Ag | AgNO₃ | Ag: impedance, and what the cell does inside

The impedance spectrum of 10 mM silver nitrate between two silver electrodes 20 µm apart, against theory, and the small-signal response inside the cell at the frequency you choose: each ion's voltage and the salt concentration oscillating across the cell, slowed down to watch. Slide the frequency from salt diffusion's regime to the ohmic one. (More about the model below.)

About this simulation

The cell is the one in the saturation demo, at rest: a strictly neutral electrolyte, Ag⁺ plated and stripped at the electrodes, NO₃⁻ blocked at both. Impedance asks how it answers a small sinusoidal voltage, δV cos ωt. The answer is linear, so driftlet solves for it directly at each frequency, as complex amplitudes over the whole cell: δX(x) for every quantity X, with the oscillation δX(x, t) = Re[δX(x) e^iωt]. That's what the diagrams draw, per 10 mV of excitation (it scales with the excitation), with the phase ωt advancing slowly whatever the frequency. The dotted lines are each quantity's envelope, ±|δX(x)|, which the oscillation sweeps between.

The level diagram uses the species voltage V_i = μ̄_i/(z_iF): an ion's electrochemical potential per unit charge, in volts. Each ion moves down its own μ̄_i, so its current flows down slopes in its own V_i (Ag⁺ moves downhill, NO₃⁻ uphill). Each electrode holds Ag⁺ level with its own silver, so δV_Ag⁺ is 0 at the left and the excitation at the right. At high frequency the salt has no time to move: both ions' voltages slope straight across the cell like the potential in a resistor, carrying current in proportion to their mobilities, and the impedance is just the bulk resistance R_b. The salt then piles up and drains only within a diffusion length √(D/ω) of each electrode, where nitrate is stopped. At low frequency that length outgrows the cell: the salt gradient spans it, nitrate's own voltage flattens (it carries no current at the electrodes, so in the end none anywhere), and Ag⁺ carries all the current, by drift and diffusion together. Its added resistance is the diffusion arc's width, R_d.

The Nyquist plot (each frequency a point, Z′ against −Z″) shows the crossover as a finite-length Warburg arc: R_b + R_d tanh√(iωτ)/√(iωτ), with τ = (L/2)²/D the salt's diffusion time, dashed. There's no double-layer arc at the top of the range, because this electrolyte is strictly neutral: it has no diffuse layers to charge. The Lissajous figure is the same impedance in time: the current against the voltage over a cycle, a line with slope 1/|Z| when they're in phase and an ellipse when the current leads.

Reading small-signal transport rail by rail is Jamnik and Maier's picture (J. Jamnik and J. Maier, Phys. Chem. Chem. Phys. 3, 1668 (2001)): an exact equivalent circuit with a rail for each species' μ̄_i/(z_iF), resistors along it for transport, chemical capacitances from it to a φ rail for storage, and dielectric capacitances along φ. The diagrams here are the voltages on those rails. In this strictly neutral electrolyte the dielectric capacitances are zero (no net charge can be stored), and what remains is the transmission line of salt diffusion behind the Warburg arc.