Binary copolymer composition equation

The monomer reactivity ratios r and r2 can be determined from the experimental conversion-composition data of binary copolymerization using both the instantaneous and integrated binary copolymer composition equations, described previously. However, in the former case, it is essential to restrict the conversion to low values (ca. < 5%) in order to ensure that the feed composition remains essentially unchanged. Various methods have been used to obtain monomer reactivity ratios from the instantaneous copolymer composition data. Several procedures for extracting reactivity ratios from the differential copolymer equation [Eq. (7.11) or (7.17)] are mentioned in the following paragraphs. Two of the simpler methods involve plotting of r versus r2 or F versus f. ... 

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... 

In the derivation of copolymer composition equation, Eq. (7.11), we considered only the rates of the four possible propagation steps in a binary system. However, the overall rate of copolymerization depends also on the rates of Initiation and termination. In deriving an expression for the rate of copolymerization in binary systems the following assumptions will be made [25] (a) rate constants for the reaction of a growing chain depend only... 

Even in the first publications concerning the copolymerization theory [11, 12] their authors noticed a certain similarity between the processes of copolymerization and distillation of binary liquid mixtures since both of them are described by the same Lord Rayleigh s equations. The origin of the term azeotropic copolymerization comes just from this similarity, when the copolymer composition coincides with monomer feed composition and does not drift with conversion. Many years later the formal similarity in the mathematical description of copolymerization and distillation processes was used again in [13], the authors of which, for the first time, classified the processes of terpolymerization from the viewpoint of their dynamics. The principles on which such a classification for any monomer number m is based are presented in Sect. 5, where there is also demonstrated how these principles can be used for the copolymerization when m = 3 and m = 4. 

In Refs. [173-176] it was suggested to use the weight composition distributions instead of the molar ones and the results of their numerical calculation for some systems were reported The authors of Ref. [177] carried out a thorough theoretical study of the composition distribution and derived an equation for it without the Skeist formula. They, as the authors of Ref. [178], proposed to use dispersion of the distribution (5.3) as a quantitative measure of the degree of the composition inhomogeneity of the binary copolymers and calculated its value for some systems. Elsewhere [179-185] for this purpose there were used other parameters of the composition distribution. In particular the discussion of the different theoretical aspects of the binary copolymerization is reported in a number of reviews by Soviet authors [186-189], By means of numerical calculations there were analyzed [190-192] the limits of the validity of the traditional assumption which allows to ignore the instantaneous component of composition distribution of the copolymers produced at high conversions. 

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]. 

Since only four copolymer compositions were studied and particularly since the maximum trimethyl comonomer content available was only 20%, it is difficult to precisely determine the three binary interaction parameters, suggested by the "copolymer model", Equations 1 and 2, from the observed variation of B with trimethyl comonomer content in the PEC, Figure 8. It is none the less interesting, however, to qualitatively assess the... 

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). 

Copolymer composition can be predicted for copolymerizations with two or more components, such as those employing acrylonitrile plus a neutral monomer and an ionic dye receptor. These equations are derived by assuming that the component reactions involve only the terminal monomer unit of the chain radical. This leads to a collection of N x N component reactions and x 1) binary reactivity ratios, where N is the number of components used. The equation for copolymer composition for a specific monomer composition was derived by Mayo and Lewis [74], using the set of binary reactions, rate constants, and reactivity ratios described in Equation 12.13 through Equation 12.18. The drift in monomer composition, for bicomponent systems was described by Skeist [75] and Meyer and coworkers [76,77]. The theory of multicomponent polymerization kinetics has been treated by Ham [78] and Valvassori and Sartori [79]. Comprehensive reviews of copolymerization kinetics have been published by Alfrey et al. [80] and Ham [81,82], while the more specific subject of acrylonitrile copolymerization has been reviewed by Peebles [83]. The general subject of the reactivity of polymer radicals has been treated in depth by Jenkins and Ledwith [84]. 

Substitution and rearrangement of this equation yields the polymer composition equation 3.35. Thus it is possible to estimate reactivity ratios for binary copolymers from sequence distributions measured by NMR. 

This is a three-part book with the first part devoted to polymer blends, the second to copolymers and glass transition tanperatme and to reversible polymerization. Separate chapters are devoted to blends Chapter 1, Introduction to Polymer Blends Chapter 2, Equations of State Theories for polymers Chapter 3, Binary Interaction Model Chapter 4, Keesome Forces and Group Solubility Parameter Approach Chapter 5, Phase Behavior Chapter 6, Partially Miscible Blends. The second group of chapters discusses copolymers Chapter 7, Polymer Nanocomposites Chapter 8, Polymer Alloys Chapter 9, Binary Diffusion in Polymer Blends Chapter 10, Copolymer Composition Chapter 11, Sequence Distribution of Copolymers Chapter 12, Reversible Polymerization. 

The statistics of binary copolymers k one of the most advanced parts of polymer kinetics. All tte necessary equations for the copolymer composition and monomer distribution in the simple four-reaction scheme (t,2 for the pmultimate effect (3,4) and the pen-penultimate effea (5) wer obtained by the kinetic method and by an approach based on the Markov chain tteory (6). Some of the recent papers in this field are very sophisticated (5,7, S) and the theoretical achievements often far exceed the experimental possibilities for testing them. 

Equation relating the instantaneous composition of a binary copolymer to the monomer reactivity ratios and the ratio of instantaneous monomer eoncentrations. 

In some cases, one is interested in the structures of complex fluids only at the continuum level, and the detailed molecular structure is not important. For example, long polymer molecules, especially block copolymers, can form phases whose microstructure has length scales ranging from nanometers almost up to microns. Computer simulations of such structures at the level of atoms is not feasible. However, composition field equations can be written that account for the dynamics of some slow variable such as 0 (x), the concentration of one species in a binary polymer blend, or of one block of a diblock copolymer. If an expression for the free energy / of the mixture exists, then a Ginzburg-Landau type of equation can sometimes be written for the time evolution of the variable 0 with or without flow. An example of such an equation is (Ohta et al. 1990 Tanaka 1994 Kodama and Doi 1996)... 

Equation (7.18) may be used to calculate the instantaneous composition of copolymer as a function of feed composition for various monomer reactivity ratios. A series of such curves are shown in Fig. 7.1 for ideal copolymerization, i.e., r r2 = 1. The term ideal copolymerization is used to show the analogy between the curves in Fig. 7.1 and Aose for vapor-liquid equilibria in ideal liquid mixtures. The vapor-liquid composition curves of ideal binary mixtures have no inflection points and neither do the polymer-composition curves for random copolymerization in which riV2 = 1. Such monomer systems are therefore called ideal. It does not in any sense imply an ideal type of copolymerization. 

Elemental analysis (EA) is a convenient method for determination of copolymer and blend composition if one homopolymer contains an element not present in the second one. For example, EA can be properly used to quantify nitrogen in copolymers containing acrylonitrile units and oxygen in polymeric surfactants such as poly(oxy-alkylene). Therefore, for a binary system, every element can be balanced according to the following equation ... 

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