Vonk model

The details of the lamellar morphology, such as crystalline layer thickness l and amorphous layer thickness /, are quantitatively evaluated using SAXS [18]. For example, they are conveniently derived from the one-dimensional correlation function, that is, Fourier transform of SAXS curves, assuming an ideal lamellar morphology without any distribution for and l [19]. More detailed information on the lamellar morphology can be obtained by fitting theoretical scattering curves (or theoretical one-dimensional correlation functions) calculated from some appropriate model to SAXS curves (or Fourier transform of SAXS curves) experimentally obtained. The Hosemann model in reciprocal space [20] and the Vonk model in real space [7,21] are often employed for such purposes. 

The Vonk model is characterized by an infinite number of alternating units of crystalline and amorphous layers. The one-dimensional correlation function y(x) derived from this model is expressed as... 

Another model for the electron density gradient was proposed by Blundell and analyzed by Vonk it consists of a linear density change in the interface [21, 28]. In this model, called the geometric linear model, the smoothing function is of rectangular type (Fig. 19.10) and its Fourier transform is given by... 

Important morphological parameters such as the long period (I), crystal thickness (Ic), and amorphous layer thickness (la) of semipolymer melts and blends can be determined using SAXS via two different approaches. In the first approach, standard models such as the Hosemarm-Tsvankin [23] and the Vonk-Kortleve [24,25] for lamellar stacks are fitted to data obtained for the SAXS profile. The second approach is based on performing a Fourier transform for the SAXS profile to produce a one-dimensional correlation function, y(z) (which is Fourier transform of the measured I(q) in SAXS) or an interphase distribution function (IDF) in real space. 

Dijkstra, M., Roelofsen, H., Vonk, R.J. and Jansen, R.C., Peak quantification in surface-enhanced laser desorption/ionization by using mixture models. Proteomics, 6, 5106-5116 (2006). 

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