Rates of polymerization

In order to simplify a kinetic expression for the rate of polymerization, it is necessary to assume that both the k and fc, are independent of the size of the radical, as well as the equal-reactivity assumption of the growing radicals. Very small radicals are more reactive than the propagating radicals with longer chains, but this effect is not important due to the fact that the effect of size vanishes at the dimer or trimer size.[13] The rate of polymerization is expressed by the following equation  

This mathematical treatment indicates that in the early stages of polymerization the rate of reaction is expected to be proportional to the square root of the initiator concentration, assuming/is independent of monomer concentration. This assumption is acceptable under the condition of the high initiator efficiencies, but with very low efficiencies, / may become proportional to [M], leading to the rate proportional to The finding that the rate of chain 

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 in accordance with Eq. 3-32. The temperature 

Er and can then be obtained from the slope and intercept, respectively, of a plot 

the activation energy for thermal initiator decomposition, is in the range 120-150 kJ mol for most of the commonly used initiators (Table 3-13). The Ep and E, values for most monomers are in the ranges 20-40 and 8-20 kJ mol , respectively (Tables 3-11 and 3-12). The overall activation energy Er for most polymerizations initiated by thermal initiator decomposition is about 80-90 kJ mol . This corresponds to a two- or threefold rate increase for a 10°C temperature increase. The situation is different for other modes of initiation. Thus redox initiation (e.g., Fe with thiosulfate or cumene hydroperoxide) has been discussed as taking place at lower temperatures compared to the thermal polymerizations. One indication of the difference between the two different initiation modes is the differences in activation energies. Redox initiation will have an Ed value of only about 40-60 kJ mol which is about 80 kJ mol less than for the thermal initiator decomposition [Barb et al., 1951]. This leads to an Er for redox polymerization of about 40 kJ mol , which is about one half the value for nonredox initiators. 

For a purely photochemical polymerization, the initiation step is temperature-independent Ed — 0) since the energy for initiator decomposition is supplied by light quanta. The overall activation for photochemical polymerization is then only about 20 kJ mol . This low value of Er indicates the Rp for photochemical pol)merizations will be relatively insensitive to temperature compared to other polymerizations. The effect of temperature on photochemical polymerizations is complicated, however, since most photochemical initiators can also decompose thermally. At higher temperatures the initiators may undergo appreciable thermal decomposition in addition to the photochemical decomposition. In such cases, one must take into account both the thermal and photochemical initiations. The initiation and overall activation energies for a purely thermal self-initiated polymerization are approximately the same as for initiation by the thermal decomposition of an initiator. For the thermal, self-initiated polymerization of styrene the activation energy for initiation is 121 kJ mol and Er is 86 kJ mol [Barr et al., 1978 Hui and Hamielec, 1972]. However, purely thermal polymerizations proceed at very slow rates because of the low probability of the initiation process due to the very low values (10 -10 ) of the frequency factor. 

To determine the effect of temperature on the molecular weight of the polymer produced in a thermally catalyzed polymerization where transfer reactions are neghgihle, one must consider the ratio kp/ kdk,), since it determines the degree of polymerization (Eq. 3-98). The variation of this ratio with temperature is given by 

Values of kt (whether ktc or ktd) are usually in the range of 10 -10 L/mol-s. Though these values are orders of magnitude greater than kp, polymerization to high molecular weight still occurs because the concentration of radical species is very small (due to low values of kd) and because, as shown later, the polymerization rate is proportional to kp and inversely proportional to 

A radical chain polymerization is started when the initiator begins to decompose according to Eq. (6.3) and the concentration of radicals in the system, [M ], increases from zero. The rate of termination or disappearance of radicals, being proportional to [M ]- [cf. Eqs. (6.17)-(6.19)], is thus zero in the beginning and increases with time, till at some stage it equals the rate of radical generation. The concentration of radicals in the system then becomes essentially constant (or steady ), as radicals are formed and destroyed at equal rates. This condition, described as steady-state assumption or steady-state approximation , can thus be described by the following two equations  

The steady-state approximation is a very useful assumption since, as shown below, it allows one to eliminate the inconvenient radical concentration term [M ] and find an expression for it in terms of known or measurable parameters. The validity of steady-state approximation has been shown experimentally in many polymerizations (see Problem below). 

Problem 6.1 Experimentally, it is observed that, except in the very earliest stages where the extent of reaction is insignificant, the loss of monomer is accounted for, quantitatively, by the formation of the polymer (Allcock and Lampe, 1990). Justify on this basis the steady-state approximation that all free radicals present in a polymerizing system are at steady-state concentrations. 

Let [M]o be the initial concentration of the monomer. Since all the monomer molecules that have reacted must be. contained either in the propagating radicals or in the inactive polymer product, the stoichiometry requires that (AUcock and Lampe, 1990)  

Equations (6.12) and (6.17)-(6.19) are based on the assumption that both the propagation and termination rate constants are independent of the size of the radical. This assumption faciUtates the derivation of a kinetic expression, as shown below, for the overall rate of polymerization, which can be tested experimentally. There is also a considerable experimental evidence to justify the above assumption (Allcock and Lampe, 1990). 

---

Similarly, the addition of low quantities of vinyl or polyvinylthiazoles in the synthesis of aromatic polyesters increases the rate of polymerization (315). 

According to the mechanism provided by reactions (5.F) and the analysis given by Eq. (5.8), the rate of polymerization is dependent upon the following ... 

The rate of polymerization is thus first order in t NHj order in... 

Figure 6.3 shows some data which constitute a test of Eq. (6.26). In Fig. 6.3a, Rp and [M] are plotted on a log-log scale for a constant level of redox initiator. The slope of this line, which indicates the order of the polymerization with respect to monomer, is unity, showing that the polymerization of methyl methacrylate is first order in monomer. Figure 6.3b is a similar plot of the initial rate of polymerization—which essentially maintains the monomer at constant con-centration—versus initiator concentration for several different monomer-initiator combinations. Each of the lines has a slope of indicating a half-order dependence on [I] as predicted by Eq. (6.26).

The mechanistic analysis of the rate of polymerization and the fact that the separate constants individually follow the Arrhenius equation means that... 

Equation (6.32) allows us to conveniently assess the effect of temperature variation on the rate of polymerization. This effect is considered in the following example. 

Using typical activation energies out of Tables 6.2-6.4, estimate the percent change in the rate of polymerization with a 1°C change in temperature at 50°C for thermally initiated and photinitiated polymerization. 

Finally we recognize that a 1°C temperature variation can be approximated as dT and that (dRp/Rp) X 100 gives the approximate percent change in the rate of polymerization. Taking average values of E from the appropriate tables, we obtain E j = 145, E = 16.8, and Ep = 24.9 kJ mol . For thermally initiated polymerization... 

Note that the initiator decomposition makes the largest contribution to E therefore photoinitiated processes display a considerably lower temperature dependence for the rate of polymerization. 

As with the rate of polymerization, we see from Eq. (6.37) that the kinetic chain length depends on the monomer and initiator concentrations and on the constants for the three different kinds of kinetic processes that constitute the mechanism. When the initial monomer and initiator concentrations are used, Eq. (6.37) describes the initial polymer formed. The initial degree of polymerization is a measurable quantity, so Eq. (6.37) provides a second functional relationship, different from Eq. (6.26), between experimentally available quantities-n, [M], and [1]-and theoretically important parameters—kp, k, and k. Note that the mode of termination which establishes the connection between u and hj, and the value of f are both accessible through end group characterization. Thus we have a second equation with three unknowns one more and the evaluation of the individual kinetic constants from experimental results will be feasible. 

In the preceding section we observed that both the rate of polymerization and the degree of polymerization under stationary-state conditions can be interpreted to yield some cluster of the constants kp, kj, and k j. The situation is summarized diagramatically in Fig. 6.4. The circles at the two bottom corners... 

If the light source is switched on and off and held for long periods of equal duration in either light or darkness, then the radical concentration in the system will consist of an alternation between the situation described in Figs. 6.5a and b. Because we have specified that the duration of each phase is long, the net behavior is essentially a series of plateaus in which the illumination is either Iq or zero and the radical concentration is either [M], or zero, with brief transitions in between. This is illustrated in Fig. 6.5c. The concentration of radicals is consistent with Iq, but is present only half of the time hence the rate of polymerization is only half what it would be for the same illumination operating continuously. 

Thus if we were to compare the rate of polymerization with intermittent illumination relative to that with continuous illumination, but under otherwise identical conditions, we would observe the following limits for equal periods of light and dark ... 

The rate of polymerization under conditions where the period of illumination is comparable to f is obtained by integrating Eq. (6.24) in the following form ... 

The three-step mechanism for free-radical polymerization represented by reactions (6.A)-(6.C) does not tell the whole story. Another type of free-radical reaction, called chain transfer, may also occur. This is unfortunate in the sense that it complicates the neat picture presented until now. On the other hand, this additional reaction can be turned into an asset in actual polymer practice. One of the consequences of chain transfer reactions is a lowering of the kinetic chain length and hence the molecular weight of the polymer without necessarily affecting the rate of polymerization. 

It is apparent from these reactions how chain transfer lowers the molecular weight of a chain-growth polymer. The effect of chain transfer on the rate of polymerization depends on the rate at which the new radicals reinitiate polymerization ... 

If the rate constant kj is comparable to kp, the substitution of a polymer radical with a new radical has little or no effect on the rate of polymerization. If kj hp, the rate of polymerization will be decreased by chain transfer. 

An interesting situation is obtained when the catalyst-solvent system is such that the initiator is essentially 100% dissociated before monomer is added and no termination or transfer reactions occur. In this case all chain initiation occurs rapidly when monomer is added, since no time-dependent initiator breakdown is required. If the initial concentration of catalyst is [AB]o,then chain growth starts simultaneously at [B"]q centers per unit volume. The rate of polymerization is given by the analog of Eq. (6.24) ... 

We shall consider these points below. The mechanism for cationic polymerization continues to include initiation, propagation, transfer, and termination steps, and the rate of polymerization and the kinetic chain length are the principal quantities of interest. 

Chain transfer reactions to monomer and/or solvent also occur and lower the kinetic chain length without affecting the rate of polymerization ... 

Applying the Arrhenius equation to Eq. (6.116) shows that the apparent activation energy for the overall rate of polymerization is given by... 

In the same research described in Example 6.4, the authors measured the following rates of polymerization ... 

In a series of experiments at 60 C, the rate of polymerization of styrene agitated in water containing persulfate initiator was measuredt for different concentrations of sodium dodecyl sulfate emulsifier. The following results were obtained ... 

C with AIBN and measured the initial rates of polymerization for the ... 

Rate of polymerization. The rate of polymerization for homogeneous systems closely resembles anionic polymerization. For heterogeneous systems the concentration of alkylated transition metal sites on the surface appears in the rate law. The latter depends on the particle size of the solid catalyst and may be complicated by sites of various degrees of activity. There is sometimes an inverse relationship between the degree of stereoregularity produced by a catalyst and the rate at which polymerization occurs. 

Chain transfer is an important consideration in solution polymerizations. Chain transfer to solvent may reduce the rate of polymerization as well as the molecular weight of the polymer. Other chain-transfer reactions may iatroduce dye sites, branching, chromophoric groups, and stmctural defects which reduce thermal stabiUty. Many of the solvents used for acrylonitrile polymerization are very active in chain transfer. DMAC and DME have chain-transfer constants of 4.95-5.1 x lO " and 2.7-2.8 x lO " respectively, very high when compared to a value of only 0.05 x lO " for acrylonitrile itself DMSO (0.1-0.8 X lO " ) and aqueous zinc chloride (0.006 x lO " ), in contrast, have relatively low transfer constants hence, the relative desirabiUty of these two solvents over the former. DME, however, is used by several acryhc fiber producers as a solvent for solution polymerization. 

When initiator is first added the reaction medium remains clear while particles 10 to 20 nm in diameter are formed. As the reaction proceeds the particle size increases, giving the reaction medium a white milky appearance. When a thermal initiator, such as AIBN or benzoyl peroxide, is used the reaction is autocatalytic. This contrasts sharply with normal homogeneous polymerizations in which the rate of polymerization decreases monotonicaHy with time. Studies show that three propagation reactions occur simultaneously to account for the anomalous auto acceleration (17). These are chain growth in the continuous monomer phase chain growth of radicals that have precipitated from solution onto the particle surface and chain growth of radicals within the polymer particles (13,18). 

The monomer recovery process may vary ia commercial practice. A less desirable sequence is to filter or centrifuge the slurry to recover the polymer and then pass the filtrate through a conventional distillation tower to recover the unreacted monomer. The need for monomer recovery may be minimized by usiag two-stage filtration with filtrate recycle after the first stage. Nonvolatile monomers, such as sodium styrene sulfonate, can be partially recovered ia this manner. This often makes process control more difficult because some reaction by-products can affect the rate of polymerization and often the composition may vary. When recycle is used it is often done to control discharges iato the environment rather than to reduce monomer losses. 

Eor some uses, higher molecular weight polymer consisting of 150—200 repeat units is required. Such polymer usually is prepared by soHd-state polymerization in which pellets are heated under an inert atmosphere to 200—240°C. The 2G is removed continuously. The rate of polymerization depends on particle size, end group composition, and crystallinity (65). 

Aromatic radical anions, such as lithium naphthalene or sodium naphthalene, are efficient difunctional initiators (eqs. 6,7) (3,20,64). However, the necessity of using polar solvents for their formation and use limits their utility for diene polymerization, since the unique abiUty of lithium to provide high 1,4-polydiene microstmcture is lost in polar media (1,33,34,57,63,64). Consequentiy, a significant research challenge has been to discover a hydrocarbon-soluble dilithium initiator which would initiate the polymerization of styrene and diene monomers to form monomodal a, CO-dianionic polymers at rates which are faster or comparable to the rates of polymerization, ie, to form narrow molecular weight distribution polymers (61,65,66). 

Itaconic acid, anhydride, and mono- and diesters undergo vinyl polymerization. Rates of polymerization and intrinsic viscosities of the resulting homopolymers ate lower than those of the related acrylates (see Acrylic ester polymers) (8,9). 

Stage II Growth in Polymer Particles Saturated With Monomer. Stage II begins once most of the micelles have been converted into polymer particles. At constant particle number the rate of polymerization, as given by Smith-Ewart kinetics is as follows (27) where is the... 

During Stages II and III the average concentration of radicals within the particle determines the rate of polymerization. To solve for n, the fate of a given radical was balanced across the possible adsorption, desorption, and termination events. Initially a solution was provided for three physically limiting cases. Subsequentiy, n was solved for expHcitiy without limitation using a generating function to solve the Smith-Ewart recursion formula (29). This analysis for the case of very slow rates of radical desorption was improved on (30), and later radical readsorption was accounted for and the Smith-Ewart recursion formula solved via the method of continuous fractions (31). 

The units oi sp are /mofh U Initial rate of polymerization is calculated from and the concentration of AIBN using the following equation ... 

---