Two Porous Spheres

Consider two interacting ion-penetrable porous spheres 1 and 2 having radii Oj and respectively, at separation R between their centers Oi and O2 (or, at separation H = R — ai — a2 between their closest distances (Fig. 13.6) [1-3]. 

The linearized Poisson-Boltzmann equations in the respective regions are 

The derivative of ij/ being taken along the outward normal to the surface of each sphere. 

DONNAN POTENTIAL-REGULATED INTERACTION BETWEEN POROUS PARTICLES 

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Note that the following exact expression for the electrostatic interaction between two porous spheres (spherical polyelectrolytes) for the low charge density case has been derived [5,6] (Eq. (13.46)) ... 

A catalyst bed contains a uniform mixture of equivolume porous spheres and cubes with two pairs of opposite faces sealed to diffusion. Accordingly the cubes may be regarded as slabs with diffusion through two faces. The spheres have a diameter of 0.02 ft, so cubes of the same volume have edges of 0.01614 ft. 

Consider the double-layer interaction between two parallel porous cylinders 1 and 2 of radii and a2, respectively, separated by a distance R between their axes in an electrolyte solution (or, at separation H = R ai—a2 between their closest distances) [5]. Let the fixed-charge densities of cylinders 1 and 2 be and Pfix2. respectively. As in the case of ion-penetrable membranes and porous spheres, the potential distribution for the system of two interacting parallel porous cylinders is given by the sum of the two unperturbed potentials... 

Volume of solution or mixture, also particles, column, and porous medium London attractive energy between two molecules or particles Coarse particle volume Fine particle volume Carrier liquid volume London attractive energy per unit area between two infinite flat plates London attractive energy between two identical spheres Elution volume, Eq. (4.7.3)... 

Repulsive potential energy between two identical spheres of same charge Volume of voids in porous medium Dimensionless applied voltage,... 

As to their structure, micro- and nanospheres can be of two types i) micelles formed from copolymers and ii) porous spheres in micrometer size or in the colloid size range. 

Two early theoretical models to rationalize this result were pursued the porous-sphere model of Debye and Bueche [1948], in which spherical beads representing the monomers are distributed uniformly in a spherical volume, and the more realistic pearl necklace model, proposed by Kuhn and Kuhn [1943], in which the beads are linked together by infinitely thin linkages. For each of these models, the principal challenge was to describe the flow of solvent around and within the volume occupied... 

The barrier device consist of three layers—porous metal plate and foam ceramics and porous metal plate, which is similar to the sandwich, as the fig 1 shown. The holes in the two porous metal plate have different center and diameter, and each hole facing the polluted air is filled by a hollow metal spheres to avoid mixing of fresh air and polluted air. As fig 2(a) and fig 2(b) shown, the fig 2(a) is the closed barrier door and the other is open. 

As described above, molecular and supramolecular templating methods have been very successfully applied for the syntheses of a large variety of microporous systems (e.g., zeolites, AlPOs) and mesoporous systems, respectively. Such approaches allow the design of the pore structure inside a crystal or particle. The simultaneous control about the pore structure at the micro and the macro scale during the formation process can be realized by the combination of at least two strategies. The combination of molecular templating with a sol-gel processing approach results in porous spheres [204]. 

Porous spheres can be prepared by leaching one glassy phase from a two-phase spherical product, or by processes similar to those used for hollow microspheres whereby the surface layer never forms in a fuUy continuous manner, i.e., the blowing bubbles are exposed at the surface. 

As an example of composite core/shell submicron particles, we made colloidal spheres with a polystyrene core and a silica shell. The polar vapors preferentially affect the silica shell of the composite nanospheres by sorbing into the mesoscale pores of the shell surface. This vapor sorption follows two mechanisms physical adsorption and capillary condensation of condensable vapors17. Similar vapor adsorption mechanisms have been observed in porous silicon20 and colloidal crystal films fabricated from silica submicron particles32, however, with lack of selectivity in vapor response. The nonpolar vapors preferentially affect the properties of the polystyrene core. Sorption of vapors of good solvents for a glassy polymer leads to the increase in polymer free volume and polymer plasticization32. 

Figure 8.11. Diffuse reflectance absorption spectra of a strongly fluorescent sample (1,6-diphenylhexatriene adsorbed on porous alumina) (a) conventional measurement w ith monochromatic irradiation and detection via an integrating sphere (b) measurement in a fluorimeter with two monochromators. Reaction spectra during Irons - cis photoisomerization are also given (adapted from Ref. 26).

Two types of structures were seen a needle type bundled structure pointing inward towards the center of the sphere and surface bumps. The needle type structures overall were less than 10 pm in length and less than 1 pm in thickness. To confirm that these structures corresponded to sodium alanate. elemental mapping of the alanate s sodium and aluminum constitutes was made. As can be seen in Fig.5, the presence of intense sodium distribution was noted on the needle structures while the both the surface bumps and needle structures showed the presence of aluminum distribution. These results confirmed spheres filling with the alanate. While these needle structures are not usual for sodium alanate. it is speculated that the porous silica could have provided a nucleation surface for these unique structures formation. Nevertheless, since the samples were exposed to ambient air prior to the... 

With hysteresis loops of Type HI, the two branches are almost vertical and nearly parallel. Such loops are often associated with porous materials which are known to have very narrow pore size distributions or agglomerates of approximately uniform spheres in fairly regular array. More common are loops of Type H2, where the pore size distribution and shape are not well defined. This is attributed to the difference in adsorption and desorption mechanisms occurring in ink-bottle pores, and network effects. The Type H3 hysteresis loop does not show any limiting adsorption at high relative pressures and is observed in aggregates and macroporous materials. Loops of Type H4 are often associated with narrow... 

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