2 Theoretical Background
2.4 Influences on Rate Coefficients
2.4.2 Concentration
For ideal polymerization kinetics, rate coefficients are considered to be independent of the concentrations of compunds. Often, this is assumed for real polymerizations as well, but for both diffusion-controlled and chemically-controlled polymerization reactions, the rate coefficients may vary significantly with concentration.
‡
exp V p
A A R T
(2.24)
‡
exp EA V p
k A R T
(2.25)
A different composition obviously leads to a different viscosity. Hence, all diffusion-controlled rates (termination, initiator efficiency, inhibition and catalyzed chain transfer) are affected. Following eq. (2.22) and eq. (2.21) their rate coefficients increase and decrease with fluidity. Sometimes a small change of one component has a large impact on viscosity.
Less obvious is the concentration dependence of chemically controlled rate coefficients. Initiator decay can be influenced a lot by other components in a hardly predictable way, e.g., rate of decomposition of sodium persulfate is increased by a factor of up to seven in the presence of acrylic acid, but depending on concentration and ionization of monomer it can also be decreased.[56]
A special case, which will be discussed in greater detail, is the rate coefficient of propagation, kp. In the late 90ies it was begun to measure propagation rate coefficients for polymerizations in aqueous solution by PLP–SEC (pulsed laser polymerization size exclusion chromatogrophy) – a method superior over the older rotating sector technique. It has been found that kp depends on monomer concentration. Several explanatory approaches were made for these astonishing results:
First, water-soluble monomers like acrylic acid and methacrylic acid (two of the earliest examined monomers) tend to associate with each other forming a variety of different dimers up to oligomers. Changes in the solvent to monomer ratio necessarily lead to different amounts of the various associations of monomer. Under the assumption that these monomer associations show different reactivities, the rate has to depend on monomer concentration.[57] This would mean that reactivity in polar organic solvents, e.g., ethanol or dimethyl sulfoxide changed in a similar way as in water, but this is not the case.[58] This theory has been discarded.
Second, the “local” concentration may be differ from overall concentration. Usually, it is assumed that overall monomer concentration is identical to the “local” monomer concentration in close proximity to the radical centre. If overall and “local” monomer concentrations are different, following eq. (2.4) kpwill appear higher than the same factor as the “local” concentration is higher as the overall concentration. However, in case of polymerizations in aqueous solution, this assumption requires an enormously large difference – a factor of ten. At low monomer concentrationsalmost all monomer molecules would have to be situated in the direct vicinity of macroradicals. As a consequence, the reaction solution consists of a few radicals with associated monomer molecules dissolved in almost pure water.[59] In addition polymer in the reaction mixture does not influence kp.[9] If the polymer collected monomer from the
solution to achieve the elevated “local” concentration, additional polymer would reduce the measured kp.This theory is now considered dismissed.
Third, the corresponding reaction is chemically controlled and thus the rate coefficient can be described by the Eyring equation, eq. (2.26), which assumes the reactants to go through a transition state (TS) as the highest point of the “pass”.[60,61]
If it is a genuine kinetic effect, it can be explained by this equation.
Q stands for the partition functions of species, ‡ denotes the transition state. is the transmission coefficient (1 or less), h the Planck constant, and
E
0 the zero-point energy difference between educts and transition state.The transmission coefficient is independent of the concentrations of the components in the reaction mixture. Thus, there remain only two possibilities. Either the partition functions are influenced consequently shifting the Arrhenius prefactor (compare eq. (2.23)) or the zero-point energy difference and the activation energy (compare eq. (2.23)), respectively. Detailed examination of the temperature dependence of kpof MAA has shown that EA is almost in
sensitive towards a variation of monomer content within a large concentration range and it is primarily A that varies.[59] Consequently, the partition functions have to be influenced by the solvent environment. Gilbert et al.[62] calculated that the effect is due to different extents of hindrance to internal rotation (vibration with an activation energy in the order of magnitude of a rotation) in the transition state (TS) structure for propagation. The solvent molecules in the surrounding area of the activated complex may impose a hindrance to the internal rotation of the activated complex depending on how strong they are attached and how big they are. The stronger intermolecular interactions of the activated complex with an environment that basically consists of monomer molecules result in a lower mobility of side groups and thus lead to a reduced pre-exponential factor towards higher monomer content.[9,59].
The same group[63] has found in a newer investigation again through calculation that a different solvent field causes a different activation energy. They found for the
propagation of AA that the activation energy in toluene representing non-polar solvents is as in the gas phase while it is considerably reduced in a water environment. The level of accuracy, however, was not sufficient for quantitative accuracy. The variation of EAwas ascribed to better resonance stabilization of the TS in the polar solvent, and better mixing of the molecular orbitals of the reactants, assisting in the transfer of electrons from the monomer to the growing chain.
In another calculation of the polymerization of MAA and AA the experimental finding was confirmed that the rate acceleration of both polymers in water is mainly due to entropic rather than electrostatic effects. Degirmenci et al. also calculated the difference of the kps of MAA and AA arises mainly from steric hindrance of the methyl group and not from difference in electronic structure.[64]
For AAm, calculations that compare propagation in gas phase with those in aqueous phase conclude that activation energy is reduced.[65] Experimental results for 1-vinylpyrrolidin-2-one[66] and N-vinylformamide,[67] suggest as well that EA varies with solvent content, although not in a way that could explain the dependency of kp on cM.
Overall, the influence on kp in aqueous solution is mostly based on an alteration of the entropy of the transition state, but there is an also a smaller effect on EA. In subchapter 5.1.2, this is discussed in detail including new results for kp.