Rotating-sector method

The rotating sector method requires the introduction of the parameter defined 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  

The ratio kpjk, can be obtained from Eq. 3-157 from 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.l The individual rate constants kp and k, can be determined by combining kpjk, with kpjl obtained from Eq. 3-25. Thus, the objective of the rotating sector method is the measurement of Xj. 

the average polymerization rate increases by a factor of (1 -I- as the cycle (pulse) frequency increases from a low value to a very high value compared with I/t.  

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 ] j at the end of each hght period of duration t and a constant radical concentration [M ]2 at the end of each dark period of duration t = rt. The two radical concentrations are related by 

To obtain the steady-state polymerization rate Rp) is first 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) versus log t. Alternately, one can plot the data as the rate ratio Rp/ Rp)oo since this ratio is related to Rp/(Rp) 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 fit is obtained. The displacement on the abscissa of one plot from the other yields log since the abscissa for the theoretical plot is log t — log r.  

The maximum and minimum values of the ratios [M ]j/[M ]S and [M ]2/[M ]S can be calculated from Eqs. 3-160 and 3-161 for given values of r and t/xs. The average radical concentration [M-] over a cycle or several cycles is given by 

Kinetics at low yields gives only the ratio K( pp)/kl, not the individual rate constants. Thus, one of the two rate constants must be determined by another method. Such a method uses the average lifetime of a chain t, which is defined by 

If termination results through recombination, then (20-72) is transformed, with insertion of the expression for the free radical concentration in relation to the overall rate [(20-55)], into 

if T can be determined, then the ratio kp/k pp can be evaluated from the experimentally obtained values of [M] and and subsequently kp and k pp can be determined from kinetics using the ratio K pp /kp. 

The average lifetime of the chain can be determined from the interval up to the attainment of the steady state. Normally, this interval is too short for direct observation (see Section 20.2.3), but if radical production is limited locally or in time, then the slowing down of the rate of polymerization can be determined. In photochemically initiated polymerizations, for example, a sharply collimated beam of radiation can be passed continuously through the polymerization vessel (aperture method). For this method, however, the rate of diffusion of the polymer must be known. 

The rotating sector interrupts the steady state. Radicals are formed during the period of illumination in the period of darkness their concentration decreases due to termination reactions. The ratio of dark to light periods is r. If the illumination time is much longer than the average lifetime t, then, after an induction period, the full rate of polymerization will 

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The rate of initiation was measured directly and radical lifetimes were determined using the rotating sector method. The following results were obtained.f... 

Even though the rates of initiation span almost a 10-fold range, the values of k, show a standard deviation of only 4%, which is excellent in view of experimental errors. Note that the rotating sector method can be used in high-pressure experiments and other unusual situations, a characteristic it shares with many optical methods in chemistry. 

The lifetime of polystyrene radicals at 50 C was measuredt by the rotating sector method as a function of the extent of conversion to polymer. The following results were obtained ... 

From this we can see that knowledge of k f and Rf in a conventional polymerization process readily yields a value of the ratio kp fkt. In order to obtain a value for kf wc require further information on kv. Analysis of / , data obtained under non-steady state conditions (when there is no continuous source of initiator radicals) yields the ratio kvlkx. Various non-stcady state methods have been developed including the rotating sector method, spatially intermittent polymerization and pulsed laser polymerization (PLP). The classical approach for deriving the individual values of kp and kt by combining values for kp kx. with kp/k, obtained in separate experiments can, however, be problematical because the values of kx are strongly dependent on the polymerization conditions (Section... 

A superficially related dependence of on the medium has been observed by Norrish and Smith working with methyl methacrylate, and by Burnett and Melville with vinyl acetate. Rates in poor solvents are high, and determination of by the rotating sector method reveals what appears to be a decrease in kt in the poor solvents. This apparent decrease in kt accounts for the increased rate of polymerization. Actually, precipitation of the polymer seems to be responsible for the effect. The growing radicals become imbedded in precipitated droplets, presumably of very small size. The termination reaction is suppressed owing to isolation of the chain radical in one droplet from that in another. This gel effect is fairly common in systems yield-... 

Few kinetic studies of reactions of alkyl radicals with tin hydrides other than Bu3SnH have been reported. Studies of the reactions of the tert-butyl radical with Me3SnH and Ph3SnH were performed by the rotating sector method,80 but an error in absolute values exists in that method as judged by differences in rate constants for reactions of Bu3SnH with alkyl radicals... 

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,... 

The PLP-SEC method, like the rotating sector method, involves a non-steady-state photopolymerization [Beuermann, 2002 Beuermann and Buback, 2002 Komherr 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 significant chain transfer is present. The first laser pulse generates an almost instantaneous burst of primary radicals at high... 

One of the early specialized techniques used for the study of radical reactions was the rotating sector method. The use of this technique (Fig. 6) for determining reaction kinetics was demonstrated by Melville for the gas phase polymerization of methyl methacrylate, and later by Bartlett and Swain for the liquid phase reaction, and by Carlsson and Ingold for tin hydride reductions. ... 

Optical cells can also be used to investigate the kinetics of radical polymerization reactions under high pressure by means of the rotating sector method. Again, the apparatus is presented in Chapter 4.3.4. An example of the method for the evaluation of individual rate constants in radical polymerization of ethylene is given below. 

The rotating-sector method was applied to determine the individual rate constants of chain propagation and chain termination of the radical polymerization of ethylene [23,24]. The photo-initiator was diphenyldisulfide. First, the overall rate of polymerization was measured under steady illumination at pressures of 50 - 175 MPa and 132 - 199°C (Fig. 3.3-9). It increases first steeply and then less steeply with increasing pressure. At 175 MPa the rate of polymerization is ten times higher than at the low pressure of 50 MPa. 

Tedder and Walton (964) report a combination rate constant of 5.3 x 10 11 cm3 molcc 1 see -1 by a rotating-sector method... 

The rotating sector method has proved to be of immense value for certain types of work. It has been used to study chain reactions, in which chain termination steps depend on a different power of the radical concentration from the chain propagating steps. It has also been used to obtain the absolute rates of certain... 

The rotating sector method has also been used to investigate the second order decomposition of biacetyl at room temperature at 4358 A... 

The principle of the rotating sector method has been described many times and we will not discuss it in detail133. Briefly it is as follows (see also Chapter 1, Volume 1). 

Hammond and coworkers (147,349,374) have developed a rotating sector method to measure by ESR the bimolecular decay of radicals produced by photolysis. Calvin and coworkers (148) have also studied the kinetics of ESR signal decay by intermittent UV-irradiation of charge-transfer complexes. A CAT was used by both groups to... 

The only investigations specifically of the termination process appear to be those of McDowell and Sharpies [62]. By combining measurements of the rate of initiation in alcohol retarded systems with determinations of the chain length by the rotating sector method, a value ot defined by... 

The rate coefficients and the Arrhenius parameters of the propagation reaction (20) were determined also by the rotating sector method Since the assign-... 

Pritchard et as well as Dodd came to the conclusion that the rate coefficients obtained by the rotating sector method might be highly erratic due to the oversimplification of the mechanism assumed. Thus, the Arrhenius parameters determined by this method should be accepted only with reserve. 

Hassel, O., and Viervoll, H. Electron diffraction investigation of molecular structures. II. Results obtained by the rotating sector method. Acta Chem. Scand. 1, 149-168 (1947). 

The elementary rate constants were calculated from ratio kp/kt, obtained from the polymerization rate and initiation rate and the ratio kp/kt, estimated from the lifetime of the radical determined by the rotating sector method. The mean lifetime of the propagating radical and derived rate constants for methacrylates are shown in Tables 7—8. The variation of the propagation rate constant for methyl methacrylate with solvents is in accordance with the result obtained by Bamford et al.2 at 25 °C. Since the largest and the smallest kp value for phenyl methacrylate differ by a factor of 1.6 and for methyl methacrylate by a factor of 1.4, the estimation of the rate constants must be performed under experimental conditions in which the accumulated error is so small as to permit a distinction of the difference. Therefore, particular attention was given to the constancy of the reaction temperature ( 0.001 °C), constancy of light source, purity of monomers and solvents, and reproducibility of observed values and to the retention of the square wave in the rotat-... 

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