Adsorbent fractal dimensions

The currently useful model for dealing with rough surfaces is that of the selfsimilar or fractal surface (see Sections VII-4C and XVI-2B). This approach has been very useful in dealing with the variation of apparent surface area with the size of adsorbate molecules used and with adsorbent particle size. All adsorbate molecules have access to a plane surface, that is, one of fractal dimension 2. For surfaces of Z> > 2, however, there will be regions accessible to small molecules... 

The monolayer amount adsorbed on an aluminum oxide sample was determined using a small molecule adsorbate and then molecular-weight polystyrenes (much as shown in Ref. 169). The results are shown in the table. Calculate the fractal dimension of the oxide. 

Because of its strong coupling with MW, its good adsorbent properties towards organic molecules [12], and its layer structure which enables it to form intercalated compounds [13], graphite has great potential in MW-assisted synthetic applications in organic chemistry, despite its weak fractal dimension (D x 2) [14]. 

There are two molecular probe methods available for the determination of surface fractal dimension. One is the multiprobe method (MP method),83,84,87-100 which uses several kinds of multiprobe molecules with different molecular sizes and requires the number of adsorbed molecules to form a monolayer Nmoao for each probe molecule. If the probe molecule is varied through a series of spheres with radius rm, the surface fractal dimension is given by Eq. (7) ... 

It is reported116,117 that as more adsorbed layers are built up, the interface between the adsorbent and the adsorbed molecules becomes smooth, and hence the surface fractal dimension would no longer describe the interface but would describe the adsorbed molecule agglomerates. Also, Eq. (9) is only valid when the adsorbed layer exceeds monolayer coverage. Therefore, for the correct calculation, dFSP should be determined from the linear... 

Lee et al. s study also investigates the hydrophilicity of the heterocatalyst. They mention that the highly acidic surface of the material is more hydrophobic than the pure titanium oxide surface. They theorize that this is because the acidic surface results in fewer adsorbed OH ions and thus a weaker interaction with water. As expected, this increased hydrophobicity leads to an increase in the stability of dispersions of nanoscale powders of this material. Saltiel et al. showed that WOs-coated titanium oxide powders were much more stable than their uncoated counterparts. Even after agglomeration, the agglomerates of the coated powders were more porous than those of pure titanium oxide (the coated powders had a fractal dimension of 1.55 while the pure titanium oxide powders had a fractal dimension of 1.60). 

Laszlo7 pointed out that solids of fractal dimension D near 2.0 are usually more efficient catalysts than those of D near 3.0. An adsorbed species diffusing across the surface of a catalyst will find a target much more quickly (i.e., catalysis will be more efficient) in space of dimension near 2 than of D near 3. We find, for example, that catalytic activities of variously prepared activated charcoals increase as we go from D = 3.0 to D = 1.9. 

Here, Nmono is the number of adsorbed molecules to form a monolayer for each probe molecule and surface fractal dimension determined by using the MP method. The probe molecules need not to be spherical, provided they belong to a homologous series for which the ratio [linear extent rm]2 to molecular cross-sectional area Ac is the same for all members, i.e., an isotropic series. In this case, Eq. (13) turns into... 

At low relative pressures p/p0 or thin adsorbate films, adsorption is expected to be dominated by the van der Waals attraction of the adsorbed molecules by the solid that falls off with the third power of the distance to the surface (FHH-regime, Eq. 3a). At higher relative pressures p/p0 or thick adsorbate films, the adsorbed amount N is expected to be determined by the surface tension y of the adsorbate vapor interface (CC-regime, Eq. 3b), because the corresponding surface potential falls off less rapidly with the first power of the distance to the surface, only. The cross-over length zcrit. between both regimes depends on the number density np of probe molecules in the liquid, the surface tension y, the van der Waals interaction parameter a as well as on the surface fractal dimension ds [100, 101] ... 

The monolayer amount Nm can be estimated from a classical BET-plot and hence the film thickness z can be obtained directly from the adsorbed amount N if the surface fractal dimension ds is known. 

Fan et. al. [74] determined surface area by nitrogen gas adsorption and stated that the fractal dimension could be determined by adsorbing different sized molecules on similar surfaces or, keeping the adsorbate the same, but changing the size of the adsorbent. They adopted the latter proeedure by using between sieve fractions. In the first case the specific surface S is related to the fractal dimension D f by equation (2.50) and in the second the relationship given in equation (2.51) applies. 

For smooth surface, the parameter S is assumed to be equal to -1/3, while for a fractal surface, S is a function of the surface fractal dimension, D. If the van der waals attractive forces are dominant between adsorbent and adsorbate, then S is equal to (Ds-3)/3 [9]. For higher surface coverage where the adsorbent-adsorbate interface is... 

Table I. The specific surface area, pore diameter, and fractal dimension of the sludge adsorbent with H3P04 at various activation conditions.

In this study, the BET surface areas and pore diameters of the sludge adsorbents are 0.2 - 499 m /g and 19 - 188 A. From the point of view of the specific surface area and pore diameter, it was concluded that the optimum activation condition for the sludge adsorbent with 7MKOH and steam occurs at 7001C for Ihr.From the analysis of fractal dimensions(D) using the FHH theory, the D values could be used to describe the structure... 

In the second method, the particles do not have to be fractionated. However, several adsorbates, whose molecules have different cross-sectional area, have to be used. " The surface fractal dimension is then determined from Eq. (8) ... 

N2 adsorption is also used to estimate the micro pore volume and the pore size distribution (see e. g. Glasauer et al. 1999) whieh ean be derived from a plot of adsorbed N2 vs. the thiekness of a statistieal monolayer, t, whieh is a function of the relative gas pressure (t-plot method). Mereury porosimetiy serves the same purpose (Celis et al. 1998). N2 adsorption isotherms have also been used to determine the fractal dimensions of Fe oxide particles (c. f.. Celis et al. 1998 Weidler et al. 1998). 

The adsorbed layer is thicker and bound less strongly for the real chain (since for weak adsorption 0 < < 1) because it pays a higher confinement penalty than the ideal chain. The excluded volume interaction of real chains make them more difficult to compress or adsorb than ideal chains. These scaling calculations can be generalized to adsorption of a polymer with general fractal dimension Ifir. 

The adsorbed layer thickness for a polymer with general fractal dimension Xfu is derived in Problem 3.18 ... 

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