Linear copolymers equation

This closure property is also inherent to a set of differential equations for arbitrary sequences Uk in macromolecules of linear copolymers as well as for analogous fragments in branched polymers. Hence, in principle, the kinetic method enables the determination of statistical characteristics of the chemical structure of noncyclic polymers, provided the Flory principle holds for all the chemical reactions involved in their synthesis. It is essential here that the Flory principle is meant not in its original version but in the extended one [2]. Hence under mathematical modeling the employment of the kinetic models of macro-molecular reactions where the violation of ideality is connected only with the short-range effects will not create new fundamental problems as compared with ideal models. 

Attenpts to Analyze Complex Polymers Using SEC Detector Technology. For linear copolymers, multiple detectors and, more recently, diode array UV/vis spectrophotometers have been used in attempts to overccxne the above analysis problems. The basic idea is to provide more than one detector response so that the polymer concentration and the number of properties will together equal the number of detector responses (Figure 4). This provides the same number of equations as the number of unitnowns (5,6). 

When the statistical moments of the distribution of macromolecules in size and composition (SC distribution) are supposed to be found rather than the distribution itself, the problem is substantially simplified. The fact is that for the processes of synthesis of polymers describable by the ideal kinetic model, the set of the statistical moments is always closed. The same closure property is peculiar to a set of differential equations for the probability of arbitrary sequences t//j in linear copolymers and analogous fragments in branched polymers. Therefore, the kinetic method permits finding any statistical characteristics of loopless polymers, provided the Flory principle works for all chemical reactions of their synthesis. This assertion rests on the fact that linear and branched polymers being formed under the applicability of the ideal kinetic model are Markovian and Gordonian polymers, respectively. 

So to estimate the copolymer composition solving the system of the linear algebraic equations ... 

The instantaneous copolymer composition X generally doesn t coincide with the monomer feed composition x from which the copolymer was produced. Such a coincidence X = x can occur only under some special values of monomer feed composition x, called azeotropic . According to definition these values can be calculated in the case of the terminal model (2.8) from a system of non-linear algebraic equations ... 

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

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]. 

In order to enhance the reactivity of aluminum porphyrins (J ) especially towards C02 in the copolymerization with epoxide (Equation 4), the effect of addition of an amine or phosphine as a possible sixth ligand to the aluminum porphyrin was examined. The enhancement in reactivity by the addition of a tertiary amine such as N-methyl-imldazole was actually observed for the epoxlde-C02 reaction. The product, however, was a cyclic carbonate (JO), not a linear copolymer. On the other hand, J he addition of trlphenylphosphlne was very effective in the formation of an alternating copolymer from epoxide and... 

The empirical DiBenedetto equation was developed in the late 1960s to mathematically relate Tg and conversion (Nielson 1969 DiBenedetto 1987). Excellent theoretical treatises on the Tg-conversion relationship can be found in Pascault and Williams (1990), Hale et al. (1991), and Venditti and Gillham (1997). Venditti and Gillham (1997) developed an equation based on thermodynamic considerations put forth by Couchman and Karasz (1978) to predict Tg versus mole fraction of constituents of a linear copolymer ... 

Instead of the homo- and crosspropagation equilibrium constants and the equilibrium comonomer concentrations, we can use the experimentally determined ratios of couespond-ing sequences of linear copolymer. For instance, for dyad model copolymerization, the following equation can be obtained ... 

Equation [24] (after rearrangement) can be used for computing the maaocyclization equilibrium constants in copolymerization on the basis of the determined equilibrium concentrations of macrocydes and linear copolymer miaostmcture. 

When the cydization equilibrium constants in copolymerization are known, the same eqn [24] can be used to predict the equilibrium concentrations of macrocydes provided the equilibrium composition and microstmcture of linear copolymer is known. When the properties of the equilibrium linear copolymer cannot be determined, but the equilibrium constants of macrocydization and copolymerization are known, the prediction of the equilibrium concentrations of macrocydes can still be accomplished, but only by formulation and solving the set of equations, taking into accormt besides eqn [23] the mass balance equations for comonomer units in linear and cydic fractions. 

The glass transition temperature of a random copolymer usually falls between those of the corresponding homopolymers since the copolymers will tend to have intermediate chain stiffness and interchain attraction. Where these are the only important factors to be considered a linear relationship between Tg and copolymer composition is both reasonable to postulate and experimentally verifiable. One form of this relationship is given by the equation... 

Figure 12 were superimposable on those for detector 2. Therefore, when the plot shown in Figure 14 is linear over the range of compositions involved in the sample, then (according to equations (1-4) ) the composition of the sample is the same at each retention volume. If the variation with retention volume is negligible the copolymer can then possibly be treated as is a homopolymer in GPC interpretation. In particular, intrinsic viscosity measurements could then lead to estimates of molecular weight via the universal calibration curve. 

A linear phenylene-ethylene copolymer is obtained by a Wurtz-type condensation of 1,2-dibromoethane and 1,4-dibromobenzene. A cross-linked variation can be obtained on addition of 1,3,5-tribromobenzene to the reaction mixture. Equation 119 illustrates the reductive scission of a cyclic ether, catalyzed by such polymers in the presence of lithium, followed by quenching with cyclohexanone . 

Various methods have been used to obtain monomer reactivity ratios from the copolymer composition data. The most often used method involves a rearrangement of the copolymer composition equation into a form linear in the monomer reactivity ratios. Mayo and Lewis [1944] rearranged Eq. 6-12 to... 

Even with the Kelen Tudos refinement there are statistical limitations inherent in the linearization method. The independent variable in any form of the linear equation is not really independent, while the dependent variable does not have a constant variance [O Driscoll and Reilly, 1987]. The most statistically sound method of analyzing composition data is the nonlinear method, which involves plotting the instantaneous copolymer composition versus comonomer feed composition for various feeds and then determining which theoretical plot best fits the data by trial-and-error selection of r and values. The pros and cons of the two methods have been discussed in detail, along with approaches for the best choice of feed compositions to maximize the accuracy of the r and r% values [Bataille and Bourassa, 1989 Habibi et al., 2003 Hautus et al., 1984 Kelen and Tudos, 1990 Leicht and Fuhrmann, 1983 Monett et al., 2002 Tudos and Kelen, 1981]. 

If material is neo-Hookean, its Mooney-Rivlin plot ought to give a horizontal line and hence yield C2 = 0. Thus one is tempted to consider that nonzero C2 must be associated in one way or another with the deviation of a given material from the idealized network model, and it is understandable why so many rubber scientists have concerned themselves with evaluating the C2 term from the Mooney-Rivlin plot of uniaxial extension data. However, the point is that a linear Mooney-Rivlin plot, if found experimentally, does not always warrant that its intercept and slope may be equated to 2(9879/,) and 2(91V/9/2), respectively. This fact is illustrated below with actual data on natural rubber (NR) and styrene-butadiene copolymer rubber (SBR). 

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... 

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