Particles scattering factors

C, the fourth parameter, represents the relationship between the first cumulant and the particlescattering factor. For values of 1/F( ) < 10, the double logarithmic plot of the first cumulant against the reciprocal particle-scattering factor yields a straight line, and the exponent v is related to the initial slope C oiF/q D, against by the equation... 

I(q) is the intensity at wave vector q, (bjjr-bp) is a contrast factor arising from the difference in scattering lengths of deuterated and protonated species, M is molecular weight of the deuterated polymer, c is concentration in gm/ml, S(q) is a particle scattering factor, and A contains machine constants, detector efficiency, and other fixed quantities. For the purpose of the current study, S(q) is the quantity of significance, and it is given by... 

Rayleigh ratio of scattering intensity at scattering angle 0 particle scattering factor = normalized molecular structure factor... 

Therefore the derived equation for the particle scattering factor simultaneously gave an equation for the radius of gyration which is... 

Here P(q) is the particle scattering factor and q = (47i/Z0) sin (0/2) is the scattering vector. The value of P reflects the specific size and shape of the polymer particle. This parameter has been calculated and tabulated for many different kinds of idealized colloidal and macromolecular structures (Burchard, 1994 Evans, 1972 Tanford, 1961). 

Mw is the weight average molecular weight and Pz(q) the z-average of the particlescattering factor the particle-scattering factor of an x-mer in the ensemble is given by the expression in the brackets of Eq. (B.1S). 

In the last chapter, equations were derived for the particle-scattering factor, the mean-square radius of gyration, the diffusion coefficient and the first cumulant of the dynamic structure factor. All these have the common feature that, for homopolymers at least, they can be written in the following form ... 

For long rays, one has 0 as 1,1 - tp = b2q2/6, and Tb and Pi can be neglected compared to P2. This leads to the following equation for the particle-scattering factor of regular stars... 

Equation (D.2) shows 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... 

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. ... 

Fig. 18. Reciprocal particle-scattering factors of star-molecules with polydisperse rays, where f denotes the number of rays per molecule. The same functions are obtained also for the ABC-type polycondensates, where nb denotes the number of branching points per molecule. The case f = 1 or nb = 0 is identical to linear chains obeying the most probable lengths distribution. It also represents the scattering behaviour of randomly branched f-functional polycondensates 1...

Table 2. Particle scattering factors of some selected models... 

The diminishing effect of the upturn due to polydispersity1551 may be demonstrated with an example. Let us consider a mixture of two linear chains with molecular weights Mi and M2 and particle-scattering factors Pi(q2) and P2(q2), respectively. The total particle-scattering factor is then given by... 

Fig, 20. Above Particle-scattering factors Pj (q) and P2 (q)for two linear chains, where (S2)2 = 9 (S2)i, and the particle-scattering factor of a mixture of both chains with w,M] = w2M2. Below The corresponding reciprocal particle-scattering factors. Note the downturn for P2(q) at low q2. The dotted line indicates the initial slope defined by (S2) of the mixture... 

This leads to the conclusion that polydisperse Unear chains cannot be distinguished from randomly branched chains using only the shape of their scattering curves. Indeed, when the link probabilities are expressed in terms of the mean-square radius of gyration, the particle-scattering factor is given in both cases by... 

The complete balance of the upturn by the polydispersity is only obtained for random branching processes. Often the reaction is impeded by serious constraints, or the primary chains before cross-linking are monodisperse. Then the resultant final molecular-weight distribution is narrower than in the random case, and the characteristic upturn as a result of branching, develops again. A strange coincidence in behavior is observed with star-molecules, where the rays are polydisperse, and with the ABC-type polycondensates. In both cases the particle-scattering factors can be expressed as ... 

Figure 23 shows as examples the reciprocal particle-scattering factors for f = 2, 3 and 6. Again, the upturn with increasing branching develops. One realizes from Eq. (D.14) that Px (q2) becomes independent of the number of functional groups per monomer if f is large. 

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