Relaxation time distributions

The T2 distribution has applications as diverse as petroleum geology and bread making. It distribution has been applied in the petroleum industry for many years to characterize rock cores to obtain pore size distributions in well-logging operations. Rock cores from oil wells are filled with water or oil. The NMR CPMG echo train is acquired in a TD-NMR instrument and the T2 distribution is obtained. This is essentially a mirror image of pore size distribution, as water in small pores is more restricted it is less mobile (short T. Water in large pores has more freedom to move (long Tj). 

Dedicated benchtop NMR analyzers for a variety of applications are available. These include an analyzer to determine fluoride in toothpaste quantitatively, such as the MQC from Oxford Instruments, a 23 MHz benchtop NMR, and another to determine water droplet size distribution in oil/water emulsions. Fluoride is often added to toothpaste as sodium fluoride or sodium mono-fluorophosphate to prevent tooth decay. The fluorine analyzer can determine fluorine at the level of a few hundred ppm. Toothpaste is squeezed into a glass sample tube and the quantitative determination of fluorine takes less than 1 min. The NMR method uses no solvents or reagents and is independent of the sample color and clarity, unlike the colorimetric methods and other instrumental methods such as ion chromatography (IC) that are used for this purpose. In the water droplet size distribution analyzer, droplets as small as 0.25 pm can be measured. The shelf life and palatability of products such as margarine, mayonnaise, salad dressings, and soft cheese depend on the size of 

Studies of rock cores from oilfields are used in the petroleum industry to assess the hydrocarbon content of rock strata and the ease with which the oil can be recovered. The GeoSpecT NMR Rock Core Analyser, a 2 MHz TD-NMR from Oxford Instruments, can provide information on the fluids in oil- and water-saturated rock including oil viscosity and clay-bound water, porosity, fluid distribution, and permeability. Rock cores can be studied at high temperatures and pressures and the instrument can provide diffusion and imaging information. 

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The velocity gradient leads to an altered distribution of configuration. This distortion is in opposition to the thermal motions of the segments, which cause the configuration of the coil to drift towards the most probable distribution, i.e. the equilibrium s configurational distribution. Rouse derivations confirm that the motions of the macromolecule can be divided into (N-l) different modes, each associated with a characteristic relaxation time, iR p. In this case, a generalised Maxwell model is obtained with a discrete relaxation time distribution. 

However, T2 is sensitive to the molecular interactions of spins and dependent on the molecular environment [60]. Thus, T2 may overlap for different components in certain materials and this technique alone may not be sufficient to identify the components. The relaxation time distributions are often broad, e.g., in meat [21], thus making it more difficult to associate the relaxation time constants with the components. 

The relaxation time for each pore will still be expressed by Eq. (3.6.3) where each pore has a different surface/volume ratio. Calibration to estimate the surface relaxivity is more challenging because now a measurement is needed for a rock sample with a distribution of pore sizes or a distribution of surface/volume ratios. The mercury-air or water-air capillary pressure curve is usually used as an estimator of the cumulative pore size distribution. Assuming that all pores have the same surface relaxivity and ratio of pore body/pore throat radius, the surface relaxivity is estimated by overlaying the normalized cumulative relaxation time distribution on the capillary pressure curve [18, 25], An example of this process is illustrated in Figure 3.6.5. The relationship between the capillary pressure curve and the relaxation time distribution with the pore radii, assuming cylindrical pores is expressed by Eq. (3.6.5). 

Fig. 3.6.6 Relaxation time distribution for different air/water saturations [20].

The concept of a T2 cut-off that partitions the relaxation time distribution between the pores which can be displaced and those that cannot does not always apply. An exception is when there is significant diffusional coupling between the micropores that retain water at a high capillary pressure and the macropores in close proximity to the microporous system [26, 27]. A spectral BVI model or a forward model has been suggested to interpret these systems [30, 31, 53]. 

NMR has proven to be a valuable tool for formation evaluation by well logging, downhole fluid analysis and laboratory rock characterization. It gives a direct measure of porosity as the response is only from the fluids in the pore space of the rock. The relaxation time distribution correlates with the pore size distribution. This correlation makes it possible to estimate permeability and irreducible water saturation. When more than one fluid is present in the rock, the fluids can be identified based on the difference in the fluid diffusivity in addition to relaxation times. Interpretation of NMR responses has been greatly advanced with the ability to display two distributions simultaneously. 

The dynamic mechanical experiment has another advantage which was recognized a long time ago [10] each of the moduli G and G" independently contains all the information about the relaxation time distribution. However, the information is weighted differently in the two moduli. This helps in detecting systematic errors in dynamic mechanical data (by means of the Kramers-Kronig relation [54]) and allows an easy conversion from the frequency to the time domain [8,116]. 

Fig. 10 Relaxation time distribution function/(r) describing the dielectric dispersion in relaxor PMN. The short timescale maximnm describes the glassy-type dynamics, whereas the long timescale part refers to the polar clnster dynamics. The same featnres are obtained in PMN, PLZT, and SEN relaxors...

The linear viscoelastic properties are often expressed in terms of an auxiliary function, the relaxation time distribution, H(x) H(x)dlnx is the portion of the initial modulus contributed by processes with relaxation times in the range lnt, InT + dlnt ... 

The mean times t and tw will be called the number-average and weight-average relaxation times of the terminal region, and tw/t can be regarded as a measure of the breadth of the terminal relaxation time distribution. It should be emphasized that these relationships are merely consequences of linear viscoelastic behavior and depend in no way on assumptions about molecular behavior. The observed relationships between properties such as rj0, J°, and G and molecular parameters provides the primary evidence for judging molecular theories of the long relaxation times in concentrated systems. 

The relaxation time distribution of the bead-spring models is discrete. The spectrum is... 

Other viscoelastic properties also acquire new characteristics at high concentration. In undiluted systems the long time end of the relaxation time distribution remains approximately Rouse-like for chains with molecular weight below Mc, as suggested by the agreement with reduced Rouse moduli in Fig. 5.2. 

The theory is not detailed enough to predict either numerical coefficients or the relaxation time distribution. Examination of the dimensional argument suggests that the form in Eq.(6.63) is probably not uniquely determined by the theory. 

Theoretical retardation and relaxation time distributions ought to obey Eq. (3.5). 

In reality, the data on isothermal contraction for many polymers6 treated according to the free-volume theory show that quantitatively the kinetics of the process does not correspond to the simplified model of a polymer with one average relaxation time. It is therefore necessary to consider the relaxation spectra and relaxation time distribution. Kastner72 made an attempt to link this distribution with the distribution of free-volume. Covacs6 concluded in this connection that, when considering the macroscopic properties of polymers (complex moduli, volume, etc.), the free-volume concept has to be coordinated with changes in molecular mobility and the different types of molecular motion. These processes include the broad distribution of the retardation times, which may be associated with the local distribution of the holes. 

The recommended procedure outlined above yields a value of (r) for each measured relaxation function. Other standard relaxation techniques measure different invariants of the relaxation time distribution. In order to compare results, the relationship between the various techniques must be determined38,40). 

In Refs. 80 and 81 it is shown that the Mittag-Leffier function is the exact relaxation function for an underlying fractal time random walk process, and that this function directly leads to the Cole-Cole behavior [82] for the complex susceptibility, which is broadly used to describe experimental results. Furthermore, the Mittag-Leffier function can be decomposed into single Debye processes, the relaxation time distribution of which is given by a mod-... 

Jaeger, F., Grohmann, E., and Schaumann, G. E. (2006). ll NMR relaxometry in natural humous soil samples Insights in microbial effects on relaxation time distributions. Plant Soil 280, 209-222. 

Dielectric spectroscopy was also used by the same group in order to study the local and global dynamics of the PI arm of the same miktoarm star samples [89]. Measurements were confined to the ordered state, where the dynamics of the PI chain tethered on PS cylinders were observed in different environments since in the SIB case the faster moving PB chains are tethered in the same point as the PI arm. The distribution of segmental relaxation times were broader for SI2 than SIB. The effect was less pronounced at higher temperatures. The PI normal mode time was found to be slower in SIB, when compared to SI2 although both arms had the same molecular weight. Additionally, the normal mode relaxation time distributions of the PI chains tethered to PS cylinders in the miktoarm samples were narrower than in P(S-h-I) systems of lamellar structure. 

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