Segment densities

Altematively, tire polymer layers may overlap, which increases tire local polymer segment density, also resulting in a repulsive interaction. Particularly on close approach, r < d + L, a steep repulsion is predicted to occur. Wlren a relatively low molecular weight polymer is used, tire repulsive interactions are ratlier short-ranged (compared to tire particle size) and the particles display near hard-sphere behaviour (e.g., [11]). 

FIG. 12 Segment density profile as function of the distance from the wall Z for flexible (empty symbols) and semi-rigid (full symbols) living polymer chains at T = 0.4 [28]. The fractional occupancy of lattice sites by polymer segments is shown for the layers in the left half of the box. Dashed lines are guides for the eyes. 

The Alexander model and its descendants impose strong restrictions on the allowed chain configurations within the tethered assembly. The equilibrium state thus found is subject to constraints and may not attain the true minimum free energy of the constraint-free system. In particular, the Alexander model constrains the segment density to be uniform and all the chain ends to be at the same distance from the grafting surface. Related treatments of curved systems retain only the second... 

Fig. 9. Comparison of the analytical SCF model [56] with the full numerical SCF calculation [53] for the segment density profile in flat, grafted layers at various surface densities (o is the fraction of the maximum possible surface coverage of grafted ends). The analytical profile is parabolic to its tip, while the numerical calculation shows that the density at the periphery of the layer drops off exponentially...

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... 

The discussion here follows the more quantitative treatment given by Debye and Bueche, although we seek to avoid their model consisting of a sphere throughout which the segment density is uniform, and beyond which it is zero. 

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, expressed in number per unit volume, within a given volume element dV,... 

However, these results are only valid for freely interpenetrating coils, a molar mass independent screening length , and a uniform segment density. 

The free energy of a polymer consists of the entropic term and the internal energy of the segmental interactions, U, which represents the excluded-volume effect, and can be expanded as a power series of the segment density p (Eq. 1) ... 

The theory of polymer adsorption is complicated for most situations, because in general the free energy of adsorption is determined by contributions from each layer i where the segment density is different from that in the bulk solution. However, at the critical point the situation is much simpler since the segment density profile is essentially flat. Only the layer immedia-... 

Several experimental parameters have been used to describe the conformation of a polymer adsorbed at the solid-solution interface these include the thickness of the adsorbed layer (photon correlation spectroscopy(J ) (p.c.s.), small angle neutron scattering (2) (s.a.n.s.), ellipsometry (3) and force-distance measurements between adsorbed layers (A), and the surface bound fraction (e.s.r. (5), n.m.r. ( 6), calorimetry (7) and i.r. (8)). However, it is very difficult to describe the adsorbed layer with a single parameter and ideally the segment density profile of the adsorbed chain is required. Recently s.a.n.s. (9) has been used to obtain segment density profiles for polyethylene oxide (PEO) and partially hydrolysed polyvinyl alcohol adsorbed on polystyrene latex. For PEO, two types of system were examined one where the chains were terminally-anchored and the other where the polymer was physically adsorbed from solution. The profiles for these two... 

The segment density profile obtained by s.a.n.s. is normalized in the form,... 

COSGROVE ET Al.. Adsorbed Polymer Segment Density Profiles... 

Figure 2. Experimental segment density profiles for PEO adsorbed on PS...

In a previous paper (15) the segment density of PVA adsorbed on PS latex in water was presented and it was noted that H Cgans was at the extremity of the s.a.n.s. profile. Calculating <5 assuming a value of a of 0.5 nm gives 13 nm in contrast to the experimental value of 18 nm. The discrepancy here is much smaller than in the case of PE0. This effect is difficult to interpret without further theoretical work but may be attributable to the fact that the PVA chain is less flexible than PEO and that the block structure (PVA is a random block copolymer of vinyl acetate. 12%, and vinyl alcohol) makes the formation of tails less likely. 

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... 

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