Nonstationary behavior

Most electrochemical systems show some nonstationary behavior due, for example, to growth of surface films, changes in concentrations of reactants or products in the electrolyte, or changes in surface reactivity. As discussed in Section 21.3.4, the issue is not whether a system is perfectly stationary, but, rather, whether the system has chamged substantially during the course of the impedance measurement. The Kramers-Kronig relations are particularly useful for identification of artifacts introduced by nonstationary behavior. These artifacts are most visible at low frequencies, but can be seen at all frequencies if the system change is sufficiently rapid. 

Remember 21.2 Bias errors in impedance measurements can arise from instrument artifacts, parts of the measured system that are not part of the system under investigation, and nonstationary behavior of the system. 

---

Both naturally aging253 and electrically301 or thermally302 stressed anodic oxides exhibit characteristic nonstationary behavior. One can distinguish the following transient effects ... 

Nonstationary behavior Platinum on YSZ exhibits pronounced hysteretic effects, suggesting that passage of current can alter either the kinetics of the reaction or the dominant reaction pathway itself. As we saw in section 3.6 (and will again in... 

In addition to the possibility of multiple transport paths, our understanding of reaction mechanisms on LSM is further complicated (as with platinum) by pronounced nonstationary behavior in the form of hysteresis of inductive effects. These effects are sometimes manifest as the often-mentioned (but little-documented) phenomenon of burn-in , a term used in development circles to describe the initial improvement (or sometimes decline) of the cathode kinetics after a few hours or days following initial polarization (after which the performance becomes relatively stable). As recently reported by McIntosh et al., this effect can improve the measured impedance of a composite LSMA SZ cathode by a factor of 5 7relative to an unpolarized cathode at OCV." ° Such an effect is important to understand not only because it may lead to insight about the underlying electrode kinetics (and ways to improve them), but also because it challenges the metrics often used to assess and compare relative cell performance. 

Pronounced nonstationary behavior. Numerous workers have reported significant hysteresis and/or irreversibility in the behavior of LSM. This factor is important to be aware of since it is possible for two labs studying the same types of electrodes under similar conditions to arrive at completely different conclusions, depending on the exact history of fabrication and testing. Hopefully as workers move forward, these effects will provide additional clues as to the reaction mechanism dominating under specific conditions. 

As an application of the method of file separation of variables, we consider the nonstationary behavior in the generalized, fractional version of the Brownian harmonic oscillator with the parabolic potential... 

Figure 1. The creation, evolution, and detection of wave packets. The pump laser pulse pump (black) creates a coherent superposition of molecular eigenstates at t — 0 from the ground state I k,). The set of excited-state eigenstates N) in the superposition (wave packet) have different energy-phase factors, leading to nonstationary behavior (wave packet evolution). At time t = At the wave packet is projected by a probe pulse i probe (gray) onto a set of final states I kf) that act as a template for the dynamics. The time-dependent probability of being in a given final state f) is modulated by the interferences between all degenerate coherent two-photon transition amplitudes leading to that final state.

Nonstatistical properties, such as nonstationary behavior and multiple scales of time and distance for evolution, in systems of many degrees of freedom. 

In the third part, those contributions are collected which discuss nonergodic and nonstationary behavior in systems with many degrees of freedom, and seek new possibilities to describe complex reactions, including even the evolution of living cells. 

Broaden the frequency range As described in Section 19.5.3, an insufficient frequency range will reduce the ability to identify system characteristics by regression. Typically, an increase in frequency range is constrained at high frequencies by instrument limitations and at low frequencies by nonstationary behavior. 

Bias errors are systematic errors that do not have a mean value of zero and that cannot be attributed to an inadequate descriptive model of the system. Bias errors can arise from instrument artifacts, parts of the measured system that are not part of the system under investigation, and nonstationary behavior of the system. Some types of bias errors lead the data to be inconsistent with the Kramers-Kronig relations. In those cases, bias errors can be identified by checking the impedance data for inconsistencies with the Kramers-Kronig relations. As some bias errors are themselves consistent with the Kramers-Kronig relations, the Kramers-Kronig relations cannot be viewed as providing a definitive tool for identification of bias errors. 

If a single model were to be regressed to all spectra showing nonstationary behavior, the resulting residual error would include contributions from the drift between scans as well as the lack of fit of the model, instrumental bias errors, and stochastic errors, i.e.,... 

Solution Low-frequency inductive features are commonly seen in impedance spectra for PEM fuel cells. Makharia et al. suggested that side reactions and intermediates involved in the fuel cell operation can be possible causes of the inductive loop seen at low frequency. However, such low-frequency inductive loops could also be attributed to non-stationary behavior, or, due to the time required to make measurements at low frequencies, nonstationary behavior could influence the shapes of the low-frequency features. 

The philosophy described here and embodied in Figure 23.1 cannot always be followed to convergence. Often the hypothesized model is inadequate and cannot reproduce the experimental results. Even if the proposed reaction sequence is correct, surface heterogeneities may introduce complications that are difficult to model. The accessible frequency range may be limited at high frequency for systems with a very small impedance. The accessible frequency range may be limited at low frequency for systems subject to significant nonstationary behavior. The ex-... 

The prior discussion appears to be rather academic In general, one does not a priori know the asymptotical behavior for small scales of an observed process. Also, finite sampling restrains the investigation of small scales. Finally, one often is interested in nonstationary behavior on scales large compared to the sampling time. 

The same set of data is also analyzed by assuming a stationary response but the identification process is not converging for the third mode. This is not siuprising by observing Figure 4.13, which shows no peak of the third mode of the building in its Fourier amplitude spectrum. Furthermore, the model parameters of the other modes are also biased. Therefore, consideration of the nonstationarity of the response is important when there is obvious evidence for the response to exhibit nonstationary behavior [285]. 

With their combination of complex kinetics and thermal, convective and viscosity effects, polymerizing systems would seem to be fertile ground for generating oscillatory behavior. Despite the desire of most operators of industrial plants to avoid nonstationary behavior, this is indeed the case. Oscillations in temperature and extent of conversion have been reported in industrial-scale copolymerization (57). 

Finite-difference techniques are useful in elucidating time-dependent behavior in such layers, for example the time-dependent rotating disk voltammogram in Fig. 6 10 (68,79) nonstationary behavior has also been observed elsewhere. " ... 

A model was developed to cope with nonstationary behavior observed in Fig. 6.10 see Fig. 6.11 for an example of the resulting simulations/ which shows the effect of rotation rate on the current. While results from the simulation exhibit curves similarly shaped to those found in practice/ there are quite a number of parameters that are not accurately known. Such unknown effects as the layer homogeneity on the rate of electron exchange and the swollen layer thickness... 

---