Flory-Stockmayer equation

The reaction is somewhat more complicated when stoichiometry on an equivalent basis is different from 1 1. In this case, the more general Flory-Stockmayer equation reads. 

This article shows how successfully the cascade branching theory works for systems of practical interest. It is a main feature of the Flory-Stockmayer and the cascade theory that all mentioned properties of the branched system are exhaustively described by the probabilities which describe how many links of defined type have been formed on some repeating unit. These link probabilities are very directly related to the extent of reaction which can be obtained either by titration (e.g. of the phenolic OH and the epoxide groups in epoxide resins based on bisphenol A206,207)), or from kinetic quantities (e.g. the chain transfer constant and monomer conversion106,107,116)). The time dependence is fully included in these link probabilities and does not appear explicitly in the final equations for the measurable quantities. 

Can the system be reacted to complete conversion without gelation If not, what is the extent of conversion of the acid functionality at the gel point calculated from (a) the Carothers equation and from the statistical approaches of (b) Flory-Stockmayer and (c) Macosko-Miller ... 

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. 

Moreover, Stockmayer has established that Flory s equation 3.1-115... 

For example, in Chapter 6, to begin with three parameters, p (shear stress), e (shear strain), and E (modulus or rigidity), are introduced to define viscosity and viscoelasticity. With respect to viscosity, after the definition of Newtonian viscosity is given, a detailed description of the capillary viscometer to measure the quantity t follows. Theories that interpret viscosity behavior are then presented in three different categories. The first category is concerned with the treatment of experimental data. This includes the Mark-Houwink equation, which is used to calculate the molecular weight, the Flory-Fox equation, which is used to estimate thermodynamic quantities, and the Stockmayer-Fixman equation, which is used to... 

One further point should be emphasized. When an epoxy reacts with a carboxyl group, an hydroxyl group is formed, see reaction (4), Thus it could be argued that in this esterification reaction, the vernonia oil has a functionality of six (or 4.8 if it is assumed that the epoxy groups are randomly distributed). These values are also shown in Table I. Of these calculations, the Flory-Stockmayer trifunctional value fits best. Use of equation (7) suggests an actual functionality of 3.61. 

Stockmayer 25 subsequently developed equations relating to branched-chain polymer size distributions and gel formation, whereby branch connectors were of unspecified length and branch functionality was undefined. An equation was derived for the determination of the extent of reaction where a three-dimensional, network ( gel ) forms this relation was similar to Flory s, although it was derived using another procedure. Stockmayer likened gel formation to that of a phase transition and noted the need to consider (a) intramolecular reactions, and (b) unequal reactivity of differing functional groups. This work substantially corroborated Flory s earlier studies. 

As in so many things in this field, if you want to work through the arguments yourself, you cannot do better than go to Flory— see Principles of Polymer Chemistry, Chapter EX. Stockmayer s equation illustrates the point we wish to make with dazzling simplicity as f the number of branches, increases, the polydispersity decreases. Thus for values of/equal to 4, 5 and 10, the polydispersity values are 1.25, 1.20 and 1.10, respectively. Note also that for / = 2, where two independent chains are combined to form one linear molecule (Figure 5-28), the polydispersity is predicted to be 1.5. Incidentally, an analogous situation occurs in free radical polymerization when chain termination is exclusively by combination. 

The recent theoretical treatment proposed by Miller, Brant and Flory (56) of polypeptide chains containing glydde and aminoacid side chains with branched arbon atoms indicates, as mentioned above that a small number of glycine residues markedly reduces the characteristic ratio, but that a small number of branched side chains does not exert much influence. The figure obtained by Tanford et al., even considering its experimental error of 10% (which is normal), and its origin in the application of the Stockmayer and Fixman equation to low molecular weight samples, is well within theoretical predictions and provides a conclusive test of these. 

One problem of Jacobson and Stockmayer s interpretation is observed in the cases of very small but unstrained rings, where much higher concentrations of rings were formed than were predicted. Flory and Semiyen ( ) explained this deviation by suggesting that not only did two termini have to meet within a volume Vg in order to establish a bond, they also had to approach each other from a specified direction. This direction was specified by a solid angle fraction 6a)/4iT. They explained that this term should appear in the entropy expression for process 1 in the inverse form, i.e. 4Tr/6o). In process 2, if the chains were sufficiently long, there would be no correlation between the probability for two termini of a molecule to meet within Vg and to approach within the solid angle 6a). In this case, the term 6a)/4ir would be valid for inclusion into the entropy term for process 2 and, hence, when the entropies for processes 1 and 2 were summed, these terms would cancel (and the equation for AS(3) from Jacobson and Stockmayer s theory should be valid). However, for short chains, the probability of approach... 

The Stockmayer distribution is an extension of the Flory distribution for copolymers. When Equation 5.4 is integrated over all chemical compositions, it is reduced to... 

Comparison of equations (4.27) and (4.28) shows that the Flory theory in near 0-solvents predicts a numerical coefficient (2.6) that is too large by most a factor of 2. Stockmayer (1955 1960) has therefore suggested that Flory s original equation should be arbitrarily adjusted so that... 

This is exactly the equation given already by Flory and Stockmayer [15, p. 373]. 

These condensation reactions can be modeled based on simple statistical approaches. One of the first approaches for predicting gelation or the gel point of complex mixtures was developed by Flory (23) and Stockmayer (24) (see equation 2). 

Let us start to explore some implications of Stockmayer s distribution it is another fundamental equation in polymer science and can be derived from the analytical solution of the copolymerization mechanism described in Table 2.11. Its derivation is long and tedious and not really required here it is enough to realize that it reflects the MWD and CCD of polymer made according to the copolymerization mechanism shown in Table 2.11 at a given time instant. The same considerations made for Flory s distribution apply to Stockmayer s distribution they will not be repeated here. 

Among the three equations we have discussed, only the Mark-Houwink equation is exact that is, no assumptions have been made. The Flory-Fox and Stockmayer-Fixman equations are both based on their models. 

At the gel point, when in the course of a three-dimensional polymerization reaction a network first appears, its mass is negligible practically all of the polymerizing material still exists as monomers, dimers, trimers, and larger groups which are not bound to the network. As the reaction proceeds after the gel point, more and more of these loose groups become attached to the network, so that its relative mass (the gel fraction ) gradually increases and approaches unity. Approximate equations for the increase in gel fraction with extent of reaction have been given by Flory (1941) and Stockmayer (1943). 

Amu (17) critically reviewed the effect of salts on the solubility of poly(ethylene oxide) in water together with intrinsic viscosity measurements and Flory theta conditions. From this work, Amu estimated the theta temperature of poly(ethylene oxide) in water to be 108.5°C. Amu presented data in terms of the Stockmayer-Fixman equation (18),... 

In the original formula of Flory, a = 2.6(3/27t). Stockmayer suggested that the choice of a = 4/3(3/2ti) gives the exact result of first order perturbation theory near 6 temperature and the result of equation (23) in good solutions. With this choice of a, equation (24) is called the modified Flory formula which can be derived in a systematic way. Many approximate closed expressions for a exist in the literature which are adequately reviewed. ... 

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