Radical stationary

Tip 9 Radical stationary state hypothesis. A few practical steps to check for the validity of the (quasi-) stationary (steady-) state hypothesis (QSSH or simply SSH) for radicals are as follows (i) determine the rate of change (with time) of the total radical concentration (ii) find the maximum rate in (i) (c) divide (ii) by the rate of initiation and (iv) if the ratio is much less than unity, the QSSH (or QSSA, A here stands for assumption) is valid. 

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... 

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. 

For an initiator concentration which is constant at [l]o, the non-stationary-state radical concentration varies with time according to the following expression ... 

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 radical concentration has the stationary-stage value given by Eq. (6.23). 

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... 

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. 

The total radical concentration under stationary-state conditions can be similarly obtained ... 

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. 

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 ... 

Purified ligninase H8 produced by P. chrysosporium in stationary cultures oxidized pyrene to pyrene-1,6- and pyrene-l,8-quinones in high yield, and experiments with showed that both quinone oxygen atoms originated in water (Figure 8.25). It was suggested that initial one-electron abstraction produced cation radicals at the 1 and 6 or 8-positions (Hammel et al. 1986), whereas in... 

Each radical transfers an electron to the colloidal particles. One colloidal particle can accept and store a large number of electrons until the aqueous solvent is reduced. Under stationary conditions, a certain number x of electrons reside on a particle, thus producing a negative potential sufficiently high for Hj evolution (n = agglomeration number) ... 

The above result was used as a ground-stone of the well known kinetic method of detection which was initially proposed by Myasnikov [75] more than 30 years ago. Above paper dealt with experimental comparison of the change of relative concentration of CH3 radicals in gaseous phase using the stationary values of electric conductivity and initial rate of its change. The experiment yielded perfect coincidence of the measured values. Using methyl radicals as example of adsorption it was established that the resolution of this method was better than 10 particles per cubic centimeter of the ambient volume [75, 76]. 

Simple analysis of the motion of mapping point through the phase trajectory (Fig. 2.13) indicates that X represents the stationary concentration of chemisorbed radicals corresponding to a given external conditions. 

In two limiting cases differing in the values of stationary concentration of chemisorbed radicals and initial electric conductivity of adsorbent the expression (2.94) acquires the following shape ... 

In case when K " > K" and Ny < Ny < Ny2, i.e. in case of average volume concentrations of radicals and recombination mechanism of their heterogeneous annihilation we arrive at the following expression characterizing the stationary electric conductivity of adsorbent... 

We should note that we used Ns as Nsaax (which is the concentration of chemisorbed radicals at the moment of activation of the source of radicals) in expressions (2.105) and (2.106). This means that we have assumed that the process of chemisorption is already stationary by this moment of time. 

Thus, the rigorous solution of kinetic equation describing the change in electric conductivity of a semiconductor during adsorption of radicals enables one to deduce that information on concentration of radicals in ambient volume can be obtained measuring both the stationary values of electric conductivity attained over a certain period of time after activation of the radical source and from the measurements of initial rates in change of electric conductivity during desactivation or activation of the radical flux incident on the surface of adsorbent, i.e. 

---