Particle-scattering function

Equation (D.2) shows pn as the Fourier transform of the pair distance distribution W (rn) for a path with its one end at r = 0 (the root) and the other at r (n-th generation). The particle scattering factor 

For the discussion of the properties of the static structure factors, it is often more convenient to write the scattering functions in terms of a space correlation function y(r)4wr2dr. 

The space correlation function y (r) is closely related to the distribution of the units over the various shells (N(n)). The main difference consists of the fact that 4 Tty (r) r2 dr measures the number of units in a shell at distance r from a given unit while (N(n)) represents the number of units in the n-th-shell, where nothing needs to be known about the distance of the shell from the origin. This distance is taken into consideration by the extra factor tpn(r) in Eq. (D.T). Both functions, the space correlation function y (r) and the population number (N(n)), have their advantages and disadvantages. It should be mentioned, however, that (N (n)) is much easier and more directly calculated by application of the cascade procedure than the space correlation function, which can be obtained 

The space correlation function has been extensively used by Kratky and Pored148 149) for monodisperse particles and was also introduced by Pekeris150 and by Debye151 for a description of spatial inhomogeneities in condensed matter. y(r) always has the properties 

In Fig. 17 to 19 the particle-scattering factors for some regularly branched and some polydisperse molecules are shown in plots of P2(q2)-1 as function of q2 (S2)z (see also Table 2). The curves demonstrate clearly that branching causes an upturn while polydis-persity tends to balance the influence of branching34,90.  

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A full treatment of particle scattering functions has been written by Kratochvil30) and the modes of effecting the limits indicated in Eqs. (37) and (42) will be described in a later Section (III.4). 

Here the symbols R, P and G denote respectively the Rayleigh ratio, particle scattering function and instrument scattering reading. It is possible to take other angles such as 60° and 120°, which are also symmetrical about 90°. However, the angles 45° and 135° are most frequently selected, and the widely used Brice-Phoenix photo-... 

Fig. 25. Typical forms of reciprocal particle scattering functions for systems containing supermolecular structure30, 108).- (a) small individual macromolecules and a small amount of supermolecular structure sometimes allowing extrapolation to unity (and hence I/M) from moderate -high 0, as indicated, (b) large individual macromolecules or small individual macromolecules but high content of supermolecular structure...

According to Nagai (5), the particle-scattering function P(0) of an interrupted helical polypeptide dissolved in a single-component solvent may be written... 

The optical constant K is a function of the refractive index and the refractive index increment, and P (0) is the particle scattering function which depends on internal interference. This function is influenced by the particle shape and is less than 1 for molecules large compared with... 

Calculation of the autocorrelation function proceeds by noting that the single particle scattering function, F- (q, t), is simply the Fourier transform of G- (x ), the Van-... 

If a multiangle light-scattering instrument is used, the mean-square radius of gyration (Rg )i at each elution volume can also be obtained from the particle scattering function... 

Methods that calculate average polymer properties without the DRI are significant because they circumvent complications that arise from the measurement of the interdetector volume between the light-scattering and concentration detectors 10, 11). It is necessary, however, that each local signal,, be divided by the computed particle scattering function, P 6)i. 

Particle-scattering functions for random coils and spheres are indistinguishable at values of Z near unity (Figure 2). Likewise, the radii of random coils and spheres are also similar in the range of Z values less than 1.2 (Figure 3). This is reflected in the similarity of radii calculated for random coils and spheres (columns 7 and 8 of Table II). The largest difference between radii calculated for these two shapes is 8.5% for... 

Figure 2. Reciprocal of particle-scattering functions for random coils (-------) and spheres (------).

In Fig. 4.16 the particle scattering function P(0) is plotted for random coils, spheres, and rods as a function of respective parameters. 

Kratochvil P. Particle Scattering Functions In Huglin MB, editor. Light Scattering by Polymer Solutions. London Academic Press 1972. p 333. 

Figure 4. Concentration dependence of C/Re of PCS AI-DI fraction treated with CNBr in 0.6M GdnCl. Optical constant k — 6.82 X 10 mL g cm, particle scattering function P(0) = 0.0675.

The only reasonably plausible interpretation which reconciles this evidence in a self-consistent fashion appears to be that summarized in Fig. 6. This model argues that chain expansion at low ionic strength and the attendant increase in congestion in the solution lead to cooperative intermolecular interactions, viz. an ordering tendency between near neighbors. As described above, it appears based on the known structural features of PGS that a reasonably accurate determination of the particle scattering function P(K) can be obtained by extrapolation at each angle to c = 0. This permits a determination of S(K) via eq. (4). These S(K) data permit a rationalization of... 

The Mazur function can be used to calculate R ) and (R ) (n = 2,3,...), but its usefulness extends no more than that. Calculation of such important global polymer properties as mean-square radius of gyration, particle scattering function, intrinsic viscosity, and diffusion coefficient needs an analytical expression for l+,j(R), the distribution function for the distance R between specified beads i and j. We find no a priori reason for this function to be approximated by the Mazur function. Knowledge about Wij has increased in recent years mainly on the basis of computer simulation experiments. Some of the typical results from such work are described below. 

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