Segment density distribution

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. 

Viscosity at low shear rates is obtained by substituting the results of Eqs. (6.8) and (6.9) into Eq. (6.6), and evaluating V from the segment density distribution of undisturbed random coils. 

Aharoni, S. M. Segmental density distribution of linear amorphous polymer molecules. J. Macromol. Sci.-Phys. B7,73-103 (1973). 

Hoeve44,45) extended his theory further by considering not only interactions between the train segments but also interactions among the loops, and found that the latter lead to a decrease in the number of possible conformations of adsorbed polymer chains. He assumed that the segment density distribution in any loop is uniformly expanded in one dimension by a factor of at as a result of loop-loop interactions. The volume fraction of segments at a distance z > 6 is then given by... 

The segment density distribution q6(z) for a single tail of size i is given by Hesselink39,40 as... 

Hesselink51 assumed that the conformation of macromolecules adsorbed on the interface consists of one train and two tails. The number of tails with i segments, n(i), is calculated from the partition function for the tail-train conformation, and the normalized segment density distribution p7(z) in tails is derived by methods analogous to those used in Hesselink s derivation of the segment density distribution in loops. The result is... 

From Eq. (B-113) the segment density distribution p(ir) and the adsorption isotherm are calculated as follows ... 

As has been depicted in Fig. 1, various conformations are possible for adsorbed polymers, depending on polymer-polymer, polymer-solvent, and polymer-interface interactions and the flexibility of polymers. To determine experimentally the conformation of adsorbed polymers only adsorption isotherm data are insufficient. The average thickness of the adsorbed polymer layer, the segment density distribution in this layer, the fraction of adsorbed segments, and the fraction of surface sites occupied by adsorbed segments must be measured. Recently, several unique techniques have become available to measure these quantities. 

Fig. 1.17 Segment density distribution of a dendrimer molecule according to the dense-shell model (schematic)...

Radial segment density distribution of flexible dendrimers... 

The excess part is approximated as a functional of solvent density distribution ps(q) and the average segment-density distribution Pnxiq)- The latter is related to polymer density pp(Q) via a transform as... 

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