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time maintains the radical concentration at approximately a constant value and is equivalent to polymerization under constant illumination at an intensity 1= 1 r times that actually used [Flory, 1953; Walling, 1957]. The ratio of the average polymerization rate Rp 1 at very short cycle time to the steady-state rate is given by
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Rp 1 1 Rp s 1 r 1=2 3-159
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Thus, the average polymerization rate increases by a factor of 1 r 1=2 as the cycle (pulse) frequency increases from a low value to a very high value compared with 1=ts . The mathematical treatment of pulsing illumination has been described [Briers et al., 1926]. After a number of cycles, the radical concentration oscillates uniformly with a constant radical concentration M 1 at the end of each light period of duration t and a constant radical concentration M 2 at the end of each dark period of duration t0 rt. The two radical concentrations are related by
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tanh 1     M 1 M 2 t tanh 1 ts M s M s 3-160
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M s M s rt M 2 M 1 ts 3-161
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The maximum and minimum values of the ratios M 1 = M s and M 2 = M s can be calculated from Eqs. 3-160 and 3-161 for given values of r and t=ts . The average radical concentration M over a cycle or several cycles is given by
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where the rst integral (for the light period) is given by Eq. 3-160 and the second integral (for the dark period) is given by Eq. 3-161. Evaluation of the integrals in Eq. 3-162 yields
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 ! M ts M 1 = M 2 M 1 = M s r 1 1 1 ln M s t 1 M 1 = M s 3-163
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Using Eq. 3-163 with Eqs. 3-160 and 3-161, one can calculate the ratio M = M s as a function of t=ts for a xed value of r. A semilog plot of such data for r 3 is shown in Fig. 3-11. To obtain ts the steady-state polymerization rate Rp s is rst measured at constant illumination (no pulsing). Then the average rate Rp is measured as a function of r and t. The data are plotted as the rate ratio Rp = Rp s versus log t. Alternately, one can plot the data as the rate ratio Rp = Rp 1 since this ratio is related to Rp = Rp s through Eq. 3-159. The theoretical plot (e.g., Fig. 3-11) for the same r value is placed on top of the experimental plot and shifted on the abscissa (x axis) until a best t is obtained. The displacement on the abscissa of one plot from the other yields log ts since the abscissa for the theoretical plot is log t log ts . The experimental determination of ts allows the calculation of kp , kt , ktr , and kz . kp =kt and 1=2 kp =kt , obtained from Eqs. 3-157 and 3-25 (non-steady-state and steady-state experiments, respectively), are combined to yield the individual rate constants kp and kt . Quantities such as
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Fig. 3-11 Semilog plot of 2[M]/[M]s versus t=ts . After Matheson et al. [1949] (by permission of American Chemical Society, Washington, DC).
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Rt and M s are calculated from Eqs. 3-22 and 3-23, respectively. ktr and kz are calculated from the values of chain transfer and inhibition constants. The measurement of ts also allows an evaluation of the validity of the usual steady-state assumption in radical chain polymerization from the relationship [Flory, 1953]
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M Rp tanh t=ts M s Rp s 3-164
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The time required for M and Rp to reach their steady-state values is calculated as 65, 6.5, and 0.65 s, respectively, for ts values of 10, 1, and 0.1 s. Thus, in the usual polymerization the steady-state assumption is valid after a minute or so at most. 3-8c PLP-SEC Method
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An examination of the literature, including the Polymer Handbook [Brandrup et al., 1999], often shows fairly wide disparities in the values of various rate constants. Much of this is due to experimental dif culties with the rotating sector method as well as the fact that different workers studied polymerizations under widely different reaction conditions. There has been a concerted effort in recent years to bring some order to the situation. A group of polymer scientists organized under the auspices of the IUPAC Working Party on Modeling of Kinetics and Processes of Polymerization has made good progress in this direction by standardization of the reaction conditions for measuring rate constants and the use of the PLPSEC method. Unlike the rotating sector method, the PLP-SEC method allows the measurement of kp directly without the need to couple it to the termination rate constant. The PLP-SEC method, like the rotating sector method, involves a non-steady-state photopolymerization [Beuermann, 2002; Beuermann and Buback, 2002; Kornherr et al., 2003; Nikitin et al., 2002]. Under pulsed laser irradiation, primary radicals are formed in very short times ($10 ns pulse width) compared to the cycle time ($1 s). The laser pulse width is also very short compared to both the lifetimes of propagating radicals and the times for conversion of primary radicals to propagating radicals. The PLP-SEC method for measuring kp requires that reaction conditions be chosen so that no signi cant chain transfer is present. The rst laser pulse generates an almost instantaneous burst of primary radicals at high