Radial segment density distribution

Radial segment density distribution of flexible dendrimers... 

Fig. 9.7. The distance dependence of the nomalized segment density distribution function for 1, an exponential function, 2, a radial Gaussian function and 3, a constant segment density function (after Smitham and Napper, 1979).

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... 

This result is the analog of the swelling equation (XIII-39) for a network. It rests principally on the assumption that the intramolecular interactions of the segments with one another are the same as would obtain for a cloud of particles, not connected to each other, but having the same radial distribution as the average radial density distribution occurring for the molecule made up of a linear sequence of particles. 

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. 

Figure 2.10 (a) The normalized end-to-end distance probability distribution function for a Gaussian chain. The radius of gyration Rg and the hydrodynamic radius R are shown in units of the root mean square end-to-end distance R. (b) Sketch of the long-ranged segment density profile falling inversely with the radial distance from the center of the chain. 

Fig. 13 Radial density profiles for A and B segments left ordinate), and radial distribution of the degree of dissociation normalized to the dissociation of an A segment in the bulk, a r)/ah 1 (right ordinate), for Aioo soo = 10, and pH — p a = 1...

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