Corrosion: a mixed potential and the Evans diagram

Iron in aerated 0.5 M NaCl, behind a layer of still water (Nernst's picture of the diffusion layer that stirring, or natural convection, leaves next to the metal: thicker for stiller water): left to itself it settles at the corrosion potential, where its own dissolution (Fe → Fe²⁺ + 2e⁻) and the reduction of dissolved oxygen (O₂ + 2H₂O + 4e⁻ → 4OH⁻) pass equal currents, and it corrodes at that current. The Evans diagram, each reaction's current against potential, shows where; driving the iron away from it gives the polarization curve you'd measure. Vary how still the water is, and the potential. (More about the model below.)

About this simulation

Two reactions share one piece of metal, so they share its Fermi level, and with no current leaving the metal, the electrons one makes the other must take. The iron settles where they balance, the corrosion (or mixed) potential E_corr (Wagner and Traud, 1938), between the iron's own redox level and oxygen's, in equilibrium with neither: iron goes on dissolving at the corrosion current i_corr, and oxygen goes on being reduced at the same rate. On the Evans diagram (E against log|i|, in the form Wagner and Traud's theory gives it) that's where the iron's anodic line crosses oxygen's cathodic curve. Oxygen's curve comes down its Tafel line and bends over to a plateau, its limiting current 4FDc/L, set by how fast it diffuses in through the still layer. Where the crossing falls on that plateau, as it does for a layer of half a millimetre, the corrosion rate is oxygen's supply, and stirring the water (a thinner layer) speeds corrosion as 1/L; for the thinnest layers here the crossing moves up the bend, and the kinetics share the control.

Driven below E_corr, the iron's dissolution dies away while oxygen's plateau carries on: that is cathodic protection. 0.1 V below E_corr the iron dissolves at about 1% of its free rate (the readout's second line gives it for the potential you choose), for a protection current of about oxygen's limit; about 0.15 V below, the Fermi level passes the iron's own redox level and iron would plate rather than dissolve (Pourbaix's immunity). Driven above, the iron's current grows tenfold every 60 mV (its Tafel slope, 2.3RT/(2(1 − α)F) with α = ½). That is also how i_corr is measured, with care. The iron's own line, extrapolated back to E_corr, gives it within a few percent, but no experiment sees that line: the measured curve is the net current, and a straight line fitted to it gives anywhere from half to 1.6 times i_corr depending on where you fit (near E_corr oxygen's plateau current takes away from it; higher up the solution's potential at the face bends it). Add the plateau back and fit close in, 25 to 75 mV above E_corr, and it comes within 5%. The polarization resistance does better: R_p = dE/di at E_corr, through Stern and Geary's i_corr = B/R_p (1957), with B the anodic Tafel slope over 2.3 alone (the cathode is diffusion-limited), gives it within 3%. The textbook construction starts each line from its couple's equilibrium potential: the iron's is −0.59 V here (with the bulk's 1 µM Fe²⁺, Pourbaix's 10⁻⁶ convention), off this diagram's left edge, and oxygen's +0.80 V, far above its top.

The lines are closed forms, worked out apart from the solver: Butler–Volmer for each reaction at the iron's face, with the Fe²⁺ the iron makes held back by diffusion through the layer (its back reaction sees the surface's Fe²⁺, not the bulk's), and oxygen's supply through the layer in series with its kinetics. The points are the simulation's net current, and the readout gives both corrosion potentials and currents: they agree to 0.1 mV and five digits, and the cathodic side agrees everywhere. Well above E_corr the simulation runs below the closed form (9% by 0.1 V up), all of it the solution's potential at the face, 2.4 mV there: about half ohmic drop through the layer, and half a diffusion potential from the FeCl₂ built up at the iron diffusing out (Fe²⁺ is slower than Cl⁻). With that taken out the two agree to 10⁻⁴. (By 0.15 V up it's 32% below.) The simulation is the more realistic curve here: it has what a reference electrode in the bulk would see.

What the model leaves out matters. The profiles below are its, and in a real cell they couldn't last: with Fe²⁺ and OH⁻ together at the face, ferrous hydroxide would be 10⁵ to 10⁶ times past its solubility, and dissolved oxygen oxidises Fe²⁺ in seconds at the pH there. So in reality rust forms within the layer, and the oxygen it takes there never reaches the metal; at most that brings the iron's rate down to two thirds of oxygen's supply. Below about −0.7 V vs SHE, on real steel, water's own reduction to hydrogen joins oxygen's, which is where the window stops. Also left out: films that slow the iron (passivity), chloride's pitting, and an iron surface that isn't one uniform electrode. Its rate constants are illustrative, chosen to put E_corr in the range measured for steel in sea water (−0.35 to −0.45 V vs SHE), here at its negative end. Oxygen's reduction is written as one four-electron step with α = 0.125, which gives the 118 mV per decade of a rate-determining first electron. The dissolved oxygen is 0.21 mM, salted out of air-saturated 0.5 M NaCl. The potentials are on an ideal-solution scale, from table μ°: E°(Fe²⁺/Fe) is −0.41 V here, where tables give −0.44 V; and the solution's conductivity is the ideal one, a third above real 0.5 M NaCl's.

The level diagram is the mixed potential itself: the iron's Fermi level, and the two couples' redox levels across the still layer, drawn in volts, so electron energy is down the page (an electron's voltage, μ̄/(−F), is the electrochemist's electrode potential; φ, the dotted line, is the SHE's level here). Electrons leave the iron's redox level for the Fermi level (dissolution) and go on up the page, down in energy, to oxygen's (reduction); the Fermi level sits between the two, where the flows match. Below it, the concentrations across the layer: oxygen drawn down toward the iron, Fe²⁺ and OH⁻ building up at it.