Volume Packing Fraction

Table 4.29 Packing volume fractions for various types of atomic packing...

The two-phase composite of microspheres and voids only (i.e. packed spheres without any binding resin) has also a minimum limit. Cubic packing (volume fraction 0.53) is the lower limit, below which the structure is not selfsupporting any more, and some resin is required to fill in between the spheres. S). 

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). 

A similar study was conducted by Poslinski et al. (36) on the effect of a bimodal size distribution of solids. They confirmed the findings of Chong et al. (28) in that the relative shear viscosity can exhibit a minimum for a plot of relative viscosity versus volume percent of small particles in total solids. Moreover, the primary normal stress also exhibited a minimum. Poslinski et al. showed that the relative viscosity can be predicted from the knowledge of the maximum packing volume fraction of the bimodal solids systems. 

Spatial configuration or fillers in different media Maximum packing volume fraction... 

Table 5.9. Maximum packing volume fraction, ( )m, of some fillers calculated by dividing tamped density by specific density of filler...

To model the elastic properties of dispersions of soft particles, we consider a dispersion of N spheres in a periodic box, as shown in Fig. 6. The particles are either monodisperse with radius R or polydisperscd with a Gaussian distribution around a mean radius R. The concentration of particles is above the random close-packed volume fraction of 0c = 0.64 so that the particles are jammed together and form facets at contact. The contacts are assumed to be purely repulsive and frictionless and hence exert only a normal repulsive force at contact. The total elastic energy stored in the structure is the summation of the pairwise contact energies. Even at the highest volume fraction at near-equilibrium conditions, i.e., without flow, deformation of a particle is no more than 10% of its radius. Thus, the particle deformation is small compared to the size of the undeformed sphere and the contacts obey the Hertzian contact potential given by (1). 

Petti (1994) showed the importance of silica-particle shape and maximum packing volume fractions on the chemoviscosity and spiral flow of highly filled epoxy-resin moulding compounds. Ultrahigh concentrations of silica could be used using the correct filler. 

Maximum Packing Volume Fraction, for Various Arrangements of Monodisperse Spheres... 

Experience has shown structured fluids to be more difficult to manufacture, due to the complexity of their rheological profiles. In addition to elasticity, dilatancy, and rheopexy, certain structured fluid compositions may exhibit solid-like properties in the quiescent state and other flow anomalies under specific flow conditions. For emulsions and solid particulate dispersions, near the maximum packing volume fraction of the dispersed phase, for example, yield stresses may be excessive, severely limiting or prohibiting flow under gravity, demanding special consideration in nearly all unit operations. Such fluids pose problems in... 

Figure 13.14. Estimated maximum packing volume fraction m of randomly dispersed cylindrical particles, Om for spheres, and geometrical percolation threshold pc for ellipsoids of biaxial symmetry. A =height/diamctcr for cylindrical fibers and thickness/diameter for cylindrical platelets. Af=(c/a), where c is the length of the ellipsoid along its axis of symmetry and a=b is the the length of the ellipsoid in the normal direction, for ellipsoidal particles.

Concentrated emulsions or high internal phase emulsions (HIPE) are systems in which the volume fraction of the dispersed phase is larger than about 0.74, which is the close-packing volume fraction of monodispersed hard spheres. The dispersed soft entities of a concentrated emulsion are no longer spherical. They deform into polyhedra separated by thin films of continuous phase. The structure is thus analogous to a conventional gas-liquid foam with low liquid content. The structure, properties, stability, and applications of highly concentrated emulsions were recently reviewed by Cameron and... 

In Eq 7.8 ( ) is the maximum packing volume fraction, and [q] is the intrinsic viscosity. The computed curves are shown in Figure 7.2. To calculate these dependencies ( ) = 0.8 and [rj] = 2 were assumed. The six points of intersection represent the iso-viscous conditions for the dispersions of liquid 1 in 2 and liquid 2 in 1, or in other words, the conditions for phase inversion. 

In Eq 7.25 ( ) is the maximum packing volume fraction. Thus, the magnitude of the shielding function My) depends on the reduced volume fraction, = ( ) / ( ). At low concentration, —> 0, the shielding factor vanishes and Einstein s relation is recovered. However, at high concentration, —> 1, the shielding function and relative viscosity both... 

In Figure 7.7, the plots of r vs. < > calculated from Simha s Eq 7.24, Mooney s Eq 7.28, and Krieger-Dougherty s Eq 7.8 are compared with the empirical curve-htted relation, Eq 7.5. For all the relations, the intrinsic viscosity [t]] = 2.5 was used. However, to optimize the fit, different values for the maximum packing volume fraction, ( ) = 0.78, 0.91, and 0.62 respectively, had to be used. Detailed analysis of Thomas data made it possible to conclude... 

The relationships between 17 and ( ) have been derived for suspensions of monodispersed hard spheres in Newtonian liquids. However, most real systems are polydispersed in size, and do not necessarily consist of spherical particles. It has been found that here also Simha s Eq 7.24, Mooney s Eq 7.28, or Krieger-Dougherty s Eq 7.8 are useful, provided that the intrinsic viscosity and the maximum packing volume fraction are defined as functions of particle shape and size polydispersity. For example, by allowing ( ) to vary with composition, it was possible to describe the vs. ( ) variation for bimodal suspensions [Chang and Powell, 1994]. Similarly, after values... 

There are numerous theories based on structural models of suspensions [Mikami, 1980]. Wildemuth and Williams [1984] considered that the maximum packing volume fraction, ( ), is a function of normalized shear stress, CTij = cTjj / M, where M is a numerical parameter. The authors derived the relation ... 

For the relative viscosity of emulsions, in the absence of deformation and coalescence, Eqs 7.24-7.30 may also be used, provided that the intrinsic viscosity is calculated from Eq 7.50 and that the maximum packing volume fraction is treated as an adjustable parameter, dependent on the interphase. This pragmatic approach has been successfully used to describe [r]] vs. (() variation for such complex systems as industrial lattices (at various stages of conversion), plastisols and organosols. 

Vrc y,y r 11 o - volume fraction of dispersed and matrix phase, respectively - volume fraction of the crosslinked monomer units - volume fraction of phase i at phase inversion - maximum packing volume fraction - percolation threshold - shear strain and rate of shearing, respectively - viscosity - zero-shear viscosity - hrst and second normal stress difference coefficient, respectively... 

When the concentration increases, terms higher than linear have to be included in Eq. (16.3). For suspensions of spherical particles a monotonic increase was observed and predicted in the full range of 0 < (p< < max, where < max is the maximum packing volume fraction experimentally, max = 0.62 for monodispersed hard spheres and... 

FIGURE 16.2 Maximum packing volume fraction for disks versus the aspect ratio. Lines are calculated, while the points are experimental. (See the text.)... 

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