Distribution segmental density

In order to render the expression for d AFa) in a usable form, it remains to evaluate pk and pi. We have already pointed out that the average segment density of a molecule will be greatest at the center of gravity and that it will decrease smoothly as the distance 5 (Fig. 114,a) from the center is increased. While the distribution will not be exactly a Gaussian function of s, it may be so represented without introducing an appreciable error in our final result, which can be shown to be insensitive to the exact form assumed for the radial dependence of the segment density. Hence we may let... 

With reference to an actual polymer molecule we should, of course, speak of the potential at a point within the molecule, since the potential will decrease radially from its center in the manner dictated by the spatial distribution of the fixed charges (which like the segment density, may often be approximated by a Gaussian distribution) and that of the counter-ions. For the purpose of the present qualitative discussion, however, we refer merely to the potential finside the molecule. 

Segment density profiles and hydrodynamic thickness measurements have been made for polyethylene oxides adsorbed on polystyrene latex. Comparison with theoretical models shows that the hydro-dynamic thickness is determined by polymer segments (tails) at the extremity of the distribution. It is also concluded that the sensitivity of the s.a.n.s. experiment precludes the measurement of segments in this region and that the experimental segment density profiles are essentially dominated by loops and trains. 

The distribution of segment density normal to the surfaces is an important configurational property which serves to characterise the structure of the interfacial region. It is described in terms of the mean fraction of segments of the chain in each of... 

Figure 5. The distribution of chain segment density, at 6 0, in an interfacial region with M - 6.

It is now well recognised (6,2,0) that such a division of the free energy is valid only if the segment density distribution is independent of the mean concentration in the interfacial region. Figures 5, 6 and 7 show that the distributions are markedly affected by the changes in concentration. This casts a serious doubt on the plausibility of the fundamental assumption underlying these theories. 

The S parameter is a function of the segment density distribution of the stabilizing chains. The conformation, and hence the segment density distribution function of polymers at interfaces, has been the subject of intensive experimental and theoretical work and is a subject of much debate (1). Since we are only interested in qualitative and not quantitative predictions, we choose the simplest distribution function, namely the constant segment density function, which leads to an S function of the form (11) ... 

Therefore, a fit of the scattering data not only gives information on an average size, Rg but also information on the segment density distribution within the dendrimers. 

Figure 11.10 is a plot of the average local segment density distribution for... 

Most, but not all of the terminal units are near the outside of the dendrimer at any given time. The SAXS studies [17] of the segment density distribution have shown that there is an abrupt transition region at the outside of large PAM AM dendrimers. The combination of these two factors suggests that the terminal functionalities of dendrimers are accessible from the outside and available for chemical reactions such as attachment to surfaces, mounting of a catalyst, or for use as a crosslink junction. 

Dendrimers are nearly spherical in shape when compared to conventional linear polymers [ 17,34], The lower generation dendrimers have internal segment density distributions similar to star molecules, but even the smallest are more compact than stars. As the generation number increases, the sphere-like characteristics increase and the largest dendrimers form a population of spheres with uniform internal density (roughly half dendrimer units-half solvent) [47], sharp interfaces on the outside and a small polydispersity in shape. 

Experimental data on branched polystyrenes were used to test the theory. There are some discrepancies in particular the parameter A varies from one polymer to another, especially for comb-shaped polymers y> also varies. It is suggested that the reason for the discrepancies in A may be the failure of the assumption of a Gaussian distribution of segment density. 

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