Rate of Radical Chain Polymerization

Equation 3-32 describes the most common case of radical chain polymerization. It shows the polymerization rate to be dependent on the square root of the initiator concentration. This... 

Table 6.2 shows the general range of values of the various concentrations, rates, and rate constants pertaining to the above kinetic scheme. These values are typical of radical chain polymerizations. 

Kinetic analysis of this type of system is possible. The basic equations for the kinetics of radical-chain polymerization are derived by assuming a low steady-state concentration of reacting radicals. Under these conditions, the rate of termination equals that of initiation ... 

Viscosity is an important factor during ultrasound-induced bulk polymerizations as the long polymer chains formed upon reaction cause a drastic increase in the viscosity of the reaction mixture, thereby hindering cavitation and consequently reducing the production rate of radicals. Precipitation polymerization forms a potential solution to this problem, because a constant viscosity and hence a constant radical formation rate can be maintained. In this perspective, high-pressure carbon dioxide is an interesting medium as most monomers have a high solubility in CO2, whereas it exhibits an anti-solvent effect for most polymers. ... 

It is appropriate at this point to briefly discuss the experimental procedures used to determine polymerization rates for both step and radical chain polymerizations. Rp can be experimentally followed by measuring the change in any property that differs for the monomer(s) and polymer, for example, solubility, density, refractive index, and spectral absorption [Collins et al., 1973 Giz et al., 2001 McCaffery, 1970 Stickler, 1987 Yamazoe et al., 2001]. Some techniques are equally useful for step and chain polymerizations, while others are more appropriate for only one or the other. Techniques useful for radical chain polymerizations are generally applicable to ionic chain polymerizations. The utility of any particular technique also depends on its precision and accuracy at low, medium, and high percentages of conversion. Some of the techniques have the inherent advantage of not needing to stop the polymerization to determine the percent conversion, that is, conversion can be followed versus time on the same reaction sample. 

The kinetic chain length v of a radical chain polymerization is defined as the average number of monomer molecules consumed (polymerized) per each radical, which initiates a polymer chain. This quantity will obviously be given by the ratio of the polymerization rate to the initiation rate or to the termination rate, since the latter two rates are equal. 

Five different types of rate constants are of concern in radical chain polymerization—those for initiation, propagation, termination, chain transfer, and inhibition. The use of polymerization data under steady-state conditions allows the evaluation of only the initiation rate constant kd (or kt for thermal initiation). The ratio kp/k J2 or kp/kl can be obtained from Eq. 3-25, since Rp, Rj, and [M] are measurable. Similarly, the chain-transfer constant k /kp and the inhibition constant kz/kp can be obtained by any one of several methods discussed. However, the evaluation of the individual kp, k ktr, and kz values under steady-state conditions requires the accurate determination of the propagating radical concentration. This would allow the determination of kp from Eq. 3-22 followed by the calculation of kt, kIr, and kz from the ratios kp/ltj2, ktr/kp, and kz/kp. 

Table 3-10 shows the values of the various concentrations, rates, and rate constants involved in the photopolymerization of methacrylamide as well as the range of values that are generally encountered in radical chain polymerizations. For the methacrylamide case, the experimentally determined quantities werei ,-, (Rp)s, [M], [I], kp/ltj1, ts, and kp/kf. All of the other... 

Consider the activation energy for various radical chain polymerizations. For a polymerization initiated by the thermal decomposition of an initiator, the polymerization rate depends on the ratio of three rate constants kp (kj/k,) 2 in accordance with Eq. 3-32. The temperature dependence of this ratio, obtained by combining three separate Arrhenius-type equations, is given by... 

Radical chain polymerizations are characterized by the presence of an autoacceleration in the polymerization rate as the reaction proceeds [North, 1974], One would normally expect a reaction rate to fall with time (i.e., the extent of conversion), since the monomer and initiator concentrations decrease with time. However, the exact opposite behavior is observed in many polymerizations—the reaction rate increases with conversion. A typical example is shown in Fig. 3-15 for the polymerization of methyl methacrylate in benzene solution [Schulz and Haborth, 1948]. The plot for the 10% methyl methacrylate solution shows the behavior that would generally be expected. The plot for neat (pure) monomer shows a dramatic autoacceleration in the polymerization rate. Such behavior is referred to as the gel effect. (The term gel as used here is different from its usage in Sec. 2-10 it does not refer to the formation of a crosslinked polymer.) The terms Trommsdorff effect and Norrish-Smith effect are also used in recognition of the early workers in the field. Similar behavior has been observed for a variety of monomers, including styrene, vinyl acetate, and methyl methacrylate [Balke and Hamielec, 1973 Cardenas and O Driscoll, 1976, 1977 Small, 1975 Turner, 1977 Yamamoto and Sugimoto, 1979]. It turns out that the gel effect is the normal ... 

One difference in the use of these equations for radical chain polymerizations compared to step polymerizations is the redefinition of p as the probability that a propagating radical will continue to propagate instead of terminating. The value of p is given as the rate of propagation divided by the sum of the rates of all reactions that a propagating radical may undergo... 

Consider the polymerization of styrene initiated by di-t-butyl peroxide at 60°C. For a solution of 0.01 M peroxide and 1.0 M styrene in benzene, the initial rates of initiation and polymerization are 4.0 x 10 11 and 1.5 x 10 7 mol L 1 s 1, respectively. Calculate the values of (jkj), the initial kinetic chain length, and the initial degree of polymerization. Indicate how often on the average chain transfer occurs per each initiating radical from the peroxide. What is the breadth of the molecular weight distribution that is expected, that is, what is the value of Xw/Xnl Use the following chain-transfer constants ... 

Most radical chain polymerizations show a one-half-order dependence of the polymerization rate on the initiation rate Ri (or the initiator concentration [I]). Describe and explain under what reaction conditions [i.e., what type(s) of initiation and/or termination] radical chain polymerizations will show the following dependencies ... 

A case classically associated with radical chain polymerization for which a (pseudo)steady state is assumed for the concentration of active centers this condition is attained when the termination rate equals the initiation rate (the free-radical concentration is kept at a very low value due to the high value of the specific rate constant of the termination step). The propagation rate, is very much faster than the termination rate, so that long chains are produced from the beginning of the polymerization. For linear chains, the polydispersity of the polymer fraction varies between 1.5 and 2. 

The gel or Trommsdorff effect (11) is the striking autoacceleration of the vinyl polymerization reaction as the viscosity of the monomer-polymer solution increases. Chain termination involving the recombination of two free radicals becomes diffusion controlled and this results in a decrease in the rate of termination. The concentration of active free radicals therefore increases proportionally. To sum up the gel effect the rate of Vazo catalyst initiation increases with temperature the rate of propagation or polymerization increases with the viscosity and the rate of termination of the growing polymer chains decreases with the viscosity. This of course also results in an increase in the molecular weight of linear polymers, but this has no practical significance when crosslinking is part of the reaction. 

The mechanism can be best understood within the framework of the conventional theory of radical chain kinetics, provided that certain of the usual simplifying assumptions are omitted. A solution is given to the problem of steady-state polymerization rate as a function of monomer and initiator concentration, taking into account termination reactions of primary radicals and recombination of geminate chains arising from the same initiation event. This model is shown to account for the kinetic data reported herein. With appropriate rate constants it should be generally applicable to radical polymerizations. 

Now the ratio A p(M)//c<(M-) = fcp(M)(M-)//c (M-) can be seen to be the ratio of the rate at which monomer is converted into polymer to the rate of termination of radical chains. If termination occurs by recombination, then this ratio is just one-half the average number of monomer units per final polymer chain, which we may represent by n, the mean chain length, or mean degree of polymerization. This permits us to write for the stationary radical concentrations... 

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