Integral pore distribution

Thomas, A. J., Durrheim, H. H., Alport, M. J. Validation of filter integrity by measurement of the pore-distribution function. Pharm Tech 16 32-43 (1992). 

Typical pore size distribution data are shown in Fig. 5, where the integral penetration of mercury into the pores is plotted as a function of applied pressure. The calculated pore diameters in angstroms are shown across the top. The integral curve clearly shows a bimodal pore distribution with mean pore diameters at 20,000 and 50 A. The latter is at the lower limit of the technique. A nitrogen desorption isotherm is required to obtain an accurate measure in the region below 100 A. 

Ignoring specific interaction effects and the influence of solvent on molecular-coil size, the usual plot of log M vers. vE is equivalent to the integral pore-size distribution in an arbitrary scale [M=pore size ve =/(pore size)]1. For a given substrate with a molecular weight M the available part of the pore volume is a/vj. Functional... 

Equation (44) [or Eq. (46)] allows one to determine the neck radius distribution provided that the void radius distribution is known from independent experiments. For example. Fig. 27 shows the integral radius distributions for necks and voids, 4>(r) and F(r) [Eqs. (32) and (33)], obtained for a model porous structure formed by a dense random package of glass balls having diameters of =250 /xm (S8). The account of interconnection of various pores is seen to lead to an essentially different distribution of necks over radii compared to the independent cylinder model the distribution shifts toward small radii and its slope is less steep. It is also seen that the neck radius distribution is significantly affected by the overlapping of the void and neck radius distributions. 

FIGURE 8.31 Integral pore size distributions in (a) air-dry fibers and (b) fibers incubated with water for 1 h. (Mikhalovsky, S.V., Gun ko, V.M., Bershtein, V.A. et al., A comparative study of air-dry and water swollen flax and cotton fibres, RSC Adv., 2, 2868-2874, 2012. Adapted with permission from The Royal Society of Chemistry.)... 

An example for the integral and differential pore distribution of an activated carbon (Norit R1 Extra) determined at BAM Berlin with a commercial mercury porosimeter is shown in Figures 1.3, 1.4. 

Figure 1.4 shows the pore size distribution of as-polymerized and various stretch-aligned films. The bulk density (0.40 g cm of the pristine film, calculated from the integrated pore volume and the fibril density (1.154 g cm [30], agrees with the value... 

The geometric structure of porous materials can be characterized by a poro-gram the integral pore size distribution function (PSDF), which describes the distribution of pore volume versus pore radii V and r or the differential PSDF (dV/dr), r. 

Bechhold [85] was the first to use Eq. (4) to evaluate pore sizes by measuring the pressure necessary to blow air through a water-filled membrane whose contact angle was assumed to be zero. The bubble-point method allows determination of the maximum pore size present in the pore distribution, corresponding to the minimum pressure necessary to blow the first observed air bubble. This method is frequently used as quality control to check the integrity of filters. 

If flux measurements are started when the relative vapor pressure of the eon-densable gas equals unity, all the pores in the membrane should be elosed (i.e., filled with condensed liquid), avoiding any diffusive flux of the noneondensable gas through the membrane. When the relative vapor pressure is slightly below unity the liquid contained in the largest pores starts to vaporize diereby opening these pores. The Kelvin equation allows correlation of die relative pressure with the size of the pores opened to flux (Kelvin radius, ). The measured flow of the noncondensable gas can be easily translated in terms of pore number onee die appropriate gas transport model is taken into account. By decreasing steadily die relative pressure until all the pores are opened, both the differential and integral pore size distributions can be obtained. 

Equation (2) permits to calculate the portion 0 (x) of free pores at desorption. On the base of the comparison of this value obtained from experimental data with the theoretical value (l) we can determine the portion p>(/o (x)) of overcritical pores. So the integral pore size distribution function is equal to... 

By using the interpolation formulae (4) for Oc(P) obtain for the integral pore size distribution function the final equation ... 

The determination of the evolution of the permeability of these rocks during acidizing is necessary when attempting to predict the evolution of the skin (Equation 2). Previous studies (6) have tried to model the shift of the pore size distribution due to acid attack. Then, permeability profiles were computed by integrating the contributions to the overall flow of each of the rock pores, all over the considered volume of rock. The main limitation of this method lies in the disregarding of the spatial correlation between rock pores. 

The values of the effective permeabilities vary over orders of magnitude, and this corresponds to the different results of the models. Furthermore, as discussed in various papers,the effective permeability of Natarajan and Nguyen (curve e) varies significantly over a very small pressure range, although they state that their capillary-pressure equation mimics data well. With respect to the various equations, the models that use the Leverett J-function all have a similar shape except for that of Berning and Djilali (curve a), who used a linear variation in the permeability with respect to the saturation. The differences in the other curves are due mainly to different values of porosity and saturated permeability. As mentioned above, only the models of Weber and Newman (curve d) and Nam and Kaviany (curve f) have hydrophobic pores, which is why they increase for positive capillary pressures. For the case of Weber and Newman, the curve has a stepped shape due to the integration of both a hydrophilic and a hydrophobic pore-size distribution. 

In Silicalite. A variety of papers are concerned with sorption of methane in the all-silica pentasil, silicalite. June et al. (87) used a Metropolis Monte Carlo method and MC integration of configuration integrals to determine low-occupancy sorption information for methane. The predicted heat of adsorption (18 kJ/mol) is within the range of experimental values (18-21 kJ/ mol) (145-150), as is the Henry s law coefficient as a function of temperature (141, 142). Furthermore, the center of mass distribution for methane in silicalite at 400 K shows that the molecule is delocalized over most of the total pore volume (Fig. 9). Even in the case of such a small sorbate, the channel intersections are unfavorable locations. 

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