SANS Cross-sections

Using a mixture of suitable amounts of deuterated and protonated solvent molecules, in general it is possible to achieve zero contrast, ApM = 0, for one component M of the system. This technique of contrast matching and variation is of essential importance for the investigation of complex multi-component systems such as A-B block-copolymer aggregates and the associated wax molecules. 

The contribution to the scattering intensity from some object described by its scattering length density distribution Ap(r) is  

With neutrons it is comparatively easy to obtain the scattering data on an absolute scale normaUzed to the sample volume. This is done normally by a comparison with scatterers where the absolute cross-section is known. For solutions, V sampie is taken as the volume of the dissolved polymer  

We first consider the cross-section for one object where we have  

Here Vo is the volume of the object. Furthermore we have assumed that the object is homogeneously decorated with a constant contrast factor, Ap. If N of these objects are located at random positions and random orientations in a solution, then all interferences between the objects average out to zero and their scattering intensities add up to N dao/dQ. A given polymer volume V in solution that splits into N aggregates yields Vo = V/N. Wth that we have  

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SANS experiments have indicated that blends of high density (linear) and long-chain branched low density polyethylenes (HDPE/LDPE) are homogeneous in the melt, though the components may separate on slow cooling due to the difference in crystallization mechanisms [43]. The semicrystalline blends form effectively two-phase systems in the solid state, and it was shown [43,44] that the Debye-Bueche (DB) [45,46] model was appropriate to describe the morphology, with a SANS cross section of the form... 

TABLE 23.6. Comparison of measured and calculated values of the absolute SANS cross section at Q = 0 for carbon-polyethylene composite materials. 

Fig. 29 Upper panels. Neutron scattering intensities obtained from a CPCl/Hex nematic micellar solution in the (qv,qvv)-plane at different shear rates (a) j =0.94 s , (b) y=34.5 s ,and (c) 7= 250 s . The nematic phase was made in deuterated water at concentration c = 35.2wt.% and molar ratio [Hex]/[CPC1] = 0.49 [301], Lower panel. Shear rate dependence of the orientational order parameter y (Eq. 5) obtained from the SANS cross-sections shown in the upper panel. The continuous line is a guide for the eyes. The increase of the order parameter was interpreted in terms of a transition between the tumbling and the flow-alignment regimes [290,303]...

Fig. 6.7 SANS pattern from a junction-labelled PEP-PEE triblock copolymer at different temperatures. Dashed lines cross-section from the PEP label. (Reprinted with permission from [284]. Copyright 2002 EDP Sciences)...

Berkowitz, J. Atomic and Molecular Photoabsorption, Absolute Total Cross Sections. Academic Press San Diego, 2002 1-350. 

Pacific Distance along the cross-section (km) San Bernardino... 

The temperature dependences of the small-angle neutron scattering (SANS) from solutions and networks of poly(N,N -diethylacrylamide) or from copolymer of DEAAm and MNa (xMNa = 0.05) in deuterated water were measured [41]. Experimental dependences of the effective scattering cross-section... 

The southward shift of the jet stream core can be seen in more detail on the atmospheric cross sections presented in Figure 4. The cross sections extend from Medford (MFR) southward across the coastal range to Oakland (OAK) and onward to Vandenberg (VBG), San Nicolas Island (NSI), and San Diego (SAN). The partial soundings from Los Angeles (LAX) were used when available. 

Example 12.1. The CMC of C12E7 is 0.083 mM at room temperature. By SANS and dynamic light scattering the mean hydrocarbon core radius was found to be 1.70 nm at a surfactant concentration of 2 mM [532], The mean aggregation number is 64. If we divide the total surface area of the core by the number of surfactants, we get the area per molecule at the core radius. It is 47r (1.7 nm)2 /64 = 0.57 nm2. The cross-sectional area of polyethylene oxide is below 0.2 nm2. So, more than half the core area is exposed to aqueous or at least to a polar medium. 

The base metal and the welded joint have been thermally treated, cooling down in water from the initial temperature 1050°C. The welded joint has been scanned by a thin neutron beam to obtain the SANS data for base and welded metal. The cross sections showed three fractions of particles with gyration radii Rgi 20 nm, Rg2 9 nm and Rg3< 1 nm (point like defects). The amount of these defects in base metal was larger by factor 5, as compared to welded metal (see Fig. 8). 

Figure 8. SANS-scan over 08H18N10T austenite stainless steel welded and base metal forward cross section I0i (per cm3) for large defects Rgl 20nm, colour scale in units of cm"1. Right and left pieces are base metal and joint metal in between (coupled triangles).

C.J. Powell, Inner-shell Ionization cross-sections, in J.R. Michael, P. Ingram (Eds.), Microbeam Analysis, Sun Francisco Press, San Francisco, 1990, pp. 13-20. 

Comparison of the molecular length of CAB (ca. 38 A) and the characteristic lengths calculated from SAS data (vide ante) indicate that molecular pairs are involved in the columns of the solid state while, in gels, association of swollen columns might be involved. Micrographs of freeze-fractured and etched CAB gels (Fig. 17) show a 3-D network of fibrous bundles. The dimensions of the rectangular cross sections of the nontwisted fibers in dodecane, 209 x 104 A. and the twisted ones in 1-octanol, 263 x 82 A [481, correspond approximately to the cross-sectional areas determined by SANS in which a circular cross section model was employed. 

Fig. 12. The cross sections and chemisuries of the BMC Refloquant test element for blood y-GT analysis. (Reprinted, with permission, from Methods in Enzymologf, Vol. 137. San Diego, CA Academic, 1988.)...

The zero angle differential scattering cross-section for the completely inter-diffused couple is calculated from equations 1 and 3 to be roughly 1100 cm The SANS intensity from the blend will approach this at a rate governed by the respective inter-diffusion... 

The shift to higher temperature of the 3 peak in SAN with respect to aPS homopolymer can be understood by considering Equation (6). It is expected that Cfl will take the same value in SAN and aPS because the van der Waals cross-sectional area is not affected. Therefore, owing to the decrease in the mass... 

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