Fineman-Ross

The traditional method for determining reactivity ratios involves determinations of the overall copolymer composition for a range of monomer feeds at zero conversion. Various methods have been applied to analyze this data. The Fineman-Ross equation (eq. 42) is based on a rearrangement of the copolymer composition equation (eq. 9). A plot of the quantity on the left hand side of eq. 9 v.v the coefficient of rAa will yield rAB as the slope and rUA as the intercept. 

The experimental composition data are unequally weighted by the Mayo-Lewis and Fineman-Ross plots with the data for the high or low compositions (depending on the equation used) having the greatest effect on the calculated values of r and r2 [Tidwell and Mortimer, 1965, 1970]. This often manifests itself by different values of r and r2 depending on which monomer is indexed as Mi. 

The r values obtained by the Fineman-Ross procedure (3) are listed in Table II. Reactivity toward the butadiene peroxy radical increases in the order cumene (0.14), sec-butylbenzene < Tetralin (1.00) < styrene (1.5) < butadiene (3.3), in reasonable agreement with previous efforts. Russell and Williamson (21) found cumene 0.10 and 0.40 as reactive as... 

These considerations lead to many important conclusions. If 0 < Ka < °°, then ri ri. This means that in this case of constant template concentration, [T], ri can be computed using conventional procedure (for instance according to Kellen-Tiidos or Fineman-Ross). However, ri value depends on the concentration of the template. 

The composition of the copolymer was determined by either NMR analysis at 90 MHz according to the equations derived by Mochel (21) or by infrared. (22) The agreement of these methods was 2% when applied to copolymer taken to 100% conversion. The reactivity ratios were calculated according to the Mayo-Lewis Plot (13,15), the Fineman-Ross Method (14), or by the Kelen-Tudos equation.(16,17,18) The statistical variations recently noted by 0 Driscoll (23), were also considered. 

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... 

For a detailed analysis of monomer reactivity and of the sequence-distribution of mers in the copolymer, it is necessary to make some mechanistic assumptions. The usual assumptions are those of binary, copolymerization theory their limitations were discussed in Section III,2. There are a number of mathematical transformations of the equation used to calculate the reactivity ratios and r2 from the experimental results. One of the earliest and most widely used transformations, due to Fineman and Ross,114 converts equation (I) into a linear relationship between rx and r2. Kelen and Tudos115 have since developed a method in which the Fineman-Ross equation is used with redefined variables. By means of this new equation, data from a number of cationic, vinyl polymerizations have been evaluated, and the questionable nature of the data has been demonstrated in a number of them.116 (A critique of the significance of this analysis has appeared.117) Both of these methods depend on the use of the derivative form of,the copolymer-composition equation and are, therefore, appropriate only for low-conversion copolymerizations. The integrated... 

Fig. 19. Determination of copolymerization parameters by the Fineman-Ross method, a = y, b = n.

The Fineman-Ross method uses a more conventional plotting procedure, rearranging the copolymer equation into the following form (Equation 6-7),... 

Both the Mayo-Lewis and the Fineman-Ross methods rely on linearizing the copolymer equation. It has been shown that... 

B. Given the molecular weight of vinyl phenol is 120 and styrene is 104, calculate the reactivity ratios describing this copolymerization using the Fineman-Ross method. 

By Fineman-Ross method and confimed by the Mortimer-Tidwell (22) non-linear least square computed method. 

Determine rj and V2 for the monomer pair by the Fineman-Ross method. Answer ... 

Kelen and Tudos [17] refined the Fineman-Ross linearization method by introducing an arbitrary positive constant a into Eq. (7.34) to spread the data more evenly so as to give equal weighing to all data points. Their results are expressed in the form... 

In the aforesaid Mayo-Lewis and Fineman-Ross methods, the experimental composition data are unequally weighted as, for example, at low [M2] in Eq. (7.28) or low [Ml] in Eq. (7.29) the experimental data have the greatest influence on... 

Figure 7.6 Plot according to the Fineman-Ross method (data from Problem 7.7).

By Fineman-Ross method and confirmed by the Mortimer-Tidwell (22) non-linear least square computed method. b By assuming that the product of reactivity ratio equals unity and using the relationship rj = Fi fp/Fp fi where Fj and / denote the mole fractions of isobutylene in the copolymer and charge, respectively, and Fp and fp those for (3-pinene. c By Fineman-Ross method. 

Using the data in Table III, the general copolymer composition equations, and the Fineman-Ross procedure (16), the reactivity ratios were estimated at rx — 0.40 and 0.42 and r2 = 0.78 and 0.75 with mean values of 0.41 and 0.77 respectively. Values of rx = 0.406 and r2 = 0.773 were calculated by a computer program based on least squares. 

Early methods such as the Intersection, " and Fineman-Ross" methods do not give equal weighting to the experimental points such that there is a non-lincar dependence of the error on the composition. Consequently, these methods can give en oneous results. 

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. 

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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