Volume fraction critical

The polyamides are soluble in high strength sulfuric acid or in mixtures of hexamethylphosphoramide, /V, /V- dim ethyl acetam i de and LiCl. In the latter, compHcated relationships exist between solvent composition and the temperature at which the Hquid crystal phase forms. The polyamide solutions show an abmpt decrease in viscosity which is characteristic of mesophase formation when a critical volume fraction of polymer ( ) is exceeded. The viscosity may decrease, however, in the Hquid crystal phase if the molecular ordering allows the rod-shaped entities to gHde past one another more easily despite the higher concentration. The Hquid crystal phase is optically anisotropic and the texture is nematic. The nematic texture can be transformed to a chiral nematic texture by adding chiral species as a dopant or incorporating a chiral unit in the main chain as a copolymer (30). 

Because of the rotation of the N—N bond, X-500 is considerably more flexible than the polyamides discussed above. A higher polymer volume fraction is required for an anisotropic phase to appear. In solution, the X-500 polymer is not anisotropic at rest but becomes so when sheared. The characteristic viscosity anomaly which occurs at the onset of Hquid crystal formation appears only at higher shear rates for X-500. The critical volume fraction ( ) shifts to lower polymer concentrations under conditions of greater shear (32). The mechanical orientation that is necessary for Hquid crystal formation must occur during the spinning process which enhances the alignment of the macromolecules. 

Figure 2. Critical volume fraction ( ) ploltcd vs. for k = Nb/Na = 2 (circles) and k = 3 (squares).

Figure 3. Estimates of the critical volume fraction cj) ,r of monomers as a function of inverse chain length. Broken curve represents a fit of the form < )cni = (1.1126 + 1.3. From Wilding et al. ...

Understanding the critical volume fraction (t)cric is more subtle (Fig. 2). In accord with theoretical predictions we find that (ticric = 1) holds only when even the... 

Once interpenetration occurs the resistance to deformation increases markedly, so for example we would expect compaction of a sediment to become limited, as would further concentration in a filter press. It is worth emphasising the point that this is a simplistic approach, as prior to interpenetration the clusters undergo structural rearrangements changing their fractal index at a critical volume fraction. A typical data set for yield stress is shown in Figure 6.16.19... 

Hydraulic resistance of membrane Velocity of i th component, cm/sec Critical volume fraction... 

However, Eq. (8) leads to the paradox that overlapping does not begin until a critical volume fraction of 2.5. In order to overcome this, somewhat modified equations have been put forward by Simha (10)... 

The above molecular thermodynamic model for polymer systems has been widely tested by comparing with simulation results (Yang et al., 2006a Xin et al., 2008a). Figure 8 shows the comparisons between predicted critical temperature and critical volume fraction for binary polymer solutions at different chain lengths of with the... 

Figure 8 Chain-length dependence of the reduced critical temperature and the critical volume fraction. Square and triangle MC data solid line this work dot-dashed line this work with X = 0 dash line Flory-Huggins s theory dotted line Freed theory (Yang et al., 2006a).

All listed effects have an influence on the granule size and shape as well as on the distances between granules. This helps in understanding why experimental values of the critical volume fraction of metal granules xc strongly differ from the calculated ones in the framework of classical percolation theory [35], in particular, for In granules on top of an Si02 surface xc — 0.82... 

Figure 5.2. Free-energy change of mixing for rods and solvent molecules. Free energy change (AGIRT) of (A) solid phase associated with transfer of a solute molecule (macromolecule) from the liquid to the solid state as a function of solute volume fraction (V2) for low (Z = 10) and high (Z = 200) axial ratios and (B) liquid phase as a function of solute volume fraction in the presence (Xi = 0.1) and absence (Xi = 0) of interactions between solute molecules. The diagrams show that separation of solute and solvent molecules occurs spontaneously for high axial ratios above a critical volume fraction and that the free energy of the solvent is raised by inter-molecular interactions.

In Eq. 5, [rj] is the intrinsic viscosity of the dispersed phase and Pm is the maximum packing volume fraction (in most cases, Pm = 1 - Per, Per is the critical volume fraction or percolation threshold). 

If the cylinders are at some angle j> to one another, then there is a large volume surrounding the rod where the approaching rod s centre-of-mass enters. If, however, the rods are oriented parallel to one another there is little volume that is excluded. This gives rise to an effective potential of the form IkTDL2(sin fi). It leads in a virial expansion to a critical volume fraction anisotropic liquid crystalline phase... 

As anticipated in Fig. 5a), the critical volume fraction ( )crit(D) gets shifted to smaller values, for a case where the walls preferentially attract monomers of the other kind (i.e. B-monomers when denotes the average volume fraction of A). One also sees that % it gets more depressed the smaller D is, i.e. confinement makes the polymerjmxture more compatible. 

All work reviewed so far in this subsection concerns thin films with neutral surfaces, but we feel that the general scaling description Eqs. (129)-(133), Fig. 23) should also apply to thin films with symmetric surfaces that both favor the same component (say B, cf. Fig. 5) relative to the other. The additional feature, not present in Fig. 22, then is a shift of the critical volume fraction critCD) with thickness. Scaling considerations [216,224,225] predict for this shift... 

This problem can be initially overcome using a volume ratio instead of a lattice site ratio, expressing the percolation thresholds as critical volume fractions [36, 4(M2], Nevertheless, the influence of the particle size of the components on the percolation threshold cannot be explained using a volume fraction that is, from this point of view, tablets with the same excipient volume are equivalent independent of their particle size. A first qualitative study of the influence of particle size on the percolation threshold [43] demonstrated that this is in clear disagreement with experimental data. 

Table 4.1 Critical volume fraction, 0, for various aspect ratios of particles, after Refs. 4 and 57...

Viscosity-volume fraction curves Fig.2 shows the viscosity as a function of volume fraction of particles. The results are typical of those usually obtained with concentrated dispersions (6), showing a rapid increase in viscosity above a critical volume fraction of the dispersed phase. When the volume fraction reaches the so-called packing fraction, g (see Discussion Section), the viscosity reaches a very high value, gp may be obtained from a plot of versus and extrapolation to... 

Unfortunately, the percolation approach is also not really able to predict accurately the critical volume fraction in real composites, because so many different factors, like filler shape, size, distribution and particle agglomeration, come into play. Lux (1993) has reviewed the various percolation models that have been proposed in theoretical treatments of the problem. 

Beyond the percolation limit, the bridging network is more concentrated. Below the critical volume fraction, no continuously bridging networks are formed and the viscosity is low. As shown in Figure 12.7, this bridging network breaks up as the shear rate increases, giving different viscosities at different shear rates. As a result, this gives low and high shear limit viscosities observed at steady state for concentrated poljmier solutions and concentrated particulate suspensions (discussed later). 

At low ij), the cmve is linear. At high values of < >, there is a critical value, where no fiirther shrinks e takes place, corresponding to liquid just filling the pores at the leatherhard point. This critical volume fraction, <, occurs when the mechanical properties of the particle network is sufficiently rigid to resist the compressive capillary pressure. The liquid expansion of a ceramic green body, a, is defined by... 

FIGURE 14JS Liquid expansiaii as a function of solids volume fraction with critical volume fraction where the particles are in cMitact... 

It is clear that the introduction of a critical volume fraction is a step toward dealing with percolation on a continuum. To this end Zallen and Scher (1971) considered the motion of a classical particle in a random potential, V r), and introduced a function, which detines the fraction of space accessible to particles of energy E. The connection with percolation is in the fact that, for energies such that 4>(E) > c(.E), there are infinitely extended volumes of allowed (V < T) space. The critical value c is identified with 0.15 for d = 3, and delocalized states appear above c ... 

The lattice model was introduced by Flory (30-31) who calculated the free energy of the system from principles of statistical mechanics. Flory found that a collection of rods becomes metastable with respect to an ordered phase at a critical volume fraction, Vp, of rods that is dependent on the axial ratio,... 

The diamter of the PBG a-helix has been estimated variously to be between 15-25 A as noted above (25). Values of d used in the determination of persistence len s are listed in the solvent categories of Table I. Values of Vp calculated from Equation 1 using x = L/d are invariably too low for the solvents of this study. Better agreement with the lattice theory has been reported for critical volume fractions of PBG in dimethylformamide and m-cresol (32). Experimental volume fractions for liquid crystal formation of PM in dioxane are lower, however, than those calculated from Flory s theory (33). PBG is known to undergo extensive... 

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