Relaxation in heterogeneous systems

In heterogeneous systems, the spin-lattice relaxation behaviour (T and/or is often found not to conform to a simple single exponential, but rather to exhibit behaviour that can be represented as a superposition of exponential processes. If we consider the simple case of a mixture of several discrete homogeneous components, each conforming to exponential relaxation as previously described, then it follows that the overall relaxation behaviour of the mixture can be described by 

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Before attempts are made to explain the origin of the effects reported here, other examples of complex relaxation must be considered, to determine whether a generalised approach to water proton relaxation in heterogeneous systems can be developed. 

Specific Features of Solvent Relaxation in Heterogeneous Systems... 

Magnetization transfer (MT) or cross-relaxation spectroscopy, also termed Z-spectroscopy by Grad and Bryant (1990), has been extensively used to obtain information on the spectral Uneshape of protons in macromolecules as well as on relaxation in heterogeneous systems (Calucd and Forte 2009). A MT spectrum was obtained by performing saturation transfer (ST) experiments at many discrete frequencies and plotting the ratio of the steady-state water signal with and without RF saturation as a function of the saturation frequency offset. For spatially homogeneous samples,... 

R. Lamanna, On the inversion of multicomponent NMR relaxation and diffusion decays in heterogeneous systems. Concepts Magn. Reson., 26A(2), 78-90 (2005). 

However, a single Tt is generally observed in heterogeneous systems partly because it is difficult to resolve T2 components unless they differ by at least a factor of two. Deuterium T2 relaxation times in crosslinked resins, also exhibit only a singleexponential relaxation process, indicating that different 2H in the same molecule have essentially the same spin-lattice relaxation times44). 

There a been a number of interesting applications of the framework developed in the studies of the simple ions were MD simulations of the quadrupolar relaxation has been performed on counterions in heterogeneous systems. Studies of a droplet of aqueous Na embedded in a membrane of carboxyl groups [54], showed that the EFG was strongly effected by the local solvent structure and that continuum models are not sufficient to describe the quadrupolar relaxation. The Stemheimer approximation was employed, which had been shown to be a good approximation for the Na ion. Again, the division into molecular contributions could be employed to rationalize the complex behavior in the EFG tensor. Similar conclusions has been drawn from MD simulation studies of ions solvating DNA... 

Despite advanced methods of crystal growth, sohds still contain lattice defects at concentrations around lO -lO " cm, and typical concentrations of defects and impurities in commercial samples are around 10 cm . The latter concentration is usually much greater than the concentration of photogenerated free charge carriers in solids under moderate photoexcitation. Consequently, defects are expected to play an important role in photoexcitation and relaxation processes in heterogeneous systems. In fact, defects create a local distortion of the periodic potential in the solid s lattice. 

Halle, B., Carlstrom, G., Anderson, T., Wennerstrom, H., and Lindman, B. NMR of water nuclei in heterogeneous systems-relaxation theory and oxygen-17 data from aqueous solutions of proteins, polyelectrolytes and micelles. Biophysics of Water, E Franks, ed., Wiley, New York, pp. 221, 1982. 

The previous subsections have considered homogeneous solutions and gels. We now address water relaxation in heterogeneous particulate suspensions, pastes and powders. We focus on the starch-water system since starch is widely used as a thickener, texturizer and emulsification aid in the food industry. 

A considerable amount of valuable information was nevertheless provided by solid state NMR over many years, and the progress of the technique has been charted by periodic reviews of the area [2-5]. In the last decade however, there have been developments that have significantly increased the information that can be extracted. Among these have been the development of a much greater understanding of the way spin diffusion affects the observed relaxation behaviour in heterogeneous systems, and the use of experimental techniques that allow the manipulation and detection of the magnetisation on the basis of chemical shift (either chemical shift or chemical shift via cross-polarisation). These have effectively added a new dimension to the classic relaxation techniques. 

The thermally stimulated relaxation in heterogeneous dielectrics, consisting of relaxing components and exhibiting interfacial relaxation, presents a special problem, which has been solved exactly only for bilayer systems [2]. Here, the accumulation of space charge at the interfaces may cause anomalous depolarization effects (currents of reversed polarity). The same is true for samples measured with air gap or with one-sided electrodes (this latter is frequently used for corona charged samples). 

Ionic relaxation It comprises ionic conductivity and interfacial or space-charge polarization. Ionic conductivity is the dominant mechanism at low frequencies and high temperamres. Interfacial polarization occurs when charge carriers are trapped at interfaces in heterogeneous systems or when they are trapped by the electrode surface (electrode polarization (EP)). 

Up to now it has been tacitly assumed that each molecular motion can be described by a single correlation time. On the other hand, it is well-known, e.g., from dielectric and mechanical relaxation studies as well as from photon correlation spectroscopy and NMR relaxation times that in polymers one often deals with a distribution of correlation times60 65), in particular in glassy systems. Although the phenomenon as such is well established, little is known about the nature of this distribution. In particular, most techniques employed in this area do not allow a distinction of a heterogeneous distribution, where spatially separed groups move with different time constants and a homogeneous distribution, where each monomer unit shows essentially the same non-exponential relaxation. Even worse, relaxation... 

Dynamic parameters for heterogeneous systems have been explored in the liquid, liquid like, solid like, and solid states, based on analyses of the longitudinal or transverse relaxation times, chemical exchange based on line-shape analysis and separated local field (SLF), time domain 1H NMR, etc., as summarized in Figure 3. It is therefore possible to utilize these most appropriate dynamic parameters, to explore the dynamic features of our concern, depending upon the systems we study. 

This picture is usually known as the heterogeneous scenario. The distribution of relaxation times g (In r) can be obtained from < (t) by means of inverse Laplace transformation methods (see, e.g. [158] and references therein) and for P=0.5 it has an exact analytical form. It is noteworthy that if this scenario is not correct, i.e. if the integral kernel, exp(-t/r), is conceptually inappropriate, g(ln r) becomes physically meaningless. The other extreme picture, the homogeneous scenario, considers that all the particles in the system relax identically but by an intrinsically non-exponential process. 

In the case under consideration different physical structures were realized due to the formation of the polymer network in the surface layers the filler surface, as usually happens in filled systems. As is known79, this induces considerable changes in the structure of the material. It is also possible that in these conditions a more defective network structure is formed. These results show that even the purely physical factors influencing the formation of the polymer network in the interface lead to such changes in the relaxation behavior and fractional free-volume that they cannot be described within the framework of the concept of the iso-free-volume state. It is of great importance that such a model has been devised for a polymer system that is heterogeneous yet chemically identical. 

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