Small-Angle X-Ray and Neutron Scattering

Scattering intensities of sufficient magnitude are only observed when the absolute values of the refractive index increments Y = dnjdc are more than 0.05 cm g. To a rough approximation, refractive index increments increase linearly with the refractive index of the solvent, the slope being given by the partial specific volume vi of the solute according to the Gladstone-Dale rule 

In general, refractive index increments of polymer solutions rarely exceed 0.2 cwl jg. For solutions where c = 0.01 g/cm therefore, even in the most favorable circumstances, the difference in the refractive index for solution and solvent is only 0.002 unit. To determine molar masses to 2%, it is necessary to know the refractive index increments to within 1 %, since they appear squared in Equation (9-76). The difference in refractive indices must therefore be known to better than 2 X 10. Because of temperature fluctuations during individual measurements, the refractive indices of solution and solvent are not measured separately instead the difference is measured directly in special differential refractometers. 

The theory of light scattering applies to all wavelengths. Consequently, it is also valid for small-angle X-ray scattering (SAXS) and neutron scattering (SANS). The form of Equation (9-76) remains the same in each case. Only the 

9 Determination of Molar Mass and Molar Mass Distributions 

According to Guinier, the scattering function for small-angle X-ray scattering measurements can be approximated by 

It is possible to evaluate M, A2 and (s )z from SAXS and SANS data in essentially the same way as for light scattering. By using the Guinier approximation defined by Equation (3.137) and the corresponding SAXS and SANS equivalents to Equation (3.143), (s )z can be determined without having to assume a molecular shape. However, the Guinier expression is satisfactory only when 1 and so its use to evaluate 

TABLE 3.6 Values of the reciprocal scattering vector q calculated from Equation (3.136) for some typical A, 0 combinations  

Synchrotron radiation sources enable X-ray radiation to be selected from wavelengths in the range 0.06 A 0.3nm, whereas in laboratory instruments the Cu Ka line (A = 0.154nm) is most commonly used. Since (s )z is invariably greater than 5nm for polymers, the Guinier region generally is inaccessible by SAXS and model assumptions are required in order to evaluate 

The use of thermal neutrons X about 0.1 nm) direct from a nuclear reactor for SANS imposes similar restrictions to those encountered with SAXS. However, by reducing the velocities of the thermal neutrons it is possible to produce cold neutrons X about 1 nm) which enable (s )up to about 20 nm to be determined by SANS in the Guinier region. Such measurements therefore complement similar measurements made by light scattering for which the lower limit of is about 15 nm using 

X = 488.0 nm. In order to gain the necessary contrast (i.e. A/ ) for SANS, use is made of the large difference between the neutron scattering lengths of hydrogen ( H) atoms and deuterium ( H) atoms. Thus, either a deutero-solvent or a deutero-solute is used. Since the and atoms act as individual point scatterers, it is possible to determine s Yz for specific blocks in molecules of block copolymers. This is achieved by preparing block copolymers in which only one block is of deutero-polymer and measuring the SANS in an appropriate solvent. 

9 Determination oj Molecular Weight, Molecular-Weight Distribution 

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See, for example, Feigin L. A. and Svergun D. I., Structure Analysis by Small-Angle X-ray and Neutron Scattering, Plenum Press, New York, 1987. 

Holt, C., de Kruif C.G., Tuinier, R., Timmins, P.A. (2003). Substructure of bovine casein micelles by small-angle X-ray and neutron scattering. Colloids and Surfaces A Physicochemical and Engineering Aspects, 213, 275-284. 

Kozielski, F., Svergun, D., Zaccai, G., Wade, R. H., and Koch, M. H. (2001). The overall conformation of conventional kinesins studied by small angle x-ray and neutron scattering. / Biol. Chem. 276, 1267-1275. 

As in binary surfactant-water systems considered previously, two constraints on the geometry of the surfactant interface are active a local constraint, which is due to the surfactant molecular architecture, and a global constraint, set by the composition. These constraints alone are sufficient to determine the microstructure of the microemulsion. They imply that the expected microstructure must vary continuously as a function of the composition of tile microemulsion. Calculations show - and small-angle X-ray and neutron scattering studies confirm - that the DDAB/water/alkane microemulsions consist of a complex network of water tubes within the hydrocarbon matrix. As water is added to the mixture, the Gaussian curvature - and topology -decreases [41]. Thus the connectivity of the water networks drops (Fig. 4.20). 

Rutherford backscattering spectroscopy Scanning electron microscopy Secondary ion mass spectroscopy Single crystal X-ray diffraction Small angle X-ray and neutron scattering Spark source mass spectrometry Transmission electron microscopy Voltametry... 

Polymer conformations are studied by various scattering experiments (light, small-angle X-ray and neutron scattering). These techniques are based on the contrast between the polymer and the surrounding media (solvent in the case of polymer solutions and other polymers in the case of polymer melts or blends). The contrast in light scattering arises from differences in refractive index between polymer and solvent, and the scattered intensity is proportional to the square of the refractive index increment dn/dc [see Eq. (1.86)]. 

Recent small-angle x-ray and neutron scattering studies revealed detailed information on the stmcture of a variety of micelles [94,95]. According to these studies, the thickness of the Stem layer is 6-9 A for cationic cetyltrimethylam-monium bromide (CTAB) micelles and anionic sodium dodecyl sulfate (SDS) micelles, whereas the palisade layer is about 20 A thick for neutral Triton X-100 (TX-lOO) micelles [94]. The radius of the dry, hydrophobic core of TX-lOO is 25-27 A and thus the overall radius of TX-lOO micelle is about 51 A The... 

Most of the research effort on the ionomers has been devoted to only a small number of materials, notably the ethylenes the styrenes, the rubbers(9)5 and those based on poly(tetra-fluoroethylene), the last of which is the subject of the present volume. As a result of these extensive investigations, it has become clear that the reason for the dramatic effects which are obsverved on ion incorporation is, not unexpectedly, the aggregation of ionic groups in media of low dielectric constant. Small angle X-ray and neutron scattering, backed up by a wide range of other techniques, have demonstrated clearly the existence of ionic... 

Small-angle X-ray and neutron scattering Chemical Methods Pore morphology, roughness of pore surface, and pore size distribution 27, 28... 

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