Stationary state, radical polymerization

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... 

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. 

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 ... 

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 ... 

Very rapid initiations are known, manifested by an instantaneous start to the polymerization after which the number of active centres is not further increased. Polymerizations with slow initiation are also quite frequent, starting only after some inhibition and/or induction period. In the course of these polymerizations, the concentration of active centres is not usually constant. A stationary state is not excluded, of course but it occurs much less frequently than with radical polymerizations. 

When the concentracion of active centres does not change during propagation, we speak of stationary polymerization. All previous and the majority of the presently valid analyses of radical polymerization kinetics assume stationary (or, better, pseudo-stationary [20, 21]) states. The conditions under which the stationary state assumption is a valid and useful approximation, can be expressed as follows [22] ... 

With growing viscosity, the diffusion rate of propagating macromolecules decreases, the probability of their effective collision is diminished, and thus also the rate of bimolecular termination. As the rate of initiation is not appreciably affected by increasing viscosity, in a certain phase some radical polymerizations pass from a stationary to a non-stationary state in which the number of radicals increases. The polymerization is appreciably accelerated. This situation is called the gel effect or the Norrish-Tromsdorff effect [55]. As the change in the rate of termination contributes considerably to the gel effect, it will be discussed in detail in Chap. 6, Sect. 1.3. 

Equation (58) indicates that an increase in initiatior concentration will not enhance the rate of polymerization. It can be used for estimating the molecular mass of the polymer assuming, of course, the absence of transfer. The ratio N/q corresponds to the mean time of polymer growth and molecular mass is equal to the product of the number of additions per unit time and the length of the active life time of the radical, kpN/e. An increase in [I] also means a higher value of q, and thus a shortening of the chains. As in Phase II, the polymerized monomer in the particles is supplemented by monomer diffusion from the droplets across the aqueous phase a stationary state is rapidly established with constant monomer concentration in the particle. The rate of polymerization is then independent of conversion (see, for example the conversion curves in Fig. 7). We assume that the Smith-Ewart theory does not hold for those polymerizations where the mentioned dependence is not linear [132], The valdity of the Smith-Ewart theory is limited by many other factors. 

For ideal radical polymerization to occur, three prerequisites must be fulfilled for both macro- and primary radicals, a stationary state must exist primary radicals have to be for initiation only and termination of macroradicals only occur by their mutual combination or disproportionation. The rate equation for an ideal polymerization is simple (see Chap. 8, Sect. 1.2) it reflects the simple course of this chain reaction. When the primary radicals are deactivated either mutually or with macroradicals, kinetic complications arise. Deviations from ideality are logically expected to be larger the higher the concentration of initiator and the lower the concentration of monomer. Today termination by primary radicals is an exclusively kinetic problem. Almost nothing has been published on the mechanism of radical liberation from the aggregation of other initiator fragments and from the cage of the... 

Active centres of ionic polymerizations do not usually decay by mutual collisions as the radical centres. The stationary state, when it exists at all, results from quite different causes, mostly specific to the given system. Therefore the kinetics of ionic polymerizations is more complicated and its analysis more difficult. The concentration of centres cannot usually be calculated. On the other hand, ionic systems with rapid initiation give rise to the kinetically very simple living polymerizations (see Chap. 5, Sect. 8.1). 

For radicals of a certain degree of polymerization in the stationary state, the generation rates are given as... 

Rates of radiation induced polymerizations are normally determined by dilatometric [85] or gravimetric [84] experiments. Some of the first quantitative results from cyclopentadiene [86] and a-methylstyrene [87] were obtained by competitive kinetic methods, based on the retarding effect of ammonia and amines. This approach tends to yield maximum values for Rp. More recently, however, a procedure combining stationary state kinetic and conductance measurements has been described [88, 89], and further refined [85]. Because the ions generated by 7-ray irradiation have a transient existence, the kinetic treatment leads to expressions which are very similar to those derived for homogeneous free radical polymerizations [90]. A simplified version of the kinetic scheme is as follows ... 

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... 

Another aspect49 is the initial presence of persistent species in nonzero concentrations [Y]o, and it will be discussed more closely in section IV. In the absence of any additional initiation, the excess [Y]o at first levels the transient radical concentration to an equilibrium value [R]s = A[I]o/[Y]o. This is smaller than that found without the initial excess and lowers both the initial conversion rate and the initially large PDI. Further, it provides a linear time dependence of ln-([M]o/[M]), which is directly proportional to the equilibrium constant. Later in the reaction course, [Y] may exceed [Y]0 because of the self-termination, then [R] is given by eq 18. If there is additional radical generation, the first stages will eventually be replaced by a second stationary state that was described above. Further effects are expected from a decay or an artificial removal of the persistent species that increases the concentration of the transient radicals and the polymerization rate (see section IV). Radical transfer reactions to polymer, monomer, or initiator have not yet been incorporated in the analytical treatments. 

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... 

When a system composed of PS-TEMPO (P-N) adduct and monomer styrene is heated to a sufficiently high temperature where the adduct dissociation takes place, the concentration of P and N will start to increase from 0 up to the equilibrium values, determined by K. Since P° /P biradical termination will continually reduce [P° ] relative to [N ], polymerization will eventually stop when [N ] reaches sufficiently high value. However, if initiation also takes place in the system, the newly formed radicals will combine with N preventing its accumulation and a stationary state for both [N ] and [P ] will be reached. Setting d P ]/dt = 0 in Eq. (P6.42.5) and = 0 in Eq. (P6.42.6) one then... 

The kinetic treatment of these polymerization reactions depends upon the establishment of stationary state equations for the rates of formation and disappearance of all the transitory intermediates. The form of the expressions derived for the rate of the main reaction depends largely upon the mode of chain ending, and the constants entering into the formulae are those characterizing initiation, propagation, and termination respectively. Special means may be employed for the study of some of these constants in isolation, whereby rather complicated relations can be unravelled. For example, the reaction may be excited photoohemically, in which case the rate of initiation is calculable from the number of quanta of light which are absorbed. This method can be applied with ease to those polymerization reactions which are started by radicals formed, for example, in the photolysis of aldehyde ... 

A growing chain is deactivated when it reacts with another chain to form a dead macromolecule. The recombination of two growing monoradicals is an example of this. Termination reactions destroy active centers both the rate of polymerization and the degree of polymerization are lowered. The deactivation through reaction of two free radicals in one of the reasons why ionic polymerizations are faster than free radical polymerizations. The deactivation reaction between two free radicals has a small activation energy, and therefore occurs very rapidly. Thus, the concentration of growing free radicals is very low in the stationary state... 

The stationary concentration of P and X are determined by different mechanisms. [P ] is determined by the balance of the initiation rate R and the termination rate ki [P ] [2]. This is the same as in conventional free radical polymerization systems. [X ] is determined, however, from the equilibrium equation shown for process I. It depends, therefore upon the equilibrium constant K and on the concentration of the adduct [P-X] and [P ] [289]. The rate of polymerization during the stationary state is... 

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