Weibullian function

Generally, to fit the observed FID, a series of exponential functions (Equation (1)) are used because the distribution of dipole interaction is expressed by Lorentzian function. This is true for the solution, melt and amorphous phases of the polymers. Actually, a PE melt with low MW exhibits a single exponential curve.14-17 The shape of the relaxation curve of amorphous molecular motion still retains the combined exponential types on cooling. On the other hand, Weibullian functions (Equation (2))6 18 are also applicable for the phase with partially restricted motion such as the interfacial phase.19 Therefore, it is reasonable to introduce the exponential and Weibullian functions as the amorphous relaxation ... 

Weibullian/sine function is introduced. Applications of this method to polyethylenes prepared by various conditions successfully explain the morphology and physical properties. Crystallisation mechanism of nylon 46 by high-resolution NMR in the solid state is also introduced. 

The introduction of Gaussian function in Equation (3) is based on the prediction by Pake23 for crystals of small molecules, such as 1,2-dichloroethane, where the dipole pairs are isolated from each other. However, in the polymeric materials, not only the dipole interactions between the nearest neighbours but also those between the proton pairs at a longer distance contribute to the FID profile shape. Therefore, the distribution of the dipole interaction may be distorted. This is expressed by the introduction of Weibullian form,24 as follows ... 

It has been known for some time that the FIDs of such systems often decay in a way that is well represented for the most part by a Weibullian [60,61]. Arguments based on consideration of correlation functions suggested that the FID of high molecular weight polydimethyl siloxane in the melt should decay for the most part with a Weibullian power of between 1.25 and 1.5 [62], and the existence of residual static dipolar interactions in these systems was confirmed by the existence of the pseudo-solid echo [63]. This reference forms part of a much larger body of work on such systems by Cohen-Addad and co-workers which it is beyond the scope of this chapter to cover in any detail, but interested readers are directed to literature such as [64] and [65]. 

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