Relaxation analysis

The relatively low entry level instrumentation cost and the relatively simple experimental methods associated with GARField - both comparable to a standard bench-top relaxation analysis spectrometer as commonly used by the food industry, for example, for water/fat ratio determinations - offer potential advantages to the industrial based user. Indeed, the overwhelming majority of the applications development work described here has been carried out in collaboration with major multi-national industrial corporations such as ICI Paints, Unilever and Uniqema, with industry sponsored research laboratories and associations such as Traetek, and with a range of small-medium sized enterprises. 

The usefulness of NMR in such analysis is because the proton spin-relaxation time constants are different for different components, such as water, liquid fat and solid fat. For example, the signal from solid fat is found to decay rapidly while the liquid signals decay much slower. This phenomenon is the basis for an NMR technique to determine the solid fat content [20], However, as the relaxation time constant of water, for example, could depend on its local environment, such as protein concentration, it may overlap with that of oil and other components. As a result, it could be difficult to formulate a robust and universal relaxation analysis. It... 

The reaction is assumed to occur in a constant volume system. A relaxation analysis of the type employed for the first-order reaction leads to the following analog of 5.1.49. 

The binding of cations to heparin derivatives was also investigated by NOE and T2 relaxation analysis.21,22 The detailed effect of the presence of different cations was analysed, by differentiating the sulphation pattern of the several GAGs under analysis. NMR along with other techniques allowed concluding that the sequence alone does not define the conformation and flexibility for this class of molecules. Indeed, the associated cations must also be considered. Interestingly, the presence of particular cations... 

Auxiliary discrete dynamical systems and relaxation analysis 130... 

We summarize results of relaxation analysis and describe the algorithm of approximation of steady state and relaxation in Section 4.3. After that, several examples of networks are analyzed. In Section 5 we illustrate the analysis of dominant systems on a simple example, the reversible triangle of reactions A2<- A3 A. This simplest example became very popular for the lumping analysis case study after the well-known work of Wei and Prater (1962). The most important mathematical proofs are presented in the appendices. 

In this subsection, we summarize results of relaxation analysis and describe the algorithm of approximation of steady state and relaxation process for arbitrary reaction network with well-separated constants. 

To all intents and purposes, the Hopf prediction is the exact result. It is not difficult to construct an intersection between the maximum and minimum which is a stable stationary state. For instance, with y = 0.02, k = 0.12, and /i = 0.2 we have 0X = 1.042, 9C = 2400, and 0SS = i/k = 5/3. The corresponding phase plane nullclines are shown in Fig. 5.10, together with a trajectory spiralling in to the stationary-state intersection. The trace of the Jacobian matrix is negative for this solution (tr(J) = —4.1 x 10 2) indicating its local stability. This is not, however, a particularly fair test of the relaxation analysis because the parameters /i and k are not especially small. In the vicinity of the origin (where is small) both approaches converge. 

T2-relaxation analysis may be used to study the effect of ingredient composition on the properties of water in whippable emulsions16. In food systems non-exponential relaxation curves are often found. This can be accounted for by the presence of 2, 3 or more recognizable components representing species of hydrogen atoms with different mobility51. 

Complex dielectric susceptibility data such as those in Figure 15.6 provide a detailed view of the dynamics of polar nanodomains in rls. They define relaxation frequencies, /, corresponding to the e (T) peak temperatures Tm, characteristic relaxation times, r = 1/tu (where uj = 2nf is the angular frequency), and a measure of the interaction among nanodomains as represented by the deviation of the relaxation process from a Debye relaxation. Analysis of data on pmn and other rls clearly shows that their dipolar relaxations cannot be described by a single relaxation time represented by the Debye expression... 

The a relaxation analysis can be performed using the Havriliak-Negami model [70], The parameters corresponding to this analysis are summarized in Tables 2.9 and 2.10. 

Returning to the question of relaxation parameters, the relationship between NOEs and proton-proton distances is well established51 and can be worked out, at least semiquantitavely, when a full matrix-relaxation analysis is considered. 

The NMR-MOUSE is a portable NMR sensor which works in highly inhomogeneous magnetic fields. Because of field inhomogeneity NMR spectroscopy of the chemical shift is not readily possible, but relaxation times and parameters of translational motion can be measured by echo techniques. These are the most important NMR parameters which are exploited for contrast in imaging. Unless fluids are investigated field inhomogeneities are essentially no obstacle for relaxation analysis [80], because molecular motion by translational diffusion is absent. 

Ligand Solution conformation Charge/tautomeric state Solution dynamics Bound conformation Pharmacophore models 1D/2D NMR Chemical shift Line shape/relaxation analysis TrNOE All of the above, including 1D/2D of multiple ligands... 

Such a difference of the amorphous characteristic was accountable by NMR relaxation analysis. Figure 10 compares the FIDs for a series of solution-crystallised materials prepared from different prior polymer concentrations.40... 

In NMR work, spin-lattice relaxation measurements indicated a non-exponential nature of the ionic relaxation.10,11 While this conclusion is in harmony with results from electrical and mechanical relaxation studies, the latter techniques yielded larger activation energies for the ion dynamics than spin-lattice relaxation analysis. Possible origins of these deviations were discussed in detail.10,193 196 The crucial point of spin-lattice relaxation studies is the choice of an appropriate correlation function of the fluctuating local fields, which in turn reflect ion dynamics. Here, we refrain from further reviewing NMR relaxation studies, but focus on recent applications of multidimensional NMR on solid-ion conductors, where well defined correlation functions can be directly measured. 

The rigorous but unfortunately mathematically difficult approach to the problem of ionic clouds around moving ions is to seek the asymmetrical distribution functions and then work out the implications of such functions for the electric fields developed among moving ions. A simpler approach will be followed here. This is the relaxation approach. The essence of relaxation analysis is to consider a system in one state, then perturb it slightly with a stimulus and analyze the time dependence of the system s response to the stimulus. (It will be seen later that relaxation techniques are much used in modern studies of the mechanism of electrode reactions.)... 

Solution dynamics Line-shape/relaxation analysis... 

The temperature dependence of Tj (Fig. 13) points to a noticeable effect of the nature of adsorbent (phenolic oligomer) on the properties of adsorbed water. Firstly, the value of Ti (1-1.2 s) is nearly half that of free water (in 0.5 1). Secondly, relaxation curves sharply differ in 20- and 400-oersted fields. In the 20-oersted field the dependence Ti = f (T) is stepwise and the steepest part is observed near temperatures corresponding to the phase transition water - ice . The authors suggest that the minimum observed between 0 to -2 °C is connected with the dispersion of the relaxation time distribution. In order to confirm this assumption a classical relaxation analysis using deuterated water and the temperature dependence of longitudinal relaxation time is required. 

The relationship between NOEs and proton-proton distances is well established and can be worked out, at least semiquantitavely and also quantitatively, when a full matrix relaxation analysis is considered. The detailed study of the conformation and dynamics of a tetrasaccharide related to the LeX antigen provides a good example of this approach.215 NOE intensities are sensitive to the respective conformer populations, and that therefore, an indication of the population distribution when these molecules are free in solution and even in the protein-bound state may be obtained by focusing on key interresidual NOEs.216... 

A typical ECG is shown in Figure 4.4, where the P wave corresponds to atrial depolarization and contraction the QRST complex corresponds to atrial repolarization and relaxation and the onset of ventricular depolarization and contraction and the T wave represents ventricular repolarization and relaxation. Analysis of the size and shape of the waves can indicate abnormalities of the heart. 

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