Stationary state, radical

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 ... 

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... 

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... 

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... 

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. 

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. 

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 ... 

Let us compare the ratio of radicals in oxidized 2-propanol and cyclohexanol at different temperatures when oxidation occurs with long chains and chain initiation and termination do not influence the stationary state concentration of radicals. The values of the rate constants of the reactions of peroxyl radicals (kp) with alcohol and decomposition of the alkylhydroxy-peroxyl radical (k ) are taken from Table 7.4 and Table 7.5. 

A different situation in the oxidation of these two alcohols is seen. The hydroperoxyl radical is the main chain propagating species in oxidized 2-propanol the portion of alkylhydroxy-peroxyl radicals in this reaction is less than 2.5%. In oxidized cyclohexanol, on the contrary, the stationary state concentrations of both radicals are close and both of them take important part in chain propagation. 

Apply the stationary-state hypothesis to the free radicals CH and C2H50 to derive the rate law for this mechanism. 

A simple kinetic analysis of reactions 15.5 and 15.6 leads to equations 15.7 (where the stationary state assumption was applied to the acyl radical concentration) and 15.8. 

On applying the stationary state (i.e. rate of initiation = rate of termination), the authors deduced that the radical concentration [R ] and propagation rate, Rp, were given respectively by ... 

This decomposition is usually first-order and only a fraction, f, of the radicals are utilised in step (62). Chain termination via step (64) and/or (65) is now an inherent feature of the reaction scheme. Often, the stationary state assumption can be applied to the A (as a group) so that... 

By applying the stationary state treatment to the formation of radicals, a value for [RJ could be obtained. Substitution of this expression into Eqn. 24 resulted in the final form of the deposition rate. 

In order to estimate kinetic constants for elementary processes in template polymerization two general approaches can be applied. The first is based on the generalized kinetic model for radical-initiated template polymerizations published by Tan and Alberda van Ekenstein. The second is based on the direct measurement of the polymerization rate in a non-stationary state by rotating sector procedure or by post-effect in photopolymerization. The first approach involves partial absorption of the monomer on the template. Polymerization proceeds according to zip mechanism (with propagation rate constant kp i) in the sequences filled with the monomer, and according to pick up mechanism (with rate constant kp n) at the sites in which monomer is outside the template and can be connected by the macroradical placed onto template. This mechanism can be illustrated by the following scheme ... 

From stationary-state treatment, the concentration of the radical intermediate [X-]ss is expressed as (Ia = intensity of absorption 4> = quantum yield)... 

---