One layer, two layers, etc. An introduction to the EIS study of multilayer electrodes. Part 1: Theory

Low-frequency impedance of an ion-exchange membrane system

“…Note that the hyperbolic tangent element, tanh˛/˛, acts an important if not the key role (see also Section 5.3) in impedance behavior of IEM. This element is well known in electrode impedance works; it is discussed in the papers by Ho et al [51], Franceschetti and Macdonald [52], and in more recent publications by Jacobsen and West [49], Freger [50], Diard et al [53]. It determines the behavior of a "finite thickness porous Warburg" or "O element" [50,54,55] in an electrochemical system, which is a finite layer open for diffusion; the concentration at the outer boundary is by definition unperturbed [50,54,55].…”

“…We have mentioned above the role that this function acts in electrode impedance spectroscopy. In a multilayer system, the contribution of each layer, more precisely, of each O-element is related to this function, the argument of which,˛, is a function of the thickness of this layer (ı) and of the current transfer species diffusion coefficient (D) there [48][49][50][51][52][53][54][55], see Eq. (34).…”

Mathematical solutions of comprehensive variations of a transmission-line model of the theoretical impedance of porous electrodes

“…Nevertheless, when z C elements in all layers are zero, the impedance of a Z-type connection can be exceptionally calculated: the rightmost layer can be calculated using the equation for "openopen" condition of Z-type, then the second layer can be calculated using the equation for "open-Q" condition of Z-type by substituting the previous result for the Q value, and then the same is done for the following all layers. Although it is for the diffusion impedance (Warburg impedance), details are described in literature [21,22]. However, for a T-type connection, or even for a Z-type connection when z C element is not zero, this technique can not be used.…”

Re-examination of the potential-step chronoamperometry method through numerical inversion of Laplace transforms. I. General formulation and numerical solution

“…We extended the mass-transfer-function formalism to bi-and multi-layer systems [28,29]. Electrochemical reactions at polymerfilmed metal electrodes [30][31][32] as well as insertion reactions in bi-layer films of host materials [28] belong to the class of multilayer electrochemical systems with intricate formulation of the relevant mass-transfer function. Whatever the electrode geometry, the boundary condition(s) and the possible presence of coupled homogeneous reaction(s), the main problem for computing chronoamperograms and sampled-current voltammograms by the Laplace transform method is to invert the relevant LT in Eq.…”

“…Some inverse transforms pertaining to finite-space (restricted) diffusion conditions can be derived from similar problems of diffusion with surface evaporation effects dealt with in the textbook by Crank [33]. Other inverse transforms can be obtained from our previous works on electrochemical insertion reactions [34,35] and polymer-filmed metal electrodes [30][31][32].…”