Diffusion impedance Limitations

The electrolyte resistance Re is added in series with the previous impedance. If the electrochemical reaction is mass-trai3sport limited, the previous equivalent circuit is still valid, but the Faradaic impedance includes a diffusion impedance as described in Chapter 11. 

Theoretical developments show that it is possible to deduce hydrodynamic information from the limiting current measiuement, either in quasi-steady state where /(f) cx py t) or, at higher frequency, in terms of spectral analysis. In the latter case, it is possible to obtain the velocity spectra from the mass-transfer spectra, where the transfer function between the mass-transfer rate and the velocity perturbation is known. However, in most cases, charge transfer is not infinitely fast, and the analysis also requires knowledge of the convective-diffusion impedance, i.e., the transfer function between a concentration modulation at the interface and the resulting flux of meiss under steady-state convection. 

At first glance, it may not be obvious that such an approach should work. It is well known, for example, that the impedance spectrum associated with an electrochemical reaction limited by the rate of diffusion through a stagnant layer (either the Warburg or the finite-layer diffusion impedance) can be approximated by an infinite number of RC circuits in series (the Voigt model). In theory, then, a measurement model based on the Voigt circuit should require an infinite number of parameters to adequately describe the impedance response of any electrochemical system influenced by mass transfer. 

In the case where only oxygen diffusion transport limitations are considered and proton transport limitations are neglected, the linear impedance response prescribes a perfect semicircle in the complex plane without showing a linear branch in the high-frequency limit. The response given by Eq. (110) is, thus, an exclusive feature of proton transport limitations, which, thereby, provides a feasible tool for their characterization. 

Figure 5.11 Representation of diffusion impedance spectra of diffusion processes limited by a Nemst diffusion layer, 182 D, D/S = 30 s . (a) Nyquist plot and (b) Bode plot.

At the minimal c.d., Bp, typical diffusion control is observed consistent with the local plateau of the current-voltage curve. A diffusion term is associated with a transport limitation across the passive layer, whereas the inductive loop at the low-ffequency end visualizes the contribution of a deca5ung film protection at increasing potentials. Application of a diffusion impedance equation to a finite-thickness layer (5 = 3 nm) given by... 

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. ... 

EHie to the assumption of semi-infinite diffusion made by the Warburg impedance for the derivation of the diffusion impedance, it predicts that the impedance diverges from the real axis at low frequencies. The DC impedance of the diffusion-limited electrochemical cell would be infinitely large. The Warburg impedance can be represented by a semi-infinite transmission line (TLM) composed of capacitors and resistors (Figure 5-6) [1, p. 59]. 

Higher-density currents result in water mass-transport limitations and the appearance of additional low-frequency impedance relaxation. This diffusion impedance due to water transport within the polymer electrolyte morphology is often represented by the finite diffusion processes affecting both the anodic (Zq ) and cathodic (Z ) processes (Figure 12-16B). An increase in anodic impedance is often related to partial drying out of the anode/membrane interface. 

In a packed column, however, the situation is quite different and more complicated. Only point contact is made between particles and, consequently, the film of stationary phase is largely discontinuous. It follows that, as solute transfer between particles can only take place at the points of contact, diffusion will be severely impeded. In practice the throttling effect of the limited contact area between particles renders the dispersion due to diffusion in the stationary phase insignificant. This is true even in packed LC columns where the solute diffusivity in both phases are of the same order of magnitude. The negligible effect of dispersion due to diffusion in the stationary phase is also supported by experimental evidence which will be included later in the chapter. 

Isothermal Infiltration. Several infiltration procedures have been developed, which are shown schematically in Fig. 5.15.P3] In isothermal infiltration (5.15a), the gases surround the porous substrate and enter by diffusion. The concentration of reactants is higher toward the outside of the porous substrate, and deposition occurs preferentially in the outer portions forming a skin which impedes further infiltration. It is often necessary to interrupt the process and remove the skin by machining so that the interior of the substrate may be densified. In spite of this limitation, isothermal infiltration is used widely because it lends itself well to simultaneous processing of a great number of parts in large furnaces. It is used for the fabrication of carbon-carbon composites for aircraft brakes and silicon carbide composites for aerospace applications (see Ch. 19). 

For the investigation of charge tranfer processes, one has the whole arsenal of techniques commonly used at one s disposal. As long as transport limitations do not play a role, cyclic voltammetry or potentiodynamic sweeps can be used. Otherwise, impedance techniques or pulse measurements can be employed. For a mass transport limitation of the reacting species from the electrolyte, the diffusion is usually not uniform and does not follow the common assumptions made in the analysis of current or potential transients. Experimental results referring to charge distribution and charge transfer reactions at the electrode-electrolyte interface will be discussed later. 

Polycondensation of highly viscous polyesters in the melt phase is limited. The removal of the volatile by-products becomes more difficult due to diffusion inhibited by the increased viscosity of higher-IV polyesters. In addition, undesirable side reactions due to thermal degradation impede the growth of the molecular chains. As a consequence, the reaction rate decreases and decomposition reactions dominate, thus resulting in a decrease in the melt viscosity [2], As it is able to address these limitations, SSP has become the method of choice and is therefore so popular. 

---