Time and frequency dependences

We are now in a position to use our relaxation equations to express the time and frequency dependences of the capacitive charging and so of the complex permittivity. Starting with the charge as a function of time  

The capacitive (real) and conductive (imaginary) components of the complex permittivity are exactly the same shape as those for the complex compliance that were presented in Chapter 10. 

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In the previous section we have discussed the relation between the time- and frequency-dependent friction and viscosity in the normal liquid regime. The study in this section is motivated by the recent experimental (see Refs. 80-87) and computer simulation studies [13,14, 88] of diffusion of a tagged particle in the supercooled liquid where the tagged particle has nearly the same size as the solvent molecules. These studies often find that although the fric-... 

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. 

Quantum control methods make use of the time- and frequency dependence of the external laser field, usually assuming that the spatial dependence of the coupling between two electronic energy surfaces of a molecule is constant. In this work we ask what may be the influence of the spatial dependence of the coupling and can it be also used for steering molecular transitions ... 

Cao et al. [80] have also discussed time and frequency dependent solutions of the generation collection equation for the case where the cell is illuminated from the substrate side. Like Solbrand et al., they assume that electron transport occurs predominantly by diffusion at relatively low light intensities where the electric field arising from reconfiguration of electrons is small. The expression derived for the response to a light step shows that the rise of the photocurrent is multi-exponential. 

Saturation of an optical transition ( hole-burning ) and subsequent analysis of the time and frequency-dependence of the recovery of the ground-state species has become a well-known technique for the study of the picosecond photophysics of radiationless transitions in stable molecules, transient species, and laser dyes in particular. " ... 

Experimentation with step-scan interferometry in electrochemistry began in the early 1990s (cf Ref. [23]), and interest has grown steadily ]24, 29-31, 51-54]. Step-scan FTIR spectroscopy provides a means to investigate time- and frequency-dependent processes. Measurements are hmited to reversible systems. However, a great deal of insight can be gained into the molecular transformations that accompany the external perturbation [181-183]. 

With increasing interest in time-resolved impedance measurements but also with the demand of parallel measurements, fast methods based on time domain approach move more and more into the focus. Although time and frequency domain are well defined, they are often not clearly presented. Especially, when the impedance spectrum changes with time, a joint analysis in terms of time and frequency dependence is often accompanied by uncertainties in wording. 

As pointed out by de Levie, however, the most important weakness in the model is the assumption that the current distribution is normal to the macroscopic surface, that is a neglect of the true current distribution. For a rough surface, the lines of electric force do not converge evenly on the surface. The double layer will therefore be charged unevenly, and the admittance will be time and frequency dependence. 

Comparison of the forms of equations 58 to 61 with equations 21 to 23 of Chapter 9 and equations 23 and 24 of Chapter 3 shows that the time and frequency dependence correspond to a generalized Maxwell model as in the Rouse theory and its various modifications, but here the spring constants (or discrete contributions to the relaxation spectrum) are not necessarily all equal they are proportional to the concentrations of the various types of strands, v e. The molecular weight does not enter explicitly, but it may be expected that the higher the molecular weight the greater the concentrations of strands which find it difficult to leave the network and hence have large values of the time parameter... 

Having examined the nature of the temperature and pressure dependence of the relaxation and retardation times, we now turn attention to the details of the time and frequency dependence of the basic viscoelastic functions and their correlation with chemical structure. Each zone of time scale represents a separate problem. The one most characteristic of polymers is the subject of this chapter, the transition from rubberlike to glasslike consistency, where the moduli increase and the compliances decrease by several powers of 10 as a function of time or frequency, as illustrated in Chapter 2. 

Wang M, Winter HH, Auemhammer GK (2013) Time and frequency dependent rheology of reactive silica gels. J Coll Int Sci 413 159... 

Due to the generation of charge carriers, conjugated polymers become electrically conductive when irradiated with UV/visible light. The quantum yield for charge carrier generation is increased by additives, such as fullerene compounds. With the aid of time-resolved THz spectroscopy, the time- and frequency-dependent complex conductivity with its real and imaginary parts can be measured. THz radiation is absorbed by the polymer due to the dielectric... 

If our function of time and the corresponding values in the frequency domain are continuous, we will write them as h(t) and H(f). If either is available only at discrete times or frequencies, we will write them as hj (the value at time tf and Hj (the value at frequency) ). In some places it is helpful to emphasize the time and frequency dependence by writing these as hftj) and Hj(fj). 

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