Zero angle, scattering

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 

We recall that Ab is the difference between the scattering length of a solute molecule and the equivalent volume of the solvent. Therefore the ratio Abjvj is equal to the difference Ap in the scattering length density p of the solute and solvent 

For a dilute polymer solution it is customary to express the concentration in terms of c (mass of solute per unit volume) defined as 

From the theory of dilute polymer solution5 it is known that the osmotic pressure can be expressed as a function of concentration in a virial expansion 

To be able to measure the osmotic pressure n, a semipermeable membrane that permits passage of the solvent molecules but not the solute molecules is needed. This can, in practice, be realized only when there is a large disparity between the sizes of the solute and solvent molecules, as in a solution of a polymer in a small-molecule solvent. However, the existence of osmotic pressure can be envisioned, at least mentally, with any kind of solution, such as a solution of two small-molecule liquids or a miscible blend of two polymers. Equation (6.6) is thus valid for any two-component (amorphous) system, as long as it is in equilibrium and classical thermodynamics is applicable to it. For applications to these general cases, it is more convenient if Equation (6.6) is reformulated in terms of the free energy of mixing and no explicit reference to osmotic pressure is made in it. 

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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

Figure 4.16 Transmission of zero angle scattered light through a slab of particles.

In the limit of dipole scattering, the zero-angle scattering Jones matrix for a system of anisotropic particles is simply found by extending equation (4.66) to the case were the polarizability is a tensor. In this case,... 

This corresponds to (f + d" ) lv(0), except that for the present pourposes we took into account the dispersionof f by an additional factor 1.4. N is the number of resonant atoms which form a definite structure, e.g. as metal binding sites a macromolecule. The ratio of resonant zero angle scattering to fluorescence Op in forward direction is... 

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. 

As neutron scattering measures the fluctuation in the scattering length density of a system, it is an ideal tool to monitor the presence and growth of concentration fluctuations in a polsrmer/polymer mixture. In fact, the inverse of the zero-angle scattering, l q=0), is proportional to the second derivative of the free energy of the mixture with respect to the blend concentration, which is the osmotic compressibility of a mixture. Theories of cooperative phenomena show that /" (q=0) (T - (60) and thus the determination of extrapo-... 

Fig. 8. The inverse of the zero-angle scattering of a pol5Tner blend (d-PS/PVME) as it approaches the phase separation temperature. Note the deviation from mean-field behavior just below 140 °C. Reprinted with permission from Ref 61. Copyright (1987) by the American Physical Society.

In equation (67) F is the excess zero-angle scattering amplitude (over the solvent amplitude) for a particle of diameter (T , and Bj is the intraparticle interference factor defined by... 

Here are the zero angle scattering amplitudes on the polarized nuclei for neutrons polarized parallel and antiparallel to the A x / axis, respectively, z is the target length, and N is the density of nuclei in the target. 

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. 

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