Stationary response

Equation (101) is very difficult to solve in general. However, we may simplify it very much if we consider the physical situation of interest experimentally, one turns on the field at time t = 0 and waits for a sufficiently long time, so that all transient effects have disappeared, before measuring the stationary response of... 

In this section, we will show that the stationary responses obtained at microelectrodes are independent of whether the electrochemical technique employed was under controlled potential conditions or under controlled current conditions, and therefore, they show a universal behavior. In other words, the time independence of the I/E curves yields unique responses independently of whether they were obtained from a voltammetric experiment (by applying any variable on time potential), or from chronopotentiometry (by applying any variable on time current). Hence, the equations presented in this section are applicable to any multipotential step or sweep technique such as Staircase Voltammetry or Cyclic Voltammetry. 

In the case of microcyclinders and microbands, fG,micro is time dependent (Table 2.3) and only a pseudo-stationary response can be achieved. This is because all the microelectrode dimensions have to fall in the range of the microns to attain a true steady state. The expressions for the pseudo-stationary current-potential responses when the diffusion coefficients of species O and R fulfills Dq = Dr are ... 

Note that, on the basis of Eqs. (2.164), (2.169), and (2.174), it can be concluded that for all these electrode geometries for which it is possible to achieve a true stationary response, Eq. (2.165) could be used by changing / crosphere-ss by the corresponding stationary limiting current for the geometry considered. Therefore, in all the previous cases it is fulfilled,... 

In this section, microdisc electrodes will be discussed since the disc is the most important geometry for microelectrodes (see Sect. 2.7). Note that discs are not uniformly accessible electrodes so the mass flux is not the same at different points of the electrode surface. For non-reversible processes, the applied potential controls the rate constant but not the surface concentrations, since these are defined by the local balance of electron transfer rates and mass transport rates at each point of the surface. This local balance is characteristic of a particular electrode geometry and will evolve along the voltammetric response. For this reason, it is difficult (if not impossible) to find analytical rigorous expressions for the current analogous to that presented above for spherical electrodes. To deal with this complex situation, different numerical or semi-analytical approaches have been followed [19-25]. The expression most employed for analyzing stationary responses at disc microelectrodes was derived by Oldham [20], and takes the following form when equal diffusion coefficients are assumed ... 

Note that from Eq. (3.194a) under transient conditions or from Eq. (3.205a) corresponding to a stationary response, it is easily deduced that the variation of the potential with ta /. PlaIle - /Cat-Plane) / cat, plane at.plane as ... 

In the case of disc electrodes, a similar behavior to that observed for spherical ones is obtained (although in this case the diffusion layer thickness is an average magnitude), whereas for band or cylinder electrodes, the diffusion layer thickness is always potential dependent, and no constant limit is achieved, even for very small values of the electrode characteristic dimension, confirming the impossibility of these electrodes achieving a true stationary response [5, 8, 16, 29, 30]. 

From Eqs. (5.92)-(5.94), it is clear that K°phe ss < x°phe < xplane, that is, the maximum value of the dimensionless rate constant is that corresponding to a planar electrode (macroelectrode). For smaller electrodes, /c(sphc decreases until it becomes identical to the value corresponding to a stationary response, xpphe ss. In practice, this means that the decrease of the electrode size will lead to the decrease of the reversibility degree of the observed signal. It can be seen in the CV curves of Fig. 5.14, calculated for k ) = 10 eras 1 and v = 0.1 Vs-1, that the decrease of rs causes an increase and distortion of the dimensionless current similar to that observed for Nemstian processes (see Fig. 5.5), but there is also a shift of the curve toward more negative potentials (which can be clearly seen in Fig. 5.14b). 

In summary, although the construction of micro-ITIES is, in general, simpler than that of microelectrodes, their mathematical treatment is always more complicated for two reasons. First, in micro-ITIES the participating species always move from one phase to the other, while in microelecrodes they remain in the same phase. This leads to complications because in the case of micro-ITIES the diffusion coefficients in both phases are different, which complicates the solution when nonlinear diffusion is considered. Second, the diffusion fields of a microelectrode are identical for oxidized and reduced species, while in micro-ITIES the diffusion fields for the ions in the aqueous and organic phases are not usually symmetrical. Moreover, as a stationary response requires fDt / o (where D is the diffusion coefficient, r0 is the critical dimension of the microinterface, and t is the experiment time), even in L/L interfaces with symmetrical diffusion field it may occur that the stationary state has been reached in one phase (aqueous) and not in the other (organic) at a given time, so a transient behavior must be considered. 

Equation (6.42) clearly shows that the CV stationary responses of disc and spherical electrodes hold the same equivalence relationship as that observed for a simple charge transfer process ... 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

It is also observed that the location of the CV at the potential axis is not affected by /cv. Indeed, when a stationary response is reached, the half-wave potential of the I-E curve coincides with the formal potential of the redox couple, as in the case of a reversible charge transfer reaction. 

The electrode size is another important factor to be considered since it affects the magnitude of the diffusive transport, as shown in Fig. 7.14 for totally irreversible processes. At planar and spherical electrodes significant differences are found between double pulse and multipulse modes, with the discrepancy diminishing when the electrode radius decreases, since the system loses the memory of the previous pulses while approaching the stationary response. Thus, the relative difference in the peak current of a given double pulse technique and the corresponding multipulse variant is always smaller than 2 % when... 

When the environment is not stationary, response functions such as x,M, (t, t ) and Xvx(t. t1) depend separately on the two times t and t7 entering into play, and not only on the time difference or observation time z t f. However, the observation time continues to play an essential role in the description. Hence, it has been proposed to define time- and frequency-dependent response functions as Fourier transforms with respect to x of the corresponding two-time quantities [5,6,58]. The time f, which represents the waiting time or the age of the system, then plays the role of a parameter. 

If only the stationary response signals are considered, the layers labelled by 2 and S should be omitted in the analysis of the plots. Some difference between the layers S can be seen. If the layer 1 is analysed, the differences between the target odours are much more evident. However, the recognition of the odour is much more reliable if the parameters of all the layers are considered in the characterisation of the odour. 

When a gas interacts with SWCNTs, the adsorbed molecules act as dopants, which possibly influences the SWCNT transport properties. These dopants can shift the Fermi level of SWCNTs or can modify the SWCNT band structure due to the orbital hybridisations between the analytes and the electron cloud of SWCNTs. If the response is assumed to be proportional to the fractional monolayer coverage of molecules adsorbed on the SWCNTs, then the stationary response is given by Equation [10.2] and the response time is given by Equation [10.3] ... 

In the case of stationary response, the auto-correlation function of the stochastic process x governed by Equation (3.1) depends only on the time difference t and it is given by [161,249] ... 

It is well known that the stationary response x is a zero-mean Gaussian random process with the auto-correlation function ... 

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]. 

This approximation resolves the computational difficulty encountered in the direct exact formulation that requires repeated computations of the solution of linear simultaneous algebraic equations and determinants of the matrices with huge dimensions. The efficiency in the approximated expansion is gained by the appreciation that the conditioning information can be truncated within one period of the system only. For linear systems, the expressions for the reduced-order likelihood function p(yi, yj, - - -, yNp W, C) and the conditional PDFs p(.yn 0, yn-Np, yn-Np+1, , y -i, C) are available since they are Gaussian and the correlation functions are known in closed forms regardless of the stationarity of the response. For stationary response, the method is very efficient in the sense that evaluation of all the conditional PDFs p(ynW, yn-Np,yn-Np+i,, y -i, C) requires the inverse and determinant of two relatively small matrices only. 

NLFR is a quasi-stationary response of a nonlinear system to a periodic (sinusoidal or cosinusoidal) input, around a steady state. One of the most convenient tools for treating nonlinear FRs is the concept of higher-order FRFs [52], which is based on Volterra series and generalized Fourier transform. This concept will be briefly presented below. 

This function is directly related to the amplitude and phase of the quasi-stationary response to a... 

Thus, an electrochemical step could be seen as irreversible if at least one IR in the system would also proceed irreversibly. On the other hand, however, the necessary condition for electrochemical irreversibility is the irreversibility of the chemical IVR it does not explicitly manifest itself in the stationary response of the system with irreversible electrochemical steps. 

Continuous cycling does not change the reversible response unless high monomer concentrations (> 0.1 M) are used, in which case, after an induction period of stationary response, a couple of additional... 

The fully non-stationary response of elastoplastic structures is calculated from the formulas of Sec. 3 by subdividing the time-axis into equidistant intervals, assuming (t, w ) to vary linearly within the... 

In the course of the present study we have discovered that different dynamic systems under different combination of random excitations may have the same probability distribution for stationary response. To Illustrate, consider... 

Yong, Y. and Lin, Y. K., Exact stationary response solutions for second order nonlinear systems under parametric and e cternal white-noise excitations, J. of Appl. Mechanics, 54 (1987) 414-418. 

The effects of the statistical uncertainties of the loading and system parameters on the mean exceedance rate of a particular threshold are investigated for a linear SDOF-system with viscous damping. For this purpose the structural loading is described by the well-known stationary Kanai-Tajimi-earthquake-model. The analysis is simplified by utilizing an approximate solution for the threshold-crossing rate, for which the error with respect to the exact solution is shown to be small. Each of the parameters involved in the expression for the mean exceedance rate of the stationary response of the structure is considered a random variable. The respective effects of the statistical uncertainties of the parameters on the threshold-crossing rate, as expressed by the first- and second moments, are shown explicitely in the numerical examples. 

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