Particle maximum packing

Cm particle maximum packing concentration, volume fraction... 

For large amounts of fillers, the maximum theoretical loading with known filler particle size distributions can be estimated. This method (8) assumes efficient packing, ie, the voids between particles are occupied by smaller particles and the voids between the smaller particles are occupied by stiH smaller particles. Thus a very wide filler psd results in a minimum void volume or maximum packing. To get from maximum packing to maximum loading, it is only necessary to express the maximum loading in terms of the minimum amount of binder that fills the interstitial voids and becomes adsorbed on the surface of the filler. 

DpopE related to SpQpp, which is in turn related to the closeness of packing of the powder. The number of particles adjacent to a given particle is represented by The maximum packing density for monosize spheres occurs at hexagonal close packing, where = 12 and = 0.2595 for... 

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] ... 

The bulk density of a powder is calculated by dividing its mass by the volume occupied by the powder (Abdullah Geldart, 1999). Tapped bulk density, or simply tapped density, is the maximum packing density of a powder achieved under the influence of well-defined, externally applied forces (Oliveira et al., 2010). Because the volume includes the spaces between particles as well as the envelope volumes of the particles themselves, the bulk and tapped density of a powder are highly dependent on how the particles are packed. This fact is related to the morphology of its particles and such parameters are able to predict the powder flow properties and its compressibility. 

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. 

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 ... 

Under CP VC conditions, the pigment particles are at a maximum packing density, and the interstices are completely filled with binder. With smaller amounts of binder, the interstices are incompletely filled. The CPVC thus represents a pigment concentration boundary at which abrupt changes in the properties of the film occur. 

The concentration of molecules is usually assumed to equal the number of these molecules Nt in unit volume, c, — N-J V. In the lattice-gas model, the concentration of the component / is characterized by the dt — NJN value, which is the ratio between the actual number of particles in some volume and the maximum possible number of the same particles closely packed in the same volume. So, one then has d( — ct v0, where v0 — V/N. 

The measured bulk volume of 0,17 g of the quartz particles is 0.15 cc. One can calculate the maximum packing density of the bed (0.11/0.15 = 0,73) if the solid volume of the quartz wool is neglected. Accounting for the solid volume of the quartz wool results in a lower packing density, but it is difficult to determine the solid volume of the quartz wool fibers. 

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. 

TABLE 12.3 Effect of Particle Size Distribution on the Maximum Packing Density"... 

Pastes are ceramic suspensions with particle volume fractions near the maximum packing value for the particular particle size distribution. 

Small-size starting particles with an appropriate size distribution that lead to a maximum packing density and final fired density are essential in the sintering step to form dense membranes. The required sintering temperatures are lower because of the active state of the Hne particulate materials used. For example, when particles of an average size of 5 nm made by the alkoxide approach are used to make stabilized zirconia, 1450X instead of the usual 2(X)0X is all that is necessary to produce fully dense material [Mazdiyasni et al., 1%7]. The amount of additives such as the stabilizers affects densification of the final solid electrolyte membranes as well. Generally there is an optimum amount of stabilizer for maximum densification. Excess addition actually can lead to lower densification. 

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

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