Equation, copolymerization

Copolymerization equations for systems of more than two monomers have been derived, and several experimental studies of copolymerizations involving three monomers have been reported. Six reactivity ratios are required for treatment of the composition in a three-compo-... 

Styrene-SQ., Copolymers. I would now like to discuss two systems which illustrate the power of C-13 nmr in structural studies. The first is the styrene-SO system. As already indicated, this is of the type in which the chain composition varies with monomer feed ratio and also with temperature at a constant feed ratio (and probably with pressure as well.) The deviation of the system from simple, first-order Markov statistics, —i.e. the Lewis-Mayo copolymerization equation—, was first noted by Barb in 1952 ( ) who proposed that the mechanism involved conplex formation between the monomers. This proposal was reiterated about a decade later by Matsuda and his coworkers. Such charge transfer com-... 

Multiplication by [M2] gives what are generally referred to as the copolymerization equations, Equations 7.17 and 7.19, which gives the copolymer composition without the need to know any free radical concentration, and which gives the composition of the growing polymer as a function of monomer feed (Equation 7.19). 

Division of both the numerator and denominator by [Mi][M2] gives the second form, Equation 7.19, of the copolymerization equation, but in terms of the composition of the feed (x) on the composition of the copolymer (n) as shown below ... 

Equation 6-12 is known as the copolymerization equation or the copolymer composition equation. The copolymer composition, d M /d Mi, is the molar ratio of the two monomer units in the copolymer. monomer reactivity ratios. Each r as defined above in Eq. 6-11 is the ratio of the rate constant for a reactive propagating species adding tis own type of monomer to the rate constant for its additon of the other monomer. The tendency of two monomers to copolymerize is noted by r values between zero and unity. An r value greater than unity means that Mf preferentially adds M2 instead of M2, while an r value less than unity means that Mf preferentially adds M2. An r value of zero would mean that M2 is incapable of undergoing homopolymerization. 

The copolymerization equation can also be expressed in terms of mole fractions instead of concentrations. If/i and/2 are the mole fractions of monomers M2 and M2 in the feed, and F and F2 are the mole fractions of M2 and M2 in the copolymer, then... 

Equation 6-15 gives the copolymer composition as the mole fraction of monomer Mi in the copolymer and is often more convenient to use than the previous form (Eq. 6-12) of the copolymerization equation. 

Although the derivation above involves the steady-state assumption, the copolymerization equation can also be obtained by a statistical approach without invoking steady-state conditions [Farina, 1990 Goldfinger and Kane, 1948 Melville et al., 1947 Tirrell, 1986 Vollmert, 1973]. We proceed to determine the number-average sequence lengths, h and h2, of monomers 1 and 2, respectively. h is the average number of Mj monomer units that follow each other consecutively in a sequence uninterrupted by M2 units but bounded on each end of the sequence by M2 units. h2 is the average number of M2 monomer units in a sequence uninterrupted by Mj units but bounded on each end by Mi units. 

The copolymerization equation has been experimentally verified in innumerable comonomer systems. The copolymerization equation is equally applicable to radical, cationic, and anionic chain copolymerizations, although the r and r2 values for any particular comonomer pair can be drastically different depending on the mode of initiation. 

The copolymerization equation describes the copolymer composition on a macroscopic scale, that is, the overall composition of a copolymer sample produced from a comonomer feed. This leaves unanswered two details concerning composition copolymer. The first concerns the exact arrangement of the two monomers along the polymer chain. Although the... 

The determination of the reactivity ratios requires a knowledge of the composition of the copolymers made from particular monomer mixtures numerous analytical methods are available (see Sect. 2.3.2). In principle, it is possible to calculate and r2, using Eq. 3.18, from the composition of only two copolymers that have been obtained from two different mixtures of the two monomers M and M2. However, it is more precise to determine the composition of the copolymers from several monomer mixtures and to calculate, for each individual experiment, values of r2 that would correspond to arbitrarily chosen values of r from the rearranged copolymerization equation ... 

Under the condition that the reaction capability is only affected by the nature of the last monomer unit of the growing polymer chain end (terminal model, Bernoulli statistics), the copolymerization equation can be transformed according to Kelen and Tudos ... 

Because Eq. 3.26 is based on the differential form of the copolymerization equation, it is strongly valid only for infinitely low conversions, but this cannot be realized in real life. For higher conversions one has to start with an integrated form of the copolymerization equation. Fortunately, Kelen and Tiidos developed an elegant method of iteration. It allows the use of the earlier suggested method without the loss of graphical clearness. 

If there are no complicating reactions and if conversions are small, the copolymerization equation (13)... 

Let us consider first the copolymerization of a divinyl monomer bearing independent vinyls of the same reactivity with a monovinyl monomer the reactivity of which may be different. The vinyl-divinyl copolymerization can be described by the normal copolymerization equation ... 

The copolymerization equation is valid if all propagation steps are irreversible. If reversibility occurs, a more complex equation can be derived. If the equilibrium constants depend on the length of the monomer sequence (penultimate effect), further changes must be introduced into the equations. Where the polymerization is subjected to an equilibrium, a-methylstyrene was chosen as monomer. The polymerization was carried out by radical initiation. With methyl methacrylate as comonomer the equilibrium constants are found to be independent of the sequence length. Between 100° and 150°C the reversibilities of the homopolymerization step of methyl methacrylate and of the alternating steps are taken into account. With acrylonitrile as comonomer the dependence of equilibrium constants on the length of sequence must be considered. 

Considering the terpolymerization of CPT-SO2-AN system as a binary copolymerization of CPT-SO2 complex and free acrylonitrile, the copolymerization equation can be derived as follows, assuming a fast equilibrium. 

For the above reactions, it is assumed that the reactivity of the propagating radical is dependent only on its terminal radical unit. However, the rate of addition of a monomer to the growing radical depends on the type of monomer in the penultimate position. The importance of the penultimate effects has not been widely investigated. As a result, it is assumed that the simple copolymerization equations given above are valid. 

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