Chain transfer rate equations

The term Cs refers to solvent used in polymerization or any component specially added as chain transfer agent. Equation (6.122), often referred to as the Mayo equation, shows the quantitative effect of various transfer reactions on the number average degree of polymerization. Note that the chain transfer constants, being ratios of rate constants with same dimensions, are dimensionless quantities dependent on the types of both the monomer and the substance causing chain transfer as well as on the temperature of reaction. Various methods can be employed, based on the Mayo equation, to determine the values of the chain transfer constants. (See Problems 6.20 and 6.21 for some of these methods.)... 

A Smith-Ewart case 1 behavior in 1) can be observed during emulsion polymerization of monomers with a high chain-transfer rate constant such as vinyl chloride because the monomer radicals have a high tendency to escape from the particles. Especially for vinyl chloride emulsion polsunerization, Ugelstad and Hansen (113) derived the following equation 18 to calculate n, which predicts dependence, h a... 

It can thus be easily inferred that chain transfer may alter the properties of the pol5uneric product in an undesirable way, or—in contrast—may be used advantageously to specifically reduce the molecular weights obtained in a specific polymerization process or to introduce specific end groups. The first case discussed above (which refers to the kinetic concept of chain transfer) is assumed for the derivation of the Mayo equation, which is frequently used to derive chain transfer rate coefficients. This specific case does not alter the overall rate of the polymerization system, because it does not change the overall free radical concentration, but rather influences the molecular weight distribution. 

This equation gives the fundamental correlation of the number-average degree of poljunerization with the rate of pol5unerization and the various chain transfer coefficients. Equation 91 constitutes the basis for determining the various chain transfer coefficients. 

If a particularly low molecular mass polymer is required or if a reactor s pressure rating is too low to operate at very high temperatimes, an additional chain-transfer agent can be employed. This reduces the molecular mass by increasing the chain-transfer rate constant above that for monomer alone in the above equation. As a result DP decreases, depending on the quantity of chain transfer agent used. 

This suggests that polymerizations should be conducted at different ratios of [SX]/[M] and the molecular weight measured for each. Equation (6.89) shows that a plot of l/E j. versus [SX]/[M] should be a straight line of slope sx Figure 6.8 shows this type of plot for the polymerization of styrene at 100°C in the presence of four different solvents. The fact that all show a common intercept as required by Eq. (6.89) shows that the rate of initiation is unaffected by the nature of the solvent. The following example examines chain transfer constants evaluated in this situation. 

This equation can be solved numerically to give values of Clr and Ctr.404 For reversible addition-fragmentation chain transfer (RAFT) (Scheme 6.5), the rate constant for the reverse reaction is defined as shown in eq. 22 ... 

The general equation for the gel effect index, equation (la) which incorporates chain transfer, was used in those cases where there was not a good agreement between model predictions and experimental data. The same values of and (derived from the values of and C2 found at high rates) were used in the integration of equation (1) and the value of the constant of chain transfer to monomer, C, was taken as an adjustable parameter and used to minimize tfie error of fitting the time-conversion data by the model. 

LDPE polymerization reaction consists of various elementary reactions such as initiation, propagation, termination, chain transfer to polymer and monomer, p-scission and so forth [1-3], By using the rate expression of each elementary reaction in our previous work [4], we can construct the equations for the rate of formation of each component. 

The Mayo equation (Equation 6.42) that gives positive slopes when the data is plotted (such as Figure 6.3) is the reciprocal relationship derived from the expression cited earlier. The ratio of the rate of cessation or termination by transfer to the rate of propagation is called the chain transfer constant (Cs). 

Values of Ca from eqn. (113) can be used in eqn. (83) to obtain instantaneous values for P at particular times (when chain transfer reactions are unimportant). If the initiation rate is taken as constant and no chain transfer reactions occur, then, using equations given by Tadmor and Biesenberger [62], it can be shown that... 

For any specific type of initiation (i.e., radical, cationic, or anionic) the monomer reactivity ratios and therefore the copolymer composition equation are independent of many reaction parameters. Since termination and initiation rate constants are not involved, the copolymer composition is independent of differences in the rates of initiation and termination or of the absence or presence of inhibitors or chain-transfer agents. Under a wide range of conditions the copolymer composition is independent of the degree of polymerization. The only limitation on this generalization is that the copolymer be a high polymer. Further, the particular initiation system used in a radical copolymerization has no effect on copolymer composition. The same copolymer composition is obtained irrespective of whether initiation occurs by the thermal homolysis of initiators such as AIBN or peroxides, redox, photolysis, or radiolysis. Solvent effects on copolymer composition are found in some radical copolymerizations (Sec. 6-3a). Ionic copolymerizations usually show significant effects of solvent as well as counterion on copolymer composition (Sec. 6-4). 

Despite the complex interaction between the components of a catalyst recipe, for example consisting of catalyst, co-catalyst, electron donors (internal and external), monomers, chain-transfer agents such as hydrogen, and inert gases and the catalyst support, the local polymer production rate rate (polymerization rate) in a given volume, Rp (kg polymer hr"1), can often be described by a first-order kinetic equation with respect to the local monomer concentration near the active site, cm (kgm"3), and is first order to the mass of active sites ( catalyst ) in that volume, m (kg) ... 

A few very important points have been neglected by some authors in their evaluation of chain transfer constants by means of kinetic measurements. Frequently, a retardation of the overall rate is to be observed in the presence of chain transfer agents. A correct value of the chain transfer constant can result only if the reactions which lead to this retardation are properly considered in the kinetic scheme. In addition, the equation which one must use to calculate the chain transfer constant depends on the type of molecular weight average which is measured. Failure to... 

In this case, deuteration has little effect on the polymerization rate since k3 (chain transfer constant) does not appear in equation (8). But there will be an increase in molecular weight compared to vinyl acetate since the chain transfer constant for tri-deuterovinyl acetate is smaller than that of vinyl acetate. 

Other reactions may be taken into consideration, with an effect on polymer structure, namely the formation of short- and long-chain branches. A complete list of reactions in S-PVC polymerization may be found in Kiparissides et al. [5]. On the above basis kinetic equations may be written. To keep it simple the chain transfer, back-biting and inhibition reactions are disregarded, while termination is considered to occur only by disproportionation. The elementary reaction rates for initiator decomposition and free radicals generation are as follows ... 

A more formal derivation of Equation 4-39, useful when we get to chain transfer, is obtained starting from v = rjre (Remember the steady-state assumption, r, = r, ) To obtain the number average degree of polymerization instead of the kinetic chain length we then use the rate of dead chain formation instead of rt (Equations 4-40). 

The "ideal" concept of emulsion polymerization was built on the assumption that the monomer was water insoluble and that in the absence of chain transfer, the number average degree of polymerization, Xj can be related to the rate processes of initiation and propagation by the steady-state relationship Xjj = 2 Rp/Rj. Since Ri and Rp are both constant and termination is assumed to be Instantaneous during the constant rate period described by Smith-Ewart kinetics, the above equation predicts the generation of constant molecular weight polymer. Data has been obtained which agrees with Smith-Ewart but there is... 

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