Critical porosity

The effect that compression force can have on the disintegration efficiency seems, therefore, largely dependent on the mechanism of the disintegrant action. The effectiveness of swelling or structure recovery may well be dependent on attaining a compression force that achieves a critical porosity in the matrix. On the other hand, the capillary uptake of liquid, which is a necessary precursor to these mechanisms could be compromised if the tablet matrix is compressed to a porosity too low. 

The effects of varying critical porosity on the concentrations of incompatible elements in the extracted melt are displayed in Fig. 3.2. The incompatible element concentrations in the extracted melt increase with the decrease in the critical porosity, owing to more efficient removal of incompatible elements at lower critical porosity. 

It has to be kept in mind, however, that the Heckel equation (Equation 20.9) is only valid for medium pressures but fails at low pressures. This is owing to the fact that there is a critical porosity C, where a reaction force on the punch exerting the presscBBi be measured. This point leads to... 

The estimated critical porosity is 0.3691 0.0541, considering a 95% confidence interval (P = 0.05). This range corresponds to a dextromethorphan hydrobromide content of between 23 and 36% w/w. 

Critically consolidated. If a powder is sheared sufficiently, it will obtain a constant density or critical porosity e for this consolidation normal stress Gc- This is defined as the critical state of the powder, discussed below. If a powder in such a state is sheared, initially the material will deform elastically with shear forces increasing linearly with displacement or strain. Beyond a certain shear stress, the material will fail or flow, after which the shear stress will remain approximately constant as the bulk powder deforms plastically Depending on the type of material, a small peak may be displayed originating from differences between static and dynamic density. Little change in density is observed during shear, as the powder has already reached the desired density for the given applied normal consolidation stress a . 

The effective diffusion coefficient depends on porosity, e, and lattice coordination number, t , as shown in Figure 4.21. Since diffusion between any two lattice sites is assumed to depend only on the molecular diffusion coefficient, A(91) is equal to 1/t. The tortuosity depends on both porosity and lattice coordination number (Figures 4.18 and 4.21). For a given coordination number, the tortuosity increases without bound at the critical porosity. 

In Figure 9.14b, the fraction accessible porosity is plotted versus the total porosity for Bethe lattices of coordination numbers 3 and 7. For all coordination numbers, a is zero for porosities less than the critical value. The critical porosity is indicated for each coordination number by the intercept of the curve with the x-axis. Above the critical porosity, rises sharply. In this transition region, the infinite, lattice-spanning cluster is growing and incorporating pores and smaller pore clusters that are isolated at lower porosities. At high porosities, 0a becomes equal to unity, indicating that all the pores are members of the infinite cluster. 

Even below the critical porosity, pore clusters exist. The clusters are finite in size and do not span the lattice. In the transition region—as the infinite cluster is incorporating more of these finite clusters—finite clusters still exist. For the Bethe lattice, the mean size of these finite clusters, S, depends on the lattice-filling probability (i.e., porosity) [45] ... 

Figure 9.14 Fraction accessible porosity and mean cluster size for a Bethe lattice. Fraction accessible porosity and mean cluster size for Bethe lattices with f = 3 (solid lines) and 7 (dashed lines). The fraction of accessible porosity (b), or the fraction of porosity that is part of an infinite cluster, is plotted versus the total porosity. The mean cluster size (a) exhibits a singularity at the critical porosity.

Crete surface to the bulk of the concrete. Permeability is high (Figure 1.6) and transport processes like, e. g., capillary suction of (chloride-containing) water can take place rapidly. With decreasing porosity the capillary pore system loses its connectivity, thus transport processes are controlled by the small gel pores. As a result, water and chlorides will penetrate only a short distance into concrete. This influence of structure (geometry) on transport properties can be described with the percolation theory [8] below a critical porosity, p, the percolation threshold, the capillary pore system is not interconnected (only finite clusters are present) above p the capillary pore system is continuous (infinite clusters). The percolation theory has been used to design numerical experiments and apphed to transport processes in cement paste and mortars [9]. 

Electrodes consisting of supported metal catalysts are used in electrosynthesis and electrochemical energy conversion devices (e.g., fuel cells). Nanometer-sized metal catalyst particles are typically impregnated into the porous structure of an sp -bonded carbon-support material. Typical carbon supports include chemically or physically activated carbon, carbon black, and graphitized carbons [186]. The primary role of the support is to provide a high surface area over which small metallic particles can be dispersed and stabilized. The porous support should also allow facile mass transport of reactants and products to and from the active sites [187]. Several properties of the support are critical porosity, pore size distribution, crush strength, surface chemistry, and microstructural and morphological stability [186]. 

Both these formulae were initially proposed for ceramics and in equation (6.3) the value of Q was estimated for tensile strength to be between 4 and 7. It has been observed that for values of porosity P below a certain value P equations (6.2) and (6.3) are in close agreement with experimental data. However, at higher porosities for P > P it is advisable to introduce the notion of critical porosity P, corresponding to the strength approaching zero and to use the following equations proposed by Schiller (1958) ... 

Again, for small porosities (f) 0, which implies (f) (p(,) accordance with Equation (95) is guaranteed. In contrast to the Hasselman relation (111) the physical interpretation of in relation (115) as a critical porosity is in principle admissible. As before, under the assumption of spherical pores we can set [ ] = 2, which reduces the number of adjustable fit parameters from two to one. 

Figure 7. Relative tensile modulus of porous ceramics (measured values, predictions and master fit) HS upper bound (thin solid curve), predictions for spherical pores (thin dotted special case of the Spriggs relation Eq. (110), thin dashed Coble-Kingeiy Eq. (117), thick solid modified exponential relation Eq. (114)), experimentally measured values (squares alumina, diamonds ZTA, triangles ATZ, circles zirconia, empty potato starch as a pore-forming agent, full com starch as a pore-forming agent) and master curve (thick dotted curve obtained by fitting with the Pabst-Gregorova relation, Eq. (121), critical porosity 0.729).

By far the best prediction is achieved with the modified exponential relation (114). We emphasize that tliis is, similar to the HS upper bound and the Coble-Kingery prediction, an unbiased a priori prediction without the need for fitting or input parameters of any kind. It is solely based on the assumption that the pores are spherical or isometric. In this case, if the intrinsic tensile modulus is allowed to vary, i.e. if we consider as an adjustable parameter to be determined by fitting according to relation (113), the result is [ ] = 2.09, which is a value very close to 2. Similarly, when a critical porosity is introduced as a fit parameter, i.e. the Mooney-type relation (115) is used with = 2, this critical porosity... 

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