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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 1=2 2 constant kd (or ki for thermal initiation). The ratio kp =kt or kp =kt can be obtained from Eq. 3-25, since Rp , Ri , and [M] are measurable. Similarly, the chain-transfer constant ktr =kp and the inhibition constant kz =kp can be obtained by any one of several methods discussed. However, the evaluation of the individual kp , kt , 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 , ktr , and 1=2 kz from the ratios kp =kt , ktr =kp , and kz =kp . Measurements of kp were performed by the rotating sector method and its variations until the late 1980s [Flory, 1953; Odian, 1991; Walling, 1957]. In the late 1980s advances in pulsed laser technology and size exclusion chromatography resulted in the development of a technique called pulsed laser polymerization size exclusion chromatography (PLP-SEC) [Beuermann and Buback, 2002; Beuermann et al., 1997, 2000; Buback et al., 1986, 1992,



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The first experiment on the Casimir force was done by Sparnaay in 1957, nine years after Casimir's paper was published. Unfortunately, the experimental error was far too large for this to be a decisive confirmation of Casimir's prediction. Since then, the experimental accuracy has improved immensely. But there are still very few experiments in comparison with the number of theoretical papers on this subject. For a list of the experimental papers on the Casimir force, see the recommended reading at the end of the chapter. Last but not least, we promised to show you the physical reason why the regularization trick works. To do so, we must calculate first the mode spectral density for both free space and for the parallel-plate case. The mode spectral density is the function p(w) of the frequency w that gives the number of modes having a frequency between wand w + dJ.,;. So, in terms of the mode spectral density, the zero-point energy (without including the cutoff) is given by





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1995; Olaj et al., 1987, van Herk, 1997, 2000]. Both the rotating sector and PLP-SEC methods are considered below. Both methods involve polymerizations under non-steady-state conditions. Most of the rate constants available in the literature were obtained using the rotating sector method, although the method is no longer in general use. The rotating sector method uses polymerization rates to obtain kp , whereas the PLP-SEC method uses polymer molecular weights. 3-8b Rotating Sector Method

(2.164)

The rotating sector method requires the introduction of the parameter ts, de ned as the average lifetime of a growing radical under steady-state conditions. The radical lifetime is given by the steady-state radical concentration divided by its steady-state rate of disappearance:

(2.165)

ts M s 2kt M 2 s 1 2kt M s 3-156

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Changing the integration variables on the right-hand side of (2.165) from Cartesian to spherical, we find that

ts kp M 2kt Rp s 3-157

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The ratio kp =kt can be obtained from Eq 3-157 from ts and the rate of polymerization under steady-state conditions (The subscript s in Eq 3-157 and subsequent equations in this section refers to steady-state values; the absence of s denotes non-steady-state values) The indi1=2 vidual rate constants kp and kt can be determined by combining kp =kt with kp =kt obtained from Eq 3-25 Thus, the objective of the rotating sector method is the measurement of ts The most recent versions of the rotating sector method use a pulsed laser as the light source for a photopolymerization involving the alternating of light and dark periods The original rotating sector experiment used a nonlaser light source, and the equivalent of pulsing was achieved by rotating a sector or disk in between the polymerization system and the light source [Nagy et al, 1983].

Pulsing was achieved by cutting a portion out of the sector The rotating sector method requires a study of the polymerization rate as a function of the ratio r of the length t0 of the dark period to the length t of the light period The cycle time is the sum of t0 and t These parameters are varied experimentally by varying the laser pulse width and/or frequency Two situations occur depending on the cycle time in relation to ts If the cycle time is very long compared to ts (slow on off), the observed Rp will be equal to Rp s during the entire light period and zero during the entire dark period The reason for this is that the times for the radical concentration to reach steady-state or to decay to zero are small compared to times t and t 0 , respectively.

(2.166)

The ratio of the average polymerization rate Rp over one complete cycle to the steady-state rate is given by.

Substituting (2.166) in (2.165), we see that we can take PFS(W) as given by38 PFS(W)

Rp 1 Rp s 1 r 3-158

(2.167)

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