Zero-angle scattering intensity

Ig(0) zero angle scattered intensity for solid-like scattering... 

Fig. 6.23 Logarithmic plots of the correlation length ( ) and zero-angle scattering intensity (/(0)) as a function of temperature reduced with respect to the Lilshitz temperature Tlp) for a blend of PE and PEP homopolymers with a PE-PEP diblock (details as Fig. 6.22) at a copolymer volume fraction

According to Equation (4.43), for a single-component liquid the zero-angle scattering intensity 7(0), that is, the intensity of scattering I(q) extrapolated to q - 0, is proportional to the mean square fluctuation ((ANv)2) in the number Nv of atoms present in a macroscopic volume v. In an equilibrium liquid (or an amorphous polymer above Tg) the mean square fluctuation ((ANv)2) is related to the isothermal compressibility of the liquid according to (4.32), so that 7(0) is given by... 

This is the desired expression relating the osmotic pressure to the free energy of mixing. Equation (6.6) for the extrapolated zero-angle scattering intensity is then written as... 

The intensity of zero-angle scattering, discussed in the previous section, reflects the concentration fluctuations present in the system on a macroscopic scale and is directly related to its thermodynamic properties, such as the free energy of mixing. 

Fig. 2 Guinier plot (ln(/(q)) versus q ) on the data sets of small q range used for the determination of both values of the zero-angle scattering intensity /(O) and the gyration radius R. (a) and (b) correspond the scattering data in Figs, la and b, respectively...

In these equations Va, Vb, and vq are the molar volumes of the network repeating unit, Nc is the number of repeating units in the network between cross-links, x is the Flory-Huggins parameter, (p is the volume fraction of the network, (p is the network volume fraction in relaxed state, usually taken to be the value at which the network was formed and Nb is the number of repeat units in the linear chain. Constants A and B follow from the rubber elasticity theory, usually A = 1 and B = 2//c, where/c is the functionality of the crosslinks, kn is the constant that determines the amount of contrast between the two components and the radiation type, and S(0) is the zero-angle scattering intensity. 

If the scattering intensity can be measured with sufficient accuracy down to very small angles, the Rayleigh ratio at zero angle may be... 

Figure 7 Concentration dependence of the reduced light scattering intensity extrapolated to zero angle for PS-S03Li-40 in cyclohexane at 27.5°C. The solid line represents the model calculation with 3 10 (1/mol) and N 12. The dashed line indicates M] = 49,000.

When the particles are large enough to display angular dissymmetry, Eq. (9) is only valid when 0 = 0, as P(0) = 1 for the angle. The experimental scattering intensities can only be analyzed as a function of the observation angle as it is impossible to measure the scattered intensity at zero angle [9]... 

This equation implies a double dependence of scattering intensities on concentration and observation angle [9]. By extrapolating the scattering data for each concentration to zero angle, the second virial coefficient, which is related to thermodynamic properties, may be measured [9,10,15-18],... 

Experimental Method and Results. Using Equations 11 and 12, it is possible to calculate the Kc/Ro values at different angles and concentrations from the Ye readings. They are calculated from the recorder readings after correction for the transmittance of a set of neutral filters in order to minimize the variations of the light intensities on the photomultiplier. Also, the scattered intensity of the solvent is subtracted. The scattered intensities were measured at four concentrations to determine the solute effects. For the simultaneous extrapolation to both zero concentration and zero angle, we used the method of Zimm (24). 

Kinetic Light-Scattering Method. Isochronous Interpolation. When high-activity samples of endocellulase are used, the reaction proceeds so quickly that, since the measurements of scattered intensity at different angles are not performed at the same extent of reaction, the extrapolations to zero angle and the subsequent calculations are erroneous. For this reason Kratochvil et al. (27) have proposed an isochronous interpolation method, whereby the Kc/Re values are plotted against sin21 (0/2) + kft. As in the double-extrapolation method of Zimm, the value of k may be chosen arbitrarily in order to space the experimental data. 

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... 

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