Fractal adsorber

A fractal adsorbing surface has the same stmctural features at length scales that are comprised between the inner cut-off limit and the upper cut-off limit r. Its irregularity is described by the surface fractal dimension A. The fact that the complexity of the adsorbing surface is captured by a single number is extremely convenient and appealing, and allows for a great simplification of the theoretical description of adsorption phenomena. 

As before, it is convenient to designate a home particle on the fractal adsorber. We now consider an arbitrary point at a distance r from the home point. Despite the absorption we expect some of the walker s tracks to remain at r, as shown in Fig. 8.4. The density of such walkers relative to the initial density is the probability that the walker at r has not been removed by adsorption. This probability is the probability that the random walk representing its past has not intersected the fractal. This is just the g(r) discussed in the last section. The fractals in this case are the adsorbing fractal, with dimension D, and the random walk, with dimension 2. As we saw above, this g r) depends on the fractal dimensions. If D 4- 2 is less than 3, the two are mutually transparent g r) 1. Virtually all the walkers in the pervaded volume of the fractal never touch the fractal. Their density u is thus virtually unaffected u r) Uq. But if D-l-2 is greater than 3, the two are mutually opaque, and g r) is substantially smaller than 1 for most points r within the absorbing fractal. All connected fractals have D > 1 and thus show opaque behavior. For all such fractals, in-... 

A fractal surface of dimension D = 2.5 would show an apparent area A app that varies with the cross-sectional area a of the adsorbate molecules used to cover it. Derive the equation relating 31 app and a. Calculate the value of the constant in this equation for 3l app in and a in A /molecule if 1 /tmol of molecules of 18 A cross section will cover the surface. What would A app be if molecules of A were used ... 

Many of the adsorbents used have rough surfaces they may consist of clusters of very small particles, for example. It appears that the concept of self-similarity or fractal geometry (see Section VII-4C) may be applicable [210,211]. In the case of quenching of emission by a coadsorbed species, Q, some fraction of Q may be hidden from the emitter if Q is a small molecule that can fit into surface regions not accessible to the emitter [211]. 

We have considered briefly the important macroscopic description of a solid adsorbent, namely, its speciflc surface area, its possible fractal nature, and if porous, its pore size distribution. In addition, it is important to know as much as possible about the microscopic structure of the surface, and contemporary surface spectroscopic and diffraction techniques, discussed in Chapter VIII, provide a good deal of such information (see also Refs. 55 and 56 for short general reviews, and the monograph by Somoijai [57]). Scanning tunneling microscopy (STM) and atomic force microscopy (AFT) are now widely used to obtain the structure of surfaces and of adsorbed layers on a molecular scale (see Chapter VIII, Section XVIII-2B, and Ref. 58). On a less informative and more statistical basis are site energy distributions (Section XVII-14) there is also the somewhat laige-scale type of structure due to surface imperfections and dislocations (Section VII-4D and Fig. XVIII-14). 

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. 

The time evolution of the mean size of CdS and ZnS nanoparticles in water/AOT/ -heptane microemulsions has been investigated by UV-vis spectrophotometry. It was shown that the initial rapid formation of fractal-hke nanoparticles is followed by a slow-growing process accompanied by superficial structural changes. The marked protective action of the surfactant monolayer adsorbed on the nanoparticle surface has been also emphasized [230,231],... 

T. Zavada, R. Kimmich 1999, (Surface fractal probed by adsorbate spin-lattice relaxation dispersion), Phys. Rev. E 59, 5848. 

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) ... 

The thickness of the layer of the adsorbed molecules is the characteristic distance scale for fractal surface. (3) Van der Waals attraction forces between solid/gas interactions and the liquid/gas surface tension forces are contributed to the grand potential of the system. 

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... 

Here, the actual thickness of adsorbed molecule layers /ad is obtained by multiplying n by adsorbed molecule diameter a0. The actual thickness range in which the fractal geometry is satisfied represents the length-scale cutoff range of fractality.116,118 Here, the value of the lower (inner) cutoff length tadjmfa is equal to... 

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). 

The fractality index a obtained from analysis of SANS curves correlates with the adsorption capacity of carbon materials for unconjugated bilirubin adsorbed from HSA solution in micro-column single-pass experiments (Table 29.2) [10]. 

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. 

Sampling in surface-enhanced Raman and infrared spectroscopy is intimately linked to the optical enhancement induced by arrays and fractals of hot metal particles, primarily of silver and gold. The key to both techniques is preparation of the metal particles either in a suspension or as architectures on the surface of substrates. We will therefore detail the preparation and self-assembly methods used to obtain films, sols, and arrayed architectures coupled with the methods of adsorbing the species of interest on them to obtain optimal enhancement of the Raman and infrared signatures. Surface-enhanced Raman spectroscopy (SERS) has been more widely used and studied because of the relative ease of the sampling process and the ready availability of lasers in the visible range of the optical spectrum. Surface-enhanced infrared spectroscopy (SEIRA) using attenuated total reflection coupled to Fourier transform infrared spectroscopy, on the other hand, is an attractive alternative to SERS but has yet to be widely applied in analytical chemistry. 

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... 

The thickness of the layer of the adsorbed molecules is the characteristic distance scale for fractal surface. 

Table 1. As the relative pressure increases, the thickness of adsorbed molecule layers on the pore surfaces increases as well, and then the pore filling process caused by capillary condensation occurs first in the small pores simultaneously with the multilayer adsorption on the larger pores. For specimen III, the value of 4d,max is significantly larger than those values for the other specimens, which is ascribed to the fact that as a result of the pore filling process in the larger macropores the adsorbed volume starts to increase abruptly only near the saturation vapor pressure in the gas adsorption isotherm. From the above results, they suggested that 4d,max / 4d,mm is closely related to rmax, that is, the larger rmax, the wider ranges the length-scale of the fractal regime in value.

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