Potential and current distributions of one-dimensional galvanic corrosion systems

Corrosion Science 52 (2010) 455–480

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Corrosion Science journal homepage: www.elsevier.com/locate/corsci

Potential and current distributions of one-dimensional galvanic corrosion systems Guang-Ling Song * Chemical Sciences and Materials Systems Laboratory, Research and Development Center, GM, Mail Code: 480-106-212, 30500 Mound Road, Warren, MI 48090, USA

a r t i c l e

i n f o

Article history: Received 21 June 2009 Accepted 7 October 2009 Available online 6 November 2009 Keywords: Galvanic corrosion Modeling Linear system One dimension

a b s t r a c t Many practical galvanic corrosion problems can be simplified into a one-dimensional mathematical equation. In this study, theoretical expressions for galvanic potentials and currents of one-dimensional systems are deduced and some critical parameters involved in the potential and current equations are systematically discussed. The developed analytical approach is then applied to some practical galvanic corrosion cases, such as a steel–Al joint exposed to bio-fuel E85, a galvanic couple separated by a passive spacer and a scratched organic coating. It is demonstrated that the analytical approach can examine the reliability of a computer modeled galvanic corrosion process and help understand the mechanism behind the computer modeled galvanic corrosion behavior. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Galvanic corrosion damage is common to components with mixed metals [1–5]. In some cases a multi-phase metallic material suffers from a severe corrosion attack simply due to the galvanic effect between its different phases [6–8]. Moreover, in a sacrificial cathodic protection system, the protection current in effect is also a galvanic current [9,10]. In theory, the degree and the distribution of galvanic corrosion damage can be expressed as an overall galvanic current and a distribution of the galvanic current density. Therefore, reliably estimating the galvanic current density or its distribution of a practical galvanic system is of great significance. Some efforts have been made in this area since the 1950s [11–22]. It has been generally accepted that [23]: (1) Laplace equation can apply as a governing relationship for the potential distribution in a galvanic couple system; (2) linear kinetics can be assumed to simplify the mathematic problems if corrosion rates are very low; (3) coupled metals may be treated as infinite or semi-infinite surfaces for simplified mathematic analyses; and (4) there is a tendency of taking a numerical approach to estimate the galvanic current distribution for complicated practical systems. Recently, an attempt was made to estimate the galvanic current distribution of a Mg alloy in contact with steel by using a boundary element model (BEM) approach with non-linear polarization curves as boundary conditions [24] and it was found that a boundary condition closely matching a non-linear polarization curve yielded a more accurate or reliable calculated distribution of galvanic current density curve than a linear boundary condition [25].

* Tel.: +1 586 986 1339; fax: +1 586 986 9204. E-mail address: [email protected] 0010-938X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.corsci.2009.10.003

Unfortunately, due to the complexity in geometric shape of some mixed-metal components, predicting the galvanic current and its distribution over a complicated component is actually difficult. Sometimes, it is impossible to have an analytical solution to the governing equation for a galvanic corrosion system that has a complicated geometric shape and/or contains more than one galvanic couple. In fact, apart from some very simple linear systems, no theoretical potential or current expression has been deduced so far for a complicated galvanic system. The galvanic current for a complicated galvanic system may be predicted by computer modeling. However, the modeling is in nature a ‘‘computer experimental process”. It can only demonstrate the influence of the variation of input parameters on the output results. However, to fully understand a system, clearly defined analytical relationships between experiential parameters and experimental phenomena are essential. Thus, computer modeling cannot replace the merit of an analytical solution. The author believes that many practical galvanic corrosion systems can be reasonably simplified into a one-dimensional model, of which an analytical description of the galvanic current will be possible. At least two types of galvanic systems can be treated in one dimension (referring to Fig. 1): (1) a fine tube containing electrolyte where its cross-section area is far smaller than its length, and (2) a surface covered by a thin electrolytic film where the thickness of the film is much smaller than the length of the electrolyte coverage. In these two systems, the radius (r) of the tube or the thickness (d) of the electrolyte film is so small as to be neglected. Also, the perimeter of the tube or the width of the electrolytic film does not need to be considered. These two types of simplified onedimensional models represent a large number of practical galvanic corrosion systems, such as a panel with a organic coating being affected by electrolyte, a panel exposed in atmosphere with some

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Nomenclature a b c C m m0 m0 mi r S y d i0 im ig iag ibg icg iac g iab g ibc g ip If Iaf Iaf0 Iafa Ibf Ibf0 Ibfb Icf Icf0 Icfc IF IaF IaFa

length of material (a), which can the length of a material of the electrolyte on the material surface length of material (b) inserted between material (a) and material (c) length of material (c), which can the length of a material of the electrolyte on the material surface the inner perimeter of a studied tube system length of studied electrolyte over material length of studied electrolyte over the left ‘‘dead end” of a multi-piece joined system length of studied electrolyte over the right ‘‘dead end” of a multi-piece joined system length of studied electrolyte over the ith ‘‘open ends” piece of a multi-piece joined system inner radius of a metal tube containing electrolyte cross-section area of electrolyte over a material surface or in a metal tube width of electrolytic film over a material surface thickness of electrolytic film over a material surface the current flowing into the thin liquid film from the left end of the section the current flowing out of the thin liquid film from the right end of the section galvanic current flowing into or out the thin liquid film at a boundary galvanic current flowing into or out the thin liquid film over material (a) at a boundary galvanic current flowing into or out the thin liquid film over material (b) at a boundary galvanic current flowing into or out the thin liquid film over material (c) at a boundary galvanic current flowing between the thin liquid films over materials (a) and (c) at their interface galvanic current flowing between the thin liquid films over materials (a) and (b) at their interface galvanic current flowing between the thin liquid films over materials (c) and (b) at their interface polarization current flowing into or out the thin liquid film at a boundary current density flowing in the liquid film along the material surface current density flowing in the liquid film along the material (a) surface current density flowing in the liquid film along the material (a) surface at the left end current density flowing in the liquid film along the material (a) surface at the right end current density flowing in the liquid film along the material (b) surface current density flowing in the liquid film along the material (b) surface at the left end current density flowing in the liquid film along the material (b) surface at the right end current density flowing in the liquid film along the material (c) surface current density flowing in the liquid film along the material (c) surface at the left end current density flowing in the liquid film along the material (c) surface at the right end Faradic current density in the direction vertical to the material surface Faradic current density over the material (a) surface in the direction vertical to the material surface Faradic current flowing in the liquid film along the material (a) surface at the right end

IbF

Rp

Faradic current density over the material (b) surface in the direction vertical to the material surface Faradic current flowing in the liquid film along the material (b) surface at the left end Faradic current flowing in the liquid film along the material (b) surface at the right end Faradic current density over the material (c) surface in the direction vertical to the material surface Faradic current flowing in the liquid film along the material (c) surface at the left end the potential relative to a reference electrode the potential over material (a) relative to a reference electrode the potential over material (b) relative to a reference electrode the potential over material (c) relative to a reference electrode the corrosion potential or open-circuit potential relative to the reference electrode corrosion potential of the material (a) relative to a reference electrode corrosion potential of the material (c) relative to a reference electrode corrosion potential of the material (b) relative to a reference electrode the potential in the electrolytic film relative to the potential of material the potential in the electrolytic film over material (a) relative to the potential of material the potential in the electrolytic film over material (b) relative to the potential of material the potential in the electrolytic film over material (c) relative to the potential of material the potential in the electrolytic film relative to the potential of material (a) at boundary x = 0 the potential in the electrolytic film relative to the potential of material (b) at boundaries x = 0 the potential in the electrolytic film relative to the potential of material (c) at boundary x = 0 the potential in the electrolytic film relative to the material potential at the left end of the section the potential in the electrolytic film relative to the potential of material (a) at boundary x = a the potential in the electrolytic film relative to the potential of material (b) at boundaries x = b the potential in the electrolytic film relative to the potential of material (c) at boundary x = c the potential in the electrolytic film relative to the material potential a the right end of the section polarization resistance

Rap

polarization resistance of material (a)

Rbp

polarization resistance of material (b)

Rcp

polarization resistance of material (c)

Rs

solution resistance solution resistivity in the one-dimensional system polarization resistivity in one-dimensional system solution resistivity in a conventional three-dimensional system polarization resistivity in a conventional three-dimensional system polarization resistivity of material (a) in the one-dimensional system the polarization resistivity of material (b) in the onedimensional system

IbF0 IbFb IcF IcF0 E Ea Eb Ec Ecorr Eacorr Eccorr Ebcorr

W Wa Wb Wc Wa0 Wb0 Wc0

W0 Waa Wbb Wcc

Wm

qs qp q0s

q0p qap qbp

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G.-L. Song / Corrosion Science 52 (2010) 455–480

qcp L La Lb

the polarization resistivity of material (c) in the onedimensional system characteristic distance characteristic distance of material (a) characteristic distance of material (b)

condensed liquid on its surface, a panel with water splashed over its surface, a water or oil pipeline, a hemmed edge of a vehicle body closure with moisture accumulated in the hem crevice, a fuel or brake oil pipeline in a vehicle, coolant and oil paths in an engine block, etc. In this paper, we will first deduce general theoretical potential and current expressions for a uniform one-dimensional system. To tackle practical galvanic corrosion problems, after the mathematical modeling is successfully applied to a galvanic couple with simply an anode and a cathode joined together, it further extended to a more complicated galvanic system with three different metals joined together, which represents more realistic galvanic corrosion cases in practice. The theoretical galvanic current expressions are then used to analyze the effect of a passive isolator on the galvanic corrosion of a galvanic couple, the possible corrosion damage to an Al engine cylinder exposed to a new bio-fuel E85, and the corrosion risk along a scratch made in an organic coating.

2. One-dimensional primary potential and current densities Fig. 1 schematically shows a one-dimensional tube and flat surface. They are not necessarily straight or flat in practice. The simplification is made under the following conditions: (1) The cross-section area (S) of the electrolyte must be much smaller than its studied length (m). (2) The state variables (e.g. surface potential) over the cross-section along either dimension perpendicular to the length direction are relatively uniform or constant compared with those along the length direction. For a metal surface under a thin electrolytic film, when the thickness (d) of the electrolytic film is very small (d ? 0), the state

Lc W Wa Wb Wc

characteristic distance of material (c) distribution characteristic number distribution characteristic number of material (a) distribution characteristic number of material (b) distribution characteristic number of material (c)

variables or parameters (e.g. corrosion potential or polarization resistance) will not vary significantly along the width (y). For a tube, this means that, if radius (r) is very small, i.e., r ? 0, there will be no need for considering the variation of the state variables or parameters over the cross-section. In such a one-dimensional system, the polarization resistivity qp and solution resistivity qs have their definitions (qp = polarization resistance  length, and qs = solution resistance/length) and units (X cm1 and X cm) different from those in a three-dimensional system (see Appendix A). Because of the above simplifications, the theoretical current and potential for a one-dimensional system can be deduced as follows. 2.1. General potential and current of a one-dimensional system It is assumed that the distribution of potential over a given section of a one dimensional system along x-coordinate is a curve as shown in Fig. 2. In this simplest case, the Faradic reaction at the interface between the electrolyte and the metal is linearly controlled by the potential W in the electrolytic film (W is the electrolyte potential relative to the corrosion potential Ecorr of the metal). The potential distribution can be easily obtained [26] (see Appendix B):

E ¼ Ecorr  A exp

x

 x  B exp  L L

ð1Þ

where

sffiffiffiffiffiffi

L¼

qp qs

ð2Þ

Correspondingly, we have expressions for Faradic current density IF and non-Faradic flowing current density If:

IF ¼ 

A

qp

exp

x B  x exp   L L qp

x  x A B þ exp exp  L L qs L qs L x  x A B þ pffiffiffiffiffiffiffiffiffiffiffi exp  ¼  pffiffiffiffiffiffiffiffiffiffiffi exp L L qs qp qs qp

ð3Þ

If ¼ 

ð4Þ

where IF is a curve similar to E in shape, and If has a shape similar to the differential of E (see Fig. 2). IF is in the direction perpendicular to the metal/electrolyte interface whilst If is parallel to the interface. IF flowing into the electrolyte and If flowing from left to right in this study are defined to have a positive value. 2.2. ‘‘Dead end” and ‘‘open ends” systems

Fig. 1. Schematic illustration of two typical one-dimensional systems: (a) small tube and (b) thin electrolyte coverage.

Based on the above general potential and current expressions, the potential and current expressions for systems with controlled boundary conditions can be obtained. In practice, this can be a case where a piece of sheet or tube metal has a thin electrolyte film over its surface, but the electrolyte or the metal is not infinite in length. When a current or potential is applied through the electrolyte at one end, the current If flowing in the electrolyte will always be zero at the other end (the ‘‘dead end”). If potentials or currents are applied at both ends, that will be a case of ‘‘open ends”.

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G.-L. Song / Corrosion Science 52 (2010) 455–480

Fig. 2. Schematic illustration of a one-dimensional system and corresponding potential and current distributions.

For a piece of metal (c) having length c, if a potential or current is applied at the left end x = 0, then at the other boundary end x = c, which is a ‘‘dead end”, the flowing current density Icf should be zero:

 Icf ¼ Ic x¼c ¼ Icfc ¼ 0

ð5Þ

Moreover, the applied potential in the electrolyte is Wc0 at boundary x = 0. By substituting the boundary conditions into the general E, IF and If expressions, the following expressions for this system can be deduced (see Appendix C):

  wc0 cosh xc c  L ; w ¼ cosh Lcc c

06x6c

  wc cosh xc c  c L ; 0 6 x 6 c Ec ¼ Eccorr  0 cosh Lc !   wc0 cosh xc c c L  ; 0 6 x 6 c IF ¼  c qp cosh Lcc !   sinh xc wc0 c L  ; 0 6 x 6 c ffic Icf ¼  pffiffiffiffiffiffiffiffiffiffi qs qp cosh Lcc wc0   wcc ¼ wc jx¼c ¼ cosh Lcc   wc0 tanh Lcc  Icf0 ¼ Icf x¼0 ¼ pffiffiffiffiffiffiffiffiffifficffi

qs qp

ð6Þ ð7Þ ð8Þ ð9Þ ð10Þ

ð11Þ

where

sffiffiffiffiffiffi c

L ¼

qcp qs

ð12Þ

where qcp is the polarization resistivity of metal (c). Wc0 is the potentials of the electrolyte in this system at the left end x = 0. Eccorr is the corrosion potential of metal (c) relative to a reference electrode. Correspondingly, the Faradic and flowing current densities IcF and Icf over the metal can be obtained according to Eqs. (3) and (4). Wcc is the potential of the electrolyte at the other end (x = c). Icf0 is the current at boundary x = 0 where the boundary potential is applied. The distributions of the potential and current densities are shown in Fig. 3. For a piece of metal (a) having a length ‘‘a”, if a current or potential is applied at its right end, then If = 0 at x = 0 which is a ‘‘dead end”. Similarly, the potential and current equations for this system can be obtained:

x  x a cosh La  ; 0 6 x 6 a ¼ w a La cosh Laa   wa cosh Lxa a ; 0 6 x 6 a Ea ¼ Eacorr  a cosh La !   a wa cosh Lxa a  ; 0 6 x 6 a IF ¼  a qp cosh Laa wa ¼ waa cosh

ð13Þ ð14Þ ð15Þ

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G.-L. Song / Corrosion Science 52 (2010) 455–480

Fig. 3. Schematic illustration of a one-dimensional system (c) with controlled boundary condition at its left end and corresponding current and potential distributions.

Ifa

waa ffi ¼  pffiffiffiffiffiffiffiffiffiffi a

!

sinh

qs qp cosh

 IaFa ¼ IaF x¼a ¼

  x La

 ; a La

wa  aa p

q

 waa ffi tanh Iafa ¼ Iaf x¼a ¼  pffiffiffiffiffiffiffiffiffiffi a

qs qp

06x6a

ð16Þ

ð17Þ  a La

ð18Þ

If two boundary conditions Wb0 and Wbb are known, we can easily work out the potential and current distributions:

   wbb  wb0 exp  Lbb x   exp b wb ¼ b L 2 sinh Lb    wb0 exp  Lbb  wbb x   þ exp  b ; L 2 sinh Lbb

where

sffiffiffiffiffiffi La ¼

qap qs

ð19Þ

where qap is the polarization resistivity of metal (a); Waa is the potential of the electrolyte applied at boundary x = a. Wa0 is the potential of the electrolyte at the other end x = 0. Eacorr is the corrosion potential relative to a reference electrode. The distribution of potential and currents described by Eqs. (13)–(16) are shown in Fig. 4. For a section of metal having length b (Fig. 5), if both of its boundary conditions are known, e.g. at the left end, the potential b of the electrolyte is Wb0 and current is i0 ; and at the right end, b b Wb and ib , respectively, then it is an ‘‘open ends” system. It should be noted that only two of these boundary conditions are indepenb b dent. When Wb0 and Wbb are fixed, i0 and ib will be dependent on b b W0 and Wb , and vice verse.

06x6b

   wbb  wb0 exp  Lbb x   E ¼  exp b L 2 sinh Lbb    wb0 exp Lbb  wbb x   þ exp  b ; 0 6 x 6 b L 2 sinh bb b

ð20Þ

Ebcorr

ð21Þ

L

Ibf

    2 b b  wb exp b  wb b 0 b 1 4wb  w0 exp  Lb x Lb     ¼  qffiffiffiffiffiffiffiffiffiffiffi exp b  b b L 2 sinh Lb 2 sinh Lb qs qbp 

x ð22Þ  exp  b ; L

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G.-L. Song / Corrosion Science 52 (2010) 455–480

Fig. 4. Schematic illustration of a one-dimensional system (a) with controlled boundary condition at its right end and corresponding current and potential distributions.

IbF

  2 b b  b 1 4wb  w0 exp  Lb x   ¼  b exp b b qp L 2 sinh Lb   3 b  b w0 exp Lb  wbb x 5   þ exp  b ; 0 6 x 6 b L 2 sinh bb

ð23Þ

L

where

sffiffiffiffiffiffi b

L ¼

qbp qs

ð24Þ

where qbp is the polarization resistivity of the metal in section b, and Ebcorr is the corrosion potential of material (b) relative to a reference electrode. The potential and current distributions in this system follow exponential functions and are schematically shown in Fig. 5. Potentials and current densities change more dramatically in a region close to a boundary and less significantly farther away from a boundary. In summary, as long as the polarization currents or polarization potentials at boundaries or ends are known, the potential and current distributions over a metal surface can be determined. The current equations deduced from the above systems suggested that both the Faradic current and the non-Faradic

flowing current change largely at a boundary (or ‘‘open end”) where a boundary potential or current is applied. The changing amplitude decreases as the distance from the boundary increases. At a boundary (or ‘‘dead end”) where no potential or current is applied, the Faradic current and flowing current become linearly dependent on distance. Moreover, it can be theoretically deduced that the applied overall polarization current of a ‘‘dead end” or ‘‘open ends” system is a linear function of the applied potential (see Appendix D), which is consistent with the linear assumption in this study. It should be noted that the above ‘‘dead end” and ‘‘open ends” system can comprise a complicated galvanic corrosion system. The potential and equation equations deduced for these elemental systems lay a foundation for deducting potential and current equations of complicated galvanic systems.

3. Galvanic current, potential and their distributions If one of the above metals is in contact with other one or two different metals, they will together form a galvanic system due to their different electrochemical activities. In this case, the external polarization current becomes an internal current flowing between two adjacent metals in connection. This flowing current is also known as a galvanic current of the system.

G.-L. Song / Corrosion Science 52 (2010) 455–480

461

Fig. 5. Schematic illustration of a section with controlled boundary conditions at both ends and corresponding current and potential distributions.

3.1. A joint of two dissimilar metals If there are two pieces of metals (a) and (c) joined up (see Fig. 6), each metal will be equivalent to a ‘‘dead end” single piece metal as a discussed above, and the current ig flowing out of the electrolytic c film over metal (a) should be equal to the current ig into the electrolytic film over metal (c) across the point interface: a

ac

c

ig ¼ ig ¼ ig ¼ ip

a6x6aþc IaF

ð25Þ

or a

  "  # pffiffiffiffiffiffi ðEacorr  Eccorr Þ qcp tanh Laa cosh xac Lc   i Ec ¼ Eccorr þ hpffiffiffiffiffiffi ;   pffiffiffiffiffiffi   cosh Lcc qap tanh Lcc þ qcp tanh Laa

c

ig  ig ¼ 0

ð26Þ

Meanwhile, for these two pieces of metals, the potential Ea of metal (a) should be equal to the potential Ec of metal (c) at their joint interface:

  Ea x¼a ¼ Ec x¼0

ð27Þ

With the above relationships, the potential and current of the system can be obtained (see Appendix E):

  "  # pffiffiffiffiffiffi ðEccorr  Eacorr Þ qap tanh Lcc cosh Lxa a a a ; E ¼ Ecorr þ hpffiffiffiffiffiffi   pffiffiffiffiffiffi  i qc tanh aa þ qa tanh cc cosh La p

06x6a

L

p

ð29Þ

"    # ðEccorr  Eacorr Þ tanh Lcc cosh Lxa a ; ¼ pffiffiffiffiffiffihpffiffiffiffiffiffi   pffiffiffiffiffiffi  i qap qcp tanh Laa þ qap tanh Lcc cosh La 06x6a

ð30Þ

"    # ðEa  Eccorr Þ tanh Laa cosh xac La   i ; IcF ¼ pffiffiffiffiffiffihpffiffiffiffifficorr   pffiffiffiffiffiffi   ffi cosh Lcc qcp qap tanh Lcc þ qcp tanh Laa a6x6a Iaf

ð31Þ

"    # ðEccorr  Eacorr Þ tanh Lcc sinh Lxa a ; ¼ pffiffiffiffiffihpffiffiffiffiffiffi   pffiffiffiffiffiffi  i qs qcp tanh Laa þ qap tanh Lcc cosh La 06x6a

ð32Þ

"    # ðEa  Ec Þ tanh Lac sinh xac Lc   i ; Icf ¼ pffiffiffiffiffihpffiffiffiffiffifficorr corr  pffiffiffiffiffiffi   cosh Lcc qs qap tanh Lcc þ qcp tanh Laa

L

ð28Þ

a6x6aþc

ð33Þ

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G.-L. Song / Corrosion Science 52 (2010) 455–480

Fig. 6. Schematic illustration of two coupled systems (a) and (b) and corresponding current and potential distributions.

  ðEccorr  Eacorr Þ ac ig ¼ Iaf x¼a ¼ Icf x¼a pffiffiffiffiffihpffiffiffiffiffiffi.   pffiffiffiffiffiffi.  i qs qap tanh Laa þ qcp tanh Lcc ðEccorr  Eacorr Þ  .   pffiffiffiffiffiffii qcp tanh Laa tanh Lcc þ qap

 IaF0 ¼ IaF x¼a ¼ pffiffiffiffiffiffihpffiffiffiffiffiffi

qap

ðEacorr  Eccorr Þ  .   pffiffiffiffiffiffii qp tanh Lcc tanh Laa þ qcp

 IcF0 ¼ IcF x¼a ¼ pffiffiffiffiffiffihpffiffiffiffiffiffi c a

qp

ð34Þ

ð35Þ

‘‘dead end”, and material (b) that is inserted between metals (a) and (c) can be treated as a sectional system with ‘‘open ends”. Similarly, the potentials and flowing currents over different metals must be equal at their joints. These will lead to their theoretical potential and current expressions (see Appendix F):

" #   cosh Lxa MEbc þ HEba  ;  Eba a GH  MN qp cosh Laa

IaF ¼  ð36Þ

The potential and current distributions of this system are schematically displayed in Fig. 6. Basically, the potential and current curves over these two joined metals have the same characteristics as those before they are joined together. Furthermore, their Faradic currents are always negative to each other according to equations of IaF and IcF . This suggests that these two metals are always anodic to cathodic each other.

ð37Þ

  bc  þHEba  MEGHMN exp  Lbb xa   IbF ¼ exp Lb 2qbp sinh Lbb   ba  MEbc þHEba þGEbc exp Lbb  NEGHMN GHMN xa    exp  b ; L 2qbp sinh Lbb NEba þGEbc GHMN

a6x6aþb

3.2. Joints of three pieces of metals in series If there are three pieces of materials (a), (b) and (c) joined in series as such that (a) is joined to (b) which is joined to (c) (see Fig. 7), then materials (a) and (c) will be equivalent to a system having a

06x6a

IcF

NEba þ GEbc ¼   Ebc GH  MN

ð38Þ !

aþb6x6aþbþc

  cosh xabc Lc  ; qcp cosh Lcc ð39Þ

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G.-L. Song / Corrosion Science 52 (2010) 455–480

Fig. 7. Schematic illustration of a thin electrolytic film covering metal (a) joined to metal (b) joint joined to metal (c) and corresponding current and potential distributions.

"

IaF

#   sinh Lxa MEbc þ HEba ba  ; ¼  E pffiffiffiffiffiffiffiffiffiffiaffi GH  MN qs qp cosh Laa

06x6a

ð40Þ

  bc  þHEba  MEGHMN exp  Lbb xa   qffiffiffiffiffiffiffiffiffiffiffi exp Ibf ¼ Lb 2 qs qbp sinh Lbb   ba  MEbc þHEba þGEbc exp Lbb  NEGHMN GHMN xa   qffiffiffiffiffiffiffiffiffiffiffi exp  b ; þ L 2 qs qbp sinh Lbb

Icf ¼ 

NEba þ GEbc  Ebc GH  MN

¼

IbF0

  sinh xabc c pffiffiffiffiffiffiffiffiffifficffi L  c  ; qs qp cosh Lc

6aþbþc " #    tanh Laa MEbc þ HEba ab ffi  Eba pffiffiffiffiffiffiffiffiffiffi ig ¼ Iaf x¼a ¼  GH  MN qs qap

¼

IbFb aþb 6x



¼

 IbF 

¼

 IbF 

ð41Þ !

Icf0

NEba þ GEbc ¼  Ebc GH  MN



Icf x¼aþb

" #  1 MEbc þ HEba IaFa ¼ IaF x¼a ¼  a  Eba qp GH  MN

NEba þGEbc GHMN

a6x6aþb

bc ig

x¼a

¼

 x¼aþb

MEbc þ HEba GH  MN

1

qbp

¼

1

qbp

!

  tanh Lcc pffiffiffiffiffiffiffiffiffifficffi

qs qp

ð44Þ

ð45Þ

!

NEba þ GEbc GH  MN

ð46Þ !

NEba þ GEbc  Ebc GH  MN

ð47Þ !

ð42Þ

 1 IcF0 ¼ IcF x¼aþb ¼  c

ð43Þ

where M, N, G and H are functions of systematic parameters of the three materials as defined in Appendix F. Fig. 7 schematically

qp

ð48Þ

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G.-L. Song / Corrosion Science 52 (2010) 455–480

presents the potentials and currents of this system. All the three connected materials can actually affect one another in their potential and current distributions. For example, currents over material (a) are not only influenced by material (b) that is in direct connection with it, but also dependent on the parameters of material (c) which is not directly in contact with metal (a). This is termed as a ‘‘remote” effect in this study.

The methodology adopted to deduce the potentials and currents for the galvanic systems with two joints and three pieces of materials is straightforward; simply let the flowing currents and potentials of the two materials in direct connection be equal at their joint point. This will generate a sufficient number of equations to determine the unknown parameters of these galvanic systems. In fact, the methodology can be extended to work out potentials and current densities for systems containing more than three pieces of materials joined together in series. For a multiple-material system, in addition to the left and right ‘‘dead end” pieces m0 and mn, there are n-1 ‘‘open ends” pieces between m0 and mn (see Fig. 8). In this case, the polarization currents flowing in/out these ‘‘dead end” and ‘‘open ends” pieces still following Eqs. (11), (18), (97) and (98):

x¼0

  w0b tanh mL00 qffiffiffiffiffiffiffiffiffiffiffi ¼

 0 in ¼ Inf  P n1 x¼

mi

 wn0 mn ffin tanh ¼  pffiffiffiffiffiffiffiffiffiffi qs qp La

i¼0

  i i0 ¼ Iif  P i1 x¼

mj

ji¼0

i ib

¼



 Ibf  P i x¼ mj ji¼0

ð49Þ

qs q0p

wi0 cosh ¼ qffiffiffiffiffiffiffiffiffiffiffi

  mi Li

ð50Þ

 wib  

ð51Þ

  wi0  wib cosh mLii   ¼ qffiffiffiffiffiffiffiffiffiffiffi qs qip sinh mLii

ð52Þ

qs qip sinh

mi Lb

W00  W10 ¼ E0corr  E1corr W1b  W20 ¼ E1corr  E2corr 

3.3. A complicated one-dimensional system

  0 ib ¼ I0f 

Parameters mi, Li, qs and qi are constants that are known or can be measured, so in total 2n variables wi0 and wib are involved in Eq. (53). They are all simple linear equations. The number of equations is n. At the same time, we know:

i1 Wbi1  Wi0 ¼ Ecorr  Eicorr

ð54Þ

 n2 Wn2  Wn1 ¼ Ecorr  En1 b 0 corr n Wn1  Wn0 ¼ En1 b corr  Ecorr

Equation set (54) provides additionally n linear equations. Therefore, all the potentials wi0 and wib can be worked out and have their analytical expressions. Correspondingly, the galvanic potentials and currents over each piece of the element can be obtained. In other words, the galvanic potential and current distributions over a multi-piece system will have analytical expressions. Although the expressions are more complicated than those for a three-piece system, it is a great advantage to analytically calculate a theoretical galvanic potential or current directly based on system parameters, rather than to simulate them through a numerical modeling process. It should be noted that this multi-piece system can be easily extended to a more complicated non-uniform system. For example, a non-uniform system can be divided into many small pieces and each of the small pieces can be regarded as a uniform system. Therefore, the complicated non-uniform system becomes a multi-piece system. Due to the complexity of potential and current expressions of a multi-system, in the following sections, only 2 and 3 piece joint systems are discussed. However, the conclusions hold for a complicated system as well. 4. Parameters and critical values

They should be equal between every two adjacent pieces: 0

1

1

2

The above deduced galvanic potential and current equations are actually determined by many parameters, such as the polarization resistivity, solution resistivity, length of materials, etc. Among these parameters, some can affect the characteristics of potential and current distributions more significantly than the others.

ib ¼ i0 ib ¼ i0  i1

ib

i

¼ i0

 n2

¼ i0

n1 ib

n in

ib

n1

¼

ð53Þ 4.1. System determination parameters A galvanic system has three different types of parameters: (1) system geometric parameters, such as the lengths of systems a, b and c; (2) material parameters, Eacorr ; Ebcorr ; Eccorr ; qap ; qbp ; qcp , and

Fig. 8. Schematic illustration of a multi-piece system (complicated system).

G.-L. Song / Corrosion Science 52 (2010) 455–480

(3) environmental electrolyte parameter qs. It should be noted that in this study, qs is not a pure solution property. It also contains geometric parameters y, r and/or d (Appendix A). According to the above general potential and current equations, the influence of these parameters on galvanic potential and current distributions can be summarized as follows. The geometric parameters only affect the amplitudes of galvanic potentials and currents. Some of the material parameters, such as the corrosion potentials, will determine the amplitude of galvanic potentials and currents, while others, such as qap ; qbp and qcp , will have an influence on potential and current distributions. The environmental electrolytic parameter qs can significant affect the potential and current distributions as well as their amplitudes. Generally speaking, an increase in the difference between the corrosion potentials of the metals will lead to an increase in galvanic current density, but an increase in electrolytic resistivity or polarization resistivity of the metals will result in a decrease in galvanic currents. It should be stressed that the dependence of galvanic potentials and currents on the parameters is not straightforward. Due to an interaction between the metals, sometimes the influence is complicated, particularly for a system with more than two pieces of materials connected in series.

4.2. Distribution parameter Among those parameters determining the potentials and currents of a galvanic system, there are two special parameters that have a critical influence on the potential and current distributions: (1) the material polarization resistivity qp and (2) the environmental electrolyte resistivity qs. L is a square root of their ratio. A system with a larger L will have a potential and current changing relatively less significantly with distance. The influence of the L value on the current distributions of a galvanic system is clearly illustrated in Fig. 9. In fact, the ratio of polarization resistivity over solution resistivity has been termed as ‘‘Wagner Number” and used to predict the galvanic current distribution of a 3-D infinite galvanic system [11,27,28]. ‘‘Wagner Number” has a unit of distance. The only difference between ‘‘Wagner Number” and parameter L defined in this study is the square root involved in L. This is because ‘‘Wagner Number” and L are defined in different dimensional systems. According to Eqs. (77) and (79), we have:

Fig. 9. Theoretical Faradic and flowing currents (IF and If) over an anode (a)– cathode (c) couple with La  Lc. The curves are produced using corresponding equations in Section 3.

sffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi q0p S q0p d qp ¼ ¼ 0 qs qs y q0s

465

sffiffiffiffiffiffi L¼

ð55Þ

which means that L is actually a combination of the traditional ‘‘Wagner Number” and the electrolyte film thickness. L can be treated as a one-dimensional ‘‘Wagner Number”. It has also a unit of distance, and can be regarded as a measure of the effective distribution distance within which significant galvanic current or potential is distributed on the metal surface [29]. So, it is termed as ‘‘effective distribution distance” in this study. If a galvanic current distribution is concerned, L is actually the distance from a couple joint to a point where the slope of the galvanic current distribution curve at the junction point intersects the x-coordinate (see Figs. 2–4). Within this distance, the galvanic current or potential of a system decreases most dramatically, and after that, the change becomes steady. For an infinite system, L is a useful measure for the distance that a galvanic potential or current is distributed. However, in a system with a limited length, this measure becomes invalid, particularly when the length of the system is shorter than L. For example, in Fig. 9, when Lc > c, exponential distribution of IcF cannot be well displayed in the entire cathode surface. There is a possibility that two galvanic systems can have the same L due to their similar qs and qp, but their current and potential distributions may appears to be different because of their different material length limitations. An example is given in Fig. 10. In fact, variable x in the above potential and current expressions is always restricted within the length of a corresponding metal. According to the galvanic potential and current expression in Section 3, a potential or current distribution determined by exponential functions is eventually determined by the ratio of x/L, not by x or L alone. It is quite possible in mathematics that two galvanic systems which have different L values show similar potential and current distribution curves in different x scales. To eliminate the influence of material length on a potential or current distribution curve, x should be transformed into a non-dimensional variable X. Let

x x ¼ mL  ¼ WX L m

ð56Þ

then

Fig. 10. Theoretical Faradic and flowing currents (IF and If) over an anode (a)– cathode (c) couple with length a  length c. The curves are produced using corresponding equations in Section 3.

466

x m L W¼ m

X¼

G.-L. Song / Corrosion Science 52 (2010) 455–480

ð57Þ ð58Þ

where m is material length, like a, b and c; L can be La, Lb and Lc; and W is a new distribution parameter, which can be Wa, Wb and Wc for materials (a), (b) and (c). After the transformation, X and W become dimensionless. In mathematics, X will always range from 0 to 1 and W is purely a curve ‘‘distribution characteristic number”, which specifies the dependence of galvanic potential or current on distance x. W is a combination of geometric effect, material effect and electrolytic effect (L only represents the effect of material and environmental properties). A large W means that potentials or currents vary significantly with X, whereas they change slowly with X if W is small. On a transformed X-coordinate, different systems should have a similar potential or current curve if their ‘‘distribution characteristic numbers” are the same. Fig. 11 shows the Faradic and flowing currents (IF and If) over an anode (a)–cathode (c) couple with different La and Lc but the same Wa and Wc. The currents appear to be mainly distributed within about 0.3 of the lengths over the anode and cathode from their joint point, even through the anode and cathode have very different lengths, polarization resistivity, and L values. According to the value of ‘‘distribution characteristic number” W, galvanic corrosion systems can be easily classified into different groups. 4.2.1. W ? 0 There are at least three possibilities that parameter W can be very small: (1) Small m (e.g. a, b or c) or m ? 0: This can be a case that the material length of a studied system is much smaller than the effective distribution distance. (2) Small qs or qs ? 0: This can be an electrolyte having very good conductivity, such as concentrated salt solution, sea water, deicing salt mixed with melted snow, concentrated acidic or alkaline solutions. (3) Large qp (e.g. qap ; qbp ; qcp ) or qp ? 1: This refers to a difficult Faradic reaction. In theory, the polarization resistivity will be very large when the material is not active, or in a passive state with a stable passive film protection, or coated with an inert corrosion resistant film, or the Faradic reaction at the metal surface is controlled by a diffusion step.

Fig. 11. Theoretical Faradic and flowing currents (IF and If) over an anode (a) and cathode (c) couple with different La and Lc but the same Wa and Wc. The curves are produced using corresponding equations in Section 3.

A combination of the above three cases can also lead to a very large W value. 4.2.2. W ? 1 W can be very large at least in the following three cases: (1) large m (e.g. a, b or c); (2) large qs; and (3) small qp (e.g. qap ; qbp ; qcp ). (1) Large m (e.g. a, b or c) or m ? 1: This can be a case when the material length of a studied system is much larger than the effective distribution distance. (2) Large qs or qs ? 1: This can be a system in some electrolytes, such as extremely diluted solutions, tap water, rain water, condensed water, pure water, some special coolants, fuels, brake fluids, anti-corrosion oils and detergents. Sometimes, such a system may not be noticed. For example, an organic coating that has taken up sufficient water after being exposed in service environments for a long time. While a metal coated with such a coating is in a dry environment, the high resistivity coating will act as an electrolytic film for the coated metal. (3) Small qp (e.g. qap ; qbp ; qcp ) or qp ? 0: Some active metal surfaces without any surface protection can have a very fast Faradic reaction. Sometimes, a passive metal after transpassivation, reactivation or suffering from pitting corrosion or crevice corrosion can also have very small polarization resistivity. A combination of the above three cases can also lead to a very small W value. 4.3. Simplified potential and current equations in some extreme cases When W ? 0 or 1, the theoretical potential and current expressions in Section 3 can be simplified (see Appendix H). In these cases, the potential and current curves will have some characteristics that can be utilized to estimate the corrosion damage of a galvanic system in practice. For a system with small W, the potential and current expressions become linear equations according to Appendix H.1. The curves in Figs. 12 and 13 are produced using corresponding equations in Section 3. They show that flowing current densities Iaf ; Ibf and Icf are linearly dependent on distance x in the system with large distribution numbers Wa, Wb and Wc. This is in agreement with the linear distribution of flowing current densities as predicted by those simplified equations in Appendices H.1.1 and H.1.2. For a system of two joined dissimilar metals with small Wa and c W , according to Appendix H.1.1 the portion of the potential difference distributed over a metal is proportional to the polarization resistance of this metal. The flowing current densities in the electrolyte are linearly dependent on the distance. In addition to the corrosion potential difference of these two metals, the sum of polarization resistances of these two metals also governs the flowing current and Faradic reaction current. The currents are reciprocally proportional to the material length and the total polarization resistance. These potentials and currents become independent from the resistivity of the electrolyte in this case due to small Wa and Wc values. The galvanic current between these two metals become proportional to both the corrosion potential difference and the reciprocal of the sum of polarization resistances. This is consistent with the simple conventional galvanic equation that is normally used to predict galvanic current between two metals [29]. Similarly, the maximum galvanic current densities of metals (a) and (c) at the joint are also simply determined by the corrosion potential difference and the sums of polarization resistances and lengths of these two metals. A larger corrosion potential difference

G.-L. Song / Corrosion Science 52 (2010) 455–480

Fig. 12. Theoretical potentials and currents (IF and If) over an anode (a)–cathode (c) couple with small Wa and Wc. The curves are produced using corresponding equations in Section 3.

and smaller polarization resistance of the metals will result in a higher maximum galvanic current density. Increasing the length of a metal will decrease its maximum current density. If the two or three dissimilar metals joined in series have large W values, then the galvanic potentials and currents will have sinh(x/L) and cosh(x/L) distributions (see Appendix H.2 or H.2.1 and Appendix H.2.2). In the region next to a joint, they can be further simplified into exponential distributions. In fact, the potentials and currents vary significantly within a narrow region adjacent to the joints. The galvanic potentials and currents will become smaller in regions far away from the joints. Figs. 14 and 15 show the galvanic potentials and currents of a system with large W values, in which the currents are nearly zero and potentials are constant in most regions except that they vary dramatically within a very narrow region next to the joints. Moreover, according to Appendices H.2.1 and H.2.2, the galvanic current between two adjacent metals are dependent on the polarization resistivity of both metals and the conductivity of the electrolyte, while the maximum galvanic current density at the joint is purely determined by the polarization resistivity of the adjacent metals and the corrosion potential difference, but independent from electrolyte film conductivity. If a piece of large W metal is in contact with a small W metal, then the galvanic potentials and currents over the couple will be

467

Fig. 13. Theoretical potentials and currents (IF and If) over a system of three pieces in series with small Wa, Wb and Wc. The curves are produced using corresponding equations in Section 3.

asymmetric (referring to Appendix H.3), which has been shown in Fig. 9. When a small W material is inserted between two other materials, an interesting prediction about galvanic current densities can be made according to Appendix H.4. The galvanic currents at joints ‘‘a/b” and ‘‘b/c” are equal and independent from the corrosion potential of metal (b). In fact, Wb ? 0 means b ? 0 or qbp ! 1 or qs ? 0. In any of these cases, the flowing galvanic current consumed by the Faradic reaction over metal (b) can be neglected, and all the galvanic current flowing into metal (b) from joint ‘‘a/b” will flow out from joint ‘‘b/c”. Hence, the galvanic currents at these two joints are nearly equal. Since the total Faradic current consumed by the system is negligible, the galvanic potential will have nothing to do with the total galvanic current. For the same reason, the Faradic current or the corrosion potential of metal (b) will have a very limited influence on the potential and current distributions over metals (a) and (b). However, the corrosion potential of metal (b) can significantly affect the potential and currents over metal (b). Fig. 16 shows the currents and potentials of such a system. In case that a large W material is insert between two small W materials (referring to Appendix H.5), the galvanic current at the connection points will be not only dependent on the potential difference of the adjacent metals, but also on their polarization resistivity and the electrolyte resistivity. The maximum galvanic current densities are also mainly determined by the adjacent material parameters. This is because the large W metal between the two small W materials effectively consumes the currents from the

468

G.-L. Song / Corrosion Science 52 (2010) 455–480

Fig. 14. Theoretical potentials and currents (IF and If) over an anode (a)–cathode (c) couple with large Wa and Wc. The curves are produced using corresponding equations in Section 3.

Fig. 15. Theoretical potentials and currents (IF and If) over a system of three pieces in series with large Wa, Wb and Wc. The curves are produced using corresponding equations in Section 3.

5. Case studies small W materials, and thus the separated small W metals have no interaction. Corresponding potential and current distributions of this system are shown in Fig. 17.

4.4. Galvanic current characterization parameters In practice, engineers are concerned about two aspects of galvanic damage to a system: (1) the overall damage (quantified by a total galvanic current) and (2) the damage intensity (quantified by a maximum galvanic current density). The total galvanic current and the maximum galvanic current density also characterize a galvanic current distribution curve or the galvanic effect of a system, and thus they can be taken as two important galvanic effect characterization parameters. Obviously the total galvanic current ig can be easily obtained according to the equations deduced earlier. A maximum Faradic current density only occurs at a joint of a galvanic system. m Therefore, Im F0 or I Fm for material (m) (m can be a, b or c) with a length of ‘‘m” represents a maximum current density. In this study, galvanic m effect characterization parameters refer to ig, Im F0 and I Fm . It should be stressed that both a positive (anodic) and negative (cathodic) Faradic currents can have a damaging effect on a coupling metal. A positive Faradic current certainly results in galvanic corrosion damage to the metal, while a significantly large negative Faradic current density can give rise to an over-protection of a metal, which is also a type of damage.

The developed galvanic potential and galvanic current equations are useful tools in analyzing galvanic corrosion. Many practical cases can have a suitable theoretical system as classified above to present, and their galvanic potential and current distributions can be easily predicted by using the corresponding theoretical equations. In this section, some examples are provided to show how the theoretical equations can be utilized to address specific galvanic corrosion issues.

5.1. Experimentally measured galvanic current densities A comparison between theoretically predicted and experimentally measured galvanic current densities is presented in Fig. 18. The experimentally measured points are obtained from reference [29]. In that reference, the galvanic current densities are denoted as Ig, but they are actually IF according to the definition in this paper. It should be noted that the anodic polarization curve of Mg (AZ91D) is nearly linear while none of the other three metals (Zn, Al and steel) has a perfect linear catholic polarization curve in 5 wt% NaCl solution [29]. As the catholic polarization curve of Zn is relatively better than Al and steel in terms of linearity, the experimentally measured current densities of the Mg–Zn couple are used in this study to compare with theoretically predicted galvanic current densities.

G.-L. Song / Corrosion Science 52 (2010) 455–480

Fig. 16. Theoretical potentials and currents (IF and If) over a system of three pieces in series with large Wa and Wc but small Wb. The curves are produced using corresponding equations in Section 3.

In Fig. 18 the theoretically predicted anodic and catholic galvanic current density curves (the solid lines) are calculated according to the galvanic current density expression in Section 3 using some parameters estimated from the polarization curve of Mg (AZ91D) immersed in the same 5 wt% solution in an electrolyte cell [29]. It appears that there is some reasonable agreement between the predicted curves (solid lines) and the experimental points. At least, it is correctly predicted that the anodic and cathodic galvanic current densities are asymmetrically distributed over the Mg and Zn surfaces; the anodic galvanic current densities in the region next to the MgjZn joint are much higher but decrease much faster with the distance from the joint on the Mg surface than the catholic galvanic current densities on the Zn surface (see the points and solid curves in Fig. 18). However, evident deviation of the experimental points from the theoretical curves (solid lines) is also seen in Fig. 18. This is understandable. The analytical equations deduced in previous sections are based on some assumptions, such as constant polarization resistivity qp, constant solution resistivity qs and stable corrosion potential Ecorr. In practice these assumption requirements cannot be strictly met. The constant polarization resistivity assumption requires that the qp of a metal does not change with distance, which means that at least the polarization curve of the metal is linear. Unfortunately, a practical metal cannot have a strictly linear the polarization curve over a wide potential range. Quite often, qp is current density dependent. It is usually smaller in a region next to the couple joint

469

Fig. 17. Theoretical potentials and currents (IF and If) over a system of three pieces in series with small Wa and Wc but large Wb. The curves are produced using corresponding equations in Section 3.

where galvanic current densities are higher, whereas in a region with lower current densities far away from the joint it becomes larger. Such a non-constant qp over a metal is contradictory to the assumption and can result in a theoretically predicted galvanic current density different from its corresponding experimental point. In this study, the relatively significant discrepancy over the Zn surface than over the Mg surface may be ascribed to the worse linearity of the catholic polarization curve of Zn than the anodic polarization curve of Mg (AZ91D). The anodic polarization curve of Mg is nearly linear and thus its qap can be reasonable assumed to be constant (independent from distance) over the Mg surface. For Zn, its catholic polarization curve is obviously non-linear. In this study, only a small section (close to be linear) of its catholic polarization curve is used to estimate qcp and Eccorr . The estimated qcp and Eccorr may only represent its polarization resistivity and corrosion potential in a particular surface area. In other surface areas, they may actually be significantly different from those estimated ones. Therefore, if these estimated qcp and Eccorr are used to calculate the theoretical galvanic current curves of the Mg–Zn couple, a significant difference will result between predicted and measured results. It is also required by the constant qs assumption that the electrolyte film has a uniform composition, concentration and thickness over the entire galvanic couple. However, due to a nonuniform surface tension force over the surface in practice, the electrolyte film may be much thin or disappear in some surface areas. Also, the Faradic reactions on the cathode and anode surfaces can

470

G.-L. Song / Corrosion Science 52 (2010) 455–480

Fig. 18. Theoretically predicted distribution of galvanic current density and experimentally measured galvanic current densities of a Mg–Zn galvanic couple under the ASTM B-117 standard salt spray condition. The experimental points are from Ref. [29]. *qcp and Ecorr are estimated from the linear region of the catholic polarization curve of Zn immersed in 5 wt% solution [29]. **Trial values assuming that the qcp and Ecorr become smaller and more positive under a salt spray condition than under an immersion condition.

cause a change in composition and concentration of the electrolyte film. Particularly in the region adjacent to the anodejcathode joint, the corrosion products from the anode and cathode may interact each other and lead to a more complicated change in the electrolyte. The variation in electrolyte thickness only affects qp. A change in composition or concentration of the electrolyte will influence not only qp but also qs. It has been demonstrated that sometimes there is a ‘‘short-cut” effect on a galvanic couple, which is a result of non-uniform distribution of the electrolyte over a galvanic corrosion couple under a salt spray condition [29]. Therefore, it is believed that violation of the constant qs assumption should also be one of the causes of the discrepancy between the experimental points and the theoretical curves (solid lines) of the Mg–Zn couple (Fig. 18). In addition to the above errors introduced by unsatisfactory assumption requirements, the different conditions for measuring the polarization curves and for measuring the galvanic current densities could also contribute to the discrepancy between the experimental points and theoretical curves. The galvanic current densities were measured under a salt spray condition, whereas the polarization curves were obtained under an immersion condition [29]. One of the most important differences between immersion and spray is the supply of oxygen, which can significantly affect a cathodic polarization curve of a conventional metallic material. It is believed that the real catholic polarization curve of Zn should have much higher cathodic current densities under a salt spray condition than under an immersion condition. In other words, the qcp and Eccorr of Zn during the galvanic current measurement in a salt spray chamber should be much larger and more positive, respectively, than those estimated from its polarization curve measured in an electrolyte cell. Evidently smaller theoretical galvanic current densities will be predicted if the qcp and Eccorr estimated from the polarization curve of Zn are used in this case, which is exactly what are presented in Fig. 18 (the solid lines lower than the points). It should be noted that the influence of oxygen on anodic dissolution of Mg is insignificant [30–32] and hence the qap and Eacorr of Mg (AZ91D) estimated from its polarization curve in the electrolyte cell should be reasonable for theoretical prediction of the galvanic current densities of the Mg–Zn couple in the salt spray chamber. Nevertheless, the inaccurate qcp and Eccorr can actually influence the prediction of the galvanic current densities not only over Zn, but also on the Mg (AZ91D) surface according to the discussion in Section 3. Therefore, the theoretically pre-

dicted curves (solid lines) deviate from the experimentally measured points on both Mg and Zn surfaces. It is believed that the theoretical prediction can be significantly improved if more accurate and reasonable parameters qp, qs and Ecorr are used. This is demonstrated in Fig. 18. The dash curves in the figure are theoretically predicted galvanic current densities using a smaller trial qcp and a more positive trial Eccorr based on the fact that sufficient oxygen supply can accelerate the cathodic process of Zn underneath a thin electrolyte film under a salt spray condition. One can see that the dash curves match the experimental points much better than the solid curves on both Zn and Mg sides of the Mg–Zn couple. 5.2. ‘‘Linear superposition” It was reported that in a ‘‘steel–AZ91D–steel” galvanic system the galvanic current density over the AZ91D is equal to the sum or addition of the galvanic current densities of AZ91D in two separated ‘‘steel–AZ91D” and ‘‘AZ91D–steel” systems based on computer numerical analysis [24]. The phenomenon was termed as ‘‘linear superposition” in that paper. However, strictly speaking, there is a possibility that the current density of AZ91D in the ‘‘steel–AZ91D–steel” system were accidently close to the addition or sum of the current densities of AZ91D in the ‘‘steel–AZ91D” and ‘‘AZ91D–steel” systems, and simply because that the differences between the current density and the sum are not significant, they were mistaken as being equal. Now with the theoretical Faradic current density expression in this study, the so-called ‘‘linear superposition” phenomenon can be analytically examined. We can assume that a = c, La = Lc, Wa = Wc, where materials (a) and (c) refer to steel and material (b) represents AZ91D. The origin point of x-coordinate is set at the left end of the ‘‘steel–AZ91D– steel” system. For the couple of ‘‘steel–AZ91D” on the left in the system, the galvanic current density over AZ91D can be expressed as follows according to Eq. (31):



b=a

IF

  3 2 Eacorr  Ebcorr tanhðW a Þ cosh xab b L 5; i4 qffiffiffiffiffiffi ¼ pffiffiffiffi hpffiffiffiffiffiffi coshðW b Þ qbp qap tanhðW b Þ þ qbp tanhðW a Þ a6x6aþb

ð59Þ

For the couple ‘‘AZ91D–steel” on the right in the system, x-coordinate should be right shifted for a distance of ‘‘a” to quote Eq. (30):

G.-L. Song / Corrosion Science 52 (2010) 455–480

Ib=c F

   3 2 Eccorr  Ebcorr tanhðW c Þ cosh xa Lb 5 i4 qffiffiffiffiffiffi ; ¼ pffiffiffiffi hpffiffiffiffiffiffi b b b c c b qp qp tanhðW Þ þ qp tanhðW Þ coshðW Þ a6x6aþb

ð60Þ

The sum or superposition of these two current densities is: b=c Ib=a F þ IF

  h    i þ cosh xa Eacorr  Ebcorr tanhðW a Þ cosh xab Lb Lb i qffiffiffiffiffiffi ¼ pffiffiffiffi hpffiffiffiffiffiffi b b a qp qap tanhðW Þ þ qbp tanhðW Þ coshðW b Þ

ð61Þ

This is a symmetric galvanic system, in which Eacorr ¼ Eccorr ; Eba ¼ Ebc ; Wb0 ¼ Wbb ; M ¼ N; and G ¼ H. Hence, Eq. (38) can be simplified:

  n    o  sinh xab Eacorr  Ebcorr tanhðW a Þ sinh xa b b L L  b o ; IbF ¼  pffiffiffiffi npffiffiffiffi pffiffiffiffiffiffi qbp qbp sinðW b Þ tanhðW a Þ þ qap sinh2 W2 a6x6aþb

ð62Þ

471

may be explained according to the theoretical galvanic corrosion models developed above. The fuel and combustion systems of a car includes various pipelines and connections. They can be treated as a one-dimensional system. When fuel is taken in through the valves prior to combustion in the cylinder bores of a engine block, there should be a very thin liquid fuel deposited on the surface of the valves. Therefore, around the in-take valve, a one-dimensional system is formed. E85 has conductivity around the order of lS/cm (MX cm) [33]. For gasoline with 0% addition of ethanol is normally considered as an insulator which has very low conductivity (can be as low as 108 lS/cm). Therefore, the corroding area in a fuel system or engine block head can be considered as a large W galvanic system. For simplicity, if only a connection with two different metals is considered, then Eq. (184) (referring to Appendix H.2.1) can be used:

 c  E  Eacorr ac ig  pffiffiffiffiffihcorr pffiffiffiffiffiffi pffiffiffiffiffiffii qs qap þ qcp

ð65Þ

Thus, b=c Ib=a F þIF

h

IbF

   inqffiffiffiffiffiffi  b o pffiffiffiffiffiffi þcosh xb cosh xab qbp sinhðW b ÞtanhðW a Þþ qap sinh2 W2 Lb Lb    ohpffiffiffiffiffiffi i qffiffiffiffiffiffi ¼n sinh xab sinh xa qap tanhðW b Þþ qbp tanhðW a Þ coshðW b Þ Lb Lb

ð63Þ

Obviously, b=c Ib=a F þ IF

IbF

–1

ð64Þ

  i.e., Ib=a þ Ib=a cannot always be equal to IbF . In other words, the F F ‘‘linear superposition” is not a common phenomenon. In theory, the galvanic current of AZ91D in the galvanic system ‘‘steel– AZ91D–steel” cannot be simply expressed as the sum of the galvanic current densities of AZ91D in two separate galvanic systems ‘‘steel–AZ91D” and ‘‘AZ91D–steel”. As discussed in Section 3.2, for a ‘‘a–b–c” joint system, material (a) can even be influenced by material (c) and the distribution of IF on it is also determined by the parameters of the indirectly connected remote metal (c) in the system. However, this ‘‘remote” effect by an indirectly connected remote material is not considered in the ‘‘a–b” and ‘‘b–c” systems. Therefore, when the galvanic current densities of two simple couples (‘‘a–b” and ‘‘b–c”) are added together without considering the ‘‘remote” effect, the ‘‘remote effect” will not be involved in the sum of the these two current densities. A lack of the ‘‘remote” effect in the sum of the galvanic current densities should be responsible for the inadequacy of the superposition principle in the ‘‘a–b–c” system. This case study shows that computer modeling or numerical analysis without analytical verification in some cases can be misleading, and the analytical approach developed in this study can help understanding computer simulated results. 5.3. Corrosion by bio-fuel E85 Bio-fuel is an important alternative to gasoline and has been used in the automotive industry. Currently, the commercial biofuel, ‘‘E85” (85 vol% ethanol + 15 vol% gasoline) produced by several oil companies, are available in petrol stations in North America and Europe. However, some corrosion problems associated with the E85 have been found in fuel systems and aluminum engine cylinder heads. It is unclear what corrosion mechanisms are involved in the damage. Nevertheless, galvanic corrosion cannot be excluded, as mixed metals are used in these systems. If a galvanic corrosion mechanism is assumed, then some corrosion phenomena

The equation indicates that the overall galvanic corrosion damage is determined by the corrosion potentials, the polarization resistivity of the connected metals and the electrolyte resistivity. There is a lack of information about the polarization resistivity and the corrosion potentials of metals in E85. Generally speaking, the difference in corrosion potential of two different metals cannot exceed 5 V. Hence, the influence of the corrosion potentials on the overall galvanic corrosion can be assumed to be insignificant in this case. Al can react with ethanol fuel [34], implying that its polarization resistivity may be lower in ethanol than in gasoline. As mentioned above, the resistivity of E85 is more than eight orders of magnitude lower than that of gasoline. According to Eq. (184), this will result in over four orders of magnitude greater overall galvanic corrosion damage if the other influences are not considerable, which may explain the identified corrosion damage in a fuel system and an Al engine cylinder head when E85 is used, whilst no significant corrosion damage is observed if gasoline is used. With regard to the maximum galvanic damage at the joint of two different metals, Eq. (185) predicts (referring to Appendix H.2.1):

 c  E  Eacorr IaFa  pffiffiffiffiffiffihcorr pffiffiffiffiffiffi pffiffiffiffiffiffii qap qcp þ qap

ð66Þ

i.e., the maximum galvanic corrosion rate is mainly determined by the polarization resistivity of the coupled metals and the difference of their corrosion potentials. Particularly, the polarization resistivity of a corroding anode is much more critical. This means that in the in-take valves of an engine block head, improving the polarization resistance of the Al alloy engine block head can more effectively reduce the maximum galvanic corrosion damage than increasing the resistance of the steel seat in theory. 5.4. A passive spacer between a galvanic couple It is a common practice in the automotive industry to use a passive spacer to isolate a cathode from an anodic part in order to reduce the galvanic corrosion. The isolator can be a passive metal, e.g. Al alloy, but more effectively an insulator, such as rubber, nylon, or teflon. It is generally accepted in the industry that the spacer should be wider than 5 mm in order to effectively eliminate the galvanic corrosion attack to Mg alloy parts joined with other metals. This practice came from Hawke’s publication [27], in which it was reported that the galvanic corrosion indicated by the weight loss of diecast AZ91D plates was linearly dependent on the insulating spacer thickness (width), and based upon this the linear dependence the galvanic corrosion was predicted to be eliminated

472

G.-L. Song / Corrosion Science 52 (2010) 455–480

if the spacer was thicker (wider) than 4.8 mm. Song et al. [29] questioned this prediction and experimentally demonstrated that the galvanic current of AZ91D in contact with steel under the ASTM B-117 standard salt spray condition was not linearly decreasing with the spacing width of an insulator and the galvanic current was found to be still as high as 22 lA/cm2 after the insulating spacer was up to 9 cm wide. The non-linearly decreasing galvanic current with spacer width was understood by using a schematic diagram illustration [29]. Now with the developed galvanic corrosion theory in this study, the effect of an insulating spacer on a galvanic current density can be further analytically examined. In a salt spray chamber, there should be a thin salt electrolyte film on specimen surfaces. The studied spacer was an insulator [29], the polarization resistivity should be infinite. Let AZ91D be material (a), steel be material (c) and the insulator spacer be material (b), then we have qbp ! 1 and Wb ? 0. This is a typical galvanic system with a small W material between two different metals. Eq. (222) (referring to Appendix H.4) can be used to describe the overall galvanic current: ab

bc

ac

ig  ig  ig 

Eccorr  Eacorr  pffiffiffiffia pffiffiffiffic

q q pffiffiffiffiffi pffiffiffiffiffi qs b qs þ tanhðWp a Þ tanhðWp c Þ

ð67Þ

ab

where ig is non-linearly dependent on the spacer width b. This theoretically denies the linear dependence of galvanic corrosion rate on a spacer width [27]. In other words, the existing practice of using washers wider than 5 mm to mitigate the galvanic corrosion of Mg alloy parts in the automotive industry may not be sufficient under the ASTM standard salt spray environment. To further support Song et al’s theoretical non-linear argument [29], the reliability of the dependence of galvanic current on spacer width measured by Song et al. [29] is examined as follows. Eq. (222) (referring to Appendix H.4) can be rewritten into:

1 ac  ig



qs

Eccorr  Eacorr



bþ

pffiffiffiffiffi

qs

"

pffiffiffiffiffiaffi

qp

Eccorr  Eacorr tanhðW a Þ

pffiffiffiffifficffi þ

qp

#

tanhðW c Þ

ð68Þ

which is a straight line and the slope is associated with the electrolyte resistivity and the corrosion potentials of the galvanic system. Fig. 19 presents such a dependence of the galvanic current measured by Song et al. [29] on the insulator width. It does display a nearly straight line. It has been measured [29] that the corrosion potentials of steel and AZ91D are around 650 mV  Ag/AgCl and 1530 mV  Ag/AgCl, respectively. The slope of the straight line in Fig. 19 is measured to be 0.153 mA1. Therefore, the electrolyte resistivity can be calculated to be 134.64 X cm from the slope and the corrosion potentials. The width of the tested specimen is already known to be 1.5 cm [29]. If the thickness of the sprayed 5 wt% NaCl solution over the specimen is assumed to be around 700 lm, then according to Eq. (77), the solution resistivity can be estimated, which can be converted into conductivity 82,525 lS/ cm. It has been experimentally measured that this solution has conductivity of 79,500 lS/cm [25]. Considering the change in composition of the solution due to dissolution of Mg into it and the error in estimating the thickness of the solution on the Mg-steel couple surface under the salt spray condition, the estimated value from Eq. (68) is very close to the experimental measured one. This suggests that the effect of the insulating spacer on galvanic current reported by Song et al. [29] is reasonably reliable. It appears that the theoretical model is useful in verifying experimental results. 5.5. A scratch in an organic coating Organic coating is one of the most popular corrosion protection measures in practice. It is common knowledge that if an organic coating is damaged locally, e.g. scratched, it should be repaired

ac

Fig. 19. Dependence of 1=ig on an insulating spacer width. The data points extracted from the literature (Fig. 8) [29].

immediately. Otherwise, severe local corrosion attack may occur. The localized corrosion attack along a scratch area can be ascribed to the galvanic effect between the exposed substrate along the scratch and the surrounding organic coating. If we take the direction perpendicular to the scratch on the coating surface as the x-coordinate, and assign the scratched section as material (b) and the coating on both sides of the scratch as materials (a) and (c), then a scratched coating can be treated as a one-dimensional system with three pieces in connection. If the scratch is made in the middle of a coated specimen, then a = c, and the system can be further simplified. Normally, an organic coating itself is nearly an insulator before degradation. It has no electrochemical activity. This means that in a coated area, the polarization resistivity is infinite. When the specimen is polarized, the resistivity of the coating will be the polarization resistivity of the coated specimen, which could be over 1014 X cm2. A scratch made by a sharp knife can be as narrow as 50 lm. If the scratched coating specimen is exposed in a salt spray environment, the polarization resistivity of the exposed substrate metal, such as a magnesium alloy, could be only 6.7 X cm2 (see Fig. 18). At the same time, it has been measured that the solution resistance of the salt is about 12.5 X cm (800,000 lS/cm) [25]. Therefore, if the width of the coated area is 1 cm, we can work out the distribution characteristic numbers Wa  Wc  1/(1014/12.5)1/ 2  3.5  107 and Wb  50  104/(6.7/12.5)1/2 = 6.8  103 for the coated and scratched regions, respectively. Both Wc (or Wa) and Wb are very small, i.e., the system has small Wa, Wb and Wc. We can use equations in Appendix H.1.2 to analyze the overall galvanic effect of the coating on the scratch and the maximum galvanic corrosion of the scratched area:

! Rbs Ebcorr  Eccorr  c  Rcp Rcp þ Rap þ Rbs Rp þ Rap þ Rbs   Rap þ Rbs Ebc þ Rcp Eba   IbFb   qbp Rcp þ Rap þ Rbs

bc ig

Eccorr  Eacorr

ð69Þ

ð70Þ

It should be noted that material (a) is the same as material (c), so they can be further simplified into: bc

ig   IbFb  

! Rbs Ebcorr  Eccorr Rcp 2Rcp þ Rbs Ebcorr  Eccorr

qbp

ð71Þ ð72Þ

These two equations suggest that the directions of the total galvanic current and the maximum galvanic current density are determined

473

G.-L. Song / Corrosion Science 52 (2010) 455–480

by Ebcorr and Eccorr . A coated area normally has a more positive corro  bc sion potential than a scratched area Eccorr > Ebcorr . Hence, ig and IbFb are positive, flowing from the scratch to the coating. In other words, the scratched area is anodic to the coated area, suffering from a galvanic corrosion attack by the coating. bc The ig equation also indicates that the galvanic current from the scratched area is only dependent on the solution resistance over the scratched region and the polarization resistance over the coated area. A larger coated area will have a smaller Rcp according to Eq. (74) (referring to Appendix A), and consequently lead to a large overall galvanic current over the scratched area. The IbFb equation suggests that the maximum galvanic current density is mainly determined by polarization resistivity of the scratched region in addition to the potential difference between the scratched region and the coated area. Other parameters have no significant influence on this maximum current density. 6. Summary A one-dimensional galvanic system can represent many practical galvanic corrosion cases. Theoretical equations developed in this study for galvanic potentials, current densities and overall galvanic corrosion damage as well as the maximum galvanic damage intensity of typical galvanic systems can be used to analyze many real galvanic corrosion problems. The theoretical analytical prediction for galvanic corrosion by this analytical approach provides an insight into a galvanic corrosion system. With developed theoretical equations, the influence of system parameters on galvanic corrosion can be clearly defined.

where S is the cross-section area of the electrolyte over the metal surface; C is the inner perimeter of the tube system and y is the width of the electrolyte coverage over a flat surface. q0s and q0p have their normal units X cm and X cm2, respectively. They are different from those (qs and qp) in the one-dimensional system. Appendix B Based on Fig. 2, there is:

IF ¼ W=qp where

W ¼ ðE  Ecorr Þ

W ¼ Ecorr  E

ð82Þ

Meanwhile, If can be expressed (see Fig. 2) as:

If ¼ 

1 dW

ð83Þ

qs dx

At x, the current density flowing into a give elemental electrolyte should be equal to the current density flowing out of it (see Fig. 2):

IF dx ¼ ðIf jxþdx  If jx Þ ¼ dIf

IF ¼ 

1

  dw dx 

qs

xþdx

   dw dx 

dx

x

2

¼

d w dx

¼

It has a common solution:

S ¼ pr 2

w ¼ A exp

S ¼ dY

ðfor a flat surfaceÞ

ð74Þ

The solution resistivity qs and polarization resistivity qp in this onedimensional system are:

Rs m qp ¼ mRp

qs ¼

ð76Þ

qp ¼

q0s S

q0p c

qp ¼

y

x  x þ B exp  L L

ð87Þ

Constant A and B need to be determined with boundary conditions. According to Eq. (82), the potential and currents of the metal relative to a reference electrode can be obtained.

ðfor a tubeÞ

Appendix C According to Eq. (83), there is

 dw ¼0 dx x¼c

ðfor a flat surfaceÞ

ð88Þ

This acts as one of the boundary conditions for Eq. (86). The other boundary condition is the applied potential Wc0 at the other end x = 0:

wc jx¼0 ¼ wc0

ð89Þ

ð77Þ

With these boundary conditions, constants A and B can be easily determined and the potential and current distributions obtained:

ð78Þ

A¼

    wc0 exp  Lcc wc exp  Lcc wc0 2c ¼ c  c ¼ 0 c 1 þ exp Lc exp Lc þ exp  Lc 2 cosh Lc

ð90Þ

B¼

      wc0 exp 2c wc exp Lcc wc exp Lcc c  c 0  c ¼ 0   L2c ¼ 1 þ exp Lc exp Lc þ exp  Lc 2 cosh c

ð91Þ

or

q0p

ð86Þ

ð75Þ

where Rs is the overall or total solution resistance over the whole region m;Rp is the polarization resistance over the whole region m. Solution resistivity qs and polarization resistivity qp are defined as the solution resistance and the polarization resistance of the metal per unit length, respectively. Both qs and qp are assumed to be constants, independent of the position x in the system. They have units X cm1 and X cm, respectively. Their relationships with the conventional solution resistivity q0s and conventional polarization resistivity q0p in a three-dimensional system are:

qs ¼

ð85Þ

qs dx2

qs w qp

The cross-section area of the fine tube and thin electrolyte film over a flat surface can be written as

ð73Þ

1 d w

According to Eqs. (80) and (85), we have 2

ðfor a tubeÞ

ð84Þ

With Eqs. (83) and (84) can be rewritten:

2

Appendix A

ð81Þ

where E is a potential of the metal relative to a reference electrode. Ecorr is the corrosion potential of the metal, which is also relative to the same reference electrode. As W is relative to the potential of the metal, not relative to the potential of the electrolyte or a reference electrode in the electrolyte, its value is negative with respect to the potential relative to a reference electrode (E  Ecorr). Therefore,

Acknowledgment The author thank Dr. Blair Carlson for his help in editing this document.

ð80Þ

ð79Þ

L2

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G.-L. Song / Corrosion Science 52 (2010) 455–480

Therefore, Wc and corresponding Ec ; IcF and Icf equations can be obtained according to the general potential and current equations deduced earlier. Appendix D When a ‘‘dead end” is on the right, Icf0 represents the current flowing into the electrolyte through the left end of the system from outside, which can also be regarded as an overall polarization current ip applied from outside, i.e.:

   wc tanh cc ip ¼ Icf0 ¼ Icf x¼0 ¼ 0pffiffiffiffiffiffiffiffiffifficffiL

ð92Þ

qs qp

In theory, the sum of IF over the surface should be equal to the polarization current ip from outside. When ip is flowing into the electrolyte from left to right, it is considered to be positive. In this case, If has to cross the electrolyte/metal interface and flow into the metal, which is negative. Hence, we have: c

ip ¼ iF

ð93Þ

Eq. (93) can be confirmed by integrating the Faradic current density IcF over the entire metal surface. Eq. (92) can also be rewritten as:

wc0

¼

c iF

"    # pffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi exp cc þ exp  cc ip qs qcp L  L c   ¼   qs qp exp Lcc  exp Lcc tanh Lcc

ð94Þ

This is a linear equation between applied potential W and current ip. If the ‘‘dead end” is on the left, then the applied potential at the boundary and the responding current Iaf also have a linear relationship as described by Eqs. (92) and (94):

a

 waa tanh La pffiffiffiffiffiffiffiffiffiffiaffi ¼ Iaf x¼a ¼ Iafa

a

ð95Þ

qs qp

pffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi ip qs qap a a a   wa ¼ iF ¼ qs qp ¼  tanh Laa

ð96Þ

In an ‘‘open ends” system, at both boundaries, the current flowing into and out of the section can be expressed as:

 

a

c

The galvanic current ig or ig between these two pieces of metals can also be regarded as the polarization currents of metal (a) and metal (c), respectively. If these two metals are jointed at x = 0, they a c can be treated as two ‘‘dead end” metals polarized by ig and ig , respectively. According to Eqs. (11) and (95), these are:

 a qs qp La  wc0 c c ffi tanh c ig ¼ ip ¼ Icf0 ¼ pffiffiffiffiffiffiffiffiffiffi c qs qp L waa a ffi tanh ig ¼ ip ¼ Iaf0 ¼  pffiffiffiffiffiffiffiffiffiffi a

ð102Þ ð103Þ

Substitute them into Eq. (25), then

wc0 ac ffi tanh ig ¼ pffiffiffiffiffiffiffiffiffiffi c

qs qp

  c waa a ffi tanh a ¼  pffiffiffiffiffiffiffiffiffiffi c a qs qp L L

ð104Þ

which can be simplified into:

  pffiffiffiffiffiaffi qp tanh Lcc waa a ffic c ¼  pffiffiffiffiffi qp tanh La w0

ð105Þ

This equation suggests that the potential in the electrolyte over metals (a) and (c) are always in opposing direction, i.e., they must be negative to each other. According to Eq. (82), we have:

Ea ¼ Eacorr  Wa c 0

ip ¼ iF ¼

Appendix E

ð106Þ

Ec ¼ Eccorr  Wc Eacorr

ð107Þ Eccorr

where and are the corrosion potentials of metal (a) and metal (c), respectively. Therefore, Eq. (27) can be rewritten as:

Waa  Wc0 ¼ Eacorr  Eccorr

ð108Þ

Combination of Eqs. (105) and (108) yields:

waa waa  wc0 Eacorr  Eccorr ¼ c 1 ¼ w0 wc0 wc0   pffiffiffiffiffiffi   pffiffiffiffiffiaffi qp tanh Lcc þ qcp tanh Laa   ¼ pffiffiffiffiffiaffi qp tanh Lac

ð109Þ

which allows us to work out Waa and Wc0 , respectively:

  b i0 ¼ Ibf 

wb0 cosh ¼ qffiffiffiffiffiffiffiffiffiffiffi

 wbb  

ð97Þ

waa

pffiffiffiffiffiffi   Eccorr  Eacorr qap tanh Lcc   pffiffiffiffiffiffi   ¼  pffiffiffiffifficffi qp tanh Laa þ qap tanh Lcc

ð110Þ

  b ib ¼ Ibf 

  wb0  wbb cosh Lbb   ¼ qffiffiffiffiffiffiffiffiffiffiffi qs qbp sinh Lbb

ð98Þ

 a pffiffiffiffiffiffi   Ecorr  Eccorr qcp tanh Laa  c  pffiffiffiffiffiffi   wc0 ¼  pffiffiffiffiffiaffi qp tanh Lc þ qcp tanh Laa

ð111Þ

x¼0

x¼b

b Lb

qs qbp sinh



b Lb

b

The total Faradic reaction current iF should be equal to the current density flowing out of the electrolyte minus the current density flowing into the electrolyte over the section at both ends: b

b

b

iF ¼ ib  i0

ð99Þ

Boundary potentials can also be determined if the boundary currents are known:

qffiffiffiffiffiffiffiffiffiffiffi wb0 ¼

qs qbp sinh

¼

L

2 qffiffiffiffiffiffiffiffiffiffiffi

wbb

 8 b < b

qs qbp sinh 2

9 b b b b = ib  i0 ib þ i0  þ   :1  cosh b 1 þ cosh Lbb ; Lb

 8 b < b L

9 b b b b = ib  i0 ib þ i0     :1  cosh b b ; 1 þ cosh Lb Lb

ð100Þ

ð101Þ

  The signs of Wc0 and Waa depend on Eacorr  Eccorr . For example, if a c c a then W0 > 0 and Wa < 0, and consequently Ecorr < Ecorr , IaF > 0 and IcF < 0, respectively, according to Eqs. (8) and (15), suggesting that metal (a) is an anode and metal (c) a cathode. With the obtained Waa and Wc0 , the potentials and current densities can be expressed according to corresponding Eqs. (6)–(9), (11), (13)–(16), (18), (20)–(23), (97) and (98) developed earlier. It should be noted that if the origin point of the x-coordinate is set at the left end of metal (a), then a x-coordinate shift (+a) should be considered in writing equations for metal (c) that has its left end sitting at x = a. This means ‘‘x” should be replaced with ‘‘x  a” in all the equations for system (b). Therefore, there are

 c pffiffiffiffiffiffi   "  # Ecorr  Eacorr qap tanh Lcc cosh Lxa a i wa ¼  hpffiffiffiffiffiffi   pffiffiffiffiffiffi   qcp tanh Laa þ qap tanh Lcc cosh La

ð112Þ

475

G.-L. Song / Corrosion Science 52 (2010) 455–480

"  # Eacorr  Eccorr qcp tanh La cosh xac c c L   w ¼  hpffiffiffiffiffiffi   pffiffiffiffiffiffi  i cosh Lcc qa tanh cc þ qc tanh aa 

pffiffiffiffiffiffi

p

p

L

pffiffiffiffiffiaffi

a

ð113Þ

q sinh b p

L

With them, Ea ; Ec ; IaF ; IcF ; Iaf ; Icf ; IaF0 and IcF0 can all be obtained.

qp

M ¼ qffiffiffiffiffiffi

  b Lb

qp

  b Lb

q sinh

Appendix F

G ¼ 1 þ M cosh

From the established equations earlier, we have galvanic cura c rent ig and ig in relation with metals (a) and (c) according to Eqs. (102) and (103). In the meantime, according to Eqs. (97) and b b (98), we have expressions for currents i0 and ib flowing into and out of the electrolyte over metal (b) at its left and right ends. At the joints ‘‘a/b” and ‘‘b/c”, the galvanic current flowing from metal (a) should be equal to that into metal (b) and that from metal (b) equal to that into metal (c):

H ¼ 1 þ N cosh

ab

a

b

ð114Þ

bc

b

c

ð115Þ

ig ¼ ib ¼ i0 Therefore,

   wb cosh b  wb 0 b a Lb   ¼ qffiffiffiffiffiffi qp b La b qp sinh Lb    wb  wb cosh b 0 b wc0 c Lb   pffiffiffiffifficffi tanh c ¼ qffiffiffiffiffiffi qp b L b qp sinh Lb waa ffi tanh  pffiffiffiffiffi a

pffiffiffiffiffiaffih waa

qp wbb  wb0 cosh

¼ qffiffiffiffiffiffi

qbp sinh

  b Lb

pffiffiffiffifficffih

qp wb0  wbb cosh

wc0 ¼ qffiffiffiffiffiffi

  b Lb

q sinh b p



ð118Þ

a La

 i b Lb

ð119Þ

c

Let

¼



ð120Þ bc

¼ E ¼ E

qp

q sinh b p

  b Lb

ba

wbb ¼

NE þ GE GH  MN bc

wb0

bc

ba

ME þ HE ¼ GH  MN

waa ¼

MEbc þ HEba  Eba GH  MN ba

wc0 ¼

b Lb

ð133Þ

tanh

c

ð134Þ

La

Lc

then Eqs. (118) and (119) become:

 

b waa ¼ M wbb  wb0 cosh b L  

b wc0 ¼ N wb0  wbb cosh b L

ð135Þ ð136Þ

Substitute

Waa ¼ Wb0  Eba

ð137Þ

bc

ð138Þ

c 0

b b

ð123Þ

W ¼W E

ð124Þ

which come from Eqs. (120) and (121) into Eqs. (135) and (136), then they can be written into:

bc

NE þ GE  Ebc GH  MN

where

ð122Þ

 

a

tanh

qp

qbp sinh

ð121Þ

where Eacorr ; Ebcorr and Eccorr are known or measurable. Therefore, there are four parameters Waa ; Wb0 ; Wbb and Wc0 which can be determined using four available relationships (118)–(121) (refer to Appendix G):

bc

pffiffiffiffiffiaffi pffiffiffiffifficffi



ð132Þ

Appendix G

Lc

cb

ð131Þ

Correspondingly, the values and distributions of ig (Eq. (97)), ig (Eq. (98)), Iaf (Eq. (9)), Icf (Eq. (9)), Ibf (Eq. (22)), IaF (Eq. (8)), IcF (Eq. (8)), IbF (Eq. (23)), Ea (Eq. (7)), Ec (Eq. (7)) and Eb (Eq. (21)) can all be obtained according to the equations developed earlier.

N ¼ qffiffiffiffiffiffi

Ebcorr

ð130Þ

    Ebc cosh xabc Lc c cosh Lc

NEba þGEbc GHMN

wb0  waa ¼ Ebcorr  Eacorr ¼ Eba Eccorr

ð129Þ

Lb

  bc  þHEba  MEGHMN exp  Lbb xa   w ¼ exp Lb 2 sinh Lbb   ba bc  MEbc þHEba þGE exp Lbb NEGHMN GHMN xa   þ exp  b L 2 sinh Lbb

M ¼ qffiffiffiffiffiffi

wbb

ð128Þ

Lb  b

NEba þGEbc GHMN

We also know at the left and right ends of metal (b), the potentials have the following relationships following the thought of Eq. (108):

wc0

ð127Þ

Lc

ab

b Lb

tanh

c

" #   cosh Lxa MEbc þ HEba    Eba GH  MN cosh Laa

wc ¼

 i

tanh

wa ¼

ð117Þ

They can be rearranged:

tanh  b

La

With the determined Wb0 and Wbb , Waa and Wc0 will be known according to Eqs. (120) and (121). Due to the shift of the origin position of x-coordinate, variable x in the above equations needs to be replaced with a shifted x-coordinate.

b

ð116Þ

ð126Þ

pffiffiffiffifficffi

N ¼ qffiffiffiffiffiffi b p

ig ¼ ia ¼ i0

a

tanh

ð125Þ

 

b wb0 1 þ M cosh b ¼ Mwbb þ Eba L  

b wbb 1 þ N cosh b ¼ Nwb0 þ Ebc L

ð139Þ ð140Þ

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G.-L. Song / Corrosion Science 52 (2010) 455–480

Let

G ¼ 1 þ M cosh H ¼ 1 þ N cosh

 b b

L b Lb

where

ð141Þ ð142Þ

then

M Eba wb0 ¼ wbb þ G G MN b NEba b wb H ¼ w þ þ Ebc G b G

ð143Þ ð144Þ

thus

NEba þ GEbc GH  MN

wbb ¼

ð145Þ

Similarly,

N Ebc ¼ wb0 þ H H MN b MEbc b w0 G ¼ w þ þ Eba H 0 H MEbc þ HEba wb0 ¼ GH  MN

wbb

ð146Þ

MEbc þ HEba ¼ Eba GH  MN NEba þ GEbc wc0 ¼  Ebc GH  MN

according to Eq. (76). Rap and Rcp are the total polarization resistance of metals (a) and (c), respectively. H.1.2. A joint of three pieces of materials If a system containing three pieces of metals joined in series is considered, then simplified equations for Waa ; Wb0 ; Wbb and Wc0 can be developed by using the approximations in Appendix I. If the origin point or the x-coordinate is set at the left end of metal (a), then the x-coordinate shift for metals (b) and (c), must be taken into account and we have:

Ea  Eacorr  waa ¼ Eacorr 

b

Ebcorr

Ebcorr

    Rap Eacorr  Eccorr þ Rbs Ebcorr  Eacorr Rcp þ Rap þ Rbs

ð150Þ

 wb ¼



Ec  Eccorr  wc0 ¼ Eacorr 



 Eacorr  Eccorr Rap



Rcp þ Rap   Eacorr Rcp

Eccorr

Rcp þ Rap þ Rbs a6x6aþb

    Rcp Eccorr  Eacorr þ Rbs Ebcorr  Eccorr Rcp þ Rap þ Rbs

    Rap Eccorr  Eacorr þ Rbs Ebcorr  Eacorr   IaF   a ¼ qp qap Rcp þ Rap þ Rbs

Ec  Eacorr  corra Rp þ Rcp  a  Ecorr  Eccorr a   IFa   a Rc þ Rap  c p  E  Eacorr  IcF0   corr c Rap þ Rcp

ð151Þ ð152Þ ð153Þ

 Rcp þ Rbs Eba þ Rap Ebc   IbF   b ¼ qp qbp Rcp þ Rap þ Rbs   qbs Eacorr  Eccorr   ðx  aÞ; a 6 x 6 a þ b  qbp Rcp þ Rap þ Rbs

ð156Þ ð157Þ ð158Þ ð159Þ

ð163Þ

ð164Þ

ð165Þ



wb

    Rcp Eccorr  Eacorr þ Rbs Ebcorr  Eccorr   IcF   c ¼  qp qcp Rcp þ Rap þ Rbs wc0

ð166Þ

ð167Þ

    Rap Eacorr  Eccorr þ Rbs Ebcorr  Eacorr   IaF   a ¼  x; qp qap Rcp þ Rap þ Rbs waa x

06x6a

ð154Þ ð155Þ

ð162Þ

  Rcp þ Rbs Eba þ Rap Ebc

waa

Ec  Ecorr  Rap þ Rcp  a  c E  Ecorr  IaF   corr a Rcp þ Rap  c  E  Eacorr  IcF   corr c Rap þ Rcp  a  E  Eccorr x   ; 06x6a IaF   corr a Rcp þ Rap  c  E  Ea ðx  a  cÞ  Icf  corr  corr ; a6x6aþc c Rap þ Rcp ac ig

ð161Þ

c

ð149Þ

H.1.1. A joint of two dissimilar metals In the case of two different metals joined together, by utilizing the approximations in Appendix I, we have:



q

  Rb Ea  Ec xa  s c corr a corr ; b Rp þ Rp þ Rbs

When W ? 0, i.e., Wa ? 0 and Wc ? 0, the approximations as described in Appendix I can be employed to simplify the galvanic potential and current distribution equations.

E 

c p

E 

H.1. Small W systems

Eacorr

ð160Þ

a

ð148Þ

Appendix H

a

Rcp ¼

qap

ð147Þ

Thus,

waa ¼

Rap ¼

Ibf

ð168Þ

 Rcp þ Rbs Eba þ Rap Ebc   x;  c  Rp þ Rap þ Rbs qbp Rcp þ Rap þ Rbs Eacorr  Eccorr



a6x6aþb

ð169Þ

    r cp Eccorr  Eacorr þ Rbs Ebcorr  Eccorr   ðx  a  b  cÞ; Icf   qcp Rcp þ Ras þ Rbs aþb6x6aþbþc !  Ea  Ec Rbs Ebcorr  Eacorr ab ig ¼ Iaf xa   ccorr a corrb  Rap Rcp þ Rap þ Rbs Rp þ Rp þ Rs

ð170Þ

ð171Þ

477

G.-L. Song / Corrosion Science 52 (2010) 455–480

!

 Ec  Ea Rbs Ebcorr  Eccorr bc ig ¼ Icf x¼aþb   ccorr a corrb  Rcp Rcp þ Rap þ Rbs Rp þ Rp þ Rs

IaFa

IbF0

IbFb



ð172Þ



    r ap Eacorr  Eccorr þ Rbs Ebcorr  Eacorr    qap Rcp þ Ras þ Rbs

ð173Þ



  Rcp þ Rbs Eba þ Rap Ebc    qbp Rcp þ Rap þ Rbp

ð174Þ

¼

IaF x¼a

¼

 IbF  x¼a

¼

 IbF  x¼aþb





 Rap þ Rbs Ebc þ Rap Eba    qbp Rcp þ Rap þ Rbp

 IcF0 ¼ IcF x¼aþb 

rcp



Eccorr

ð175Þ

  þ  Ebcorr  Eccorr   qcp Rcp þ Ras þ Rbs Eacorr



Rbs

ð176Þ

where

Rbs ¼ bqs

ð177Þ

according to Eq. (75). Rbs is the total electrolyte resistance along metal (b). H.2. Large W systems a

c

When W ? 1, i.e., W ? 1 and W ? 1, the approximations as described in Appendix J and be employed to simplify the galvanic potential and current distribution equations. H.2.1. A joint of two dissimilar metals By utilizing the approximations in Appendix J and setting the origin point at the left end of system (a), we have:



pffiffiffiffiffiffi

 2  q x pffiffiffiffifficffi pffiffiffiffiffiaffi Ea  Eacorr þ expðW a Þ cosh a ; qp þ qp L 06x6a Eccorr

Eacorr

a p

ð178Þ

 pffiffiffiffiffiffi  2 Eacorr  Eccorr qcp xac pffiffiffiffiffiaffi pffiffiffiffifficffi Ec  Eccorr þ expðW c Þ cosh ; qp þ qp Lc a6x6aþc ð179Þ IaF

   2 Eccorr  Eacorr x h  pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffii expðW a Þ cosh a ; L a c a qp qp þ qp

06x6a

a6x6aþc    2 Eccorr  Eacorr x a h If  pffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffii expðW a Þ sinh a ; L c a qs qp þ qp

Ec  Eacorr IaFa  pffiffiffiffiffiffihcorr pffiffiffiffiffiffi pffiffiffiffiffiffii qap qcp þ qap

ð185Þ

 a  E  Eccorr IcF0  pffiffiffiffiffiffihcorr p ffiffiffiffiffi ffi p ffiffiffiffiffiffii qcp qap þ qcp

ð186Þ

H.2.2. Three pieces of materials In this case, similarly we can also have simplified expressions for Waa ; Wb0 ; Wbb and Wc0 , and then set the origin point of x-coordinate at the left end  of metal (a)  to obtain: a

E 

06x6a

ac ig

 Eccorr  Eacorr h  pffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffii qs qap þ qcp

IcF

ð192Þ

    2 Ebcorr  Eacorr a x   pffiffiffiffiffiqffiffiffiffiffiffi pffiffiffiffiffiffi exp  a sin a ; L L qs qbp þ qap 06x6a

IbF

ð191Þ

    2 Ebcorr  Eccorr c xabc  pffiffiffiffiffiffiqffiffiffiffiffiffi pffiffiffiffiffiffi exp  c cosh ; L Lc qcp qbp þ qcp aþb6x6aþbþc

Icf

ð184Þ

ð189Þ

ð190Þ

 1 6 Eb  Eccorr xab ffiffiffiffiffiffi pffiffiffiffiffiffi exp IbF   qffiffiffiffiffiffi 4qcorr Lb qbp qbp þ qcp 3  Eb  Eacorr xa 7 ffiffiffiffiffiffi pffiffiffiffiffiffi exp  b 5; a 6 x 6 a þ b þ qcorr L qbp þ qcp

ð183Þ



ð188Þ

2

   2 Eacorr  Eccorr xac c i Þ sinh expðW ; Icf  pffiffiffiffiffihp ffiffiffiffiffiffi pffiffiffiffiffiffi Lc qs qap þ qcp a6x6aþc

ð187Þ

    2 Ebcorr  Eacorr a x IaF  pffiffiffiffiffiffiqffiffiffiffiffiffi pffiffiffiffiffiffi exp  a cosh a ; L L a b a qp qp þ qp

Iaf

ð182Þ

pffiffiffiffiffiffi   2 qap Ebcorr  Eacorr a x qffiffiffiffiffiffi pffiffiffiffiffiffi þ exp  a cosh a ; L L b a qp þ qp

 pffiffiffiffiffiffi  2 qcp Ebcorr  Eccorr c qffiffiffiffiffiffi pffiffiffiffiffiffi exp  c Ec  Eccorr þ L qbp þ qcp  xabc cosh ; aþb6x6aþbþc Lc

ð181Þ 06x6a

Eacorr

06x6a  qffiffiffiffiffiffi  qbp Ebcorr  Eccorr xac b b exp E  Ecorr  qffiffiffiffiffiffi pffiffiffiffiffiffi Lb qbp þ qcp  qffiffiffiffiffiffi  qbp Ebcorr  Eacorr xa exp  b ; a 6 x 6 a þ b þ qffiffiffiffiffiffi pffiffiffiffiffiffi L qbp þ qap

ð180Þ

   2 Eacorr  Eccorr xac c i Þ cosh ; expðW IcF  pffiffiffiffiffiffihp ffiffiffiffiffiffi pffiffiffiffiffiffi Lc qcp qap þ qcp



2  1 6 Ebcorr  Eccorr xab   pffiffiffiffiffi 4qffiffiffiffiffiffi pffiffiffiffiffiffi exp qs Lb qbp þ qcp 3  Eb  Eacorr xa 7 ffiffiffiffiffiffi pffiffiffiffiffiffi exp  b 5; a 6 x 6 a þ b  qcorr L qbp þ qap

ð193Þ

ð194Þ

    2 Ebcorr  Eccorr c xabc   pffiffiffiffiffiqffiffiffiffiffiffi pffiffiffiffiffiffi exp  c sin ; L La qs qbp þ qcp aþb 6 x 6 aþbþc

ð195Þ

478

G.-L. Song / Corrosion Science 52 (2010) 455–480

 Eb  Eacorr ab qffiffiffiffiffiffi pffiffiffiffiffiffi ig ¼ Iaf x¼a  pffiffiffiffifficorr qs qbp þ qap

ð196Þ

 Eb  Eccorr bc qffiffiffiffiffiffi pffiffiffiffiffiffi ig ¼ Icf x¼aþb   pffiffiffiffifficorr qs qbp þ qcp

ð197Þ

This is an extreme case when Wb ? 0, but Wa – 0 and Wc – 0. Utilizing the approximations in Appendix I, we can work out the galvanic potentials and current densities as well as the overall galvanic currents and maximum current densities:

 Eb  Eacorr qffiffiffiffiffiffi pffiffiffiffiffiffi IaFa ¼ IaF x¼a  pffiffiffiffiffifficorr qap qbp þ qap

ð198Þ

wa 

  IbF0 ¼ IbF 

ð199Þ

x¼a

Eb  Eacorr qffiffiffiffiffiffi pffiffiffiffiffiffi   qffiffiffiffiffifficorr qbp qbp þ qcp

H.4. A small W material between two other materials

¼



w 

Eb  Eccorr qffiffiffiffiffiffi pffiffiffiffiffiffi  pffiffiffiffiffifficorr qcp qbp þ qcp

ð201Þ

 c  Ecorr  Eacorr Ec  Eccorr  c pffiffiffiffiffiffiffiffiffiffiaffi Rp þ q s q p

qcp



a

ðW c Þ

þ b tanhL

cosh

c



ðW c Þ

xabc ; c L ð213Þ

Ea Ec x  corr corr pffiffiffiffi pffiffiffiffic cosh a ; pffiffiffiffiffiaffi qap qp L a pffiffiffiffiffi b qp sinh ðW Þ qs þ tanh W a þ tanh W c ð Þ ð Þ ð214Þ

a6x6aþb

cosh

ð215Þ

c p

6aþc

ð205Þ

aþb6x6aþbþc

ð216Þ

 Ea  Eccorr x  corr pffiffiffiffia pffiffiffiffic sinh a ; qp qp pffiffiffiffiffi pffiffiffiffiffi L a qs sinhðW Þ b qs þ tanhðW a Þ þ tanhðW c Þ

06x6a

06x6a

Ec  Eacorr ðx  a  cÞ ¼  corrpffiffiffiffiffiffiffiffiffiffi ffi ðx  a  cÞ; q Rcp þ qs qap c wc0

ð204Þ

Ec Ea  corr corrpffiffiffiffi pffiffiffiffic

pffiffiffiffiffiffi qa q pffiffiffiffiffi b qcp sin ðW c Þ qs þ tanh Wp a þ tanh Wp c ð Þ ð Þ

xabc ; Lc

Iaf  

  Rcp Eccorr  Eacorr Ec  Eacorr   ¼  corr pffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffi ffi ffi qcp Rcp þ qs qap c Rcp þ qs qap

   2 Eccorr  Eacorr x a ffia expðW Þ sinh ; Iaf   pffiffiffiffiffiffiffiffiffiffi qs qp þ Rcp La Icf  

ð212Þ

2 La 3 a ba bc þbE þ tanhL W c Eba a c a E tanh W 1 E E ð Þ ð Þ corr corr þ ðxaÞ5; IbF  b 4 a c a c qp bþ tanhL W a þ tanhL W c bþ tanhL W a tanhL W c ð Þ ð Þ ð Þ ð Þ

ð203Þ

06x6a

ð206Þ

Ibf 

ð217Þ

Eccorr  Eacorr  pffiffiffiffia pffiffifficffi pffiffiffiffiffi pffiffiffiffiffi q q qs b qs þ tanhðWs a Þ þ tanhðWs c Þ ba

a

c

ba L L Ebc þ bE þ tanhðW c E tanhðW a Þ Þ i  qffiffiffiffiffiffih ðx  aÞ; La Lc qbp b þ tanhðW a þ Þ tanhðW c Þ

a6x ð207Þ

Icf  

 Ec  Eacorr Eccorr  Eacorr ab ig ¼ Iaf x¼a  pcorr ffi ffiffiffiffiffiffiffiffiffiffiaffi qcp ¼ pffiffiffiffiffiffiffiffiffiffi qs qap þ Rcp qs qp þ c

ð208Þ

 pffiffiffiffiffi c  qs Ecorr  Eacorr ffi IaF x¼a   a p ffiffiffiffiffi c pffiffiffiffiffi qp qs þ Rp qap

ð209Þ

 Ec  Eacorr IcF x¼a    corr pffiffiffiffiffiffiffiffiffiffi ffi c Rcp þ qs qap

ð210Þ

a6x6aþb

ð218Þ

  Eccorr  Eacorr sinh xabc La  pffiffiffiffi pffiffiffiffic ; qa q pffiffiffiffiffi pffiffiffiffiffi qs sinhðW c Þ b qs þ tanhðWp a Þ þ tanhðWp c Þ

ð219Þ

aþb 6 x 6 aþbþc

The distributions of potentials and currents are highly asymmetric.

ð211Þ

c

aþb6x6aþbþc

IcF  

  pffiffiffiffiffi 2 q Ec  Ea x   a psffiffiffiffiffi corr c pcorr ffiffiffiffiffiaffi expðW a Þ cosh a ; qp qs þ Rp qp L wc0

ba

06x6a

ð202Þ

Rcp

IcF ¼ 

ðW c Þ

06x6a

 pffiffiffiffiffiffiffiffiffiffiffi  2 qs qap Eccorr  Eacorr x þ pffiffiffiffiffiffiffiffiffiffiaffi expðW a Þ cosh  a ; c Rp þ qs qp L

06x6a

IaF

1 þ b tanhL

IaF  

If Wa ? 1 and Wc ? 0, after mathematical approximations, we have:

E 

þ b tanhL

Lc Eca b sinh ðW c Þ

c

H.3. A joint of a large W material and a small W material

Eacorr

c

ðW a Þ

 x ; La

þ bE þ tanhL W c Eba Eacorr  Eccorr ð Þ þ ðx  aÞ; a c La Lc b þ tanh W a þ tanh W c b þ tanhL W a þ tanhL W c ð Þ ð Þ ð Þ ð Þ

ð200Þ

The simplified equations predict reasonable results.

a

a

La Ebc tanh ðW a Þ

b

w  IcF x¼aþb

1 þ b tanhL

cosh

a6x6aþb

 Eb  Eccorr qffiffiffiffiffiffi pffiffiffiffiffiffi IbFb ¼ IaF x¼aþb   qffiffiffiffiffifficorr qbp qbp þ qcp IcF0

La Eac b sinh ðW a Þ

 ab ig ¼ Iaf x¼a  

Eacorr  Eccorr  pffiffiffiffia pffiffiffiffic

q q pffiffiffiffiffi pffiffiffiffiffi qs b qs þ tanhðWp a Þ þ tanhðWp c Þ

 bc ig ¼ Icf0 þ Icf x¼aþb 

Eccorr  Eacorr  pffiffiffiffia pffiffiffiffic

q q pffiffiffiffiffi pffiffiffiffiffi qs b qs þ tanhðWp a Þ þ tanhðWp c Þ

ð220Þ

ð221Þ

479

G.-L. Song / Corrosion Science 52 (2010) 455–480

ab

bc

ac

ig  ig  ig 

Eccorr  Eacorr  pffiffiffiffia pffiffiffiffic

q q pffiffiffiffiffi pffiffiffiffiffi qs b qs þ tanhðWp a Þ þ tanhðWp c Þ

ð222Þ

 Ea  Eccorr IaFa ¼ Iaf x¼a   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficorr pffiffiffiffia pffiffiffiffic

q q pffiffiffiffiffi b qap tanhðW a Þ b qs þ tanhðWp a Þ þ tanhðWp c Þ 2



 Ibf  x¼a

IbF0

¼

IbFb

  ¼ Ibf 

IcF0

c

3

bc ba L L 1 tanhðW a Þ E þ bE þ tanhðW c Þ E 5  b4 La Lc qp b þ tanhðW a þ Þ tanhðW c Þ

x¼aþb

¼

ba

a



IcF x¼aþb

ð223Þ

2 a 3 bc bc ba L Lc 1 4tanhðW a Þ E þ bE þ tanhðW c Þ E 5  b La Lc qp b þ tanhðW a þ Þ tanhðW c Þ Ec  Eacorr   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficorr pffiffiffiffia pffiffiffiffic

q q pffiffiffiffiffi b qcp tanhðW c Þ qs þ tanhðWp a Þ þ tanhðWp c Þ

ð224Þ

ð225Þ

ð226Þ

In the case that the metals (a) and (b) have extremely large W, e.g. Wa ? 1 and Wc ? 1, the above potential and current equation can be further simplified:

 2L x wa  expðW a Þ cosh a ; 0 6 x 6 a bþL þL L  a ba La Ebc þ bE þ Lc Eba E  Ec þ corr a corrc ðx  aÞ; wb  a c bþL þL bþL þL a6x6aþb  2Lc ðEccorr  Eacorr Þ xabc a ; expðW Þ cosh wc  b þ La þ Lc Lc aþb6x6aþbþc  2La ðEa  Ec Þ x IaF   a corr a corr expðW a Þ cosh a ; 0 6 x 6 a c L qp ðb þ L þ L Þ a

ðEacorr  a

Eccorr Þ c

ð227Þ

ð228Þ

ð229Þ ð230Þ

H.5. A large W material between two small W materials This is an extreme case when Wa ? 0 and Wc ? 0. Utilizing the approximations in Appendix I, we can work out the galvanic potentials and current densities as well as the overall galvanic currents and maximum current densities:

Rap Eba Ea  Eacorr  qffiffiffiffiffiffiffiffiffiffiffi qs qbp þ Rap

ð242Þ

qffiffiffiffiffiffiffiffiffiffiffi  qs qbp Ebc xab exp Eb  Ebcorr  qffiffiffiffiffiffiffiffiffiffiffi Lb qs qbp þ Rcp qffiffiffiffiffiffiffiffiffiffiffi  qs qbp Eba xa  qffiffiffiffiffiffiffiffiffiffiffi exp  b ; a 6 x 6 a þ b L qs qbp þ Rap

ð243Þ

Rcp Ebc Ec  Eccorr  qffiffiffiffiffiffiffiffiffiffiffi qs qbp þ Rcp

ð244Þ

Eba  Iaf   qffiffiffiffiffiffiffiffiffiffiffi a qs qbp þ Rap

ð245Þ

IbF

2 qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi  b bc qs qbp Eba 1 6 qs qp E xab q ffiffiffiffiffiffiffiffiffiffi ffi þ ¼  b 4qffiffiffiffiffiffiffiffiffiffiffi exp qp Lb qs qbp þ Rcp qs qbp þ Rap 

xa exp  b ; a6x6aþb L

ð246Þ

Ebc  Icf   qffiffiffiffiffiffiffiffiffiffiffi c qs qbp þ Rcp

ð247Þ

x Eba qffiffiffiffiffiffiffiffiffiffiffi Iaf   a q qb þ Ra

ð248Þ

ba

IbF  

La Ebc þ bE þ Lc Eba Ea  Ec  b corr a corr c ðx  aÞ; a c b qp ðb þ L þ L Þ qp ðb þ L þ L Þ

a6x6aþb  2Lc ðEc  Ea Þ xabc c c ; Þ cosh expðW IF   c corr a corr qp ðb þ L þ Lc Þ Lc aþb6x6aþbþc  2ðEacorr  Eccorr Þ x a a IF   Þ sinh expðW ; a6x6aþb qs ðb þ La þ Lc Þ La

ð231Þ

ð232Þ ð233Þ

ba

IbF  

Eacorr  Eccorr La Ebc þ bE þ Lc Eba ðx  aÞ; a c  qs ðb þ L þ L Þ qbp ðb þ La þ Lc Þ

a6x6aþb  2ðEccorr  Eacorr Þ xabc c ; IcF   a c expðW Þ sinh c qs ðb þ L þ L Þ L aþb6x6aþbþc  Eacorr  Eccorr ab pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ig ¼ Iaf x¼a   bqs þ qs qap þ qs qcp  Eccorr  Eacorr bc pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ig ¼ Icf x¼aþb  bqs þ qs qap þ qs qcp  Ea  Eccorr pffiffiffiffiffiffiffiffiffiffiffi ffi IaFa ¼ IaF x¼a   pffiffiffiffiffiffiffiffiffiffiacorr b qs qp þ qap þ qap qcp p ffiffiffiffiffi ffi pffiffiffiffiffiffi ffiffiffiffiffi p  qap Ebc þ b qs Eba þ qcp Eba   pffiffiffiffi IbF0 ¼ IbF    pffiffiffiffi pffiffiffiffi  x¼a qbp b qs þ qap þ qcp pffiffiffiffiffiaffi bc pffiffiffiffiffiffi pffiffiffiffiffi  qp E þ b qs Ebc þ qcp Eba   pffiffiffiffi  IbFb ¼ IbF  pffiffiffiffi pffiffiffiffi  x¼aþb qbp b qs þ qap þ qcp IcF0

 ¼ Ic 

F x¼aþb



Ec  Eacorr pffiffiffiffiffiffiffiffiffiffiffi ffi  pffiffiffiffiffiffiffiffiffifficcorr c a c b s p þ p þ p p

qq

q

qq

ð234Þ

s

p

2 qffiffiffiffiffiffiffiffiffiffiffi  qs qbp Ebc 1 xab 6 b IF ¼  qffiffiffiffiffiffiffiffiffiffiffi 4qffiffiffiffiffiffiffiffiffiffiffi exp Lb qs qbp qs qbp þ Rcp 3 qffiffiffiffiffiffiffiffiffiffiffi  qs qbp Eba xa 7 exp  b 5; a 6 x 6 a þ b  qffiffiffiffiffiffiffiffiffiffiffi L qs qbp þ Rap

ð249Þ

 xabc Ebc qffiffiffiffiffiffiffiffiffiffiffi Icf   c q q b þ Rc

ð250Þ

  ab ig ¼ Ibf0 ¼ Ibf 

ð251Þ

s

ð235Þ ð236Þ

p

x¼a

p

p

Eb  Eacorr ffiffiffiffiffiffiffiffiffiffiffi  qcorr qs qbp þ Rap

ð237Þ ð238Þ ð239Þ

  bc ig ¼ Ibfb ¼ Ibf 

 Eb  Eacorr  ffiffiffiffiffiffiffiffiffiffiffi IaFa ¼ IaF x¼a   qcorr a qs qbp þ Rap

ð240Þ ð241Þ

x¼aþb

Eb  Eccorr ffiffiffiffiffiffiffiffiffiffiffi   qcorr qs qbp þ Rcp

IbF0

¼



 IbF 

x¼a

 pffiffiffiffiffi b qs Ecorr  Eacorr ¼ pffiffiffiffiffi qffiffiffiffiffiffi qbp qs þ qbp Rap

ð252Þ

ð253Þ

ð254Þ

480

IbFb

G.-L. Song / Corrosion Science 52 (2010) 455–480

  ¼ IbF 

x¼aþb

 pffiffiffiffiffi b qs Ecorr  Eccorr ¼ pffiffiffiffiffi qffiffiffiffiffiffi qbp qs þ qbp Rcp

 Eb  Eccorr  ffiffiffiffiffiffiffiffiffiffiffi IcF0 ¼ IcF x¼0   qcorr c qs qbp þ Rcp

ð255Þ

ð256Þ

Appendix I When W ? 0, there are:

m x !0 !0 L   L x m m x exp !1þ !1þ exp L L L  m  x L x m !1 !1 exp  exp  L L L mL x m x ! ! sinh sinh L  L  L m L m x x sinh  ! sinh  ! L L L L m x cosh !1 cosh !1 L   L m x !1 cosh  !1 cosh  L m x L x m tanh ! ! tanh L  L  L m L m x x ! ! tanh  tanh  L L L L

  c 1 c exp ! 2 Lc Lc     a 1 a b 1 b cosh  a ! exp a cosh  b ! exp b 2 2 L L L L   c 1 c cosh  c ! exp c 2 L L    a b c tanh b ! 1 tanh c ! 1 tanh a ! 1 L L L    a b c tanh  b ! 1 tanh  c ! 1: tanh  a ! 1 L L L

cosh

References

ð257Þ

where m can be a, b, or c, the length of system (a), (b) or (c). L can be La, Lb, and Lc. W can be Wa, Wb or Wc. Appendix J When W ? 1, there are:

a b c !1 !1 !1 b La Lc L    a b c exp a ! 1 exp b ! 1 exp c ! 1 L L L    a b c exp  b ! 0 exp  c ! 0 exp  a ! 0 L L L     a 1 a b 1 b sinh a ! exp a sinh b ! exp b 2 2 L L L L   c 1 c sinh c ! exp c 2 L L     a 1 a b 1 b sinh  a !  exp a sinh  b !  exp b 2 2 L L L L   c 1 c sinh  c !  exp c 2 L L     a 1 a b 1 b cosh b ! exp b cosh a ! exp a 2 2 L L L L

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