Stationary state of radicals

Io denotes the concentration of the initiator quantitatively converted into living polymers). Since u and v are two different variables, not related by the conventional equation describing the stationary state of radical co-polymerization, i.e. k2)v[S] k ufpMeS], the above three equations have to be identical, i.e. they have to represent only one relation of u to v. This is possible only if... 

This important equation shows that the stationary-state free-radical concentration increases with and varies directly with and inversely with. The concentration of free radicals determines the rate at which polymer forms and the eventual molecular weight of the polymer, since each radical is a growth site. We shall examine these aspects of Eq. (6.23) in the next section. We conclude this section with a numerical example which concerns the stationary-state radical concentration for a typical system. 

The propagation of polymer chains is easy to consider under stationary-state conditions. As the preceding example illustrates, the stationary state is reached very rapidly, so we lose only a brief period at the start of the reaction by restricting ourselves to the stationary state. Of course, the stationary-state approximation breaks down at the end of the reaction also, when the radical concentration drops toward zero. We shall restrict our attention to relatively low conversion to polymer, however, to avoid the complications of the Tromms-dorff effect. Therefore deviations from the stationary state at long times need not concern us. 

The superpositioning of experimental and theoretical curves to evaluate a characteristic time is reminiscent of the time-tefnperature superpositioning described in Sec. 4.10. This parallel is even more apparent if the theoretical curve is drawn on a logarithmic scale, in which case the distance by which the curve has to be shifted measures log r. Note that the limiting values of the ordinate in Fig. 6.6 correspond to the limits described in Eqs. (6.46) and (6.47). Because this method effectively averages over both the buildup and the decay phases of radical concentration, it affords an experimentally less demanding method for the determination of r than alternative methods which utilize either the buildup or the decay portions of the non-stationary-state free-radical concentration. 

In this example the number of micelles per unit volume is exactly twice the stationary-state free-radical concentration hence the rates are identical. Although the numbers were chosen in this example to produce this result, neither N nor M are unreasonable values in actual emulsion polymerizations. 

Even though the catalyst may be only partially converted to H B", the concentration of these ions may be on the order of 10 times greater than the concentration of free radicals in the corresponding stationary state of the radical mechanism. Likewise, kp for ionic polymerization is on the order of 100 times larger than the sum of the constants for all termination and transfer steps. By contrast, kp/kj which is pertinent for the radical mechanism, is typically on the order of 10. These comparisons illustrate that ionic polymerizations occur very fast even at low temperatures. 

Assuming the stationary state for radicals and long chains, derive an expression for rate of disappearance of X2 in halogenation, Scheme 10, p. 498. 

A steady-state (also named stationary-state) approximation on the concentrations of CH3 radicals and CuCH2+, namely, 9[CH3]/9f = 3[CuCH2 - ]ldt = 0, gives a stationary concentration of radicals ... 

A block diagram of the lamp circuit is shown in Fig. 38. Each pulse produced 10" -10" % decomposition. They were fired at the rate of 30-40 pulses per second, sufficient to produce a quasi-stationary state. The radical concentration following each pulse falls to the same value [J ]d prior to the pulse (Fig. 39). A consideration of the rate of formation of CjHg and CH leads to an expression from which and ki5 may be determined by varying the length of the dark period t. Paired pulses with a varying t should lead to direct evidence for the participation of hot radical reactions. 

The expressions for the evolution of the degree of polymerization and the polydispersity index for polymerizations in the stationary state of constant radical concentrations are50... 

Figure 14 shows the concentrations of all species in a log—log-representation as functions of time for the parameters given above. As qualitatively expected, the decay of Y leads to an intermediate stationary state of the radical concentrations. This state is entered at the time to = l/6ky, and for ky = 3 x 10 3 s this is at to = 56 s. It breaks down at the much larger time t 1.2 x 107 s = 3300 h. This may seem surprising, because the natural lifetime of Y is only 1 /ky = 330 s. The explanation is that the unstable persistent species is present in this form only for a small fraction of time and stays essentially incorporated in the dormant chains where it does not decay. This time fraction is approximately equal to [Y]/[I]o = 0.1% for the region where [Y] is constant (Figure 14). 

Apart from the qualitative evidence mentioned above which is in favor of a ferric ferrous change, it has been shown recently (8) that an expression which is, under certain conditions identical with Andersen s empirical Eq. (VIII) can actually be obtained from the free radical mechanism discussed above by taking into account the six Eqs. (1), (2), (4), (5), (6), and (7). Treating these equations for the stationary state of OH, HO2, and of the ferrous/ferric system one obtains ... 

Another general problem, the development of an algorithm for the construction of kinetic models for the quasi-stationary state of the evolution of non-equilibrium chemical system, is solved by the method of linear routes as simple cycles of a graph assigned to sets of elementary reactions and intermediate substances (see Chapter 2). A general algorithm for construction of kinetic models for the linear catalytic and un-branched radical-chain processes, including a free radical polymerization, is proposed. 

Plot of stationary states of the 2,4,5-triphenylimidazyl radical (AU554) as a function of residence time + points for conditions of figure 1 o points for conditions of figure 2... 

We present here the first experimental demonstration of photochemical bistability in an open reactor. This bistable reaction results from the non-linear properties of a photochromic system the dimer of the triphenylimidazyl radical in chloroform. Hysteresis is observed on the plots of the stationary states of the system over a wide range of flow rates. Within this region, the system is bistable and can be made to flip from one state to the other by an external manipulation. One of the stable states is characterized by a high concentration of violet radicals 2 while in the other the violet radicals are replaced by highly fluorescent compounds. Mechanistic studies showed that this bistability was due to a positive feedback loop. This was thought to arise from the screening effect of the violet radicals 2 with respect to the irradiation of the triphenyl imidazole 3 in combination with an inhibition of the violet radicals 2 by the products of photolysis of triphenylimidazole 3. 

The equations are especially valid for thermo-oxidative degradation, as in photo-oxidation the role of hydroperoxides is less important, their curve of formation showing a maximum an improvement of the theory for other situations is also required. For the kinetics of hydroperoxide decomposition in a general case no stationary state for radical concentration is postulated. The induction period tends to a finite value when the initial ROOH concentration tends towards zero. The rate of hydroperoxide decomposition in the PE/PP blends is an increased function of the PP content. 

Polymer propagation steps do not change the total radical concentration, so we recognize that the two opposing processes, initiation and termination, will eventually reach a point of balance. This condition is called the stationary state and is characterized by a constant concentration of free radicals. Under stationary-state conditions (subscript s) the rate of initiation equals the rate of termination. Using Eq. (6.2) for the rate of initiation (that is, two radicals produced per initiator molecule) and Eq. (6.14) for termination, we write... 

Instead of using 2fk j [I] for the rate of initiation, we can simply write tliis latter quantity as Rj, in which case the stationary-state radical concentration is... 

When results are compared for polymerization experiments carried out at different frequencies of blinking, it is found that the rate depends on that frequency. To see how this comes about, we must examine the variation of radical concentration under non-stationary-state conditions. This consideration dictates the choice of photoinitiated polymerization, since in the latter it is almost possible to turn on or off—with the blink of a light—the source of free radicals. The qualifying almost in the previous sentence is actually the focus of our attention, since a short but finite amount of time is required for the radical concentration to reach [M-] and a short but finite amount of time is required for it to drop back to zero after the light goes out. 

Suppose the radical concentration begins at zero when the light is first turned on at t = 0 (unprimed t represents time in the light primed t, time in the dark). The radical concentration then increases toward the stationary-state value during the time of illumination. We have already encountered in Example 6.2 the expression which describes the approach of [M-] to The equation is... 

We saw in the last chapter that the stationary-state approximation is apphc-able to free-radical homopolymerizations, and the same is true of copolymerizations. Of course, it takes a brief time for the stationary-state radical concentration to be reached, but this period is insignificant compared to the total duration of a polymerization reaction. If the total concentration of radicals is constant, this means that the rate of crossover between the different types of terminal units is also equal, or that R... 

Acrylamide polymerization by radiation proceeds via free radical addition mechanism [37,38,40,45,50]. This involves three major processes, namely, initiation, propagation, and termination. Apart from the many subprocesses involved in each step at the stationary state the rates of formation and destruction of radicals are equal. The overall rate of polymerization (/ p) is so expressed by Chapiro [51] as ... 

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