Diameter, pore

Distribution of Pore Diameter. Pore diameters of the PS layer formed under a given set of conditions have a distinct distribution. Normal, log-normal, bimodal, fractal, and nonuniform distributions have been found for the PS formed under different conditions. 

Illumination during formation of PS on p-Si has been found to affect the distribution of pore diameter it increases the amount of the smaller nanocrystals, while reducing the amount of larger crystals. For the PS formed under an illuminated substrate, the relative amount of small crystals is found to increase with reduction of light wavelength.  

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Fig. XVII-27. Nitrogen adsorption at 77 K for a series of M41S materials. Average pore diameters squares, 25 A triangles, 40 A circles, 45 A. Adsorption solid symbols desorption open symbols. The isotherms are normalized to the volume adsorbed at Pj = 0.9. (From Ref. 187. Reprinted with kind permission from Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)...

Anotlier important modification metliod is tire passivation of tire external crystallite surface, which may improve perfonnance in shape selective catalysis (see C2.12.7). Treatment of zeolites witli alkoxysilanes, SiCl or silane, and subsequent hydrolysis or poisoning witli bulky bases, organophosphoms compounds and arylsilanes have been used for tliis purjDose [39]. In some cases, tire improved perfonnance was, however, not related to tire masking of unselective active sites on tire outer surface but ratlier to a narrowing of tire pore diameters due to silica deposits. 

The limiting cases of greatest interest correspond to conditions in which the mean free path lengths are large and small, respectively, compared with the pore diameters. Recall from the discussion in Chapter 3 that the effective Knudsen diffusion coefficients are proportional to pore diameter and independent of pressure, while the effective bulk diffusion coefficients are independent of pore diameter and inversely proportional to pressure. 

Thus, when the pore diameters are sufficiently small or the pressure suffi-e B... 

The Knudsen diffusion coefficients are given by equations (2.11) in which K. is independent of pressure and proportional to pore diameter a, so that we can write... 

The limiting form of the flux equations for large pore diameters or high pressure is best approached starting from equations (5.7) and (5.8). 

This determines the total flux at the li/nic of viscous flow. Equations (5.18 and (5.19) therefore describe the limiting form of the dusty gas model for high pressure or large pore diameters -- the limit of bulk diffusion control and viscous flow,... 

Consideration of the behavior of equations (5.26) and (5,27) when the pressure or the pore diameter becomes Large illustrates the care which must... 

It ls not surprising chat such a relation should hold at the Limit of Knudsen diffusion, since Che Knudsen diffusion coefficients are themselves inversely proportional to the square roots of molecular weights, but the pore diameters in Graham s stucco plugs were certainly many times larger chan the gaseous mean free path lengths at the experimental conditions. 

When the mean free paths are long compared with all pore diameters, condition (i) above determines the form of the flux relations completely in isothermal systems, and no further modelling is required if is regarded... 

Let us now assemble the complete set of dimensionless parameters for the problem. These are set out in Table 11.1, where the last column indicates the nature of their dependence on the external pressure p, the mean pore diameter and the pellet radius a. Symbols ft and 0... 

At Che opposite limit, where Che density Is high enough for mean free paths to be short con ared with pore diameters, the problem can be treated by continuum mechanics. In the simplest ease of a straight tube of circular cross-section, the fluid velocity field can easily be obtained by Integrating Che Nsvler-Stokes equations If an appropriate boundary condition at Che... 

These calculations lend theoretical support to the view arrived at earlier on phenomenological grounds, that adsorption in pores of molecular dimensions is sufficiently different from that in coarser pores to justify their assignment to a separate category as micropores. The calculations further indicate that the upper limit of size at which a pore begins to function as a micropore depends on the diameter a of the adsorbate molecule for slit-like pores this limit will lie at a width around I-So, but for pores which approximate to the cylindrical model it lies at a pore diameter around 2 5(t. The exact value of the limit will of course depend on the actual shape of the pore, and may well be raised by cooperative effects. 

Adsorbent Pore diameter, nm Particle density, g/cm Specific area, mVg Apphcations... 

Micropore Diffusion. In very small pores in which the pore diameter is not much greater than the molecular diameter the diffusing molecule never escapes from the force field of the pore wall. Under these conditions steric effects and the effects of nonuniformity in the potential field become dominant and the Knudsen mechanism no longer appHes. Diffusion occurs by an activated process involving jumps from site to site, just as in surface diffusion, and the diffusivity becomes strongly dependent on both temperature and concentration. 

However, the size of the pores is the most important initial consideration because, if a molecule is to be adsorbed, it must not be larger than the pores of the adsorbent. Conversely, by selecting an adsorbent with a particular pore diameter, molecules larger than the pores may be selectively excluded, and smaller molecules can be allowed to adsorb. 

Typical pore size distributions for these adsorbents have been given (see Adsorption). Only molecular sieve carbons and crystalline molecular sieves have large pore volumes in pores smaller than 1 nm. Only the crystalline molecular sieves have monodisperse pore diameters because of the regularity of their crystalline stmctures (41). 

The search for a suitable adsorbent is generally the first step in the development of an adsorption process. A practical adsorbent has four primary requirements selectivity, capacity, mass transfer rate, and long-term stabiUty. The requirement for adequate adsorptive capacity restricts the choice of adsorbents to microporous soUds with pore diameters ranging from a few tenths to a few tens of nanometers. 

A surprisiagly large number of important iadustrial-scale separations can be accompHshed with the relatively small number of zeoHtes that are commercially available. The discovery, characterization, and commercial availabiHty of new zeoHtes and molecular sieves are likely to multiply the number of potential solutions to separation problems. A wider variety of pore diameters, pore geometries, and hydrophobicity ia new zeoHtes and molecular sieves as weU as more precise control of composition and crystallinity ia existing zeoHtes will help to broaden the appHcations for adsorptive separations and likely lead to improvements ia separations that are currently ia commercial practice. 

Fig. 4. Surface of a polysulfone ultrafUtration hoUow-fiber membrane spun with poly-(vinylpyrrohdinone) (3). Surface pore diameter is 0.2—0.4 p.m.

Fig. 3. Microporous membranes are characterized by tortuosity, T, porosity, S, and their average pore diameter, d. (a) Cross-sections of porous membranes containing cylindrical pores, (b) Surface views of porous membranes of equal S, but differing pore size.

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