Distribution of Pores

All the simulations presented previously were carried out assuming the presence of one or multiple uniform pores of the same geometry. However, in real systems, there are many different pores of different geometry. 

The simplest method of taking into account the distribution of pores of different sizes is to use the transmission line ladder network (Fig. 9.4,9.14,9.17, or 9.19) and use different values for the parameters ri, r, and Ci or interfacial impedances z-, and calculate the total admittance by the addition of the admittances of the small pore elements. Such a method was used, for example, by Macdonald et al. [446,447] and Pyun et al. [448]. Although such a model can be used to simulate impedance spectra assuming changes in parameters with the position in the pore, it is difficult to obtain the pore parameters from the experimental spectra. 

Another method is to assume a certain distribution of pore parameters. Song et al. [449 52], in a series of papers, considered the distribution of pore parameters for electrodes in the absence of electroactive species, i.e., for purely capacitive electrodes. De Levie s equation (9.7) is applicable to individual pores, but for different pores different values of the parameter A are obtained. The dimensionless penetrability parameter, a, was defined as 

---

A procedure that is more suitable for obtaining the actual distribution of pore sizes involves the use of a nonwetting liquid such as mercury—the contact angle on glass being about 140° (Table X-2) (but note Ref. 31). If all pores are equally accessible, only those will be filled for which... 

Below the critical temperature of the adsorbate, adsorption is generally multilayer in type, and the presence of pores may have the effect not only of limiting the possible number of layers of adsorbate (see Eq. XVII-65) but also of introducing capillary condensation phenomena. A wide range of porous adsorbents is now involved and usually having a broad distribution of pore sizes and shapes, unlike the zeolites. The most general characteristic of such adsorption systems is that of hysteresis as illustrated in Fig. XVII-27 and, more gener-... 

For a second active carbon, AG, the DR plot was convex to the logio(p7p) This carbon was believed from X-ray results to have a wider distribution of pores. It was found that the isotherms of both benzene and cyclohexane could be interpreted by postulating that the micropore system consisted of two sub-systems each with its own Wq and and with m = 2 ... 

As pointed out earlier (Section 3.5), certain shapes of hysteresis loops are associated with specific pore structures. Thus, type HI loops are often obtained with agglomerates or compacts of spheroidal particles of fairly uniform size and array. Some corpuscular systems (e.g. certain silica gels) tend to give H2 loops, but in these cases the distribution of pore size and shape is not well defined. Types H3 and H4 have been obtained with adsorbents having slit-shaped pores or plate-like particles (in the case of H3). The Type I isotherm character associated with H4 is, of course, indicative of microporosity. 

In a typical amorphous adsorbent the distribution of pore size may be very wide, spanning the range from a few nanometers to perhaps one micrometer. Siace different phenomena dominate the adsorptive behavior ia different pore size ranges, lUPAC has suggested the foUowiag classification ... 

Fig. 26. Screen filters contain pores of a uniform size and retain all particulates greater than the pore diameter at the surface of the membrane. Depth filters contain a distribution of pore sizes. Particulates entering the membrane are trapped at constrictions within the membrane. Both types of filters are rated 10...

Adsorption. Although several types of microporous soHds are used as adsorbents for the separation of vapor or Hquid mixtures, the distribution of pore diameters does not enable separations based on the molecular-sieve effect. The most important molecular-sieve effects are shown by crystalline zeoHtes, which selectively adsorb or reject molecules based on differences in molecular size, shape, and other properties such as polarity. The sieve effect may be total or partial. 

Characterization. The proper characterization of coUoids depends on the purposes for which the information is sought because the total description would be an enormous task (27). The foUowiag physical traits are among those to be considered size, shape, and morphology of the primary particles surface area number and size distribution of pores degree of crystallinity and polycrystaUinity defect concentration nature of internal and surface stresses and state of agglomeration (27). Chemical and phase composition are needed for complete characterization, including data on the purity of the bulk phase and the nature and quaHty of adsorbed surface films or impurities. 

There are many complications with interpreting MWCO data. First, UF membranes have a distribution of pore sizes. In spite of decades of effort to narrow the distribution, most commercial membranes are not notably sharp. What little is known about pore-size distribution in commercial UF membranes fits the Poisson distribution or log-normal distribution. Some pore-size distributions may be polydisperse. 

The graphite microstmcture is assumed to contain a log-normal distribution of pores. Under these circumstances, for a specific defect, the probability that its length falls between c and c+dc is f(c)dc, with f(c) defined as ... 

How does the structure of the media, including the size, shape, geometry, and distribution of pores, influence the transport characteristics for a given solute ... 

A question of practical interest is the amount of electrolyte adsorbed into nanostructures and how this depends on various surface and solution parameters. The equilibrium concentration of ions inside porous structures will affect the applications, such as ion exchange resins and membranes, containment of nuclear wastes [67], and battery materials [68]. Experimental studies of electrosorption studies on a single planar electrode were reported [69]. Studies on porous structures are difficult, since most structures are ill defined with a wide distribution of pore sizes and surface charges. Only rough estimates of the average number of fixed charges and pore sizes were reported [70-73]. Molecular simulations of nonelectrolyte adsorption into nanopores were widely reported [58]. The confinement effect can lead to abnormalities of lowered critical points and compressed two-phase envelope [74]. 

Natural rocks seldom have a single pore size but rather a distribution of pore sizes. If all pores are in the fast-diffusion limit, have the same surface relaxivity and have no diffirsional coupling, then the pores will relax in parallel with a distribution of relaxation times that corresponds to the distribution of the pore sizes. The magnetization will decay as a sum of the exponentials as described by Eq. (3.6.4). 

The relaxation time for each pore will still be expressed by Eq. (3.6.3) where each pore has a different surface/volume ratio. Calibration to estimate the surface relaxivity is more challenging because now a measurement is needed for a rock sample with a distribution of pore sizes or a distribution of surface/volume ratios. The mercury-air or water-air capillary pressure curve is usually used as an estimator of the cumulative pore size distribution. Assuming that all pores have the same surface relaxivity and ratio of pore body/pore throat radius, the surface relaxivity is estimated by overlaying the normalized cumulative relaxation time distribution on the capillary pressure curve [18, 25], An example of this process is illustrated in Figure 3.6.5. The relationship between the capillary pressure curve and the relaxation time distribution with the pore radii, assuming cylindrical pores is expressed by Eq. (3.6.5). 

In addition, mercury intrusion porosimetry results are shown together with the pore size distribution in Figure 3.7.3(B). The overlay of the two sets of data provides a direct comparison of the two aspects of the pore geometry that are vital to fluid flow in porous media. In short, conventional mercury porosimetry measures the distribution of pore throat sizes. On the other hand, DDIF measures both the pore body and pore throat. The overlay of the two data sets immediately identify which part of the pore space is the pore body and which is the throat, thus obtaining a model of the pore space. In the case of Berea sandstone, it is clear from Figure 3.7.3(B) that the pore space consists of a large cavity of about 85 pm and they are connected via 15-pm channels or throats. 

The BET surface of ACC, oxidized ACC and Pt/ACC were 1300, 680, and 580 m2g" respectively. Surprisingly, the distribution of pore radius in the three samples exhibited 4 sharp peaks centered at the same position at 0.37, 0.55, 0.75, 0.95 nm, respectively (Table 1). Therefore, neither the NaOCl oxidizing treatment, nor the metal loading modified the micropore size. However, the peak heights decreased in the series ACC ACC(oxidized) > Pt/ACC resulting in a decrease of the differential volumes dV/dr given in Table 1. Therefore, the... 

Table 1 Distribution of pore radius and pore volume.

Nitrogen adsorption isotherms were measured with a sorbtometer Micromeretics Asap 2010 after water desorption at 130°C. The distribution of pore radius was obtained from the adsorption isotherms by the density functional theory. Electron microscopy study was carried out with a scanning electron microscope (SEM) HitachiS800, to image the texture of the fibers and with a transmission electron microscope (TEM) JEOL 2010 to detect and measure metal particle size. The distribution of particles inside the carbon fibers was determined from TEM views taken through ultramicrotome sections across the carbon fiber. 

In a given porous medium there will be a distribution of pore-body and pore-throat sizes. If a lamella exits a constriction into a pore body whose ratio, R /R, is less than... 

---