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Compressed Exponential Response Function Arising From a Continuous Distribution of Gaussian Decays – Distribution Characteristics


A compressed exponential function (CEF) is shown to be represented by a distribution of Gaussian functions. The properties and characteristics of the distribution are detailed and discussed, and are illustrated with reference to the change in the NMR spectroscopy proton Hahn echo relaxation response that takes place in a composite material (prepared by melt compounding a mixture of graphite nanoparticles and pyromellitic anhydride modified polypropylene carbonate (PPC)) during aging at 90 °C. The results show that both the width and the average value of the spin‐spin relaxation rate distribution decrease during aging, suggesting that the molecular motion becomes less constrained and less heterogeneous.

An exponential function S(x;q) = exp [−(s*x)q], with x being a (dimensionless) independent variable and s* and q (1 < q < 2) being constants, is denoted a compressed exponential function (CEF) and is represented by a sum of Gaussian functions, that is, , which, in the limit of infinite n, becomes the Laplace transform of I(s). The dependence of I(s) on q is illustrated.

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Authors:   Eddy W. Hansen, Xiaoliang Gong, Qun Chen
Journal:   Macromolecular Chemistry and Physics
Year:   2013
Pages:   n/a
DOI:   10.1002/macp.201200715
Publication date:   31-Jan-2013
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