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.

Fig. XI-7. Volume fraction profile of 280,000-molecular-weight poly(ethylene oxide) adsorbed onto deuterated polystyrene latex at a surface density of 1.21 mg/m and suspended in D2O, from Ref. 70.

Chemical shift imaging (CSI) was used to monitor the oil volume fraction during the mixing process. Figure 4.5.13 shows normalized volume fraction profiles along the vertical center-line (see Figure 4.5.3) at different times. The mixing time is expressed in strain units as y = tV/(R0 - i)> where t is the time. One revolution of the outer cylinder corresponds to 9.83 strain units. The initial condition (y = 0)... 

The next step is to generate all possible and allowed conformations, which leads to the full probability distribution F). The normalisation of this distribution gives the number of molecules of type i in conformation c, and from this it is trivial to extract the volume fraction profiles for all the molecules in the system. With these density distributions, one can subsequently compute the distribution of charges in the system. The charges should be consistent with the electrostatic potentials, according to the Poisson equation ... 

Figure 13. The overall density (volume fraction) profile for DMPC bilayers is shown here. Apart from the distribution of the overall DMPC molecules, the density distribution of the head-group units (including the choline group, the phosphate group and the oxygens of the glycerol unit), and the end groups of the lipid tails are also indicated. In addition, the free-volume profile and the water profile are depicted...

Figure 14. Volume-fraction profiles of parts of the DMPC molecules for lipids that have the head group at positive coordinates (continuous lines) and at negative coordinates (dashed lines). The centre of the bilayer is positioned at z — 0. The phosphate group, the nitrogen of the choline group and the CH3 groups of the tail ends, as well as the other hydrocarbon units, are indicated...

Figure 28. Accessible free-volume fraction profile across a DPPC bilayer for solutes of different sizes (diameters indicated in nm 0.0 corresponds with the total free volume). Results of Marrink et al. [132]. Redrawn by permission of the American Institute of Physics...

FIGURE 8.20 Soot volume fraction profiles for premixed flames. See Table 8.7 (from Harris and Weiner [74]). 

Figure 2.6 (a) Volume-fraction profiles for the polymer backbone (full line), the redox sites (dashed line), and the whole redox polymer (dot-dashed line) predicted by the molecular theory (Section 2.5) fora single layer of PAH-Os on a thiolated gold electrode, (salt... 

There are a number of assumptions made in the model that are questionable, and these are probably responsible for the discrepancy between the predicted and observed behaviour of the slurries. For example, the steric contribution has been calculated assuming that the adsorbed layer has a well-defined thickness. For adsorbed polymers this is unlikely to be the case, as the volume fraction profile of the polymer will decay gradually as a function of distance from the surface. Furthermore, it was assumed that the effective ionic strength in the adsorbed polymer layer is the same as in solution. However, this also is unlikely since one of the main components of the solution ionic strength is the polymer itself, and unadsorbed polymer will be excluded from the adsorbed layer. Finally, the connectivity of the charged groups on the polymer was not considered, so its contribution to ionic strength may have been overestimated. 

Fig. 3.38 (a) Neutron reflectivity profile for a PS-PEO diblock (M = 15 kg mol-1,1.5% PEO) end-adsorbed from d-toluene onto quartz (Field et al. 1992a). The symbols indicate measured values, whilst the full line is a fit to a parabolic volume fraction profile, (b) Models for the density profile. The parabolic function was found to give the best fit to the data. 

Fig. 6.37 Interfacial volume fraction profiles calculated for a ternary blend of a PS-PB block copolymer with PS and PB homopolymers in a good solvent (Noolandi and Hong 1982). The diblock has N — 600 and f = The homopolymers have infinite molecular weight. The solid lines are the volume fractions of homopolymer (A = PS) (B = PB), the dashed lines indicate the volume fractions of PS and PB blocks of the diblock. The dots correspond to the total volume fractions of the A and B components and the position is measured in units of a segment length a = 6.95 A.

Fig. 6.44 Calculated lamellar volume fraction profiles for a blend of diblocks for % = 0.2 (Shi and Noolandi 1994). The long diblock has Nt = 500 and f = 0.5. The short diblock has N, = 50. (a) Pure long diblock (b)-(d) blends with a short diblock (, = 0.05). (b) /, = 0.3, (c) /, = 0.5 and (d) ft = 0.7. The long chain profiles are shown as solid lines, and the short chain profiles are plotted as dashed lines. Changes in the lengths of the horizontal axes correspond to the changes in the domain size with added short chains.

Fig. 6.51 Volume fraction profiles determined from modelling reflectivity profiles for blends of a PS-PMMA diblock (M = 91.5kgmol, = 0.53) with 5wt% of rfPS...

Apart from yielding information about the film thickness and the nature of the interface, it is also possible to determine the volume fraction profiles of components present in a particular layer. For example, in the case of an adsorbed polymer... 

Fig. 7c Volume fraction profiles (t>coex(z)> coex(z) at phase coexistence. From Flebbe et al. [58]...

Fig. 11. Polymer volume-fraction profiles calculated by Scheutjens and Fleer (1980) and plotted as logfep,) vs. i. Parameters are given in the inset.

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

Volume fraction profile during constant rate period (i.e., boundary layer mass or heat transport is the rate determining step) ... 

From the gas pressure which includes the partial pressure profile, the temperature profile, and local capillary pressure which is a fvmction of the volume fraction profile and the liquid radius of curvature profile inside the green body, the average stress can be determined. The partial pressure profile is determined by the flux, J, and the effective pore diffusion coefficient, as follows ... 

The problem is to relate v (z) to the surface potential - v (0) or the surface charge density a° = a(O)) and the volume fraction profiles of the components. Early versions t-2) of a polyelectrolyte adsorption model neglected the volume of the small Ions and solved (numerically) the Poisson-Boltzmann equation 13.5.6). A more sophisticated, yet simpler, approach was proposed by Bflhmer et al. who accounted for the Ion volume by adopting a multilayer Stem model, see fig. 5.17. This Is a straightforward extension of the monolayer Stern model discussed in sec. 3.6c. The charges of the ions and the segments are assumed to be located on planes in the centres of the lattice layers. The lattice is thus con-... 

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