Function chord distribution

The multidimensional chord distribution function (CDF) for oriented materials (in particular useful for the study of materials with uniaxial orientation, i.e., fibers) (Sect. 8.5.5)... 

Computational methods combined with a novel approach in the application of scattering physics were recently employed by Barbi et al. in a synchrotron SAXS study of the nanostructure of Nafion as a function of mechanical load. A new method of multidimensional chord-distribution function (CDF) analysis was used to visualize the multiphase nano-... 

Complete information about the specimen would be available only by tomographic methods with a stepwise rotation of the sample (see e.g. Schroer, 2006) or using inherent symmetry properties of the sample. Under the assumption of fibre symmetry of the stretched specimen around the tensile axis, from the slices through the squared FT-structure the three-dimensional squared FT-structure in reciprocal space can be reconstructed and hence also the projection of the squared FT-structure in reciprocal space. The Fourier back-transformation of the latter delivers slices through the autocorrelation function of the initial structure. Stribeck pointed out that the chord distribution function (CDF) as Laplace transform of the autocorrelation function can be computed from the scattering intensity l(s) simply by multiplying I(s) by the factor L(s) = prior to the Fourier back-... 

Figure 14.25. Reconstructed SAXS patterns (oriented scattering) of two UDP MFC materials and their respective chord distribution functions. Fibril axis is vertical. The CDF function is presented in absolute values (both positive and negative faces in one image)...

The evaluation of the chord-distribution function h is easily demonstrated when the volume V has a simple shape. As an example, consider a slab of width a (see Fig. 7.11). Let chords of length a make angle 0 with the x axis, and let /i = cos d. Then for any chord a with inclination we have Q n = cos m and with dQ = dfi (7.228) reduces to... 

Stribeck, N. and Fakirov, S. (2001) Three-dimensional chord distribution function SAXS analysis of 48 the strained domain stmcture of a poly(ether ester) thermoplastic elastomer. Macromdecules, 34, 7758-7761. 

In addition to item 2, carry out edge enhancement in order to visualize structure by means of the chord distribution function (CDF), z(r), and interpret or fit it. 

Stribeck N and Fakirov S (2001) Three-Dimensional Chord Distribution Function SAXS Analysis of the Strained Domain Structure of a Poly(ether ester) Thermoplastic Elastomer, Macromolecules 34 7758-7761. 

Figure 1.8 Dynamic load-reversal mechanical test of hard-elastic iPP film at a strain rate of c 10" s". The following are shown as a function of elapsed time t top - elongation middle - long period L (solid line), lateral extension of a sandwich made of two crystalline lamellae (broken line), and strength of the chord distribution function (dotted line) bottom - tensile stress. Vertical bars indicate zones of strain-induced crystallization (black) and relaxation-induced melting (gray). Stribeck et al. [32]. Reproduced with permission of John WUey...

The multidimensional chord distribution function (CDF) is an advancement of the interface distribution function. CDF is adopted to the study of highly anisotropic materials. The advantage of the CDF is that the only required assumption is a multiphase structure [46, 60], This assumption is correct for most of polymeric materials. Thus the structure of the material is revealed in real space without any adjusting parameters. The main prerequisites for computing the CDF are fiber-symmetr y and completeness of the data. 

Multidimensional chord distribution functions [13, 14] (CDF) visualize the nanostructure of the studied materials and its evolution in real space. Figs. 4.8, 4.9, and 4.10. The general course of the peak intensities in the CDFs is identical to the development observed in the SAXS patterns. The materials exhibit an initial increase of peak intensities that is followed by a decrease for higher strain. 

Fig. 4.8 Tensile test of TPU 205. Chord distribution functions (CDF) computed from SAXS. The pseudo-color fiber patterns z r 2,r ) show the region —50nm < ri2, rs < 50nm. Straining direction is vertical. Pattern intensities on a logarithmic scale. Cross-head speed is 2 mm/s. A white border indicates the region used in Figs. 4.12, 4.13, and 4.14...

Fig. 4.10 Tensile test of TPU 235. Chord distribution functions (CDF) computed from SAXS. The pseudo-color fiber patterns z(n2, show the region —50nm < r 2, < 50nm. Straining...

Fig. 5.4 Absolute values z(fn, rs)] of chord distribution functions (CDF) of PP, a blend (+MMT) and two composites (+lcMMT, +hcMMT) as a function of the local strain s. Straining direction is vertical. The images are on the same logarithmic scale. In a repetitive pseudo-color representation the images show the region —60 nm < rn, f3 < 60 nm of the patterns computed from SAXS data by a special Fourier transform...

Figure 12. Bristle of neat EM400. Chord distribution function (CDF, logarithmic scale) computed from 2D SAXS pattern. Fiber direction is vertical, (a) Positive contours describing the domains from the nanostructure, (b) Negative contours describing the lattice properties 22]...

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