Linear diffusion impedance. General expression and applications

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Abstract

Linear diffusion in thin film electrodes can be interpreted in terms of a matrix relationship. A general expression for the mass-transfer function and the diffusion impedance can be written, irrespective of the boundary conditions at the film surfaces. This expression applies, for example, to bounded or restricted diffusion in single-layer or multilayer electrodes and also to boundary conditions leading to resistive, capacitive or CPE interfacial behaviour.

Introduction

The EIS method is used widely to investigate electrochemical reactions with species diffusing in thin film materials or filmed-metal electrodes, such as hydrogen absorption into metals and alloys, proton or lithium ion insertion in oxide host materials and electron transfer (hopping) in polymer coated electrodes. A general expression for the linear diffusion impedance can be derived, irrespective of the diffusion boundary conditions at the film surfaces. This expression covers many practical cases, as pointed out in this paper.

Section snippets

General expression for the linear diffusion impedance

We consider linear diffusion of a species in a plane sheet of thickness L, according to the nomenclature of Crank [1]. This species is produced at the surface of abscissa x=0 (input side) and then diffuses in the sheet (0≤xL). Different boundary conditions can be envisaged at the surface of abscissa L (output side), depending on whether this surface is permeable to the diffusing species or not.

c(x, t) and J(x, t) are the concentration and diffusion flux of the species at a distance x from the

Application to practical cases

The general expression in Eq. (7) covers many practical cases. First, bounded diffusion conditions are given by M(L, u)=0 and therefore:Z*(u)=tanh (iu)1/2/(iu)1/2which corresponds in Fig. 1 to short-circuited output terminals (ZL=0). The well known impedance diagram is plotted in Fig. 2A, using the complex plane representation, i.e. −X″(u) versus X′(u) where X′(u) and X″(u) are respectively the real and imaginary parts of the complex number X(u) which denotes either Z*(u) or M(0, u).

Eq. (8) and

Conclusions

Once written in dimensionless form, the linear diffusion impedance for a thin film electrode, Z*(u)=Zmt(u)/Rmt, is equal to the mass-transfer function, M(0, u), defined on the input side of the electrode. M(0, u) and therefore Z*(u) are related in a simple way to the mass-transfer function expression on the output side, M(L, u), irrespective of the boundary conditions at the two surfaces. Setting u=ωL2/D, we have derived in this paper the general dimensionless form of the diffusion impedance as:Z*u

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