Capillary pore sizes

The pore radius determined from infiltration kinetics can be reconciled to the experimentally determined radii from SEM and porosimetry, by assuming a two-pore-size model (pore neck and pore bulge), instead of a single capillary pore-size [Dullien etal., 1977 Einset, 1996]. The schematic diagram for this two-pore-size model is shown in Fig. 5.2(b). 

The drawbacks of SC and IM injections include potentially decreased bioavailability that is secondary to variables such as local blood flow, injection trauma, protein degradation at the site of injection, and limitations of uptake into the systemic circulation related to effective capillary pore size and diffusion. The bioavailability of numerous peptides and proteins is, for example, markedly reduced after SC or IM administration compared to their IV administration. The pharmacokine-tically derived apparent absorption rate constant is thus the combination of absorption into the systemic circulation and presystemic degradation at the absorption site. The true absorption rate constant ka can then be calculated as ... 

Normally, the moisture sorption-desorption profile of the compound is investigated. This can reveal a range of phenomena associated with the solid. For example, on reducing the RH from a high level, hysteresis (separation of the sorption-desorption curves) may be observed. There are two types of hysteresis loops an open hyteresis loop, where the final moisture content is higher than the starting moisture content due to so-called ink-bottle pores, where condensed moisture is trapped in pores with a narrow neck, and the closed hysteresis loop may be closed due to compounds having capillary pore sizes. 

Fig. 13.4 Dependence of pore diameters on fiber diameters for PAN-based nonwovens (a) geometric pore size (b) capillary pore sizes, largest pore sizes bubble-point pores) and average pore sizes mean flow pores) [77] (Reprinted with permission from Ref. [66]. Copyright 2010, Elsevier)...

Pore size is directly related to the filtration performance. Unlike the sieve, electrospun nanofibrous filters do not have regular pores. The effective maximum pore size of electrospun filters is reflected as the maximum size of particles which can pass through the interconnected tortuous path of the membrane. In practice, the maximum pore size and mean flow pore size are measured by means of a capillary porometer [76], Hussain and co-workers correlated the mean fiber diameter and pore size of ENMs [77], In Fig. 13.4, they found that both values of bubble-point pore size and mean flow pore size could be positively correlated with fiber diameters. The average correlation ratio of the geometric pore size (diameter) to the fiber diameter amounted to about 6. The number became about 5 for the mean capillary pore size (mean flow pore size) and about 9 for the largest capillary pore... 

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-... 

Brunauer and co-workers [211, 212] proposed a modelless method for obtaining pore size distributions no specific capillary shape is assumed. Use is made of the general thermodynamic relationship due to Kiselev [213]... 

At the present time there exist no flux relations wich a completely sound cheoretical basis, capable of describing transport in porous media over the whole range of pressures or pore sizes. All involve empiricism to a greater or less degree, or are based on a physically unrealistic representation of the structure of the porous medium. Existing models fall into two main classes in the first the medium is modeled as a network of interconnected capillaries, while in the second it is represented by an assembly of stationary obstacles dispersed in the gas on a molecular scale. The first type of model is closely related to the physical structure of the medium, but its development is hampered by the lack of a solution to the problem of transport in a capillary whose diameter is comparable to mean free path lengths in the gas mixture. The second type of model is more tenuously related to the real medium but more tractable theoretically. 

As already indicated in Section 3.1, the study of mesoporous solids is closely bound up with the concept of capillary condensation and its quantitative expression in the Kelvin equation. This equation is, indeed, the basis of virtually all the various procedures for the calculation of pore size... 

It must always be borne in mind that when capillary condensation takes place during the course of isotherm determination, the pore walls are already covered with an adsorbed him, having a thickness t determined by the value of the relative pressure (cf. Chapter 2). Thus capillary condensation occurs not directly in the pore itself but rather in the inner core (Fig. 3.7). Consequently the Kelvin equation leads in the first instance to values of the core size rather than the pore size. The conversion of an r value to a pore size involves recourse to a model of pore shape, and also a knowledge of the angle of contact 0 between the capillary condensate and the adsorbed film on the walls. The involvement of 0 may be appreciated by consideration... 

In calculations of pore size from the Type IV isotherm by use of the Kelvin equation, the region of the isotherm involved is the hysteresis loop, since it is here that capillary condensation is occurring. Consequently there are two values of relative pressure for a given uptake, and the question presents itself as to what is the significance of each of the two values of r which would result from insertion of the two different values of relative pressure into Equation (3.20). Any answer to this question calls for a discussion of the origin of hysteresis, and this must be based on actual models of pore shape, since a purely thermodynamic approach cannot account for two positions of apparent equilibrium. 

When the relative pressure falls to pj/p", the second group of pores loses its capillary condensate, but in addition the film on the walls of the first group of pores yields up some adsorbate, owing to the decrease in its thickness from t, to t. Similarly, when the relative pressure is further reduced to pj/p°, the decrement (nj-Wj) in the uptake will include contributions from the walls of both groups 1 and 2 (as the film thins down from tj to fj), in addition to the amount of capillary condensate lost from the cores of group 3. It is this composite nature of the amount given up at each step which complicates the calculation of the pore size distribution. 

The time is perhaps not yet ripe, however, for introducing this kind of correction into calculations of pore size distribution the analyses, whether based on classical thermodynamics or statistical mechanics are being applied to systems containing relatively small numbers of molecules where, as stressed by Everett and Haynes, the properties of matter must exhibit wide fluctuations. A fuller quantitative assessment of the situation in very fine capillaries must await the development of a thermodynamics of small systems. Meanwhile, enough is known to justify the conclusion that, at the lower end of the mesopore range, the calculated value of r is almost certain to be too low by many per cent. 

The limits of pore size corresponding to each process will, of course, depend both on the pore geometry and the size of the adsorbate molecule. For slit-shaped pores the primary process will be expected to be limited to widths below la, and the secondary to widths between 2a and 5ff. For more complicated shapes such as interstices between small spheres, the equivalent diameter will be somewhat higher, because of the more effective overlap of adsorption fields from neighbouring parts of the pore walls. The tertiary process—the reversible capillary condensation—will not be able to occur at all in slits if the walls are exactly parallel in other pores, this condensation will take place in the region between 5hysteresis loop and in a pore system containing a variety of pore shapes, reversible capillary condensation occurs in such pores as have a suitable shape alongside the irreversible condensation in the main body of pores. 

The capillary retention forces in the pores of the filter cake are affected by the size and size range of the particles forming the cake, and by the way the particles have been deposited when the cake was formed. There is no fundamental relation to allow the prediction of cake permeabiUty but, for the sake of the order-of-magnitude estimates, the pore size in the cake may be taken loosely as though it were a cylinder which would just pass between three touching, monosized spheres. If dis the diameter of the spherical particles, the cylinder radius would be 0.0825 d. The capillary pressure of 100 kPa (1 bar) corresponds to d of 17.6 pm, given that the surface tension of water at 20°C is 12.1 b mN /m (= dyn/cm). 

Important physical properties of catalysts include the particle size and shape, surface area, pore volume, pore size distribution, and strength to resist cmshing and abrasion. Measurements of catalyst physical properties (43) are routine and often automated. Pores with diameters <2.0 nm are called micropores those with diameters between 2.0 and 5.0 nm are called mesopores and those with diameters >5.0 nm are called macropores. Pore volumes and pore size distributions are measured by mercury penetration and by N2 adsorption. Mercury is forced into the pores under pressure entry into a pore is opposed by surface tension. For example, a pressure of about 71 MPa (700 atm) is required to fill a pore with a diameter of 10 nm. The amount of uptake as a function of pressure determines the pore size distribution of the larger pores (44). In complementary experiments, the sizes of the smallest pores (those 1 to 20 nm in diameter) are deterrnined by measurements characterizing desorption of N2 from the catalyst. The basis for the measurement is the capillary condensation that occurs in small pores at pressures less than the vapor pressure of the adsorbed nitrogen. The smaller the diameter of the pore, the greater the lowering of the vapor pressure of the Hquid in it. 

Capillary Suction Processes. The force needed to remove water from capillaries increases proportionately with a decrease in capillary radius, exceeding 1400 kPa (200 psi) in a 1-p.m-diameter capillary. Some attempts have been made to use this force as a way to dewater sludges and cakes by providing smaller dry capillaries to suck up the water (27). Sectors of a vacuum filter have been made of microporous ceramic, which conducts the moisture from the cake into the sector and removes the water on the inside by vacuum. Pore size is sufficiently small that the difference in pressure during vacuum is insufficient to displace water from the sector material, thus allowing a smaller vacuum pump to be effective (126). 

A microscopic description characterizes the structure of the pores. The objective of a pore-structure analysis is to provide a description that relates to the macroscopic or bulk flow properties. The major bulk properties that need to be correlated with pore description or characterization are the four basic parameters porosity, permeability, tortuosity and connectivity. In studying different samples of the same medium, it becomes apparent that the number of pore sizes, shapes, orientations and interconnections are enormous. Due to this complexity, pore-structure description is most often a statistical distribution of apparent pore sizes. This distribution is apparent because to convert measurements to pore sizes one must resort to models that provide average or model pore sizes. A common approach to defining a characteristic pore size distribution is to model the porous medium as a bundle of straight cylindrical or rectangular capillaries (refer to Figure 2). The diameters of the model capillaries are defined on the basis of a convenient distribution function. 

A multidimensional system using capillary SEC-GC-MS was used for the rapid identification of various polymer additives, including antioxidants, plasticizers, lubricants, flame retardants, waxes and UV stabilizers (12). This technique could be used for additives having broad functionalities and wide volatility ranges. The determination of the additives in polymers was carried out without performing any extensive manual sample pretreatment. In the first step, microcolumn SEC excludes the polymer matrix from the smaller-molecular-size additives. There is a minimal introduction of the polymer into the capillary GC column. Optimization of the pore sizes of the SEC packings was used to enhance the resolution between the polymer and its additives, and smaller pore sizes could be used to exclude more of the polymer... 

Figure 12.7 Cliromatograms of a polycarbonate sample (a) microcolumn SEC ti ace (b) capillary GC ti ace of inti oduced fractions. SEC conditions fused-silica (30 cm X 250 mm i.d.) packed with PL-GEL (50 A pore size, 5 mm particle diameter) eluent, THE at aElow rate of 2.0ml/min injection size, 200 NL UV detection at 254 nm x represents the polymer additive fraction ti ansfeired to EC system (ca. 6 p-L). GC conditions DB-1 column (15m X 0.25 mm i.d., 0.25 pm film thickness) deactivated fused-silica uncoated inlet (5 m X 0.32 mm i.d.) temperature program, 100 °C for 8 min, rising to 350 °C at a rate of 12°C/min flame ionization detection. Peak identification is as follows 1, 2,4-rert-butylphenol 2, nonylphenol isomers 3, di(4-tert-butylphenyl) carbonate 4, Tinuvin 329 5, solvent impurity 6, Ii gaphos 168 (oxidized). Reprinted with permission from Ref. (14).

This concept may be invoked to account for electrolyte formation in microcracks in a metal surface or in the re-entrant angle formed by a dust particle and the metal surface. More importantly, it can also explain electrolyte formation in the pores of corrosion product and hence the secondary critical humidity discussed earlier. Ferric oxide gel is known to exhibit capillary condensation characteristic and pore sizes deduced from measurements of its adsorptive capacity are of the right order of magnitude to explain a secondary critical relative humidity as70 7o for rusted steel . 

The small pore size and the uniform distribution result in capillary forces which should allow wicking heights and thus battery heights of up to 30 cm. Due to the cavities required for gas transfer and under the effect of gravity, the electrolyte forms a filling profile, i.e., fewer cavities remain at the bottom than at the top. Therefore with absorptive glass mats a rather flat battery... 

The prime requirements for the separators in alkaline storage batteries are on the one hand to maintain durably the distance between the electrodes, and on the other to permit the ionic current flow in as unhindered a manner as possible. Since the electrolyte participates only indirectly in the electrochemical reactions, and serves mainly as ion-transport medium, no excess of electrolyte is required, i.e., the electrodes can be spaced closely together in order not to suffer unnecessary power loss through additional electrolyte resistance. The separator is generally flat, without ribs. It has to be sufficiently absorbent and it also has to retain the electrolyte by capillary forces. The porosity should be at a maximum to keep the electrical resistance low (see Sec. 9.1.2.3) the pore size is governed by the risk of electronic shorts. For systems where the electrode substance... 

The nitrogen physisorption isotherm and pore size distributions for the synthesized catalysts are shown in Figs. 3 and 4. The Type IV isotherm, typical of mesoporous materials, for each sample exhibits a sharp inflection, characteristic of capillary condensation within the regular mesopores [5, 6], These features indicate that both TS-1/MCM-41-A and TS-l/MCM-41-B possess mesopores and a narrow pore size distribution. 

The principle underlying surface area measurements is simple physisorb an inert gas such as argon or nitrogen and determine how many molecules are needed to form a complete monolayer. As, for example, the N2 molecule occupies 0.162 nm at 77 K, the total surface area follows directly. Although this sounds straightforward, in practice molecules may adsorb beyond the monolayer to form multilayers. In addition, the molecules may condense in small pores. In fact, the narrower the pores, the easier N2 will condense in them. This phenomenon of capillary pore condensation, as described by the Kelvin equation, can be used to determine the types of pores and their size distribution inside a system. But first we need to know more about adsorption isotherms of physisorbed species. Thus, we will derive the isotherm of Brunauer Emmett and Teller, usually called BET isotherm. 

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