Model for Cylindrical Pores

The derivation for the modified HK model for pores with cylindrical geometry (Rege and Yang, 2000) follows the same procedure as that for the slit-pore 

Note that int is a mathematical function that truncates the decimal part of a value and retains only the integer part. 

The first layer of molecules will be enclosed by a layer of molecules belonging to the sorbent material (e.g., oxide ions). As it is assumed that molecules in a layer 

One can write the equation for the energy potential for the ith layer i 1) as follows  

The maximum number of molecules of diameter d that can be accommodated with their centers along the circumference of a circle of diameter D is given by N = int[p/sin (d/D) for d D and N = for d D. For the layer of adsorbate molecules within the pore, the molecules lie along a circle of diameter 2[L — do — i — I ) (/i], and hence the maximum number of molecules of diameter dA that are present in a horizontal cross section of the i layer (when dA 2[L — do — (i — l) f/t] can be written as  

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Pore size has been evaluated through the DFT method, using the NLDFT adsorption branch model for cylindrical pore [10],... 

Over the past two decades, different types of HK models have been developed depending upon the pore geometry. The original HK model discussed slit-shaped pores (Horvath and Kawazoe, 1983), whereas models for cylindrical pores (Saito and Foley, 1991), and spherical pores (Cheng and Yang, 1994) have also been proposed. The basic framework for all the different HK models is the same ... 

FIG. 14 PSD of MCM-41 calculated using the original and Rege-Yang-corrected HK models for cylindrical pores with and without Cheng-Yang (CY) correction. (From Ref. 57.)... 

Since they all necessitate a knowledge of the value of r, and of both r and either directly or indirectly, all as a function of p p°, these data are given in tabular form for reference (Table 3.2). If required, intermediate values of t may be obtained to sufficient accuracy by graphical interpolation, and the corresponding values of r can be calculated with the Kelvin formula. The values of r refer to the most commonly used model, the cylindrical pore, so that r " = r + t. The values of t are derived from the standard nitrogen isotherm for hydroxylated silica and though the values do differ... 

Filtration of Liquids Depending on the specific electrochemical reactor type, the filtration rate of a liqnid electrolyte throngfi tfie separator should be either high (to secure a convective snpply of snbstances) or very low (to prevent mixing of the anolyte and catholyte). The filtration rate that is attained under the effect of an external force Ap depends on porosity. For a separator model with cylindrical pores, the volnme filtration rate can be calcnlated by Poiseuille s law ... 

The pore shape affects the pressure of mercury intrusion in ways not contemplated by the usual Washbum-Laplace or Kloubek-Rigby-Edler models. These models have been developed for cylindrical pores and correctly account for the penetration of mercury in the cylindrical pores of MCM-41. The uneven surface of the cylindrical pores of SBA-15 is responsible for a significant increase of the pressure of mercury intrusion and, thereby, for a corresponding underevaluation of the pore size if the classical pressure-size correlations are applied. 

Fig. 4. Schematic diagram of the layered model for a pore (47). The two nuclear spins diffuse in an infinite layer of finite thickness d between two flat surfaces. The M axes are fixed in the layer system. The L axes are fixed in the laboratory frame, with Bq oriented at the angle P from the normal axis n. The cylindrical polar relative coordinates p, (p, and z are based on the M axis. The smallest value of p corresponding to the distance of minimal approach between the two spin bearing molecules is 5.

Based on the above general principles, quite a number of models have been developed to estimate pore size distributions.29,30,31-32,33 They are based on different pore models (cylindrical, ink bottle, packed sphere,. ..). Even the so-called modelless calculation methods do need a pore model in the end to convert the results into an actual pore size distribution. Very often, the exact pore shape is not known, or the pores are very irregular, which makes the choice of the model rather arbitrary. The model of Barett, Joyner and Halenda34 (BJH model) is based on calculation methods for cylindrical pores. The method uses the desorption branch of the isotherm. The desorbed amount of gas is due either to the evaporation of the liquid core, or to the desorption of a multilayer. Both phenomena are related to the relative pressure, by means of the Kelvin and the Halsey equation. The exact computer algorithms35 are not discussed here. The calculations are rather tedious, but straightforward. 

In using this approach to obtain the pore radius or pore width, it is necessary to assume (i) a model for the pore shape and (ii) that the curvature of the meniscus is directly related to the pore width. The pore shape is generally assumed to be either cylindrical or slit-shaped in the former case, the meniscus is hemi-... 

The model isotherm for each pore size class was calculated by methods described previously [9], modified to account for cylindrical pore geometry. These calculations model the fluid behavior in the presence of a uniform wall potential. Since the silica surface of real materials is energetically heterogeneous, one must choose an effective wall potential for each pore size that will duplicate the critical pore condensation pressure, p, observed for that size. This relationship is shown in Figure 2. The Lennard-Jones fluid-fluid interaction parameters and Cn/kg were equal to 0.35746 nm and 93.7465 K, respectively. 

In order to evaluate the solute diffusion coefficient in the stationary phase, Ds, and the equilibrium partition coefficient, Keq, a model for the pore is required. A simple model where the pore is considered to be an infinitely long cylinder and the solute is a rigid sphere has been shown to be adequate in describing the elution process (25). The intrapore diffusivity, Ds, was estimated from the hydrodynamic theory of hindered diffusion for spherical solutes in cylindric pores (19) ... 

The above model qualitatively illustrates one of the reasons for an increase in resistance, namely, that pores reduce the volume in which the current can flow. Quantitatively, however, the situation is more complex, because the pores will attain a charge in order to direct the current flow around them. The charge distribution will, in general, depend upon a pore s shape and whether or not it is isolated from other pores (i.e. has neutral material in between). Juretschke et al. [6] have shown that, for cylindrical pores that are parallel to the magnetic field (h direction) and... 

The frequency dispersion of porous electrodes can be described based on the finding that a transmission line equivalent circuit can simulate the frequency response in a pore. The assumptions of de Levi s model (transmission line model) include cylindrical pore shape, equal radius and length for all pores, electrolyte conductivity, and interfacial impedance, which are not the function of the location in a pore, and no curvature of the equipotential surface in a pore is considered to exist. The latter assumption is not applicable to a rough surface with shallow pores. It has been shown that the impedance of a porous electrode in the absence of faradaic reactions follows the linear line with the phase angle of 45° at high frequency and then... 

Pore systems of solids may vary substantially both in size and shape. Therefore, it is somewhat difficult to determine the pore width and, more precisely, the pore size distribution of a solid. Most methods for obtaining pore size distributions make the assumption that the pores are nonintersecting cylinders or slit-Uke pores, while often porous solids actually contain networks of interconnected pores. To determine pore size distributions, several methods are available, based on thermodynamics (34), geometrical considerations (35-37), or statistical thermodynamic approaches (34,38,39). For cylindrical pores, one of the most commonly applied methods is the one described in 1951 by Barrett, Joyner, and Halenda (the BJH model Reference 40), adapted from... 

Like any idealized model, the cylindrical pore model has many limitations, knowledge of which is essential for proper interpretation of experimental data. In particular, the following should be kept in mind ... 

The PSD was calculated using the DFT method using the MiCTomeritics DFT applied to silica gel 200DF with model of cylindrical pores and another DFT version developed for the model of voids between spherical particles randomly packed in agglomerates (Gun ko 2007, Gnn ko et al. 2007f). 

An overall adsorption equation derived within the framework of DFT (Do et al. 2001) improved to be used for different pore models (Gun ko et al. 2007f) and nonlocal DFT (NLDFT Quantachrome Instruments software, version 2.02) with the model of cylindrical pores in silica was applied to calculate the PSDs (PoSD). The differential distribution functions (j/v(l p)6 l p Vp) were converted to incremental PoSDs (IPSDs, 0() p) at EO ( p) Vp) (Figure 1.189). 

FIGURE 4.27 (a) Nitrogen adsorption-desorption isotherms and (b) pore size distributions of Si-60 and carbosils (NLDFT with the model of cylindrical pores, some PSD are displaced along the F-axis for better view). 

Notes The specific surface area (in mVg) and the pore volumes (in cmVg) of nanopores (Sna o and at radius for the model of cylindrical pores for LiChrolut EN (or half-width for carbon adsorbents) R<1 nm, mesopores (S eso and V eso) at 1 < / < 25 nm and macropores (5 acro and V acro) at / > 25 nm were determined by integration of the fs(R) and/v(/ ) functions, respectively. The fractal dimension Djj2 and D 2o values were determined from the nitrogen and water adsorption isotherms. The Aw value determines the average error of the model of pores due to roughness of the pore walls. 

The PSD functions were calculated using the equation proposed by Nguyen and Do (1999) for carbon adsorbents and modified to study adsorbents with cylindrical and slit-shaped pores and voids between spherical nonporous particles or certain mixtures of these pores (Gun ko et al. 2008e). The nitrogen desorption data were utilized to compute the PSD functions using the model of cylindrical pores for LiChrolut EN and other polymeric adsorbent and slit-shaped pores for carbon adsorbents. 

Besides the simplest model of cylindrical pores and rigid sphere molecules, theoretical treatments have been given for parallel, randomly positioned planes and/or randomly oriented planes models, with several types of molecular shape (ref. 10). However, these models still seem to be too artificial to be examined on an experimental basis. Interestingly enough, the Laurent Killander s model based on the Ogston s theory (ref. 35) (OLK model) should be counted as the most realistic version for polymeric gels, in the framework of Biddings et al (ref. 10). 

The PSD for the cylindrical pore model can now be easily determined by using either the HK Eq. 4.23, or the HK-CY Eq. 4.24. The algorithm for determining pore size by using the corrected HK equation for cylindrical pores was also given by Rege and Yang (2000). 

Pore siie is calculated fiom Kelvin equation for cylindrical pore model ... 

A simple model for the pore volume or the void space in a porous material is to assume it to be composed of a collection of cylindrical pores of radius r. Then a volume of a liquid that does not wet the pore wall surfaces can be forced under pressure to fill the void space. This liquid is invariably mercury because it has a high surface tension, thus the Hg penetration (or porosimetry) method is used to determine pore volumes and the pore size distribution of larger pores, i.e., those with radii larger than about 10 nm. The relationship between pore size and applied pressure. Pap, is obtained by a force balance, that is, the force due to surface tension is equated to the applied force ... 

Table 1 Values calculated from the cylindrical pore model for mean pore spacing and pore density (number of pores per unit volume of sample). The numbers in brackets are the percentage increase between successive values. Mean particle diameter 0.24 pm. Green samples.

The variant of the cylindrical model which has played a prominent part in the development of the subject is the ink-bottle , composed of a cylindrical pore closed one end and with a narrow neck at the other (Fig. 3.12(a)). The course of events is different according as the core radius r of the body is greater or less than twice the core radius r of the neck. Nucleation to give a hemispherical meniscus, can occur at the base B at the relative pressure p/p°)i = exp( —2K/r ) but a meniscus originating in the neck is necessarily cylindrical so that its formation would need the pressure (P/P°)n = exp(-K/r ). If now r /r, < 2, (p/p ), is lower than p/p°)n, so that condensation will commence at the base B and will All the whole pore, neck as well as body, at the relative pressure exp( —2K/r ). Evaporation from the full pore will commence from the hemispherical meniscus in the neck at the relative pressure p/p°) = cxp(-2K/r ) and will continue till the core of the body is also empty, since the pressure is already lower than the equilibrium value (p/p°)i) for evaporation from the body. Thus the adsorption branch of the loop leads to values of the core radius of the body, and the desorption branch to values of the core radius of the neck. 

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