Fast time scale

Many of the fiindamental physical and chemical processes at surfaces and interfaces occur on extremely fast time scales. For example, atomic and molecular motions take place on time scales as short as 100 fs, while surface electronic states may have lifetimes as short as 10 fs. With the dramatic recent advances in laser tecluiology, however, such time scales have become increasingly accessible. Surface nonlinear optics provides an attractive approach to capture such events directly in the time domain. Some examples of application of the method include probing the dynamics of melting on the time scale of phonon vibrations [82], photoisomerization of molecules [88], molecular dynamics of adsorbates [89, 90], interfacial solvent dynamics [91], transient band-flattening in semiconductors [92] and laser-induced desorption [93]. A review article discussing such time-resolved studies in metals can be found in... 

For comparison, the calculated linear and 2D spectra using ft = 12.3 cm-1 and 6 = 52°, which correspond to an a-helical structure (see the contour plot Fig. 19) for the isotopomer Ala -Ala-Ala are shown in Figure 21. The observed spectra for Ala -Ala-Ala are strikingly different from the calculated spectra for a molecule in an a-helical conformation. We emphasize here an important point In contrast to the NMR results on oligo(Ala), in which averaging of different backbone conformations might be present because measurements are made on a time scale that is slow compared to that of conformational motions, these vibrational spectroscopy results are detected on a very fast time scale (Hamm et al, 1999 Woutersen and Hamm, 2000, 2001). This rules out conformational averaging. 

The lifetime detection techniques are self-referenced in a sense that fluorescence decay is one of the characteristics of the emitter and of its environment and does not depend upon its concentration. Moreover, the results are not sensitive to optical parameters of the instrument, so that the attenuation of the signal in the optical path does not distort it. The light scattering produces also much lesser problems, since the scattered light decays on a very fast time scale and does not interfere with fluorescence decay observed at longer times. 

Light absorption is usually quite fast (time scale = 1-10 femtoseconds), and various physical measurements can be used to characterize the properties of intermediates that are formed along the reaction coordinate. This strategy was introduced by Porter who later shared the Nobel Prize in Chemistry with Eigen and Norrish for their germinal contributions to fast reaction kinetics. See Chemical Kinetics... 

Below roughening, pronounced lattice effects show up in the simulations, as in the case of wires. The meandering of the top(bottom) steps and the islanding on the top(bottom) terrace leads to slow and fast time scales in the decay of the amplitude. The profile shapes near the top(bottom) broaden at integer values of the amplitude and acquire a nearly sinusoidal form in between. Again, these features are not captured by the continuum theory. For evaporation kinetics, continuum theory suggests that the decay of the profile amplitude z scales like z t,L) = where g =... 

In Eq. (15) we have introduced a fast time scale or frequency co0 and a life time x. I is the unit and J the operator corresponding to a Jordan block representation. 

The value of ( used here corresponds to 10 3 of its value in room temperature aqueous solutions. [80] On the one hand, using such weak friction improves the sampling efficiency in the simulations and does not affect equilibrium structural properties. On the other hand, the dynamical properties that we observe may be different from those probed by SM-FRET techniques, which would not be able to resolve conformational dynamics on such fast time scales. Thus, the relevance of the following analysis of dynamical properties relies on the assumption that increasing the friction will not significantly alter our main conclusions. It is interesting to note in this context that the folding mechanism in similar models has been observed to be relatively insensitive to the value of the friction coefficient. [81]... 

Short-range hole transfer reactions occur on a very fast time-scale (kEX = 109-1012 s-1). 

Moving now to the fast dynamics of this reaction system, we define the fast time scale r = t/e, obtaining... 

Similarly, a representation of the fast dynamics in the limit e —> 0 is obtained in the stretched fast time scale r = t/e as... 

As mentioned before, obtaining an explicit variable separation for the system in Equation (2.36) requires a nonlinear coordinate transformation. The fact that k(x) = 0 in the slow time scale t and k(x) 0 in the fast time scale r indicates that the functions fcj(x) should be used in such a coordinate transformation as fast variables. Then, it can be shown (see, e.g., Kumar and Daoutidis 1999a) that a coordinate change of the form... 

On introducing the stretched fast time scale r = tje and considering the limit —0 in Equation (2.11), we also obtain the following description of the fast dynamics ... 

Remark 3.2. We can regard the developments above from a converse perspective. Namely, if we consider the model of each individual unit (preserving the input and output flow structure of the process) in the fast time scale t, we can write a simplified model for unit i in the form... 

The above time-scale decomposition provides a transparent framework for the selection of manipulated inputs that can be used for control in the two time scales. Specifically, it establishes that output variables y1 need to be controlled in the fast time scale, using the large flow rates u1, while the control of the variables ys is to be considered in the slow time scale, using the variables us. Moreover, the reduced-order approximate models for the fast (Equation (3.11)) and slow (the state-space realization of Equation (3.16)) dynamics can serve as a basis for the synthesis of well-conditioned nonlinear controllers in each time scale. 

Tikhonov s theorem (Theorem 2.1) indicates a further requirement that must be fulfilled by the controllers in the fast time scale in order for the time-scale decomposition developed above to remain valid, these controllers must ensure the exponential stability of the fast dynamics. From a practical point of view, this is an intuitive requirement one cannot expect stability and control performance at the process level if the operation of the process units is not stable. 

The distributed control objectives for this process involve the stabilization of the individual unit holdups (Mr, Me, and Mb), which, according to our prior analysis, should be addressed in the fast time scale. The design of the distributed controllers for the stabilization of the three holdups can easily be achieved, using the large flow rates F, D, and V as manipulated inputs and employing simple proportional controllers - note that only these three flow rates (i.e., the components of u1) affect the fast dynamics. More specifically, the proportional control laws... 

Specifically, the individual units of the process exhibit a fast dynamics in the fast time scale the response times in the fast time scale are typically of the order of magnitude of the time constants of the individual units. In the fast time scale, the dynamic coupling between the units (induced by the recycle stream) is weak and can be ignored. 

The interactions between units do, however, become significant over long periods of time processes with recycle exhibit a slow, core dynamic component that must be addressed in any effective process-wide control strategy. This chapter presented an approach for systematically exploiting this two-time-scale behavior in a well-coordinated hierarchical controller design. The proposed framework relies on the use of simple distributed controllers to address unit-level control objectives in the fast time scale and a multivariable supervisory controller to accomplish process-wide control objectives over an extended time horizon. 

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