Cylindrical pores

Consider a cylindrical pore of length L and radius a containing a symmetric (z z) type electrolyte at the number concentration hq. Let the internal wall of 

The electric potential inside the pore is calculated by following exactly the same procedure as in the above cases. The potential is given by the Poisson-Boltzmann equation (Equation 3.13). For symmetric electrolytes, the valencies z are the same for both the anions and cations, except for the sign. Also, it turns out that, for both types of ions, io = no is a good approximation. 

In general, this equation needs to be solved numerically to get the spatial variations of the electric potential and charges (Gross and Osterle 1968). Therefore, as usual, we make the Debye-Hiickel approximation by linearizing the 

The net charge density p r) inside the pore follows from Equations 3.12 and 3.71. With the Debye-Htickel approximation used in getting Equation 3.73 (i.e., linearization of the exponentials in Equation 3.12), p r) is — Q K - r), 

The value of the ratio of the Debye length to the pore radius, pertinent to an experiment, is obtained from the radius of the nanopore and the electrolyte concentration (Table 3.1). For the typical salt concentrations used in singlemolecule electrophysiology experiments, the ion cloud is close to the wall of the pore, its thickness being given by the Debye length. The interior of the pore near the pore axis has no net charge due to the uniform distribution of counterions and coions. 

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Microscopic analyses of the van der Waals interaction have been made for many geometries, including, a spherical colloid in a cylindrical pore [14] and in a spherical cavity [15] and for flat plates with conical or spherical asperities [16,17]. 

The pore systems of solids are of many different kinds. The individual pores may vary greatly both in size and in shape within a given solid, and between one solid and another. A feature of especial interest for many purposes is the width w of the pores, e.g. the diameter of a cylindrical pore, or the distance between the sides of a slit-shaped pore. A convenient classification of pores according to their average width originally proposed by Dubinin and now officially adopted by the International Union of Pure and Applied Chemistry is summarized in Table 1.4. 

Fig. 3.7 Cross-section, parallel to the axis of a cylindrical pore of radius r , showing the inner core of radius i and the adsorbed film of thickness t.

In a cylindrical pore the meniscus will be spherical in form, so that the two radii of curvature are equal to one another and therefore to r (Equation (3.8)). From simple geometry (Fig. 3.8) the radius r of the core is related to r by the equation... 

Fig. 3.8 Relation between r of the Kelvin equation (Equation (3.20)) and the core radius r for a cylindrical pore with a hemispherical meniscus 6 is the angle of contact.

Fig. 3.11 Capillary condensation in cylindrical pores, (a) Cylinder closed at one end, B. The meniscus is hemispherical during both capillary condensation and capillary evaporation, (h) and (c) Cylinder open at both ends. The meniscus is cylindrical during capillary condensation and hemispherical during capillary evaporation. Dotted lines denote the...

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. 

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... 

All values rounded lo nearest 0-05 A. Cylindrical pores arc assumed. 

A procedure involving only the wall area and based on the cylindrical pore model was put forward by Pierce in 1953. Though simple in principle, it entails numerous arithmetical steps the nature of which will be gathered from Table 3.3 this table is an extract from a fuller work sheet based on the Pierce method as slightly recast by Orr and DallaValle, and applied to the desorption branch of the isotherm of a particular porous silica. 

Fig. 4.9 Enhancement of interaction potential in (i) a slit-shaped pore between parallel slabs of solid, (ii) a cylindrical pore in a block of solid. 0/0 is plotted against d/r (see text). (Reduced from a diagram of Everett...

As would be expected, the enhancement of potential in cylindrical pores turns out to be considerably greater than in dits, as curve (ii) of Fig. 4.9 clearly demonstrates. At R/r = 2 the enhancement is more than 50 per cent, and it is still appreciable when R/r = 3 (R = radius of cylinder). The calculations show that at radii in excess of R = 1086ro, the single minimum (comparable with Fig. 4.8(c)) develops into a ring minimum (i.e. two minima are present in any axial plane, cf. Fig. 4.8(a)). 

Figure 9.15 Schematic illustration of size exclusion in a cylindrical pore (a) for spherical particles of radius R and (b) for a flexible chain, showing allowed (solid) and forbidden (broken) conformations of polymer.

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.]...

Fig. 3. Microporous membranes are characterized by tortuosity, T, porosity, S, and their average pore diameter, d. (a) Cross-sections of porous membranes containing cylindrical pores, (b) Surface views of porous membranes of equal S, but differing pore size.

Fig. 4. Diagram of the two-step process to manufacture nucleation track membranes, (a) Polycarbonate film is exposed to charged particles in a nuclear reactor, (b) Tracks left by particles are preferentially etched into uniform cylindrical pores (8).

Pjjydro — 9exceeds the pressure required to drive it into the pores, which for a cylindrical pore is as foUows ... 

Dpi is smaller than the diffusivity in a straight cylindrical pore as a result of the random orientation of the pores, which gives a longer diffusion path, and the variation in the pore diameter. Both effects are commonly accounted for by a tortuosity factor Tp such that Dp = DjlXp. In principle, predictions of the tortuosity factor can be made if the... 

Since theoretical calcination of effectiveness is based on a hardly realistic model of a system of equal-sized cylindrical pores and a shalq assumption for the tortuosity factor, in some industrially important cases the effectiveness has been measured directly. For ammonia synthesis by Dyson and Simon (Ind. Eng. Chem. Fundam., 7, 605 [1968]) and for SO9 oxidation by Kadlec et aJ. Coll. Czech. Chem. Commun., 33, 2388, 2526 [1968]). 

Consider first penetration into a cylindrical pore. An estimate of the extent of penetration can be obtained by equating the back pressure of trapped air to the capillary driving pressure. Then the distance. r penetrated into a pore of length / and radius r is then ... 

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

FIG. 2 Geometry for a simulation of a cylindrical pore in contact with a bulk-like aqueous phase. A, V, P, and W denote the aqueous phase, the excluded volume, the pore wall, and the confining walls, respectively. 

Any real sample of a colloidal suspension has boundaries. These may stem from the walls of the container holding the suspension or from a free interface towards the surroundings. One is faced with surface effects that are small compared to volume effects. But there are also situations where surface effects are comparable to bulk effects because of strong confinement of the suspension. Examples are cylindrical pores (Fig. 8), porous media filled with suspension (Fig. 9), and thin colloidal films squeezed between parallel plates (Fig. 10). Confined systems show physical effects absent in the bulk behavior of the system and absent in the limit of extreme confinement, e.g., a onedimensional system is built up by shrinking the size of a cylindrical pore to the particle diameter. 

FIG. 8 A colloidal fluid confined within a cylindrical pore. 

The porous materials that offer the narrowest possible pore size distribution are those that have cylindrical pores of uniform diameter penetrating the entire medium without branching. Branching gives polymer molecules in the junctions extra conformational entropy. An agglomerate of tiny pieces of these porous materials, interlaced with larger voids (much larger than the pore size), should also be chosen. 

The other type of porous glass that has cylindrical pores is mesoporous silicate (MPS) (14,15). The advantage of MPS is in its feasibility to make a small pore diameter, typically below 10 nm. A columnar-phase liquid crystal, formed from surfactant molecules with a long alkyl chain tail and silicate molecules, is calcined to remove hydrocarbons. At the end, a hexagonal array of straight and uniform cylindrical holes is created in a crystalline order. MPS is not available commercially either. 

Two macromolecular computational problems are considered (i) the atomistic modeling of bulk condensed polymer phases and their inherent non-vectorizability, and (ii) the determination of the partition coefficient of polymer chains between bulk solution and cylindrical pores. In connection with the atomistic modeling problem, an algorithm is introduced and discussed (Modified Superbox Algorithm) for the efficient determination of significantly interacting atom pairs in systems with spatially periodic boundaries of the shape of a general parallelepiped (triclinic systems). 

Figure 3. Partition coefficient of freely jointed chains between the bulk solution and a cylindrical pore. The chains have different numbers of mass-points (n) and different bond lengths, and are characterized by the root-mean-square radius of gyration measured in units of the pore radius. See text for details.

Figure 4. The most CPU time-consuming portion of the scalar version of the Monte Carlo program for the evaluation of chain partitioning between the bulk solution and a cylindrical pore, in the scalar form (left) and the explicitly vectorized form (right).

The effects of various pore-size distributions, including Gaussian, rectangular distributions, and continuous power-law, coupled with an assumption of cylindrical pores and mass transfer resistance on chromatographic behavior, have been developed by Goto and McCoy [139]. This study utilized the method of moments to determine the effects of the various distributions on mean retention and band spreading in size exclusion chromatography. 

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