Dependence on pore radius

Fig. 6.1 Distribution of cations and anions in pores of a cation-exchanger membrane depends on pore radius which decreases in the sequence A- B— C. In case C the membrane becomes permselective. (According to K. Sollner)... 

According to Coelingh (143) and Brunauer (144), it is possible to subdivide silica gels into four groups, depending on pore radius ... 

The diffusion coefficient in the transition regime, Dp i, can depend on composition, as a result of the second term on the right-hand side of Eqn. (9-22), i.e., the resistance to molecular diffusion. The diffusion coefficient in the transition regime also depends on the pore radius, since Dp depends on pore radius. 

The simple geometry of the model opens the way to transform the fundamental problem of permeability dependence on pore radius into the problem of dependence on specific internal surface. With this step, a permeability estimate from logs becomes possible. 

The influence of interconnection effects is diagrammatically illustrated on the example of a simple system consisting of one wide capillary of radius pj and two capillaries of radius pj (see Fig. 1). Capillary condensation in cylindrical capillary of radius p occurs at one value of relative pressure x+ (X=P/Ps) and desorption at another value of relative pressure x- The values x+ and X- depend on pore radius p, moreover X-(p)>x+(P) In this inequality the capillary hysteresis on the level of one capillary is displayed. It is conditioned by the difference of the mechanisms of capillary condensation and desorption. Capillary condensation occurs by means of spontaneous filling at the moment of the loss of adsorption film stability on the internal surface of capillary. This process is not reversible. Desorption occurs at the moment of equilibrium meniscus formation on the open end of capillary. 

Substitution of Equation 8.11 into 8.10 gives the dependence of the effective diffusivity on pore radius for gases as... 

Fig.3. The theoretical dependence of positronium lifetime on pore radius for - RP-2 -RP 8 - RP-18. Full points - LiChrosorbs RP, open points - LiChrosorbs RP after burning off Triangle - lifetime and pore radius for amorphous silica gel Si-100. The size of symbols is longer than experimental error.

The gas-liquid permporometry combines the controlled stepwise blocking of membrane pores by capillary condensation of a vapor, present as a component of a gas mixture, with the simultaneous measurement of the free diffusive transport of the gas through the open pores of the membrane. The condensable gas can be any vapor provided it has a reasonable vapor pressure and does not react with the membrane. Methanol, ethanol, cyclohexane and carbon tetrachloride have been used as the condensable gas for inorganic membranes. The noncondensable gas can be any gas that is inert relative to the membrane. Helium and oxygen have been used. It has been established that the vapor pressure of a liquid depends on the radius of curvature of its surface. When a liquid is contained in a capillary tube, this dependence is described by the Kelvin equation, Eq. (4-4). This equation which governs the gas-liquid equilibrium of a capillary condensate applies here with the usual assumption of a=0 ... 

The sealing capacity of a rock under hydrostatic conditions is determined by the minimum hydrocarbon-water displacement pressure of the rock, which depends on the radius of the largest connected pore throats in the rock and the oil-water and gas-water interfacial tensions, and in addition on the densities of groundwater and hydrocarbons accumulating in the adjacent reservoir rock. The maximum height of an oil or gas column that can accumulate below a seal is given by Equation 4.17 (Section 4.1.3)... 

There exists a trade-off between the expense at which protons are available and the ease with which they move in different pore regions. Proton concentration decreases from the surface to the center, but their mobility is highest in the pore center and low in the proximity of anionic groups. The distributions of proton density and mobility within the pore determine the dependence of proton conductivity on pore radius. This dependence was studied in detail in Refs. 40, 43. Increase of the pore radius shifts the balance from surface-type to bulk-type conductivity. 

When the pore diameter is small compared with the mean free path, the mode of gaseous diffusion takes on quite another aspect. Instead of colliding with its own type, the molecule will collide much more frequently with the wall. The molecule s progress down its pore will thus depend on pore geometry as well as on the physical characteristics of the diffusing substance. For a straight cylindrical pore of radius d, Knudsen showed that Fick s law could be used with a diffusion coefficient... 

Fig. 3. Dependence of diffusion coefficient in pores on pore radius and on total pressure, for gases only. The values are calculated for an average gas having a bulk diffusion coefficient of 0.33 cm. /second at 1 atm. pressure. Curves calculated from ...

The dependences of pore volume and pore surface-area on pore radius for an active mass formed from 3BS paste on lead—antimony grids are presented in Fig. 10.19a and 10.19b, respectively. 

The function h r, r ) accounts for the possibility that due to hysteresis effects, the interface does not advance completely to pores with capillary radius with increasing S, it depends on the detailed topology of the pore space and, thus, demands more detailed microstructure characterization. T is a factor of order 1, depending on pore geometry and wetting properties. 

Both the Poiseuille and Knudsen models are generally valid in the absence of air in the pores of the membrane, and both show a large dependence of membrane flux on pore radius. In these three models the vapour flux is related to the total pressure difference between the two sides of the membrane and the membrane porosity. 

Fig. 14. Dependence of surface area and pore volume on pore radius.

This equation shows that the vapor pressure P of a droplet depends on its radius and on its surface tension y With analogous reasoning, we can obtain an equation for the vapor pressure of a liquid contained in a pore of radius r. 

Another method used to treat the problem of thermal conductivity of porous Si is the phonon hydrodynamic approach. Alvarez et al. (2010) applied this method to the analysis of thermal conductivity of porous Si, considered as a solid matrix with a random inclusion of small insulating spheres, and they explored the effect of the pore size on the effective thermal conductivity of the material. They also predicted that the thermal conductivity of porous Si depends not only on porosity but also on pore radius. It is lower for higher porosity and for smaller pore radius. This was attributed to phonon ballistic effects. 

FIGURE 8.6 Dependence of pore specific surface area on pore radius for TVEX with following TBP/DVB content, % 1-50/10 2-50/25 3-50/45. 

Note that the mean radius P of the pore forms a perfect spherical meniscus and that this radius will depend on the radius of the tube, and contact angle between liquid and solid tube through the following relation ... 

To determine the distribution of pores with diameters smaller than 20 nm, a nitrogen desorption technique is employed which utilizes the Kelvin equation to relate the pore radius to the ambient pressure. The porous material is exposed to high pressures of N2 such that P/Po 1 and the void space is assumed to be filled with condensed N2, then the pressure is lowered in increments to obtain a desorption isotherm. The vapor pressure of a liquid in a capillary depends on the radius of curvature, but in pores larger than 20 nm in diameter the radius of curvature has little effect on the vapor pressure however, this is of little importance because this region is overlapped by the Hg penetration method. 

All these methods lead to a set of parameters (membrane thickness, pore volmne, hydraulic radius) which are related to the working (macroscopic) permselective membrane properties. In the case of liquid permeation in a porous membrane, macro- and mesoporous structures are more concerned with viscous flow described by the Hagen-Poiseuille and Carman-Kozeny equations whereas the extended Nernst-Plank equation must be considered for microporous membranes in which diffusion and electrical charge phenomena can occur (Mulder, 1991). For gas and vapor transport, different permeation mechanisms have been described depending on pore sizes ranging from viscous flow for macropores to different diffusion regimes as the pore size is decreased to micro and ultra-micropores (Burggraaf, 1996). 

---