Effective pore radius

Effect of the value of the contact angle 6 of mercury on the calculated value of pore radius at different values of applied pressure P... 

Fig. 3J0 Plot of cumulative pore volume against logarithm of r the effective pore radius, (o) For charcoal AY4 A by mercury intrusion O by capillary condensation of benzene, (b) For zinc chloride carbon AYS A by mercury intrusion O by capillary condensation of benzene x by capillary condensation of benzene, after mercury intrusion followed by distillation of mercury under vacuum at temperature rising to 350°C. (Courtesy...

One other cause of hysteresis remains to be mentioned. As was pointed out earlier (p. 177) the contact angle may be different as the mercury is advancing over or receding from a solid surface, and it depends also on the chemical and physical state of the surface the mercury may even react with the surface layer of the solid to form an amalgam. A change in 9 of only a few degrees has a significant effect on the calculated value of pore radius (cf. Table 3.15). 

It depends only on J sJkj A, which is a dimensionless group known as the Thiele modulus. The Thiele modulus can be measured experimentally by comparing actual rates to intrinsic rates. It can also be predicted from first principles given an estimate of the pore length =2 . Note that the pore radius does not enter the calculations (although the effective diffusivity will be affected by the pore radius when dpore is less than about 100 run). 

The permeability coefficients and molecular radii are known. The effective pore radius, R, is the only unknown and is readily calculated by successive approximation. Consequently, unknown parameters (i.e., porosity, tortuosity, path length, electrical factors) cancel, and the effective pore radius is calculated to be 12.0 1.9 A. Because the Renkin function [see Eq. (35)] is a rapidly decaying polynomial function of molecular radius, the estimation of R is more sensitive to small uncertainties in the calculated molecular radius values than it is to experimental variabilities in the permeability coefficients. The placement of the perme-ants within the molecular sieving function is shown in Figure 9 for the effective... 

Table 5 Permeability Coefficients of the Paracellular Pathway and Estimation of the Effective Pore Radius and Molecular Restriction Factor for the Caco-2 Cell Monolayer... 

From these data one can calculate the effective radius of the pores through which solutes diffuse across the junctional strands (Fig. 16). At day 3, the pore radius was —5.5 A. This correlates with pore radii of —10 A for dog alveolar epithelium (Taylor and Gaar, 1970) and 5 and 8 A for rabbit and bullfrog gallblad-... 

Figure 16 Correlation of effective pore radius of the paracellular route of rat alveolar epithelial cell monolayers with transepithelial electrical resistance and time in culture. Pore radii were calculated from the data shown in Figure 14.

Hence, Tct is seen to increase with pore density and pore radius. However, a problem appears at a porous substrate when thin films are to be deposited during metallization to form interconnections, thin-film capacitors, etc.335 Sputtered material falls deep into the pores, which affects the planarity of the deposited layer and the electrical resistivity of the oxide layer underneath.335 To cope with this effect, the porous oxide should be padded by inorganic (A1203 and Si02) or organic (polyimide, negative photoresist) layers. 

When the hydrogen pressure is 1 atm, and the temperature is 77 °K, the experimentally observed (apparent) rate constant is 0.159 cm3/ sec-g catalyst. Determine the mean pore radius, the effective diffusivity of hydrogen, and the catalyst effectiveness factor. 

If we assume a quasi-cylinder pore structure of the electrode material as in Fig. 1, an average effective pore radius r can be evaluated from the relationship r = 2V/A, where V is the total pore volume, and A is the total pore surface that can also be determined using the DFT method (see also [5]). Then the migration coefficients determined as shown in Fig. 5 can be plotted vs. the pr2 product - see, e.g., Fig. 7 for five electrodes, which were made of various porous carbons produced by Skeleton Technologies. 

Fig 1- Effect of the polymerization temperature on the pore size distribution. At the highest temperature (T+8) the average pore radius is 400 nm while at the lowest T the pores are much larger with an average pore radius of 850 nm... 

There are few studies in literature reporting pure gas permeabilities as well as separation factors of mixtures. Vuren et al. (1987) reported Knudsen diffusion behavior of pure gases for y-alumina membranes with a mean pore radius of 1.2 nm. Separation experiments with a 1 1 H2/N2 mixture showed, that the theoretical Knudsen separation factor [of 3.7, Equation 6.4)j for this mixture could be obtained (Keizer et al. 1988 see also Figure 6.2). In Figure 6.2, the effect of process parameters is also demonstrated. The separation factor is a function of the pressure ratio over the membrane, which is the ratio of the pressure on the permeate-side to that on the feed-side. For pressure ratios approaching unity, which means the pressure on both sides of the... 

Surface diffusion has been extensively studied in literature. An overview of experimental data is given in Table 6.1. Okazaki, Tamon and Toei (1981), for example, measured the transport of propane through Vycor glass with a pore radius of 3.5 nm at 303 K and variable pressure (see Table 6.1). The corrected gas phase permeability was 0.69 m -m/m -h-bar, while the surface permeability was 0.55 m -m/m -h-bar, and so almost as large as the gas phase permeability (Table 6.1). It is clear from Table 6.1, that the effects of surface diffusion, especially at moderate temperatures, can be pronounced. At higher temperatures, adsorption decreases and it can be expected that surface diffusion will become less pronounced. 

Effects of annealing are also observed on the water transport properties. Both the diffusional permeability and the permeability under osmotic pressure decrease in comparison with the sample before annealing (Table IV). Moreover, the g ratio also decreases, which, in terms of the equivalent pore radius, means that the membrane becomes tighter upon annealing. 

Figure 3. Separation factor-particle diameter behavior computed from the pore-partitioning model showing the effect of the Hamaker constant at a low eluant ionic strength (O.OOl M). Other parameters are = 0.60, interstitial capillary radius = l6 fim, pore radius = fim,...

This convolution equation takes into account the fact that Kqpc, the distribution coefficient, is a function of both the effective solute radius (f) and the pore radius (r). Even if all the pores of a gel have the same diameter, solutes of... 

R is the effective mean radius of macromolecules dissolved in the eluent d is the effective mean pore radius... 

In addition to the packed bed acting as an ultrafilter, the porous frits used at both ends of the column may act as very effective filtering devices. Thus a 2-vim porosity frit would have an average pore radius of 1 lun. Because of the tortuosity and relatively wide pore-size distribution present in frits, it would be safe to assume that it contains much smaller crevices which can entrap macromolecules. 

The case of transport through microporous membranes is different from that of macroporous membranes in that the pore size approaches the size of the diffusing solute. Various theories have been proposed to account for this effect. As reviewed by Peppas and Meadows [141], the earliest treatment of transport in microporous membranes was given by Faxen in 1923. In this analysis, Faxen related a normalized diffusion coefficient to a parameter, X, which was the ratio of the solute radius to the pore radius... 

In real porous solids, the pores are not straight, and the pore radius can vary. Two parameters are used to describe the diffusion path through real porous solids the void fraction, e, defined as fhe ratio of pore area to fofal cross-sectional area, and the tortuosity, r, which corrects for the fact that pores are not straight. The resulting effective diffusivity is then... 

Figure 9.12 contains sketches for several different models of pores that will be useful in our discussion of capillary condensation. Figure 9.12a is the simplest, attributing the entire effect just described to variations in pore radius with the depth of the pore. That is, when liquid first begins to condense in the pore, the larger radius Ra determines the pressure at which the adsorption-condensation occurs. Once the pore has been filled and the desorption-evaporation branch is being studied, the smaller radius Rd determines the equilibrium pressure. Although bottlenecked pores of this sort may exist in some cases, this model seems far too specialized to account for the widespread occurrence of hysteresis. 

Gas adsorption data may be analyzed for the distribution of pore sizes. What is generally done is to interpret one branch of the isotherm and use an appropriate equation to calculate the effective pore radius at a given pressure. The amount of material adsorbed or desorbed for each increment or decrement in pressure measures the volume of pores with that effective radius. 

It will be noted that the importance of the correction for surface conductivity increases as Rc decreases and vanishes as Rc - oo. Equation (54) also suggests that the numerical evaluation of ks may be accomplished by studying electroosmosis in a set of capillaries identical in all respects except for variability in Rc. Finally, the expansion of Equation (50) to Equation (54) in correcting for surface conductivity explicitly assumes a cylindrical capillary. Experiments made with porous plugs cannot be corrected for surface conductivity by Equation (54), but the qualitative conclusion that the effect of surface conductivity increases as the pore radius decreases is valid in this case also. 

The pores in these materials can be so small that they exert constraining effects on chains contained within them. For example, no glass transition was observed for PS in a mesoporous hexagonal silica [243] with a pore radius of 28 A [144]. 

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