Chain polymerization steady-state condition

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. 

The average degree of polymerization P of a polyolefin (its molecular mass), produced under steady-state conditions, with a non-living process, is determined by the ratio between propagation rates and chain release rates (Equation (1)). 

Termination reactions cannot be eliminated in radical polymerizations because termination reactions involve the same active radical species as propagation therefore, eliminating the species that participates in termination would also result in no polymerization. Termination between active propagating species in cationic or anionic processes does not occur to the same extent because of electrostatic repulsions. Equation (1) represents the rate of polymerization, Rp, which is first order with respect to the concentration of monomer, M, and radicals, P, while Eq. (2) defines the rate of termination, Rt, which is second order with respect to the concentration of radicals. To grow polymer chains with a degree of polymerization of 1000, the rate of propagation must be at least 1000 times faster than the rate of termination (which under steady state condition is equal to the rate of initiation). This requires a very low concentration of radicals to minimize the influence of termination. However, termination eventually prevails and all the polymer chains produced in a conventional free radical process will be dead chains. Therefore they cannot be used in further reactions unless they contain some functional unit from the initiator or a chain transfer agent. 

The second step of initiation [Eq. (8.83)], being slower than the first [Eq. (8.82)], is rate-determining for initiation (unlike in the case of free-radical chain polymerization) and so though the amide ion produced upon chain transfer to ammonia can initiate polymerization it is but only at a rate controlled by the rate constant, ki, for initiation. Therefore, this chain transfer reaction may be considered as a true kinetic-chain termination step and the application of steady-state condition gives Eq. (8.90). 

The determination of the various rate constants (ki, kp, kt, kts, ktr) for cationic chain polymerization is much more difficult than in radical chain polymerization (or in anionic chain polymerization). It is convenient to use Rp data from experiments under steady-state conditions, since the concentration of propagating species is not required. The Rp data from non-steady-state conditions can be used, but only when the concentration of the propagating species is known. For example, the value of kp is obtained directly from Eq. (8.143) from a determination of the polymerization rate when [M J is known. The literature contains too many instances where [M" "] is taken equal to the concentration of the initiator, [IB], in order to determine kp from measured Rp. (For two-component initiator-coinitiator systems, [M" ] is taken to be the initiator concentration [IB] when the coinitiator is in excess or the coinitiator concentration [L] when the initiator is in excess.) Such an assumption holds only if Ri > Rp and the initiator is active, i.e., efficiency is 100%. Using this assumption without experimental verification may thus lead to erroneous results. 

Homopolymerization. Nonionic Monomers. The chain-growth polymerization of vinyl monomers occurs via a three-part mechanism initiation, propagation, and termination. Under steady-state conditions, the rate of polymerization (R ) is defined (21-24) by equation 1 ... 

At the start of the polymerization, the rate of formation of radicals greatly exceeds the rate at which they are lost by termination. However, [M"] increases rapidly and so the rate of loss of radicals by termination increases. A value of [M ] is quickly attained at which the latter rate exactly equals the rate of radical formation. The net rate of change in [M ] is then zero and the reaction is said to be under steady-state conditions. In practice, most free-radical polymerizations operate under steady-state conditions for all but the first few seconds. It is, therefore, quite satisfactory to assume steady-state conditions and to set d[M ]/d/ = 0 in Equation (1.3). This yields the following equation for the steady-state concentration of chain radicals ... 

In view of the comnents above it is not surprising that solution polymerization dominates most discussions of polymerization kinetics, e.g. in ref. 8 the extensive discussion includes four sentences on solid-state polymerization, which is dismissed as only of academic interest. The application of diacetylenes as time-tenqperature indicators seems likely to change this. The basic ideas of conventional discussions are, however, relevant even in the solid state. The polymerization is vie% as three distinct processes, (a) initiation, (b) propagation and (c) termination of the polymer chain. The details of these processes depend on the type of polymerization reaction involved. The polymerization kinetics are then deduced by solving the rate equations for the three processes. In order to do this it is usual to assume steady-state conditions, when the population of propagating polymer chains remains constant as a result of an exact balance between the rates of initiation and termination (8). 

The kinetic chain length i is the average number of monomers that react with an active center from its formation until it is terminated. It is given by the ratio of the rate of polymerization to the rate of initiation and under steady-state conditions where Rt = / , ... 

Because of the respective values of rate constants of termination (kt 10 to 10 L-mor -s ) and propagation (kp 10 to 10" L mor -s at 60°C) reactions, it is recommended to work with particularly low instantaneous concentrations in free radicals ([RM ] 10 M), in order to favor propagation over termination reactions. It is difficult to measure such low value of [RM ], except by using a spectrometric technique as sensitive as electron spin resonance (ESR). Assuming that the number of active chains remains constant—which is true only during short intervals of time—, one can calculate the rate of polymerization even if [RM ] is experimentally inaccessible and thus unknown. This assumption implies that the rate of appearance of RM is equal to their rate of disappearance, which corresponds to steady-state conditions one can accordingly write Ri = Rt, which corresponds to... 

Equilibrium studies under anaerobic conditions confirmed that [Cu(HA)]+ is the major species in the Cu(II)-ascorbic acid system. However, the existence of minor polymeric, presumably dimeric, species could also be proven. This lends support to the above kinetic model. Provided that the catalytically active complex is the dimer produced in reaction (26), the chain reaction is initiated by the formation and subsequent decomposition of [Cu2(HA)2(02)]2+ into [CuA(02H)] and A -. The chain carrier is the semi-quinone radical which is consumed and regenerated in the propagation steps, Eqs. (29) and (30). The chain is terminated in Eq. (31). Applying the steady-state approximation to the concentrations of the radicals, yields a rate law which is fully consistent with the experimental observations ... 

The steady-state assumption is not unique to polymerization kinetics. It is often used in developing the kinetics of many small-molecule reactions that involve highly reactive intermediates present at very low concentrations—conditions that are present in radical chain polymerizations. The theoretical validity of the steady-state assumption has been discussed [Kondratiev, 1969] and its experimental validity shown in many polymerizations. Typical polymerizations achieve a steady-state after a period, which may be at most a minute. 

According to Bodenstein, for a chain reaction in the steady state, the number of radicals formed and disappearing in a given time must be the same. This applies to most addition polymerizations, at least in the region of low conversion. Under these conditions v, and v, may be equated ... 

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. 

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