In Chapter 2 (in the section “Interface Potential Difference and Half-Cell Potential”), the equilibrium half-cell potential for the metal reaction, E′M, was defined relative to potentialsφ as follows:
E M
Mo M H H
′ = φ′( − φ′ +)− φ′( − φ′ +)
2 (Eq 4.1)
E′ = φ′M ( Mo − φ′M+)− φ∆ SHE (Eq 4.2) where primes indicate values at equilibrium,φ′Mo is the potential in the metal,φ′M+ is the potential in the solution near the metal surface, and φ′H2 andφ′H+ have similar meanings relative to the standard hydrogen electrode (SHE). In Chapter 3 (in the section “Charge-Transfer Polar-ization”), the definition was written in general terms to encompass nonequilibrium conditions:
EM
Mo Mm SHE
= φ( − φ +)− φ∆ (Eq 4.3)
In these prior discussions, only the metal reaction was under consider-ation. The equivalent general definition for species in solution (XX+and X) capable of undergoing reduction or oxidation at the metal surface is:
EX
Xo Xx SHE
= φ( − φ +)− φ∆ (Eq 4.4)
Thus, the E values (relative interface potential differences, or interface potentials) represent differences in potentials across the metal/solution interface minus the potential difference across the standard hydrogen reference electrode interface. The E values are physically measured by attaching one lead of an electrometer to the metal, the other lead to a ref-erence electrode in the solution and very close to the metal surface (a point discussed further in Chapter 6). If the positive electrometer lead is
connected to the metal, the sign of the electrometer read-out will pro-vide the correct sign for E. In practical measurements, the SHE is gener-ally not employed. Rather, for convenience, another reference electrode such as the saturated calomel electrode (SCE) or the saturated Ag/AgCl electrode might be employed. When this is done, the measured potential relative to a given reference electrode is Emeas,ref, which is related to E by the expression (see the section “Interpretation of Charge-Transfer Polarization from Experiment” in Chapter 3):
E = Emeas,ref+ Eref (Eq 4.5)
where Erefis the potential of the reference electrode relative to the SHE (Table 2.2 in Chapter 2 provides selected Erefvalues).
Under conditions of steady-state corrosion, during which net oxida-tion is occurring at a given anodic site (M→ Mm++ me) and net reduc-tion at a given cathodic site (XX++ xe→ X), the potentials at the an-odic site and cathan-odic site, respectively, are given by:
′′ = ′′φ − ′′φ − φ
where the double primes indicate the steady-state corrosion condition,
′′
φMo and φX′′o represent the potentials of the metal at the anodic and cathodic sites, respectively, andφM′′m + and φX′′x + represent the poten-tials in the solution at the anodic and cathodic sites, respectively. In order to more clearly associate the potentialsφ in Eq 4.6 and 4.7 with either the metal or solution, and either the anodic or cathodic sites, the following changes in designations will be introduced: φM,a= φM′′o, φS,a= φM′′m +,φM,c= φX′′o, andφS,c= φX′′x +, where the subscripts M and S refer to the metal and solution, and the subscripts a and c refer to the anodic and cathodic sites. With these designations, Eq 4.6 and 4.7 be-come:
E″M= (φM,a–φS,a) –∆φSHE (Eq 4.8)
and
E″X= (φM,c–φS,c) –∆φSHE (Eq 4.9)
With reference to Fig. 4.1, since the corrosion process is taking place, E″Xat the cathodic site has to be greater than E″Mat the anodic site such that conventional current (Icorr) flows in the metal from the higher potential site (cathode) to the lower potential site (anode); electrons
flow in the opposite direction. The driving potential difference respon-sible for the corrosion process is (E″X– E″M), a positive quantity. From Eq 4.8 and 4.9:
(E″X– E″M) = (φS,a–φS,c) – (φM,a–φM,c) (Eq 4.10) However, since the metal is an excellent electrical conductor, differ-ences in potential within the metal are generally negligible (i.e., (φM,a–φM,c)≈ 0). Therefore:
(E″X– E″M) = (φS,a–φS,c) (Eq 4.11) In Eq 4.11, (E″X– E″M) is positive since E″X at the cathodic site is greater than E″Mat the anodic site. Thus, the potential in the solution at the anodic site,φS,a, is greater than the potential in the solution at the cathodic site,φS,c, which is consistent with the overall electrochemical corrosion circuit. It follows that the driving potential difference for con-ventional current flow (Icorr) in the solution is:
∆φS= (φS,a–φS,c) (Eq 4.12)
with the current flowing from the higher potential site (anode) to the lower potential site (cathode). Within the solution, the potential will
de-Fig. 4.1 Schematic representation of measurements of potentials along a path from anode to cathode area on a corroding surface
crease continuously fromφS,aat the anodic site to φS,cat the cathodic site.
It is only possible to physically measure the quantities E″M, E″X, and
∆φS, where the∆φSmeasurement is between any two points in the solu-tion. With reference to Eq 4.8 and 4.9 for E″Mand E″X, the quantities φM,a≈ φM,cand∆φSHEare constants, but unknown constants. In Eq 4.8 and 4.9, let the constant quantities (φM,a–∆φSHE) and (φM,c–∆φSHE) be represented by k, where k is another unknown constant:
(φM,a–∆φSHE) = (φM,c–∆φSHE) = k (Eq 4.13) Then, upon rearrangement:
φS,a= (k – E″M) (Eq 4.14)
φS,c= (k – E″X) (Eq 4.15)
and the potential difference in the solution (Eq 4.12) becomes:
∆φS= (φS,a–φS,c) = (k – E″M) – (k – E″X) (Eq 4.16) Since it is apparent from Eq 4.16 that the unknown constant k, regard-less of its value, will always cancel, it is convenient to define k as zero.
Then, from Eq 4.14 and 4.15:
φS,a= –E″M (anode) (Eq 4.17)
φS,c= –E″X (cathode) (Eq 4.18)
or, in general:
φS= –E″ (Eq 4.19)
In order to illustrate the above principles, with reference to Fig. 4.1, assume that E″M(anode) = –350 mV(SHE) and E″X(cathode) = –250 mV (SHE). Since (E″X– E″M) is a positive quantity (+100 mV), corro-sion will occur. Furthermore,φS,a(anode) = +350 mV, and φS,c (cath-ode) = +250 mV. Under these conditions, with the use of a SHE refer-ence electrode and assuming a semicircular current path in the solution, experimental measurements with an electrometer—with the positive (high, red) and negative (low, black, common) leads connected as shown—will indicate the potential values shown in Fig. 4.1. In the solu-tion, the potential will vary from +350 mV at the anode to +250 mV at the cathode. In Fig. 4.1, cross sections of constant-potential (iso-potential) surfaces are schematically represented as dotted lines at 20 mV increments.