The Rate of Polymerization

Making this type of assumption may seem a bit dubious to you, but it s often the type of thing you have to do in science. In the end, the validity of any assumption you make must stand the test of experimental confirmation. So let s see how well this and also Flory s assumption, that reactivity is independent of chain length, works. The equation for the rate of polymerization is obtained by substituting the expression for the concentration of radical species into the equation for the rate of polymerization (Equations 4-29). 

The final term we want to consider contains all the rate constants. In particular, you 

FIGURE 4-17 Graph of Rp versus [M][/],/z (plotted from the data listed in P. J. Rory s book, Principles of Polymer Chemistry, for the polymerizations involving methyl methacrylate and styrene). 

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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... 

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 ... 

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 ... 

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). 

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). 

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