Small amplitude perturbations

Mullins and Sekerka (17) were the first to construct continuum descriptions like the Solutal Model introduced above and to analyze the stability of a planar interface to small amplitude perturbations of the form... 

Another advantage of the small amplitude techniques, one that is less generally recognized, is that they can be combined with one of the d.c. voltammetric methods. In other words, the small amplitude perturbation,... 

It may be emphasized here that such an elaboration is possible for any small amplitude perturbation technique. It is only necessary to explicitize either the first-order current or the first-order interfacial potential, corresponding to the type of perturbation, to be able to derive expressions for 7q, 1 and AEl. So, the treatment is also useful to estimate the error due to second-order non-linearity in the step methods. However, a separate measurement of the second-order effect can only be done with (sinusoidal) a.c. perturbation. In Table 5, the explicit expressions for SF pertaining to the four methods mentioned in Sect. 2.4.1 are given in such a way that the connection between them is clearly shown. 

Equation (108) is universal for any small amplitude perturbation method and may, in principle, be used as the starting point for the derivation of a response vs. time relation for a given perturbation function. It is easily verified [53] that substitution of s = ico into any expression of an operational impedance or admittance delivers the complex impedance or admittance as they are defined in Sect. 2.3.1. 

As before (see Sects. 4 and 5), the derivation of the electrochemical response in the case of small-amplitude perturbation is possible on the basis of the formal expansion of eqns. (166e—g) in first-order Taylor terms... 

In all cases of second- or higher-order coupled homogeneous reactions, it is a severe problem that products of concentrations are present in the rigorous mass transfer equations. Consequently, straightforward Laplace transformation is not feasible. However, already in 1951 Gerischer (146] pointed to the fact that linearization of such terms is permitted if one confines the treatment to apply only to small amplitude perturbations. For example, a product cAcB will be written as... 

Numerical integration of equations (2) and (3) with initial values for X,Y on the limit cycle and with one of the rate constants oscillating as in equation (4) or (5) may result in a transition of the X,Y trajectory across the separatrix towards the stationary state. The occurrence of a transition is dependent on the parameters g, u) and 0. For extremely small amplitude perturbations (g - -0), the trajectory continues to oscillate close to the limit cycle. As g is increased, however, transitions may occur. The time taken for a transition is then primarily a function of the frequency of the perturbation. The time from the onset of the oscillating perturbation to the time at which the trajectory attains the lower steady state (At) is plotted in Figure 3 as a function of with all other parameters held constant. The arrow marks the minimum value for At which occurs when the frequency of the external perturbation exactly equals that of the unperturbed limit cycle itself. The second minimum occurs at the first harmonic of the limit cycle. Qualitatively similar results are obtained when numerical integration is carried out with differing values for g and 0. 

Whereas the charge-injection method is a small-amplitude perturbation method in which measurement is conducted during open-circuit decay, we now discuss a different open-circuit measurement, in which the initial overpotential is high, in the linear Tafel region. The equations we need to solve are similar to Eqs. 9K and lOK, except that the value of the current in Eq. lOK is that corresponding to the linear Tafel region, namely... 

When a small amplitude perturbation AE exp j(ot) is applied to the interface around the polarization potential, the corresponding current response, A// exp(j( t), is obtained by differentiating the equations describing the value of the faradaic current and mass transport. By eliminating the terms in exp(jfi)t) on the two sides of the following relationship, one has the following when there is a transport limitation by diffusion ... 

Also of significance is that initial instability of a thin film in accordance with the above mechanism does not inevitably lead to film rupture. The analysis, like all others in this chapter, is based on linear stability theory, and hence is valid only for small amplitude perturbations. It has been observed experimentally that at low surfactant concentrations instability of a film some tens of nanometers in thickness does produce rupture. But for many surfactants it is found that above a critical concentration, the instability leads to formation of black films which are only slightly thicker than the total length of two surfactant molecules (She-ludko, 1967). These black films can be very stable and are a major factor in foam stabihty. 

The small amplitude perturbation provoked by imposing an AC current is handled by using the mentioned property of the convolution to be transformed into an ordinary product... 

G. Bernine, "Small-amplitude perturbation theory for two dimensional surfaces, in light... 

These expressions are particularly useful for the determination of the transfer coefficient ac by small amplitude perturbation methods such as a.c. impedance. 

Another methodology consists in applying small amplitude perturbations where the steps or sweeps are sufficiently small to yield a Unear current-potential relationship (Figure 11.15). This approach is at the heart of electrochemical impedance methods. 

Figure 11.15 A large amplitude perturbation, here a potential sweep to a mass transport-controlled region, yields a nonlinear current-potential relationship. In contrast, a small amplitude perturbation, a few mV between and yields a linear relationship between current and potential.

Of course one can fit the reaction order by choosing an appropriate density of function for the surface states but there are no measurements that give information about their energetic distribution in mesoporous Ti02 (Peter, L.M., private communication). Applying a method which involves some form of small amplitude perturbation the analysis of data is even more complicate because dw/ dt is not zero. Actually, the electron movement in these mesoporous films is not understood. 

In this section, the impedance of the GDL is ignored. The first part of Equation 5.94 establishes a small-amplitude perturbation at the CCL/GDL interface (at i = 1). The second part of Equation 5.94 means zero proton current at x = 1. Equations 5.95 express zero oxygen flux through the membrane and zero concentration perturbation at the CCL/GDL interface, respectively. Note that the disturbance of the overpotential, can be applied on either side of the CCL the linearity of the system (5.91) and (5.92) guarantees that the solution would be the same in both cases. 

As in the previous sections, the superscript 0 marks the steady-state solution to the system (5.135) through (5.137), and the superscript 1 marks the small-amplitude perturbations. 

Equation 5.165 is linear thus, an equation for the small-amplitude perturbation... 

Most generally, this problem has no analytical solution and must be solved numerieally unless the linearized form of these equations is used. A linearization procedure is allowed by using a small-amplitude perturbation NE i) so as to neglect die nonlinear terms (degree higher than one) in the Taylor expansion of S, S and /j around the mean steady state ofthe system. It is known from linear system theory that under these eonditions the response A/(t) is proportional to the perturbation NE t). The dynamie behavior of the electrode at this partieular polarization point is completely described by its eomplex impedanee Z (/ ) = A (/(b) / A/(/(b) in the frequency domain where A (/(b) and /(Jo) are the Fourier transforms of A (t) and A/(t). 

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