Diffusion-controlled current with memory

The diffusion equation complicated by a delay of a concentration flux, J, from the formation of a concentration gradient, ∂c/∂x, was formulated in the context of electrochemical measurements. In contrast with the Fick’s first law, J = −D∂c/∂x, the flux at a short time is known to be delayed owing to a finite propagation speed of the gradient, called the memory effect or the second sound for thermal diffusivity. The modified Fick’s law contained the second time-derivative of the concentration multiplied by the relaxation time, τ, additive to the conventional diffusion equation. It was applied to chronoamperometry. The current–time curve was smoother than that for the Cottrell equation. The current at a short time was almost constant owing to the rate-determining step of the propagation velocity, (D/τ)1/2, and then decays obeying the Cottrell equation. This variation was similar to the curve mixed with the Butler–Volmer kinetics. The relaxation time was estimated from the period during which a diffusing particle can recognize the concentration gradient by collision with the nearest diffusing particle. The propagation velocity was of the order of some cm s−1, which is similar to the maximum values of the measurable charge transfer rate constant.