Carother’s equation

A step polymerization yields high molecular weight polymer only after very high conversion is reached (Fig. 1) as described by Carother s equation [Eq. (2)] relating the number average degree of polymerization (DP ) to conversion (p). 

Figure 3.23 Carother s equation and synthesis of telechelic oligomers by a nonstoichoimetric polycondensation method.

By combining the above expression with Carother s equation and solving for [M], one obtains... 

There were some previous reports on the preparation of the PPBIs in PPA that resulted in low IV polymers or incomplete imidazole ring closure [ 102, 103]. In the present study, the initial polymerizations using diadd monomers without further purification gave PPBIs with IVs less than l.OdLg", as shown in Table 2. Based on the Carother s equation, monomer purity and accurate stoichiometry are crucial to obtain high IV polycondensation polymers. Therefore, a detailed study of the monomer purity and purification method was carried out on aU the diadd monomers. DSC scans were employed to monitor the relative purity of the monomers. They confirmed the improved monomer purity after purification. Table 2 shows that all diadd monomers polymerized to yield high IV PPBIs after appropriate purification of starting materials and careful manipulation of polymerization conditions. 

The next point discussed by Flory was the relationship between DP (Flory frequently used x as symbol) and conversion. He formulated Eq. (4.2), later called Carother s equation, but he also formulated Eq. (4.3). As will be shown below, Eqs. (4.2) and (4.3) do not give the same result under aU circumstances. Anyway Eq. (4.2) demonstrates that syntheses of high molar mass polymers via polycondensation requires extremely high conversions (>98 %) in contrast to chain-growth polymerizations. On the basis of Eq. (4.2), Mn was defined according to Eq. (4.4). 

According to Carother s equation, the critical extent of reaction Pq is... 

What amounts to invoking the shortsightedness principle has become standard practice in polymerization kinetics where, for most purposes, the rate coefficients of step or chain growth and termination are taken to be independent of the lengths of the polymer chains (see Sections 11.2.1, 11.3.1, 11.4.1,and 11.5.1). In fact, the principle is the basis of Carother s approach to step-growth polymerization kinetics leading to his equations 11.6 and 11.7. 

Carcass method Carnot cycle Carothers Casson s equation Catalyst (general)... 

If Carothers s assumption that reactivity is equal and random reactivity holds. Equation 1.92 still equals the general form for step polymerization writtm in Equation 1.62. However, if a chemical system is designed such that k tf. Equation 1.92 can be rewritten in a simple form, which now resembles the rate of evolution for a chainwise system (see Eq. 1.33). 

To be kinetically useful we need to put this equation entirely in terms of cq. Replacing the N s in the Carothers equation by c s gives c = co(l — p). Substituting this value for c in the integrated expression ... 

While the theory of Flory underestimates the gel point, the theory of Carothers overestimates it. For example, for the case of a trifunctional and bifunctional reaction system presented above, at equimolar reactive groups, while Flory s theory forecasts pg = 0.707, Carothers theory forecasts a value of pgei=0.833, while the experimental pgei is 0.765-0.775. Recently, however [18,19], thermodynamic reasons why the two formulas should be combined came to Kght. This led to the proposal of a very simple equation affording much higher precision than each theory alone. This simple equation also gave much better precision of all the very complex and difficult to use theories devised and presented since Flory s 1942 one ... 

Statistical methods were developed for prediction of gelaticMi [7]. These actually predict gelation at a lower level than does the Carothers equation shown above. As an example, we can use a reaction of three monomers, A, B, and C. We further assume that the functionality of two monomers,/A and/s, is equal to two, while that of/c is greater than two. The critical reaction conversion can then be written as ... 

In the mid 1960 s, Prochaska was granted several patents for the preparation of monomeric cyclic aromatic carbonates with 7- or 8-membered rings, formed from 2,2 -biphenols (Equation 7).l The monomeric cyclic carbonates could be prepared either directly, via phosgenation, or using a vacuum distillation/depolymerization technique similar to that developed by Carothers for the preparation of cyclic aliphatic esters and... 

The first synthesis of f-caprolactone was reported by Carothers(1). He also investigated its polymerization under the influence of heat cmd catalysts. Since then the polymerizations of this as well as that of other lactones were studied by many researchers. Throughout the 1950 s to the 1970 s the polymer formation and its properties were the subject of several investigations in our laboratories(2-5). Union Carbide is presently the commercial producer of the monomer and of a series of polymers which range in molecular weights from 500 to 40,000. The starting f-caprolac-tone is produced by the peracetic acid oxidation of cyclohexanone as shown in Equation (I). 

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