Small-angle scattering of interacting particles. II. Generalized indirect Fourier transformation under consideration of the effective structure factor for polydisperse systems

Abstract

The indirect Fourier transformation (IFT) is the method of choice for the model-free evaluation of small-angle scattering data. Unfortunately, this technique is only useful for dilute solutions because, for higher concentrations, particle interactions can no longer be neglected. Thus an advanced technique was developed as a generalized version, the so-called generalized indirect Fourier transformation (GIFT). It is based on the simultaneous determination of the form factor, representing the intraparticle contributions, and the structure factor, describing the interparticle contributions. The former can be determined absolutely free from model assumptions, whereas the latter has to be calculated according to an adequate model. In this paper, various models for the structure factor are compared, e.g. the effective structure factor for polydisperse hard spheres, the averaged structure factor, the local monodisperse approximation and the decoupling approximation. Furthermore, the structure factor for polydisperse rodlike particles is presented. As the model-free evaluation of small-angle scattering data is an essential point of the GIFT technique, the use of a structure factor without any in¯uence of the form amplitude is advisable, at least during the ®rst evaluation procedure. Therefore, a series of simulations are performed to check the possibility of the representation of various structure factors (such as the effective structure factor for hard spheres or the structure factor for rod-like particles) by the less exact but much simpler averaged structure factor. In all the observed cases, it was possible to recover the exact form factor with a free determined parameter set for the structure factor. The resulting parameters of the averaged structure factor have to be understood as apparent model parameters and therefore have only limited physical relevance. Thus the GIFT represents a technique for the model independent evaluation of scattering data with a minimum of a priori information.

Multiresponsive Reversible Gels Based on Charge‐Driven Assembly

“…ABA triblock copolymers with negatively charged end blocks and a water-soluble neutral middle block (PSPMA 28 -PEO 230 -PSPMA 28 ) were synthesized by atom transfer radical polymerization (see the Supporting Information). [23][24][25][26] The molecular structure is given in Scheme 1 a. Poly(allylamine hydrochloride) 160 (PAH 160 ) was used as oppositely charged homopolymer (Scheme 1 b).…”

“…To investigate the coassembly of PSPMA 28 -PEO 230 -PSPMA 28 with PAH 160 in dilute solution, dynamic light scattering (DLS) titrations were used. We define a charge composition variable f + [Eq.…”

“…The complexes that are formed in these dilute solutions are flowerlike micelles with a hydrodynamic radius of approximately 20 nm. When PSPMA 28 -PEO 230 -PSPMA 28 and PAH 160 are mixed at higher concentrations, a highly viscous and transparent gel is formed spontaneously (movie available as Supporting Information). This result indicates that an interconnected micellar network is indeed formed.…”

“…Excellent agreement between experimental SAXS data and fits can be achieved by using a form factor for polydisperse (Gaussian) homogeneous spheres, in combination with an effective structure factor for the higher concentrations (Figure 3). Generalized indirect Fourier transform [27][28][29] analysis indicates that the scattering objects, the micellar cores, have a spherical shape with a radius of gyration R g = (8.1 AE 0.4) nm. For samples with more than 4 % (w/w) polymer, the structure factor becomes more pronounced, exhibiting a peak for the gel samples at q % 0.21 nm À1 .…”

Structures of the cylindrical and vesicular micelles of an P4VP-longer asymmetric PS-b-P4VP

“…The shape of p(r) resembles that of a vesicle with a monolayer wall. [26][27][28][29][30][31][32][33] The occurrence of minima in p(r) signifies the existence of regions of electron density with alternating signs. The alternate signs of the electron density can happen because of the low electron density in the middle of the vesicle.…”