Fineman-Ross equation

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. 

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... 

Copolymerization constants were determined by kinetic method [63] and calculated by the Fineman-Ross equation [64]... 

G is then plotted against F, or G/F against l/F, in the Fineman-Ross equations (see Figure 22-4) ... 

Figure 22-4. Graphical evaluation of linear copolymerization equations after Fineman-Ross [Equation (22-25)] and Kelen-Tiidos [Equation (22-28)], for the free radical copolymerization of styrene with methyl methacrylate in bulk at 6(f C.

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 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 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... 

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. 

Copolymer-reactivity ratios obtained from the feed and copolymer composition data with linearized equations, as in the Fineman-Ross procedure, do not allow proper weighting of the experimental data, and cannot provide a proper estimate of the precision of the parameters, which, being interdependent, have joint confidence limits. Computer-based methods for determining reactivity ratios have been summarized and non-linear least squares methods described.. Errors in the dependent variables were included by Yamada... 

If such effects occurs, the classical copolymerization equation n = (r x 1)/ r /x + 1)) gives abnormal value for, at least, one of the reactivity ratios (for instance, Tb < 0,...) and the Fineman-Ross plot is not a straight line on the whole range of monomer composition (x), but looks like the curve of Fig. 1,... 

Sometimes, it occurs that the Fineman-Ross plot shows that experiments do not fit the theoretical straight line corresponding to a penultimate effect in the extreme range of composition of monomer feed. This fact might indicate the influence of the more remote units. For instance, such an occurence is encountered when copolymerizing styrene-acrylonitrile and vinyl chloride-vinyl acetate systems. Calculations analogous to those mentionned above may be performed with the equation proposed by G. E. Ham [7] for pen-penultimate effects, which allows the determination of the reactivity ratios (with adjunction of some more assumptions). We performed these types of calculation for the two systems for... 

Using Eq. (32) and feed and terpolymer composition data, a copolymerization composition diagram can be drawn, compared with the theoretical curves, and the coefficients of the Mayo-Lewis equation, riK and / 2jK estimated (Table 10.23). Fineman-Ross plots may also be used to estimate the Mayo-Lewis coefficients. These dimensioned apparent or modified reactivity ratios deviate from the true reactivity ratio values the more greatly the equilibrium constants differ from unity. 

Independent of the analytical methods the copolymerization parameters derived from the Mayo-Lewis equation have some major drawbacks, but do enable a part of the information contained in each copolymerization experiment to be extracted, namely the integral composition of the copolymer. A series of experiments is necessary to obtain the copolymerization parameters. Furthermore, using this approach it cannot be decided whether the application of the first-order Markov model is valid or not. One may even obtain a nearly perfect Fineman-Ross plot, where other methods show that a first-order Markov model cannot be applied. 

These P values represent the fraction of all the mi (or m2) sequences formed by n members. F is a function of the ratio f between the molar concentration of mi and m2 in the copolymer and can be obtained from the copolymerization equation written by Fineman and Ross (24) ... 

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