Pores pore dimensions

Now, in principle, the angle of contact between a liquid and a solid surface can have a value anywhere between 0° and 180°, the actual value depending on the particular system. In practice 6 is very difficult to determine with accuracy even for a macroscopic system such as a liquid droplet resting on a plate, and for a liquid present in a pore having dimensions in the mesopore range is virtually impossible of direct measurement. In applications of the Kelvin equation, therefore, it is almost invariably assumed, mainly on grounds of simplicity, that 0 = 0 (cos 6 = 1). In view of the arbitrary nature of this assumption it is not surprising that the subject has attracted attention from theoreticians. 

This simple model illustrates how the fraction K and, through it, Vj are influenced by the dimensions of both the solute molecules and the pores. For solute particles of other shapes in pores of different geometry, theoretical expressions for K are quantitatively different, but typically involve the ratio of solute to pore dimensions. 

Figure 9,16 Comparison of theory with experiment for rg/a versus K. The solid line is drawn according to the theory for flexible chains in a cylindrical pore. Experimental points show some data, with pore dimensions determined by mercury penetration (circles, a = 21 nm) and gas adsorption (squares, a= 41 nm). [From W. W. Yau and C. P. yidXont, Polym. Prepr. 12 797 (1971), used with permission.]...

If the solute size is approximately the (apparent) membrane-pore size, it interferes with the pore dimensions. The solute concentration in the permeate first increases, then decreases with time. The point of maximum interference is further characterized as a minimum flux. Figure 4 is a plot of retention and flux versus molecular weight. It shows the minimum flux at ca 60—90% retention. 

If the solute size is greater than the pore dimensions, the solute is retained by mechanical sieving. 

These tetrahedra are arranged in a number of ways to give the different zeohtes. The stmctures are unique in that they incorporate pores as part of the regular crystalline stmctures. The pores have dimensions of the order of molecular dimensions so that some molecules fit into the pores and some do not. Hence the zeohtes are molecular sieves (qv), and they are apphed in industrial separations processes to take advantage of this property. Some zeohtes and their pore dimensions are hsted in Table 2. 

Table 2. Zeolites and Their Pore (Aperture) Dimensions ...

Fig. 15. Pore dimensions of 2eoHtes and critical dimensions of hydrocarbons (80).

The factors to consider in the selection of cross-flow filtration include the cross-flow velocity, the driving pressure, the separation characteristics of the membrane (permeability and pore size), size of particulates relative to the membrane pore dimensions, and the hydrodynamic conditions within the flow module. Again, since particle-particle and particle-membrane interactions are key, broth conditioning (ionic strength, pH, etc.) may be necessary to optimize performance. 

Bead Composition (% agarose) Size exclusion pore dimension° (nm) Fractionation range Maximum Maximum operating linear velocity (cm/hr) ... 

Type Bead size djo (/xm) Size exclusion pore dimension (nm) Exclusion limit protein (M,) Fractionation range protein (/V ,) pH stability (long term/ short term)... 

Table 4 shows //(ApH) at the 7 mAh stage. We can see that //(ApH) is very small for CMD and nonporous manganese dioxide (NMD). This is very reasonable since the pore diameter of CMD and NMD is much bigger than that of EMD. This is why EMD for ZnCl2 cells is usually produced at high current, in order to increase its pore dimensions. 

Figure 10.1 Framework types of some well-known zeolites with their specific pore size and pore network dimensions [5].

Synge, RLM, Experiments on Electrical Migration of Peptides and Proteins Inside Porous Membranes Influences of Adsorption, Diffusion, and Pore Dimensions, Biochemical Journal 65, 266,1957. 

Because the pore dimensions in narrow pore zeolites such as ZSM-22 are of molecular order, hydrocarbon conversion on such zeolites is affected by the geometry of the pores and the hydrocarbons. Acid sites can be situated at different locations in the zeolite framework, each with their specific shape-selective effects. On ZSM-22 bridge, pore mouth and micropore acid sites occur (see Fig. 2). The shape-selective effects observed on ZSM-22 are mainly caused by conversion at the pore mouth sites. These effects are accounted for in the hydrocracking kinetics in the physisorption, protonation and transition state formation [12]. 

In industry, the emphasis is mainly on developing an active, selective, stable and mechanically robust catalyst. To accomplish this, tools are needed which identify those structural properties that discriminate efficient from less efficient catalysts. All information that helps to achieve this is welcome. Empirical relationships between those factors that govern catalyst composition (e.g. particle size and shape, and pore dimensions) and those that determine catalytic performance are extremely useful in catalyst development, although they do not always give fundamental insights into how the catalyst operates on the molecular level. 

Figure 3.26. Kinetic diameters of some important organic molecules. For reference the pore dimensions of some common zeolites are shown (Van de Graaf et al., 1998).

The situation becomes quite different in heterogeneous systems, such as a fluid filling a porous medium. Restrictions by pore walls and the pore space microstructure become relevant if the root mean squared displacement approaches the pore dimension. The fact that spatial restrictions affect the echo attenuation curves permits one to derive structural information about the pore space [18]. This was demonstrated in the form of diffraction-like patterns in samples with micrometer pores [19]. Moreover, subdiffusive mean squared displacement laws [20], (r2) oc tY with y < 1, can be expected in random percolation clusters in the so-called scaling window,... 

For displacements shorter than the mean pore dimension, (z2) < a, where flow velocities tend to be spatially constant and homogeneously distributed, Brownian diffusion is the only incoherent transport phenomenon that contributes to the hydrodynamic dispersion coefficient. As a direct consequence, the dispersion coefficient approaches the ordinary Brownian diffusion coefficient,... 

For polyatomic gases in porous media, however, the relaxation rate commonly decreases as the pore size decreases [18-19]. Given that the relaxation mechanism is entirely different, this result is not surprising. If collision frequency determines the Ti, then in pores whose dimensions are in the order of the typical mean free path of a gas, the additional gas-wall collisions should drastically alter the T,. For typical laboratory conditions, an increase in pressure (or collision frequency) causes a proportional lengthening of T1 so the change in T, from additional wall collisions should be a good measure of pore size. 

On this basis the porosity and surface composition of a number of silicas and zeolites were varied systematically to maximize retention of the isothizolinone structures. For the sake of clarity, data is represented here for only four silicas (Table 1) and three zeolites (Table 2). Silicas 1 and 3 differ in their pore dimensions, these being ca. 20 A and 180 A respectively. Silicas 2 and 4, their counterparts, have been calcined to optimise the number and distribution of isolated silanol sites. Zeolites 1 and 2 are the Na- and H- forms of zeolite-Y respectively. Zeolite 3 is the H-Y zeolite after subjecting to steam calcination, thereby substantially increasing the proportion of Si Al in the structure. The minimum pore dimensions of these materials were around 15 A, selected on the basis that energy-minimized structures obtained by molecular modelling predict the widest dimension of the bulkiest biocide (OIT) to be ca. 13 A, thereby allowing entry to the pore network. 

Change in wettability or reduction in pore dimensions by adsorption (organics with large molecular weight). 

Ordinary or bulk diffusion is primarily responsible for molecular transport when the mean free path of a molecule is small compared with the diameter of the pore. At 1 atm the mean free path of typical gaseous species is of the order of 10 5 cm or 103 A. In pores larger than 1CT4 cm the mean free path is much smaller than the pore dimension, and collisions with other gas phase molecules will occur much more often than collisions with the pore walls. Under these circumstances the effective diffusivity will be independent of the pore diameter and, within a given catalyst pore, ordinary bulk diffusion coefficients may be used in Fick s first law to evaluate the rate of mass transfer and the concentration profile in the pore. In industrial practice there are three general classes of reaction conditions for which the bulk value of the diffusion coefficient is appropriate. For all catalysts these include liquid phase reactions... 

---