Concentration of Living Polymers

Frequently the spectral features of the different species participating in propagation are closely similar. Whenever this is the case, variation of their relative concentrations leaves the spectrum virtually unaltered. This is often implicitly assumed. However, it is important to realize that examples to the contrary are known. In fact, the differences of spectral features of different species were effectively utilized in establishing their nature and proving their existence in the investigated systems (see, e.g.280 ). 

In conclusion, the spectral methods are most useful and reliable if proper care is exerted in their applications. They are invaluable whenever it is desired to continuously monitor the concentration of living polymers. Partial destruction of living polymers, or any other changes in their nature taking place in the course of polymerization are then readily detected, especially when the absorbance is measured over a sufficiently wide spectral range. 

Some irreversible reactions quantitatively convert living polymers, but not dead ones, into unique products, detection of which is easily and quantitatively achieved. It is immaterial whether such a reaction proceeds with only one of the inter-convertible species or with all of them, as long as the inter-conversion is fast. The concentration of the resulting product measures then the total concentration of living polymers, whatever their state. For example, addition of Michler ketone to solutions of living poly-styrene converts their carbanionic groups into an alcohol readily oxidized to an intensely colored dye. The conversion is quantitative and applicable for analytic usage24.  

Various titration techniques are based on the same principle. The agent used in titration has to react irreversibly and quantitatively with living polymers, but not with dead ones. The end-point is determined by a sharp change in some convenient property of the system, e.g. by the disappearance of the color of living polymers, provided that the dead polymers are colorless. 

An unconventional technique was developed by Schulz and his co-workers261. It calls for determination of number average degree of polymerization, DP , at various degrees 

---

Similarly, at low concentration of living polymers, when the dimers, D, are dominant,... 

The break-seal on A is then crushed, the sample to be investigated diluted with the solvent in C, and the optical density of the resulting solution determined in the appropriate optical cell. The solution is then transferred to conductivity cell H, and its resistance is measured. The optical density is redetermined, and thereafter about two thirds of the solution is transferred to C. The solvent from C is distilled into the chilled ampoule G and used to dilute the residual solutions left in H. The conductivity and the optical density of this solution are determined as described previously thereafter, two thirds is again transferred to C, and the remaining one third is diluted by the above-described procedure. In this way the conductivities are determined for decreasing concentrations of living polymer, so that the molar conductivity A can be calculated as a function of [living polymer] down to about 10" M. 

In the presence of another electrolyte, sharing a common counterion with the growing polymer, the above relation must be modified (26). For example, the addition of Na+,BPh4 to living sodium polystyrene, —S",Na+, retards its polymerization, and the propagation constant is then independent of the concentration of living polymers—viz.,... 

Another assumption is that the rate of fonnation of J j from the initiator is instantaneous and that at time i = 0 the initial concentration of live polymer is i io " /o- This assumption is veiy reasonable for this initiation mechanism. Under the latter assumption the mole balances become... 

At a constant concentration of living polymers, the pseudo-first order rate constants for each monomer addition remain the same whether the polymers are or are not uniform. However, their rate of dissociation into a specified monomer is smaller in the system discussed here than in that involving the appropriate uniform polymers. While each uniform polymer may dissociate into its monomer, only a fraction of non-uniform polymers, i.e. those possessing the appropriate ending, is capable of degrading into that monomer. Therefore, the stationary concentrations of the respective monomers are given by their true equilibrium concentrations multiplied by the mole fractions of those with the required ending, viz. 

Finally, a quantitative conversion of initiator into living polymers is assumed in many studies. Provided that termination and other destructive reactions are absent, the initial concentration of the initiator is equal to the concentration of living polymers. Presence of a residual initiator in a non-quantitative initiation is readily detected. 

Polymerization of styrene initiated in cumyl-methyl ether by cumyl sodium is hardly affected by variation of concentration of living polymers or by the addition of sodium tetraphenyl boride269. This is not surprising since the dielectric constant of that solvent is also very low, 3.7 at 20 °C, although higher than of dioxane (2.2). The bimolecular propagation constant at -20°C is 1 M 1s 1, a value expected for a propagation of contact ion-pairs. 

Propagation constants, k , of the polystyryl ion-pairs in THP were derived, like in the other studies, by extrapolating the plots of kp vs. l/c[a to infinite concentration of living polymers, or by suppression of the dissociation of ion-pairs of living polymers caused by the addition of appropriate tetraphenyl boride salts. The kinetic and conductometric results of all of these studies, summarized in Table 8, show a fair agreement between the data reported by all the investigators. 

Fig. 47. Plot of the observed overall propagation constant, kp, of cesium polystyryl in THF vs. 1/CLW (CL concentration of living polymers). - results of Ldhr and Schulz29 - results of Bhattacharyya et al.123) O - results of Shimomura et al.26 . Note their self-consistency...

In conclusion, the effect of high concentration of electrolytes upon the rate of anionic propagation deserves exploration. If it is real, it affects extrapolation and may lead to too high values of k . The only evidence to the contrary is provided by Dainton andlvin2685, who determined kp of sodium polystyrene in THP at constant concentration of living polymers but variable concentration of sodium tetraphenyl-boride. Their results are shown in the self-explanatory Fig. 48. It might be beneficial to have similar data for other solvent systems. 

The proposed mechanism could be tested. It predicts a change in the reaction order from A at low concentrations of living polymers to A at their higher concentrations, i.e. the inequality... 

Therefore, the zeroth moment is simply the total number (or concentration) of living polymer chains [Eq. (68)]. 

For a batch reactor, the monomer is added once and can be lost only by polymerization. Similarly, a balance on the total concentration of live polymer chains, P, can be written as ... 

---