Steady-state approximation free radical polymerization

Photoinitiated free radical polymerization is a typical chain reaction. Oster and Nang (8) and Ledwith (9) have described the kinetics and the mechanisms for such photopolymerization reactions. The rate of polymerization depends on the intensity of incident light (/ ), the quantum yield for production of radicals ( ), the molar extinction coefficient of the initiator at the wavelength employed ( ), the initiator concentration [5], and the path length (/) of the light through the sample. Assuming the usual radical termination processes at steady state, the rate of photopolymerization is often approximated by... 

The molecular weight distribution and the average molecular weight in a free-radical polymerization can be calculated from kinetics. The kinetic chain length v is defined as the average number of monomers consumed per number of chains initiated during the polymerization. It is the ratio of the propagation rate to the initiation rate (or the termination rate with a steady-state approximation) ... 

As with free radical polymerization, to express the rate of polymerization in terms of measurable terms, one can approximate a steady state for the growing chain end, which implies that the rate of initiation equals the rate of termination, thus Ri = Rf, and... 

This relation is of fundamental importance in free-radical polymerization since the kinetic chain length decreases with an increase in the rate of initiation. Thus an attempt to accelerate polymerization by adding more initiator will produce a faster reaction but the polymer will have shorter chains. This can also be seen as a consequence of the steady-state approximation in a linear chain reaction since the rate of termination is equal to the rate of initiation and, if the rate of termination increases to match the rate of initiation, the chains must necessarily be shorter. 

The specificity of the reaction mechanism to the chemistry of the initiator, co-initiator and monomer as well as to the termination mechanism means that a totally general kinetic scheme as has been possible for free-radical addition polymerization is inappropriate. However, the general principles of the steady-state approximation to the reactive intermediate may still be applied (with some limitations) to obtain the rate of polymerization and the kinetic chain length for this living polymerization. Using a simplified set of reactions (Allcock and Lampe, 1981) for a system consisting of the initiator, I, and co-initiator, RX, added to the monomer, M, the following elementary reactions and their rates may be... 

Problem 6.2 Experimentally, it is found that, except in the very earliest (and generally negligible) stages of the reaction, the loss of monomer is accounted for quantitatively by the appearance of the polymeric product. Justify on this basis the steady-state approximation that all free radicals present in a polymerizing system are at steady-state concentrations. 

The most important difference between a living ionic polymerization which has no termination or transfer mechanism and free-radical or ionic processes that do have termination or chain transfer steps is that the distributions of the degrees of polymerization are quite different. The distribution function can be derived by a kinetic approach due to Flory [6], which is analogous to that used earlier for free-radical reactions (see Problem 6.44). However, in the present case with no chain termination the simplifying steady-state approximation cannot be used. 

C. Spade and V. Volpert, On the steady state approximation in thermal free radical frontal polymerization, Chem. Eng. Sci., 55 (2000), pp. 641-654. 

The kinetics of cationic polymerizations are considerably more complex than those of the free-radical polymerizations, and kinetic data is difficult to interpret becanse of several reasons (92,138-140). For example, in cationic polymerizations the identity and proximity of the coimterion has a marked effect on the reactivity of the active center. An active center that is encumbered by a closely associated counterion has a dramatically lower reactivity (typically an order of magnitnde lower) than an active center that is separated from the counterion. As described in the section on photoinitiation, this consideration has lead to the development of large, nonnucleophilic coimterions however the reactivity of a cationic active center still depends on the proximity of the counterion. Therefore, at any given time, a variety of propagating species may exist, ranging from ion pairs to separated ions. For this reason, an effective propagation rate constant which inclndes contributions from all propagating species is usually adopted, as described below. Secondly, unlike free-radical polymerizations, the steady-state approximation for active center concentration is not valid since the cationic active centers are not reactive toward one another, and the rate of active center... 

The previous sections address the kinetics for each of the processes involved in free radical polymerization, as well as the overall polymerization process. A steady-state approximation was used to determine the overall rate of polymerization and the chain length distribution. Practically, there are many exceptions to these approximations, including nonsta-tionary polymerization and dead-end polymerization [50, 51], which are treated in more detail elsewhere. 

The data are from a free radical polymerization of butyl acrylate (BA) in butyl acetate. When fractional monomer conversion reached 0.4, an extra amount of azobisisobutyro-nitrile (AIBN) initiator was added (initiator boost). Its effect can be immediately seen in the rapid drop of as the quasi-steady state approximation (QSS A) predicts for kinetic chain length (e.g., see Chapters 1 and 5) which is the molecular weight of chains being produced at any instant in a free radical reactions. It is proportional to the concentration of monomer to that of initiator. Hence, the addition of initiator causes the instantaneous chain length, and hence to fall. 

In order to simplify the kinetic scheme a steady-state approximation has to be made. It is assumed that under steady-state conditions the net rate of production of radicals is zero. This means that in unit time the number of radicals produced by the initiation process must equal the number destroyed during the termination process. If this were not so and the total number of radicals increased during the reaction, the temperature would rise rapidly and there could even be an explosion since the propagation reactions are normally exothermic. In practice it is found that the steady-state assumption is usually valid for all but the first few seconds of most free radical addition polymerization reactions. 

The quasi-steady hypothesis is used when short-lived intermediates are formed as part of a relatively slow overall reaction. The short-lived molecules are hypothesized to achieve an approximate steady state in which they are created at nearly the same rate that they are consumed. Their concentration in this quasi-steady state is necessarily small. A typical use of the quasi-steady hypothesis is in chain reactions propagated by free radicals. Free radicals are molecules or atoms having an unpaired electron. Many common organic reactions such as thermal cracking and vinyl polymerization occur by free-radical processes. There are three steps to a typical free-radical reaction initiation, propagation, and termination. 

The rate of formation of ions is a factor of approximately 10-100 times smaller than that for free radicals. Conversely, the recombination constants are about 100 times larger for ions than for free radicals. Thus, it follows that the steady state concentration of ions is about 100 times smaller than that for free radicals. Consequently, the majority of radiation-initiated polymerizations proceed by a free radical mechanism. 

Again the steady state with its general approximations is assumed in which the concentrations of the reactants, such as the monomer, free radicals, and transfer agent, do not vary with time. Hence, in equation 83 the number of polymerized monomer units can be substituted with the rate of polymerization and the numbers of end groups by the rate of their formation. 

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