 0.56 0.53 0.51 0.48 0.46 0.43 0.41
electrode surface. In other words, such non-linear factors as migration and variation of diffusion coefficients with concentration do not influence the shape of the pit. However, these factors influence the time to achieve a given pit size by the quantities D~ and Cs which enter in the definition of the characteristic time to. In Table 1 values of Doo and Cs are presented for Fe dissolution in NaCI solutions of different concentrations. The quasi-concentration Cs was calculated according to equations (10) and (14). The values of D~ that coincide with DI~ according to equation (13) were calculated by Walden's rule (D1/~ = constant) based on known viscosity values/~ of aqueous NaCI solutions. 2° If the boundary condition (11) will hold at the electrode surface the transition to dimensionless variables leads to a large number of dimensionless parameters. In this case the calculations were performed for a specific system: dissolution of Fe in 1 M NaC1. For this situation equations (2) and (12) were solved with boundary conditions (6), (7) and (11). Equation (11) is approximated by a Tafel kinetic expression is = io exp
(13zlF(V
-
~bs)/RT),
(20)
where io = 10 -8 A cm -2 and fl = 0.4 are assumed. 24
NUMERICAL METHOD The solution of the problems formulated above was obtained by the method of finite differences using a rectangular non-uniform grid spacing (Fig. 2). The spatial intervals close to the pit mouth in direction x and r, &x and Ar, were usually chosen from the condition &x = Ar = ro/5. However, for the model with an infinitely thin passive layer the corresponding step along the x-axis &x = d = ro/500 was chosen. The grid spacing steps were enlarged sharply far from the pit mouth outside the pit in order to save computer time. We consider that the position of the metal/electrolyte interface of a corrosion pit is known for a given time t (for example, curve I, Fig. 2). The position of this boundary is defined by the coordinates of its crossing points, marked by numbers 110, with the coordinate axes. Methods for the solution of Laplace's equation and for
The determination of the shape of developing corrosion pits
,._,6
Ax' r"
1717
\
l " - ~ " ~ ~.~
:,0 I" 10~, ,
xT I~ Fig. 2.
12
"/iV/, "//I/.//
r0
3_d 2-
A v]
Cross section through the axis of symmetry of a pit with defined grid system.
the solution of the more general equation (12) for the arbitrary two-dimensional region are well known from literature. 25 For the solution of these equations a locally one-dimensional method was applied for this work. 26 When the solution is found, the current density at the metal surface can be calculated numerically using one of the relations (16). The value of the angle between the surface normal n and the x-axis can be found for point i of the border using the relation tan (n, x) = (Xi+l - xi)(ri - r i - l ) 2 + (Xi - x i - 1 ) ( r i + l - ri)2 ( r i - ri-l)(ri+l - ri)(ri+l - ri-l)
(21)
For the right point of the phase boundary (i = 10, Fig. 2), tan(n,x) = (xi x i ) / ( r i - 1 - ri) will hold. As it follows from equation (2), the coordinates of points lying on an arbitrary grid line parallel to the x-axis (points 1,2, 3, 4, 6, 7), when the phase boundary shifts from position I to II for time t + At, can be found from the relation
X i - l)/(ri -- r i - l), and for the left point (i = 1), tan(n,x) = (xi + 1 -
Xu
= xl -
Kvi~At/cos(n,
x)
(22)
The position of the points of the new phase boundary which are located on grid lines parallel to the r axis can be found in principle with the help of interpolation. The determination of a new position of the interface corresponding to the solution to equation (2) can lead to a decrease in accuracy for the region where cos(n, x) ~ 0, i.e. in the region where the boundary is parallel or almost parallel to the x axis. This situation occurs close to the point of undercutting of the isolating passive layer (point 10) for limiting current conditions. For this region it is convenient to shift all points which lay on the grid lines parallel to r according to equation (23) which is analogous to equation (22) ru = rI -
KvisAt/cos(n,
r)
(23)
1718
G. Engelhardt and H.-H. Strehblow
It is convenient to use equation (22) in the region where the angle between the xaxis and the surface normal n is less or equal ~r/4 (tan(n,x) -< 1), but equation (23) in the region where this angle is larger than :r/4. Thus on Fig. 2 points 1, 2, 3, 4, 6 are shifted along x-axis by equation (22), but points 8, 9, 10 along the r-axis by equation (23). The position of points 5, 7, 8, 9 on the new boundary are found by linear interpolation using the known coordinates of the rest of the boundary points. After determination of the new position of the interface the boundary problem is solved for the new region. The further changes are calculated with the same procedure. When this algorithm is applied, it is useful to keep in mind the following two details. The solution of the boundary problem which is found for time t can serve as a good initial approximation for the solution of time t + At, i.e. for the new position of the pit surface. The concentration values at the grid's points which belong for time t to the metallic phase but for t + At to the solution can be obtained by interpolation. These points are marked by * on Fig. 2. The distance Ax* (or ArT) may be very small. This situation may lead to a significant error for the calculated flux which is proportional to 1/Ax*. In this situation it is reasonable to change AX*oby a small quantity Axo if Ax* < &,Co.For the present work AXo = ro/250 was chosen.
THE RESULTS OF N U M E R I C A L CALCULATIONS Figure 3 presents the change of the metal/electrolyte interface of a closed pit during anodic dissolution, obtained from the numerical solution of the boundary problem (19), for the case of an infinitely thin barrier layer. According to these calculations the surface form can be approximated by an ellipsoid of revolution
100 ~
._o ~
N
-- 50
2
1
I 0
1
2
4
R (dimensionless) Fig. 3. The shape of a closed pit with normalized coordinates at different dimensionless times. Dashed curve represents results of experiments.16 Initial form of the pit is a plane disk.
The determination of the shape of developing corrosion pits
1719
around the x-axis for the related values of dimensionless time r > 5, i.e. the equation for any cross section through the x-axis can be approximated by the following equation of an ellipse X 2 / H 2 + R2/L 2 = 1
(24)
The time dependence of the axes of this ellipse for r > 5 can be approximated with an accuracy better than 10% by H = brl/3; L = a + b r 1/3,
(25)
where a = 0.5 and b = 0.75. From equation (25) follows that the surface of the dissolving pit can be approximated for r ~ o0 by a sphere. For r > 150 the axes H and L differ less than 10%. Figure 3 presents the experimental results for the shape of a cavity found during diffusion controlled dissolution of GaAs. 16 At time r = 13 the depth H of a pit calculated according to equation (25) coincides with the experimental value. The agreement of calculated and experimental results for the pit shape and size confirms the applicability of equations (24) and (25) for the determination of the shape of closed pits during electrochemical or chemical dissolution with diffusion-controlled conditions. For the majority of electrochemical experiments the dependence of the characteristic pit size (depth h, mouth radius rp) on time t can be expressed by simple relation h (or rp) -- tg
(26)
Registered values of g equal approximately 1/3, 1/2, 2/3 and 1.0.11"27-30 It follows from equation (25) that the power g equals to 1/3 during dissolution of a closed pit for current limited conditions. In the case of one-dimensional pits g = 1/2 is obtained for this condition. 31 According to equation (25) the transport limited current l,m through a closed pit with half-ellipsoid shape is described by equation (27): I,m = (dVpit/dt)/Kv = /lim.~ (1 nt- 0 . 8 8 8 / T 1/3 "{,- 0 . 1 4 8 / r 2 / 3 ) ,
(27)
where Vpit = 2:rhl2/3 is the volume of the pit and Ilim,~ -- 0.884r3o/gvto the current flowing through it at t ~ ~. Relation (27) can be compared with experimental results. For a one-dimensional pit one obtains I ~ 1~kit. Calculations have shown that in the region 0 -< d* -< 1 the form of the pit practically does not depend on the passive layer thickness. The shape of the cavity can be described by equations (24) and (25), but the parameter b becomes a linear function of dimensionless isolation d*: b = 0.75 - 0.12d*;0-< d* -< 1
(28)
Although the pit shape depends on its initial form, this influence is smoothed at
1720
G. Engelhardt and H.-H. Strehblow 5O
_o
2
tO
"5 t-
X
0
1
2
3
R (dimensionless) Fig. 4.
The shape of a closed pit with normalized coordinates at different dimensionless times. Initial form of the pit is a hemisphere.
t ~ ~. Figure 4 shows the growth of a closed, initially hemispherical pit which changes to an ellipsoidal form which then transfers to spherical form at t ~ ~. If the metal dissolution is charge transfer controlled and does not depend on transport phenomena, the local current density is constant with time. For this situation it is convenient to introduce the dimensionless time parameter t* = and rewrite equation (2) in the following dimensionless form:
(-isKv/ro)t
aA ~
b-r-
(aAI2
(29)
1 + \0R/
According to equation (29) the pit shape is determined only by the initial conditions. For example, if the initial pit shape is a hemisphere with radius R = 1, one obtains the following solution of equation (29) A=N/(I+t*) 2-R2;
R-
(30)
i.e. the pit keeps its hemispherical form for its further growth. If the initial pit has the form of a cone inclined to the passivated surface with an angle a (Fig. 5) it will remain a cone for any later stage of growth described by the linear function A = t*/cos a + (1 - R) tan (a);
R -< 1 +
t*/sin(a)
(31)
X
Metal
v r
Passive film / Electrolyte Fig. 5.
Schematic diagram of a pit with cone geometry.
The determination of the shape of developing corrosion pits 3
-
2-
.~
1721
1000 s
X 0
I
~
I
1
2
3
\i 4
IN
J
5
6
R (dimensionless) Fig. 6. The shape of closed pit with normalized coordinates for Fe dissolution in 1 M NaCI at different times. Initial form of the pit is a plane disk with radius ro = 30pm. V = 0.55 V.
If we assume the growth of a plane defect within a barrier layer one has to follow the limiting transition a ~ 0 for equation (31) (see Fig. 5). For this case with undercutting of a passivated surface by AL = L - 1 = t*/sin a--~ oo with a--~ 0 a stable pit growth is evidently not possible. In the opposite case, when the initially plane defect growths under mass-transfer control, AL has a finite value for any time. Accordingly, one can expect that in the intermediate case of mixed kinetics the sub-etching under the barrier layer Al will be larger than in the case of transport-limited current densities for the growth a pit with the same depth. This suggestion is confirmed by numerical calculations carried out with the above mentioned method for Fe dissolution in 1 M NaCI at V = 0.55 V. (Fig. 6). This can be also explained by the use of equation (23). For diffusion-controlled conditions the current density where an electrode meets an insulator is either infinite (at a > 90°) or zero (at a < 90 °) unless a right angle is formed 32 (see Fig. 1). Thus, for these conditions, the motion of the interface boundary occurs so that the right angle (a = 90 °) is established by the etching process. For mixed kinetics a is not rectangular, and cos(n,r) becomes less than 1, which leads to the increase of the subetching Al. Figure 7 illustrates the influence of the initial shape of an open pit on its further
3 o o
"0
~ M
2
1
I 0
1
2
3
4
5
R (dimensionless)
Fig. 7. The shape of an open pit with normalized coordinates for Fe dissolution in 1 M NaCI. Initial form of the pit is a hemisphere (1) and a plane disk (2). r o = 10pm. V = 0.55 V.
1722
G. Engelhardt and H.-H. Strehblow
2 o e-
.o
X 0.625 v o
1
2
3
R (dimensionless) Fig. 8.
The shape of an open pit with normalized coordinates for Fe dissolution in 1 M NaCI at different applied potential. Of value ro = 30/Lm.
development. For the initial shape a hemisphere or a plane disk are assumed, both with the same radius r o -- 10 ~tm. In both cases corrosion pits tend to become more shallow as they increase in size, which shows that the dissolution rate is higher at the sides of the pit than at its bottom. When the pit depth h gets three times larger than the initial radius r o the ratio of the pit radius r to its depth r/h equals 1.15 for an initial hemisphere and 1.6 for an initial disk. In Ref. 10 is has been shown experimentally that for pitting of Ni in 0.5 M NaCl this ratio does not exceed 1.25 for pits up to 100 /~m diameter if saturation is not reached in the cavity. Our calculations show that if saturation is not reached in the cavity and the initial shape of pits was a hemisphere this ratio does not exceed 1.2 for Fe dissolution in 1 M NaCl for pits up to 60 ~tm diameter. This result shows that the assumption of an initially hemispherical pit shape is more realistic compared to a planar form. Figure 8 shows the influence of the electrode potential V on the form an initially hemispherical pit when its depth has increased twice of its initial value. The calculated form and size of a pit depend strongly on a possible supersaturation. For example, in Ref. 10 it is suggested that some supersaturation occurs for pitting of Ni. However, it is reasonable to assume for the general case a saturation limit. For this work the calculations were performed in the following way. In any point of the pit surface it was verified if the surface concentration of FeCl 2 exceeds the saturation level, i.e. 4.105 M for the case of Fe dissolution in 1 M NaC1 (see the table). If so, the boundary condition (9) is used for this point, if not, the boundary condition (11) is applied. For V -- 0.625 V (Fig. 8) saturation is reached on the electrode surface for R -1.1. From Figs 9 and 10, one may deduce that supersaturation is an important factor for the calculation of the current density and the shape of the pit. For example, the current density at the bottom of the pit may decrease with the increase of the electrode potential V if the concentration does not exceed uniformly the saturation value at the whole pit surface. This may be explained the following way. The increase of V leads to an increase of the metal ion flux at the sides of the pit wall where the saturation limit is not yet reached. This effect will cause a general increase of the metal concentration inside the pit. As the surface concentration at the bottom of the pit has reached saturation and therefore is constant, the concentration gradient at the bottom will decrease together with the current density at this point. Of course, the
The determination of the shape of developing corrosion pits
. . . . . . . . . °°... °°... -. 0.65 V "'.... 0.65 V'"-.
4
0.625
r..) --
0.6
0
v
V
I
I
I
I
25
50
75
100
Angle (degrees) Fig. 9. Local concentration of Fe 2+ along the bottom of a hemispherical pit at different applied potentials (bottom 0°, edge 90°). Dashed curve was calculated with the assumption of a supersaturation. Of value r o = 30/zm.
1.5 -0.65 v
1.0
"'"""
-
E
<
o,,
0
o/
25
50
laov
75
100
Angle (degrees) Fig. 10. Local current density of the bottom of a hemispherical pit for different applied potentials (bottom 0°, edge 90°). Dashed curve was calculated with the assumption of a supersaturation. Of value ro = 30 pm.
1723
1724
G. Engelhardt and H.-H. Strehblow
~
l
0
1
2
3
4
5
R (dimensionless) Fig. 11.
The shape of a partially closed pit with normalized coordinates. Initial form of the pit is a hemisphere.
total current flow within the pit must increase with the electrode potential V till saturation is reached all over the dissolving pit surface. Dissolution within a pit for limiting current conditions and with zero sub-etching of the barrier layer (Fig. lb) should not occur for simple mathematical reasons. This is a consequence of the properties of the solution of Laplace's equation with a local current density i --~ ~ at the pit edge. One possibility to describe the boundary problem for this condition is to assume that the pit is partially open because of a remaining part of the barrier layer. For example, it can be supposed that the passive layer is mechanically destroyed or dissolved if the sub-etching exceeds some critical value A/1 with a remaining subetching of Al2. Figure 11 shows the shape of a partially open, initially hemispherical pit when its radius increases to three and to five times of its initial value. The calculations were performed with Al 1 = 0.4r o and Al2 = 0.2r o. If, for example, the pit radius increases from r o -- 10/~m to r -- 50~tm, the ratio of radius of the pit to its depth equals 2.6. Ratios of 2.75 were observed for shallow pits growing under masstransport control on stainless steel. 10 For this condition the value of Al 1 equals only 4 /~m which is a small fraction of the pit radius. It seems reasonable that the remaining passive layer is dissolved by the aggressive pit electrolyte. High C1- concentrations favor fast dissolution of Fe-oxides. 18
CONCLUSIONS A simple method was developed to calculate the shape and the size of growing corrosion pits and to demonstrate the concentration distribution of metal ions and the electrical potential within the cavities. It was shown that the surface of a closed pit growing under mass-transfer control in the absence of convection can be described with the help of a general expression which depends only on the initial geometry of the system but not on its physical properties as, for example, diffusion coefficients or charges of species. These expressions were determined for the cases of initially plane or hemispherical pits. For sufficiently large times they can be approximated by the equation by an ellipsoid of revolution. The method was used to simulate the growth of open and closed corrosion pits on iron in 1 M NaCI in the case that the current density at the surface is determined by a Tafel kinetic expression. It was shown that the shape of these pits depends on their initial form and on a possible supersaturation of the system. The calculation of the
The determination of the shape of developing corrosion pits
1725
shape of open pits with limiting current conditions requires assumptions for the limits of the dissolution rates at the contact of the active metal and the passivated surface. The finite difference techniques for the displacement of the boundary of a pit described in this paper is absolutely independent of mass-transport problems. It can be used jointly with an arbitrary finite difference method describing transport phenomena in systems with the fixed boundary including chemical reactions and non-steady-state processes. Acknowledgements--The support of Dr G. Engelhardt by the Deutsche Forschungsgemeinschaft during his research stay at the Heinrich-Heine-Universit~it Diisseldorf is gratefully acknowledged.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.
S. M. Sharland, Corros. Sci. 27,289 (1987). J. R. Galvele, J. electrochem. Soc. 123,464 (1976). H. S. Isaaks, J. electrochem. Soc. 120, 1456 (1973). J. V. Tester and H. S. Isaaks, J. electrochem. Soc. 122, 1438 (1975). R. Alkire, D. Ernsberger and D. Damon, J. electrochem. Soc. 123,458 (1976). H. W. Pickering and R. P. Frankenthal, J. electrochem. Soc. 119, 1297 (1972). D. G. Atea and H. W. Pickering, J. Appl. Electrochem. 11,453 (1981). J. Newman, D. N. Hanson and K. Vetter, Electrochim. Acta 829, 22 (1977). M. W. Verbrugge, M. W. Baker and J. Newman, J. electrochem. Soc. 140, 2530 (1993). J. N. Harb and R. C. Alkire. J. electrochem. Soc. 138, 2594 (1991). S. Szklarska-Smialowska, In Localized corrosion, p. 312. NACE (1974). T. R. Beck and E. A. Grens, J. Electrochem. Soc. 116, 117 (1969). H.S. Karslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edn, Oxford Press, London (1959). J. A. McGeough, Principles o f Electrochemical Machining, Chapman and Hall, London (1974). A. D. Davydov and J. Kozak, High-Rate Electrochemical Shaping, Nauka, Moscow (1990). H. K. Kuiken, J. J. Kelly and P. H. L. Nottin, J. electrochem. Soc. 133, 1217 (1986). B. Chatterjee, Precision Engineering 8, 131 (1986). K. J. Vetter and H.-H. Strehblow, In Localized Corrosion, p. 240. NACE (1974). G. R. Engelhardt, A. D. Davydov and V. S. Krylov, Soviet Electrochemistry 13, 1103 (1979). G. R. Engelhardt, G. N. Zaidman, G. S. Suvorova, G. S. Domente and E. A. Vysokova, Soviet Surface Engineering and Applied Electrochemistry, No. 6, p. 1 (1987). H.-H. Strehblow, Werk. Korros. 35,437 (1984). B. P. Pillay and J. Newman, J. electrochem. Soc. 140,414 (1993). G. R. Engelhardt and H.-H. Strehblow, J. electroan. Chem. 365, 7 (1994). K. J. Vetter, Electrochemical Kinetics, Academic Press, New York (1967). S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington (1980). A. R. Mitchel and D. D. Griffits, The Finite Difference Method in Partial Difference Equations, John Wiley & Sons, Chichester (1980). F. Forchamer and H. J. Engel, Werk. Korros. 20, 1 (1969). N. Sato, T. Nakayawa, K. Kudo and M. Sakashita, In Localized Corrosion, p. 447. NACE (1974). J. Tougek, Werk. Korros. 25,496 (1974). J. Tougek, Corrosion 33,193 (1977). G. R. Engelhardt, A. D. Davydov, T. B. Zhukova and H.-H. Strehblow, Soviet Electrochemistry 28, 199 (1992). J. Newman, Electrochemical Systems, Prentice-Hall, Inc., Englewood Cliffs, NJ (1973).