Reactivity ratios estimation

While sequence distributions are usually subject to more experimental noise than composition data, this is often outweighed by the greater information content. In principle, reactivity ratios can be estimated from a single copolymer sample. The consistency in reactivity ratios estimated with eqs. 45 and 46 for copolymers prepared with different monomer feed compositions and/or obtaining the same result from cqs. 50 and 51 (4 aab—Ai ab) and cqs. 52 and 53 (r aba-Aiba) arc... 

Reactivity Ratios Estimation Based on Copolymer Composition Data... 

The thorough treatment of the experimental data does allow one to obtain reliable values of the reactivity ratios. The results of such a treatment are presented in Table 6.3 for some concrete system let us form a notion about an accuracy of the reactivity ratios estimations. The detailed analysis of such a significant problem in the case of the well-studied copolymerization of styrene with methyl methacrylate is reported in Ref. [227]. Important results on the comparison of the precision of rj, r2 estimates by means of different methods are presented by O Driscoll et al. [228]. Such a comparison of six well-known linear least-squares procedures [215-218,222,223] with the statistically correct non-linear least-squares method leads to the conclusion that some of them [216, 217, 222] can provide rather precise rls r2 estimates when the experiment is properly planned. 

The use of a continuous stirred tank reactor permits one to apply the instantaneous copolymer equation for reactivity ratios estimation. 

There are two ways to improve the accuracy of reactivity ratios estimates. The first one is to carry out ejq)etiments at the optimal comonomer feed composition. The intuitive approach is to carry out ejq)etiments at compositions that are equally distributed over the entire composition range. This method is very frequently fotmd in literature. For getting a first impression of the value of the reactivity ratios, this approach is very well suited. However, once initial estimates of reactivity ratios are available, experiments can be carried out at compositions where the sensitivity toward changes in reactivity ratios is maximal. Tidwell and Mortimer" derived expressions for these comonomer feed compositions. They did this exercise for the TM and came up with the expressions shown in eqns [37] and [38], where /21 and f22 are the fractions of monomer 2 in the reaction mixtrue that are most suitable for the accurate determination of reactivity ratios ... 

Reactivity ratios for the 7V-vinylphthalimide (molecule 1)-styrene (molecule 2) system were measured, and foundt to be ri = 0.075 and I2 = 8.3. Use these values to estimate values of Q and e for 7V-vinylphthalimide then estimate the parameters rj and 12 for system in which molecule 2 is vinyl acetate. 

Epichlorohydrin Elastomers without AGE. Polymerization on a commercial scale is done as either a solution or slurry process at 40—130°C in an aromatic, ahphatic, or ether solvent. Typical solvents are toluene, benzene, heptane, and diethyl ether. Trialkylaluniinum-water and triaLkylaluminum—water—acetylacetone catalysts are employed. A cationic, coordination mechanism is proposed for chain propagation. The product is isolated by steam coagulation. Polymerization is done as a continuous process in which the solvent, catalyst, and monomer are fed to a back-mixed reactor. Pinal product composition of ECH—EO is determined by careful control of the unreacted, or background, monomer in the reactor. In the manufacture of copolymers, the relative reactivity ratios must be considered. The reactivity ratio of EO to ECH has been estimated to be approximately 7 (35—37). 

The rates of addition to the unsubstituted terminus of monosubstituted and 1,1-disubstiluted olefins (this includes most polymerizable monomers) are thought to be determined largely by polar Factors.2 16 Polymer chemists were amongst the first to realize that polar factors were an important influence in determining the rate of addition. Such factors can account for the well-known tendency for monomer alternation in many radical copolymerizations and provide the basis for the Q-e, the Patterns of Reactivity, and many other schemes for estimating monomer reactivity ratios (Section 7.3.4). 

Numerical approaches for estimating reactivity ratios by solution of the integrated rate equation have been described.124 126 Potential difficulties associated with the application of these methods based on the integrated form of the Mayo-kewis equation have been discussed.124 127 One is that the expressions become undefined under certain conditions, for example, when rAo or rQA is close to unity or when the composition is close to the azeotropic composition. A further complication is that reactivity ratios may vary with conversion due to changes in the reaction medium. 

Terminal model reactivity ratios may be estimated from the initial monomer feed composition and the dyad concentrations in low conversion polymers using the following relationships (eqs. 45, 46). 

Similarly, penultimate model reactivity ratios can be estimated from initial monomer feed composition and triad concentrations using eqs. 50-53. 

The solvent in a bulk copolymerization comprises the monomers. The nature of the solvent will necessarily change with conversion from monomers to a mixture of monomers and polymers, and, in most cases, the ratio of monomers in the feed will also vary with conversion. For S-AN copolymerization, since the reactivity ratios are different in toluene and in acetonitrile, we should anticipate that the reactivity ratios are different in bulk copolymerizations when the monomer mix is either mostly AN or mostly S. This calls into question the usual method of measuring reactivity ratios by examining the copolymer composition for various monomer feed compositions at very low monomer conversion. We can note that reactivity ratios can be estimated for a single monomer feed composition by analyzing the monomer sequence distribution. Analysis of the dependence of reactivity ratios determined in this manner of monomer feed ratio should therefore provide evidence for solvent effects. These considerations should not be ignored in solution polymerization either. 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

Mayo-Lewis Binary Copolymeriration Model. In this exeimple we consider the Mayo-Lewis model for describing binary copolymerization. The procedure for estimating the kinetic parameters expressed as reactivity ratios from composition data is discussed in detail in our earlier paper (1 ). Here diad fractions, which are the relative numbers of MjMj, MiMj, M Mj and MjMj sequences as measured by NMR are used. NMR, while extremely useful, cannot distinguish between MiM and M Mi sequences and... 

Applications of the method to the estimation of reactivity ratios from diad sequence data obtained by NMR indicates that sequence distribution is more informative than composition data. The analysis of the data reported by Yamashita et al. shows that the joint 95% probability region is dependent upon the error structure. Hence this information should be reported and integrated into the analysis of the data. Furthermore reporting only point estimates is generally insufficient and joint probability regions are required. 

The estimation of the two parameters requires not only conversion and head space composition data but also physical properties of the monomers, e.g. reactivity ratios, vapor pressure equation, liquid phase activity coefficients and vapor phase fugacity coefficients. 

The product of the reactivity ratios can be used to estimate the copolymer structure. When the product of the reactivity ratios is near 1, the copolymer arrangement is random when the product is near zero, the arrangement is alternating when one of the reactivity ratios is large, blocks corresponding to that monomer addition will occur. 

The NMR analysis (21) of the chemical composition for copolymers from various monomer feed ratios at fairly low conversion are shown in Table IV. The results were then used to estimate the reactivity ratios for the diene monomers under the conditions employed. Various published methods of calculating monomer reactivity ratios have been examined. These include the once popular but now somewhat out of favor Fineman-Ross method... 

Various methods have been applied to estimate their reactivity ratios which are tabulated in Table V. Typical values at 20°C are rg = 2.64 and r-j-= 0.404. Preliminary evidence was reported that suggests the copolymerization is more selective at lower temperatures. 

With the exception of [64], the majority of copolymerizations has been carried out with non-recrystallized DADMAC. Although, there is no evidence that the monomer purity markedly influences the reactivity ratios of Table 5, a general influence on the rate of polymerization should be taken into account. The majority of analytical methods require removal of the monomers before the copolymer composition can be determined. For this reason, HPLC has been shown to provide estimates of reactivity ratios with more narrow confidence intervals [70]. Due to the differences between rx and r2, particularly at higher DADMAC contents in the monomer feed, it is quite challenging to maintain a low conversion of AAM and a constant monomer feed composition. 

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