Copolymerization of Four Monomers

In this case according to Fig. 3 a phase space is found to be a tetrahedron, inside which an azeotrope of one of the eight types, presented in Fig. 8, may be located. In order to know which of them is to be realized one should find the signs of coefficients at, a2, a3 of characteristic equation (5.11) and also their combinations  

When P 0, using Table 5.5 one can immediately point out the type of the inner azeotrope, and when p 0, only the node (N) and saddle (S) in this table are replaced with the focus (F) and saddle-focus (SF), respectively. The sign of Ind (X ) of this azeotrope is opposite to the sign of a3 and is determined by indexes of the boundary SPs (see Table 5.6) in accordance with the rule of azeotropy (5.18) which in the case of tetrapolymerization yields  

The types of SPs located at the apexes and edges of the tetrahedron can be established at once according to the right part of Table 5.6 without any additional calculations. For the first of the above SPs in each j-th point it is sufficient to indicate a number q of the parameters a exceeding unity. The type of the four-component azeotrope, located at the edge (kl) of the tetrahedron is unambiguously characterized by the types of the two azeotropes, which correspond to it in three-component systems (ikl) and (jkl) represented by the tetrahedron faces 

X = Di(jkli)/D(jkl) (5.13). The sign of the latter is exactly the same as Ind(x ) when the three-component azeotrope in the system (jkl) is the node or focus, and its sign is opposite to the sign of lnd(x ) when the above azeotrope is the saddle. The separatrix surface inside the tetrahedron surely passes through the azeotrope located on its face when in the ternary system corresponding to this face such an azeotrope is the saddle. 

Using the expression (5.22) together with Tables 5.5 and 5.6 on the base of the general principles reported in Sect. 5.2 one can carry out an exhaustive classification of the four-component systems as it has been already done for terpolymerization in Sect. 5.3. However, when the forth monomer is added, the number of the system types increases from 7 (see Fig. 6) to 41 (see Fig. 9) and that is a reason why the results of the complete theoretical analysis cannot be represented in the framework of this review. Without appealing to the classification and using only the algorithm described in Sect. 5.2 one may present a phase portrait of any concrete four-component system and hence predict the qualitative character of its dynamic behavior before the computer calculations of trajectories x(p) are performed. 

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Chen J, Asano M, Yamaki T, Yoshida M (2006) Preparation and characterization of chemically stable polymer electrolyte membranes by radiation-induced graft copolymerization of four monomers into ETFE films. J Membr Sci 269 194-204... 

Note Copolymers that are obtained by copolymerization of two monomer species are sometimes termed bipolymers, those obtained from three monomers terpolymers, those obtained from four monomers quaterpolymers, etc. 

In more conventional copolymerization, where both monomers are present at the start of reaction, standard analysis methods are available which were developed for free radical initiation. It is first necessary to enquire whether the techniques will be applicable to anionic systems containing long lived active species. Four propagation steps can be recognized in the copolymerization of two monomers Ml and M2 ... 

Under the copolymerization of more than two monomers Eqs. (5.3) cannot be integrated explicitly, and in order to determine the system trajectories one should need the numerical calculations. Examples of such calculations of the conversional change of composition and structure characteristics of the terpolymers have been reported in Refs. [195-200]. One should pay special attention to Ref. [200] where the programs for the computer realization of such calculations are presented. Under the copolymerization of four or more monomers, the composition drift with the conversion was calculated [7,8] only within the framework of the simplified terminal model described above in Sect. 4.6. 

The kinetic copolymerization models, which are more complex than the terminal one, involve as a rule no less than four kinetic parameters. So one has no hope to estimate their values reliably enough from a single experimental plot of the copolymer composition vs monomer feed composition. However, when in certain systems some of the elementary propagation reactions are forbidden due to the specificity of the corresponding monomers and radicals, the less number of the kinetic parameters is required. For example, when the copolymerization of two monomers, one of which cannot homopolymerize, is known to follow the penultimate model, the copolymer composition is found to be dependent only on two such parameters. It was proposed [26, 271] to use this feature to estimate the reactivity ratios in analogous systems by means of the procedures similar to ones outlined in this section. 

An analogous analysis of the solution copolymerization of styrene with acrylonitrile in toluene leads [283] to the same conclusion concerning the choice of the kinetic model as for the bulk copolymerization of these monomers. The applicability of the penultimate model (2.3) was also convincingly proved [283] for the given system, and the estimated values of its four parameters (2.4) (see Table 6.8) were found to be slightly different from the ones obtained in the bulk copolymerization [283], The experimental values of the fractions of all six triads, determined by means of NMR, in the solution copolymerization products, practically do fit the theoretical plots of the triad fractions vs conversion, which were calculated on the basis of the kinetic parameters presented in Table 6.8. 

In many systems, however, the analysis of the cationic copolymerization of heterocyclic monomers is complicated by two factors (1) at least some of the homo- and cross-propagation reactions may be reversible (2) redistribution of the sequences of comonomers within the chain may occur as a result of chain transfer to polymer. Therefore, the conventional treatment involving four irreversible propagation steps is rarely applicable in cationic ring-opening copolymerization. Instead, the diad model should involve four reversible reactions, i.e., eight rate constants... 

Free radical copolymerization involves initiation, propagation and termination, just like homopolymerization. With one or two exceptions, initiation and termination are pretty much the same in these polymerizations and it is the propagation step that gives copolymerization its special character. For the copolymerization of two monomers, there are four possible propagation steps (Figure 6-6). If the monomer at the terminal position of the growing chain radical happens to be a monomer of type 1, M, then it can either add another monomer, or It can add an M2 monomer. The same two possibilities apply if the chain end is an M2-... 

The conventional [Eq. (7.77)] and simplified [eq. (7.81)] terpolymeriza-tion equations can be used to predict the composition of a terpolymer from the reactivity ratios in the two-component systems M1/M2, M1/M3, and Ms/Ms- The compositions calculated by either of the terpolymerization equations show good agreement with the experimentally observed compositions. Neither equation is found superior to the other in predicting terpolymer compositions. Both equations have been successfully extended to multicomponent copolymerizations of four or more monomers [30,31]. 

In the copolymerization of two monomers there are four separate propagation steps. In the present case, Ml and M2 refer to ethylene and propylene, respectively, and Mf and ftff refer to growing polymer molecules with ethylene (M,) or propylene (M2) residues at the active end. 

In the copolymerization of two monomers, and M2, the four propagation steps possible are... 

Anionic copolymerizations have been investigated by applying the classical Mayo-Lewis treatment which was originally developed for free-radical chain reaction polymerization [198]. The copolymerization of two monomers (Mj and M2) can be uniquely defined by the following the four elementary kinetic steps in Scheme 7.21, assuming that the reactivity of the chain end (Mj" or ) depends only on the last unit added to the chain end, that is, there are no penultimate effects. 

In multimonomer polymerizations, the number of possible reactions increases rapidly with the number of monomers. For example, for the copolymerization of two monomers under the assumption of terminal model kinetics, four propagation steps must be considered (see Eq. 6.1). Similarly, for the termination step (chemically controlled), three reactions are possible ... 

By designating the two monomers as M, and Mj and their corresponding chain radicals as M] and M2, the four propagation reactions and the associated rate equations in the copolymerization of two monomers may be written as follows ... 

Copolymerization, on the other hand, may take place in a bulk phase. For copolymerization of a monomer. Ml, and a surfactant, M2, to occur, the reactivity ratios should preferably be ri < 1 and r2 < 1, where r and V2 are defined as r =k /k 2 and V2 = 22/ 2with knm being the rate constants for the four possible ways in which the monomer can add, shown as follows ... 

In the case of a copolymerization of two monomers A and B, the propagation step consists of at least four different reactions ... 

From a chemical point of view, polymers can be classified according to their chemical composition. First, the number of chemically different monomer units, from which the macromolecule is constructed, is considered. A homopolymer is derived from only one species of monomers. This could be the actual starting reactant or a hypothetical monomer if the homopolymer is formed by a modification of another homopolymer (e.g. by polymer analogue reaction). The homopolymer shows only one repeating unit in contrast to a copolymer. A polymer derived from more than one species of monomer is termed copolymer. Copolymers that are obtained by copolymerization of two monomer species are sometimes termed bipolymers, those obtained from three monomers terpolymers, those obtained from four monomers quaterpolymers, etc. [96IUP]... 

Experiments indicated that the rate constants ( y) are independent of the chain length and the rate of monomer addition depends on the nature of the monomer and the growing chain end. Only four equations are, therefore, required to describe the copolymerization of two monomers. 

Polymerization equilibria frequently observed in the polymerization of cyclic monomers may become important in copolymerization systems. The four propagation reactions assumed to be irreversible in the derivation of the Mayo-Lewis equation must be modified to include reversible processes. Lowry114,11S first derived a copolymer composition equation for the case in which some of the propagation reactions are reversible and it was applied to ring-opening copalymerization systems1 16, m. In the case of equilibrium copolymerization with complete reversibility, the following reactions must be considered. 

Anionic copolymerization of lactams presents an interesting example of copolymerization. Studies of the copolymerization of a-pyrrolidone and e-caprolactam showed that a-pyrrolidone was several times more reactive than e-caprolactam at 70 °C, but became less reactive at higher temperatures due to depropagation210 2U. By analyzing the elementary reactions Vofsi et al.I27 concluded that transacylation at the chain end occurred faster than propagation and that the copolymer composition was essentially determined by the transacylation equilibrium and the acid-base equilibrium of the monomer anion together with the usual four elementary reactions of the copolymerization. 

The rate of copolymerization in a binary system depends not only on the rates of the four propagation steps but also on the rates of initiation and termination reactions. To simplify matters the rate of initiation may be made independent of the monomer composition by choosing an initiator which releases primary radicals that combine efficiently with either monomer. The spontaneous decomposition rate of the initiator should be substantially independent of the reaction medium, as otherwise the rate of initiation may vary with the monomer composition. 2-Azo-bis-isobutyronitrile meets these requirements satisfactorily. The rate Ri of initiation of chain radicals of both types Ml and M2 is then fixed and equal to 2//Cd[7], or twice the rate of decomposition of the initiator I if the efficiency / is equal to unity (see Chap. IV). The relative proportion of the two types of chain radicals created at the initiation step is of no real importance, for they wall be converted one into the other by the two cross-propagation reactions of the set (1). Melville, Noble, and Watson presented the first complete theory of copolymerization suitable for handling the problem of the rate. The theory was reduced to a more concise form by Walling, whose procedure is followed here. 

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