Estimation of Reactivity Ratios

Methods for evaluation of reactivity ratios comprise a significant proportion of the literature on copolymerization. There are two basic types of information that can be analyzed to yield reactivity ratios. These are (a) copolymer composition/convcrsion data (Section 7.3.3.1) and (b) the monomer sequence distribution (Section 7.3.3.2). The methods used to analyze these data are summarized in the following sections. 

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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. 

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. 

The most generally useful methods and the only statistically correct procedures for calculating reactivity ratios from binary copolymerization data involve nonlinear least squares analysis of the data or application of the error in variables (EVM) method. Effective use of either procedure requires more iterations than can be performed by manual calculations. An cfHcicnl computer program for nonlinear least squares estimates of reactivity ratios has been published by Tidwell and Mortimer [13]. The EVM procedure has been reported by O Driscoll and Reilly [14]. 

The traditional methods for the estimation of reactivity ratios (28, 29) and their more recent offsprings (30 32) are based on the transformation of equation 1 into one of the following forms ... 

The free-radical copolymerization of acrylamide with three common cationic comonomers diallyldimethylammonium chloride, dimethyl-aminoethyl methacrylate, and dimethylaminoethyl acrylate, has been investigated. Polymerizations were carried out in solution and inverse microsuspension with azocyanovaleric acid, potassium persulfate, and azobisisobutyronitrile over the temperature range 45 to 60 C. The copolymer reactivity ratios were determined with the error-in-variables method by using residual monomer concentrations measured by high-performance liquid chromatography. This combination of estimation procedure and analytical technique has been found to be superior to any methods previously used for the estimation of reactivity ratios for cationic acrylamide copolymers. A preliminary kinetic investigation of inverse microsuspension copolymerization at high monomer concentrations is also discussed. 

Gloor, P. Estimation of Reactivity Ratios Using the Error-in-Variables Method and Data Collected from a Continuous Stirred Tank Reactor, MIPPT-Report, McMaster University, Hamilton, Ontario, Canada, 1987. 

Tip 12 Copolymerization, reactivity ratios, and estimation of reactivity ratios. In a binary copolymerization of monomers and M2, reactivity ratios r and r2 are important parameters for calculating polymerization rate, copolymer composition, and comonomer sequence length indicators (see Chapter 6 for basic equations and further information). 

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 ... 

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). 

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. 

Residence time and reactivity are strongly correlated through equation (7.2.9). This is true for seawater composition since Whitfield and Turner (1979) showed a rather good correlation between oceanic residence times and seawater-crustal rock partition coefficients which are taken as a measure of element reactivity in the ocean. Actually, a better estimate of reactivity is given by oceanic suspensions, so Li (1982) suggested to use pelagic clay-seawater concentration ratios as a proxy to partition coefficients. 

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... 

Other graphical [219, 220] and computing [221-223] procedures of r r2 estimation also deserve to be mentioned despite their rather limited application. Using the computer one can estimate the reactivity ratios via minimization of the modulus of the deviation of the observed copolymer composition from the computed one. Nevertheless, this procedure performed in Refs. [219-223] has the same disadvantages as any other aforementioned linear least-squares procedures. 

Numerous reports are available [19,229-248] on the development and analysis of the different procedures of estimating the reactivity ratio from the experimental data obtained over a wide range of conversions. These procedures employ different modifications of the integrated form of the copolymerization equation. For example, intersection [19,229,231,235], (KT) [236,240], (YBR) [235], and other [242] linear least-squares procedures have been developed for the treatment of initial polymer composition data. Naturally, the application of the non-linear procedures allows one to obtain more accurate estimates of the reactivity ratios. However, majority of the calculation procedures suffers from the fact that the measurement errors of the independent variable (the monomer feed composition) are not considered. This simplification can lead in certain cases to significant errors in the estimated kinetic parameters [239]. Special methods [238, 239, 241, 247] were developed to avoid these difficulties. One of them called error-in-variables method (EVM) [239, 241, 247] seems to be the best. EVM implies a statistical approach to the general problem of estimating parameters in mathematical models when the errors in all measured variables are taken into account. Though this method requires more information than do ordinary non-linear least-squares procedures, it provides more reliable estimates of rt and r2 as well as their confidence limits. 

The kinetic copolymerization models, which are more complex than the terminal one, involve as a rule no less than four kinetic parameters. So one has no hope to estimate their values reliably enough from a single experimental plot of the copolymer composition vs monomer feed composition. However, when in certain systems some of the elementary propagation reactions are forbidden due to the specificity of the corresponding monomers and radicals, the less number of the kinetic parameters is required. For example, when the copolymerization of two monomers, one of which cannot homopolymerize, is known to follow the penultimate model, the copolymer composition is found to be dependent only on two such parameters. It was proposed [26, 271] to use this feature to estimate the reactivity ratios in analogous systems by means of the procedures similar to ones outlined in this section. 

The application of NMR spectroscopy data to estimate the reactivity ratios is regarded to be very promising [272]. The Q and e values of the Alfrey-Price scheme may be immediately calculated analyzing the shifts of the corresponding bands in carbon-NMR spectra Such data obtained for more than fifty pairs of monomers are tabulated in Ref. [273]. A quite different method based on the application of the trivial expressions ... 

Thus, how should block copolymers between styrene and a vinyl ether be prepared Starting with styrene or with a vinyl ether In the former system, the propagating styryl cation is intrinsically more reactive but present at much lower concentration. A rough estimate of the ratio of cation reactivities is = 103 but the ratio of carbocations concentrations is = I0 S. Thus, the ratio of apparent rate constants of addition is 10-2. Macromolecular species derived from styrene should add to a standard alkene one hundred times slower than those derived from vinyl ethers. Thus, one cross-over reaction St - VE will be accompanied by =100 homopropagation steps VE - VE. Therefore, in addition to a small amount of block copolymer, a mixture of two homopolymers will be formed. Blocking efficiency should be very low, accordingly. 

Another important recent contribution is the provision of a good measurement of the precision of estimated reactivity ratios. The calculation of independent standard deviations for each reactivity ratio obtained by linear least squares fitting to linear forms of the differential copolymer equations is invalid, because the two reactivity ratios are not statistically independent. Information about the precision of reactivity ratios that are determined jointly is properly conveyed by specification of joint confidence limits within which the true values can be assumed to coexist. This is represented as a closed curve in a plot of r and r2- Standard statistical techniques for such computations are impossible or too cumbersome for application to binary copolymerization data in the usual absence of estimates of reliability of the values of monomer feed and copolymer composition data. Both the nonlinear least squares and the EVM calculations provide computer-assisted estimates of such joint confidence loops [15]. 

As mentioned in Chapter 1, ethylene is always the more reactive olefin in systems used to produce copolymers involving a-olefins (LLDPE and VLDPE). An important process consideration for copolymerizations is the reactivity ratio. This ratio may be used to estimate proportions needed in reactor feeds that will achieve the target resin. However, fine tuning is often required to obtain the density or comonomer content desired. Reactivity ratios were discussed previously (Chapter 2) in the context of free radical polymerization of ethylene with polar comonomers. Reactivity ratios are also important in systems that employ transition metal catalysts for copolymerization of ethylene with a-olefins to produce LLDPE. Discussions of derivations and an extensive listing of reactivity ratios for ethylene and the commonly used a-olefins are provided by Krentsel, et al. (1). 

The data provide an estimate of the ratio of reactivity to oxidative attack of branch points compared with linear hydrocarbon chains. The butyl C2 carbon resonance intensity at 23.4 ppm (Figure 2) decreases from 9.7 to 6.6 per 1000 CH2 upon absorption of 53 ml-g"1 of oxygen. Oxidative cleavage at an n-butyl (or longer) branch point occurs as follows (15,16,17) (the tertiary alkoxy radical having been generated by steps parallel to those shown above) ... 

The error-in-variables method was used to estimate the reactivity ratios. This method was developed by Reilly et al. (57, 58), and it was first applied for the determination of reactivity ratios by O Driscoll, Reilly, and co-workers (59, 60). In this work, a modified version by MacGregor and Sutton (61) adapted by Gloor (62) for a continuous stirred tank reactor was used. The error-in-variables method shows two important advantages compared to the other common methods for the determination of copolymer reactivity ratios, which are statistically incorrect, as for example, Fineman-Ross (63) or Kelen-Tiidos (64). First, it accounts for the errors in both dependent and independent variables the other estimation methods assume the measured values of monomer concentration and copolymer composition have no variance. Second, it computes the joint confidence region for the reactivity ratios, the area of which is proportional to the total estimation error. 

Having established the importance of reactivity ratios, it falls to the researcher to have to estimate their values. Given the number of statistical tools and computational devices available over the past several decades, one would expect this to be straightforward. However, there has been resistance to using proper parameter estimation techniques and the reader is advised to exercise caution when using reactivity ratios found in the literature [117]. A good practice is to consider reevaluating these from... 

The next step in the protocol answers the question about what is the best method to estimate the reactivity ratios. Historically, because of its simplicity, linearization techniques such as the Fineman-Ross, Kelen-Tudos, and extended Kelen-Tudos methods have been used. Easily performed on a simple calculator, these techniques suffer from inaccuracies due to the linearization of the inherently nonlinear Mayo-Lewis model. Such techniques violate basic assumptions of linear regression and have been repeatedly shown to be invalid [117, 119, 126]. Nonlinear least squares (NLLS) techniques and other more advanced nonlinear techniques such as the error-in-variables-model (EVM) method have been readily available for several decades [119, 120, 126, 127]. 

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