Maximum packing fraction

Apart from chemical composition, an important variable in the description of emulsions is the volume fraction, outer phase. For spherical droplets, of radius a, the volume fraction is given by the number density, n, times the spherical volume, 0 = Ava nl2>. It is easy to show that the maximum packing fraction of spheres is 0 = 0.74 (see Problem XIV-2). Many physical properties of emulsions can be characterized by their volume fraction. The viscosity of a dilute suspension of rigid spheres is an example where the Einstein limiting law is [2]... 

This fitted the data well up to volume fractions of 0.55 and was so successful that theoretical considerations were tested against it. However, as the volume fraction increased further, particle-particle contacts increased until the suspension became immobile, giving three-dimensional contact throughout the system flow became impossible and the viscosity tended to infinity (Fig. 2). The point at which this occurs is the maximum packing fraction, w, which varies according to the shear rate and the different types of packings. An empirical equation that takes the above situation into account is given by [23] ... 

Aluminum composites. See also Aluminum-filled composites maximum packing fraction of, 70 25-26 spatial charge carriers in, 70 22 weight gain during relative humidity aging, 70 24... 

Maximum outlet temperature, 13 253 Maximum packing fraction (MPF), 10 25-26... 

The intrinsic viscosity is the Einstein value [rj] = 2.5 and the packing fraction cpm(0) is that in the low shear limit. As the volume fraction approaches the maximum packing fraction, the viscosity rapidly... 

Figure 6.7 Plot of the stress-dependent packing fraction

The final density of the compact is less than the maximum packing fraction of the particles, PF ax [cf. Eqs. (4.8) and (4.67)], due to frictional forces at particle contacts that retard particle sliding The effectiveness of the compaction process is quantified... 

Oil absorption is a very simple technique which when carefully applied can give a useful guide to the packing ability of fillers [83]. This determines the amount of a selected oil that is needed to just form a continuous phase between the filler particles when they are subjected to a certain mixing procedure. This is a good guide to the maximum packing fraction of filler that is likely to be achievable in a polymer matrix, especially if the oil used is chosen to have a similar polarity to that of the polymer to be used. 

Fig. 20. Dependence of the maximum packing fraction on the specific surface area of the filler in PP/CaC03 composites...

A large number of empirical modifications to this expression have been proposed which model the viscosity of a liquid containing moderate concentrations of spherical particles [5] These include Mooney [6], Maron-Pierce [7] and Krieger-Dougherty [8] expressions which take into account the maximum packing fraction of the particles, and where interaction effects are absent, and can be represented by the general form ... 

In these Equations, G is the modulus of the syntactic foam, G0 is the modulus of the polymer matrix, v0 is Poisson s ratio of the polymer matrix, and 9 is the maximum packing fraction of the filler phase. For uniform spheres, 9 0.64 (see Sect. 3.6). The volume fraction of spheres in the syntactic foam is 9sph. The slope of the G/G0 vs. 9sph curve depends strongly upon whether or not G/G0 is greater or less than 1.0. The slope is negative if the apparent modulus of the hollow spheres is less than the modulus of the polymer matrix. 

Determination of the Maximum Packing Fraction of Polydisperse Fillers and Optimization of the Composition of Composites. .. 142... 

It proved later that it is very convenient to relate the volume fraction of the disperse phase in dispersions to its maximum packing fraction (

large number of equations containing this parameter. Among these a different form of Eq. (17) can be mentioned ... 

The values of the maximum packing fraction cpmax for each narrow filler fraction were found experimentally by filling the voids in it with an inert low-molecular liquid (water, ethanol). 

The concept of the free volume of disperse systems can also be correlated with the change in the structure of the composite of the type solid particles — liquid — gas during its compaction. In that case the value of the maximum packing fraction of filler (p in Eq. (80b) remains valid also for systems containing air inclusions, and instead of the value of the volume fraction of filler, characteristic for a solid particles — liquid dispersion-system solid particles — liquid — gas should be substituted. This value can be calculated as follows the ratio of concentrations Cs x g/Cs, to the first approximation can be substituted by the ratio of the densities of uncompacted and compacted composites, i.e. by parameter Kp. Then Eq. (80b) in view of Eq. (88), for uncompacted composites acquires the form ... 

For reliable application of the free volume concept of disperse systems one must have dependable methods of determination of the maximum packing fraction of the filler tpmax. Unfortunately, the possibility of a reliable theoretical calculation of its value, even for narrow filler fractions, seems to be problematic since there are practically no methods available for calculations for filler particles of arbitrary shape. The most reliable data are those obtained by computer simulation of the maximum packing fraction for spherical particles which give the value associated with possible particle aggregation, so that they are probable for fractions of small particle size. Deviations of particle shape is nearly cubic. At present the most reliable method of determination of [Pg.142]

At the optimum point corresponding to the maximum packing fraction, e1 = e11, since the porosity value at this point does not depend on whether fine or coarse particles form the mixture skeleton. The Equation (90) permit calculation of the porosity values for a mixture of fractions of any composition. A similar procedure can be used for calculating the porosity of mixtures of three or more fractions. A laboratory check of these Equations (90) confirmed good agreement between the calculation forecast and the experimental determination of porosity coefficients and wide variety of combinations of narrow filler fractions. 

FIGURE 12,18 Maximum packing fraction,, as a fimction of the geometric standard... 

Wildemuth, C. R. and Williams, M. C. 1984. Viscosity of suspensions modeled with a shear-dependent maximum packing fraction. Rheol. Acta 23 627-635. 

Nielsen s modification, that takes in to account the maximum packing fraction of filler particles, is given by ... 

Flow of any concentrated suspension will become impossible when the solid particles can form a continuous three-dimensional network of contacts throughout the sample. This so-called maximum packing fraction 4> depends mainly on the particle size distribution and the particle shape. Broader particle size distributions result in lower values of 4>m, because the smaller particles can fill the gaps between the bigger ones, and a deviation from spherical shape results in lower values of 4>m due to steric hindrance of packing. Also flocculation will result in a decrease in the value of 4>in, because the individual floes are only loosely packed. 

Van Houten [33] has shown that the maximum packing fraction obtained from the rheology of concentrated alumina suspensions is predictive for the maximum wet packing fraction that can be obtained in colloidal ceramic processing. 

Note that the viscosity increase with filler addition depends on the properties of filler such as its maximum packing fraction (Figure 9.2). With an increased volume fraction, viscosity increases but this increase would not be the same for example, for three different grades of calcium carbonate. What causes this is difference in their maximum packing fraction. The rate of viscosity increase depends on the ratio ())/())m where ()) is volume fraction of filler added and(j)m is the maximum packing fraction. 

With solid-in-liquid dispersions, such a highly ordered structure - which is close to the maximum packing fraction (q> = 0.74 for hexagonally closed packed array of monodisperse particles) - is referred to as a soHd suspension. In such a system, any particle in the system interacts with many neighbours and the vibrational amplitude is small relative to particle size thus, the properties of the system are essentially time-independent [30-32]. In between the random arrangement of particles in dilute suspensions and the highly ordered structure of solid suspensions, concentrated suspensions may be easily defined. In this case, the particle interactions occur by many body collisions and the translational motion of the particles is restricted. However, this reduced translational motion is less than with solid suspensions - that is, the vibrational motion of the particles is large compared to the particle size. Consequently, a time-dependent system arises in which there will be both spatial and temporal correlation. 

For more concentrated suspensions (q> >0.2), the sedimentation velocity becomes a complex function of At > 0.4, a hindered settling regime is usually entered whereby all of the particles sediment at the same rate (independent of size). A schematic representation for the variation of v with is shown in Figure 9.12, which also shows the variation of relative viscosity with rp. It can be seen from these data that v decreases exponentially with increase in approaches zero when cp approaches a critical value (the maximum packing fraction). The relative viscosity shows a gradual increase with increase in cp such that, when cp = the relative viscosity approaches infinity. 

The maximum packing fraction cp. can be easily calculated for monodisperse rigid spheres. For an hexagonal packing

random packing (Pp = 0.64. The maximum packing fraction increases with polydisperse suspensions for example, for a bimodal particle size distribution (with a ratio of 10 1), 0.8. 

As can be seen from Figure 10.24, v decreases with the increase in (p and ultimately approaches zero when (p exceeds a critical value, monodisperse hard-spheres ranges from 0.64 (for random packing) to 0.74 for hexagonal packing, but exceeds 0.74 for polydisperse systems. For emulsions which are deformable,

[Pg.189]

Catastrophic inversion is illustrated in Figure 10.33, which shows the variation of viscosity and conductivity with the oil volume fraction (f. As can be seen, inversion occurs at a critical maximum packing fraction. 

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