Spherically Symmetric Gels

The kinetics of swelling gels is successfully described in the collective diffusion of equation which was discussed in Sect. 3.1. In the case of the swelling of 

Thus T is proportional to the square of the gel radius and the inverse of the diffusion constant. 

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We should point out that Eqs. (6.2) and (6.3) are only valid for a spherically symmetric gel. In the case of gels having different kind of shape, such as a thin cylinder or a thin disk, the equation has to be modified. The fact that the... 

Another way to determine the cooperative diffusion coefficient is to measure the degree of swelling of a sample with a defined geometry as function of time. The experimental data were evaluated by a solution of the diffusion equation based on the equation of motion for a given sample geometry. In the following this is demonstrated on example of swelling of a spherically symmetrical gel (shear modulus G = 0). 

Equations (48) and (50) are valid only for a spherically symmetric gel. The specimens for swelling measurements are mostly cylinders or strips (slabs). In case of other specimen geometries, (50) has to be modified due to G 7 0. Nevertheless, the proportionality between time constant x and the square of dimension is independent of sample geometry. 

In this section, the swelling of spherical symmetric gels will be discussed. In Eq. (6), let... 

The purpose of calculating Henry s Law constants is usually to determine the parameters of the adsorption potential. This was the approach in Ref. [17], where the Henry s Law constant was calculated for a spherically symmetric model of CH4 molecules in a model microporous (specific surface area ca. 800 m /g) silica gel. The porous structure of this silica was taken to be the interstitial space between spherical particles (diameter ca. 2.7 nm ) arranged in two different ways as an equilibrium system that had the structure of a hard sphere fluid, and as a cluster consisting of spheres in contact. The atomic structure of the silica spheres was also modeled in two ways as a continuous medium (CM) and as an amorphous oxide (AO). The CM model considered each microsphere of silica gel to be a continuous density of oxide ions. The interaction of an adsorbed atom with such a sphere was then calculated by integration over the volume of the sphere. The CM model was also employed in Refs. [36] where an analytic expression for the atom - microsphere potential was obtained. In Ref. [37], the Henry s Law constants for spherically symmetric atoms in the CM model of silica gel were calculated for different temperatures and compared with the experimental data for Ar and CH4. This made it possible to determine the well-depth parameter of the LJ-potential e for the adsorbed atom - oxygen ion. This proved to be 339 K for CH4 and 305 K for Ar [37]. On the other hand, the summation over ions in the more realistic AO model yielded efk = 184A" for the CH4 - oxide ion LJ-potential [17]. Thus, the value of e for the CH4 - oxide ion interaction for a continuous model of the adsorbent is 1.8 times larger than for the atomic model. 

The synthetic schemes and structures of these initiators are shown in Figs 5.7(a) and 5.7(b). They are both AIBN-analogue initiators for free radieal polymerization. The use of another symmetric bicationic azo compound, 2,2 -azobis(isobutyramidine hydrochloride) (AIBN), has also been proven to be feasible for styrene SIP on high surface area mica powder. " However, no structural information for these SIP products has been reported. Asymmetric azo initiators in the form of silanes have also been successfully employed to free radically polymerize styrene from spherical silica gel surfaces. To the best of our knowledge, there have been no reports on a direct free radical SIP approach from surface-bound monocationic azo initiators on individual clay nanoparticles. 

Engelhardt and Mliller reported on the differences in the physical properties, such as specific surface area, specific pore volume and average pore diameter - and on the different amounts of stationary and mobile phase per unit column volume for various commercially available silica gels. If the retention for various solutes were normalized for these factors, distinct selectivities were still noticed. This could be explained by differences in the surface pH of the silicas. Irregular ones were usually neutral or weakly acidic, whereas the spherical ones were either acidic (pH ca.4) or basic (pH ca.9) (see Table 1.4).To obtain the required and optimum selectivity, the pH of silica gel can easily be adjusted. For basic compounds more symmetrical peak shapes were obtained on silica with a basic character. 

Related systems are inclusions in membranes, which could model large proteins or rigid gel-like domains. Even if these inclusions are flat and symmetric they can affect the conformation and the fluctuations of the surrounding membrane via boundary conditions. This induces effective interactions between them [39,40]. Although most of these problems have been studied mainly for planar membranes, a recent study addresses inclusions on a spherical vesicle [41]. 

The method of gel-copolymerization on the interface for obtaining gradient elements (polymeric media) of various geometrical forms has been described [43] radical-symmetric, axial-symmetric, spherical gradient polymeric media, as well as gradient wavebeam guide (PG). 

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