Echo attenuation function

Fig. 9. Echo attenuation function E q, ) as a function of q for a close-packed suspension of 9.870 im polystyrene spheres surrounded by water. The times A are 10 ms (circles), 20 ms (solid circles), 20 ms (squares), and 40 ms (solid squares). The solid lines represent fits using Eq. (110) for which the parameters are a = 3.0 /xm, b = 10.7 /xm, f = 0.1 /xm, and [Reproduced by permission from Coy and Callaghan, 1994.]...

Assuming that (5.4.42) is also valid for finite distances and times, and representing ID displacements parallel to the gradient direction by Z = zi — Zi), it can be shown that the echo attenuation function for a lattice of pores in which there is no correlation between lattice spacing and pore size is given by... 

The idea that the echo attenuation function could be inverted to yield an image of the average propagator is relatively new and of special value in... 

If such an analytical formalism is not available, the mean squared displacement, i.e., the second moment of the propagator, can be evaluated approximately from the experimental echo attenuation function by restricting oneself to the low wave number limit. The phase factor given in Eq. 2 can be expanded according to... 

In the situation where V2DA is of the same order or larger than the distance between any diffusional barriers in the system, so-called restricted diffusion is observed. In a W/O emulsion, for example, the water molecules are restricted in the extent of their diffusion by the presence of the boundaries of the water droplets. The extent of the restriction of the diffusion of the water molecules is reflected in the ratio R = E /E. An expression for the echo attenuation R-factor as a function of droplet diameter has been derived by Murday and Cotts for uniform spherical droplet sizes [7] ... 

Eq. 1 showed that in the case of unrestricted diffusion the echo attenuation value R depends upon the durations 8 and A. This is also true in the case of restricted diffusion, although in a different manner. The dependence of the R-value upon these two parameters is shown in Fig. 5. This figure clearly shows that the echo attenuation factor R steadily decreases with increasing A in the case of unrestricted diffusion, but becomes independent of this parameter in the case of restricted diffusion. It may be deduced from this figure that it is necessary to determine the parameters of the log-normal droplet size distribution R as a function of A or by measuring R as a function of 8 for a fixed large value of A. Measurement of only one R-value, at a chosen 8 or A, is not sufficient for a careful determination of the droplet size distribution in Fig. 5 a given In R-value can be found on more than one In R versus A-5/3 curve. This means that the In R-values have to be determined for different values of A and/or 8. 

Fleisher et al. [12] studied the self-diffusion of oil and water in rape seeds. The selfdiffusion of oil was found to be completely restricted. The experiments could be explained in toms of the model of diffusion within spherical droplets and a Gaussian mass distribution of the droplet radii. At the same time Van den Enden et al. [9] introduced the technique described above. It is a rapid method for the determination of water droplet size distributions in spreads by using low resolution pulsed field gradient NMR. Their method was based on the recognition that a set of echo attenuation values (R) as a function of the field gradient pulsed width, obtained under conditions where R is independent of the time allowed for diffusion, contains all the necessary information on the water droplet size distribution (see above). A log-normal distribution of water droplet sizes was assumed. 

This is directly evidenced by the echo attenuation of the proton resonance in PFG-NMR experiments of phosphonic acid functionalized oligomer [104], Only the echo of the phosphonic protons is attenuated ivhile the echo of the oligomer protons is only slightly affected by the magnetic field gradient (Fig. 23.9). The reader may recall that complexation of phosphoric acid and a basic polymer does not show any sign of this effect (see Fig. 23.7), which opens the way to the development of true single ion conductors. 

Fig. 26.3. H-difTusion coefficient measured on TSA.28H2O by the PFG-NMR technique , (a) Echo attenuation as a function of applied magnetic field gradient showing separation of the contributions from the mother liquor and the solid, (b) The resultant diffusion coefficient as a function of the reciprocal temperature the diffusion coefficient calculated from the proton conductivity measured by a.c.-impedance spectroscopy is given for comparison (see Chapters 29-31).

Stejskal and Tanner devised a quantitative relationship of the echo attenuation as a function of gradient strength, the Stejskal-Tanner equation [71] ... 

Fig. 2 The echo attenuation as a function of — 5/3) for different radii of emulsion droplets (according to Eq. (4)) with A =0.100 s,...

Fig. 44. Echo attenuation curves for polyethyleneoxide, M = 11,200, confined to PHEMA pores at 80 °C as a function of the squared wave number, k, for different diffusion times. The solid lines represent a fit of Eqs. 76-79 to the experimental data. The tube diameter is found to be a=(9 l) nm as the only fitting parameter. Other parameter values such as N=MJ853, Do=9.66x10 ° m /s, and =8.38x10 ° m were taken from the literature [180, 187]...

The reproducibility of spin-echo measurements is a sensitive function of the temperature stability of the sample during measurement. Changes in temperature affect both the unattenuated echo height (mainly via T2) and the degree of attenuation (via D). 

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