Equilibrium constant of propagation

Kinetic techniques for the determination of the equilibrium constant of propagation... 

Extrapolating the apparent to [M] —> 00 gives Kc —i.e. the equilibrium constant of propagation for a high-molecular weight polymer. The results obtained by this method agree excellently with those reported by other investigators... 

Here [M]g denotes the equilibrium concentration of the monomer and K the equilibrium constant of propagation. The equilibrium concentration of the monomer depends on the system, i.e., on the nature of monomer, of solvent and temperature, but its value is not affected by the mechanism of polymerization. Determination of [M]g over a temperature range allows us to calculate AH and AS of propagation, and its dependence on polymer concentration and the nature of solvent provides information on the solvent-polymer interaction. A thorough discussion of these topics is reported elsewhere However, it should be stressed that the above thermodynamic ramifications apply to high-molecular weight polymers and the treatment has to be modified when one deals with oligomers. ... 

Kp = equilibrium constant of propagation Ke = equilibrium constant of end-to-end cycylization Kbb, Kbb" = equilibrium constants of back-biting Kse = equilibrium constant of segmental exchange... 

The parameters indexed with a are connected with the nucleation step or other effects occurring only once per triple helix. Parameters denoted by s are related with the equilibrium constants of the propagation steps and are ordered to be independent of the position of the reacting chain segment. This implies that end effects are neglected. Since the same dependences are valid for AH° and AS, with the help of their chain length dependence we can determine AG by extrapolation up to 3 n - 2 = 0, and thus, a can be estimated it depends neither on temperature nor on the chain length. 

The polymerization kinetics have been intensively discussed for the living radical polymerization of St with the nitroxides,but some confusion on the interpretation and understanding of the reaction mechanism and the rate analysis were present [223,225-229]. Recently, Fukuda et al. [230-232] provided a clear answer to the questions of kinetic analysis during the polymerization of St with the poly(St)-TEMPO adduct (Mn=2.5X 103,MW/Mn=1.13) at 125 °C. They determined the TEMPO concentration during the polymerization and estimated the equilibrium constant of the dissociation of the dormant chain end to the radicals. The adduct P-N is in equilibrium to the propagating radical P and the nitroxyl radical N (Eqs. 60 and 61), and their concentrations are represented by Eqs. (62) and (63) in the derivative form. With the steady-state equations with regard to P and N , Eqs. (64) and (65) are introduced, respectively ... 

Molar Cyclization Equilibrium Constant (Kx) and the Rate Constant of Propagation (kp x) and Depropagation (kj x) for Cyclic x-mer at 0eC with LiO-t-Bu (4.5xl0-3M)... 

Kinetics. Details of the kinetics of polymerization of THF have been reviewed (6,148). There are five main conclusions. (/) Macroions are the principal propagating species in all systems. (2) With stable complex anions, such as PI, 5bI, and Asp, the polymerization is living tinder normal polymerization conditions. When initiation is fast, kinetics of polymerizations in bulk can be closely approximated by equation 2, wlicre/ is the specific rate constant of propagation /is time / is the initiator concentration at t = 0 and [A/], [ /], and A /] are the monomer concentrations at t = 0, at equilibrium, and at time /, respectively. 

Fig. 3. Equilibrium constant of living polystyrene propagation in cyclohexane and benzene In if = -AGj,sJRT plotted versus 1/T, Reproduced, with permission, from Bvwater and Worsfold J. Polymer Sci 58,571...

A yellow color developed when CPT and S02 were mixed, and the equilibrium constant of the molecular complex formation was measured as K = 0.0353 ( 40°C, in hexane) (2). This complex might be the intermediate in this alternating copolymerization, and it might participate both in the initiation and propagation steps of this spontaneous copolymerization. 

We now define k as the rate constant of propagation excljiding the equilibrium constant K. Hence k " "... 

CH3)3CO— is an initiator residue]. With copolymerization of free monomers, they should have observed an increasing A/B ratio according to the method used with complex propagation, A/B should remain constant. The authors observed both cases. They concluded that maleic anhydride with a monomeric donor, like styrene, yields a DA complex by a reversible reaction, with an equilibrium constant of 10-1 to 10-2 dm3 mol-1. The initiating radical is formed from the complex, and the copolymerization is in fact a terpolymerization involving the two free monomers and their complex. These authors have applied the same technique in a study of the type of radicals formed in copolymerization of maleic anhydride with vinyl sulphides. Even in this case they provided evidence of the existence of a complex. 

Initiation is apparently slower than propagation. That is, the nucleophilic-ity of vinyl ethers is higher than their basicity. Other monomers such as p-methoxy-a-methylstyrene are apparently more basic and react rapidly with acid. In addition, the equilibrium monomer concentrations of a-meth-yl styrenes are relatively high ([M] 0.2 mol/L at —30° C). Because they can not polymerize at low concentration, they are ideal monomers for model studies [12,13]. The equilibrium constants of dimerization and tri-merization are much larger than that for the formation of high polymer. Therefore, dimers and trimers can be formed below [M] although high polymers cannot. 

The reversibility of propagation, or more specifically, the position of the equilibrium as determined by the ratio of the rate constants of propagation and depropagation is also independent of the mechanism. The equilibrium monomer concentration of monosubstituted alkenes such as styrenes and vinyl ethers are so low ([M] < 10-6 mol/L) at temperatures used for carbocationic polymerizations that the reversibility of polymerization can be neglected. 

The equilibrium constants of ionization are relatively low in most polymerization systems, resulting in very small proportions of ionic species. The proportion of ionic species may be estimated from the overall polymerization rates by assuming that covalent species are inactive. In polymerization systems with a half-lifetime of monomer in the range of r 1=5 20 min to 3 hr, the concentration of propagating ionic species should... 

There are some measurements of the rates of polymerization in systems with reversible formation of covalent species. The equilibrium constants of ionization can be calculated from these kinetic data according to the procedure outlined subsequently in Section IV.D.2.a. The ionization constant depends on the strength of the Lewis acid. For example, the propagating species are almost completely ionized in polymerizations of vinyl ethers with SbCL-, BCLt-, and SnCl5- counteranions, but only partially ionized when the counteranions are I3- or Zn3-. 

Because the reactivities of ions and ion pairs are similar and only weakly affected by the structure of the counteranions, kp + or kp determined by either stopped-flow studies or y-radiated systems (cf., Section IV. 13) can be used in Eq. (75). The equilibrium constant of ionization can then be estimated from the apparent rate constant of propagation and the rate constant of propagation by carbenium ions [Eq. (77)]. For example, Kf 10-s mol-,L in styrene polymerizations initiated by R-Cl/SnCl4 [148]. Kt for vinyl ether polymerization catalyzed by Lewis acids can also be estimated by using the available rate constant of ionic propagation (kp- = 104 mol Lsec-1 at 0° C) [217], The kinetic data in Ref. 258 yields Kj == 10 3 mol - l L in IBVE polymerizations initiated by HI/I2 in toluene at 0° C and Kf 10-1 mol- -L initiated by HI/ZnI2/acetone can be calculated from Eq. (76). 

Deactivation of growing carbenium ions by reaction with sulfides is evidently very fast. Sulfonium ion formation is exothermic (AH = -40 kJ/mol) and exoentropic (AS = -74 J/mol-K) [271]. High equilibrium constants (Keq = 104 moI-1L) for sulfonium ion formation were calculated from the apparent rate constants of propagation and the rate constants of carbocationic growth. Dynamic NMR experiments of model systems with tetrahydrothiophene indicate that the bimolecular deactivation rate constant is kdeacl 106 mol-1-Lsec-1 at 0° C (AH = 20 kJ/mol, AS = -37 J-mol-K), and that activation is faster than bimolecular exchange (k act foe) [67]. 

The equilibrium constants for sulfonium ion formation were calculated from the apparent rate constants of propagation and the rate constants of carbocationic propagation [257], using Eq. (31) obtained by the combination of Eq. (30) with Eqs. (28) and (29). 

The overall polymerization rates and the apparent rate constants of propagation (/c/pp = RP/[M][I]0) for the same initiating system are, however, very different for each class of monomers. For example, the same initiating system, that will polymerize a-methylstyrene (aMeSt) in 1 h, will complete polymerization of vinyl ether within less than 1 min but would require a few days to polymerize styrene and isobutene under otherwise identical conditions. This trend is due to the equilibria between dormant and active species. In this case the apparent rate constant of propagation is the product of the rate constant of propagation (weakly depending on monomer structure) and the ionization constant (kpapp = kp + -Kf). This equilibrium constant is much higher for more stable cations derived from vinyl ethers than from aMeSt, than styrene or isobutene. 

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