Diffusion impedance Concentration gradient

Conductivity is of course closely related to diffusion in a concentration gradient, and impedance spectroscopy has been used to determine diffusion coefficients in a variety of electrochemical systems, including membranes, thin oxide films, and alloys. In materials exhibiting a degree of disorder, perhaps in the hopping distance or in the depths of the potential wells, simple random walk treatments of the statistics are no longer adequate some modem approaches to such problems are introduced in Section 2.1.2.7. 

P(02) ) /C2, respectively. The limitation on the current is due to limitation on the driving force, i.e., the concentration gradient which reaches its maximum when the low concentration can be neglected while the high concentration maintains a fixed value. Normally, the limiting current in SOFC cathodes exhibits a linear dependence on (P(02) ) However, in the past it has been reported that the diffusion limited cmrent was linear in P(02) rather than in (P(02) ) This can be ascribed to excessive diffusion impedance to the gas in the sputtered and probably quite dense Pt electrode used. ... 

Then, the two models give equivalent results. This calculation was also given by Buck without the electroneutrality hypothesis (i.e. Cp 0). The transmission line approach is often called the porous model of a conducting polymer as electrons are supposed to cross the polymer (phase 1) and ions are supposed to move into pores, filled by electrolyte, represented by the second branch of the transmission line. It is noticeable that the transmission line approach allows more complicated kinetics to be tested for a two-species problem, e.g. charge transfer in parallel to the capacity Ce(x) and C,- (x), or diffusion of the ion in the ionic pores , i.e. to introduce complex impedances instead of the real resistance p and/or p2, of the pure capacitances Ci and/or C2. It also allows position-dependent parameters to be introduced to mimic concentration gradients in the polymer [Cj(x) constant]. 

In the EHD impedance method, modulation of the flow velocity causes a modulation of the velocity gradient at the interface which, in turn, causes a modulation in the concentration boundary layer thickness. As demonstrated previously in Section 10.3.3 and Fig. 10.3 the experiment shows a relaxation time determined solely by the time for diffusion across the concentration boundary layer. Although there is a characteristic penetration depth, 8hm, of the velocity oscillation above the surface, and at sufficiently high modulation frequencies this is smaller than the concentration boundary layer thickness, any information associated with the variation of hm with w is generally lost, unless the solution is very viscous. The reason is simply that, at sufficiently high modulation frequencies, the amplitude of the transfer function between flow modulation and current density is small. So, in contrast to the AC impedance experiment, the depth into the solution probed by the EHD experiment is not a function... 

Eq. (14) allows to translate concentration into impedance profiles. Figs. 6 to 8 display some of the computed impedance maps. One wnll notice that the presence of diffusivity gradients at the surface completely modiHes the distributions. In spite of the slight differences observed in the corresponding... 

Let us consider now diffusion inside a sphere neglecting the diffusion gradient outside the sphere. Such a case might be observed for hydrogen absorption or Li intercalation into spherical particles. Diffusion inside the sphere can go only to the sphere center and is called finite-length internal spherical diffusion. In the steady state in which the impedance measurements are carried out, dc concentration inside the sphere is uniform, and no dc current is flowing. Ac perturbation causes oscillations of concentration at the sphere surface, which diffuse inside the sphere. In such a case, two boundary conditions in Eq. (4.94) are changed ... 

Diffusion resistance Zp,yy to current flow carried by electroactive species can create impedance, frequently known as the Warburg element [23, p. 376]. If the diffusion layer Lp is assumed to have an unlimited thickness within the experimental AC frequency range, than a "semi-infinite" diffusion may become the rate-determining step in the Faradaic kinetic process. In the "semiinfinite" diffusion model the diffusion layer thickness Lp is assumed to be always much smaller than the total thickness of the sample d (Lp d. The equation for the "semi-infinite" Warburg impedance Z m) is a function of concentration-driven potential gradient dV/rfC. The "semi-infinite" diffusion limitation is modeled by characteristic resistance and a Warburg infinite diffusion component Z that can be derived [8] as ... 

As the first approximation, impedance of a porous electrode can always be considered as a series combination of two processes—a mass-transport resistance inside the pores and impedance of electrochemical reactions inside the pores. De Levie was the first to develop a transmission line model to describe the frequency dispersion in porous electrodes in the absence of internal diffusion limitations [66]. De Levie s model is based on the assumption that the pores are cylindrical, of uniform diameter 2r and semi-infinite length /, not intercoimected, and homogeneously filled with electrolyte. The electrode material is assumed to have no resistance. Under these conditions, a pore behaves like a imiform RC transmission line. If a sinusoidal excitation is applied, the transmission line behavior causes the amplitude of the signal to decrease with the distance from the opening of the pore, and concentration and potential gradients may develop inside the pore. These assumptions imply that only a fraction of the pore is effectively taking part in the double-layer charging process. The RpQi i- [ohm] resistance to current in a porous electrode structure with number of pores n, filled with solution with resistivity p, is ... 

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