Higher harmonics, from nonlinearity

Faradaic rectification — When the electrode potential of the working - electrode is modulated with a sinusoidal -> alternating current the mean potential is shifted from the DC value by a small increment in many cases when the AC modulation is sufficiently large. This effect has been named faradaic rectification, it is caused by the nonlinearity of the electrode response, in particular the variation of current with electrode potential [i]. A theoretical treatment for an electrode in contact with a solution containing a redox system has been provided [ii]. It was extended to reactions where one reactant is present in its element form dissolved in the liquid metallic phase (e.g., Cd2+ + 2e -> Cd(Hg)) [iii]. An improved evaluation technique has been proposed [iv], and some inherent problems have been reviewed [v]. A variant of this method applied to -> polarography has been described [vi]. Second and higher harmonics in - AC voltammetry (polarography) [vii] also arise from this nonlinearity, and hence these techniques also have some characteristics that resemble those found in - faradaic rectification voltammetry. 

The formation of the ligament can be explained by the nonlinear theories. The nonlinear analysis of Yuen [63] and also Chaudhary and Redekopp [71] have revealed that the mode coupling results in a feedback from higher harmonics to the fundamental and vice versa. For instance, the second harmonic generates interactions between the first four harmonics only by considering the second order solution. The summation of all of the fundamentals generated by this mode... 

For weakly nonlinear systems, the contributions of the higher harmonics and higher FRFs decrease with the increase of their order. Different harmonics of the output can be estimated directly by harmonic analysis of the output signal. On the other hand, as can be seen from Equation (11.13) to Equation (11.16), the first-order function Gi(cd) corresponds to the dominant term of the first harmonic, the second-order functions G2(dominant terms of the second harmonic and the DC component, respectively, the third-order function Gs(ft), ft), to) to the dominant term of the third harmonic, etc. This fact enables estimation of different FRFs from the harmonics of the output obtained for different input amplitudes, as shown by Lee [53]. 

Let us look at the influence of nonlinearities on observed spectra for typical amplitudes of 5 and 50 mV rms, that is, amplitudes of 5v and 50v [641]. They are shown in Fig. 15.3. At small amplimdes, the nonlinear effects are negligible. However, for larger amplimdes the current observed in Tafel conditions is no longer sinusoidal and contains contributions from higher harmonics. 

Due to the nonlinearity of the I c — Eac relation also direct current component may be detected (faradaic rectification) without any analytical application. Improvement may be achieved by following the second harmonic current response. The main reason for the use of this technique ensues from the fact that the faradaic-to-charging current ratio is extremely favorable in this case. The double-layer charging current, unlike the faradaic component, does not practically involve higher harmonic contributions [4]. 

Fig. 2 Schematic depiction of the procedure for conducting LAOS experiments and analysis using FT-rheology. a Measurement of the oscillatory shear strain and shear stress response in the time domain, b Normalized frequency spectra after the Fourier transformation of the shear stress exhibit the fundamental peak at the angular frequency (d. Higher harmonics / /i with n being a positive odd integer are detected for a periodic nonlinear shear stress, c By variation of yo the transition from linear to nonlinear mechanical behavior can be observed in the increase of/ /i...

Assuming a viscosity that is nonlinear in the shear rate, Wilhehn et al. [46] used a polynomial expansion and proposed that the absolute intensity of a higher harmonic / ought to scale with the nth power of the strain amplitude for small enough deviations from the linear regime. Therefore, a normalized intensity is... 

These parameters can be used to infer the inherent nonlinear material properties of a sample as the trivial scaling for the relative intensities of the higher harmonics, 4/1 oc. yQ, is eliminated. As an example of this, the intrinsic nonlinearity parameter Q, that is derived from the third harmonic, has been shown to be useful in evaluating the topology of polymer melts [20]. 

In order to identify the influence of microstructural parameters on the nonlinear behavior of dilute emulsions, specifically on the intensities of the higher harmonics and their strain amplitude dependence, we have employed modelling approaches [36, 37], For dilute emulsions the individual droplets can be regarded as independent of each other, therefore, the measured shear stress can be obtained from a linear superposition of the matrix contribution and the contributions originating from each droplet and their interface [5, 16], as predicted by Batchelor [2] ... 

Figure 18A shows the Fourier spectra thus obtained. The real and imaginary parts correspond to the elastic and viscous components of the DOPC thin film, respectively. We can see that the spectrum is composed not only from the fundamental (coo) but also from the higher (2harmonic components. Such a trend indicates that the DOPC thin film exhibits rather large nonlinearity in the viscoelastic characteristics. 

Fig.l. Schematic representation of three common nonlinear processes that convert low-energy photon pump sources into higher-energy output SHG = Second Harmonic Generation, STPA = Simultaneous Two-Photon Absorption, UC = Upconversion. Adapted from [17]... 

Aside from CARS, a second-order nonlinear technique, such as second harmonic generation or sum frequency generation, can also be a suitable spectroscopic technique because of its higher surface selective detection [103], which is arising from... 

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