Scattering density profiles, structure factors

The normalized neutron scattering intensities, 1(Q), are related to the neutronscattering density profiles normal to the clay sheets, p(z), via the crystallographic structure factor, S(Q), as... 

Fig. la-d. Small-angle scattering from a dilute, random dispersion of membranes (vesicles), a corrected intensities or thickness factor obtained from the experimental intensity distribution 1(h) by multiplication with h. b Structure factor (amplitude function) with arbitrarily chosen signs (-k,—, +, —,).c Autocorrelation function of the electron density q(x) profile across the membrane obtained by cosine transformation of I,(h) (Eq. 5a) the insert shows the profile obtained by de-convolution. d Centrosymmetric electron density profile obtained by cosine transformation (Eq. 5b) of F,(h). From a study on lipoprotein X, an assembly of unilamellar vesicles (Ref. 84, with permission)... 

Fig. 4a-d. Scheme of diffraction patterns and Fourier transforms arising from a lamellar stack with centrosymmetrical unit cell, (a) The discrete maxima at the reciprocal spacings s = n/d sample the continuous scattering function of one lamella, which would be observed without stacking (dotted line), (b) The structure factor or amplitude function, (c) The Patterson function obtained by Fourier transformation of I,(s). (d) The electron density profile obtained by Fourier transformation of Fj(s). Adapted from Ref. 3, with permission... 

Thereby, (j) is the volume fraction of polymer in solution (as aggregates), Vc is the fraction of the crystallizable segment in the polymer, R is the lateral dimension of the lamellae (discs) and D x) denotes the Dawson function. The second term in Eq.5 arises from the polymeric structure of the brush (the "blob" scattering). P(Q) is the form factor of the density profile perpendicular to the lamellae surface including the contrast factors of the core and the brush parts and the density profiles of the polymer volume fraction. The form factor of an infinitely large plate of the thickness d considering a simple rectangular density profile is... 

Because these structure factors depend on the particular geometry of the substrate, there is no simple relationship between them and the polymer density profile (z). There is, however, one limiting case of interest, when the surface of the substrate is sharp and well-defined and when its curvature is smaller than the scattering vector q, then the surface can be considered as flat at the scale q . In this regime, the partial structure factors reach an asymptotic limit depending only on the structure of the DOlvmer layer and on the area per unit volume of the solid phase,... 

The neutron-scattering results then becomes particularly informative. One can vary systematically the scattering length density of the solvent, n by isotopic substitution, and from the corresponding intensity measurements separate the different partial structure factors. This gives access to the detailed polymer profile. 

Similarly, the correlation effects were described in terms of a paracrys-talline structure factor Eq. 41. The common wax copolymer structures were characterized in terms of an ensemble of finite-sized disks, each disk consisting of a wax- and polymer-containing lamellar similar to the description of the PEB-II/C24 or PEB-7.5/C36 two-dimensional aggregates Eq. 70. A common fit at both contrast conditions in terms of this model [14] provides a good description of the scattering profiles at 0 °C and rehable structural and density parameters of the wax-copolymer aggregates (see Table 12). 

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