F Model

Physically, it is the fraction of functional groups available to reaction (and hence equivalent to Hx in Eqs. (65) and (72)). Equation (99) reveals that for this version of the FS model the time units are nine times longer" than those from direct application of the original Smoluchowski equation. 

Only for the FS model (jci=2/3, jc2=1/3) have explicit solutions of Eqs. (102)-(104) been found. The functions Hxx, Hxy, Hyy have, as expected, a singularity at f=3 of the form (3 - f) 1. The singularity corresponds to the gel point. At tc=3, where tc is the time where the gel point is, these functions simultaneously diverge. No modifications of Eq. (88) that would be valid beyond this point are known. 

All other a-phase salts (M = K, Rb, and Tl) showed AMRO periodic in tan 0, too. In contrast to the magnetoresistance shape in the former materials no resistance maxima as expected for a 2D warped cylinder but instead sharp resistance minima were observed [289, 309, 310, 311]. Figure 4.14a shows examples of this behavior for Q -(ET)2KHg(SCN)4 and a-(ET)2RbHg(SCN)4 at R = 11T and T — 1.4K [312]. Very prominent minima periodic in tan are visible in the magnetoresistance. These minima cannot be explained by the 2D warped FS model but are attributed to Lebed-like oscillations of the ID bands [313] (see also Sect. 3.3). From the calculated band structure shown in Fig. 2.20 one can see that the FS of the a phase consists of 2D closed orbits in the Brillouin zone corner and two ID open sheets perpendicular to the a direction. The maximum amplitude of the AMRO, however, was not observed for a rotation of the field parallel to these sheets but for angles 20° and 24° towards the a direction of the M = K and M = Rb crystals, respectively (see Fig. 4.14a). 

Successive high-resolution AMRO experiments shown in Fig. 4.39 verified the proposed FS in an impressive way [376]. As mentioned in Sect. 3.3, 0-(ET)2l3 was one of the first compounds where AMRO, i. e., resistance oscillations periodic in tan (9, were observed [258]. These results, which were reproduced later [377], are understood by the warped FS model explained above. The period of the oscillations is related to the Fermi wave vector via (3.18). In the experimental data shown in Fig. 4.39 not only the previously reported fast AMROs but also slow ones (indicated by small dashes) were observed [376]. The insets of Fig. 4.39 show the peak numbers of the (a) fast and (b) slow oscillation frequency vs tan O. From the slopes for different field rotation planes fcr(0) could be constructed. The resulting two ellipsoidal FSs are in good agreement with the proposed topology of Fig. 4.37c with respect to both form and area. 

The first theory that attempted to derive the divergences in cluster mass and average radius accompanying gelation is that of Flory [52] and Stockmayer [53]. In their model, bonds are formed at random between adjacent nodes on an infinite Cayley tree or Bethe lattice (see Figure 47.7). The Flory-Stockmayer (FS) model is qualitatively successful because it correctly describes the emergence of an infinite cluster at some critical extent of reaction and... 

MIFS=FS(analyte) + FS(model phase) FS(analyte — model phase complex),... 

The increase in resistivity at narrow fine widths has been attributed to surface scattering and grain-boundary scattering. The Fuchs and Sondheimer (FS) model attributes the resistivity increase in thin and narrow fines to diffuse scattering of electrons at the exterior surfaces with a probability of 1 — p, where p is the specular scattering coefficient. The length scales in the FS model are the thickness and line width of the conductor and the mean free path A. The simplified expression for resistivity as a function of thickness (T) and linewidth (W) of the conductor is given by ... 

The difference between the FS model and percolation model is in the critical phenomenon. As summarized in Table 1, if the statistical values are normalized by the equivalent distance e(= 1 — a/a ) from the gel point (the critical point), there is a significant difference in critical index for flie FS model and percolation model. This difference reflects the difference in size distribution (see Fig. 1 [6]). The difference of the structure in flie model is reflected on the fractal dimension D of the fraction that has a certain degree of polymerization x. If the radius of a sphere that corresponds to the volume of the branched polymer fiaction with the degree of polymerization x is R, the relationship between x and R is fimm the fiactal dimension D... 

If R is replaced by the radius of gyration of the branched polymer fraction with the degree of polymerization x from Table 1, item (4), the fractal dimension D — 2.5 for the percolation model and Z) = 4 for the FS model is obtained. The FS model appears to be packed by more than a sphere (D = 3) and thus is unrealistic. This contradiction can be understood when a dendrimer is considered. The FS model grows dendritically. When a trifunctional (f = 3) monomer grows radially, there will be no space beyond the sixth generation. However, for the FS model, the branches grow theoretically even imder such conditions. In the percolation model, the entire space is predistributed to the structural units, thus no such... 

The aggregation model can be understood as a more general model of fiie FS model and percolation model from the fractal point of view, and aggregation phenomena can be found everywhere in nature. Antibody antigen reactions, colloidal suspension, and the aggregation of clouds as well as file Milky Way, all exhibit various aggregation levels and tiiey all... 

Because the average cluster size is proportional to M2it)/Mi(t), the gelation takes place at the time f, at which M2(t) becomes infinite. When the bonding probability Ky is constant, i -I- j, or ij (Table 2), the Smoluchowski equation can be solved analytically [15]. In particular, it will yield results similar to those with the FS model at Ky = ij. In the /-functional condensation polymerization system (FS model), the functionality /(/ — 2) -I- 2 of the cluster with degree of poljnnerization i is proportional to the volume. However, if the shape of the cluster (branched... 

Here, the lower limit corresponds to the surface area of the densely packed body with spatial dimension d and the upper limit corresponds to the FS model where each fiinctional group maintains complete reactivity. If w > j, sol-gel transition takes place within a limited time. In general, (a = D/d iP is the fractal dimension). The index to define the width of the cluster distribution is (see Eq. (7), t of Table 1, item (6)),... 

When domains are packed closely (D = 3), Eq. (12) is the same as the correlation function of the Debye-Busche type [20], and equivalent to the domain that consists of the random aggregates with smooth surface. The structure of the domain can be described by the structural model described in the previous section. In any case, Eq. (12) can be used for approximation. However, the fractal dimension would depend on the model used. For example, for the classical model (FS model), D = 2. 

Sierpinski gasket (there can be size distribution) forming networks as a trifiinctional structural unit. Therefore, the network as a whole can be described by the FS model D = 2) and the inside structure has the fiactal dimension of Sierpinski gasket D — 1.585. 

As the simplest model, we will start with the Flory-Stockmayer dendritic model (the FS model see Chapter 3, Section 1, Fig. 2) [45]. If the individual structural unit of the FS model is the scattering point, Debye s particle scattering can be applied [46]. This corresponds to the... 

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