Copolymerization equilibria

The random nature of the copolymerization equilibria can be considered a consequence of two concurrent entropically driven equilibria similar to reactions 2 and 3. These copolymerization equilibria, however, would involve the comonomers interacting reversibly with two different chain ends and the reversible transfer of the different comonomer units between chains. Expressed in another way, the equilibria could be written in a manner similar to the Mayo-Lewis model but with rate constants replaced by equilibrium constants, K, K12, K22, and K21, and comonomer concentrations replaced by the total concentrations of the different siloxane units in the system, M and M2, regardless of their locations in the rings or chains. 

Choosing, however, as standard states for copolymer units the infinite dilution solutions in the corresponding comonomers, we can get an alternative set of equations for bulk copolymerization equilibria ... 

By equilibrium concentration Szwarc meant a concentration of a given monomer in equilibrium with its own growing species. If there is a multiplicity of growing structures, this monomer is involved in a number of equilibria and the measured value is the copolymerization equilibrium concentration ( stationary concentration ), differing from the equilibrium concentration in homopolymerization. 

However, unlike in homopolymerization systems, the concentrations of macrocydes are not at equilibrium equal (or approximately equal for not infinitely long linear chains) to the macrocydization equilibrium constants. It stems from the fact that various compositions and microstractures of chains can usually be obtained depending on the initial monomer feed and copolymerization equilibrium constants (of homo- and crosspropagations). The only exceptions are alternating copolymers for which macrocyde concentrations conespond to macrocydization equilibrium constants like in homopolymetization systems. 

In another paper, Howell with Izu and O Driscoll related the triad model of copolymerization equilibrium constants to the microstmcture and composition of the equilibrium copolymer. Also, analysis of a multicomponent equilibrium copolymerization was presented. ... 

Concentrations of comonomers at the copolymerization equilibrium are lower than those in homopolymerizations, provided no specific interaaions/solvation play important roles in distinguishing these systems qualitatively or quantitatively. The decrease in the monomer equilibrium concentration in copolymerization stems from the decrease in the proportion of homosequences. For instance, for the dyad model of copo-lymerization, when we assume the same value for the equilibrium constant of homopropagation (no specific interactions) as in homopolymetization (cf. Scheme 2), the following equation for the equilibrium concentration of monomer A can be formulated, independently if homo- or copolymerization is considered ... 

When the dyad model of copolymerization equilibrium constants and initial conditions are known, the basic features of the equilibrium system (without cydizations) can be found by solving a set of 4t equations, where i is the number of comonomers, denoted below as A B,. .., H ... 

However, the assumption introduced by Mita that the interaction parameters for all components of the copolymerization system are equal can be questioned. The present author provided analogous analysis of the copolymerization equilibrium from the point of view of intercomponent interactions... 

Unfortunately, the authors did not discuss their results from the point of view of the copolymerization equilibrium. However, it is a bit striking that the equilibrium concentration of NS3 is almost independent of the feed composition. This may indicate that either specific intercomponent interactions influence the values of the apparent equilibrium constants or... 

The mechanism of phase separation proposed here (and also observed experimentally) involves the formation in the first stage of polymer blanks1, the globules size depends on the initial comonomers and the copolymerization conditions. In the case of slow phase separation proceeding near the thermodynamic equilibrium... 

To determine the crosslinking density from the equilibrium elastic modulus, Eq. (3.5) or some of its modifications are used. For example, this analysis has been performed for the PA Am-based hydrogels, both neutral [18] and polyelectrolyte [19,22,42,120,121]. For gels obtained by free-radical copolymerization, the network densities determined experimentally have been correlated with values calculated from the initial concentration of crosslinker. Figure 1 shows that the experimental molecular weight between crosslinks considerably exceeds the expected value in a wide range of monomer and crosslinker concentrations. These results as well as other data [19, 22, 42] point to various imperfections of the PAAm network structure. 

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. 

Another feature of the equilibrium copolymerization, shown in the next equations, is specific for ring-opening systems ... 

In the anionic copolymerization of lactams, this exchange reaction is faster than the propagation reaction and the copolymer composition is determined by this reaction and not by the propagation reaction127. A general solution of the copolymerization problem considering this equilibrium has not as yet been obtained. 

In the copolymerization of five- and six-membered oxacyclic monomers, the effective monomer concentration in the propagation reaction decreases because only the monomer in excess of equilibrium is available for copolymerization. However, it is not easy to determine the equilibrium monomer concentration in a copolymerization system. The following equilibrium is expected to exist in the copolymerization of THF. 

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. 

A detailed description of AA, BB, CC step-growth copolymerization with phase separation is an involved task. Generally, the system we are attempting to model is a polymerization which proceeds homogeneously until some critical point when phase separation occurs into what we will call hard and soft domains. Each chemical species present is assumed to distribute itself between the two phases at the instant of phase separation as dictated by equilibrium thermodynamics. The polymerization proceeds now in the separate domains, perhaps at differen-rates. The monomers continue to distribute themselves between the phases, according to thermodynamic dictates, insofar as the time scales of diffusion and reaction will allow. Newly-formed polymer goes to one or the other phase, also dictated by the thermodynamic preference of its built-in chain micro — architecture. 

Barrett and Thomas (10)proposed that these effects of differential monomer adsorption could be modeled by correcting homogeneous solution copolymerization reactivity ratios with the monomer s partition coefficient between the particles and the diluent. The partition coefficient is measured by static equilibrium experiments. Barrett s suggested equations are ... 

Using copolymerization theory and well known phase equilibrium laws a mathematical model is reported for predicting conversions in an emulsion polymerization reactor. The model is demonstrated to accurately predict conversions from the head space vapor compositions during copolymerization reactions for two commercial products. However, it appears that for products with compositions lower than the azeotropic compositions the model becomes semi-empirical. 

The temperature at which this condition is satisfied may be referred to as the melting point Tm, which will depend, of course, on the composition of the liquid phase. If a diluent is present in the liquid phase, Tm may be regarded alternatively as the temperature at which the specified composition is that of a saturated solution. If the liquid polymer is pure, /Xn —mS where mS represents the chemical potential in the standard state, which, in accordance with custom in the treatment of solutions, we take to be the pure liquid at the same temperature and pressure. At the melting point T of the pure polymer, therefore, /x2 = /xt- To the extent that the polymer contains impurities (e.g., solvents, or copolymerized units), ixu will be less than juJ. Hence fXu after the addition of a diluent to the polymer at the temperature T will be less than and in order to re-establish the condition of equilibrium = a lower temperature Tm is required. 

Fig. 137.—Equilibrium swelling ratio qm of poly-(methacrylic acid) gels prepared by copolymerizing methacrylic acid with 1, 2, and 4 percent (upper, middle, and lower curves, respectively) of divinylbenzene plotted against degree of neutralization i with sodium hydroxide. (Katchalsky, Lifson, and Eisenberg. )...

At the initial stage of bulk copolymerization the reaction system represents the diluted solution of macromolecules in monomers. Every radical here is an individual microreactor with boundaries permeable to monomer molecules, whose concentrations in this microreactor are governed by the thermodynamic equilibrium whereas the polymer chain propagation is kinetically controlled. The evolution of the composition of a macroradical X under the increase of its length Z is described by the set of equations ... 

Here Jta(x) denotes the a-th component of the stationary vector x of the Markov chain with transition matrix Q whose elements depend on the monomer mixture composition in microreactor x according to formula (8). To have the set of Eq. (24) closed it is necessary to determine the dependence of x on X in the thermodynamic equilibrium, i.e. to solve the problem of equilibrium partitioning of monomers between microreactors and their environment. This thermodynamic problem has been solved within the framework of the mean-field Flory approximation [48] for copolymerization of any number of monomers and solvents. The dependencies xa=Fa(X)(a=l,...,m) found there in combination with Eqs. (24) constitute a closed set of dynamic equations whose solution permits the determination of the evolution of the composition of macroradical X(Z) with the growth of its length Z, as well as the corresponding change in the monomer mixture composition in the microreactor. 

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