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ESR spectroscopy would clearly be the most direct method for determining kp . Measurement of the propagating radical concentration together with the steady-state polymerization rate and monomer concentration allows the calculation of kp from Eq. 3-22. However, the propagating radical concentrations are too low in most polymerizations to easily and accurately determine by ESR. Some success has been achieved by ESR, but by using reaction conditions that are usually atypical using initiation rates 1 2 orders higher than normal [Kamachi, 2002; Yamada et al., 1999]. For example, kp values of 150, 256, 510, and 4500 L mol 1 s 1 have been obtained for 1,3-butadiene (5 C), styrene (50 C), methyl methacrylate (60 C), and vinyl acetate (5 C), in reasonable agreement with those obtained by the other methods. The general use of ESR for determining kp will probably increase in the future, especially if further advances in ESR instrumentation and techniques allow measurements at the radical concentrations present in typical polymerizations.



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(3.35)

The effect of temperature on the rate and degree of polymerization is of prime importance in determining the manner of performing a polymerization. Increasing the reaction temperature usually increases the polymerization rate and decreases the polymer molecular weight. Figure 3-13 shows this effect for the thermal, self-initiated polymerization of styrene. However, the quantitative effect of temperature is complex since Rp and X n depend on a combination of three rate constants kd , kp , and kt . Each of the rate constants for initiation, propagation, and termination can be expressed by an Arrhenius-type relationship

Now then, to tum OF into 0' fI, we must multiply it on the left by U, on the right by T, and F' must be obtained from F by F' = U* FT, where T is still to be determined: UOFT





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where A is the collision frequency factor, E the Arrhenius activation energy, and T the Kelvin temperature. A plot of ln k versus 1/T allows the determination of both E and A from the slope and intercept, respectively. Values of Ep , the activation energy for propagation, and Et , the activation energy for termination, for several monomers are shown in Tables 3-11 and 3-12.

Fig. 3-13 Dependence of the polymerization rate (*) and polymer molecular weight (&) on the temperature for the thermal self-initiated polymerization of styrene. After Roche and Price [1952] (by permission of Dow Chemical Co., Midland, MI).

=O'F'.

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Note that the variations in the values of Ap , the frequency factor for propagation, are much greater than those in Ep indicating that steric effects are probably the more important factor in determining the absolute value of kp . Thus the more hindered monomers, such as methyl methacrylate and dimethyl itaconate (dimethyl 2-methylenebutanedioic acid), have lower kp and Ap values than the less hindered monomers. The Ap values in general are much lower than the usual value (1011 1013 ) of the frequency factor for a bimolecular reaction probably due to a large decrease in entropy on polymerization. The variations in At , the frequency factor for termination, generally follow along the same lines as the Ap values. The At values in general are larger than the Ap values. 3-9a-1 Rate of Polymerization

---1

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 (kd =kt 1=2 in accordance with Eq. 3-32. The temperature dependence of this ratio, obtained by combining three separate Arrhenius-type equations, is given by

(3.36)

" # " # kd 1=2 Ad 1=2 Ep Ed =2 Et =2 ln kp ln Ap RT kt At 3-168

ISIn Appendix C, we only talk about the quatemion gradient (i.e., a gradient when the coefficients of the quaternion 'R = Xo + Ln inxn describing a point in four dimensional space are all real). For Minkowsky's space-time. we use biquaternions (last section of Appendix C) (i.e., use complex coefficients, with Xo = ct and Xl = ix, X2 = iy, X3 = iz in the previous expression for'R). Then it is convenient to define the biquaternion gradient as i multiplied by the gradient defined in Appendix C. 19Sometimes the D' Alembertian is written as 0 2 , perhaps because it is composed of square derivatives (i.e., second derivatives). But we feel that the superscript 2 is an unnecessary emphasis, for a square is already a square. 20See Appendix C or [506] or 9 of Lanczos's book on mechanics [386].

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