The Cottrell Equation: Diffusion-Limited Current Decay at a Planar Electrode

What the Cottrell Equation Describes and Where It Applies

The Cottrell equation governs the current transient that follows a sudden potential step at a stationary planar electrode. Before the step, the potential sits where no faradaic reaction occurs; at t = 0 it jumps to a value where the electroactive species (say Fe(CN)₆³⁻ or Ru(NH₃)₆³⁺) is reduced or oxidized as fast as it reaches the surface. The electron-transfer kinetics become irrelevant — the surface concentration is clamped at zero — and the current is mass-transport limited.

• Regime: semi-infinite linear diffusion to a flat, unstirred electrode.
• Requirement: supporting electrolyte at ~100× excess so migration is suppressed and diffusion dominates.
• Domain: the technique is called chronoamperometry (current-vs-time at fixed potential).

It applies wherever a well-defined planar electrode meets a quiet solution: measuring diffusion coefficients, counting electrons in a reaction, sizing electroactive concentrations, and as the theoretical baseline against which spherical, cylindrical, and microelectrode behavior is compared.

Deriving It Step by Step from Fick's Laws

The derivation is a textbook solution of a partial differential equation. Diffusion in one dimension obeys Fick's second law: ∂c/∂t = D(∂²c/∂x²), where c is concentration, x is distance from the electrode, and D is the diffusion coefficient.

• Initial condition: c(x, 0) = c* everywhere (uniform bulk concentration).
• Boundary 1: c(0, t) = 0 for t > 0 — the potential step pins surface concentration to zero.
• Boundary 2: c(∞, t) = c* — the bulk stays undisturbed (semi-infinite).

Solving by Laplace transform gives the concentration profile c(x,t) = c*·erf[x/(2√(Dt))]. The current is proportional to the flux at the surface: i = nFAD(∂c/∂x)|_x=0. Differentiating the error-function profile and evaluating at x = 0 yields (∂c/∂x)|₀ = c*/√(πDt). Substituting delivers the Cottrell equation:

i(t) = nFA√D · c* / (√π · √t). The 1/√t falls straight out of the √(πDt) in the flux term.

Symbols, Constants, and a Worked Example

Every symbol carries units, and getting them consistent (cm-based) is where students stumble:

• n = electrons transferred per molecule (dimensionless)
• F = Faraday constant = 96485 C/mol
• A = electrode area (cm²)
• D = diffusion coefficient (cm²/s), typically 5×10⁻⁶ to 2×10⁻⁵ for small ions in water
• c* = bulk concentration (mol/cm³; note 1 mM = 1×10⁻⁶ mol/cm³)
• t = time after the step (s)

Worked example. For n = 1, A = 0.020 cm², D = 1×10⁻⁵ cm²/s, c* = 1×10⁻⁶ mol/cm³, at t = 1 s: numerator = 96485 × 0.020 × √(1×10⁻⁵) × 1×10⁻⁶ = 96485 × 0.020 × 3.16×10⁻³ × 1×10⁻⁶ ≈ 6.10×10⁻⁶. Denominator = √π × √1 = 1.772. So i(1 s) ≈ 3.44×10⁻⁶ A = 3.44 µA. At t = 4 s it halves to 1.72 µA — exactly the 1/√t scaling.

How It's Measured and Used in the Lab

In practice you apply the step with a potentiostat and record i vs t. The diagnostic move is to plot i against t^-1/2: a valid Cottrell process gives a straight line through the origin, and its slope equals nFA√D·c*/√π. Everyday uses:

• Diffusion coefficients: with n, A, and c* known, D = (slope · √π / nFAc*)². This is a standard way to obtain D for a redox couple.
• Electrode area: run a species of known D (e.g. ferrocyanide, D ≈ 6.5×10⁻⁶ cm²/s) to calibrate the true active area A.
• Electron count n: compare slope against a one-electron reference under identical conditions.

Deviations are informative: at very short times (<~1 ms) double-layer charging current dominates and the plot curves; at long times (seconds to minutes) natural convection and edge diffusion make the current exceed the Cottrell prediction, so the line bends upward. The clean linear window is typically ~10 ms to a few seconds.

How It Differs from Its Close Cousins

The Cottrell equation is one of a family of mass-transport expressions, each tied to a geometry or hydrodynamic condition:

• vs. Randles–Sevcik: Randles–Sevcik gives the peak current in cyclic voltammetry (i_p = 0.4463 nFAc*√(nFvD/RT), scaling with √(scan rate v)), whereas Cottrell is a fixed-potential transient scaling with 1/√t.
• vs. Levich equation: Levich describes the steady-state current at a rotating-disk electrode (i_L = 0.62nFAD^2/3ω^1/2ν^−1/6c*); forced convection replaces transient diffusion, so the current is constant, not decaying.
• vs. microelectrode steady state: at an ultramicroelectrode, radial (hemispherical) diffusion reaches a time-independent limit i_ss = 4nFDc*r, so the Cottrell term dies away and a plateau remains.

The unifying idea: Cottrell is the pure planar, unstirred, transient limit — the reference case from which spherical corrections and convective steady states are built.

Limits, Exceptions, and Why It Endures

The Cottrell equation assumes an ideal world, and its breakdowns are as instructive as its successes:

• Edge effects: real discs are finite, so diffusion at the rim is enhanced; the current runs above Cottrell at long times.
• Convection: density gradients and vibration stir the depletion layer after seconds, flattening the expected decay.
• Charging current: the capacitive spike (i_c = (ΔE/R)e^−t/RC) swamps the faradaic signal in the first sub-millisecond and must be subtracted or outrun.
• Sphericity: for a sphere, add a steady term — i = nFADc*[1/√(πDt) + 1/r].

Despite these caveats, the equation has anchored electroanalytical chemistry for over a century. It underlies the chronoamperometric and chronocoulometric methods (integrating i(t) gives the Anson equation, Q = 2nFAc*√(Dt/π)), calibrates commercial glucose sensors and clinical amperometric assays, and remains the first sanity check any electrochemist runs on a new electrode. Cottrell, who also invented the electrostatic precipitator and founded Research Corporation, left a remarkably durable equation.

Cottrell current for a 1 mM species (n = 1, A = 0.020 cm², D = 1×10⁻⁵ cm²/s) at increasing times

Time t (s)	1/√t (s^-1/2)	Cottrell current i (µA)	Diffusion layer √(πDt) (µm)
0.01	10.0	34.4	5.6
0.1	3.16	10.9	17.7
1	1.00	3.44	56.0
4	0.50	1.72	112
25	0.20	0.689	280
100	0.10	0.344	560