Intensity functions reduced

The isolated intrachain scattering is used to determine the persistent local conformation by comparison between the scattering from single chains and the experimental data. The experimental data needs to be corrected for absorption, polarisation, multiple and incoherent scattering and in addition the intensities normalised to electron units, before quantitative intensity comparisons are possible. The s-weighted reduced intensity function si(s) is a better choice for such comparisons where ... 

The structure-dependent part of the diffraction curve, the reduced intensity function, t(s), is obtained by subtracting the independent coherent scattering from the normalized experimental values (Fig. 2) ... 

The reduced intensity observed in an X-ray diffraction experiment corresponds to the sum over the different partial structure functions, each weighted by the product of the scattering factors for the two atomic species involved. In an aqueous solution of a salt MX, which contains four atomic species M, X, O, and H, the number of different pair interactions is ten and the reduced intensity function can be written ... 

Often, the integrand in equation (10.20) is called the reduced intensity function ... 

The last term in (4.5) represents the unobservable null scattering or the scattering from the sample as a whole (see Section 1.6), and it will be ignored in the discussion from now on. At this point we define the interference function (or reduced intensity function) i(q) by... 

Fig. 20. (a) Experimental, s-weighted, reduced-intensity function for random co-polyester X. The broken contours are negative and the extrusion axis is... 

Fig. 4. The s weighted reduced intensity function si (s) for poly( -butyl methacrylate) obtained by subtracting the independent scattering (dashed line) of Fig. 3 from the fully corrected data function / (.r) (full line) and weighting with the scattering vector.

Fig. 9. The s weighted reduced intensity function for a 1 mm thick sample of crepe rubber at 56°C and at 24 C.

If we "build" molecular models in a computer, the relationship between atom coordinates and the x-ray scattering intensity is straightforward and the reduced intensity function i (s) may be calculated by... 

Fig. 10. The s weighted reduced intensity functions si (s) calculated from random single chain...

Fig. 12. The s weighted reduced intensity functions si (s) calculated for a complete random chain model. The intrachain structure is that used in Fig. lOe. The interchain interactions are modeled as if they arise from randomly packed spheres. The curves are calculated using a distribution of sphere sizes. The numbers stated on the curves give the standard deviation of that distribution. The sphere diameter distribution is centered on 5.3 A. The packing density of the spheres is 0.6.

Fig. 17. Hie s wei ted reduced intensity function si (s) for ir-PoMS obtained from the data of Fig. 14 using the procedures described in Section 2.

Fig. 19. The s weighted reduced intensity function si (s,a) for an oriented sample of PaMS measured at room temperature. The sample was extruded in a plane strain compression device to an extenstion ratio of 2. a is the angle between the extension axis and the normal to the plane containing the incident and scattered paths (see ref. 12). The dashed lines represent negative values of the intensity function.

Fig. 21. The s weighted reduced intensity functions si (s) calculated for random isotactic chains of PaMS, in which the rotation states, and their distribution are according to the scheme of Sundararajan. The valence angle about the substituted skeletal carbon atom is 110, while that about the unsubstituted carbon atom is set at 110°, 122°, or 128. 122 corresponds to the value chosen by Sundararftjan,3 The full lines represent scattering functions calculated for chains in which the trans state has been assigned a value of O , while dashed lines correspond to chains for which the trans state was 10°. 

Fig. 22. The s weighted reduced intensity functions si (x) calculated for random chains of isotactic PaMS, in which the rotation states and their distribution are assigned by Sundararajan except that (a) the value of the E(gauche) - E trans) difference is varied as indicated on the curves from 0.0 to 2.0 kcal/mol, where 0.5 kcal/mol corresponds to the value selected for a random coil chain by Sundararajan, and (b) the valence angles are assigned (as described in the caption to Fig. 21) as the 110°/122 dashed line and as the 110 /128 full line.

There is a range of thermoplastics whose chemical configuration consists of phenylene units bridged by a variety of groups. Table II indicates the stracture of the polymers considered. Figure 23 shows the reduced intensity functions si (s) for these six polymers. Although there is considerable variety in the shapes and relative intensities of the peaks within these curves, they have several features in common. First, they exhibit an intense peak at s 1.5 with other weaker peaks at j = 3 and 5.5 A l. A number of these polymers have been considered in detail,and their intrachain, orientational, and spatial parameters evaluated. Here we would like to focus our attention on one particular feature also common to all these si (s) curves, namely the shoulder or feature at 5 = 1.9 A l. In particular we shall consider the scattering curve for polyphenylene sulfide (PPPS). 

Fig. 23. The reduced intensity function si s) for a range of phenylene tfaennoplastic polymers obtained in their noncrystalline state. The meaning of the abbreviations is detailed in Table II. 

Fig. 24. Hie weighted reduced intensity functions si (s) calculated for PPPS models compared with the experimental si (s) function (bottom curve). The top curve marked "1 segment" is that appropriate to a random single chain model as analyzed in ref. 34. The curve marked "2 segments" is that for a random chain, but in which additionally only pair-wise face-to-face correlations between the phenyl units of the polymer chains have been introduced. The third curve corresponds to a similar correlation but extended over three phenyl units.

Fig. 25. The s weighted reduced intensity functions si (s) calculated for parallel chain models of PPPS and compared with the experimental si (s) curve. The chains are packed on a disordered five-fold grid and contained within a spherical radius of 7.S A. The models from which the curves are obtained exhibit differing levels of rotational and longitudinal correlations as marked on the curves.

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