Kelvin radius

This equation describes the additional amount of gas adsorbed into the pores due to capillary action. In this case, V is the molar volume of the gas, y its surface tension, R the gas constant, T absolute temperature and r the Kelvin radius. The distribution in the sizes of micropores may be detenninated using the Horvath-Kawazoe method [19]. If the sample has both micropores and mesopores, then the J-plot calculation may be used [20]. The J-plot is obtained by plotting the volume adsorbed against the statistical thickness of adsorbate. This thickness is derived from the surface area of a non-porous sample, and the volume of the liquified gas. 

The procedures described so far have all required a pore model to be assumed at the outset, usually the cylinder, adopted on the grounds of simplicity rather than correspondence with actuality. Brunauer, Mikhail and Bodor have attempted to eliminate the over-dejjendence on a model by basing their analysis on the hydraulic radius r rather than the Kelvin radius r . The hydraulic radius is defined as the ratio of the cross-sectional area of a tube to its perimeter, so that for a capillary of uniform cross-section r is equal to the ratio of the volume of an element of core to... 

Use of the condition of constant meniscus volume is most appropriate when growth and dissolntion of the meniscus is comparatively slow. An alternative is to consider the Kelvin eqnilibrium condition. The Kelvin eqnation relates the eqnilibrinm meniscus curvature (also known as the Kelvin radius) to the relative vapor pressure and if Kelvin eqnilibrinm is maintained during the separation process, then the adhesive force becomes [19] ... 

Brunauer s modelless method uses pore volume and pore area not as functions of the Kelvin radius but rather as functions of hydraulic radii that he defines as ... 

Unfortunately, the Kelvin radius (r ) does not equal the actual pore radius (rp), one would like to measure. This is due to the fact that multilayer adsorption occurs, prior to the capillary condensation, resulting in a pore narrowing. Therefore, if t is the thickness of the adsorbed layer, then rp equals... 

Table 1 summarises the most important results from the investigation of metal doping. In this table the results of MAP treatment are combined with effects of firing temperature and doping. As can be seen in Table 1, y-alumina membranes with pore radii as low as 2.0 nm (Kelvin radius) may be obtained after firing at 600°C. Note that an instrumental standard error of 0.5 nm (90% reliability) is common in permporometry. This technique should therefore only be used for comparison purposes and to obtain a qualitative impression of the pore-size and pore-size distribution of the material under investigation. 

The encouraging results that were found for the 6 mol-% lanthanum doped samples could not be reproduced for the materials with 9 mol-% lanthanum doping. These samples had already a Kelvin radius of 8.6 nm before SASRA treatment. Hence no further investigations were performed with such highly-doped membranes. 

Amorphous microporous silica membranes as discussed here, consist of a macroporous a-alumina support (pore diameter -100 nm) with a mesoporous y-alumina intermediate layer (Kelvin radius of 2.5 nm) and a microporous silica top layer (pore diameter -4 A) [1,2],... 

For the preparation of tubular silica membranes, commercially available mesoporous membranes [17] are used. These tubular supports have a total length of 25 cm and are enamelled at both ends, required for a gas-tight sealing with carbon seals to the reactor, so that an effective porous length of 20 cm remains. The tube consists of 4 layers. Layer 1, 2 and 3 consist of a-alumina with a thickness of 1.5 mm, 40 and 20 im and a pore diameter of 12, 0.9 and 0.2 im respectively. Layer 4 consists of y-alumina with a thickness of 3-4 im a Kelvin radius of 4 nm. A schematic drawing of the cross-section of a mesoporous support tube is provided in Figure 4. 

In this figure the oxygen permeation is plotted as a function of the relative pressure of cyclohexane. The curve, which was obtained from the membrane before the CVI-experiment, shows a clear transition point. This is indicative of a mesoporous material, in this case with a calculated Kelvin radius of 2.5 nm. For the membrane after CVI no such curve could be obtained and no clear transition point could be observed. This behaviour is representative for microporous materials. 

Figure 7.1. Relation between the Kelvin radius rK and the pore radius rp in a cylindrical mesopore.

The overall pore volume of the bed of packed spheres is mainly determined by the. size of the inner cavities, which is in turn dependent on the particle radius, R, and the coordination number, N. The effective Kelvin radius corresponding to this major stage of capillary condensation is approximately given by the radius, r, of the inscribed sphere within the cavity (Karnaukhov 1971). 

If the pores are slit-shaped, the meniscus is assumed to be hemicylindrical. In principle, the pore width can be readily calculated from the Kelvin radius in practice,... 

In connection with quantifying possible changes in the structure of liquids due to confinement in pores, an old rule, known as Gurvitsch s rule ) may be mentioned. It states that for a variety of different liquids the total pore volume is within narrow limits identical, if computed on the basis of their bulk densities. For Instance, McKee l finds a maximum variation of 4% for a number of organic adsorptives on silica gel with a (Kelvin) radius of 1.7 nm. suggesting that only minor structural deviations take place. However, for more strongly associated liquids like water, and for narrower pores, the deviations may be larger. Then the liquid-solid interaction also starts to play its role. 

As pointed out in Section I, the pore space can generally be treated as a lattice of voids interconnected by necks in a three-dimensional network (Figs. 2 and 3). It is often possible to consider that the pore volume is concentrated in voids, whereas necks do not possess a volume of their own (Fig. 2). In the framework of this model, the filling of every void on the adsorption branch of the isotherm is determined only by the individual void characteristics and does not depend on the neck-size distribution. In particular, voids with radii lower than the Kelvin radius, r < Vp, are completely filled and those with r > Kp are filled only partly via the reversible sorption... 

As pointed out above, the desorption process is dependent both on the void- and neck-size distributions,/(r) and C Fig. 13a) and the void and neck arrangements are random (the latter term means that the probability for an arbitrary void or neck to have a given value of the radius does not depend on the sizes of the neighboring voids and necks), the desorption process is mathematically equivalent to the bond problem in percolation theory. In particular, the probability that an arbitrary void is empty at a given value of the Kelvin radius during desorption is equal to the percolation probability 9 b(zo ) for the bond problem. Thus, the volume fraction of emptied voids under desorption [1 — Udes(rp)] can be represented as the product of the fraction of pore volume that may be emptied in principle at a given value of rp [1 - Uad(rp)] by the percolation probability b(zo ), i.e.,... 

Following Zhdanov et al. (23), we will apply the percolation theory data only to those voids that can, in principle, be emptied at a given value of the Kelvin radius, i.e., to those with the radius r > rp and to their directly connected necks. By analogy with Eq. (20), we can write... 

Fig. 4.6. A typical adsorption-desorption isotherm for a mesoporous sample showing corresponding steps of adsorption [1] rp. pore radius tk Kelvin radius f f-layer thickness.

Types IVand V Capillary condensation in mesopores. Initially, a multilayer is adsorbed on the capilla walls. When the pressure is further increased, liquids droplets form preferentially at sites where the curvature fulfills the Kelvin equation. When two opposing droplets touch, the pore is filled. On desorption, pores whose radius is smaller than the Kelvin radius are emptied. The adsorption branch indicates the extent ofthe pores, and the desorption branch the size ofthe pore openings [Evertt 1976]. 

Two theoretical force curves calculated by using Eqs (25) and (26) are shown in Fig. 18. In these calculations we used a Kelvin radius of 320 nm and an interfacial tension difference of 3.3 mN/m. The measured force curves fall in between the extreme cases of full equilibrium, where the volume of the condensate is changing with distance to minimize the free energy, and the case of no change in condensate volume with separation. Long-range forces due to capillary condensation have been observed previously by Petrov et al. who foimd that water condensed between two surfaces immersed in a microemulsion. (83). Capillary condensation of sparingly soluble surfactants between surfaces close to each other in surfactant solutions has also been reported (84). 

Radius of liquid meniscus, that is the Kelvin radius, depends on the capillary diameter and wetting angle 0. It can be calculated from the formula ... 

Substituting these values into eq. (3.9-2) gives the following equation for the Kelvin radius in terms of the reduced pressure P/Pq... 

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