Potential and current steps: Cottrell and Sand

A platinum electrode in still 1 mM Fe³⁺ and Fe²⁺ (in 1 M KCl with 0.1 M HCl), stepped from rest: to a potential, where the current falls as 1/√t as the solution near the electrode runs out of Fe³⁺ (chronoamperometry, against Cottrell), or to a current, which the surface's Fe³⁺ can carry only until Sand's transition time (chronopotentiometry, against Sand). Choose the step and its size, and whether the electrode has a double layer to charge. (More about the model below.)

step η i with a double layer (20 µF/cm²) playback

About this simulation

At rest the solution has as much Fe²⁺ as Fe³⁺, so the electrode sits at the couple's standard potential E° and a step's η is its overpotential. A potential step holds the surface's Fe³⁺/Fe²⁺ ratio near θ = e^Fη/RT, and Fe³⁺ has to diffuse in from farther and farther away: the current falls as 1/√t, the straight line of slope −½ on these axes, and i√t stays constant (which is how D is measured). With Fe²⁺ present the line is Cottrell's times (1 − θ)/(1 + ξθ), ξ = √(D_Fe³⁺/D_Fe²⁺): a small step converts only part of the Fe³⁺, and past about −0.2 V it takes it all. (The textbook step starts where nothing reacts, with no Fe²⁺; this is the same law, generalised.) At first the electrode can't keep up: at t → 0 the current is F(k_f − k_b)c, the rate constants k_f = k⁰e^−αFη/RT and k_b = k⁰e^{(1−α)Fη/RT}, and it joins Cottrell's line once diffusion has slowed below that, on a time scale 1/H² with H = k_f/√D_Fe³⁺ + k_b/√D_Fe²⁺ (2 µs at −0.3 V, 5 ms at −0.1 V); it's within 5% of the line only after about ten times that, so fit Cottrell's line later than that. Its slope gives D back: here D_Fe³⁺ = 6.04 × 10⁻⁶ cm²/s, the value the model uses (D_Fe²⁺ = 7.19 × 10⁻⁶). The full line is that whole theory, Butler–Volmer kinetics and semi-infinite diffusion in closed form; the dashed one is Cottrell's line alone.

A current step draws Fe³⁺ out of the surface at a fixed rate, and its concentration there falls as √t (Sand's solution), reaching zero at the transition time τ = πD(Fc)²/(4i²), so iτ^½/c is the same at every current, whatever the kinetics. Until then the potential drifts as the surface's Fe³⁺/Fe²⁺ ratio and Butler–Volmer need; the line is Sand's surface concentrations put into Butler–Volmer. At τ nothing is left in the model to carry the current: with no double layer the potential falls away at once, toward any value at all, and the run stops there. In this cell the next thing to reduce is H⁺ (from the 0.1 M HCl), near 0 V vs SHE, which would hold the potential there; the model leaves it out.

A real electrode has a double layer, and it changes both experiments. After a potential step it takes a spike of charge first (here only microseconds, through 100 µm of solution; on a millimetre disk, in a real cell, ten microseconds or more, which would hide the first decade of this plot). At small steps it does more: its ψ speeds the reaction (the Frumkin effect, as below), so for milliseconds the current runs above the theory drawn (1.8 times at 0.1 ms and 1.1 at 10 ms, at η = −0.1 V), and a k⁰ read from the early current would come out too high by about that. After a current step it matters more still: the potential can move no faster than i/C_dl, and the charge that moving it takes, C_dlΔE, comes out of the current, so the transition is smeared out and read late: crossing E° − 0.25 V at 1.10τ at 0.2 mA/cm², 1.19τ at 0.5 mA/cm², and not by 1.3τ at 2 mA/cm². That is chronopotentiometry's old weakness. (Before τ, about half of the potential's lift above Sand's line isn't charging but the diffuse layer's own effect on the kinetics: its ψ draws both iron cations toward the plane and speeds the reaction, the Frumkin effect. The late transition is the charging.) Tick the box to see it (the double layer is driftlet's diffuse layer behind a Stern layer, uncharged at rest).

The electrode faces 100 µm of still solution, then a stirred bulk: long enough that over these times the solution is as good as infinite (natural convection would end a real Cottrell experiment after 10 or 20 s anyway), and thin enough that the current through it shifts the electrode's potential little. That shift, about 2 mV at 10 µs, puts the current up to 1.2% off the theory there (most for middling steps, where the kinetics still count); from 1 ms on they agree to 0.2%. In a real cell the uncompensated resistance is set by the cell's geometry, and is often ten times more. The solution is acidified because Fe³⁺ must be (above about pH 2 it hydrolyses and precipitates), and would be purged of oxygen, which Pt reduces in this range. The potentials are on an ideal-solution scale, from table μ°: E° = 0.77 V, where the formal potential in real 1 M HCl is nearer 0.70 V. k⁰ = 5 × 10⁻³ cm/s is illustrative, within the range reported for this couple at platinum (driftlet's k0 is that times its 1 M reference concentration, 0.05 mol/(m²·s)). Currents are cathodic positive.

The profiles below are the concentrations across the solution, simulated (thick) against the same theory (thin). The level diagram has the platinum's Fermi level and the couple's redox level, drawn in volts, so electron energy is down the page (reducing Fe³⁺, electrons go from the Fermi level up to the redox level, toward higher voltage). The redox level tilts across the depleted layer: its gap to the Fermi level at the surface is the overpotential the kinetics see, and the tilt is the concentration polarisation. φ is the SHE's level here.