Harmonic relaxations

The software of the PAT1 allows the programming of a complete sequence of experiments, such as transient and harmonic relaxations after the equilibrium of adsorption has been established. The oscillations were usually performed at various frequencies, so that sufficient data for the dilational rheology were available. A typical result, as an example, is shown for / -LG in Figure 5, where... 

Figure 5 Typical dynamic experiment consisting of adsorption kinetics, transient and harmonic relaxations for a ft-LG concentrations oflx 10 s mol/l in buffer at 22°C...

The software driven apparatus allows different types of area changes step and ramp type, square pulse and trapezoidal as well as sinusoidal area deformations. The construction ensures that area changes are almost isotropic. Area changes used in transient and harmonic relaxation experiments are of the order of 1 to 5%. The surface tension response measured via the Wilhelmy balance has an accuracy of better than 0.1 mN/m. 

There are transient and harmonic perturbations of the interfacial area. As it was shown by Loglio et al. [144] the theoretical basis is the same and therefore transient relaxations correspond also to a certain characteristic frequency. For harmonic relaxation processes there is a phase difference between the generation of the oscillation and the response function which is a measure of the exchange of matter. 

A particular way of presenting harmonic relaxation data is the plot of surface tension changes versus the corresponding area changes. As one can see in Fig. 4.45 an ellipse results the tilt angle and the thickness of which contain the rheological information. 

Transient as well as harmonic relaxation experiments give aecess to the dilational rheology of the studied interface or film (211). The definition of the dilational elasticity E is given by the relation ... 

Note that in contrast to nematics Tj is controlled by the helical pitch Po = 2nlqo and not by cell thickness d. At a strong field, the distortion involves several harmonics with number m and wavevectors q = 2nm/Po and each harmonic relaxes with its... 

Fourier transformation as a tool to analyse harmonic relaxation studies.472... 

First experiments on drop shapes were done a long time ago [34, 35, 36] using standard photo techniques. Although, at that time the efforts were huge, the results were much less accurate than those obtained from force measurements. Now, due to modem video technique there is quite a number of commercial instruments available on the market. The accuracy of most is comparable. However, the variety of measurement features is very dilTerent. Only the instruments which can control the size of the drop or bubble on-line during the measurement are suitable for more advanced studies. Transient or harmonic relaxation studies, as described in the monograph by Dukhin et al. [37], cannot be done properly without this important flmction. Hence, these devices are not able to provide interfacial rheological data of interfacial layers. 

Although no direct comparison with other commercial products are given we can state that the instrument PATl discussed here has the best features in respect to interfacial rheology studies. It provides a comfortable function generator for any type of transient and harmonic relaxation studies and also the theoretical tools to analyse trapezoidal and sinusoidal relaxation experiments. The on-line control of the interfacial area changes is very accurate and the oscillations performed in the range between 0.01 and 0.2 Hz are ideally smooth sinusoidal functions in contrast to experiments performed with other instruments. 

Next, consider the relaxation of a diatomic molecule, essentially a single oscillator of frequency two interacting with its thermal environment, and contrast its behavior with a polyatomic molecule placed under similar conditions. The results of the simple harmonic relaxation model of Section 9.4 may give an indication about the expected difference. The harmonic oscillator was shown to relax to the thermal equilibrium defined by its environment, Eq. (9.65), at a rate given by (cf. Eq. (9.57))... 

Castro A, Sitzmann E V, Zhang D and Eisenthal K B 1991 Rotational relaxation at the air-water interface by time-resolved second-harmonic generation J. Phys. Chem. 95 6752-3... 

The role of two-phonon processes in the relaxation of tunneling systems has been analyzed by Silbey and Trommsdorf [1990]. Unlike the model of TLS coupled linearly to a harmonic bath (2.39), bilinear coupling to phonons of the form Cijqiqja was considered. In the deformation potential approximation the coupling constant Cij is proportional to (y.cUj. There are two leading two-phonon processes with different dependence of the relaxation rate on temperature and energy gap, A = (A Two-phonon emission prevails at low temperatures, and it is... 

Curve C of Fig. 6-15(a) corresponds to /a 0.1, and curve C of Fig. 6-15(b) to p = 10. It is observed that in the first case the curve O is almost a circle (as in a harmonic oscillator) while in the second case it has very special features. In fact from very high velocity (y — sc), the solution drops very rapidly to zero (at the point where it cuts the x-axis). At this point the relaxation interval begins, the representative point moving near the x-axis with a very small velocity after that its velocity increases rapidly to a considerable value, and soon. 

In NMR theory the analogue of the relation (1.57) connects the times of longitudinal (Ti) and transverse (T2) relaxation [39]. In the case of weak non-adiabatic interaction with a medium it turns out that T = Ti/2. This also happens in a harmonic oscillator [40, 41] and in any two-level system. However, if the system is perturbed by strong collisions then Ti = T2 as for y=0 [42], Thus in non-adiabatic theory these times differ by not more than a factor 2 regardless of the type of system, or the type of perturbation, which may be either impact or a continuous process. 

Such a construction is not a result of perturbation theory in <5 , rather it appears from accounting for all relaxation channels in rotational spectra. Even at large <5 the factor j8 = B/kT < 1 makes 1/te substantially lower than a collision frequency in gas. This factor is of the same origin as the factor hco/kT < 1 in the energy relaxation rate of a harmonic oscillator, and contributes to the trend for increasing xE and zj with increasing temperature, which has been observed experimentally [81, 196]. 

In summary, the NFS investigation of FC/DBP reveals three temperature ranges in which the detector molecule FC exhibits different relaxation behavior. Up to 150 K, it follows harmonic Debye relaxation ( exp(—t/x) ). Such a distribution of relaxation times is characteristic of the glassy state. The broader the distribution of relaxation times x, the smaller will be. In the present case, takes values close to 0.5 [31] which is typical of polymers and many molecular glasses. Above the glass-to-liquid transition at = 202 K, the msd of iron becomes so large that the/factor drops practically to zero. 

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