Half-Gauss Distribution

As an example of a distribution with a smaller number of long chains than the Flory distribution, a half-Gauss distribution is chosen,... 

Fig. 47 Ultimate strength of PpPTA fibres versus the degree of polymerisation applying a half-Gauss distribution of chain lengths for various values of the diameter 2r... 

For the transformation of the macrocomposite model to a molecular composite model for the ultimate strength of the fibre the following assumptions are made (1) the rods in the macrocomposite are replaced by the parallel-oriented polymer chains or by larger entities like bundles of chains forming fibrils and (2) the function of the matrix in the composite, in particular the rod-matrix interface, is taken over by the intermolecular bonds between the chains or fibrils. In order to evaluate the effect of the chain length distribution on the ultimate strength the monodisperse distribution, the Flory distribution, the half-Gauss and the uniform distribution are considered. 

Fig. 44 Flory distribution, half-Gauss and uniform chain length distribution for an average DP of 100 monomeric units (m.u.)...

Fig. 54 Ultimate strength of PpPTA versus the degree of polymerisation for a rod diameter of 0.5 nm. Comparison of the results calculated for the Flory, the half-Gauss and the uniform distributions...

Figure 10 shows linearized DISPA plots for two simulated non-Lorentzian line shapes. For example. Figure 10a shows the DISPA deviations for a line shape consisting of an unresolved doublet of two Lorentzians of different resonant frequency. The maximum (upward) displacement from the reference line clearly occurs at frequencies less than one half-width away from the observed absorption maximum. In contrast. Figure 10b shows that for a log-Gauss distribution in relaxation time, the maximum displacement (downward this time) occurs at a frequency approximately equal to the half-width at half-maximum height of the observed absorption peak. 

Figure 10. Plots of DISPA radius squared versus square root of frequency. Frequency is measured in units of half-width at half-maximum observed absorption peak height, so that no assumptions about line shape need be made. The horizontal line is for a reference Lorentzian line shape, (a) Composite spectrum consisting of two Lorentzians of equal width and intensity, separated by0.6/r, 0.8/t, 1.0/t, and 1.2/t, respectively, (b) Composite spectrum resulting from a log-Gauss distribution in relaxation time, with distribution parameter = 0.5, 1.0, and 1.5, respectively. The (circular) DISPA plots for (a) and (b) are shown in Figures 3 and 5, respectively. [Taken from ref. 12.]...

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