Concentrated suspensions maximum packing fraction

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. 

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. 

In this way, /3 is related to the particle volume fraction in terms of the maximum packing fraction such that the separation between the particle surfaces approaches zero in the limit effective microstructure of a flowing suspension is a simple cubic (, = 0.52), or body centered cubic = 0.68), or face centered cubic = 0.74). It is therefore assumed that Eq. (9.3.9) is also applicable to other effective suspension microstructures such as the random microstructure. Equation (9.3.8) is appropriate only for high solid volume fractions ( 2 0.25) since it was developed for concentrated suspensions for which the average separation distance between two similar size particles is close to or less than the particle size. 

SENGUN, M.Z. c PROBSTEIN, R.F. 1989a. High-shear-limit viscosity and the maximum packing fraction in concentrated monomodal suspensions. PhysicoChem. Hydrodynamics 11, 229-241. 

How do these effects occur When any suspension of particles is placed next to a smooth wall, the original microstructure is locally disturbed. For a simple suspension at rest where the particles are randomly dispersed in space, the concentration of particles undergoes a damped oscillatory variation as one moves away from the wall, see figpre 16, where the concentration is at the maximum packing fraction, so that the effect is enhanced. The new distribution has two effects, first that the variation in concentration does not die out until about five particle radii away from the wall, and secondly that the average particle concentration is zero at the wall and less than average for a small distance away from the wall. 

Eq. (42) is only valid when < ><1%. Ball [26] assumed that the effect of the panicle in a concentrated suspension is the sum of the effect of the particle added sequentially. The free volume unoccupied by the particle relative to the maximum packing fraction is ... 

The dependence of viscosity on volume fraction sohds is shown in Fig. 8.88. At high particle concentrations, viscosity of the suspension increases more rapidly than predicted by the above equation due to interparticle interactions. Several empirical equations are available to relate viscosity to the solid concentration behavior of suspensions. As the volume fraction of solids is increased further, a stage will be reached where the particles will be interlocked and no flow will occur (i.e., viscosity approaches infinity). The volume fraction of sohds at which this occurs is called the maximum packing fraction and its... 

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... 

If/m is the maximum packing density of the particles, which is defined as the volume fraction at which the particles touch one another, so that flow is not possible, then the actual particle volume firaction/used in injection molding is lower than/m by 5-10 vol%. This means that in a well-dispersed suspension, the particles are separated from one another by a thin layer of polymer with a thickness of about 50 nm dming the molding, so that the mixture is able to flow. Therefore, the volume fraction of particles / is determined by the particle size and distribution and the particle shape. In practice, the volume firaction of ceramic powders is determined from viscosity measurements by using a capillary rheometer. Data for the relative viscosity, i.e., the viscosity of the mixture divided by the viscosity of the unfilled polymer versus particle concentration can be well fitted by the following equation [209] ... 

Fedors (214) took into account the maximum packing density of spheres in suspension to develop an equation for very high concentrations. The volume fraction of random dense-packed spheres is 0.63. 

Integrating this equation to any volume fraction, q>, with the boundary conditions that the elasticity is equal to the network value when there are no particles present, and that when the volume fraction reaches packed filler bed), then gives us Equation (2.64). There is an analogous equation describing the viscosity of suspensions of particles and this will be introduced in Chapter 3. When a value of 0.64 is used for the maximum filler concentration, Equation (2.64) becomes... 

Generally, when 0.1 < 0 < 0 the suspensions are considered to be concentrated and the above discussed equations do not apply. Here 0 is defined as the maximum attainable concentration and has fire following form 0 = 1 — e, where e is the void fraction or porosity, and is defined as the ratio of the void volume to that total volume. Theoretically, the value of 0 is 0.74 for equal spheres in compact hexagonal packing, but in practice it is more like 0.637 for random hexagonal packing or 0.524 for cubic packing ([79]). 

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