Polymer segment, volume fraction profile

Fig. XI-6. Polymer segment volume fraction profiles for N = 10, = 0-5, and Xi = 1, on a semilogarithinic plot against distance from the surface scaled on the polymer radius of gyration showing contributions from loops and tails. The inset shows the overall profile on a linear scale, from Ref. 65.

Figure 1. Illustration of a cap-and-gown model for the radial distribution of polymer segment volume fraction in a micelle, The micellar core is described as the cap and the diffuse outer layer as the gown. The relative sizes of the cap (radius a) and the gown (Gaussian width cr) are related by the PEO to PPO volume ratio. The water penetration profile is complementary to the polymer segment distrubution.

Fig. 19. Polymer volume fraction profiles, plotted as log[ p(z)/

In this section we discuss briefly some experimental methods for investigating adsorbed polymers. Determination of adsorbed amounts is for polymers not much different from that discussed in chapter 2. We therefore concentrate on three aspects which are typical for polymers. These are the (relative) number of segments in contact with the surface (l.e., the trains) (sec. 5.6a), the extension (thickness) of the adsorbed layer (sec. 5.6b), and the volume fraction profile normal to the surface (sec. 5.6c). 

Fig. 12 Radial volume fraction profile of polymer segments

The polymer concentration profile has been measured by small-angle neutron scattering from polymers adsorbed onto colloidal particles [70,71] or porous media [72] and from flat surfaces with neutron reflectivity [73] and optical reflectometry [74]. The fraction of segments bound to the solid surface is nicely revealed in NMR studies [75], infrared spectroscopy [76], and electron spin resonance [77]. An example of the concentration profile obtained by inverting neutron scattering measurements appears in Fig. XI-7, showing a typical surface volume fraction of 0.25 and layer thickness of 10-15 nm. The profile decays rapidly and monotonically but does not exhibit power-law scaling [70]. 

The physical significance of the experimental profile is that it is the probability that a segment of an adsorbed polymer chain is at a distance z from the interface. In order to find the volume fraction (z) at a distance z from the interface we require the mass/unit area T and the partial molar volume of the polymer V (12), where (z) is given by... 

It is usually assumed that the micellar corona is a continuous phase extending from the micellar core to the micellar radius Rm. The internal structure of the micelle can be described by a density profile as shown in Fig. 8. The micellar core is a homogeneous melt or glass of insoluble polymer blocks. For hydrophobic blocks in aqueous solutions, the polymer volume fraction in the micellar core is 0C 1. The micellar shell is swollen with water or aqueous salt solution and has a polymer segment density that is expected to decrease in the radial direction as 0(r) r-a as typical for star polymers or... 

In order to discuss a number of important quantities we consider the interfacial profile for the case of positive adsorption. Such a profile is sketched in fig. 5.6. It represents the polymer concentration dz) as a function of the distance z from the interface. The quantity c(z) is related to the volume fraction concentration profile of segments belonging to free molecules, having no contact with the surface. The excess adsorbed amount r (the amount of... 

Fig-i- Theoretical volume fraction vs. distance profile across an interface between PS and PVP polymers. A value of x = 0-12 appropriate to a temperature of 160 °C [ 105] has been used to predict an interface width a7 of 1.6 nm from Eq. (3). This profile is consistent with neutron reflectivity measurements of PS/PVP interface segment density profiles if the apparent broadening of the interface by capillary waves is taken into account [ 106]... 

The radial density profile (polymer volume fraction (Pa) in the corona of miceUe can unambiguously be used to find the aggregation number p. The sizes of core and corona are less trivially obtained. To help define the micellar dimensions we have incorporated two molecular markers at both ends of the hydrophilic block, named X2 (at the junction between A and B segments) and Xi (at the free end of the A block). The first moment < X > ... 

Thereby, (j) is the volume fraction of polymer in solution (as aggregates), Vc is the fraction of the crystallizable segment in the polymer, R is the lateral dimension of the lamellae (discs) and D x) denotes the Dawson function. The second term in Eq.5 arises from the polymeric structure of the brush (the "blob" scattering). P(Q) is the form factor of the density profile perpendicular to the lamellae surface including the contrast factors of the core and the brush parts and the density profiles of the polymer volume fraction. The form factor of an infinitely large plate of the thickness d considering a simple rectangular density profile is... 

Studies have shown that it is also important to consider a continuous segment concentration profile from the solid surface to the edge of the brush layer, rather than a simple step concentration profile of height h as implied in the above scaling analysis. For example, the theoretical self-consistent field (SCF) analysis of Milner et al. [75] predicts that, given a SCF-calculated equilibrium brush height h, the polymer volume fraction (p(z) in the brush layer follows a parabolic distribution with respect to the distance z from the surface ... 

---