Chemical slow dynamics

Multidimensional and heteronuclear NMR techniques have revolutionised the use of NMR spectroscopy for the structure determination of organic molecules from small to complex. Multidimensional NMR also allows observation of forbidden multiple-quantum transitions and probing of slow dynamic processes, such as chemical exchange, cross-relaxation, transient Over-hauser effects, and spin-diffusion in solids. 

You may wonder why we would ever be satisfied with anything less than a very accurate integration. The ODEs that make up the mathematical models of most practical chemical engineering systems usually represent a mixture of fast dynamics and slow dynamics. For example, in a distillation column the liquid flow or hydraulic dynamic response occurs fairly rapidly, of the order of a few seconds per tray. The composition dynamics, the rate of change of hquid mole fractions on the trays, are usually much slower—minutes or even hours for columns with many trays. Systems with this mixture of fast and slow ODEs are called stiff systems. 

From physical considerations, at most C equations (with C being the number of chemical components) are required in order to completely capture the above overall, process-level material balance. Thus, we can expect the dimension of the system of equations describing the slow dynamics of the process to be at most C, and the equilibrium manifold (3.12) of the fast dynamics to be at most C-dimensional. 

Figure 2.1 Adapted and reproduced with permission from J. Phys. Chem. C 2007, III, 16076-16079. Copyright 2007 American Chemical Society. Simplified Jablonski diagrams depicting the processes involved for a fluorophore in three possible situations discussed in the text (a) Standard fluorescence, (b) Surface Enhanced Fluorescence in the slow-dynamics regime (SDMEF), and (c) MEF in the ultra-fast-dynamics regime (UFDMEF).

As applications of TD-DFT, we considered density fluctuations in liquids and treinsport coefficients. We are currently trying to solve the L-D equation in a real space-time to study slow dynamics in supercooled liquids. This study may be considered to be a dynamic counterpart of the work by Dasgupta and Ramaswamy and as a similar attempt we mention the work by Lust and Vails . We expect that our L-D equation has many fields of application. One example is dynamics in molecular liquids, as noted at the end of subsection IV(A), which is a very important field in connection with chemical reactions in solutions but, at the same time, is very complex to deal with from first principles. 

The assumption that IVR is much faster than intermolecular energy relaxation considerably simplifies the description of the well dynamics. In the following discussion the molecular motion in the reactant well is taken to be completely characterized (on the relevant time scale) by the time evolution of the total molecular energy E the energies in the different modes are determined from Ej by statistical considerations. In Section VI we also present the solution of a model in which IVR is slow relative to intermolecular relaxation, though this case is probably less relevant to the chemical reaction dynamics of polyatomic molecules in solution. 

Owing to the orientation dependence that it imparts to the NMR frequency, the chemical-shift anisotropy (CSA) has proven useful not only in studies of slow dynamics but also for characterizing segmental orientation distributions and fast segmental reorientations. While static powder patterns provide this CSA information in the most accessible form, site resolution by MAS is indispensable in all but the simplest unlabeled systems. The two requirements can be combined in two-dimensional (2D) separation experiments. Recently, a robust sequence, termed separation of undistorted powder-patterns by effortless recoupling (SUPER), was introduced that makes CSA measurements under standard MAS conditions routine.28 It enables identification of functional groups and measurements of orientation distributions, segmental dynamics, and conformations. 

Although the pulse sequences used to study phase transitions are usually quite simple in the examples presented in this review (one to maximum four pulses), the interpretation may be subtle. Solid-state NMR nevertheless remains a difficult technique since quantitative interpretation of the spectra rely on a profound knowledge of the chemical composition and structure of the sample analysis of NMR results also requires a model to relate the observed NMR spectral shapes or relaxation behavior to hypothesis concerning the structure and dynamics of the atoms or molecules carrying spins. That NMR motionally average the atomic and molecular displacements that occur on a time-scale faster than 10—8 10—9s is an important point that should be considered in the interpretation of data. In particular, the difference in perception between NMR and X-ray diffraction with regard to fast and slow dynamical disorder in molecular crystals undergoing phase transitions between different polymorphs was illustrated. In fact, the interpretation of NMR data almost always needs the support of other data obtained by different techniques. Therefore, we emphasized the different complementarities with X-ray (or neutron) diffraction, IQNS and other spectroscopic methods to provide, by cross-correlation of the different data, consistent picture of the phase transition. 

Slow dynamical systems and chemical kinetics equations... 

The desired behaviour of a chemical dynamical system can be modelled by an effective system of kinetic equations in the way similar to that described in Section 3.5 for modelling the heartbeat. The method involves designing a system of differential equations having the desired slow dynamics (the proper slow surface). We should now answer the question whether application of the Tikhonov theorem to the standard kinetic system (4.27) may yield a completely arbitrary slow dynamical system (4.40b ). A partial answer to this question is provided by the Korzukhin theorem Each dynamical system of the form... 

From the Korzukhin theorem follows an important conclusion. Any dynamical systems of the form (4.58) may be regarded as those corresponding to slow dynamics of a standard kinetic system. In other words, the behaviour of dynamical systems can be modelled using chemical reactions. In particular, any of the gradient systems may be modelled in this way. As will be shown in Chapter 5, catastrophes occurring in complex dynamical systems are equivalent to catastrophes appearing in much simpler systems. The latter can be classified — these are so-called standard forms. The standard forms are of the form (4.58) and it follows from the Korzukhin theorem that they can be modelled by the standard equations of chemical kinetics (4.27), corresponding to a realistic mechanism of chemical reactions. 

When discussing the general aspects of FTNMR, we have to remember that all principal statements about Fourier methods have been introduced for a strictly linear system (mechanical oscillator) in Chapter 1. In Chapter 2, on the other hand, we have seen that the nuclear spin system is not strictly linear (with Kramer-Kronig-relations between absorption mode and dispersion mode signal >). Moreover, the spin system has to be treated quantummechanically, e.g. by a density matrix formalism. Thus, the question arises what are the conditions under which the Fourier transform of the FID is actually equivalent to the result of a low-field slow-passage experiment Generally, these conditions are obeyed for systems which are at thermal equilibrium just before the initial pulse but are mostly violated for systems in a non-equilibrium state (Oberhauser effect, chemically induced dynamic nuclear polarization, double resonance experiments etc.). 

The difference in the relaxation rates of ZQ and DQ coherences is the result of three principal mechanisms. These include the cross-correlation between the chemical shift anisotropies of the two participating nuclei, dipolar interactions with remote protons as well as interference effects due to the time-modulation of their isotropic chemical shifts as a consequence of slow mus-ms dynamics. The last effect when present, dominates the others resulting in large differences between the relaxation rates of ZQ and DQ coherences. Majumdar and Ghose have presented four TROSY-based experiments that measure this effect for several pairs of backbone nuclei. These experiments allow the detection of slow dynamic processes in the protein backbone including correlated motion over two and three bonds ". A suite of TROSY-based NMR relaxation dispersion experiments that measure the decay of DG and ZQ coherences as a... 

The curves shown in Fig. 5 not only involve the sensor dynamics, but also the dynamics of the gas mixing apparatus, the pipes, and the measurement chamber. After the control signal to increase the propane concentration, it takes some time (less than one minute) until a concentration step (smoothed by mixing and gas flow) reaches the sensor. The associated dead times are, however, much smaller than the settling times visible in Fig. 5. Therefore, the sensor investigated exhibits the slow dynamics typical of chemical sensors in general. 

Globular and membrane proteins in aqueous solution or Hpid bilayers/ biomembranes could undergo a wide range of motions at ambient temperatures. Such motions can be characterized by several types of NMR relaxation parameters, chemical exchange, dynamic interference including SRI [27—29]. Detection ranges for motions by the respective parameters are schematically illustrated in Fig. 1.1 for solution and soUd state NMR approaches. Naturally, fast motions are solely examined by solution NMR techniques for any portions of proteins. The experimental approaches to be able to detect intermediate or slow motions are obviously different between the solution and solid state NMR methods. 

The standard theoretical treatment of chemical reaction dynamics is based on the separation of the total molecular motion into fast and slow parts. The fast motion corresponds to the motion of the electrons and the slow motion corresponds to the motion of the nuclei. The theoretical foundation for the separation of the electronic and nuclear motion was first developed by Born and Oppenheimer. In this approach, the total molecular wave function is expanded in terms of a set of electronic eigenfunctions which depend parametrically on the nuclear coordinates. The expansion coefficients are the... 

Recently attention has been focused in the chemical engineering literature on stiff systems. These are sets of differential equations that contain a mixture of very fast dynamic equations and very slow dynamic... 

As far as the dynamic properties of solid systems are concerned, NMR can cover a very large dynamical range It is possible to study processes on a time scale of 10 sec (indirect detection of a reaction kinetics) to processes on a time scale of lO" " sec (slow dynamic processes like conformational changes of a molecule or slow chemical reactions) and observe directly or indirectly molecular structures and their transformations on these time scales. 

Temperature Oscillation Calorimetry A more elegant way to estimate online the overall heat transfer coefficient without any additional measurement was developed by Carloff [ 11] by the technique known as temperature oscillation calorimetry, TOC. In this approach, the unknown product UA is computed from the analysis of the sine-shaped oscillations, which are superposed on either the reactor temperature or jacket temperature. The objective is to decouple the slow dynamic of the chemical heat production from the fast dynamic variable heat transfer during the reaction. The oscillations can be achieved either by adding a calibration heater to the system or by adding a sine signal to the set point of either T or Ty Figure 7.2 shows the evolution of the reactor and jacket temperatures in a reaction calorimeter where a sine wave temperature modulation was superimposed on the reactor jacket temperature. 

When integrating a differential equation numerically, one would expect the suggested step size to be relatively small in a region in which the solution curve displays much variation and to be relatively large where the solution curve straightens out to approach a line with a slope of nearly zero. Unfortunately, this is not always the case. The DDEs that make up the mathematical models of most chemical engineering systems usually represent a collection of fast and slow dynamics. For instance, in a typical distillation tower, the liquid mechanics (e.g., flow, hold-up) is considered as fast dynamics (time constant seconds), compared with the tray composition slow dynamics (time constant minutes). Systems with such a collection of fast and slow ODEs are denoted stiff systems. 

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