Function interface distribution

If the scattering entities in our material are stacks of layers with infinite lateral extension, Eq. (8.47) is applicable. This means that we can continue to investigate isotropic materials, and nevertheless unwrap the ID intensity of the layer stack. To this function Ruland applies the edge-enhancement principle of Merino and Tchoubar (cf. Sect. 8.5.3) and receives the interface distribution function (IDF), gi (x). Ruland discusses isotropic [66] and anisotropic [67] lamellar topologies. 

Figure 8.25. The features of a primitive interface distribution function, g (x). The IDF is built from domain thickness distributions, ha (x) and hc (x), followed by the distribution of long periods, (x), and higher multi-thickness distributions...

Stribeck N (1980) Computation of the Lamellar Nanostructure of Polymers by Computation and Analysis of the Interface Distribution Function from the Small-Angle X-ray Scattering. Ph.D. thesis, Phys. Chem. Dept., University of Marburg, Germany... 

Radial) chord length distribution (CLD) (One-dimensional) interface distribution function (IDF) Image data format returned by image plate scanners... 

Lamellar morphology variables in semicrystalline pol5miers can be estimated from the correlation and interface distribution functions using a two-phase model. The analysis of a correlation function by the two-phase model has been demonstrated in detail before [30. 31]. The thicknesses of the two constituent phases (crystal and amorphous) can be extracted by several approaches described by Strobl and Schneider [32]. For example, one approach is based on the following relationship ... 

Figure Bl.9.12. The schematic diagram of the relationships between the one-dimensional electron density profile, p(r), correlation function yj(r) and interface distribution function gj(r).

This intensity can be used to calculate the correlation function (Bl.9.1011 and the interface distribution function (B 1.9.1021 and to yield the lamellar crystal and amorphous layer thicknesses along the fibre. 

A more detailed analysis of data from SAXS can be carried out by means of the interface distribution function gi(r), introduced by Ruland [30]. This function is the second derivative of the one-dimensional correlation function and... 

If enough data is acquired at large q values, and 4 can be determined using nonlinear least-squares fitting and, therefore, the ideal intensity. However, the problem arises in the selection of q limits, since the choice of both the lower and upper limits significantly affects the validity of the Porod s law. In this case, the upper limit was set to ( upp = 2, and the lower limit q was varied until the minimum area of the interference function G iq) was obtained (Eq. 19.53). Once this criterion is achieved, the interface distribution function gi(r) (Eq. 19.54) is calculated from the Eourier transform of G iq). 

Tj corresponds to the tth interface distance, and to its corresponding standard deviation. Interface distances and their distributions were obtained by means of a nonlinear least-squares fitting of the /ideaiC ) 0 to Equation (19.A.2). A weight factor of l//ideai( ) 0 was used. The periodicity or long period L is the sum of + r2- The resulting parameters are shown in Tables 19.A.1 and 19.A.2. Finally, the ideal intensity, the Lorentz intensity, the interference function, and the interface distribution function were... 

Figure 19.A.4 Interface distribution function. (See insert for the color representation of the figure.)...

Figure 19.A.8 Statistical effect on the interface distribution function.

In this model we assume the presence of two phases a crystalline phase (nanocrystal or NPs) and a semi-ordered phase (IL) [8, 28, 107, 108]. This model can be used since ILs are not considered as statistical aggregates of anions and cations but instead as a three-dimensional network of anions and cations i.e. polymeric supramolecular structures constituted from aggregates of the type [(BMI) (X),t )] [(BMI), (X)J] . In this context, the two-phase model was adopted to represent the nanocrystals dispersed in the IL [109]. Applying the interface distribution function to the experimental data, the extended molecular lengths of pure... 

Fig. 23.8. sPP crystallized at 115°C Left) Variation of the interface distribution function during a heating to the melt. The peak location gives the crystal thickness [18]. Right) DSC curves measured with different heating rates after the crystallization... 

Figwe 29 Top TEM micrograph of PET melt-crystallized at 200 °C. Bottom Temperature dependence of the characteristic lengths / and of semicrystalline PET, derived from the ID autocorrelation (CF) and interface distribution functions (IDF). The major component I2 is obtained from the difference of the computed Lb and fy The crossed line corresponds to an evaluation of /i CFfrom TEM data. With permission from Haubmge, H. G. Jonas, A. M. Legras, R. Macromolecules 20M, 37,126." ... 

Ruland W (1964) Crystallinity and disorder parameters in nylon 6 and nylon 7. Polymer 5 89-102 Ruland W (1977) Determination of the interface distribution function of lamellar two-phase systems. Coll Polym Sci 255 417... 

Another function used to obtain structure information is the one-dimensional interface distribution function, g(x) (18,19). This is simply the second derivative of fee onedimensional correlation Action, or g(x) = y"(x). This function gives the probability that two interfaces will be separated by a distance, x. In an ideal two-phase system, the phases would have constant d and L throughout fee scattering volume. The interfece distribution function would be a series of delta functions. Real polymo systems have a spread cf values of d and L. This causes g(x) to be a smooth curve wife broad peaks located at d, L-d, L, L+d, etc. The peak locations and feeir breadths can be analyzed, and it has been shown(18) that g(x) provides a more reliable estimation cf d and L than Y(x), when the material contains broad distributions of thicknesses. 

Rrojections of the SAXS pattern data from sample PEE 1000/57 according to eq 2 yield the scattering curves shown in fig 5(a). From these curves interface distribution functions (IDF) (fig 5(b)) are computed by use of eqs 5 and 6. 

Figure 4 PE Interface distribution function at the beginning and the end of an isothermal crystallization at 121 °C.

Different from s-PP, crystallites of polyethylene thicken with time. This was clearly demonstrated by time-dependent SAXS experiments carried out during an isothermal crystallization [6]. Fig. 4 shows a typical result, in a comparison of the interface distribution function K" z) obtained at the beginning and the final stage of the crystallization process. 

Figure 6 PE Interface distribution functions obtained after an isothermal crystallization at 124 °C and a subsequent cooling to 78 °C.

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