Polymerization rate laws

Rate of polymerization. The rate of polymerization for homogeneous systems closely resembles anionic polymerization. For heterogeneous systems the concentration of alkylated transition metal sites on the surface appears in the rate law. The latter depends on the particle size of the solid catalyst and may be complicated by sites of various degrees of activity. There is sometimes an inverse relationship between the degree of stereoregularity produced by a catalyst and the rate at which polymerization occurs. 

Ideal Gas Law constant, polymerization rate, mole/liter sec. 

Using the titanocene-catalyzed co-hydrogenation of cyclohexene, we have studied the kinetics of the polymerization of a number of primary silanes ( 20 ). The rate law was found to be ... 

A more active zinc catalyst has been developed containing a /3-diketiminate ancillary ligand.823 Complex (272) converts 200 equivalents rac-LA. in less than 20 min at room temperature to highly heterotactic PLA (the probability of a racemic junction between monomer units, PR = 0.90 at 25 °C and 0.94 at 0°C) of narrow polydispersity (Mn = 37,900, Mn calc = 28,800, Mw/Mn= 1.10). Kinetic studies reveal that the zinc initiators display a fractional rate law order (1.56 0.06). In accord with the observation that kR/ss > kR/RR (or ks/RR > ks/ss), the polymerization of L-LA proceeds significantly slower than rac-LA (kapp L-LA = 0.031 min-1 kapp rac-LA = 0.22 min-1).824... 

Improved control was observed, however, upon addition of benzyl alcohol to the dinuclear complexes.887 X-ray crystallography revealed that whereas (296) simply binds the alcohol, (297) reacts to form a trinuclear species bearing four terminal alkoxides. The resultant cluster, (298), polymerizes rac-LA in a relatively controlled manner (Mw/Mn=1.15) up to 70% conversion thereafter GPC traces become bimodal as transesterification becomes increasingly prevalent. NMR spectroscopy demonstrates that the PLA bears BnO end-groups and the number of active sites was determined to be 2.5 0.2. When CL is initiated by (298) only 1.5 alkoxides are active and kinetic analysis suggests that the propagation mechanisms for the two monomers are different, the rate law being first order in LA, but zero order in CL. 

Equilibrium studies under anaerobic conditions confirmed that [Cu(HA)]+ is the major species in the Cu(II)-ascorbic acid system. However, the existence of minor polymeric, presumably dimeric, species could also be proven. This lends support to the above kinetic model. Provided that the catalytically active complex is the dimer produced in reaction (26), the chain reaction is initiated by the formation and subsequent decomposition of [Cu2(HA)2(02)]2+ into [CuA(02H)] and A -. The chain carrier is the semi-quinone radical which is consumed and regenerated in the propagation steps, Eqs. (29) and (30). The chain is terminated in Eq. (31). Applying the steady-state approximation to the concentrations of the radicals, yields a rate law which is fully consistent with the experimental observations ... 

In the following four years Mark successively reported on the viscosity and molecular weight of cellulose (40), Staudinger s Law (41), high polymer solutions (42), and the effect of viscosity on polymerization rates (43). Confident of his findings, he proposed (at the same time as R. Houwink) the general viscosity equation now known as the Mark-Houwink Equation (44, 45). 

A critical survey of the literature on free radical polymerizations in the presence of phase transfer agents indicates that the majority of these reactions are initiated by transfer of an active species (monomer or initiator) from one phase to another, although the exact details of this phase transfer may be influenced by the nature of the phase transfer catalyst and reaction medium. Initial kinetic studies of the solution polymerization of methyl methacrylate utilizing solid potassium persulfate and Aliquat 336 yield the experimental rate law ... 

Nonetheless, their steady-state solutions agreed well with those obtained by direct numerical integration of the rate laws for two different sets of dimerization rate constants, and their analysis provides a rather satisfying view of the actin polymerization process. 

It is essential to characterize the reactant species in solution. One of the problems, for example, in interpreting the rate law for oxidation by Ce(IV) or Co(III) arises from the difficulties in characterizing these species in aqueous solution, particularly the extent of formation of hydroxy or polymeric species. We used the catalyzed decomposition of HjOj by an Fe(III) macrocycle as an example of the initial rate approach (Sec. 1.2.1). With certain conditions, the iron complex dimerizes and this would have to be allowed for, since it transpires that the dimer is catalytically inactive. In a different approach, the problems of limited solubility, dimerization and aging of iron(III) and (Il)-hemin in aqueous solution can be avoided by intercalating the porphyrin in a micelle. Kinetic study is then eased. 

This rate law predicts that the rate of polymerization will be suppressed by the reduction in the free actin concentration as a consequence of AX formation. 

Deoxygenated sickle cell hemoglobin (deoxyHbS), the j8-Glu-6-Val point mutant form of adult hemoglobin, appears to obey the following empirical rate law for nucleation of polymerization ... 

This active site counting methodology has been applied to the determination of initiation, propagation and termination rate laws and activation parameters for the polymerization of 1-hexene [141] catalyzed by 91 in toluene solution. 

In most cases, the metal-catalyzed ROP activity is best assessed by determining the polymerization rate constant (kp). A second order rate law is being followed by most of the catalytic lactide polymerizations ... 

Experiments in the presence of tetrahydrofuran show that the polymerization rates are always proportional to the square root of added initiator concentration (0.1% to 15% THF). 34% of the initiator is found to lead to active chains. The dependence on monomer concentration is strange. The monomer reacts according to a first order law but the rates are usually dependent on initial monomer concentration. The square root dependence on initiator concentration can be explained if reaction proceeds via free (solvated) ions in labile equilibrium with the undissociated solvated species XLi wTHF, if the latter is still the major species present. The mechanisms suggested formally explain the observed... 

Continuous mixture theory found application in industry and was, as I understand it, incorporated into some of the models the chemical and oil companies were developing. Some of the work on polymerization and coal devolatization also used the notions of continuous mixtures, but there was little development on the formal side until the work of Astarita and Ocone in the latter eighties. Their paper (AIChE J. 34 1299, 1988) introduced the idea of uniform kinetics. This allowed the time scale to be warped and results to be obtained when the underlying reactions were not of the first order. Indeed, it was shown that intrinsic kinetics (i.e. rate laws for A(x)dx) could be found that would mimic any kinetic law for the lump as a whole (AIChE J. 35 529,1989 [248, 249, 259]). 

Such products are found to undergo continued inorganic polymerization resulting from pendant hydroxy groups forming bridges between adjacent metal centers. Kinetic studies of this reaction mechanism indicate that a general rate law may be written as... 

Experiments in which initiators other than AIBN are used do not indicate any unusual effects. Polymerization rate is nearly always proportional to the square root of the initiator concentration or at least to a value between 0.5 and 0.6. Ulbricht (135) established the square root law for ammonium persulfate, AIBN and benzoyl peroxide in DMF, the rates being fastest with persulfate, slower with AIBN and slowest with the peroxide. One expects that persulfate and peroxide will be more active than AIBN in abstracting hydrogen from other components of the systems. Other initiators have been used, including UV with di-t-butyl... 

Understanding the polymerization kinetics is of fundamental importance not only for comparing different initiators but also understanding the catalytic cycle and preparing more active compounds. The polymerization generally obeys a second-order rate law, with first-order dependencies on both monomer (lactide) and... 

The use of chemical sensitizers such as benzoyl peroxide, cumene hydroperoxide, or azo-bis-isobutyronitrile, which decompose thermally to give free radicals in a convenient temperature range (i.e., 60 C to 150 C), makes it possible to study polymerizations over an extended temperature range. The form of the rate law with chemical initiations would be given by setting III = 2k (ln)< >i in Eq. (XVI.10.4). Here (In) is the initiator concentration, k I its specific rate constant of decomposition which can usually be measured independently, and is the efficiency with which its radicals initiate chains. The measure of t is subject to the difficulties already indicated in connection with the photolysis systems. ... 

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