Flory-Stockmayer theory

In the classical theory, however, the neglect of loops significantly affects the size distribution and other properties of the clusters as one approaches the gel point. Some of the critical exponents that describe these properties in the classical theory and in percolation theory near p Pc are compiled in Table 5-1 (Martin and Adolf 1991). 

When p Pc, one can define P p) to be the fraction of bonds belonging to the infinite cluster. The percolation predictions of the modulus G, the longest relaxation time r, and the viscosity rj depend on whether one uses the Rouse-Zimm (R-Z) theory, or the analogy to an electrical network (EN). The exponent for the modulus G is predicted to be greater than either of these (i.e., around 3.7) if bond-bending dominates (Arbabi and Sahimi 1988). Further details about these exponents can be found in Chapter 5 of Drinker and Scherer (1990), as well as in Martin and Adolf (1991). 

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This results in a value of d = 2.5 for bond percolation on a 3-dimensional lattice. The fractal dimension of the Bethe lattice (Flory-Stockmayer theory) is... 

Problem 5.34 Calculate the gel-point conversions for the systems cited in Problem 5.27 using the recursive approach for comparison with the corresponding values calculated according to Flory-Stockmayer theory. 

However, as In the stepwise network polymerization. It would be valuable to determine relationships between the molecular structure and rheological properties. The Flory-Stockmayer theory for the structural buildup In a network forming radical chain growth pol3nnerlzatlon (9,10) predicts a conversion for gelation which Is much less than that found experimentally (11). Analysis of experimental results has determined the cause of this deviation to be the formation of Intramolecular crosslinks, l.e. cycllza-tlon (12). 

For t/j = 1 (linear chains). Equation (11.9a) provides the correct value, d = 2, corresponding to a macromolecular coil at the 0-point (see Table 11.2). As noted previously, d = 4/3 for a percolation cluster, irrespective of the dimension of the Euclidean space (see Table 11.1) therefore, from Equation (11.9a), we obtain df= 4, which is consistent with the Flory-Stockmayer theory [60] for phantom chains. For three-dimensional space, d > 3 has no physical meaning because the object cannot be packed more densely than an object having a Euclidean dimension. It is evident that this discrepancy is due to the phantom nature of the polymer chains postulated by Cates [56] it is therefore, necessary to take into account self-interactions of chains due to which the dimension of a polymer fractal assumes a value that has a physical meaning. 

Early theoretical approaches to the gel-formation [1-4] as the Flory-Stockmayer theory do not take into account several aspects which naturally occur as the individual molecules grow to form the gel, such as cyclic bond formation, excluded volume effects and steric hinderance. The Flory-Stockmayer theory assumes that in the gelation process each bond between two individual monomeric, oligomeric or polymeric molecules is formed randomly. Thus this theory assumes point-like monomers. This apparently is not the case when already existing macromolecules are crosslinked, i.e., in vulcanization reactions as well as in copolymerization reactions of macromolecules with the functionality /> 3 with bifunctional monomers. 

This approach has been applied to the Chaenomeles polymer (82). The theory can be applied with reasonable confidence in this case as GPC shows that it possesses a symmetrical MW distribution. Measurements of P and P (Table 7.7.2) shows that the dispersivity of the polymer is 2.6. The Flory-Stockmayer theory predicts that for the chain length of this polymer the dispersivity implies that 2% of the units are branched (J units), which means that there will be 25% more T units than B units (Fig. 7.7.2), which is corroborated reasonably well by the experimental NMR value of 20%-30% (see Sect. 7.7.2.1). Therefore, two experiments support the fact that proanthocyanidin polymer chains are branched, but that the frequency of branching is low. 

As a natural consequence of the crosslinking reaction process, the density of the primary polymer differs depending on the time of this primary polymer formation. That is, in the case of the copolymerization of vinyl and divinyl monomers, the generally formed inhomogeneous crosslink formation can be regarded as a natural consequence of the mechanism of crosslink formation. This is true except for die special reaction conditions by favorable timing of the incorporation of divinyl monomer in the polymer chain (formation of pendant double bonds) and consumption of pendant double bonds (formation of crosslinks). These special reaction conditions are used by Flory as simplified conditions when the Flory-Stockmayer theory is applied to the copolymerization of vinyl and divinyl monomers. Flory s simplified conditions include die following three assumptions (1) the reactivities of the monomer and die double bonds in the polymer are all equal (2) any double bond reacts independently and (3) there will be no intramolecular reactions (cyclization) within the finite size molecules (sols). 

Flory-Stockmayer theory, 43, 122 Focal point method for refiactive index, 392 Force curves, 246-7 Formaldehyde, 99... 

The static theories of equilibrium gelation (percolation theory) and vulcanization (Flory-Stockmayer theory) are well known. But many of the interesting experiments on polymer sols and gels concern dynamics viscoelasticity), which are less universal... 

Flory-Stockmayer theory have just started, and further more refined models will emerge in the future. We will summarize current ideas and add newer results. 

The cluster is dominated by the immense fluctuations in connectivity and density and it can be shown, by simulations, that many structural elements like single connecting bonds, dangling ends, loops and blobs are critical quantities, i.e, their number diverges with a certain exponent at the thres-hold. " This is another indication that one cannot expect validity of the mean field exponents as the Flory-Stockmayer theory calculates. 

As already mentioned the classical Flory-Stockmayer theory predicts... 

In normal Euclidian space the fractal dimension is equivalent to the normal space dimension, as it must. The most well-known fractal is the linear Gaussian polymer chain, and we see immediately that it is characterized by dfo = 2, - whereas for the self-avoiding chain the mean field value dfo = 5/3 is found at once. The cluster in the classical Flory-Stockmayer theory is characterized by a fractal dimension of 4 as visualized by R 1/425,26 another example consider the percolation cluster. The number of monomers in a volume is given by... 

Zheng, Y., Cao, H., Newland, B., Dong, Y., Pandit, A., Wang, W 3D single cycUzed polymer chain structure from controlled polymerization of multi-vinyl monomers beyond flory-stockmayer theory. J. Am. Chem. Soc. 133(33), 13130-13137 (2011)... 

Fig. 2. Structure of the Bethe lattice with f = 3 (interior part of the infinite system only). Each possible bond is shown as a line connecting two monomers (dots). The Flory-Stockmayer theory assumes that each actual bond of these possible bonds is formed with probability p...

Such hyperscaling relations are also known from other phase transitions a short introduction to scaling in the case of thermal phase transitions is given in the appendix of Ref. 7. In contrast to scaling relations (Eq. (8)), the hyperscaling relation (9) involving the dimensionality d cannot be used in Flory-Stockmayer theories and similar approaches. 

Specifically for gelation, we will discuss in Sect. C.V. various modifications of the simple percolation model of Fig. 1 and check if the exponents diange. In most cases, they do not in particular, the lattice structure (simple cubic, bcc, fee, spinels ) is not an important parameter since different lattices of the same dimensionality d give the same exponents within narrow error bars. More importantly, percolation on a continuum without any underlying lattice structure has in two and three dimensions the same exponents, within the error bars, as lattice percolation. In the classical Flory-Stockmayer theory which does not employ any periodic lattice structure, the critical exponents are completely independent of the functionality f of the monomers or the space dimensionality d. But if the system is not isotropic or if the gel point is coupled with the consolute point of the binary mixture solvent-monomers , the exponents may change as discussed in Sect. D. 

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